Fundamentals of

RotatingMachinery
Diagnostics
Fundamentals of
RotatingMachinery
Diagnostics

Donald E. Bently
Chairm an ofthe Board and Chief Execut ive Officer
Bently Pressurized Bearing Comp any

With
Charles T. Hatch

Edit ed by
Bob Grissom FOR REFERENCE ONL'f I
.J

Bently Pressurized Bearing Press

, ,- . .
Ii:; PT. EMOMi
FOR
CONTROL
 Copyright © 2002 Bently Pressurized Bearing Company
All Rights Reserved.

The following ar e trademarks of Bently Nevada Corporation in the United

States and Other Count ries: Bently Nevada", Keyphasor", Proximitor", REBAM®,
Seism op robe", Velornitor", Orbit Design.

The Bently Pressurized Bearing Co. logo is a trademark of

Bently Pressurized Bearing Company in the United States and Other
Countries.

MATLAB' is a regi st ered trademark of The MathWorks, Inc.

Bently Pre ssurized Bearing Press is an imprint of

Bently Pressurized Bearing Company
1711 Orbit Way
Minden, NY, USA 89423

Phone: 775-783-4600
bpb.press@bpb-co.com
www.bpb-co. corn

Library of Congress Control Number 2002094136

ISBN 0-9 714081-0-6

Book design by Charles T. Hatch

Set in Adobe Keple r and Myriad Multiple Ma ster typefaces

Printed in Canada
First Printing
 Dedication
This book is 50 p ercent due to the brilliant work ofDr. Agnes Muszynska. Dr.
Muszynska is a m emb er of the Polish Acade my ofSciences and worked with m e
f or m ore than 18 years.

Dr. Muszynska is an excellent researcher in her own right and pioneered thefirst
correct modeling ofequations f or modern rotor dynamics. Chap ter 22 on insta-
bility illustrates mu ch of the work we did in partnership on the development of
modern rotor equations.

Donald E. Bently
 vii

Table of Contents
Acknowledgments xvii
Foreword XXI
Introduction xxvii

Fundamentals of Vibration

1 Vibration 3
The Basic Vibration Signal 4
Frequency 5
Amplitude 7
Displacement, Velocity, and Acceleration 9
The Vibration of Machines 11
Rotation and Precession 13
Free Vibration 14
Forced Vibration 16
Resonance 17
Self-Excited Vibration 19
Summary 19

2 Phase 21
What is Phase? 21
Why Is Phase Important? 22
The Keyphasor Event 23
Phase Measurement 25
Absolute Phase 26
Relative Phase 29
Differential Phase 31
Summary 31

3 Vibration Vectors 33
Unfiltered Vibration 33
Filtering and the Vibration Vector 34
Working with Vibration Vectors 38
The Slow Roll Vector 44
Summary 47
viii Fundamentals of Rotating Machinery Diagnostics

Data Plots

4 Timebase Plots 51
The Structure of a Timebase Plot 52
The Keyphasor Mark 54
Compensation of Timebase Plots 54
Information Contained in the Timebase Plot 56
Summary 66

5 The Orbit 69
The Construction of the Orbit 70
The Keyphasor Mark 72
Compensation of Orbits 74
Information Contained in the Orbit 79
The Orbit/Timebase Plot 94
Summary 95

6 Average Shaft Centerline Plots 97

The Construction of the Average Shaft Centerline Plot 98
Information Contained in the Average Shaft Centerline Plot 101
The Complete Picture: Orbit Plus Average Shaft Centerline Position 108
Summary 109

7 Polar, Bode, and APHT Plots 111

The Structure of Polar and Bode Plots 112
Slow Roll Compensation 115
Information Contained in Polar and Bode Plots 117
APHT Plots 127
Acceptance Region Plots 129
Summary 130
References 130

8 Half and Full Spectrum Plots 131

The Half Spectrum Plot 133
Technical Issues 135
The Full Spectrum 138
Spectrum Cascade Plots 148
Spectrum Waterfall Plots 150
Summary 153
 Table of Contents ix

9 Trend and XY Plots 155

Trend Plots 155
XY Plots 160
Summary 161
References 161

The Static and Dynamic Response of Rotor Systems

10 The Rotor System Model 165

Introduction to Modeling 167
Assumptions 170
The Coordinate System and Position Vector 172
Lambda (A): A Model of Fluid Circulation 175
Fluid-film Bearing Forces and Stiffnesses 179
Other Sources of Spring Stiffness 180
The Damping Force 182
The Perturbation Force 183
The Free Body Diagram 185
The Equation of Motion 186
Solution of the Equation of Motion 187
Nonsynchronous Dynamic Stiffness 189
Amplitude and Phase of the Vibration Response 189
The Attitude Angle: Rotor Response to a Static Radial Load 191
Synchronous Rotor Response 192
Synchronous Dynamic Stiffness 192
Predicted Rotor Vibration 193
Nonlinearities 198
The Benefits and Limitations of the Simple Model 198
Extending the Simple Model 200
Summary 206
References 207

11 Dynamic Stiffness and Rotor Behavior 209

What Is Dynamic Stiffness? 209
Rotor Parameters and Dynamic Stiffness 214
Synchronous Rotor Behavior 217
Synchronous Behavior Below Resonance 217
Synchronous Behavior At The Balance Resonance 218
Synchronous Behavior Above Resonance 221
x Fundamentals of Rotating Machinery Diagnostics

How Changes In Dynamic Stiffness Affect Vibration 222

Sum mary 225

12 Modes of Vibration 227

Mode Shapes 228
Forced Mode Shapes and Multimode Resp onse 236
Modal Parameters 239
Th e Measurement of Mode Shape 243
Mode Identificat ion Probes 246
Summary 248

13 An isotro p ic Stiffness 249

Anisotropic Stiffness 250
Split Resonances 253
Measured Rotor Behavior and Ani sotropic Stiffn ess 256
Probe Mounting Orientation and Measured Resp on se 262
Virtual Probe Rotation 265
Forward and Reverse Vect ors 268
Summary 270
References 272

14 Rotor Stability Analysis: The Root Locus 273

What is St ability? 274
Stability and Dynamic Stiffn ess 278
Stabili ty Analysis 280
State-Space Formulation of th e Eigenvalue Problem 286
The Root Locu s Plot 289
The Root Locu s and Amplifi cation Factors 295
Paramet er Variation and the Root Locus 302
The Root Locu s of Anisot ropic and Multimode Systems 304
The Root Locu s and the Logarithmic Decrement 308
Th e Root Locu s and the Campbell Diagram 310
Root Locus Analysis of Machine Stability Probl em s 312
Summary 312
Referen ces 314

15 Torsional and Axial Vibration 315

The Torsional View of the Rotor 316
Static and Dynamic Torsiona l Response 321
Torsional/Radial Cross Coupling 326
 Table of Contents xi

Torsional Vibration Measurement 328

Axial Vibration 332
Summary 335
References 336

16 Basic Balancing of Rotor Systems 337

Unbalance and Rotor Response 337
Vibration Transducers And Balancing 341
Balancing Methodology 342
Locating the Heavy Spot Using a Polar Plot 347
Using Polar Plots Of Velocity and Acceleration Data 349
Selecting the Calibration Weight 350
Relating Balance Ring Location To Polar Plot Location 353
Single Plane Balancing With Calibration Weights 354
Weight Splitting 361
The Influence Vector 365
The Influence Vector And Dynamic Stiffness 370
Multiple Modes And Multiplane Balancing 371
Multiplane Balancing With Influence Vectors 378
How Balancing Can Go Wrong 380
Summary 381
References 382

Malfunctions

17 Introduction to Malfunctions 385

What is a Malfunction? 385
Detection of Malfunctions 387

18 Unbalance 391
Rotor System Vibration Due To Unbalance 391
Stress and Damage 393
Other Things That Can Look Like Unbalance 394
Runout 396
Rotor Bow 396
Electrical Noise in the Transducer System 398
Coupling Problems 398
Shaft Crack 398
Loose Part or Debris 399
xii Fundamentals of Rotating Machinery Diagnostics

Rub 400
Changes in Spring Stiffness 400
Electric Motor Related Problems 400
Loose Rotating Parts 405
Summary 407

19 Rotor Bow 409

What is Rotor Bow? 409
Causes of Rotor Bow 410
Rotor Dynamic Effects of Rotor Bow 418
Thermal Bow During Operation 420
Diagnosing Rotor Bow 424
Removing Rotor Bow 425
Summary 426
References 427

20 High Radial Loads and Misalignment 429

Static Radial Loads 429
What Is Misalignment? 432
Temperature Changes and Alignment 434
Causes of Misalignment 435
Symptoms of High Radial Load and Misalignment 437
Bearing Temperature 437
Vibration Changes 439
Stresses and Wear 440
Abnormal Average Shaft Centerline Position 440
Orbit Shape 445
Rub 445
Fluid-Induced Instability 446
Summary 447

21 Rub and Looseness 449

Rub and Looseness 449
Partial Radial Rub 450
Full Annular Rub 455
Rub-Induced Forces and Spring Stiffness Changes 456
Rub and Steady State IX Vibration 458
Rub and IX Vibration During Resonance 461
Subsynchronous Vibration 462
Symptoms of Rub 467
 Table of Contents xiii

Other Malfunctions with Similar Symptoms 470

Summary 473

22 Fluid-Induced Instability 475

The Cause of Fluid-Induced Instability 476
Modes of Instability: Whirl and Whip 480
Symptoms of Fluid-Induced Instability 486
Other Malfunctions That Can Produce Similar Symptoms 488
Locating the Source of Instability 492
Eliminating Fluid-Induced Instability 492
Summary 496

23 Externally Pressurized Bearings and Machinery Diagnostics 499

Types of Bearings 499
Internally Pressurized Fluid-Film Bearings 500
Externally Pressurized Fluid-Film Bearings 501
Stiffness and Modal Damping in Fluid-Film Bearings 503
Variable Stiffness in Internally Pressurized Bearings 504
Variable Stiffness in Externally Pressurized Bearings 508
Rotor Dynamic Implication of Variable Stiffness Bearings 509
Diagnostic Implications of Variable Stiffness Bearings 512
Summary 514
References 515

24 Shaft Cracks 517

Crack Initiation, Growth, and Fracture 517
Reduction of Shaft Stiffness Due To a Crack 520
Shaft Asymmetry and 2X Vibration 522
The First Rule of Crack Detection (IX) 524
The Second Rule of Crack Detection (2X) 526
Other Malfunctions That Produce IX Vibration Changes 529
Other Malfunctions That Produce 2X Vibration 530
Design and Operating Recommendations 530
Monitoring Recommendations 531
Summary 532
xiv Fundamentals of Rotating Machinery Diagnostics

Case Histories

25 High Vibration in a Syngas Compressor Train 537

Steady State Analysis 539
Transient Data Analysis 541
Inspection and Modification of the Machine 544
Summary 548

26 Chronic High Vibration in a Draft Fan 549

The First Balancing Attempt 551
The Real Problem 556

27 A Generator Vibration Puzzle 559

Unusual Vibration Behavior 561
Data Analysis 562
Conclusions 568

28 High Vibration in an Electric Motor 569

Data Analysis 571
Diagnosis 576

29 Problems with a Pipeline Compressor 579

The Machine Train 579
Tests With Pinned Seals 582
Tests With Unpinned Seals 588
Tests With Damper Bearing 588
Additional Analysis 591
Conclusions and Recommendations 595
References 596

Appendix

Al Phase Measurement Conventions 599

The Instrumentation Convention 599
The Mathematical Convention 600
Converting Between The Two Conventions 602
Phase On Bode and APHT Plots 603
 Table of Contents xv

A2 Filtered Orbit and Timebase Synthesis 607

Timebase Synthesis 608
Orbit Synthesis 612

A3 The Origin of the Tangential Stiffness Term 615

Modeling the Pressure Wedge 615
Tra ns format io n to Stationary Coord inates 619
Reference 620

A4 SAF Calculation 621

Peak Ratio Method 623
Phase Slope Method 624
Polar Plot Method 626

A5 Vector Transforms 629

Virtual Probe Rotation 629
The Forward and Reverse Tran sform and Full Spectrum 633
References 638

A6 Eigenvalues of the Rotor Model 639

The Threshold of Instability 644
References 645

A7 Units of Measurement 647

Metric/US Customary Unit Conversions 647
Unit Prefixes 650
Unit Abbreviations 651
Force. Weight, and Mass in the US Customary System 652

AS Nomenclature 655
Upper case Roman 655
Lower case Roman 656
Upper case Greek 658
Lower case Greek 658

Glossary 661
Index 709
About the Authors 723
About Bently Nevada 725
 xvii

Acknowledgments

ANYONE WHO HAS WRITTEN A BOOK knows that it takes a great many people
to make it a success. I envisioned this book more than fifteen years ago.
Patience, faith, and support made this book possible.
During the writing of this book, I had the help of many others who provided
information or reviewed the drafts. These people helped me add depth, breadth,
and clarification to the book.
Agnes Muszynska formalized much of the mathematics of the rotor dynam-
ic model that is presented in Chapter 10. Agnes developed some of the mathe-
matics on her own; we worked together on other of the mathematical models
contained in this book.
Several technical experts within the company provided me with a great deal
of in-depth, specialized knowledge. Bill Laws' strong background in large steam
turbines helped me improve the chapter on rotor bow. Throughout this project,
Ron Bosmans and Richard Thomas have been patient teachers and excellent
guides through the world of rotating machinery. Our many debates on obscure
aspects of rotating machinery behavior have been both interesting and inform-
ative, and they provided many subtle technical details that appear in this book.
Each chapter of this book has been thoroughly reviewed by experts with
many years of experience in machinery diagnostics. Ron Bosmans and Richard
Thomas acted as primary reviewers and read every chapter. Other reviewers
included Don Southwick, Rett Jesse, Paul Goldman, Wes Franklin, Bob
Hayashida, John Winterton, Rob Bloomquist, Clair Forland, Dave Whitefield,
Craig Sever, Mike Quinlan, and Pascal Steeves. We also obtained special help
from two talented engineers, Ingrid Foster and Susan McDole; their detailed
reviews of the appendix material kept me on my toes.
xviii Fundamentals of Rotating Machinery Diagnostics

The case histories in this book originated in the field with Bently Nevada
machinery specialists, and, when finished, were reviewed by them. In recreating
these events, we read their reports and articles and, whenever possible, dis-
cussed the details with them. Peyton Swan was a valuable source of information
for the compressor problem described in Chapter 25; Peyton is also an excellent
writer, and we gratefully borrowed material from an article he wrote for ORBIT
magazine. Peyton also worked with Kevin Farrell on the generator problem
described in Chapter 27. We had several interesting discussions about the
underlying rotor dynamic mechanism that was responsible for this unusual
behavior. John Kingham supplied additional information for the draft fan prob-
lem he encountered, which is described in Chapter 26. Rob Bloomquist provid-
ed considerable detail concerning the pipeline compressor problem that is
described in Chapter 29.
We want to thank Bob Grissom, who edited this book. Bob was an instruc-
tor in Bently Nevada courses for many years, and he possesses a broad knowl-
edge of the subjects covered. During the writing and editing process, Bob
reminded us of many technical details, which made his editing very thorough.
Because of Bob's effort, this is a much better book than it would have been with-
out him.
I am in debt to Walter Evans for his teachings on root locus. I worked with
Walter at Rocketdyne in Downey, California. I also attended classes at University
of California Los Angeles where Walter taught root locus and other principles of
control theory. I have used root locus techniques extensively throughout my
career; although over the years, I thought root locus had gone out of style. About
five years ago, I was visiting at California Polytechnic University in San Luis
Obispo, California. A professor there showed me the textbook, Modern Control
Engineering (Third Edition) by Katsuhiko Ogata that was being used at the uni-
versity. The principles presented in the book rely heavily on Walter Evans'
method of root locus. I hope that today's students will find root locus as useful
in their careers as I have found it.
Finally, it is important that Charlie Hatch's name appear with me on the
cover of this book. Charlie is more than a hired gun or a professional editor; he
is also a researcher. After earning his first degree in forestry, Charlie attended
University of California Berkeley, where he earned B.S. and M.S. degrees in
mechanical engineering. After graduation, he worked at Bently Nevada
Corporation as a production engineer and later transferred to the research lab-
oratory where he worked with Agnes Muszynska and me. His first job was to
attempt to build rheologic bearings, which are oil bearings with unique mag-
netic particles suspended inside. When this approach proved not to be feasible,
Charlie then helped write a paper on the behavior of damping on flexible rotor
 Acknowledgments xix

systems. This excellent work is taught at all Bently Nevada seminars. Charlie and
I have since worked collaboratively to document several other research study
results and projects. I taught Charlie root locus methods, and he promptly
became an expert on it. It was only natural that he would be my choice to help
develop this book. In addition to collating and editing, Charlie was an inspira-
tional collaborator and contributing researcher on the many ideas that appear
in this book.

Donald E. Bently
Minden, NV
March 11, 2002
 xxi

Foreword

ROTATING MACHINERY VIBRATION ANALYSIS REQUIRES the use of principles

that are still quite unfamiliar to many mechanical engineers. These principles
are probably the least understood of those in any other field, yet are critical to
the design, operation, and diagnosis of high-speed, high -power machinery. Over
the past 100 years, misconceptions, misstatements, and mistakes in the descrip-
tion of rotor dynamics have compounded the problems.
In this age of detailed mathematical study of shaft dynamics, the rapidly
advancing technology is not being properly communicated to the practicing
engineers and engineering students in straightforward, compelling terms.
Certainly, these days, most engineers do not have the time to digest all the pub-
lished material. One of the most powerful new ideas is Dynamic Stiffness.
The vibration we measure is a ratio, the ratio of the dynamic force to the
Dynamic Stiffness of the machine. This book clearly shows how to use Dynamic
Stiffness to understand and recognize malfunction behavior. It is also a single
source for the description of the fundamental principles of rotor dynamics and
how machinery behaves. It corrects the misconceptions that have plagued the
discipline and opens new territory and routes to understanding the dynamics of
rotating machinery.
For example, in existing literature, the cross stiffness terms, K xy and Kyx ' are
treated as independent variables. We call these terms quadrature terms, which
have a very simple relationship. The "cross stiffness" is actually a tangential stiff-
ness term (quadrature term) that acts perpendicular to the direction of dis-
placement. The tangential stiffness term, DAD, is defined in basic rotor dynam-
ic parameters, which are much more useful when you 're trying to diagnose
machinery operation.
Exploring new territory is always a fantastic adventure, and never without
problems. In exploring the basic nature of rotating machinery, I regularly hit
unforeseen cliffs, swamps, or other impediments. Looking back, having solved
the problem, these pitfalls are interesting.
xxii Fundamentals of Rotating Machinery Diagnostics

Crossing into new territory, it sometimes was necessary to tread on old tra-
ditions where these traditions were wrong, or were nearly correct but had been
slightly misinterpreted. Great resistance to progress was, therefore, encountered
from people who had an incorrect view of the theory.
Since the invention of rotating machines, the pursuit of higher power out-
put has driven machine speeds higher and higher. With the breaking of the first
balance resonance "barrier" (achieved by De Laval with a steam turbine in 1895),
rotating machines were shown to be able to operate above the first balance res-
onance. However, with this new capability came a new problem for machines
using fluid-lubricated journal bearings: fluid-induced instability. Over the years,
many different methods have been developed by researchers to identify and
understand the important parameters that influence rotor stability and, so,
increase the reliability of the machinery.
Reliability is often thought to be synonymous with long, trouble-free life,
and improved reliability to mean a longer, trouble-free life. But these are not
acceptable definitions. A machine or component becomes reliable when its
operation and actions are predictable. The accuracy with which these actions
may be predicted is a true measure of its reliability. It follows, then, that reliabil-
ity can best be improved by learning as much as possible about equipment oper-
ation and using this knowledge to reduce or eliminate as many unpredictable
items as possible. Accurate predictions require accurate, meaningful data from
which analysis can be made. When you have the data necessary to make accu-
rate predictions of machine operations, you also have the data to improve
designs, extend the life of components, probably even reduce its cost and
increase its safety.
Meaningful information is the key. This book is a major step in assuring that
good data can become meaningful information through the increased knowl-
edge of the machinery specialist. It is a well-constructed foundation of the
bridge to the future.
Machinery technology is rapidly changing, and new developments are
always making their way into machines. One very promising new technology is
the externally pressurized bearing, which Bently Nevada is developing. This
bearing is an externally pressurized (hydrostatic), fluid-film bearing that can be
operated in a passive mode, a semi-active mode, or in a fully active mode. In the
passive mode, the bearing operates with a fixed design pressure and, by exten-
sion, fixed-by-design spring stiffness and damping. In the semi-active mode, the
external supply pressure can be adjusted under operator control to change the
values of stiffness and damping while the machine is operating. In its active
mode, it is capable of producing fully automatic, instantaneous changes in stiff-
ness and damping to control the rotor position in real time.
 Foreword xxiii

In June 2001, we demonstrated suppression of oil whirl by increasing bear-

ing pressure at the International Gas Turbine Show in Munich, Germany. In
August 2001, we demonstrated the suppresion of oil whip. This was the first
demonstration of a supplementary bearing in the central span of a rotating
machine.
These two successful innovations, never performed before in history, do not
solve all instability problems, but they certainly make it possible to control two
obvious problems that have presented challenges for rotating engineers for
man y decades.
This new technology promises to change the way machines respond dynam-
ically and will require changes in the way we interpret and apply machinery
data.
For example, the balance resonance is usually thought of as occurring at a
fixed operating speed, where running speed coincides with a fixed rotor system
natural frequency. With a semi- or fully active bearing, the natural frequency
and balance resonance speed now become variables under the machine opera-
tor's control. By changing the bearing spring stiffness in semi-active mode, the
balance resonance can be quickly moved to another speed, enabling the opera-
tor or machine control system to jump the resonance rapidly through the
machine during startup or shutdown. This behavior will greatly alter, even elim-
inate, the usual balance resonance signature in a polar or Bode plot.
Changes in the balance resonance speed will also affect balancing. Active
shifting of resonances will make polar plots look different, changing the way we
identify the heavy spot. If a resonance is shifted to a different speed, then heavy
spot / high spot relationships may change. For example, what was above a reso-
nance might now be below, or vice versa. Response that was out of phase might
now be in phase. Influence vectors may depend on bearing settings, and
repeatability will require simil ar bearing settings.
Changes in bearing stiffness can also change the rotor mode shape. A mode
associated with low bearing stiffness, for example, a rigid body mode, could be
modified by higher bearing stiffness to a bending mode. This change in mode
shape could change the match to the unbalance distribution, producing a
change in balance state. It is possible that the existing unbalance distribution
would become a better or poorer match to the new mode shape, and that the
rotor would have to be balanced specifically at particular bearing settings.
Some malfunctions manifest themselves as a self-excited vibration at a sys-
tem natural frequency. Because of the new, variable nature of the balance reso-
nance, this natural frequency will exist somewhere in a frequency band, which
will depend on the range of bearing settings and their effect on rotor modal stiff-
ness. Under some circumstances, the bearing will allow the operator to move the
xxiv Fundamentals of Rotating Machinery Diagnostics

natural frequency to a place where the malfunction vibration ca n no t occur. The

diagnostician will need to understand how this kind of variable-parameter bear-
ing operation will affect his or her interpretation of the data, and how it can be
used to suppress unwanted vibration.
New technology will give us awesome new opportunities and new chal-
len ges. No matter what new de velopments occur, the fundam ental principles of
rotor dynamics presented in this book will remain the same. The ma chinery diag-
nostician who has a solid foundation in the fundamentals will be able to apply
th e basic principles presented in thi s book and solve machinery problems.
 xxvii

Introduction

WHY READ THIS BOOK? If you are responsible for the maintenance or opera-
tion of industrial rotating machinery, you know that catastrophic failure of a
critical machine, large or small, can cause serious injury or death, result in the
total loss of the machine, shut down the plant for an extended period, and be a
public relations nightmare. For these reasons, it is not acceptable to wait until a
machine fails before fixing a problem; the machinery manager must take a
proactive stance. This book will give you the knowledge you need to detect prob-
lems with your machine before they cause economic losses associated with
decreased plant efficiency, unplanned downtime, damage, or a serious loss of
production.
This book will help you to understand the basic principles of machinery
behavior that are common to all machines, ranging from very large steam and
gas turbine generator sets in the power industry, to steam and gas turbine-driv-
en compressors in the petrochemical industry, to motor-driven induced draft
fans, cooling tower fans, blowers, and large and small pumps.
It will also give you a solid foundation in machinery diagnostics, the body of
knowledge and technique that is used to identify the root cause of a machine
malfunction through the use of vibration, position, and process data. Machinery
diagnostics is a science in the sense that, during the diagnostic process, a
hypothesis is formed that must be supported (or rejected) by the data and veri-
fied by inspection or corrective action. It is also an art in the sense that it
requires detection of a meaningful pattern in what is often a bewildering array
of data. Whether viewed as science or art, it first of all requires knowledge: the
diagnostician must have a solid understanding of basic rotor dynamic behavior
and of the various malfunction signal characteristics.
This book presents the fundamentals of that knowledge largely from an
intuitive and practical, rather than theoretical, point of view. It is written for any-
one who is responsible for the operation, maintenance, management, or mal-
function diagnosis of rotating machinery. It also provides an information
:xviii Fundamentals of Rotating Machinery Diagnostics

resource for those who write technical standards, or design transducers, moni-
toring systems, or software packages for rotating machinery application. Thi s
book also provides a valuable resource for the machinery designer; awareness
and application of the basic principles in this book are essential to a good,
robust machine design.
This book covers much of the material presented in Bently Nevada diagnos-
tics courses over the years. These courses have long been recognized as some of
the best in the world, but they are, by their nature, limited. This book greatly
extends the depth of the material and provides a readily available reference.
The first section of the book, Chapters 1 through 3, presents the basic con-
cepts of vibration, phase, and vibration vectors. Phase can, at first glance, be dif-
ficult to understand; because of this, it is often a neglected facet of machinery
data. This is unfortunate, because the timing information it provides is a pow-
erful tool; without phase, diagnosis becomes much more difficult, and efficient
balancing is not possible. I hope that the discussion in Chapter 2 will help clari-
fY this topic.
In vibration analysis, "vector" data is an important tool. Vibration vectors
are actually complex numbers, which simplify calculations involving amplitude
and phase. It is vital for the machinery diagnostician to understand their mean-
ing and use. Chapter 3 discusses vibration vectors in detail, and this chapter
should be thoroughly mastered. Throughout this book, vibration vectors, which
possess both amplitude and phase, appear as italic boldface, and scalars, which
possess only amplitude appear as italic.
Data must be presented in a meaningful manner, and, to enhance commu-
nication, it must conform to accepted standards. The second section, Chapters
4 through 9, discusses the many different kinds of data plots that can be creat-
ed from ma chinery data and how to construct and interpret them: timebase and
orbit plots; average shaft centerline plots; polar, Bode, and API-IT plots; spec-
trum plots; and trend and X'Y plots. Each chapter contains many examples of
data from actual machines.
The next section looks at rotor dynamic behavior, starting in Chapter 10
with the development of a basic rotor dynamic model. A result of the model is a
powerful new insight, Dynamic Stiffness, which is discussed in terms of rotor
behavior in Chapter 11. Other chapters in this section deal with modes of'vibra-
tion, the behavior of rotor systems with anisotropic stiffness, rotor stability
analysis using root locus techniques, and torsional and axial vibration. The sec-
tion ends with an introduction to balancing of rotors.
The fourth section introduces the most common rotor system malfunctions
and the signal characteristics that can be used for their detection. The malfunc-
tions include unbalan ce, rotor bow, radial loads and misalignment, rub and
 Introduction xxix

looseness, fluid-induced instability, and shaft cracks. Each chapter also lists
other malfunctions that may have similar symptoms and provides guidelines for
discriminating between them.
In the last section, several case histories show how this knowledge was
applied in the real world to solve machinery problems. The case histories are
well illustrated with data, and they discuss the sequence of thought that led to
the solution. Every effort was made to present the events and data as accurate-
ly as possible, while protecting the privacy of our customers. Thus, certain
details are fictionalized, but the data you will see is real, the problems you will
read about did happen, and the resolution of the problems were as described.
Finally, the Appendix contains additional technical information for those
who wish to pursue some topics further, as well as lists of common unit conver-
sions and a glossary of machinery diagnostic terms.
For the most part, the material in this book is presented with a minimum of
mathematics, but it cannot be avoided completely. The general reader should
have a working knowledge of algebra and basic trigonometric functions; the
advanced reader will benefit from a knowledge of differential equations, which
are used in the development of the rotor model in Chapter 10 and in some mate-
rial in Chapter 14. For those without this background, the more difficult mathe-
matics can be skipped without a loss of understanding; the key concepts are
always stated with a minimum of mathematics. It is more important to come
away with a good understanding of the basic principles than to be able to dupli-
cate a complicated derivation from memory.
This book primarily uses metric (SI) units of measurement, followed by US
customary units in parentheses. At least that was the original intent.
Unfortunately (or fortunately, depending on where you live in the world), much
of the data that is used to illustrate this book originated as US customary meas-
urements. Rather than attempt to convert all the data to metric, data plots are
presented in whatever units of measurement were used when the data was
taken. Thus, the reader will find many places in the book where the discussion
is conducted in US customary units, followed by metric. I apologize for the
inconvenience and ask for the reader's patience.
As in so many things, this book represents only a starting point; as the title
says, it presents the fundamentals of rotating machinery diagnostics. The world
of rotating machinery is extremely complex, and the science of rotor dynamics
is young; that is what makes it so interesting. No single book can possibly
address this topic in its full extent. I hope that it will help those getting started
in this field, while at the same time providing new insight and serving as a use-
ful reference for experienced practitioners. We are all in a continuous process of
learning.
Fundamentals of Vibration
 3

Chapter 1

Vibration

~BRATION IS THE PERIODIC, BACK AND FORTH MOTION (oscillation) of an

object. We encounter vibration in many different ways in our daily lives. Nearly
all musical instruments utilize the periodic vibration of mechanical elements to
make sound; for example, pianos and guitars use the vibration of a string and
connected soundboard, clarinets use the vibration of a small reed, and trumpets
use the vibration of the player's lips.
Vibration also exists in nature. The motion of the tides is an example of a
very low frequency vibration that is produced by the gravitational force of the
moon and sun. This motion is an example offorced vibration (and resonance, in
the case of the Bay of Fundy). A sudden gust of wind acting on a tall pine tree
can also produce a periodic, low frequency vibration of the tree, an example of
free vibration resulting from an initial impulse. The wind blowing on aspen
leaves produces a continuous, periodic motion of the leaves, an example of self-
excited vibration.
Machines, because of internal and external forces, also vibrate. Machinery
vibration involves the periodic motion of rotors, casing, piping, and foundation
systems, all at the same time. Usually this vibration is so small that sensitive
equipment is needed to detect it. To illustrate the small size of machinery vibra-
tion, we can compare it to the diameter of a human hair. The average diameter
is approximately 130 ~m (about 5 mil). This is an unacceptable vibration level on
some steam turbine generator sets that are the length of a house.
Vibration in machines causes periodic stresses in machine parts, which can
lead to fatigue failure. If the motion due to vibration is severe enough, it can
cause machine parts to come into unwanted contact, causing wear or damage.
 Becau..... o f t hi•• the co nt rol of ,i b. al ion i~ an imp.." a nt 1"'" n f machi,,"y
ma " 8.jl<'ment. a nd Ihe roncepu u nde. Jyin!l \ib ,at;on n'''~t be th.......ughly u n<lt-r-
, tood ~- Ihe mach ilK'")' p.of...... ion ...!. Th;s ch ...pt ... will d i""lL" the "".ic ron·
....1'1. ...ni near , 'ibration. the ,ibratio n of machin~ . ot a t;on. a nd t he on"" rom·
monly u M"d ,i b ration m..as ur..ment units.11I..n, ..... will Inl'·.-...." Io a disc ussi on
o f t h.. con pl. offr... \i b. a tion at ... na tu ral f'eque nq . forero vihratlon. a " d t ha i
m,..1 in t li n!l ma. riagE' of th.. r.."O. ",,,, na ...., ,. Fina lly. we " i ll disc" .. ..-If·
..xcited ,i b. al ion. ""he,,, a .}'<I .. m el'" i ntetnall~ I. a nd e r e nt''ll..'' to p rod ......
,i b. a lion at a nalU. a l f. eq" " O(:y.

The 8 a sk Vib , a tio n Sig n a l

A vibration tronstlua!r i. ... drtice Ihal , ,,\-...1. m<'<"ha nical mot ion int o an
..1<-'CI . o nic , i!lnal. A d i'placeme nt I. a nsd u r ("a " .... uoed 10 m"", ,,... t .... d i.<·
plac..m..nt. or po s il ",n, o f a n •••j« 1 . ......t i\ 10 the transd ul..... f or most tra n. ·
d u....... t his is. id..a lJy a t I.....t. ... o rw--d ime" . io nal mea,ur<"m..nl. If t .... o bjKt i.<

MgU'" 1-' ,1M rel.. _ i p 01. d,'plo<; nt ,,1><_ "'J""'to ~ rnoo:iotl of ..,
objo'<t_.... tho objKt _ ...._ 1 0 ~ mg ".nsduo'<. • ~"'iI- -
oge "9"'" is cr....... i"",~ A """""9 P<Wl"'" vKlOr 11<'<1) """,.......... orrul.or fr..-
QOnO(y. _ ' lG_ ~ Col"" omeg .~ 1M pIOlKt>On 01 ,"'. _ _ on ~ If"""""'"
_ J"'PJ<""""
• • is b'. . II... aetu.l1 d;,pl;oc........." oI,~ object.
 Chapter 1 Vibration 5

vibrating, the position of the object relative to the transducer will change in a
repeating pattern over time.
Figure 1-1 shows an object that is vibrating toward and away from a trans-
ducer. The different images show the evolution of the system over time. The
transducer converts the position of the box on the transducer's sensitive axis to
an output voltage, which is displayed at the top of the figure. The transducer
output voltage, or signal, is proportional to the distance from the transducer to
the object, the gap. This changing voltage signal represents the relative position
of the vibrating object versus time.
Note that the waveform reaches a maximum value on the plot when the
object is closest to the transducer and a minimum value when the object is far-
thest from the transducer.
There are two primary characteristics that we can measure on this signal,
frequency and amplitude. This signal has a simple, sinusoidal shape, and it con-
tains only one frequency. More complex (and typical) signals contain several
frequencies of vibration with different amplitudes.

Frequency
Frequency is the repetition rate of vibration per unit of time. The vibration
signal in Figure 1-1 has only one frequency. The frequency of this signal is found
by measuring the amount of time it takes to complete one cycle of vibration
(Figure 1-3). This length of time is called the period, T, and is shown in the fig-
ure. It has units of seconds per cycle of vibration. The frequency,J, has units of
cycles/second, or hertz (Hz) and is the reciprocal of the period in seconds:

1
f(Hz)=- (1-1)
T

In rotating machinery applications, we are often interested in expressing the

frequency in cycles per minute, or cpm, so that the frequency can be directly
compared to the rotative speed of the machine, measured in revolutions per
minute, or rpm. The frequency in cpm can be calculated from the period using
this expression:

f (cpm) = (f s
s
cycles ) (60__. ) = 60
min T
(1-2)
6 Fundamentals of Vibration

There are 27f radians or 360 in a circle ; this concept can be extended to say
0

that th ere are 27f radians or 360 in one cycle of vibration. Thus, the frequency
0

can also be expressed in radians/second (rad/s):

W(ra d/ S) = [27f rad] (fcYcles)=27f (1-3)

cycle s T

The frequency w (Greek lower case omega) is sometimes called the circular
frequ ency . The red vector in Figure 1-1 rotates at the circular frequency w and
completes one revolution for each cycle of vibration.
The rotating vector may seem like a complicated description for the simple
system shown in the figure. But, in single-frequency vibration in rotating
machinery, the rotor shaft moves in a two-dimensional circular or elliptical path,
and the rotating vector is a logical model of its motion. In general, w will be used
to represent the vibration frequency of the rotor system. The one-dimensional
position of a rotor can be expressed as the projection of the rotating vector onto
the axis of sensitivity of the transducer (the yellow vector in Figure 1-1).
Several terms are commonly used to describe frequency ranges in machin-
ery (Figure 1-2):

Synchronous, or lX. The same as rotor speed. The "X" is equ ivalent
to a mathematical multiplication symbol. Thus, IX can be read
as "I times rotor speed."

Nonsynchronous. Any frequency except IX.

Subsyn chronous . Any frequency less than IX. This can include sim -
ple integer ratios such as %X, %X, etc., decimal ratios such as
0.48X, 0.37X, etc. , or subharmonics (see below).

Supersynchronous. Any frequency greater than IX. This can include

simple integer ratios such as %X, %X, etc., decimal ratios such
as 1.6X, 1.8X, etc., or superharmonics (see below).

Subharmonic. A frequency less than IX that is an integer ratio with

one in the numerator: for example, Y2X, V3X, ~ X, etc.

Superharmonic. A frequency greater than IX that is an integer mul -

tiple: for example, 2X, 3X, 4X, etc .
 Chapter 1 Vibration 7

Synchronous
Subharmonics I Superharmonics
1/4X 1/3X 1/2X 1X 2X 3X 4X

Subsynchronous Supersynchronous Region

Region

Figure 1-2. Machinery vibration frequency definitions. Synchronous, or 1X,is equivalent to the
running speed of the mach ine.

Amplitude
Amplitude is the magnitude of vibration expressed in terms of signal level
(for example, millivolts or milliamps) or in engineering units (for example,
micrometers or mils, millimeters per second or inches per second, etc.).
The amplitude can be measured using several methods. One is to measure
the total voltage change from the minimum of the signal to the maximum of the
signal. This method is used for displacement signals and is referred to as double
amplitude, or peak-to-peak, abbreviated pp. In Figure 1-3, the peak-to-peak volt-
age change represents a total change in position of 120 urn (4.7 mil).
Besides being used for the simple signal shown here, peak-to-peak ampli-
tude is well-suited for the measurement and evaluation of complex vibration
signals that contain many different frequencies. Often, the machinery diagnos-
tician simply needs to know how much the machine rotor is vibrating relative to
the available diametral clearance in the machine. Peak-to-peak measurement
makes this comparison relatively easy.
The single amplitude, or peak method, abbreviated pk, measures the voltage
change from the middle of the signal to the maximum value of the signal. This
method yields an amplitude that is one half of the peak-to-peak value. Th is
method of measurement is commonly used for velocity and acceleration vibra-

Figure 1-3.Three methods of amplitude

measurement of a single-frequency (sine
wave) signal.The period , T, represents the
time to complete one cycle of vib ration.

J_
I--- T----l
l ion ';lI:na l... bul is not " Ihuitro 10 l he m..a,u"-"llK'nl of d i'J'la....llK'nt silt"al.
for th....... son ~iwn abO' .
\\"h.. n ma th"",at ically m od d inl! rolo r w,l..m b.-ha''io•. l oc ..m plitud.. of
,il>",tion g i'...n ~. a mod..l is <'<1ui'·ak nl to Ih.. p<'ak mct ho d . 1 h is can som ..-
l im.... I.. ad 10 con fu.io n " .....n Ih.. a mpl ilu d.. of di.pl......m..n t , ibral ion p. roicl -
ro by a mod.. 1 i. compa rcd th .. p<'ak-lo -p<'a k ,il>ration "",asu rcd on a mac hin...
..\ ""t h..r potcnl ial co nfusio n ca n ocrur wh..n th .. Uyna mi<: <;t iffn"". of a
m ad lln.. (a p.-ak kind o f para nt..t..r ba..-<l on a mod..l) i. cal rulatoo usmg p..a k-
lO-p<'a k " bra tion data.
II is int po n a nt 10 II<' awa... of a nd """v t.ack of th .. d,ff..... n.... bo>t"'....n Ih..
p<'a k-to- p" ak a oo p<'a k nt....hods. It i. ..... y to g<>t co n fu..-<l about ..t1.i<:h syst..m
i. lI<'ing u..-<la nd ..nd up ..,t h a n .. ITO. o f a fac lor of 1" "0_ All .ib",llo n ""'a ....
m<>nU ....o uld bE> wri lt ..n do...n " i l h th.. compl..t.. u nils of m..a. ......m..nt. M ly
... rili ng 250 ",m o r 10 mil ia "01 enough a nd is a n in. -ila tion to e mbarra.."",nl_ Is
it p<'ak. o r p"..k-Io-p.-ak? Co mple le un " s ....ouId bo> ...",ten in all nut... and <"41-
ru lations 10 a,"Oid <:onf..., ion: wrile 2.'; () Il'" 1'1' ( 10 mil pp). o r l.'ilJ Il'" I' k (\ 0 mil
p kl.
Th.. TOOf-m M n -."lfUfJre mMhntl. " hh.....1aIt"d rm", d '-"'Cr'~ Iii.- amplilu oJ.. of
" co n tin uo..., ly c ha ng inll s i/lna l "" a fo. m o f a'-""'Il"- As lh .. na m.. , u!lIl<'st il is
calculat ed by la king rh.. "'j ua", .... .t of the "'....n. , or a'....a~, of Ih.. ..Iu of
Iii.- sig nal ,.a1 .......
If. tlnd only if. II... signal is a ain.. wa.... (s ing!.. freq....n")" ). th<' noa a"' plituoJ..
will Ii.- <'qual W 0.707 l im... Iii.- petJi a ml>lit ud... a nd it " i l] lw "'lua l to O~
I, ,,,... th e ('f'tIk-lo-pMi a mpl ilude. If th .. si/(n" I,. fUJI a sin ' Iht"JI the . ms
,-alue u. ing lhis s imp l.. calcula tio n " ill '101 bo> corrt'ct. M t m hin.. vib ration
sill:nala a... not s ine ..·a' ..... Inatead. Ih"Y conta in a m ixl u of d, ff nl frcquen -
cil'S (Fil(ll'" J- t).

fi<J".. ' -I ~ llibr"'ion oqn.ot _ conta....

_,tdolle<"""- ft<oq~ n-.. ~I\OI
......, tdOl>! ""'" once _ ~ut _ '"
m.. ""'" ""It ~ (hapt~ 2~
 ( hapt'" 1 v ;o'.>tion 9

Displacement. Ve loci t y. a nd Au e le ra lio n

Di.spJacrmen' de serioc.. l he po. it ~ ln of ~ n oo;.-ct. I Wocily d.,,;crioc.. ho ....
,apidl~' the object i5 rna nging !,os il;"'" ....ith lime. and IlCCl"It>ro.lWn dO'SCrit.....
how fa sl th .. wlooty chanjW< wil h lime. Figu", 1·5 <.I1, ,,,·s a n ""';llali ng ""oou·
lurn ob..·" ....J by a d isplu ...men l l ra n<duCt'. a nd plots of d is ploc.-m<'fl t, Ioci ty.
and aCCE'l..mtion. Th.. di 'J'la.....me nl o f th .. .... nduJum i. m..as ur<'d relat i Lo I....
....rt ical. rest pos ition.
:">01.. Ihat Ih.. ""ak .....loc ity o('c u,," ..-hen l he P<'nd ulum I""'....s Ih rou gh l he
",u ical po.itio n .>nd i5 mm;n~ tow ard Ihe I ran,.,luC'(',. As Ihe ""nd ulu m , ,,,,rn '
... Ihe ..nd o f ilS mot io n an d is cIo.... t to l he Im n....u'...,- (th.. m<lU imum IN...iti...•
,-aI Ul" in Ih.. displac..me nt plot l. t h.. , -nocily is zero. morn.. n t n ~ y. T/If;' I"' nd ulum
Start5 1fIO\;fL/l. back in t .... Of'J"'",il.. d i......-t,o n ",i t h ....gat" -el"cily. Wh e n t he
""nduJum pa~ Ih' OIl¢l I.... .....rt ical pos it io n alt" in, Ih.. , Iocily . eaches in
max imum ....gat i..... .·a l..... Wh..n t he ""nduJum ",ache. th .. o p"""ile displac<-
n,..nt ..xneme. Ihe displacrm..nt is al l he mimm um o n lh .. ,Ji5p l...", n,e nl plot,
and Ihe ,..Io..ily i. aga in lero.
fkca """ tho:- ""nduJu m is d ri",n b)' gt3"ty. I .... aCCE'l..mtio n i5 l .. m " 'hen Ih..
pe nd u lum is in tho:- rt ical l'''''tion. .-\s th .. di5ploc.-m..nl approach... t he po,; -
ti, a k n.... r th .. l n.sdu....... th.. fo rce d ue to gt3" ty is act ing in such a W dV a s
to wd lho:- ,..I"dly. T hu~ t he a.......... rat ion is n"lta ti' .... and il wache. 11-" max -
im um nep t i..... ..,.]u.. ! I.... mm imu m on th.. ploll .......n Ihe displa m..nt is ma x-
im um. The n. a ' Ihe pe nd ulum 51a rt s back a"'a~' fro m l he t ran,.,I" th .. nega -
Ii,.. alu"'.atio n he<'om.... . rnalle~ h..adi"ll up I_'ard t h.. l .. ro crossing "n rh..

/ -, Di><I'o<.........

_
--
(;gu,. 1-S n-.. motJOI' at .. I"'roduIum C......... at d;sp.cemoonl (9'","nl.veIoctty (bluei.

_.Ior
iOCC...... ..,., (""''''J<').e """"" --.... M'e n-.. jeIIow doI:. -.prnen1 the "9.....
t l>e pend"'m I'O'fhDr' <hawn.
10 Fundamentals of Vibration

plot. When the pendulum reaches the opposite extreme, gravity is trying to stop
the pendulum and push it back toward the transducer. At this point, the accel-
eration reaches a maximum positive value on the plot.
For single-frequency signals (sine waves) only, such as shown in the illustra-
tion, there is a simple mathematical relationship between displacement, veloci-
ty, and acceleration:

d = Asin(wt)
v = Aw sin (wt + 90°) = d (1-4)
a = Aw2sin(wt+180 0)= -Aw2sin(wt) = v= d

where d is the displacement, v is the velocity, a is the acceleration, A is the ampli-

tude of the displacement (the maximum displacement possible, expressed as
either pk, pp , or rms), and w is the frequency of vibration in rad/s. The numbers
0 0
90 and 180 represent the relative phase, or timing between the signals. The dots
C, ..) represent the first and second derivatives with respect to time (d/dt and
d 2/dt 2 ) .
In Equations 1-4, velocity leads displacement by 90 that is, it reaches its
0
;

maximum one quarter of a cycle, or 90 before (the reason for the plus sign) the
0
,

displacement maximum. Figure 1-5 shows a set of plots of Equations 1-4; note
that the velocity maximum occurs when the displacement is zero and rising.
Acceleration leads displacement by 180 and acceleration leads velocity by
0
,

180 0
-90 = 90 In the plot, the acceleration maximum occurs when the veloc-
0 0
•

ity is zero and rising. and the displacement and acceleration change in opposite
directions.
Important note: the phase angles here are based on a mathematical defini-
tion of phase. not on phase measured by instrumentation. See Appendix 1 for a
discussion of this important difference.
As shown in Equations 1-4. the amplitude of the velocity is related to the
amplitude of displacement by a factor of w. Similarly, the amplitude of accelera-
tion is related to the amplitude of displacement by a facto r of w 2• This has
important implications for transducer selection, because the amplitude of
velocity and acceleration signals can become very small at low frequencies.
Why do we care about velocity and acceleration if we want to know the dis-
placement vibration of the machine? Inside the machine, we can mount a dis-
placement transducer on the machine casing or bearing structure and measure
the displacement of the shaft relative to the casing. On the outside of the
machine. displacement must be measured relative to something, and it is
impossible to measure the displacement of a machine casing with a displace-
ment transducer mounted on the casing. For this reason, velocity and accelera-
 Chapter 1 Vibration 11

tion transducers are used for ca sing vibration measurements on machines.

These transducers provide their own internal inertial reference, and they can be
mounted directly on the machine casing to provide casing vibration informa-
tion.
Because these transducers are used for machine monitoring, it is important
to understand the amplitude and phase relationships among displacement,
velocity, and acceleration signals. Phase will be covered in detail in Chapter 2.

The Vibration of Machines

Machine rotors rotate and, because of forces in the machine, they also move
laterally, or radially, in a plane perpendicular to the axis of the machine. This
motion is periodic and usually occurs at the same frequency as rotor speed (pri-
marily in response to unbalance), but can occur at frequencies below or above
rotor speed. Often, several frequencies of vibration are present at the same time.
To visualize rotor vibration in its simplest form, imagine a short piece of
metal rod (for example, a metal coat hanger) that is bent into a bowed shape. As
the rod is rotated about its long axis, a point on the bowed part of the rod will
move in a circle about this axis. This path of this point is called an orbit. In this
example, the point will complete one orbit for each revolution of the rod, an
example of IX vibration.
All machine components (the rotor, machine casing, piping system, and
support structures) can vibrate in several different directions. As already men-
tioned, radial vibration takes place in the plane (the XY plane) perpendicular to
the rotor axis of the machine. Axial vibration takes place in a direction parallel
to the rotor axis (the Z axis).
Angular vibration (the periodic change of the angular orientation of the
machine component) is also often present. Some angular deflections are detect-
ed by radial vibration transducers, because the angular motion has a radial
vibration component. Torsional vibration is a special case of angular vibration,
where the relative angular deflection appears as a periodic twisting of the rotor
shaft. This twisting of the shaft does not directly produce a radial vibration com-
ponent and cannot be directly detected by radial vibration transducers.
Displacement transducers are mounted on the machine casing to observe
the motion of the shaft. If the machine casing were absolutely still, then the
rotor vibration would be measured relative to an absolutely motionless trans-
ducer, and the rotor vibration signal would be referred to as shaft absolute vibra-
tion. Shaft absolute vibration refers to the motion of the shaft measured relative
to a fixed (inertial) reference frame.
Usually, however, there is some casing vibration present (Figure 1-6). This
happens because forces are acting on the case, including the dynamic forces of
I.... "; hral in !l n olor tha i .r.. lra n.miUN th roll« h th.. twa nng> m ltl th.. "*""ll
'lC ructu ....., Th..... ~'n.omic f _ = Ih.. r.~ I"ll IOVlhr-. t.., too. TIM' a mounl rJI
r"';"{l vihr.c-. df.pt>ndo 0fI1he' ... t.ln m&.~_ 011.... rotor and cao;~ m.. !IIltt-
....... oilhe' ring.., and tM oliff,...,.. ot 1M 'ltru<1"r.. lhat "'Pl_h I"'" ra.. n,l(
itwlf. I\ooa ~~ ' lbr.otion aho·~ nisl:s in nl, 1M d ispbamml
lra nod if. .. I in motion, .. nd m.. nod 1"'" i. rmn-.-d to u
tIuIfI- nolan... nbnu..... In m"dllnn " '1Ih tw ). .. ...u· pponN r ....;~ liklo
h,,- J"'"'I"'" enml"'"'""""" and rno<l.largf- "' m I" rw:s, cas.Infl "brallOII and.
Ihri .. fut d~1 tran ...,..,. molll . ... ;.. oil small mmputd III snJl
nbniI In Ih ;" raw, Iht nod .n.J'1 noIal;" "braI-. IS . ~ iIIJ'P"'D'''
mat iun Itl shaft ablloh K... I on machi....... _h I~~ ca.~ "" ooft
...pport .. " .. "It nbrilII - . roOn bot lIigniflQl nl 1"-: o.IYft no!aIn-... ....odi... d ,f·
r.... " n lrKa ..u,. from ..... ft ab<.oIuI .
C - "II n brAUnn IS Also tranonllllN I".nd from .n~ SllfTOUndl"{l pip"'fl '~
I ...,,, th-..,d> th.. pipO"{l COI>I'IoO'(1._ on I .... ..-hi..... .'U>o. ..;brat.... can bt
Ua""""' UN ;nlo Iht ft,u n.latlOll .nd thmu,d> tht buil.h"(l: t.lructur .. to othn.
'""<It<" local ......... 11 ill q"ilO' iX""'ihl.- fut OIomll tnInod"'...... on ...... matli.,.. 10
d<"l <"<1 ' lho-_ "", origln"ti"ll in .noI~ ....,,~. mathi.....
R.d.a1 \ l hralk>n i. t"'"
mO"I rom mtlnJ~ oul"l'd t~l ... ..f ' i l>noh o n. p," t ly
bo·..." il io \ ..........\ 10 " JI""' rad ial ' i l>,-al ion m.~· ", ~ ..n.1IYII ....
thO' "' 1 .ijlo iflu nl ";!mII o of \ ' Y" nn- fu r ..a..mp lr. .....1 "; l>ral lon {'li n
I>o:-rc.me If.. m.. In ""Ilti nlt " m p nd I" m on a! "hral;"n, th.. lTI06l

I-
f loJu.. 1-6 ... com po'""", d

---
'YPl'" d tNCIIO... ""'0''''''
~C-"J ...-.. ..
_ ohon l9'ft'!l i o _
_ .... "' ........... .1'otdI tef. •
_ ft _ _ .......

--
(-.g
SI\oIt l r _........ - . . .• .
- -
_ b , - ...... ~
•
I•
"'IJ.---"
" . lI/'IC>ooOII. _
1 .... .
_ " " 1I'>e_....
.......... d_.-.g _
_ "'"" _ """""'''CMl
bo taver "'aIr
f6 . . . . . . _
_ _ _ """,,",do og ""

d .... ;.,;'~""~;_'"
N,' _ ,~ ....
--- C> <:7C><:7 C>
<:7
 (hapt... I Vibralion 13

d iltic ult o f all 10 .....a ca n "'oc h d.... mcti"" I....... ~ ,,~ t h ......y lill l.. otl tsid..
ind ica tion o f it~ n l~l ..n .

ROlat ion and Pre<M sion

Roiatton is Ih.. a ngul ar mol lo n o f l he rot or a boUI ItS g...........tric .....n t.. ~ or
"'a ft c..nt ..rh n... Ro ta t iOl1 ca n Ih<"Orrtica llv ..'cu' wilho ut a ny lal..ra l nlo tion of
Ih.. ......... In l he a.....""" of a n~' c xtcma l foret"!<. a !", . ft"l1ly balan,'t'd rolOr ,,~ll
"('In in rJ a..... a round the gromcl ric cent.... of th e ~h~ ft "~tho,, t a nv cha ng.. in
po oslljon ~ " ibrat inn ) o f thai .....nt ..r 4the lofl . ide o f fi!lU'" 1-1).
l'ra"",,",,, i8 th .. lat .. rai mot io n. or ,~bral ion. o f t he ''''or 1lI,<,mel . ic ,""1 <"1'
in Itt.. Xl' I' la" '" wh ich IS l"''J''''nd icular to th.. ....i. of the r..t ..r. I' .... ...,.~ion i~ a l...
known a~ orbit ing or ,ibralion; th.. orim is a d i.p1av ..f the la te ral mo tio " o f th..
~ha ll «"" t....Ii"" and i~ ~ho...n in lijdtl l(tN' n in th .. fill'''''' I' ,...;.-..;;"n can ta k..
ria,.., ",-e" if th.. rotor is ,toPlIf.'d; it i~ complelely indcp'·nd ..nl ..f rol a t io " ( th ..
~1 'itIo> of FIgu re 1-1). Th.. orbit o f th.. rotor ca n ra nll" fro m purely c ircttl,.. I..
a .-.-rY n'ml'lica lO"d "'ap" oonta in injt: ma ny freq....nci... o f .~bra ' ion.
II i. po_ibk- fUT a " ...... 10 rolat e " , thoUl .,b.al inll- a nd il i~ p<>". ib l.. for a
rolor to "linn e wilboul rotalinjt:. Usual ly. ho......""•• both rola l ion a nd p fl'Ct'>.Sion
ocru r al th.. ""me li m...
Th .. di f{'('tiOfl ..f r..lation a nd p ......,....., ion ca " be e xp ... s\Cd a. X to 1'. o r roun -
I..TCIorl< ..·~... ~C{:W ). " ft.." Ih.. rol'" m.....·s in Itt.> po,ill"" mal he matical a ngula •
....n.... (Fill" "' 1 -8 ~ M.. lio " in lhe opl"'"il.. d ,""' tion i. n p.... .:>ed "" 1' 10 X. o r
d ockwi.. .. f{: W ).

X lo Y YtoX
F"HJ"''' 1·7 An ~ ~.. ",f >haft rota'''''' ' ·8 1Iotat.,., .rod pr«...<ion ""mrng
f "J ~ r ..
..-.:I prKM""".On trw loft. ttw stWt wm '''''''''''bOn>.On ttw Iffi .. X 10 n""'nt...·
""",rod '" ~ <""',.. voithooI '"brat ' 010<:_ , ...... ~ ... """ion, On I,," rignI" r
inq." ~ ",pI<- 01 Pu'" rot.ot>Oo1. On Irw ' 0 X (c:locI<wo",,)""9"10, mot"""
rig!1l 'tw non rotAMg ,haft ~ In"
tI'bt, ""-" t>v Itw 9'....... ..."'.an ....IT4*
or PO'" P"'C""""'"
14 Fundamentals of Vibration

The terms X to Yand Y to X are preferred over CCW and CW because, once
the coordinate system is fixed for a particular machine, the angular sense is
invariant with viewpoint. For example, if you view a machine from the drive end,
and the machine is turning X to y(CCW), and then move to the opposite end of
the machine, it will appear to be turning cwo However, the basic definition of X
to Y still holds.
When the direction of precession is the same as rotation (for example, X to
Y for both or Y to X for both), then the motion is defined as forward precession.
When the direction of precession is opposite to the direction of rotation, then
the motion is referred to as reverse precession. These concepts of forward and
reverse precession have powerful application in full spectrum and in the diag-
nosis of certain types of malfunctions.

Free Vibration
When any underdamped mechanical system is displaced from its equilibri-
um position and then released, it will oscillate (vibrate) at a frequency that is
called its damped natural frequency. In free vibration, once the system is
released, it continues to vibrate until either the vibration dies out or is re-excit-
ed.
Vibration in most mechanical systems involves the periodic conversion of
energy from potential energy to kinetic energy and back again. For example, if
you displace a pendulum from its rest position, you have increased its potential
energy. At the moment you release the pendulum, the potential energy is at a
maximum, and the velocity is zero. Thus, the kinetic energy is zero. After release,
the pendulum begins to move under the force due to gravity. As the pendulum
reaches bottom, the gravitational potential energy is zero and the kinetic ener-
gy due to motion is maximum. As the pendulum reaches the opposite extreme
of motion, the potential energy is again at a maximum, and the kinetic energy is
zero as the pendulum comes momentarily to a stop. The pendulum continues to
oscillate until frictional losses gradually remove the energy from the system.
Rotor system vibration involves the conversion of energy from the kinetic
energy of motion to the potential energy of various spring-like components.
Potential energy can be stored temporarily in rotor shaft deflection, bearing
deflection, the deformation of the machine casing, the deflection of any
attached piping system, and the deflection of the foundation system. Virtually
every element of the rotor system can act as a spring that is available for tem-
porary storage of the energy of vibration.
In free vibration, the energy is supplied by the force that caused the initial
disturbance of the system away from its equilibrium position. This can be in the
 Cha ptl'f 1 V,b. alion 15

fo. m " r a n impul... (hk.. a hamm... hlow) or a ..... fUl)(1ion, ", h... .. a sudd..n
magnil u<le ' ila n!!" ta"". pia.,.. in an appl ied fo"'<'(Figu.... 1·9 ~
Sla hlt-. r ly , i b. al ing s~"I ..m . ..• nruaJJy SlOp "ibo-a lin ll lX'ca u... ph~·s;"'a1
mechan i. m m"'.....n....!lY from Ih yst....... En.. '!!.}· lo...~ ..a n t... d u.. 10 air
.....istanu-. ' ,<co us d a mp ing. p1a st;'" d.-forma lio n. int....,a l or ..>:t"",aJ fr iet ;"".
or int.- rnal ma l..lial loM.-s jsoch a , h ~'su> rl"l ic da mpi ng).
Uns ta hl" ~.,.tt'ms pos"'ss a m'-'Cha ni.m for (u n,...rt inll ..n{'tg;o· fro m o n.. fo rm
an d (ra ns f"rring Iha l ..n....gy into l he s~"'''''m. On.... disl u rbed, u n..a blo> s~"'I .. m.
I.. nd 10 vib .a t.. " i l h i""""'smg an' l'lil ... l.. un h l ..it""r lim,t ing nonhn..arili~
rom.. inlo pla~' o . th.. S~'Sl""" LS d ..Sl fO~-..d Eu mpl" . o r u n" a bl.. t...havior a ....
f1u i<l ·'ndoc.-d instabIlity ,n 1010. 'ro, ,,m. a nd a.-rodyna m;'" Ilull ..r in a irc ra fl-.
Sre Cha pte r 14.
Rol or syatt''''. ar.. compk x m'-'Cha ni("a l . ysl..ms " i l h d ist ribllloo mass. stiIT-
n<'M, and dam p' ng. and th")' "'...".".,; ma ny d ifT.."" nl na tu ral fr"'lu" ""it's. I( is
J"_ ,bI.. fo r rota . S)'Sl.. m. 10 haw .........-3 1 nat ura l f""lu" "",,,, ..xcit oo "mull,,,,..-
ou.ly.Sc.- C....pt.... 12.
Rotor ')"1""" also ....,... torsion al na tural f""l....nci.... Tor . ion al 'ibrat ion
()("("tl rs as ro lor rna,'".,,, (ine rt ia. ) OS<"iUat.. a oo t"ist th.. ro lor sha n . T.... roIo r
. ....It ac~ as t he s pring in th.. sr't..",. storm ll iorsi""a! po U>nl ial " n"'llYd u linll
twi.ti ng. a nd th.. ttl to r rna""", un oc'!!" cycli .. a ngula r .... Ol"C! M'n. a rou oo t h"

f OfCO'd vib...no n

f\ f\ f\
v VV
e
! f_ _,...,~

f'9u<" 1-' f,.., ..,..; forced ..o-.." "n__ • m«twl"Oul sy<t is ~ ted to a
, . - impul~n wit "';bra'" (ring) il n, ........., h~y hmluaill< d _ <y\,.....
,,"ablo> th" "'brat"'" wiR"'" o"t foA:<"d ",bra,,,,,, UO\ <><:cur .. a ny fr«l uency:.nc!
tho /ofu ~.nc!~ ~ a", oonU ant
16 Fundamentals of Vibration

shaft axis. The angular velocities of the inertias store the kin etic energy of tor-
sional vibration. The variation in torsional twisting occu rs about the mean value
of shaft twist caused by the st atic torque being transmitted through the shaft.
Torsional vibration usually involves all of the coupled components of a machine
train, including gearboxes.
Free torsional vibration can occur due to sudden changes in load in the sys-
tem. For example, load switching in electrical generators can create an impulse
that can excite a torsional natural frequency of the system.

Forced Vibration
A mechanical system ca n forcibly be moved at any frequency we wish. For
example, a pendulum which is initially at rest can be moved by hand extremely
slowly. The pendulum responds to the force provided by the hand. This force can
be applied at any frequency we wish. If we increase the frequency of motion of
the hand, the pendulum will respond at that frequency. This phenomenon is
called forced vibration.
Forced vibration is caused by a periodic force acting through the Dynamic
Stiffn ess of the rotor system. Dynamic Stiffness is the combination of various
spring-like support stiffnesses and the dynamic effects of ma ss and damping.
The vibrat ion that we measure is th e ratio of the force to the Dynamic Stiffness;
thus, changes in either the force or the Dynamic Stiffness will produce a change
in vibration. This basic principle of machinery behavior is key to successful
machinery diagnostics and will be encountered throughout this book. Dynamic
Stiffness will be derived in Chapter 10 and discussed in detail in Chapter 11.
Forced vibration differs from free vibration in two aspects. First, the fre-
quency of vibration depends only on the frequency of the input force to the sys-
tem. The forcing frequency may be completely independent of the natural fre-
qu ency of the system. For line ar systems, the frequency of the forced vibration
response (the output) is th e same as the frequency of the force (the input).
Nonlinear systems can produce output that contains the fundamental forcin g
frequency and additional higher order harmonics. Rotor systems can exhibit
both linear and nonlinear behavior.
The second difference between forced vibration and free vibration is the fact
that, for a constant amplitude forcing input, the system vibration response
remains at a constant amplitude and does not decay with time (Figure 1-9). This
amplitude may be different for different forcing frequencie s, but for any partic-
ular frequency it will be constant.
Unbalance is the most co m mon force that produces vibration in rotating
machinery. The asymmetric mass distribution of the rotor produces a cen-
tripet al force (sometimes called centrifugal force) that rotates with the rotor.
 Chapter 1 Vibration 17

This rotating force causes a synchronous, or IX, forced vibration response of the
rotor.
Other examples of forced vibration include vane-pass excitation of pump
impellers and housings (at a blade- or vane-pass frequency that is an integer
multiple of running speed), turbine blade-pass excitation, or gear mesh fre-
quencies. Because of the large shaking forces that can occur in machinery,
forced vibration can extend to the surrounding piping and support structures
and, on occasion, to nearby machinery.
Axial forced vibration can occur during surge in compressors, or because of
balance piston problems or coupling problems in machine trains.
Torsional forced vibration can occur because of variations in gear geometry,
periodic rotor-to-stator contact, electrical motor slip frequency or torque irreg-
ularities, misaligned couplings, reciprocating drivers or loads, or because of
radial vibration. Radial vibration can couple into torsional vibration because the
deflection of the shaft away from the spin axis of the machine increases the
mass moment of inertia of the system, creating a torque disturbance to the sys-
tem. It is also possible for torsional vibration to produce radial vibration through
the same mechanism.

Resonance
When the forcing frequency is near the natural frequency of a mechanical
system, the vibration response amplitude can become highly amplified. This
phenomenon is called resonance.
Pushing a child on a swing is an example of resonance involving the period-
ic input of a relatively small force. The child and swing respond at the system
natural frequency, like a pendulum. The system acts like an energy storage sys-
tem, oscillating at the natural frequency. If successive pushes are timed correct-
ly (the forcing frequency is equal to the natural frequency and in phase with the
motion), each successive push on the swing puts more energy into the system,
which increases the amplitude of vibration. The amount of amplification,
expressed as the Synchronous Amplification Factor (SAF), will depend on the
effective damping of the system.
In rotor systems, the force caused by a rotating unbalance produces a syn-
chronous, IX, forcing frequency that is equal to the rotor speed. When the rotor
speed nears a rotor system natural frequency, the vibration amplitude will
increase. At the natural frequency, the rotor will reach a balance resonance (also
called a critical speed, or critical), and the vibration will reach a maximum
amplitude. As the machine speed moves beyond the rotor system natural fre-
quency, the amplitude will decrease (the amplitude plot in Figure 1-10).
 Th~ I hi~ ,.."..,nan"" g;on , im po~ nt <:ha ng in th.. l im ing. or pha....
of Ihe ~ihr..tion al... occ u'. T a n'plitud....nd ph.a bt'havior IhlOl>gh ..,.....
na n"," i. di <.cU"""" in d elai l in Chap te. 11.
\ \ "hUe t .... majo . ily o f ....Ulli ng mach in.... oper..t.. below th.. firsl bola n.,.. ......
onan.,... mo", la rge. cr itical-p ro<:<"AA machin.... orerate aho"e o n.. Of mOl"<' hllJ-
a n<..., ......"""""'" Ihus. th.. a mplificat ion o f n .tor \libral ion in ...wnanU'$ d u. ing
" ... tup or slt" .d"'.... i. a n imp",. a nl co n",,",. High \lih....l ion a t .....m a n""
opt'll" up l it.. .wn!l"f of h igh roto r " .....~ ro to r-to- SUllor w nta ct. and ......
"'{'a•. fo r t h is .........n. th.. AlIK'rican l'l-trole um Inst ilut.. (API) ..,ts recomm.. nd -
td lim its fo r max im um s~'T>ChronolL8 a mpli fical ion fac to • ..."jun for a"""pla oce
In tin ll of rotal in!! mach i ry.
l o ....ional m;ona n can al.... occur "'-h.........,r t .... frtoqueney o f a torsional
dis tu rb.",,,," i. eq ual to a lors iona l nal um frequ.. ncy o f t.... . p lem. To rs ion al
mod .., in ro tor ,~.tems le nd 10 bt' very poorly da m ped. l hu<;. lor . ionai a mplifi-
cal ion fac tors I..nd to bt' ''''Y high. Opera tion for a ny "." 'fIIh of l ime o n a 10"
sio nal .....,nance co uld produ"," w ry high to rs ional ,ib.alion a mplilud<>s and
l i kel~ mach i.... d a mage: fortu nalely. hecau... to rsion al ........, a nct"l a re w ry n... ·
row. lh is ...."'Iroccu.. in practice.
II is u oo..,i...bl.. lo o pe. al.. mach i ry al a n~ 'f"""d co. res pond ing to .. 'I .... .
a rewnance o. half o f a rewnan.,.. f""' ncy id ing operal io n o n a reso na nc..
.....m. fairl~' ob,io ... (al tho u¢t it hap pen. mo Ih.an 0 .... ,,_ Id Ih ink). bul
hecalL"'" of no nlin..a rilin p.......nt in mo, l mach inery. unbalance can p.oduce
....m.. 2X ,i b. al ion. AI..,. cenain mal funct ,o rul ca n produ"," 2X ,ib.at ion. If th..
ma<:h in.. o...·.at"" .. 1 a .,It't'dd to o lt<'-h.alf o f............nce f"", ueney. t.... n a ny
2X .ib.al ;on " i U .... ampl if by I.... ...,."na nce thai exists .l I lwice run ning
. I"....t. 1 h is a m plifICa tio n ca n bt' Lo'!lt". ,..,....ll ing in a ~i!.tnifiCllnt .......1"f2X \'il>.a-
l ion in I.... mach ine.

II ~ I
f ;g u... l -l 0 A _ pIor ~ . =o­

n~ 01 . n idNI rot""..,..om Tho: . - pIor

_ tho p/w5e log <- Owpl.. 2) .........
rot'" ~ ; m., _ pIar """"' !Io<; ..-npIi-
~ aI_.""" WfW. rotor <peed wt.....
rotor ~ ....." ' I>< rot<:< ~ nat
quon<y. tI>t _p1~_aI . b illion inc .....IT..-
'" , I>< rotor <peed """" belO"'.'l the
lm:tuency...... ~ _ ........ ~ a1'"
( ~II.
u,..
JIL--zq
 Chapter 1 Vibration 19

Self-Excited Vibration
Self-excited vibration can occur when a mechanism exists for converting
non-vibration-related energy into energy of vibration. The oscillation of aspen
leaves in a breeze is an example of this kind of vibration. The wind is the energy
source that is converted, through the mechanism of vortex shedding, into the
torsional and lateral vibration of the leaves.
On a larger scale, the destruction of the Tacoma Narrows bridge in 1940 was
produced by a similar mechanism. The wind blowing across the bridge interact-
ed with the bridge structure to produce an aerodynamic excitation of several dif-
ferent torsional modes of the bridge. Eventually, the amplitude of the torsional
vibration became so large that the bridge was destroyed.
Self-excited vibration always involves the excitation of a system natural fre-
quency by the energy conversion mechanism. So, in one sense, the phenomenon
of self-excited vibration is similar to the phenomenon of resonance. The primary
difference is in the way that energy is delivered to the vibrating system.
Resonance involves periodic forcing at a natural frequency (for example, unbal-
ance in rotors); self-excited vibration involves conversion from some other
source of energy (for example, the energy in moving air).
One example of self-excited vibration in rotor systems is fluid-induced
instability. The fluid circulation in a fluid-film bearing or seal acts as a mecha-
nism to convert some of the energy of rotation to large amplitude, subsynchro-
nous lateral vibration. The frequency of the sub synchronous vibration is a natu-
ral frequency of the system. See Chapter 22.
Another example is the subsynchronous vibration due to rub. Under certain
conditions, periodic rotor-to-stator contact (similar to a periodic impulse)
excites a rotor system natural frequency. At the rotor surface contact point,
large tangential friction forces act as a mechanism to convert some of the kinet-
ic energy of rotation to lateral vibration. At the next contact, the process repeats,
supplying more energy that makes up for any energy losses that may have
occurred. Thus, the rotor evolves into a steady state condition of self-excited
vibration. Rotor vibration in this form of rub takes place at harmonic submulti-
ples of running speed, V3X, 1;2X, %X, %X, etc. See Chapter 21.

Summary
Vibration is the periodic, back and forth motion (oscillation) of an object. A
transducer converts the vibration of an object into an electrical signal that we
can measure.
Frequency and amplitude are two of the primary measurements of the
vibration signal. (A third important measurement, phase, will be dealt with in
the next chapter.) The frequency of a signal is found by taking the reciprocal of
20 Fundamentals of Vibration

the period, the time to complete one cycle of vibration. Amplitude is normally
measured as a peak-to-peak (minimum to maximum, abbreviated pp). peak
(one-half of peak-to-peak, abbreviated pk), or root-mean-square (abbreviated
rms) value.
Displacement is usually measured as a peak-to-peak value, while velocity
and acceleration are usually measured as a peak value. For vibration at a single
frequency, displacement, velocity, and acceleration can be simply calculated
from each other. See Appendix 7 for conversion formulas.
Absolute vibration is the vibration of the rotor or casing relative to a fixed,
inertial frame. Shaft relative vibration is measured relative to whatever the
transducer is mounted on, using a transducer without an inertial reference.
Rotation and precession (vibration) are independent concepts. A machine
can rotate or precess or both.
Free vibration is initiated by a disturbance to the system. Once started, the
vibration occurs at the system natural frequency and dies away at a rate deter-
mined by damping or other frictional losses.
Forced vibration is produced by a continuous, periodic forcing of the sys-
tem. If the amplitude of the force is constant, the amplitude of vibration will also
be constant.
Resonance occurs when the forcing function is close to the natural frequen-
cy of the system. Vibration can be greatly amplified, when compared to the
response at other frequencies.
Self-excited vibration occurs when a system has a method of converting a
non-vibrating energy source to vibration energy. The self-excited vibration takes
place at a natural frequency of the system.
 21

Chapter 2

Phase

WE HAVE SEEN THAT VIBRATION CAN BE MEASURED with two parameters,

frequency and amplitude. These are good and useful measurements. However,
vibration never occurs in isolation; there is usually a root, or fundamental, cause
of vibration in a machine. The machinery specialist needs to identify the root
cause of any vibration problem, and it is often difficult to do this on the basis of
frequency and amplitude alone. More information is needed.
One piece of information that can be very useful is the timing difference, or
phase, between events. If we know the timing between a root cause and its effect,
we can use our knowledge of rotor behavior to deduce the possible root causes
of what we can measure. This gives us a powerful tool for the diagnosis of rotat-
ing machinery.
Phase is just such an essential piece of information. It really is basically a
simple concept with some complex applications. It is, as we will see, one of the
most powerful machinery diagnostic tools that we have.
In the following pages, we will define phase and try to identify some reasons
why it is an important piece of information. We will discuss a special reference
signal for measuring phase, the Keyphasor signal, and three types of phase
measurements: absolute phase, relative phase, and differential phase.

What is Phase?
Phase is another name for the relative timing between two events in differ-
ent signals. For example, in Figure 2-1, two similar vibration signals are shown.
These two signals reach the positive peaks at different times. This timing differ-
ence is referred to as a phase difference.
22 f " nda _ ntAk 01Vib,.olion

floJure 2-1 ~ "" P"""' __ ~~_

...........n.. g'- ' ""iJ"OI PNI< oa"" 011... "'"

//1 fX\
"""n
Ill .............. PNI<o the sig",,1 ha>. pho'" t
log 0I6l1' witt>'''''ll«110 "'" Ill .... ..",... ~
(Tho.., • .., 3ro" in • '~ vOotIOtl eyo: " J

\Y..I s.
In mac hin.. ry a w Hca tioo", t h.. plJa... d ilT..renc .. o r "'Iu;'-a l.,ot e,..,ot~ on d if-
f.....nt " bral ion sij\nals is ca ll.-d ,../" Ii"" 1'1"..... AbsoI" lr ph"",,, comJ"l res t....
lim i n ~ o f . n ["\'e ol on I ,-jbrat ion ..·.'-.,[orm 10 a d iffr ren t type o f ,de...1I«" ~ig·
nal. producrd lor a oO ·p""'·turn marker 011 • 5hall Both meth oo s a re corn ·
mOl1 ly u...-d. a nd both ha", the ir part icu lar applica tion...
h en thoulU' p ha "-' compa..... the timi ng of ["\'..nls, il is e~pr....,....j in unils of
degTffS o f l he "brat io n e~·d e. In machinery inMru me ntat io n. the t iming dilT....·
e OC<' i5 us ually exp ressed a s a mel ion o f Ihe com plet.. 360" vib ration C)"eI... Th i5
is a pos ili", numb<'r ..-jth ioc...as ing li me delay and i5 called possl''''' phtue lag.
For .. xam ple. in Fif(U'" 2· 1. t he It''''''n signal Ingo l he hlu.. siRnal by 60".

Wh~ Is Phase Imponanl ?

In a n automobil e engi n... th e... may b<' ' J"Irk a nd fuel. b ut if the t iming is off.
the engin.. may not ru n. [n rotor b<'havior. th,,'immg (p ha...' is jus t a. im po rtant
a ' , ' brat ion a n' I,lit ud.. ..nd r.....uency for ..ffecti,... d iagnosis o f machin.. b<'hav-
ior. It is ..I", n.",..,,,,...)" fm emcient l>ala ncinl! of mac hinery. Ba[anci ng ""'l ui....
u. to know th .. a ngu lar loca t io n of th.. u nbal.."".. (th .. h<'tJ>y spol ~ W.. ded uce
th ,. locallo n 1»' us ing pIJ..... _ a........m..nt o f th.. vibratio n .... po n o f the
machin.. comhined ..ith our knowl~ " f , ot or beha,; o r. With out p ha inf"r-
ma tion. " ... ',,' u[d h",... to I...rf" nn ma ny more ru ns to cakulate a n init ia l baI -
a n, .., w lul,o "_
When ro tors " b",le. Ihry d..nect aW>lv from the mach i" e cent..rlin... Whe"
lh.. ,; bration is I X. the po inl on th e 5hilft wh ich i. o n th .. "" t. ide oft he d ..n ",,·t·
.... sha ft i. called the hi/(h spot (Fil(U re 2-2).Th .. ti m inRof th .. rot or hi~ ~pot pas-
sall" lInde, a t ra n'l<lu<TT (lh .. po.iti .... p"'ak of di ""a""",,,nt I p"n..l" impo rta nt
in forma tion a""'it rotor briiavior. [I ean .... romp.ored to tn.. ti mi ng a t di ll.....nt
..... ~l l po.il io ns in th .. "'TTl(' mach ine. The a mpl il ud e a nd plJa.... ;n f"" oa li" n ca n
be co mbined to p rod uce a pict ure of lh .. d ,·n ,,,,lion , haJ't'. or moo.. . hape. ,,( th ..
 Chapte, 2 P...... 21

f ig " .. 2-2. Tho Mjh >pat ol _tlt_on.

rotorsIwIt_ ~ f9" ~ " - " . ~~ of
---
-.-
/ \
~ ~ of thr _ed
imogn of ~ '0101 '" ~ """""' "' .... <>rbn.
roo- 1X 0""","' 0Ib<t> only. • !opOf (yefow) on
fJ!..,- or high 'POll Sl¥
rotor tthr 0.<1..
on ~ out.o. '" ,.....
rotororbits. _
10. probe.lhr _onon
'Il< high """ po»e'> " "..

prot,.. il ot it> ~ i m um
Vg.... from.hot ,
Tw o _
,

rolor al ru nn ing .J-'d Plta_ Can " I"" he u.-cd to documenl rno<k sM "..of l he
ca sing or ,;Iruct ..... in a . imila r ma n",,,_
T.... 'ibral ion at th.. ......." r a mac hine p rob k-m always hap"..ns ..a rli..,.t in
tim... .-\5 'ih'a Lion propagat"'" a ay from th e SOU I"C<' loca l ion. il ..xl... ri ..n...... a
li m.. delay (p ha... lag). Tn .ica1Iy. the fart her fro m Ih.. ""' th .. It"'g.., l it<'
pha... lag. Th"" by m....s" rinll t he ,da tiVE' pha ... b<>no n d ifr nt axial ",..i-
l io ns in a m.....·hin.. a nd loo kin!! for t he earli...1 .i g nal. " can ""m..tim..,. d<'t<'r-
mi ne thel'>cal''''1d ",", 10 th e "'''tn'of lhe p robl..m.
A h..al thy mac hine s hould operate a nd " h rate " i th a rc>pea ta hle reuer« day
afte r day. Once th .. ...._ li" .. ' i brat ion c ha racterislic>; of a machine a... I n,,,,,n
(including chat1#' wilh load o r ot her meas urable factors ). cha ng... in ,;btaLio"
th ai b ak I"," pall ..m ind ica t.. l hat someth ing may be ...rong wilh th e ma,"' in...
Cha ng in pha... a... jusl a. importa nt a . cbang in vihra lion a m plil ude ... fn--
'1....n" . a nd on.. may e han!!e indcpcnd..ntly of I oIh Cha ng..,. in p ha .
am plitu de, or fr-..que ncy can ...am th a I someth ing i. hal', " i,,!! I.. Ih<' mach in .
F,... all o f Ihe... ...a son,;., p ha... is an impo rta nt 10..1 in ma,'hinery d iallnus -
,~

n... K..yp hnor h en!

Th.. m",I common 'ibral ion in rol or 'ys«'",s is a ,," >cialed wilh rot .. r u nbal -
a n'.... The unbala n.... act s as a on.. --<:y<..... I.....-r.....oIul i..n rutal inll f o n th ..
rotor. Th is I X forcing p rod a IX. or synchron,Jtj", .-ibrali..n n in t he
mach ine. !leeau... unbal a n is s.. com m..n. it is ....s irabl.. l0 ha, a fixed, ti m ·
inll ref......n.... sip'al so Iha t "" Ca n rna "'" pha... m..as uremenls.
An eddy curr.... t dtspLtct>rn..nt I.... o..t u"..r I",okingal a kC).....-a)· or k y ...",e.
Ih is Purp<"'-' .... rfc>ctly. Such a l ra nsd....,..,. (. ca lL.,d a K" phasoT Im""dllcer. Wh ile
a Keypha>Or t ransd......r is u.......lly a n l'ddy cu rw"t Ira "..tu...." it can be any c,l'"
of lransd ......r. as Iong a. il I" " vidn " "'...." ta IJle. " "'........ r-t " m rer........... ";gnal.
24 Fundamentalsof Vibration

Figure 2-3 shows a Keyphasor transducer ob serving a keyway and the result-
ing signal. As the leading edge of the notch passes by the transducer, the
observed distance will increase suddenly, and the transducer signal voltage will
abruptly become more negative. When the trailing edge of the notch passes by
the probe, the transducer signal voltage will return to normal. Other shaft con-
figuration s for generating the Keyphasor signal are possible.
Thi s pattern of voltage changes occurs once every revolution of the rotor
and is referred to as the Keyphasor event. The Keyphasor event can be thought
of as a timing signal. When the even t occurs, a timing clock is set to zero and
started. This event is used to measure the elapsed time between the Keypha sor
event and an event on another signal. Each time the rotor completes a revolu-
tion, the Keyphasor event occurs again, resetting the imaginary timer.
This once-per-turn event is the timing reference used by instrumentation to
measure the absolute phase of vibration signals at IX and integer multiples (2X,
3X, ...). It is also used to measure rotor speed and other important characteris-
tics of the dynamic response of the rotor.

Keyphasor event

vf
Transducer signa l

Triaaer level

V
Figure 2-3 .The Keyphasor event. When th e Keyphasor transducer
observe s a once-per-turn mark (notch or keyway), it generates a
Keyphasor (timing) event. The event is th e sudden negative change
in signal voltage when th e leading edge of th e keyway passes next
to the pro be.
 (hapte, 2 "".w 2S

11 \
f lg ~ ' e 2-. The Keyph..... ~ _ • , , , •
;J ~
• •
l"me lm l)
s r
- "e ~ The Keyph'''''' eYe~t i<
..... !i>ed ' eIe<..... e from wIW;h vibrmon
pho<e I"J .. _-.red When «><t'ItWIed
""'"' the vibt.hon ....... (bonom), « p0<>-
I
d........ blonloJdot d"'l'lolr "" ..... vobra<oon
pial (1l<1'tdy oonvrntoon) " ' " <lm-

• , , , • , • , •
p1ofiO'1 p/I.I me- t .. pIal~ meas·
o.ore p/I.I 1.>g from dot 10' M ne.-t
posi, PM"
t- ~' -j

V\ j l\
I\. IV
• , , , • , • , •
"; r

Phil", ~..s ure menl

PI>;,S(' is u to compa ... t he ,iminjl; O rN'" ..,nts in diff.."'nt ~ign.ak If a n
.......nl' '':<:U'" !a in \i m.. l lmn a ... r.....rtCC nl. I n it lag. , h.. r..f.....nCl'..w nt.
Similarly, if a n nt ocru... before . h.. r.-f nCl' nt• •h..,. i' leads t t... "'r....·
.."'... .......n t.
In " rd," '0 make mea ni" f/.ful p ha.., mea~ur......." I th...ijtnal. b..injl; u.-..d
m,,-t ."On~ist of a oinf/.I.. pri ma ry fn.qu..nc;o· com pon ' 0 •• in ' h.. ca... of t ....
K,·n >ha signal. one d ca rly iden lifia bl.. rcfc renCl' .., nl. For t his r'-"'SOn. ~i!li-
""Is a usuall~· filtcr<'d to t .... fn.<Ju..ncy o f int..",.t b..t"or.. mak"'!! Ih.. m..a. u.....
menl. alt houjth un fil....rM . illnaJ. ca n b.. u....t if th<>y a... do mina'M by " ne r...,..
qu ..ncv.
T.... conwnt ion used in mos, ,ib.alio n ",..a.",.."...'" ilL'l run ", n, ..' i" n i.< ,..
lnCa <ll re plm... lall ..~, h a po' " i''' numlx-r "...,;""'" call<'d posIliw pIw.... lall.
ror c xample. tal.. t t... vib tiOfl ";!lnal sho " in Fi!l"' " 2--4. Tile "f' i"'-' in Ille
Ke)-pha...... . ignal pro\1(k>. I l l mi ~ "'f.."'n Tt... .......nt of int.-r..... i. , h.. f,n ,
26 Fundamentals of Vibration

positive peak of the vibration sign al that occurs after (to the right of) th e
Keyphasor event. Becau se the first positive peak occurs after the Keyphasor
event, the time delay between the two events is referred to as the phase lag.
For convenience, the Keyphasor event can be combined with the vibration
signal. This is shown in the bottom signal in the figure . The blank/dot sequence
(a Bently Nevada convention) displays the Keyphasor event on the waveform. In
data plots, the precise moment of the Keyphasor event is represented by the dot.
(Oscilloscopes use a different convention, where a notch-driven Keyphasor
event is indicated by the beginning of the blank.) Remember: the period between
two Keyphasor marks represents one revolution ofthe shaft.
The numeric value of the phase lag is found by taking the measured time
delay and comparing it to the time for one cycle of vibration:

p=[t 2 -t1 ms ][360 deg ] (2-1)

T ms/cycle cycle

where P (Greek upper case phi) is the phase lag in degrees, t} is the time in mil-
liseconds at which the refe rence event takes place, t 2 is the time at which the
event of interest takes place, a nd T is the time in milliseconds for one cycle of
vibration.
Phase measurements are usually reduced to numbers between 0° and 360°.
This is automatic in many instrumentation systems. For example, a calculated
phase measurement of 395° would be reduced to 395° - 360° = 35°.
The measurement of phase using this instrumentation convention (positive
phase lag) differs from the measurement of phase using a standard mathemati-
cal convention. In the mathematical system, phase lag would be a negative num-
ber, and phase lead would be po sitive. For a discussion of this important differ-
ence, see Appendix 1.

Absolute Phase
Absolute phase is the phase angle measured from the Keyphasor event to
the first positive peak of the waveform. For IX vibration in a circular orbit, this
peak occurs when the rotor high spot is nearest the vibration transducer (the
high spot and the orbit will be discussed in more detail in a later chapter). Thus,
for IX vibration, the absolute ph ase is sometimes said to represent the phase of
the rotor high spot.
The absolute phase is found by measuring the time between the Keyphasor
event and the time of th e first positive peak (t 2 - t}), measuring the time for one
complete cycle of vibration, T, and using Equation 2-1.
 ChaPI...- 2 Phase 27

In Figu re 2-4. 1, is m..a~u,<'d as ab..ul 2.25 m,.. a nd ' 1 is a bo ul 5.0 m •. Tis 6.2.5
m. - 2.25 ms = 4.00 InS. Appl~ing Equal io" 2- 1. lbe pba.... "'. is

'" = 2.50" ph a.... la!!

In Figu re lo S. an unfille rl'd ' lbral ion .ip;nal is shown alo ng ..ll h iu IX· a nd
2X-f'h e rl'd .1b,..lio" co ml"'", enl.s. ;'Ilole Iha l lhe I X signal has one Ke~-pha.<or
..........1 per C)'ck o r " bra lion: lh" ", lh e ' , bralion rTftlue ncy is 0 lM' tim... ( I X) ro lo '
speed. The botto m signal has t>\n cycle. of ' lbral ion for eac h . haft ,....oluti...n:
the " hrat io" r""l."e ocy is t>\ice (2 X) ru nn ing 'J"-'""d. XOI.. l hat. in I.... 2X signal.
I I>.- ahsolu te pha i. m ..... ur<'d 10 the)irst po.ili.... p"ak; I I>.- ..-cond lM'a k i.
illnorl'd. ;'\ole a t lhal t he period. T. i. meas ur<'d for on ty o ne C}-cl.. ... r the 2X
'ih"'l ioll.

FOg" ... J -S 11(.11(. .nlC _ute

~_An ...,fiIt.. O<lllibtation..,... Os
_ _ iU l ~ - • .-.:ll ~-~~~
•
com~ .-.-.... ph,o,..,,_ t-- -"J20·
by <<ltr>pOofing 11 _ ' _ fmm_
~ -m Idol) to _ ""'"
PWI~ peal< 01_ .. bt.tion "q.",
I..... t
to 2)
_'"./
fa< ..... cyd.. 01,.-..
T_Not..",",-for"l
"
~~_nn... .... _ .......;
only the IitU """'.... peal< .. .ned fa
_,eph,o,.. meawoement.
28 Fun da"",nlal~ of Vitwation

fCogure l-.6 . AbYlIut.. pha... . nd """harmonic

,;gn.J~ AOsoIut.. pha... «ItlfIO( ~ _ _ on
rr, .'
1/'··..· ·=···:·,,·· ·,,· V,
_ - . ,,;gn,I, _ I....... ft"lUf'<>Cy ~ not •
........."... """"pie '" running l.p<'<"(l ('It.. ~
.. na/l ll. lX. l ll.<tt.l- Tt.. pt.. ... ~.......,'"
from ..ach """ ,~ Koy~ dol ~ .
d",",..", ,
~ lor
Th"".bsoIul" pt..... ~ .......n-
""""""""""-,, ""<"<1 ",Ix . loon /
A""'>lul .. 1'1>..", can on ly be mcu urt'd on ,-jbralin n . i!(nal. wil h .. freq u.. n ~"y
Iha l is a ha rmon ic (inl'1l..r) mult ipl.. of ru nn in!'i ~ p<'<'d. ~lJ(· h a. I X. 2X. 3:'(...I<:.
,\ Il..m plinjt 10 m....~l1 .... a bso lul .. p ha... for frequ..ocic>. Iha l a.... nol inl..W'.
m ull ipl...... o r Iha t aT" .ubs~."chronou ... ca n I..ad 10 a mbi!lllity_ For exa mpl... in
Figu r.. 2·6. a ,-jbral ion sill-nal i. Mm,." 11...1 ha~ a freq umcy . lilllftly hijth..,. lhan
IX. Jo I...." ..ing abWut.. pha", fro m ' llC<"f""Si, " '-'YPha >or rn.lrh .... ull~ in di f·
f.....m ph...... , 'a1u...... For .ign al. ,,-jlh frftJ....nd 11 1 ar .. not inl..!t". mul t ipl..s
of ru nn ing spero. at>.oluI" p ha... m....~"'<'1TI .. nls a m....ninlll........
1\01" lhal. if Ih.. i",l ru m..nt ca n no t combi.... t " "1l'ha 'KII siltnal wilh Ih..
, -ib mtion ,ignaL it is ,I ~l I"'"ihl.. to m<'a ",... ab ".,lul .. p ha by d i"f'lari nll bo lh
w",·..ro rm...Th.. lo p two p lot . in Figu re 2-4 can be us.cd 10 m " r.. lh.. a b.<.olu l..
p ha \<';lh.. ba.,c d..fin it ion of a!»oI",.. p ha... is uncha n!(nl.

f ioJu,.. l- 1 . R8.0_ pha-.. ~ <q.lIVa ......

POOnlO on sog 1oI tN, .. """'pie,,;qnal A ;,
lM ,_""" '>ogtWII A
"""" '1m< ~ /to ,
&..., ......... "'urn
rtI. all "9nal B. That • •
trmo. compa""" 10 tI>t po-riod c1...... ";1:...''''''-
~.q;...n. '0101""" pha-.. cI , 30': tee ,ot.-
"""' pha... " ~b<"d", 'S"y"" 8 !og>1.IQhaI
A 1>1 HO":

\
 O1apt... 1 I't1.M<o 19

~Iat i"e Ph ase

Relat i,... p lLa... ,~ Ih.. tim.. d..la~' 1>o-1"'....n "'lui..al..nl ......nt ~ (.,.... ks . u-ro
("ro'Isi"l!" <'te. ) on hoi... ""'l>aral.. "igna l•. a nd d' .....n·1 u... th.. K~'plLa",," ,,,..." 1.
folllUfl' 2·7 ~ho",. a n ..""m ple o fa ......ali'" p lLa... m..as " fI'm<'nt. The hoi... " b ra l io n
. ignal. have ~n fill...ed to the ""me ftequ..""y a n" ...-p nl th e displa""me nt
"bralion al d ,ffe rent 3.\iaIl.. ~ili"n. "n a T!la<'h ine {hu t Ih me tra nodlKe' o ri-
..nlat ion ). SPlect o ne o f I"" "ignal, a" a ...-fe... n..... a nd t hen Iect a ronwn,e nl
m..a~wemenl pu"'l u n it. 11" 11;", th.. p"", it.i'·e peak i~ u....t. The <'tJui"ale nt p..inl
i. loca ted on t "" ......n.. ';!(naJ, a nd the re lat i.... plLase '5 cak ulaled ,,' i"!/.
Equat ion 2-1.
Th e result is. "Signa l R Iag' ''llnal A by 130".- 11 i~ im po rta nt 10 "u te ..-hich
,illnallead~ (or tailS) the ot he, a " d try. how much. or t he d.-.crillli " " i~ nUl ,~ ,,­
red or rom plete. Xu nnal lv. th.. 11;",......,..... . ig"al ;" ..at ed las t. I" t his ""'''''. ,i1lnal
.-\. ",as t "" refe... nce !lignal. a " d ,ill"al 6 "'a ' m "ured reJat i.... IO Ihe ...re...",:.-.
For anot he, Ha mple. "'ppo... a ...Iative p lLa mea"urement is mad.. "n the
sillnal. in Figure 2-lI. Init ia lly the -.-\." ,i!,:nal is ,;ejected a s a ,..ferenc The ...Ia-
ti.... p h..,.. 'II meas u"'" a. - 6 lap .-\. by 270".- Th .. res ult ca n ta led as "6
lead<. A b,- ~I".- Eithe, illro..ed.
In uttl .., I" ma k.. a "'Iat ;..... p ha se measu...ment. th.-..e co nditio"" . oould be
met. Firs l. the 1..'0 .illnals must ha.... th.. sa me fnoqu"",'y. bu t il ca n .... a n" f ·
quen cy. nut ju"t i nlt~..,. mu ltip.... o f ru nn lnll speed. [f we a U""'pl 10 m..a.u...
Ihe , elal i"" plLa".. betw.-en . illnal, " ' th d ilTe,ent f, <'<J"" "" ;"" ..... will . obta in d if·
f..",nl rPSl.ll t. d epe"d in!(on wh..,... ..... ma ke th.. mea surem.."t. a nd th.. n ...ult ..,II
be mea n inttl.......

• •
F,gu-e 2-' EquMllmt ",""",,,,,,,,,nrs <J ~lo-
I pI\.o",_ WI-.rn "'1....1Ai>......,. .... If4'f-r·
.."" B load!. A by 'KT 1810qs A by 27(J'l.When
'":I i>uM'das._<'OC(". A ... d, Bby
271:1' (.II Ioo;rl B by 'llrI_Uch i> corr.-tt.t"<l ~
iIpIlhc.JI"'" _1m""" _ i> "llP'''P'.....
 Seco nd. n-Ia ti'.... ph a.... m<·asun- m~n t . an- mo,;t ofte n ..ppli",,1 t.. ';bration
s ignal. wit h t he sam... un its o f m...as un- m~ nt. If Wl' wa nt to rompa.... th~ ....I..ti'....
p ha o f t he roto r ';brat ion a t di ll nt ....i.... p""' it i',..,,,, "";" ''' ' uld t~pl~al l~ ~..m-
pa dispLa,...m""t. ....... ,....I ity. or tv.... a <"," ~ ra t ion signa l-<. (S,n, t he
pha ...-Iat ionship o f ocn"l"",tion a nd ,...Iocit~. alMl , l,l(' it ~ a nd d~pb me nt
a .... a1wa~. 9lr (..... f igu r... \ · 4 ~ ..... co uld co mp;.'" t h , siltnal. by appl ); n!! t he
right correctio n.)
Th ird. , 'ibration t ra nsd uce.. sh oul d ha\1' th..... me radial o rientation ift ""',
a . .. in di ff"l1'nt ax ial pla n.....
R..lat iw ph a ... m..as ur.."",nt s m n he ma d.. bctw n tra nsd oc",,., " , th d if·
f......nt ori<.'nt a tions. as lon g a . t ....y a ....in t .... sa"", plan to oct.<'rm in.. t .... d i....,-
tion of p...,.,....,ion o f a rot or.
Fo, ..""m ple. fi gu ... 2·9 s how. a n ..nd 'wwo f a rot o r sha ft th at i. p l'<'Ct"Ss i"lt
in acircula . o rb it. T h.. , ibration bignals from X a nd )' (co pla nar) t , a n sdu ,... rs a.~
al....bown wit h t he t· signal a bow th~ X s ignal th~ lWntly ~e,ada displa~ co n ·
,..,nt i" n. T.... ...-Iat i.... pha sl><" •." that X lead . Yby 9lr. (To d etenn i"" th~ <Ii......
l ion o f prt"<'E'"'ion, l he ati.-.- pha... I1'lationship u...... m u" he th~ 0 "" that i.
le"" than 1110'".) T h is mea ns t hat t h~ rote" h igh spot pa~ u nder th~ X trans·
d ll....,. 'JIlr ""fme it ""..... und .., th.. Y t ra nsduC'l" . T his . h"",." lhat th~ d i. C'ct ion
o f I' """"""i,, n u f th .. rotor is X t o ) '.

f ig u ~ 2·9 jq,I,,_ ~ .,..,do,,,,,,",,, of P'<'<'"'VOtl. T_ pootpo.-ndoc..... , ptOboos in t"- ........

pia"" _ " """ !Nt IS m<l'o'ing '" a l X,.<ir<;ula, orbL To 11«........, ~ "'" .., ,"'" of 1"""""""'-
,_ttw loading '.....,i>'e pI\aw retar,on,l"p that i> "", than tllO";,,", ~. X !<-ads Y by '". This
InN'" lhol ~ """.. hq, 'POI 1'0'_ ..... 110 ~ X ~ ~ n _ _ ...." to ~ Y probe;
""'~ ~ dol""'''''' d P'<'<'"'''''''' of "'" ,ot<w ~ X to Y
 Chapter 2 Phase 31

Differential Phase
Differential phase is a special application of relative phase measurement. It
can be used to locate the source of a machine problem, such as fluid-induced
instability. Several vibration measurements, filtered to the frequency of interest,
are taken at different axial locations in a machine. Because the frequencies are
the same, relative phase measurements can be made between the signals. The
signal with the earliest phase will be from the transducer that is mounted clos-
est to the source of the problem.
For this kind of measurement, all the transducers must have the same radi-
al mounting orientation (for example, all must be vertical or all must be 90° R
(right), etc.).
This technique can be used on vibration signals of any frequency, like those
that result from fluid-induced instability. Differential phase measurements can
be made between any two signals, as long as both signals are filtered to the same
frequency.
Differential phase measurements must be used with caution. Significant
phase changes can occur across nodal points that can produce misleading
results. (A nodal point is an axial position where shaft vibration is zero or a min-
imum.) Phase delays may be more than 360°, which could reduce to seemingly
small phase numbers.

Summary
Phase is a measure of the timing between two events. Phase measurement
requires that the signals of interest have one dominant frequency component.
For this reason, signals are usually filtered to a single frequency before phase
measurements are made.
A Keyphasor event is used as the timing reference for absolute phase meas-
urement. It is most commonly produced when a once-per-turn notch on the
shaft passes by a Keyphasor transducer.
For absolute phase, the phase is measured from the Keyphasor event to the
next positive peak of the vibration signal, and the result is expressed in degrees
phase lag. Absolute phase can only be applied to a signal with a frequency that
is an integer multiple of running speed (IX, 2X, ..., nX).
Relative phase compares the timing of equivalent events on two vibration
signals, and either one can provide the reference event. The result is expressed
as "Signal A leads (or lags) signal B by so many degrees:' Relative phase meas-
urement requires that the two signals have the same frequency, usually the same
measurement units, and that the transducers are either in the same plane or in
different planes with the same orientation.
32 Fundamentals of Vibration

Differential phase is a type of relative phase measurement where the signals

from several transducers are compared. All the transducers must have the same
mounting orientation and measure the same parameter. The vibration signal
with the earliest phase typically comes from the transducer that is closest to the
source of the vibration.
 33

Chapter 3

Vibration Vectors

IN THE PRECEEDING CHAPTERS, WE DISCUSSED vibration signals and the

mea surement of phase. Most of the discussion involved the measurement of
vibration signals that are single-frequency sine waves.
In order to make a meaningful phase measurement, a vibration signal must
contain only one (or predominantly one) frequ ency. However. since typical
machinery vibration signals contain several frequencies, th e signal must first be
filtered to a single frequency. Measurement of the amplitude and phase of the fil-
tcred signal produces a filtered response, respon se vector, or vibration vector.
Thi s vector is a powerful tool that provides the foundation for the detection of
many different machine malfunctions and is vital information for balancing.
The vibration vector is the underlying concept for all the Bode, polar, and ampli-
tude-phase-time (APHT) data plots.
We will start with a discussion of the characteristics of unfiltered vibration,
followed by a discussion of the vibration vector. Finally, we will discuss an
important special case of the vibration vector called the slow roll vector.

Unfiltered Vibration
The raw (unfiltered) vibration signal from a transducer is sometimes called
the direct vibration signal; theoretically, there is nothing in the signal path
between the transducer and the instrumentation. In practice, some modifica-
tion of the signal may occur (for example, an output from a monitor that
includes signal processing), but if the circuitry pro vides no signal processing
besides buffering (something that should always be verified), the output is
assumed to be an exact cop y of the original, unfiltered signal, including any de
offset.
~ J.- l _ Uo "' '' o<I _ ~
_iI;> l . ·_ll~iiO_ '
<....,........ oI>..o.ca.-- r:fta>ot _ ,.
___
"
.
,
.
.
.
,
.
.
.
.
b
t
~
.. t!'>r _ '--'<Y-

Th.... iD ~ tlw word wtjikn-td impbef. rn.t 110 mochf_bon of tIw US-
naI has taRn ~ in tIw ilutnunent.mon. and that il: c:ontai... a! ol in. 1ft..
quo-nq comJlO"l""l'U (WIth Mlpliludie and p/Iaw i nlaCt) th.I nllt in tIw inc0m-
ing rnn~ucn Ioig...... FiIun' 3-1 ...........n "nfilt.nood 't'ibnt lO<l si«n.oI.and its IX-
and U-fiJ!.l'«'d ~ com ponmts. A numbrr 01 Ollw.- fnoquoncy <.'OITl J-
nlMlU . ", contained in ttl. unfllt.!l'<1 .ij( n&!
Th • • mp hU>de of a n " nfittnood . ign&! CIOn I. aa:lIra l,.(y"""...,.."j in PNk-
' o- peak or peak unita r...... ("hl p t" . I ); h O"~_ u n th. u nfill.<>rftl ., bn l,o n
signal io dom ln,r"<i by a l ing!. fnoqu. ncy. II is not ~ibl. 10 rTH'. ',,", pha w
,.,lation lohl~ accunl,.(y.1'tIu. mea..,~nl ""lu i""" a l ignal WIt h a .i~. r..-
,->'
Also. llin« particular rotor bo-hav>on .. malfund:i1oc . may I. auoriiIIft;!
WIth " " - ' ~ (for nampa.. _ unbalantt/. ~ ol tlw ...l;n.
lion Iijma! ls nonnaIIy noquirftl..

FHt"';ng . nd the Vibr.liorl VKtor

VLitmns i. a.ignaJ ~nl tftilniquoo !hat. idNlly. ...;..ct. aD frNtl>t'l>ClO"l
that I", o ut.ide u.. ba ndpa ... "'fli on ofth. filt..... Tn. fill• • uwd moo.l oft. n on
.......i.....,. ",1Int1On wp;naI. ;. th. ~ film. ..hie" ~ aU !ligna! con-
1..,1 tNt .. aboo~ and ......... ..... cn>1.... lbandpoaMl 1l"'l"'f"'CY of ..... filtn. The
cnll....lTnjCH'nCJ .. u....ur wt In .. u..... runni,,!! "I'ft'd ' IXI .... . multiploo of run-
..... "f'"'d if . ....ifi<anl amounl of lnK"i.... .i bntion 0ttUn .ot ttx- tr...
qunxin (for nampW.• 5-...... pump impc'On W<>UId prod~ fMo_pooo.
~ .... P" ~ u1ion, .., l X·fiIuri"ll mi¢rt boo desir" '). 8e<:auoe- IIw _
~ clv.~ """'"' flkO'n ."I.......hall!" .djuot 1M baodpuo fnoqunocy 10
lnod< runn ing 'f'"'"d Such• fill....;. t:aIJaI. trac:tu.,fill.....nd ;. com~ "w
In rot.lms .naclllnny .pplirM""'"
Aft .... filuring.lIw vibnuoo oi~na1 i. cJ<- 10 . pun oiIW ....... at 1M band·
pa" ' ''-'''tLJoency. an d IIw .ml'liloo". . nd pha... or 1M fill......d . i,;nal can t... m N . ·
urlP<! uoing th ,. lK"niqUt'S dioo<"u.w in lhe pr" iou. d ... p1on.
Th,. am phlud,. and p""'" of lh,. filte nod signal dPOCriIw a ~ilmltiotr ....-tor.
which i. pIoll l.'d in lh,. ' ro" .t dUlW rt!spo,,~ (W} f'ki- (Figu... 3-2). A ,,,,,to. ill
.. malh"",at ical objfft l hal h.. bolh mag nitud.. and d' l'P<"lion. Th.. mag nilud<>
ul lh,. vibr. tio n •....,Ior currnpo nds to lh.. ,i b..I,on . mplil ...... (in ..-hal.......
uniu . ... con,,,nient. hul ll"llaIJy I'"' PP or mil pp for wft noIat;,,, vibration ).
n... diff'c1.ion of ..... w.etor cor~ to lh,. aboo!ut.. pn.... of 1M fillf'rt'd
vibratIOn o.gnaL

,
-- '

--gg'"
' " _ )-2 n.. .. 1><.,,,,,,, _l<It, The_conI..... It'Ie~ _ ._
~ _ ""'" m. ~ _ _'"""" "';INt,II .. plcMd '" ltw It"""""...
"'..... l iE ploroo. - . . t'""" 1$ oIgnod _ m. _ _ It-"'- The
Io<ql\dm._ ' ,...... ...-....ond ........... _ d l l ' l e _
-. _
",""_phnoed"'Iq'Iil.The~ .........-from
Il\t I; _;"Il\t ~""" _ I l \ t ~ _ d _ d m. """
 Th .. U axis o f th " I,I" n.. j~ " Iig""" with th.. mt'a""....m..nt ui~ o f Ih.. Ua n. ·
d uCE'J. In th.. f'guro>. l h" lr.ll n... lu~'t"t i~ mo unl.-d J(j from Ih.. horil o nta l. Th.. \ .
ax;'" is a1....a~.." 90" from Ih.. U u i.. in Ih.. d i.....·I' o n o ppo. il.. of ~hafl rol"l ion. It
is important to no l.. Ihal Ih .. (,V a ~"" a .... ;ntkpt'nd..nt of a ny o lh..r mach in..
coordi nat.. sysl"'" and a .... a~<t>ci al .-d wil ll e<lch transducer. Earn Iran""uCE'J has
ih o n Iransdu"". ......pon... pla n... and Ih.. "' u.....m..nl axis of lh .. lra nsduCE'J
is al ay. al ign.-d ..i lh th.. U ,,~ i . of il. ""'POn plan...
Th.. I..ngth o f th .. 'ibrat ion ,..-cto r i. "'I.uaI 10 Ill.. ampl iludE' of Ih.. r,h.."",
, i b , al i"n, Th .. a ngl.. o f lh pon... n 'Cto r ...Iati,-.. to Ih .. (J ax ,s IS Ih.. a bs<,lute
p ha... lag. "'..a'u ...... fm m t m..a. u...",..nl axis in a di reclion opfJOSue 10 II>e
di l"f"<1i..n of "'tali.. n o f I.... ma..h in.. ro lo r (FijlU '" 3·3). Thu•. d"l"'nd ing " n tho>
di 'o>cti,, " o f ....ta l'o'" th .. ,",'Clor ..a n p101 in d ifT.....nt pla""s. Xor:.. l lla l th.. po.i-
ti,... i ' a xis ;'; 1I1.....y .I at a t 'l(l". m..a.ur.-d 0Pl_it<> t h.. d iro>ct ,on o f rolallon.
To p lur: a ,i bral ;"n to r. fo il...... th..... st"ps:

I) fM r-nn in.. Ib.. anp;ular o ';"" la l ion o f th.. lra nsd uCE'J relativ.. to
~~Iu rmam in.. ' . ....-poin l Thi. ..i.U d..fin.. t he d il"f"<1 ion of th e U
I m..a.u........nt ) axi<.,

2) 1><'·Il"TTTl in.. Ib.. d i.o>ct ion of rotatio n of t l>e . 00or. TI>e po" iti'-e F
axis will be loca l.-d 90" from t .... 1/ ax is in a d il't'C'l ioo ol",,"it.. to
rolor mtalio n.

3 ) l '... th .. ab..,l ut.. ph.a'" of th.. ti" ..,oo ,i b. atio" sif!nal to lor a t..
the a n!IUla r o ri..ntalio n o f th.. ,ib.atio o '"t"Cto r, .\1.."",... I....
a ngl.. fro m the U ax i.. oppo.it.. tbe direclio" o f mtalim•. tow ard
Ihe d irl'C1 ion o f Ih.. i ' ax i..

4 ) Th.. l..ngt h of th .. ,ibmlio o , -..ct." , is th .. a"'phtud.. (If I.... fil-

t..r.-d ,ib,alio n s ignal. Typ i.... IJ~~ displ "': " "'..n t u "il5 a .... lIm pp
or mil pp. ' ....oci!)· unil8 are m m/ o pI.. Or ;n /o pI<. a..d ""'",,I...-a -
l ion u nit. a r.. g I, k.

Ik>ca u... t l>e "b,all" o ,'t'(:to t8 d..tin.. I.... ..... po n... of I.... mam inery to a
"a rio>ty o f fact o ,... il i. critical I" d" ..um..nl th is d al a u nocr a ,·ari<'t~· (If opt' ,at ·
inf! co nd ition.. On c-ril ir a l "'....hin..ry. .......... lra n""u.....rs ar.. in >ta lled al man~'
locat ion .. th .. vi!>ral'"'' d ala fro'" ..arh tw nsd .......r .houl d Ix- ......"d.-d 0'0' ''
Ih..
..nl ,re opera tinf! s pt'<'d ra ng.. d uring , Ia rt up and shut <kru.-n. I X a..d 2X v....to••
a... mo..t "",,,, m,,nly ",..uur.-d. h ut ot b...- frequ..ncy compo....n.. . ho uld Ix-
...........rn1 if thftt;.. fofri"8 funct ..... I.<Xh .. bl.Idoo P"·.. ~ l lhM is.l . Mr'
monic of runnl~ 'f'"'d.
\ 'obnlion \"Kton .... ..., monil.orN wh iJo. . modluw io. ~ ... ......-
".nl ..,....d o...nKn in .,....wn,.and Ic-.1 cond.. ....... CIII pmd~ pn-doct.lbW
m.n,n. in lftp<>nM' ~on.. but lignific.lJlt ~ _~ duo ., n \~ C1>UkI
indicltO' • ~ In ttw rnlI<il,...... ho-llth.llnnpt'<1N ~ in \mr.tion ...c-
Ion Iff' imporunl for t ho- .,.uty drt«tion of ....:nl.... mtffnll pn>bk-mo.. suoch ..
unbll.o~. rul>. inotlbilll nd w it cracko.. a nd ntnnal ploblo ,, ' .... udI ..
rouplin~ failurt, pipi"8 .u. i .and foundation dooI.. rionl ......
ThO' t Ip of 1hfo ...br."on ~ ~ ...... a poinl In lho- Ifl nodlKTf rnpon...
plan... A plot of. IIt'I ..f th poi nll corT1'Spond i n~ I.. d, n..... nl ma min.. condi-
tion. p. orid.... powt'I"ful VI d i.p1a~· of th.. .....f'<'n ... o f th.. mach in.. at lha t
tr..nod...,... loca tion . ....hri ho-. I m....hin.. io .l.art m~ up. • 1 ''P''ral ing sp<'fti. 0 1
COIll in!! down. n." l'f" t o f 1 o f sla n ul' 0 . sh utd ·n ' 1Ml lio n , t"cto r points
io t'<jui,'a1em 10 I pokir plol (s.-.. <-"hapte. 7). o ne of I m' l6t info rma li.... plots
....Ilabl.. for dllf!:"""" nll m....hin.. 'Y rond ition. A tt't o f . uch points al • ot dy
op<"f. ting . po"<'<! ("....l)I ",. 1.,) prodUCt'S an APl ff plot. a nd vib rat IOn Vt'cto"' ....
rnorulON'd dUring machi"., opt'I"llion in accepI4InCt" f1"lPOn s in ttoes.. p lot ..

' OV- J.-) 1' .... . . -....... - . . . 11I .._ . at ..... _ ~ . " .... .. _ 9(1' ""...
... ~ t! _ - - - " the III "'..' 01 _ d "'" '"'1110 ~ no
I' n. ...... oot»__ •_ ....... do":aa otl. "".... 40 .. on N (101<_, d ..........
38 Fundamentalsof Vibration

Working with Vibration Vectors

When working with vibration vectors, it is important to use a system of
notation that is both convenient and complete. Commonly, vibration vectors are
noted as

r=A L.rp (3-1)

where r is the displacement vibration vector, A is the vibration amplitude, L.
identifies the value that follows as an angle, and rp (Greek upper case phi) is the
phase angle, usually expressed as positive phase lag. This is a polar representa-
tion of the vector. An equivalent rectangular representation is given by

r=u+ jv (3-2)

where j is the square root of -1, and

u = Acosrp
v = Asinrp (3-3)

The variables u and v are the rectangular coordinates of the vector in the
transducer response plane. The u coordinate is measured along the U axis (the
transducer measurement axis), and the v coordinate is measured along the V
axis. (As we have already noted, U and V are not the same as the physical XY
coordinate system used to describe the machine.) Vibration vectors are actually
complex numbers, and the j term originates in complex number theory.
Conversion from rectangular form to polar form is performed using these
expressions:

A =.Ju 2 +v 2

arctan2(~)
(3-4)
rp =

where arctan2 represents the arctangent2 function, which takes quadrants into
account.
For example, the vibration vector in Figure 3-2 can be expressed in polar
form as
0
r = 90 /lm pp L.220° (3.5 mil pp L.220 )
" nd in Il"ctangu lu form u

r = - 69 I'm pp - J5S fII1l pp {- 2.; m~ pp - j 2.3 mil Pp J

I I.....",,,'. ('<In,,,,,,;,,n in lhe tlPPO$;t" d i...."iu ro can [pad to di tHculty. lf a 5imp l..
... ct" n1(..nl funct 'u ro is",...u 10 ~<lku lale Ih.. pha,... lag a n!!!... Ih.. "","ull ;$

..·hieh is mrorroct. TIl.. silual ion is 5hown in Figu..... 3· 4. Adding 180" produ n-s
th.. rorn-ct It'Sul,: 220". \\"hen using t he 5ta nda rd arct a ngent function. il is a
1l0 0d id..a 10 skctch Ih.. 5it uat ion. ca n-fuIlJ nol inlll h.. 5illns of the coo,d inal......
a nd ,...ri~·lh ",Il l. Engi n....rinll a nd "'-;.." lific cakulatou u th.. a rclanll..nl2
funct ion n ronwn in ll from [('("la nj(l1lar to polar coo rd inal b uI t hey pro -
duct' a ... sull I><-Iw n .t ISO"'. Wh .. n ,h .. ca kulalro ..... u11 is ....ll"l iw. add 360" 10
prod uc.. a posil " pha... lag I><-I........ n 0" a nd 360".

fiooJu'e 3" ~ .. <t.ngen' tuncl"" . nd r«tillgul¥to poIo'~.The ........ _ 1Ol i' plot-
t'" "'" d~,ent tOIi\o01'l di''''I'''''''_ the r«t."9..... cOOtdi""IH..e """"".l/1.e 0I1t">e 'I....
""d aocungenr tunc,.... roe"'" pIl . ", 1iQ""9'" 01 OO':wIloct> " """''''''t the uue pII'''' Ii9
""II'" is 12O'Wl-oen """9 the ' 1CIarl9<'"' f\.ncoon. n is. yood ON to >1«<'" "'" " n"".... to I'<'ft-
fy the ",,1cu_...-.gIe
 \,brllion ''ft."tOn oft.... .......l to boP addod ...bc l'8Clft!. nwlliplit>d. and dl'id ·
II'd. Addllion Ind ""bcraction of ho'O ,'ibration >?don can boo do<. ~ph"'.u~ or
1M rnWt ~ boP No.ilY ~aln"'lf'd u";"!lIM N'rta"llUiu lot.... Mul1 iplrilion
Ind drm.IOI'I ~akul.o lf'd ........ "I~ u.illjl tlw poWr form-
.<\ddi11Ol1 ~ <Jon. by addi"fl l'" • cornponrnu and ~ cornponrnu .tofy
Ind lhom comblnilljl tlw.......no 10 df'fi I . - wnox no.. unih of _
mml oft'" two '"l'Clon m .... 1w tlw or tlw mult W1lIIw llM'..., i~
Two .-ibnllOl'l ,?don, r , and r :- can Iw addf'd LVo.phior~ 10 pro<fuott I
........tarll >"KtOr. r J 1__ F.,..... J.-i). To do thil.,

1) Plot r , in thf' lra n !ld...-n ''' 'PO''''' pia......., lh iu t l il .at 1M ori-

".
2 ) Plot r j in " i th 1/.' (.til 11 1M origin.

4 ) [)r-~w . n....· \TC1or ftom lh<-" I'i gln to Ih<- lip of Ih.. «>py of ';ro
Th il \TC1of iJ 1M ..... ull.nl. ' ,.

- -
f . . . . }oS Glop.:: _ _ 101Oll 1',
....., I'r plal _ """ -.en ...... ,,1ftWIucet
_~. C"P\' ,,"' ''''''''''''d l'.''
II ..... ~ pd ' I_ The rew ~""'. 'r ~ N _
~_orqnl0_l,pd . ..
 The IIm,", K'al lech niq ue i. the mal h<>ma tica l "'lu i,-a lelll u f ..dd i" l1 Ih" IWO
,-...::1",... a nd
V ( . . mpo""nl "

(3-51

b am/""
.-\ ga~ tu.hi ne ro tal "" III a r to X d irec lio" a l i·GO ' pm. o..ta i~ la l en fro m s "a~·
ing \-.-I",,·ity t.s n'l<lu<'<"l" (wh ich pro.,des ah...-.]ute ca~inll mtltiun ) a nd s . ha ft . ....
ati .... di~plal"('me"t t .ansduce~ Roth tran.sdu""r~ a re mtlunl"" at 45" R. The I X.
integTa t..... casi nll .ibrat iun. r c' i. rOll t>d 10 be .w ",m PI' L 3S" ( 1.6 m d pp L r;").
The I X. ~h. ,fl ll"la l"..... brat iun. r" . i~ "",.. ", red as 30 j.Ull pp L I N 0 .2 m il pp
L IN ). Find Ih.. I X• .haft absol ut .. ' ibra tion ......,Ior. r ,.

S<J/uli,,"
The ~ha ft s m..lut " .,bra tion \-..clor is fou nd hy atld ing the u a t>d v compo"enl '
of the ("asi"l1 ••bra tiun a nd s haft ali'... vihm l'tI" '"eCto~ (l'ill" re 3·6 ). The
Imn.sdUC<'r r<'Spo n'" pla ne is s hown il h Ihe U ""is a h!tned wil h th.. transd uc·
er. m..a." r"'n<>nI &X, lkes"... ...Ia tion is r l.. X. t .... po. il'.... , . ax is i~ localt'(!
90" counl...c1od....., ( '" the U ax;", a nd Ih<- ph a,...· a n!!l.. i. meas ured in thai
d iT<"ct iun .

r c · "' '' W
r" . 10 D:N
:':-' ~_

t' I. 1))

Fig u,. H i """"bOn 01vt:oOllOll ~ 10 find ..... >Nit a l»olu!. Vlbt"'O:>n_Gtop!l" oIty.odd .....
u ""J . bsoI..... vt:oouon _IOU , . to ..... "'oft ",101_ vibtatO:>n _tOf.• r Of .odd,"" ~ and •
<amP<>"""" of NCh wetor_
42 Fundamentalsof Vibration

1) Convert both measurements to rectangular form , using Equation

3-3:

r c = (40 f.lm pp jcos Sfi"+ j (40 f.lm pp )sin 35·

r c = 33 f.lm pp + j23 f.lm pp

r sr = (30 urn pp )cos 120· + j (30 f.lm pp)sin 120·

r sr =- 15 f.lm pp + j26 urn pp

2) Add the components to get the solution in rectangular form

using Equation 3-5:

rs =(33-15) um pp-l- j(23+26) um pp

r s =18 urn pp-l- j49 f.lmpp

3) Convert to polar form usin g Equation 3-4:

A = ~(18 f.lmpp)2+ (49 f.lmppi

1> = arctan2[49 f.lm ppJ
18 f.lmpp

r. = 52 f.lm pp L70· (2.0 mil pp L70·)

Subtraction is done by subtracting the u components and v components
separately and then combining the results to define a new vector. Graphical sub-
traction is performed by adding the negative of one vector to the other. Again,
the units of measurement of the two vectors must be the same or the result will
be meaningless.
Two vibration vectors, rl ' and r 2 , can be subtracted graphically to produce a
resultant vector, r 3 (see Figure 3-7). To find r 3 = r 1 - r 2,

1) Plot r 1 in the transducer response plane with its tail at the ori-
gin.

2) Find the negative of r 2 by changing the phase angle by 180~ This

is also equivalent to multiplying r 2 by - 1.
 e hoapl e , 3 V i bi al io " V,"" IOf~ 43

3 ) rlo l - r 1 wilh II.' la ll al lhe "rigi" .

~} Corr - r l «) Ihal ils I" il i. .. I Ih.. lip of , I'

S} Dra", a " t'W ,-..elor fm m Ih.. origin I" I t... lip of Ihe ""rr " f - ' ,
Th is '''''1''' is l he ....,;" ll a " l. ' l '

T h" p ap hi"al l""h niqu" is Ih" ma th MTUll ical "'lui.-aI"" 1 of s " hlracti" lI lh "
"'-0 ,"{'('t" rf;' u a nd v rompo...." ls ;
( Hl)

Mu lli pl;"'a tiOfl o f I"''' ,"('('tOrf; is pt' rformnl mo, 1 l"asily ".i"l1 Ih" pola . for-
mal: m " hi pl)"l h.. " m plilud... a" d add Ih.. ph a.... a ngl""

(3-7 )

Di,; sion is l",rf" m lnl b~ d,,;di" l1 Ihe " mplil ude. a nd . " bl r" cl mg the pha ...
angles:

See Ihe Al' pt"ndi x for examp l... of mu h ipl;"'"l i" n a nd d",.;.,,, of ,;h ralion
,""to....

f ig<>re J-1.G'.."nc _ 1..- "'bllaetJon. To ",t>-

"""I ',!tom ," plot " 0.-.:1 ., ,,,
<tie" • .-..dur.-
... ~ p1. ..... l'loc <tie rwgatI>'e ot " and
cop)' ~ ' " (hot I"'" to! ot - ', ~ al l..... l;p 01 ' ,.
The nl ' r ~ tile ","",at !tom <tie 009 "
"' '11'01 _"
44 Fundamentals of Vibration

NOTE: For all vector operations, the phase lag angle should be expressed as
a positive number between 0° and 360~ If the calculated phase lag is negative,
add 360~ If the result is greater than 360°, subtract 360°.
Most scientific and engineering calculators can operate directly on complex
numbers and don't require conversion between polar and rectangular forms.
This vector concept extends to more than just vibration measurement. The
force due to unbalance is a rotating force vector that has a particular angular
position when the Keyphasor event occurs. And Dynamic Stiffness, a very
important concept in machinery behavior, is also expressed as a complex num-
ber, usually in rectangular form. These two vector entities, together with the
vibration vector, are fundamental to understanding the dynamic behavior of
machinery.

The Slow Roll Vector

The slow roll vector is an important application of vector subtraction. The
slow roll vector is a constant, or slowly varying, component of the vibration vec-
tor that represents nondynamic action observed by a transducer. The slow roll
vector will be different for each measurement transducer location. It originates
in mechanical effects, such as a bowed rotor or coupling problem, or in mechan-
ical or electrical runout, and it can distort and obscure the machine's dynamic
response data (Figure 3-8). Slow roll vector compensation is the technique of sub-
tracting the measured slow roll vector from the transducer vibration vector
(Figure 3-9).
To measure the slow roll vector, we must be able to find an operating condi-
tion where the slow roll is the dominant component of the vibration signal.
Since the IX dynamic response due to unbalance tends to zero at low speeds,
any IX vibration measured at these low speeds is considered to be due to
sources other than unbalance. Thus, slow roll vectors are measured in this speed
range, which is called the slow roll speed range. One guideline (and it is only a
guideline) is that the upper limit of the slow roll speed range is about 10% of the
first balance resonance speed of the machine. The slow roll speed range is best
identified using an uncompensated Bode plot (see Chapter 7) of a machine
startup or shutdown.
While noise affects all transducers, slow roll data is usually obtained only
from displacement transducers. Eddy current displacement transducers have a
frequency response that extends to dc (zero speed), while velocity and accelera-
tion transducers do not. Because of this, velocity and acceleration slow roll
response generally cannot be measured, and vibration data from these trans-
ducers is not compensated.
 -,

-_.
fC_ _ H n..
IX _ ""' _ . IX
_""' _ .... i_ _ OO ..... ,.-.-....

~._--_
~_(bW I_1N-' _
~~"""

_ lOt. Sbw",. -un un bo ~ lot

n..
.....
.. .."."""" 90 1"'' -- - - - .1<''''''- - - -1
...,. h.ittnOfllC of "'''''"l1oPO'O<l-

At- 1-9 _ ... _ <""""""",en TO

""""'""'"'''' for _"",...woct lhoo_
",. _ !rom 1N ..tItif"'" _ lOt .......-
Ul f<j bI' t........... , , _... .n.. ~onl is
troo~t<'d.dynomoc

_ ' " _t<:< (_ n ). -_.

1. _ _ "

, ~.

--
_ _
1» _
'ftct... ~

.......

.., _.
1;.., ._ i
.- _
IO~ 'S7" ~-
-
• • • tIloL _ _
--~
opood. .... _

-~
i04 .. _ -
II ....
pn>booo
• __ .... _ _ _ oil

~_
~
. ...........
_ - . . . .. lIl«IJ' ...poo>du<.-. " " iil __
I ,,
,
• I
d _ ~

...--..u...,..-.......-.
tho<... ~ ... . _ _ oil .....

___
"' _ ~ _ "' IiI ""
. ...... pn>booo ..., ,_ • d-.. _ of
.1 ,~--
iO
""'" .. ~ ..-itb . _ ......... _ _
I""""" l- IO~ !loa" ... of th .. oil ..............,j
" ........, ...nllj' <lot. ohoWd _ u"": ....h COlt·
tlun. I,... from . ..... . ho......... . much ..... ---
, I. '

-
.... . ... _
-
b.....,. iii ho.. thd J'f'>hI--
r ...n.fl" .... ,,,,,mal
m.a<hll... ,..<hoi

_-.
--_
"'I ",hbnum .. N nfWl(l "1"""1 ...... n>II <lot.
....
..... __
"90 .. 1 ' 0
...., '" irwoI>o:l \IM:I\;.......,h h'ICh _"""lI «>Ill _ _,_ _ _ ""'"
""'.....,." 1
........ FadXnto. _h . . . . . 'otlN.... ...
(0)<_-......_ ...._
...It......., <" op ' WOn ..... _ .._
....
_
_
.... _
. ..... _

.._
_ _ . . . -Il.I.. _

, ......... --..-
00loi ...... ... ___ _ . . . - .
... 01 __.. . -,. .. . _d_
"""'
__ ."",
_ -_
. ._
_-.lr- __

__..........
J . _ ..... _

-_..... _-_... ----

... ..-...
a..,.to"' _ ' - .
..... _
... , ,
. . . ... ... -0_.
, . , <OIid -..... .
00loi...... ... - - . _ ...

r _ ....... pn>booo ......... _

_ --... - . -., -
•
~

• looIf; jIO!'riod of .......... to

__ .-...- w..... " d ow-. .w,
~ _ __
,100 - . . . . ....-fo«. ... .. il I ... . -...
..- . ...o..npn,: ..... ... ~ ...."""' .
_,hot ~ ...n,-" ..
tho .......-hi-....l _ _ _ .,..od
~
"....
 Chapter 3 Vibration Vectors 47

Summary
An unfiltered, or direct, vibration signal is unchanged from the original
transducer vibration signal. It is assumed to contain all of the original frequen-
cy, amplitude, and phase content and the original de offset, if any.
Filtering removes signal content. Many machinery vibration signals are
bandpass filtered to a multiple of running speed, most often IX. The filtered sig-
nal is a sine wave with a frequency equal to the bandpass frequency of the filter.
After filtering, the amplitude and absolute phase of the signal can be measured.
A vibration vector is the combination of the amplitude and absolute phase
of a filtered vibration signal. This vector is plotted in the transducer response
plane. Because vibration vectors are complex numbers, they can be added, sub-
tracted, multiplied, and divided.
The slow roll speed range of a machine is the range of speeds where the
dynamic rotor response due to unbalance is insignificant compared to the slow
roll vector; roughly, it is below 10% of the first balance resonance speed of a
machine. The slow roll speed range is best identified using an uncompensated
Bode plot.
Slow roll compensation is the subtraction of the slow roll vector from a
vibration vector at the same measurement location. The resultant vibration vec-
tor will only reflect the dynamic response of the rotor.
Data Plots
 51

Chapter 4

Timebase Plots

THE TIMEBASE PLOT IS THE MOST FUNDAMENTAL graphic presentation of

ma chi nery dynamic data. It shows how a single parameter (most often displace-
ment, velocity, or acceleration, but also any other dynamic measurement) from
a single transducer changes on a very short time scale , typically a fraction of a
second. This is in contrast to trend plots, which display the value of a slowly
changing parameter (for example, axial position) over a much longer time scale,
typically hours to months.
A timebase plot represents a small slice of time in the vibration history of
the machine. Usually, the amount of time involves only a few revolutions of the
rotor. During this short length of time (about 17 ms for one revolution of a 3600
rpm machine), the overall behavior of the machine is not likely to change signif-
icantly. However, unfiltered timebase plots can clearly show a change in
machine response if sudden events occur in the machine or if the machine is
rapidly changing speed (such as an electric motor startup).
Timebase plots have several important uses. They have the advantage in
being able to clearly display the unprocessed output from a single transducer.
This allows us to look for noise on the signal or to detect the presence of multi-
ple frequency components. An important use of a timebase plot is to identify the
presence and timing of short term transient events.
Multiple timebase plots can allow us to establish timing relationships at dif-
ferent axial locations along the machine train. Or, the timebase plots from a pair
of XY transducers can be used to determine the direction of precession of the
rotor shaft.
To understand the timebase plot, we will discuss its structure and con-
struction, followed by an explanation of the meaning of th e Keyphasor mark.
S2 D. ' . Pion

We " i ll l ....n d i...u"," . 10'" roll romp<'n ",l;tl n a nd " ~1'..eiaJ " ppl ical ,o n of th e
.... .w fonn coml""nMlion I""hn iqu.. lh..t ca n h.- u"-'<l lo p. odu.... a Not· IX l im....
ba... plot
Finally. "'.. will <I..mo n. t ,at.. "''''. W .~ "ai n 1M la ' ge a mo unt of infor mal io n
thai exi.t. in a l imeba ... plol ••u"h a s the p.-ak,-to- peal; a ml>lim de. l he 1111 .....
•ibrat ion freq u..ncy, l he roWr . peed, l he nX . mplilmk a nd pi",,,., of a fill .....
•ignal. a nd Ihe relal i", frNj"..ncy of Ih.. m t..,......ihrat ~'n signal "'........ ru nning
· I.....d·

The Struc lureof a Timebase Plol

The l imeba ... plot i. a recta ngu la , (Cart,,"ia n ) p lol o f a pa. ..m"'... ",,,,u.
l ime l Figu re 4- 1). Time i. o n l he ho riwn tal a xi... a nd e1af""'d l im.. int "..,.
from left to ri¢'l; ,,',.. .nt. occu rri ng lat..r in li m.. " i ll b.- ttl th.. righl of ' Ii...
('wnt s. Ileca u... of the t ime ""a1('s encou nt..rnl in rotalinll mach io.-ry. l ho-
....,1...... li m.. i~ 1}I' ,eaJly d i. pla}-..d in m iD i"""tmd. Im s).
Th.. ....,a "" red pam m"'..... com...n .... from m lta lt" 10 .."!li......ring u n;l s, i. on
l he ....rt k a l u i... On R..ntly ~",,'ada l imeba.., plot .. t .... d al a i. approxima t.. 1y
vert ica lly cent .. ,...! in l he plot . a nd t ho- un it o f m.....urem ..nl p<'r vertical ""a le
t ick, ma rk is identifi.... (in the f'gu . ... l hi. is I JiIllld i' -I.

FOg..... 4- t Unfi~ ... ed and ~~"'ed U~

base pion. Tho- plot ,....,.,.. ..... tharolJe .. a
.........,"'"PO"~'" .-r hme. Ti"",.""
.1'>< t.onlon"t aQ~ int~ from ~ to
rq.t The ....-tJUl1 is ",pre;enrs tI'>r
j\
"",-,,,,<I pa .. m ,m...,..
(In <k-
IX -li"....:l
...' - ,
pIac ....... U•• nd tI'>r ", a~ ind" ....,
It"""d..1rm" to th.. sma/l lrl matI<s-
T...-...t>a", plots can be un6~... ed (,op) Of
t r-' """,,,,"""IN'"

6_ :bon""'._ t X -6~"'ed plot of .1'><

........ dat al. Tho Koypt1a«>r ma'" ndicate<
tI'>r ow...''''''',, of a l(ey,*,""" .........
H/V/\
L
o

'.
~

•~:o---c~:o---c~
•
 Chapter 4 Timebase Plots 53

The vertical position of a point on the timebase plot represents the instan-
taneous value of the measured parameter. For velocity and acceleration trans-
ducers, it represents the instantaneous value of velocity and acceleration rela-
tive to a point in free space; for displacement signals, the vertical position rep-
resents the instantaneous position relative to the probe tip.
Note that the terms peak (pk), peak-to-peak (pp), and root-mean-square
(rms) are used to describe how changes in the parameter are measured and are
not appropriate units for the vertical axis of a timebase plot. However, the signal
can swing through a range that can be measured in peak-to-peak units. In the
figure, the amplitude of the filtered signal (bottom) is about 6.0 urn (0.24 mil) pk,
12 Jlm (0.47 mil) pp, and 4.2 Jlm (0.17 mil) rms. All of these terms describe the
same signal.
In unfiltered timebase plots, digitally sampled signal voltages are first divid-
ed by the transducer scale factor to convert them to equivalent engineering
units. Then, the converted values are plotted on the timebase plot. The resulting
waveform describes the instantaneous behavior of the measured parameter
from one moment to the next.
Filtered timebase plots are constructed from the amplitude and phase of
vibration vectors. The plot is synthesized by computing a sine wave with the cor-
rect frequency, amplitude, and phase (see Appendix 2 for details). This synthe-
sis process assumes that conditions in the machine don't change significantly
over the period of time represented by the synthesized waveform. This is usual-
ly, but not always, a correct assumption.
Computer-based timebase plots display a digitally sampled waveform. The
sample rate determines the upper frequency limit of the signal that is displayed,
and the length of time over which the waveform is sampled determines the low
frequency limit. Low frequency signals will not be completely represented if the
sample length is shorter than the period of the low frequency component. For
these reasons, digitally sampled, unfiltered timebase plots are, inherently, both
low- and high-pass filtered.
A timebase plot has several important differences from the timebase display
on an oscilloscope: a basic oscilloscope displays voltage on the vertical axis,
while a timebase plot displays engineering units, such as urn, mil, mm/s, g, etc.;
the scope can display over a very long time frame; and there are subtle differ-
ences in the display and meaning of the Keyphasor mark.
54 Data Plots

The Keyphasor Mark

The blank/dot sequence on the timebase waveform is called a Keyphasor
mark. The mark represents a timing event, the Keyphasor event, that occurs
once per shaft revolution. The timing signal comes from a separate, Keyphasor
transducer and is combined with the waveform so that the timing of the
Keyphasor event can be seen clearly. The time between two Keyphasor marks rep-
resents the period ofone revolution ofthe shaft.
On all Bently Nevada plots, the Keyphasor event is shown as a blank/dot
sequence, and the dot represents the instant that the Keyphasor event occurs
(see Figure 2-4). This is different than the Keyphasor mark on an oscilloscope,
which may be a blank/bright or bright/blank sequence depending on the type of
shaft mark and the type of oscilloscope used.
The Keyphasor mark on a timebase plot adds important additional infor-
mation that will be discussed below. It can be used to measure rotative speed,
the absolute phase of an nX frequency component (n is an integer), and the
vibration frequency in orders of rotative speed.

Compensation of Timebase Plots

The primary objective of compensation is to remove unwanted signal con-
tent (noise) that is unrelated to the machine behavior that we want to observe.
This noise, electrical and mechanical runout (glitch), bow, etc., can partially or
completely obscure the dynamic information. Shaft scratches or other surface
defects create a pattern of signal artifacts that repeats every revolution. It can be
very useful to remove this noise to better reveal the important dynamic infor-
mation. In Chapter 3 we discussed one type of compensation, slow roll compen-
sation of vibration vectors. Most often, we wish to remove the effects of any IX
slow roll response that may be present in the signal so that we can see the IX
response due to unbalance.
Slow roll compensation is primarily applied to eddy current displacement
transducer data because these transducers have a significant output at slow roll
speeds. At these speeds, output from velocity and acceleration (seismic) trans-
ducers is extremely low, and there is usually no measurable slow roll signal. For
this reason, slow roll compensation is rarely, if ever, performed on seismic trans-
ducer data.
Filtered timebase plots can be slow roll compensated using a IX, 2X, or nX
slow roll vector. The slow roll vector is subtracted from the original vibration
vector, and the new, compensated vibration vector is used to synthesize the fil-
tered waveform. The end result is a filtered timebase plot that is slow roll com-
pensated.
 Chapter 4 Timebase Plots 55

1X-filtered, uncompensated
7 urn pp L84 "

Figure 4-2 . Slow roll compen sation of fil-

...c ... ~~
OJ .

tered timebase plots.The top plot is a 1X- E

OJ
u
fi lte red, uncompensated plo t.Th e bot - '"
Q.
VI
tom plot shows the same data after com- zs
pensation with a slow roll vecto r.Note
that, in this example, the amplitude is 1X-filtered, slow roll compensated
larger for the compen sated plot and th e ;;- 14~mpp L170"
absolute phase is significantly different. '0
More often , slow roll compensation w ill
E
::J.

result in a signal w ith lower am plitude. ...c

OJ
E
OJ
u
'"
Q.
VI
i5
o 20 40 60 80 100
Time (m s)

Figure 4-2 shows plots of an uncompensated (top) and compensated (bot-

tom) IX-filtered waveform. The bottom plot has been compensated by subtract-
ing the slow roll vector. 15 11m pp L 17° (0.59 mil pp L 17°). from the uncompen-
sated response vector. 7.0 11m pp L 84° (0.28 mil pp L 84°). Note that the vibration
amplitude of the compensated plot is larger, 14 11m pp L 170° (0.55 mil pp
L 1700). Subtraction of vectors can sometimes result in a larger vector. depend-
ing on the relative amplitudes and phases of the two vectors. In this example. the
slow roll vector is significantly larger than the original vibration vector. Note
also that the absolute phase is quite different between the two plots.
Another type of compensation. waveform compensation. can be applied to
the unfiltered waveform. Unfiltered timebase waveforms consist of a sequence
of digitally sampled values. One waveform. selected from the slow roll speed
range. becomes the slow roll waveform sample. Each of the slow roll sample val-
ues can be subtracted from a corresponding value in the original waveform (the
Keyphasor event can be used as a waveform timing reference). This method has
the advantage of being able to remove most. if not all. of the slow roll component
of the signal.
Waveform compensation will remove all components with frequencies up to
the Nyquist sampling frequency limit (Y2 the sampling rate). Thus. IX. 2X..... nX
(n an integer). and all subsynchronous and supersynchronous frequencies (to
the Nyquist limit) will be removed from the vibration waveform. which includes
56 Data Plots

many of the signal artifacts due to shaft surface defects. Figure 4-3 shows unfil-
tered timebase plots. with the same scale. from a machine before and after slow
roll waveform compensation. Two things are immediately clear: the compensat-
ed plot has higher vibration amplitude and the waveform is much smoother.
Most of the high frequency noise in the signal also existed in the slow roll signal;
the waveform compensation removed it.
Unfiltered timebase waveforms can also be notch filtered by compensating
with a synthesized. filtered waveform. The compensation waveform is recon-
structed from a nX-filtered vibration vector that is sampled at the same time as
the waveform to be compensated. The synthesized waveform is then subtracted
from the vibration waveform of interest.
Using this technique. you can examine a vibration signal without the pres-
ence of any IX vibration. A Not-IX waveform is created by subtracting the IX-
synthesized waveform from the original unfiltered waveform. The resultant
waveform reveals any frequency information that may have been obscured by
the IX response. This can be helpful for identifying vibration characteristics
associated with a variety of malfunctions.
Figure 4-4 shows an unfiltered timebase plot. with a combination of IX and
VzX vibration. (top). and the Not-IX version (bottom) of the same signal. Note
that the Y2X vibration. which is the dominant remaining component. is clearly
visible.
Compensation is an art as well as a science. There are many variables that
can change the compensation vector or waveform. It is possible. by using incor-
rect compensation. to produce plots that convey a wrong impression of machine
behavior. Initially. it is always best to view data without any compensation. Then.
when it is used. compensation should always be done with caution.

Information Contained in the Timebase Plot

The timebase plot has many features of a basic oscilloscope display. Before
the widespread use of computerized data acquisition systems. the oscilloscope
timebase display was a basic tool for machinery diagnosis. With the advent of
the digital vector filter and the addition of the Keyphasor mark. the capabilities
of the oscilloscope timebase display were extended. Now. computer-based data
acquisition systems have evolved to the point where they almost provide a vir-
tual oscilloscope.
Both oscilloscope timebase displays and computer-based timebase plots
can be used to make a number of measurements. The following discussion
applies primarily to timebase plots. but it can be extended to oscilloscope time-
base displays.
F. . . . ... ) _ ' lea,•• ~'"
..... e toll _11'It top
t

_ _ r.l' d_ lea . _
__-....._dOUi_
• .....-...,-.g " • .!'JO __ 11'It

•• , f"" .~........, .....

_...,.,..
.... • d"" .r~~u
dNtt,....-.....,~_

""'" «•• ~ .....,., "'" - moot

'" "'" _ . . In ~ "9~ 11'It - . . ....
mool hktly _ ... <JIil<h.

..
•
--
• •

~ ..... _~ ...
poodu<t'IOol· I'l- Jht top pIClI_ . .
........ . '4 ..- . . . . ... I X _ . . . " . _.
10< _ " __ -...1 ..
' - 1O <onWUCl "'" I ~ ~ " "
wi do"" The NoI-I . plea; "",,"<Jl'I>I ..
...... ongonoI..,... _ 0f'II' t~ I X ton-
t t ' f l 1 ~ _ ~_ P ' _

.....tty t n ~ "'blAho"_
 On.. addillo nal ...o rd o f cautio n sho ,,1d t... m..."tiannl: o n ,>scillo"",,1'ffio a
dll'ision " , ual ly m..a ns Ih.. SIM"" Ix>tw n major IIr id lin..._On Bt-ntly N." ·,,,la
I'lols, a <li" , io n is " sually Ih.. ,"".... beh> n tic k ma rk...1 h is n1n a nd d...... ..a u'"
ron fus io n. a nd it is impona nt to Ix> su r hi,'" m..a " ,...mnt t spl..m i. ho-ing
"""'-
Sing l.. l.i"",ha ... plots with K.. ~l'hasor nta rks ..an Ix> usN 10 lTK'a, u the
a ml'lit ud.. of un fill.. rw1 ,'ib.at ion; th.. ro to r s po"l'd; th .. 1'requ..ney. am plitud a nd
al....~IJ tl' ph""" " f filt... 1'<l ,·ib ral.ion: a nd th.. ...aal i,... frl'qu..ney of filt.. rcd ,i bra·
Ii" n ,......us nlU" "'.....d.
Add ition a lly. l ho> sha pe of a n " nfiJI.. rcd t im.......... sij(nal
Ca n p n,.. ..... imp"'rtant c!u..,-, I" Ih.. ho>ha'ior o f macbin.. ry.
Multir l.. l im..a", ,,.. rl,~ . un bl' "....J 10 m..as ut .. l ho> ...Iati.... pha... oft...o sifl;'
"a i, a nd. wh..n th.. s ignal' a."
fro m ",' hogonal d is pl"""m..nt I.ansdun"" t ....
d,«'('II" n " f I"""".. ion of l ho> rotOL
II.-fo .... '~ .nl.i nu ing. it i. imp" rta n t 10 r<><:a li l hat. for dyna m i.. sign als Iha t
con f"rm 10 II.-ntl]" Ne,·ada .ta"da rds. tb.. (><Kih '''' I",a k of t ho> tirnl'ba... "",....
rorm ... prt"Sf'nts th.. muim um p' ... iti'-e ~a lu .. of I" .. n,.."a ut .."",nt 1,.1ra m..t..L
l .... posi ti~e J"'Il k "'p...",.."t. t .... ma~ i mu m (><"'ith... , I"cily for ,...Joci ly l ra ns·
dIKe ..... the max im um pmit i"" aect' le"'l ion r". "'-.;..I " m..I..... a nd Ih.. m""i·
mu m positi'... p r=. ...... f" r dyn a mic p " ,... I",,,,"-Iun" "
f or d ispl" c...-ne nl ,illn"l ... I.... (>< ith Jl""k "f th.. tim..ba p lO!: a1...a~"S ....p-
......nl. t .... ..In.... t app",,,..h " f lh haft to Ih.. tr" nsd """'L 1' I X ,i bralion. lh ..
point o n Ih.. s haft whic h ia o n th.. ,,,,t,id.. of th.. d ......... t<>d sha ft i. ..allnl t ho> high
spot. Th u. Ih.. !""-, it i"" J"'Il k in a IX-fill d isp l","''''''nt s igna l "1 " ...,... "t. t ho>
"" ~ of th e hillh s pot next to Ih.. di.pla "'..nt tr" nsd UL"'L s.... Chal'l .... 2.

-~~~~~~~
f;g_ 4- ~ . MN "" 1nJ PNI<- ,•
". ~ .m pl ~on. t ...
~ plot.~ _tiul ",.IM
,-
~. 2 I'll' (0.08 m.Il/d....
, ~ rAt. ~ ••
InCR'fll<f\I hc .... (<< lOti n....
r<'<I
n f'IO'gOI
d'''''''
.ognaI_ n... n ".., 01
ot tho: 1'0"-
p" . ks 01 t ....
do"""",
)
•
•
~n tho: 5n... ~""" "'"
"'.... (.oc_ ;, th<' PNk·lo-
PN~ _p1n.n. 01 t.... _ ...
IA Vol I.A •
""" ,",,""l

• • • ,oo
 Chapter 4 Timebase Plots 59

Perhaps the most basic measurement that can be made on a timebase plot
of vibration is the amplitude. This measurement can be made on either filtered
or unfiltered plots. To measure the peak-to-peak amplitude,

1. Draw horizontal lines that just touch the most positive and neg-
ative peaks of the signal.

2. Count the number of vertical divisions between the two lines

(peak-to-peak).

3. Note the vertical scale factor (units per division) on the plot.

4. Calculate the peak-to-peak amplitude using Equation 4-1.

pp amplitude = (number of div PP)( un,its

dlV
J (4-1)

The peak amplitude is one-half of the peak-to-peak amplitude.

For example. Figure 4-5 shows an unfiltered displacement timebase plot

that was captured during the shutdown of a 10 MW steam turbine generator set.
The Keyphasor marks show that approximately three full revolutions of data are
plotted. Red horizontal lines have been drawn that touch the maximum and
minimum of the signal. The vertical scale factor is 2 um/div
To make measurement of the peak-to-peak amplitude easier, a duplicate
scale has been placed at the right of the plot, aligned with the lower measure-
ment line.
Following the procedure above, there are a little over 13 divisions between
the two measurement lines. Applying Equation 4-1. the total change is

pp amplitude = (13 div PP)( 2dlV~m J= 26 /lm pp (1.0 mil pp)

Examination of the shape of the unfiltered waveform in the figure reveals

that the vibration is predominantly IX (one large cycle of vibration per
Keyphasor event). Also, a low level of some higher order, probably harmonic,
vibration is also present. Some of the noisy appearance of the waveform may be
60 ~ 1.. PloU

d u(' 10 d !"<'l rica l ..r m",·ha oi,:..1run..ut (!dnc h) in t he sha ft. wh ich i. roo f<' , 'i. ib!('
b<'cau... of l h~ f<'lal iwly I, ,,,, I o f IX •• h.at ion that i. p ...,...nl.
T h.. " '"Yp ha w . ,j"l o c..n he '«'d 10 mea. u.e I h" rotor ,pN'd. fl (( ;..... k "J'P<'T
ca ... .. m~ga ). of lhe mad lln",

2. lkt... m i.... Ihe .-Ia p.....l tim... ~t. (el..]I.. tl he l.."", n t he dou.

3. C.. k ula t.. t .... roto . ......-d in rpm f. o m th.. f, ~Io ..ing fo.m ula.

(It,pm'·l '"' 1''' ' ' .0'11'''')

~I (m.) • mm
(4· 2)

Fo. " xa m ple , in Figur" 4 ·6. fl.'d ....rt ical lio.'" ha, lw<on el m..-n throu gh adja·
ce nt 1('>H ,ha>o' dots..-\ m"a,u ... men l ",a1~ ha n p1a•..-d he lm.- Ih.. tin.... to
h.. 111""'1.1 '"
lp th.. "'al",.,.] time be t ""n K'-'YPha ",-..ot.. ..-hio.,h ,,,,"""""1.o n..
"",~,hn ion o f th" "'a fl. r .... tim fo r on(' ...... ~"t i .." i. apl'"", in' a t.-l)' 34 m..
..\ p plyinll: Equa tio n 4· 2,

!} = [ 34' ~ms I[ ,"""• m'I[ m""in 1 " _

I , flO rp'"

_<:'!'d~~~~ _

Fill"'_ 4--6, "'",,,,,oog rpn'I GO _

'''''oN«' plot. Th< n"", ~
>UC(....... ~ .......... ~
_IS t"" t_ lot ~ 'P>OIvlion d
""' ......Il Th.. = proul or It... ~
_ numbo< or ~ utIOn> _ unO

Ii......t""
rotor ~ or the
mad..".. «<,<; ,"" ,e., lot del. ..),

• •
•
-"
 1 h... frequellCY o f a fill...,.l"d ,i b. at io" ~ill"al can I><' rn..asurl"d 0 " a l im<.>ba....
plo t- f o rn..as" re frcq u... " cy.

L 1 >J~pla~' a fill.."", I' m<.>ba.... p101 ....tJ ich ~~ al l..a.1 0 "" full
"1'cl... oh-ibra tio ". f or wry low frcqu""c i..... I hi~ ma~' " 'q u ire ... ~ .
.. ralre,"O l uh on~ ,,-ort h o f d ata. (,'In u nfill.."", plot ca n I><' used if
Ih.. . i~nal i. dom ina ll"d b~ 0 "" freq u.. oc~. )

2. Draw \l'"rt ical li n~ d llou gh two ""lui, -al..nt po int. 0 " Ih.. ~ignal
I ha t a re 0 "" eyel.. of ,ibratiOfl apart. For ua mple, u<.e , .. ro
...,."""i ng~ o. pea k..

3. lIt·t..rmin.. l h.. l'lap>o<'d tim.., " tJ ich i~ tbe pe riod . T. of th.. sign al.
(I f a nd o"J~ if l hi. i. IX ,ibral, o n, it " i ll I><' l he sa me a~ Ih.. time
1><''''....." K"n , ha;or dot .. )

~. Ca lcula t.. the freq ue ncy.I. o f , -ib rat ion w;ing tli... fn ll"" 'mg cqua-
lion. Th,~ eq ua tio " " "-" " " ,,,,, tha t lhe ",:riod ha , !>reo rn..a.u o:d
io m illiM'CO",k

(4-3)

l l -ll.... «! --,

Hg "", . ·7 "",""",,"9 _ ~

~ d a ~""Illd ~g""t l.<x"~

I'M> pol",," on tho ... -'orm that
/\
.. ~ ~ ~ .".01 6" <hi< GO"'.
t h o _ p".bi ~~....
~ m...., ~""" i< tho pm-
oct r. cf ~ ""'...... Tho f.~y J
" tho "",;poe,", 01m~ P""Od. v \.

• •
f--.
• •
I .... '.....
• .
,
• -• •
62 Data Plots

For example, in Figure 4-7, red vertical lines have been drawn through suc-
cess ive minima of the signal. A measurement scale has been placed below the
lines to help measure the period of one cycle of vibration. This time is approxi-
mately 34 m s. Applying Equation 4-3, we can calculate the frequency,f:

f = (1 CYcle)(1000 mS)(6~ s) = 1760 cpm

34 ms s min

The amplitude and absolute phase of a vibration vector can be measured

from a filtered timebase plot. The peak-to-peak amplitude is found using the
method discussed above. The absolute phase is defined as the phase lag from the
Keyphasor event to the first positive peak of the filtered vibration waveform.

1. Draw vertical lines through a Keyphasor dot and the first posi-
tive peak after the Keyphasor dot.

2. Determine the elapsed time, b..t, between these two lines. The
elapsed time is always less than the time for one complete cycle
of vibration.

3. Determine the period, T, of one cycle of vibration, using the

method described above.

4. Calculate the absolute phase , tP, of the signal using Equation 4-

4.

p= b..t (ms) [360 degj (4-4)

T (ms/ cycle) cycle

For example, in Figure 4-8, to find the peak-to-peak amplit ude, draw two
horizontal lines at the positive and negative peak of the signal. The distance
between the two lines is a little over 10 divisions. Use Equation 4-1 to find the
peak-to-peak amplitude, A:

A = (10 div PP)(

2
dlV
~m) = 20 f-lm pp (0.79 mil pp)
 Ch apt~' 4 n "",b...~ PlolS 63

Th~ p"'a k a mplit ude i. one- half of the peak·lo -p"'a k a mp litude: 10 jilll (0.39
mil) pk. ll<>cauM' Ihi. filt~ ..... . ignal i. a .ine \Va' .... the
t ime. lh e p"'ak a mplit ude. o r 7.0 jilll (o.U mil) rm• .
.m
.
a mplitud e i. 0.707

To mea....", t h~ab solule pha.... d raw w rt ical lin..,; th rough a Keyphaw , dot
and th~ tirst pmili'... peak of lhe .illnal_ The elapwd lim ':>1. i. 12.5 m... a nd I ....
pnlod. T. wh ic h i. t ~ ..ame a. in FiflUR' +-7. is Jot m• . t.: Equat ion +-4 to de ter-
mine a bsolute pha....

• • [ 12.5 m. )[ 360 d"l!)= I30'

. e ",""
Jot m . l ....d . Ie

Th u... t .... I X " b,a tion '"ffto~ r . i.

r = 20 lIm pp L I30' (0.79 mil ppL IlOl

Il<>cau ... Ihi. is a 1X-fill....e d 5ignal. eac h .ignaJ p"'a k ,e p. o:wn" t .... P<',.... !t ~
of l ....ro tor high ' pot next 10 the p.........

-,
" _ _ed

fiol!y'* .... t.lN W r'I"lg trw peak- o 10 lO

"
_
pho...
akan'9ituc1e and absolute
oIafi_~ .The
",,- j-j..
""""",e ph.... IS _ as tt>e I
riaP"'<! 11m<' from a ilrl'pha>or
r<ent to tl>e fim _ peakci
,

-
tt>e '''''......It i> nat«l trac-
tion ci. lui <)CIe.~ <ed in

• .
'
64 Data Plots

The relative frequency, in orders of running speed, is the ratio of the vibra-
tion frequency to the rotative speed. When a filtered timebase plot contains
Keyphasor marks, the frequency of the filtered vibration signal can be compared
to rotor speed:

1. Find the frequency,j, of the filtered vibration signal.

2. Find the rotor speed in the same units (Hz or cpm).

3. Divide the frequency of the vibration signal by the rotor speed.

4. Express the result in the form nX, where

n = fsignal (4-5)
frotor

The n will be a number that represents the relative frequency in orders of run-
ning speed.
For example, in Figure 4-8, the frequency of vibration is equal to rotor speed;
thus, n = 1, and the relative frequency is IX. If there were two complete cycles
of vibration per revolution of the shaft, the relative frequency would be 2X. Sub-
and supersynchronous frequency ratios are possible, such as V2X, 0,43X, %X,
%X, or 1.6X.
A useful visual analysis is to examine the progression of Keyphasor marks
across an unfiltered timebase plot. If the relative frequency is a sub- or super-
harmonic of running speed (V3X, V2X, 2X, 3X, etc.), then the Keyphasor dots will
always be in the same relative place on the waveform, from one Keyphasor dot
to the next. If the Keyphasor dots gradually shift position on the waveform, then
the vibration frequency is a more complex ratio, such as ?iX, SAX, %X, %X, or a
decimal fraction such as 0,47X or O.36X.
Figure 4-9 compares two unfiltered timebase plots, each with eight revolu-
tions of data. In the top plot, the waveform is dominated by V2X vibration (there
are exactly two Keyphasor dots for each cycle of vibration). Note that the
Keyphasor dots do not change position with time; every other Keyphasor dot
occurs at the same relative place in the waveform. This fixed pattern indicates
that the vibration frequency is a simple lin or nil ratio relative to running
speed, where n is an integer.
In the bottom plot, the relative vibration frequency is not a sub- or super-
harmonic of running speed; it is slightly less than V2X, close to 0,48X. For this
 ChaptPf 4 TImO'ba... Plot s 6S

"a , """'ry olh ..r K..~-phaso. ",,'..n t occur s a t a slip,lly d'tT..... n l pia...., in th ..
..'a, for m; th.. K"l' p hasor don plo t al d iff nl '....I i.-a1 pmil io n8. Th i' ' i 8Ual
....ha,; or u; cll'8.r ind i.-al ion thall he ... Ial i, ' ibra tion frrqu,.,,<')' '8 nol a " mpl..
int .-g..r relat ionsh ip to runntnll ' p<'<'d.
It i' 1"'. ...1>10> 10 ..... 1>:0' ins p<'Ction th ai Ih.. ' i bra tion f""' nt"}' i8 .Jillhlly I""",
Iha n 'hX. Fi•• t, pic k a K..~-phasor dol a s " ,ta rti nll ... r......I)( "'<-xl. mo>... to th..
r'gh t to on.. com" I..I.. c~"CI .. o f ' i b ral io n (Ih.. red lin.. in t he fi(tu ...). I n mo' i n!! 10
Ih.. righl e pa.. two Kory pha sor do t.. Th .. c~"CI .. of ' l b ra tion i. romple'" at th..
red ci rcl T ht>... f...... lh..... i, le.. t han on.. c~"CI .. or.i bra llOn for 1 0 .....u lu tion .
of th e sh" fl. fo r " ra tio of I""" tha n 1:2 ( Ie.. Iha n I'>X). •-\no th ..r y 10 d..te . min ..
the ralio i' 10 nOl .. Ih"t the pt'riod of,i b ral ,o n is longer than Ih.. po>riod for "'u
, haft ...m lu tio o.. Ih.....fo... the fr.-qu.....-y of t he 'ibralio n ill II"SS than o,.;X.

Fi9 y ,. 4-9 F1r1otiw- ~ 000 """ho,·

"""'ic vibr~lon.~ ynli~~ "rT~'"
UOlO) hoi. on<' ~Io of .. bra'lon for two
~ "",n.. (t>oo:> ~"'lons of thfo
"""'-I. _ , ... ~ o _ ""n.. do M1
~ p<anoon WIth..,.;,......... C)'<I<'\

lt ~ "Iock<'d")."""'.-b.. ,n.. '-.....

I'<'Q ""Y of "'" vibr;noon is ..""""" Inx
~ mon.. w ;N IflNIn kx:k<'d wi'>«'>-
....... tht" vibrat Oon is 0 "'~ Of SOJPt"fho'·
_ of ........."9 ..,...-.c1 ""'h itS 1/)1(.
l n l(. 11(.210 1(..-«. ~ "'.I..... . thfo baI-
v~:..--
;;;;;;.~ __ ~_~~ _
10m ......... INn on<' q<1o of """0!I0n for
two K.oyp/\O\Or morks._ tht" vibro'oon
It~ is .... thon lnX ro,~ X). Tht"

~ "'""" """ P""tionIt"", on<'

q<1o"'thfo ' .Tht"pan.-rnwill ...............
II''opNt d thfo, _ Ir.-q....-ncy is"" ""...
9'" ,..." '...:h itS 21311. 3141(. 01311. .. ,<.
66 Data Plots

X and Y timebase plots can be used to determine the direction ofprecession

of a rotor shaft. Determination of the direction of precession is an application of
relative phase (see Chapter 2). The plots must be constructed from data from
two, coplanar, orthogonal displacement probes. By measuring the relative phase
of the two waveforms, the direction of precession can be determined.
In Figure 4-10 a rotor shaft is observed by an XY pair of displacement
probes. By Bently Nevada convention, the Y timebase plot is displayed above the
X timebase plot. Use the positive peak of the X signal as a reference, and find the
corresponding peak on the Y signal that is less than 180° out of phase. The rela-
tive phase shows that X leads Y by 90°. Thus, the rotor first passes the X probe
and then passes the Y probe, showing that the precession of the shaft is X to Y.
Relative phase measurements can also be made between pairs of transduc-
ers in different axial locations, as long as the transducers have the same angular
orientation. One application of this is to estimate the mode shape of the rotor by
examining the timebase plots from several axially spaced transducers. The rela-
tive phase information in the plots can help establish a picture of how the rotor
is deflecting along it s length, including the approximate location of nodal po ints
(Figure 4-11). This can provide useful information for balancing or for trou-
bleshooting other machinery problems, such as coupling misalignment. See
Chapter 12 for more information on mode identification.

Summary
The timebase plot is a rectangular plot of a vibration signal from a single
transducer. Elapsed time is shown on the horizontal axis, with zero at the left
edge of the plot. The vertical axis shows the instantaneous value of the meas-
ured parameter in engineering units (urn , mil, rnm/s, g. etc .).
Timebase plots can present filtered or unfiltered vibration data. Filtered
timebase plots are synthesized from vibration vectors using a mathematical sine
function with the appropriate phase lag. Unfiltered timebase plots represent the
digitally sampled waveform from the transducer.
Keyphasor events are indicated on the plot by a blank/dot sequence. Th e
Keyphasor event, which occurs once per shaft revolution. is a timing event and
is observed by a separate transducer.
Filtered timebase plots can be compensated with synthesized. filtered wave-
forms created from vibration vectors. Unfiltered timebase plots can be compen-
sated with unfiltered waveforms (usually a slow roll waveform). or with a syn-
thesized waveform from a vibration vector. If the vibration vector is measured at
the same speed as the uncompensated vibration signal, then the resulting sub-
traction produces a Not-nX waveform. where nX represents the filtering fre-
quency relative to running speed.
 (l>apt e. 4 Timebiw Plots 67

I
j

Fi9<- ... 10 ~on d 1>"'<"'_ from Iff b ~ pIol<. Ai rn. """ ""...... n MIl
PO.. clooetoIMX~ ~ ~P""e< 1M r~. 1n "'" ~~piOt\,the ~
PHks aI m. ....fWl. _~ ~nt ltle p.t>SO~e aI the '0101' ~ 9' W>OI neI,..;,ltle
pt<lOeo.lIIow <hOI XINd. y by 9O".n..t InN'" tho, the rotOl' l< l>"'<... s.ft;I ;n on )( to Y
"'........ the .. me <'teUton .. rot.toon;t!'oJ\ <he prec..won i. lorwo'"

F ;g~ .. "'11,The ~onon 01...._

plIO'" from ~ ;n .....1y "'PO,.'ed
pIoneo. 1X-H!ered ~ rnebooe ~ from_·
~ol ~ nell the lIN'"'9' oI>ow _
ltle ~_ pho'" d~" by obout 220". The
_ _ pial on m. ,; ~ ht is ~ted on
ltle 1ft! pial for ~re ""e The rNtlve
phose _ oteo _ the tOIOIl<oppro.d--
mole!,-out d ~ .. OP\>O'~ en<:b d
the rTIO(~"". ~ rig'" body "'"~ is """""'
_ <:Ielle<:t>on ","pes ore poo.. bIe,_
more ~ ore. - to conIi.... the
<hofl defIec~ ........
68 Data Plots

Timebase signals should first be viewed without any compensation. When

necessary, compensation should be used with caution and should never be auto-
matically applied to a signal.
Single timebase plots with Keyphasor marks can be used to measure the
amplitude of unfiltered vibration; the rotor speed; the frequency, amplitude, and
absolute phase of filtered vibration; and the relative frequency of filtered vibra-
tion versus rotor speed, in orders of running speed. The shape of an unfiltered
timebase signal can provide important clues to the behavior of machinery.
Timebase plots from XY probe pairs can be used to measure the direction of pre-
cession of the rotor, and timebase plots from probes at different axial locations
can be compared to determine the mode shape of the rotor.
 69

Chapter 5

The Orbit

IN THE CHAPTER ON TIME BASE PLOTS, WE SAW how rotor dynamic motion
along the measurement axis of a single transducer can be displayed as a time
varying waveform. While the timebase plot can provide important and useful
information, it is inherently limited to one dimension of rotor motion. Since, in
any lateral plane along the rotor, the rotor moves in a two-dimensional path, or
orbit, this one-dimensional picture provided by a single transducer is not ade-
quate.
To measure this motion, a second transducer must be installed perpendicu-
lar to, and coplanar with, the first transducer. Only then will there be enough
information to observe the complete motion of the rotor in that plane. This
information could be presented on two, one-dimensional timebase plots, but it
would be even better if we could display the two-dimensional dynamic motion
of the rotor. That is the purpose of the orbit plot.
The orbit represents the path of the shaft centerline relative to a pair of
orthogonal eddy current transducers. These transducers are usually mounted
rigidly on the machine casing near a bearing; thus, the orbit typically represents
the path of the shaft centerline relative to the bearing clearance of the machine.
(Orbits can be constructed from casing vibration data, but this has limited
application. This chapter will present orbits from shaft relative (displacement)
probes.) Because of its ease of interpretation and extensive information content,
the orbit, with Keyphasor mark, is probably the most powerful single plot format
available to the machinery diagnostician.
In our discussion of the orbit, we will start with a description of how two
timebase waveforms are used to create the orbit. We will discuss the informa-
tion obtained when the Keyphasor mark is included on an orbit, and slow roll
70 Data Plots

and Not-IX compensation of orbits. Finall y, we will dis cu ss the various kinds of
information that can be obtained from an orbit.

The Construction of the Orbit

The orbit combines the timebase waveform data from two , perpendicular,
coplanar transducers to create a single plot showing the two-dimensional
dynamic motion of the shaft centerline (Figure 5-1). The orbit in the figure is
unfiltered, but is predominantly IX. The data comes from XY transducers, which
observe the motion of th e rotor. These transducers are mounted at 0° and 90° R
(relative to the reference direction, "Up:' also known as the orientation angle ref-
erence), but any transducer orientation is possible as long as the transducers are
perpendicular to each other. The signals from the transducers can be displayed
as two, independent timebase plots or can be combined to produce the orbit.
The Orbit plot format is a square, with identical horizontal and vertical
scales and scale factors. A point on the orbit is defined by a pair of X and Y val-
ues , which are obtained from the timebase waveform data. A set of values from
the sampled waveforms can create anything from a portion of the orbit to sev-
eral orbits. The center of th e orbit plot is defined by the average values of the X
and Y timebase waveforms. A Keyphasor signal acts like a strobe: the dot shows
the location of the shaft cen terline when the Keyphasor event occurs. To com-
plete the plot, a reference direction (for example, "Up" or "West"), the probe
locations, and the direction of rotation are included on the plot.
Note that the direction of rotation cannot be determined from the orbit
without additional information. The best way to determine the direction of rota-
tion is to examine the machine. The direction of rotation may be visible to the
eye, or it may be marked on the machine with an arrow. Another way is to use
slow roll orbits (with caution!), which are almost always forward; thus, knowl-
edge that the machine is at slow roll allows us to determine the direction of rota-
tion by observing the direction of precession. The direction of precession can be
determined by the blank/dot sequence on the plot. Be aware that crossed probe
wiring or an active rotor malfunction that causes reverse precession at slow roll
can render this method invalid.
Positive voltage or position changes on the timebase plot correspond to a
motion component toward the transducer. On the orbit, this positive change
always corresponds to motion toward a transducer, along its measurement axis,
regardless of the transducer orientation. Since the orbit is much smaller than
the transducer, the measurement is always made parallel to the transducer
measurement axis.
In Figure 5-1, points 1 through 5 show the progression of the shaft center-
line around its orbit. Point 1 shows the location of the shaft centerline
(crosshaiT'l) ..il..n th .. " "YPh.owr ......n t occtl N; Ih.at i", wh..n t.... I....tling nl!l" of
Ih.. K"~l'ha""r not ch pa"''''''' ....I<t to t he K..~l'h.a,or I, rollt> (~ huwn h.. ~ 1W Ik
rot or ).
l'oi nls 2 and 4 ma rk th.. farth....t a M dO>e<.l al' proach to l hoe X p robe Uk
m inim um a M ma ximu m pea k. o n th.. X tim..lM... plot ). Simila rly. po ints 3 anti
5 ma rk Ih.. fa rt hest a nd dosest app roach 10 Ih.. r p robe (lh.. min im um a nd >nax-
im um pea k>. on Ih.. n imeba«> p lOlj.
O ft..n. ' ral 9 ...1.... of ,~hrati ..n a.... plot ted on th.. omI t. In tb .. rlp.u .... Iwo
c~ I ar h in th.. timel",,,,,, r IOl" ..·hich n...a l1$ tha t th it al"" hu I"....
9 1 a M. in t his ca.... . n·... ..-. it... it..-l f during Ik ..-conti q -d .
T positi,... pea k of the tim..n.. ... r 1uI al,,·a~.,. .-..p n t. t h.. d~
approa ch of t hoe m a lt to the a ..." ",ialnl Ir a nsd" ,,,,r. For IX ,; b",t iun. t h.. poml
on th.. . ha lt " h id! i., o n ,hoe out~idE" o f ' .... d ..n..ctnl , haft i, caUnl th.. h Igh spot.
T.... P"",ti,... pea k in a IX-fil' e red d i.'pla""m..nt <ignal .-..p,,,,,,,nh II><' pa ' ''''!t'-'' of

-- ..
- . -
~v / , I~ . ,
,
~-

•
- \
,
,
,i

.... ,
••
,~
._.
Fogu"" 5-1 eon""xM" oIa" o<bt. XY ',,,,,sduc...,; oI>I.<'rw "'"' ",bra'lOlI 01 a roo:or >hoIt-A notch
a ddlfl.... _
.. "'"' "'"" (01 ~ouho") is do<E"CU'<l b\l a Keypha..,.. t'''''sduc....n.. vb """"
"""sduc..- sig...... prod",," two t""O'bo~ plot. ImO:I<Iloo) ...-.ch < ~ in,oan ""'" plot l"'.llu!_
At PG';~on 1, """" _ ~ prrm. do<O'<" ,.... oorch. _ y,aff =~ .... i> Ioca'<'<l .. tho-
fIO"IlOlI 01_ ""vphasor do.. Po.IIIOl1' 2 "" OUlJ'l 5 ""*_ <"""lotion 01_ orbif paoition
an4 _ _ plot ......... Two rru<>k/bon!; 01data ilI@"""""'inmet_... pIoto.butcfty
""" u n l;w _n on _ orbot bo<:........ ;o [hi> co...."'"' y,aft ~ ,......_ pall\.
72 Data Plots

the high spot next to the displacement transducer, and this concept extends to
the orbit. For a IX-filtered, circular orbit, the orbit represents the path of the
rotor high spot, as well as that of the shaft centerline.
Transducer mounting orientations are usually measured relative to the ref-
erence direction for the machine. For a horizontal machine, the reference direc-
tion is usually "Up:' For vertical machines, the reference direction can be any
convenient reference, for example, "North:' Bently Nevada orbits are always
plotted with the reference direction at the top of the plot. The actual transduc-
er locations are indicated on the edge of the plot. This provides a uniform visu-
al reference along the machine train, regardless of transducer mounting orien-
tation. The orbit on the plot is oriented as an observer would see it when posi-
tioned with his head in the reference direction, looking along the axis of the
machine in the viewpoint direction. (By Bently Nevada convention, this view-
point is usually from the driver toward the driven machine.)
Figure 5-2 shows two examples of an orbit display with different transducer
orientations. In both cases, the rotor orbit is the same, only the transducer
mounting orientations are different. The orbit plots show the same orbit orien-
tation relative to the "Up" reference. Note that the probe labels on the orbit plots
show the actual probe mounting orientations.
At the bottom of the figure, equivalent oscilloscope orbits are shown.
Because the XY axes of the oscilloscope on the right do not match the actual
transducer locations, the oscilloscope must be physically rotated 45° CCW to
display the orbit with the correct orientation. In this orientation, the horizontal
and vertical oscilloscope axes are aligned with the actual transducer orienta-
tions. It is important to remember that, when viewing orbits on an oscilloscope,
the XY axes of the oscilloscope must match the actual transducer mounting ori-
entations, or the displayed orbit will be rotated and will not appear as it does on
the machine.
Filtered orbit plots, like filtered timebase plots, are not constructed directly
from waveform data. A filtered timebase waveform plot is constructed (synthe-
sized) from a filtered vibration vector; a filtered orbit is constructed from the
synthesized waveforms from a pair of vibration vectors. See Appendix 2 for
details of this process.

The Keyphasor Mark

The blank/dot sequence on the orbit is called a Keyphasor mark. The mark
represents a timing event, the Keyphasor event, that occurs once per shaft revo-
lution. The timing signal comes from a separate (Keyphasor) transducer that is
mounted at a different axial location. The timing signal is combined with the
orbit so that the timing of the Keyphasor event can be seen clearly. The
 (hapl.... 5 Tn. Orl>it 71

"' "'
,~ ~ ~
C3 •
(9
"' "'
/ ":] /' "

~i .:»
."'~ -~

0..;; _

fi_ So2. _ <;>ri@n''''''''' ¥<l_D<brtplaLOn _''''' - . ~ ~ _

X p<obM . ... mou",~ '" IT _ 9fT Jl ...\oPl'<'","". ~ _1>1oO: . nd O'd~
me""" do>ploy """'" ~ "'''''' __ On I"" '9" t I"" prob<os .. ~ fI"IOI.O'l~
a' 4S· l . nd ' S' Jl ~ D<brt 1>1oO: " . """""'"ally """'~ 50 th.>lrt it. i>'OP-
orty ";gno<l with tt,e'Up. ~ ~ di<c<:';Dn. Thr =ilk>!.oope.~.
muU ~ phY"{.o'Iy rotat~ 4S· CCW to di<ploy ~ 'omo<;f orbit """,,,";on.
NoI ~ ,hat tt.. K'"Yflh..... m,,, h on I"" D<brt plot. >!>ow ,"" plot ,.,.,....,.
lIOn, _ ~ tIM' dot marl<> _ ~ ,"" bonom d........ Ii'ow ""' ........"
at tt.. bo<;J'nning of _ blan k....... powblr- o>dllosc:""" d ....oy
74 Data Plots

Keyphasor mark on the orbit shows the location of the shaft centerline at the
instant when the once -per-turn mark on the shaft passes next to the Keyphasor
probe. On circular, IX-filtered orbits, the Keyphasor dot marks the location of
the rotor high spot at the instant of the Keyphasor event. The blank/dot
sequence shows the direction of increasing time.
Figure 5-3 shows an orbiting rotor (the size of the orbit is greatly exaggerat-
ed) that is viewed by two orthogonal transducers. As the rotor rotates, the shaft
centerline also moves (precesses) along a path which defines the orbit. A
Keyphasor probe is installed to detect a once-per-turn mark on the shaft, in this
case a notch. When the leading edge of the notch passes next to the Keyphasor
probe (position 3 in the figure), the shaft centerline is located at the Keyphasor
dot on the orbit. The Keyphasor signal is like a strobe that briefly illuminates the
shaft as it travels in its orbit. Even though, in the figure, the rotor is on the oppo-
site side of the orbit from the Keyphasor probe, the notch on the shaft is in a
position to be sensed by the Keyphasor probe.
On all Bently Nevada plots, the Keyphasor event is shown as a blank/dot
sequence (as shown in Figure 5-2 in the orbit plots), and the dot represents the
instant that the Keyphasor event occurs. The Keyphasor mark on an oscillo-
scope, however, may be a blank/bright or bright/blank sequence depending on
the type of shaft event and the type of oscilloscope used. Figure 5-2 (bottom)
shows a blank/bright sequence; the beginning of the blank marks the event.
In timebase plots, the time between two Keyphasor marks represents one
revolution of the shaft. In orbit plots, the rotor moves along the path between
two Keyphasor marks during one revolution of the shaft. This path may be quite
complicated. A Keyphasor mark will be plotted every time the rotor completes
one revolution. If several revolutions of data are plotted on an orbit (Figure 5-4),
several Keyphasor marks should be visible. However, in nX-filtered orbits, where
n is an integer, successive Keyphasor marks will plot on top of each other. In the
figure, the orbit on the right is an unfiltered orbit, but consists primarily of IX
vibration. The Keyphasor dots plot almost on top of each other.
The Keyphasor mark on an orbit plot adds important information. It can be
used to determine the instantaneous direction of motion of the rotor and to
estimate the absolute phase, the vibration frequency in orders of rotative speed,
and, with multiple orbit plots, the mode shape of the rotor.

Compensation of Orbits
As we discussed in the chapter on timebase plots, the primary objective of
compensation is to remove unwanted signal content (noise) that is unrelated to
the machine behavior that we want to observe. Like timebase plots, both filtered
and unfiltered orbits can be compensated.
~ , ;y ~ , ;y ~ , ;y ~ • ;y
0 @ /~,.
....... @
•• ••
/
- ,.
I •

fiogunr Sol ~ """""' of ~ >tWI. ..""tid .. 1)(. <"""...

O<bn. Th< ~ "" ~ ox""
'"' "",n .", ,""'ft·
l . .....torn t hr ~"" pr~ ""'..,., the notell.At that ""'... ~ , .... "'""
hnr i>kx..red ... _ PO'fI"", 01_ ~ do< in "'" o<llil pk)\

/" -C/--. ~

• •
0
, I " I II

t'
f Ogy", So. Two unIi~...... o<bot< rrom .. .300 """ < ~"". [och orbot plot _
....,..,,«1
""", "",,,.,,""" cI d.>la. Th< Yprol>< .. """-"""<l .. l r L ard ,!-or X J>ftlb!' .. a!
611" R n- dir«tion' . ... ndic.'''' bI'the ........ bIock~. on t l>o ~ Th< I<'h pIo1
- . mu~~ I(.oyph,nor dots be<..,.. 01 . "'bsynclKoro>u' froquen<y comp on.,.... _
to. ftuod-indU< M ;nu.~, Th< togM 0'bI """'" P"'dom""",tIy IX to.hoviOt at .. W'l-,r
""'"" thell"tabMty .. _ Of ""'1 ........ ~ tl-or du """"'.l of i<eyph. "" <loB ,mo"""
group. ondicatng donvnant I X briI.Mor.
76 1>.>1.. PIau

FiJt.. red orb it p iOl' en IJ<' ,.]0'" roll co m ren""tl'd o~ing IX. 2X. or n X , II,. .
roll ,'-"CI0". Wh il.. COml't'n""t ion ..r a timeba... plot ""lui. .... 0""" .11....· ,,,II ' T<:-
to r. comrem,a tion of a n o rbil pl..t rt'<Ju i,..,. a pair of slo..· roll .'-"CI0~ Ea ch Ira o, -
d o...... ha s a slow roll '''''to r l ha l " subtr....tl'd from its original ,ib,alion ,..,..tor;
th .. res ulti ng pa i. o f comreo""ted 'ib mlion Vl'Clors is u"-'<l to ~~nl h""i"" I.... fiI·
lered orbil . T he e nd ....... 11 is a filtcrnl orbit plolt ha t i' slow roll <:uml'en", tl'd.
figu... :;''' .1><,...,. plo ls ..f a n uncomren""ll'd llefl ) a nd <:umren""t.,.] ( ri!(hl)
lX -fiItl' rl'd '''bil. The s l",,' roll ' '-''CI 0 ... a ... ...Ial ively la.g.. a nd in p ha... \\il h t he
vih ral'on. '" t ha t. aft er comren,;al ion, Ihe orbi t is s;gn if,u olly s mall,·r. AI.... t he
ch ang.. in I"",iti..o o r lite I\e~-phaso. do t show~ Ihal til.. p ha",' o r t he orbit has
~ignlf'C8 nlly cha n!(.-d, movi ng l he ob. ,,'t'd high S"..I 1,,,,,,1io o ah.." t 4 ,," cau n·
I..,d o c k...,.." Th is is a s ignifica nt d iff n.... IM I is i"'l",ft a ol for ..m c;"nt a nd
acrural.. bala nc in!!_
A'lOth.., IYJ"-' of comren",l io n. .." .·..fo.m co n'!",o",t ino , n o IJ<' app lil'd 10
Ih.. un filt..rnl orbit. Each un fill..rl'd tim<>ha.., ..-a...·r..rm " .....J to consl ruct a n

K_ <J.5.I mI pp a11"
' _ ll5l ml pp L HI. ·

~~ ': -~
/ /
• o •
I
G.l-.t.. ' 2Oll'P"I

flg u.., S-S SlowrolVKI",~oI. 'X_fi~"""Otbn. Tl>e

_ . . , «>Mff\K1od from t"r ~ 01-.;"'.."'" wst<", lhown_
~ pIot~ Slow ro/I wsw, 01 X _ U mil pp £ 3]. ' . "" • • U mllpp

£ 23' " . ", ,..t>Uoctod from t .... origi..... VKI"" to plod""e ~-.""
u><'d "' ~ compenWod plOl Noll' the ;;gnh... ", """I... "'" 01 the
0Ibrr ..... ~ ctwrqe .. pt>\it.... 01 the (eyptw>or dot !"I """" thon 4 ~·
<""n!_kw~ from l"r Uf>C~edpoYhOn_ ThO; <0fT<KWn p<o-
due"".n _ Wt d"pIoy; only the dyno mic re-;pon", of ~ , )'S,em
and i\ ~ ''''porI'n! for <ffiOent ..... KOJ'ale bo.......Oog.
orb it C'Ons~.ts o f a ""l""nu' o f digilall~ ... ml>ll.'d .·al For "am tra n>d"ct'•• a
~"iLahl" tJO'o'o' ro ll ....... 'd .. nni~ ....Ifflro fro m th" sl roll ' I"-".'d . a" ll". Each o f
tl><> :rJo...· ro ll "''''Ill" .·31..... ca n I><' ~"b [rac lro from its corres pondin!! ,-al.... in
tl><> o riginal a...,forms (Ih" Ke~-phasor .."..., t i. US<'d a~ a ......'-..fo rm l iming ",f·
" r""""j. Th is thod has Ihe advantag" o f "''''0\1nlt ",...1. jf nol a ll. o f [h" . Iow
roll co"'p.. ....nt ofl h" . i!!"aL
\\ -......for", corn,"'''''''li.. " ...i ll ....m"'...all co"'po nents ...i l h f""l" "nc io><. "p I..
th" "~-q" "'t ... mp ling f,"-/",, "'1' limil ( ~ th" "''''I'ling . al" ). Thus. I X. 2X. _ nX
(n a n i" t"!!". ). &",,1 all ,..J1",yn,·h n, nou, a nd s U''''''Y''mr",,, ,,,, fwq ....nci... et<>
th.. "' ~'1 ui " limit ). ,,-jil l><' ... "",,-...1 from Ih" ",hi t. 111;.. ind " d.... " '0 ' [ of th.. s;lI-
n,,1" rt,f",1" d .... ,,, . haft "u. f...,.. d ..f...1s.
Fig" ." .'H> .h,,,,.,. hn"" " ·aV('f,, nn romJ""n",li" n ....... k$. At th.. Il'ft. tw..
"""""'!"''''''''...
_roI_.
I Ii.....""... p in's from a . 1....'" lu rb i.... a," comhi nffi I" I'rod Ol'"

,
"'\...-
-
~
J I j
,- - - -, - , .~

-- -~

~J ~/
" ,.~

-~
.-

Fig..... S-6 Slaw roI -.n ~ d an orbit from .. "'......, turbo ..... ...' Irtt. two
uno:wnpoon~"'
form. _ wb<'0<~
foo"" ""'
kom ,~eng
til con""'" "'" un<"",poon_
'0 prodox.
w .............
Ofbj{ 1....., _
'~comp<'fMl..:l·~" t tew,,,, and
roI "'_

o1Jt .. rogIIl.........,h ..'" much <I. ..,...

..0 u ncom pt'n;a ted ..0.1 , ,,the r noi.~,. o rbiL To coml",tL",te th e ..rbit. oJaw ro ll
...".....-..rm ... r<> . uN racted poi nt· b~·· point from th e or igina l u ""'.ml""n ted
..." , .....-. .rtTlS _ l he rt'.ull IS a pa ir o f compt'nsa ted ...a'.....-o rm . a t the righL hieh
prod ...... t he compt'o""ted o rb it a t botto m rigllL The " >apt' o f the ..rbit i. mlK·h
. m•• " he r, IeM no isy. ..0.1 more ind ic.. ti, ... o f the d~·na mic "" llOf1"" of Ih.. rot o r.
l 'n filt.-red o rbil. ca n a1 .., b.- not ch fdt....-...:l 1»' coml",lUI-al i"ll hot h o f t he
..rigioa l wa,"<'form. .."ith a s~' nth ....ized. filte rwl ...,.,'e fo nn. r... ch COfT1pt'n",l io n
wawform is wcon. l ruet ed from a n X-f,lte red "il>"'t;o n >...ctn r tha t is ....mp led at
the me li me a. l he o rip nal ..-awf" rm . .....'" !lampled. T he Il."CQn,;tructed. nX
wa, form8 a ", th en . ub tract ed fro m the o rigina l vihrat ~ ." w..,...form.. a nd ,....
two ",sllli inl/ ..·a'.....-o rm. are u<ed lo .." ,," ruet the Nut-o X orbit.

, 00., ",,,,,,''''«1 , ""' - , ~

'----0' ~:::i
,
os'o"'"'
- -- I
aSIO"""

PJ\ [;\j\ L .
I ~
. ..0 .....
I
.s'O""" .~ ' O """

~)J
I
/
\ D

Fogu... S-1 Not-1X <Om p<'O,""",,, of on_ ", ~ l"'O u ~ o dw~ rns .... used '0
C""" nlCI ~ u~ o d orbt. 1X.........o.m\ ~ lM from .o.ouon vena!> ., """'.....
'P"'la'e """"'acl«l fr<>m the o<9n o" to ptoducett>e Not -I ~
rns and orbn ., ri ght_
Thew w~ • ...-.d orbn CCW\Wn .. ~ ""'"" ""'op1 11l T1Y=nanrog _ _ i> primonly
lf2~ fr<>m 0 nb
 Chapter 5 The Orbit 79

For example, using this technique, you can examine an orbit with all of the
IX vibration removed (Figure 5-7). A Not-IX orbit is created by subtracting two,
synthesized IX waveforms from the original unfiltered waveforms. The result-
ant orbit reveals any frequency information that may have been obscured by the
IX response. This can be helpful for identifying vibration characteristics associ-
ated with a variety of malfunctions. In the figure, note that the 1;2X vibration,
which is the dominant remaining component, is clearly visible.
The warning about compensation mentioned in the last chapter is worth
repeating here: compensation is an art as well as a science. There are many vari-
ables that can change the compensation vectors or waveforms. It is possible, by
using incorrect compensation, to produce orbits that convey the wrong impres-
sion of machine behavior. Initially, it is always best to view data without any
compensation. Then, when it is used, compensation should always be done with
caution.

Information Contained in the Orbit

An orbit plot, especially with Keyphasor marks, is a powerful diagnostic
tool. It can be used to measure the amplitude of filtered or unfiltered vibration
in any radial direction; the relative frequency of filtered vibration versus rotor
speed; the relative frequency of X versus Y unfiltered vibration; and the direction
of precession. Using the orbit, the absolute phase of filtered vibration can be
estimated. Additionally, the shape of an unfiltered or filtered orbit can provide
important clues to the behavior of machinery, highlight significant changes in
response that one-dimensional timebase plots cannot, and help identify where
a problem may be occurring in relationship to the components of the machine.
Recall that timebase signals are plotted about a mean value and contain no
de information. These plots contain only dynamic (ac) information. Because the
orbit is constructed from timebase plots, it also has no de content. The orbit
only displays the shaft motion relative to the average position; there is no infor-
mation in the orbit about the average position of the rotor. To obtain the aver-
age shaft position, use the average shaft centerline plot (Chapter 6).
Multiple orbit plots can be created from the same location at different rotor
speeds to show evolution of rotor vibration over speed; or, they can be created
from different axial locations at the same speed to show the mode shape of the
rotor.
Like the timebase plot, the orbit has many of the characteristics of a basic
oscilloscope display. Both oscilloscope orbit displays and computer-based orbit
plots can be used to make a number of measurements. The following discussion
applies primarily to orbit plots, but it can be extended to oscilloscope orbit dis-
plays.
80 Data Plots

While many software packages have the ability to process and display this
machinery vibration information, a machinery diagnostician should be able to
analyze the information without using a computer, for two reasons. First, an
oscilloscope, with an orbit display, may be the only instrument available.
Second, it is always good to verify the numerical results that come out of a com-
puter. A quick glance at the orbit plot can provide a useful sanity check on any
other numerical results.

One of the most basic measurements that can be made on an orbit plot is
the peak-to-peak amplitude of vibration. This measurement can be done on
either filtered or unfiltered plots. To measure the peak-to-peak amplitude
(Figure 5-8),

1. Select a transducer. In general, the amplitude will be different

for each transducer. In the figure, the X transducer is chosen.

2. Draw a line (the measurement axis) from the transducer loca-

tion (the X mark on the perimeter of the plot) through the cen-
ter of the plot. The line must extend well beyond the limit of the
orbit itself.

3. Construct two lines (3 and 4) that are perpendicular to the

measurement axis and tangent to the orbit at the maximum
and minimum peaks of vibration with respect to the transducer
(the red circles).

4. Measure the distance between the two tangent lines in a direc-

tion parallel to the measurement axis (a plot scale has been
added for convenience). Calculate the peak-to-peak amplitude
using Equation 5-1:

pp amplitude = (number of div PP)( u~itS)

dlV
(5-1)

The peak amplitude is one-half of the peak-to-peak amplitude.

In the figure, there are a just under 12 divisions between the two measure-
ment lines. Applying Equation 5-1, the peak-to-peak amplitude of the orbit as
viewed by the X probe is
 ChaPI... 5 ThO' Ofbil 81

ppa mpJitud.> = {12 .Ii'"PPJ[ ' '""1=

.I ...
"0 ~m pp {2.4 m il p pj

T""'n> a n> two key POints to ...me",I..., "npn u' ing this tl"Ch niqu... Fi..t. I"'"
f'Pl'k-to-pt"a k m..as u ...ment mU51 "'" ",ad.. parull"/ 10 the mM5ureme'" axis at
Ihe probe. M..a.u ring s, mpl~ , n icall~ o' h..ri7..n ta lly. in th is ClI5<'. would 1" 0-
du ct' a d iff....·nl a nd incorrt"Ct «u.
Second . t h.. pt"ak· to- p.... 1. "'pa."u....men t i' made be!».....n the lO.ngt'fll.• to the
orbit l hal a... al", .,...-p..nd ic"b r to the meas ure m..nt ax is of the pml....
Il<'tnembe, t ha I t he o rb it i" "mall e<.>mpa r<'< 1to Ih.. Ira nsd oce ' s ize. ".. Ihal thp
mea", m..nt axi.. i" alwa~. in thp .'lamp d iwction. 1t " i ll help to r.."'..ml..... t hat.
o n Ihe a le o f th.. p lO(. Ihe fa"e ..f a n ...Jdy c ur....... t t ransducer ....ould bt' a h.." .
I m (3 ft ) in dia me le ~

f;qu.., s-.a."'....u"'m.... of peal< 1O.pt' ''''pi;'

!Ud<'onon orb;t, Herfo. thO' Xt,"""'uc"' ......
men( d,own ""J<'Iher Mth perp<!lldlCUl.l'
...... ,... '" ~ , to thO' """"mum and ",;"i-
"""" pom" on ~ orbir f>ooith .....,.." 10 me x
tJ,n\dU<",).A plot ""'"' .. d'...... p".01"" '0"""
ffi8\U",menl ,>i,
lor C<lr''''''''''lCe.

•
82 Data Plots

An orbit can be used to determine the direction ofprecession of the rotor.

The Keyphasor mark on the orbit (the blank/dot sequence on Bently Nevada
orbit plots), shows the direction of increasing time, which is the direction that
the shaft is moving (the direction of precession). On oscilloscopes, however, the
sequence can be a blank/bright or bright/blank, depending on the type of oscil-
loscope and whether a notch or projection is used as the Keyphasor mark on the
shaft. When using an oscilloscope, it is always a good idea to verify the configu-
ration of the event that is being used in your equipment by examining a time-
base display.
Once the Keyphasor mark sequence is determined, it will show the direction
of increasing time. This is the direction that the rotor moves in the orbit, regard-
less of the direction of rotation. The direction of precession can be compared to
the direction of rotation to determine whether the precession is forward (pre-
cession direction same as rotation direction) or reverse (precession direction
opposite rotation direction).
On complex orbits, the rotor may undergo forward precession over part of
the orbit and reverse precession over another part of the same orbit. In the Y2X
orbits in Figure 5-9, note how inside loops maintain forward precession (left),
while outside loops show reverse precession (right).

The filtered orbit can be used to estimate the absolute phase of the two
component signals. Phase estimation will be most accurate for circular orbits,
and less accurate for elliptical orbits (Figure 5-10). This is because, in a circular
orbit, the shaft moves along the orbit with constant angular velocity (equal time
intervals and equal angles between dots). In elliptical orbits, the orbital angular
velocity is not constant (equal time intervals and unequal angles between dots).
Because phase is a timing measurement, this angular velocity variation causes
inaccuracies when trying to estimate the phase with respect to each transducer.
On IX orbits, it also causes the high spot on the shaft to oscillate as the rotor
traverses an orbit. This effect, when combined with bending, actually produces
2X stress cycling in a shaft with a IX, elliptical response.
This phase estimation technique is best used to quickly confirm the validi-
ty of phase data from some other source. Usually, a visual inspection of the orbit
with a Keyphasor mark will provide a good cross-check of the data. Foraccurate
measurement of both absolute and relative phase, the timebase plot is a better
choice.
fC'9 ~ " s·, DitKtoon el ptK.,,,,,,, in _ In tho orb<! pOol,tho "'"" - . ~om tho
blonk _ ' d ,.... doL Tho orb! plot on !tle Iofl illu<l..,,,,, tho, .... in...oo loop i< .oM-.tr;
~ 1"'0<"" ""'. A' INo toP ell!>. right <>rbil, , .... !.Nfl - . in & '<)Ur>, etdo<;io:M..
()( to Y) 6rKhOn. which is !tle w_ ... tho d,,,,,,tion cI "".bon. Thu~ tho
bw...d-A, ,.... bol1om eI tho right o>J1>iI. "" path form, . n ~ looQ,and .... wit
P""''''''''
is

- . lot &....,... .......

in ~ pre<;fl",,"-

'/•
•
•
•
• • • •
• • • •
• • • • •• • •

f ~ 5-10 Shott 0<MaI . '""J'-l.. _ rry in torcular and ",ipto:&l OIbn Tho orbn dol>
""""" tho t""""'" in PO'ition eI tho """ ,~...- on "l",,1 '""'" ~ .. o..;ng
..-do ~ _ in''''''.l tho WIt '''''''''' 18".1n • arrLto. _ ,on-
tho >!toft ,,,,,,,,..;,,,, two
st...., 0tb01&I .ngular """""'v and - . lItrou<1' """" 18" i ~ " in oqu&t ,,_

d"""""" "'pC"'.
"""""'''-In orbif>, !tle wit ,,,,,,o<Ii,,,, tw, _ "'" o<bn&I .... u....
po ", of tho 0tbiI, 'n 1hi< ..... mp... tho >!tal< orbIal . rog<Ja, """""'v _
_ iii"" in
from
5"10 61" pot in,....... Thi< ''''''-''';''''.1 • ..."...• • _rry mak.. ~ diffi<u/I ' 0 ..,.... ,. ptw ..
from hogNy",iptot. oo-bm,.
84 Data Plots

To estimate the absolute phase of a signal (Figure 5-11),

1. Be sure that the orbit is filtered to a harmonic of running speed

(IX, 2X, 3X, etc.). An unfiltered signal can be used to estimate
the phase if the orbit is dominated by a single frequency. Select
the desired transducer. In the figure, the Y transducer is select-
ed.

2. Locate the Keyphasor mark on the orbit.

3. Determine the direction of precession. This will be the direction

that the rotor is moving in the orbit and is indicated by the
blank/dot sequence (red arrow).

4. Absolute phase is the fraction of the vibration cycle, in degrees,

from the Keyphasor event to the first positive peak of the signal
with respect to the selected transducer. On the orbit, this will be
from the Keyphasor dot, in the direction of precession, to the
point on the orbit that is closest to the transducer (the red cir-
cle in the figure).

5. Draw lines from the Keyphasor dot to the center of the orbit and
from the closest approach point to the center of the orbit. The
angle between these lines is the estimate of the absolute phase.

The relative phase of the two signals can also be estimated. The relative
phase is the fraction of the vibration cycle between the point of closest approach
to one probe and the closest approach to the other probe. For a circular, IX orbit,
this will be 90 For an elliptical orbit, this number can range from nearly 0 to
0
•
0

0
nearly 180 depending on the orbit ellipticity and orientation (Figure 5-12).
,

While the orbit can provide an estimate of the relative phase, the timebase plot
is a better choice for accurate measurement.
 Clwpt. ' S 11M Omi! as

S·l l [".... ,,"9 -..,~ pi'wI".. "'lh '""Il<'<1 to

Fiog~ te
~ ytt....l<I"".. on • lX-~~~ """'. ~ . nglo
~ tI>r I\owh''''' dol .nd ,"-~" " , , _h
to tI>r proM ~od <",,10) ~ .n ~""""'~ lor " ... abYlIUI~
""'".. ell ~ Y t,....l<I"".. "'JI"III.~the m~Hu«<l
anglo .. abOut 6So. 8<o<.ouw lh. ""'" ;s "';pI"'l It-.
fQfOr ot>oIt wiI boo- moving oIong thi, PO" of It-. path
~ quickly < _ F'9~'" S-lOl . nd -...iI """""' tI>r
"""9'" in _ '""' ~. n...,. lhoI .bl.olute ~
.... boo .... . !hMl M~ ~ is OC1WIIy;about «1'.

~- . --

fiog ~ , . S·1l. [".... "ng ,olot ..... ~ .. _

_ T"'" lX-fiR<ft<l hlgl>ly
I
.11111'0001 """', • .., ""'"'" ~ """" PN'< of lhoI X ,;gr.., .._
_h.
willi • g_n. ~ h.... . nd tI>r pas; PNk ell tn. y,;gnol ;, ,hown
bkH!. """""""' ~"". Tho _ .. ph.~;n ttHs """'- .. tn. pornon
of lhoIvit>r."on <yeIfo bol...,.." ""'....... ell tt... X . nd y • .,....~ ....."""""
in rr.. do-oroon of p""'''''''''' (blank to 001, ;ncrN1oOng ''''''').In ,"- ~
plot. tn. .....,..... ph. ".. ;s ab<lut'm.1n rr.. right plot. the ' .."'..... p........
is ab<lut N.8ou...... ell l""ing is. .... _ IIhighly ~lpbUII o'M~ ~
w.. pIo{l. • .., • bon.. <001 b detennmalion 01 .....;,.. ~
86 Data Plots

The unfiltered orbit can be used to determine the reLativefrequency ofvibra-

tion versus running speed. This technique can be used for both supersynchro-
nous or subsynchronous vibration and uses the Keyphasor marks that are gen-
erated once per turn of the rotor.
To determine the relative frequency versus running speed:

1. Plot an unfiltered orbit, and select a reference transducer.

2. Start at a Keyphasor dot on the orbit. Trace a path around the

orbit, counting the number of complete cycles of vibration rela-
tive to the reference transducer, until you return to the original
Keyphasor dot. This may require several revolutions worth of
data.

3. Count the number of shaft revolutions that occur during the

cycles of vibration.

4. Express the result in the form nX, where n is a fraction and is

defined by

number of cycles of vibration

n = - - - - - - - - = ' - - - - -- - - (5-2)
number of shaft revolutions

5. Reduce the order of the fraction to the lowest common denom-

inator. For example, %X can be reduced to V2X.

Figure 5-13 shows several examples of orbits with different frequency ratios.
There are several items of note:

1. For integer ratios, the number of Keyphasor dots in the orbit is

always equal to the denominator of the frequency ratio. Both
the 1/3X and %X orbits have three Keyphasor dots. Similarly, YiX
and %X would have four.

2. Frequency ratios with one in the numerator (1I3X, V2X, IX) have
orbits with clearly defined blank regions before the Keyphasor
dots.
 3. . ~ , al" ...t h an ' n l....... «fwr t ha n ...... in I numfta-
tor <:lin poorly dotfi.....d or ot--u lO'd h1anl For
O'umplt. in I 2X ...... t. IhnO' two cornploou ~ of ,; !>ra-
h on M... ..ach In"Olution of u"" ft. Thu ... lto..r.- WIiI boo ......
.....vn.- nwrl< for"""Y 1_ ~ of ,-;hralion. lf thO' orbit has
... « ...... ~ .......J'O'W1'I&. it wiD ~ Ilf' I bla nk of 1M
"""vn.- mark dun "fl lh.. ..-coOO. ~n.. .of "'b Thu .. at
fir"'~. a 1X ] .'1: om.l ..ill look ' '0'1)' ..m doAf 10 a IX orbrt;
...........'01; thO' h;p, orb,u " ill pIIint ,,,,", !hO' h1anka of thO'
" ",-rolla.. ... m.,b "" an MriIIoocope. ~ ......."flWa~ ploni"fl
J'K~ ha.... a ~pt..,ial p.....-i>.ion 10 . ,,,id Ihi..

t::oi nl! l h~ ru in. it i.o p'"",ihle. " , Ih a 11111.. 1" . <1 '...•• I.. d.. l.. rmin.. ,i n' I,I..
f.....' k'ncy , ali" o " r ",h,. I,,, n wil h a quic k ob""n al io" "t .o n " rb il.

~ S.lJ "1bf_1r~1 _10 I\ralln9 0P0'0'<l ~ .......... 0'Il """"..... - . ....

. - _ dIlorenI ~ ,.,.,. ,tIIIl>e 10 IUrvOng """"'" The paths 01 the 2/lX.nd lX

_m.. . .
0Ibt' .... """"" lJO;Jh<I1- I O ~ the "",,, ond ~o"", dol _ .The .......b<ot
oI . b _ (j'de> yrId< the .............,. or ..... '~""n<)' ,.no. rhe .......b<ot or ~ dol'
rhe~.,.
O••• Plou

We ha"' d i....."..... I hi~ ht-ha>i " r fo. ,ib,al ion a t ~, mple inl'1!..r ral i"" Wn a t
if the ' i b ml io n fuoq" ,."cy i~ oJigh llY I""" or mon> than an int " jle, ral io? F.och
o rbil in Fi!(l'w :i- J4 ~how1< ....·..raJ ...... ol..lI.. n. o f data. The m iddl.. orb'l ,",,0""
' i bral ion thai i. ..u ct ly \!o X. l he o, n it ..n the l<.'ft ~ho .....i b, al ion th ai i. oJijlhtly
bo>low %X. and th e o rb il OIl l he righl ~hgbll y abo.... %X.
\ \1I<-n tbe frcq oc ncy is a n .. u <1 int ..ge' , a tio. .....,h K,,:,·pha.'l' " d ot pion in
the ..... m.. pla.... ...... .ry ' i bml io ll <)'<"k-: lhe K,·y ph a...... dnt~ a , e .'laid to .... "loc ked."
\ \1I<-n th e fuoq""""y i sligb lly a w or bt'l,,,,", .. .un' I'''' in t.-ge ' , al io. the
"-'1' ba,;o, dot. will m", lo..iy a rou nd t orhil. l .... di rt.'ction Iba tl b.-y m",...
..ill d"l)(>nd Oil ..... l.'1he. Ih.. " !>ratio n fuoqlt<'ncy i. a b..w o r bel..... t be f.act ional
' a tio. Ilow fas t they m",'" d.."..nd.on ho... fa ' l be fwq u.. ncy i.'\ rrom I h.. intege'
' a tio.
Wh en l he fn.q..ffiCY i....hgb l ly Ix-k a ., mpl.. in t'"!l'" . alio (le ft o rbill l h..
o,hital mo tion of th.. ,otor is ............ "" t rotor romp l..t"" I..... " f t.... \ib,a tio n
C)"(..lt> fW' .... '·ol..I'o n. T Kt>ypha",. dot OIT...... a lilt l.. ..a rli..r in tli.. <,ycl .. and
af'".-a... 10 m",,, " o...ly a lIld th .. o rbil in a d i,eclio n OJ'J"w;it.. I" tli.. d irect i" n
of ro lo , I'",, :~it ln.
Wh .. n ..... f.'"'Ju"""y i~ <.ligh tly a bow a .i mpl.. inl'1!'" mt io h igbl o rbil ). I....
orbilal mOl i"n o f . .... rotor i•• Ii¢ll~· faq.. ~ .., th.. ro to r rom" I..'''' mow or a
cyd .. " f ' i loral io n l"" reyol..t io n. Th.... th .. Kt>yphawr dol occ a lilt l.. Lat.., in
the c,-cl .. a nd al'p"an. 10 mo.....Iowly a round th .. orbit in Ih m.. d i",,-tio n a'"
th.. d irectio.. ..fpr......... i.....

'"'
. '9.... 5· 1. Kewh. .... doI bohiMof in unli _ _ n.. dR<nonof
"""'lOtI in all ~ _ . it Y te x (cIc<k,,"...~ .nd tho ......r..n \hew
suurwwo ~ <let>. ",_..n Ctbot tho _~ tr.q......cy of
voooon it oIightly ..... t t>an 112X. .,nd Ndl ' '''''''''~ ~ tiel
pIcIs in po<iIicn Ic<atO'd shr1'1Iy "'-"""" "'" dr=non of I'OtOlicrI
n.. _ _ "'-' ...ctty 112Xvi><aticn; tho Xrfpha>a dot> on
Iocl«od. Tho "'iI'" O<bn .......... b<aticn thai it oIightIy g_''''
th.tn 112X.
. nd >uca-,;_ ~ dot> _ in 1M ....... ,j;'KI;cn., tM d,~·
ben of rototicn.Tho Krwha"" de< " acn"'l h," a "'obo>.
 8«au... of II............. Mty 01 Ih•• ¥iouaI rikd. tlw oriMI can boo oupt't'_ 10 , '''''
"f'"'drum pIoc for dHrnn'ninfl"-tw\hoot ribnl ..... i' lIl> ~ 1II1f'Ct."f ratio 01. ru n-
..,.. "f'"'d. SpKtrum pIot.o ~M ontlrn:oftl limit '" h" ich if. oJrtn·
IN...... by Uw "J'UI and .... m"'" (I{ oprctnd bnrs. t
lrum _ h • opan of suo Hz wI1l ~ • .....,jution. or ~ bon ..'idIIL of
-..pliP .j(l()-Jrn,."f'l'C-

51 . ,/", ,,, ,, I.E Hz '" 15 cpm.lf. m.od\inoo i<O t.utn"'ll.ol)tA1rpm. .... X \'ibr.otion
' - . ~ 01. )f,,-, cpm..nd tl..t<JX . 'il>fation ha•• ~ oI.l:M '1"'"
nor... £""'fLOl'llcin dJffn ","' .... I~ Jt> cpm. k-u thin Uw opt"ctI,oI -.lul><>n. Thu\,
-' lit.. rnoIution, boll> ll'f'<f~ would pok>t in thr opr<1.raI lin<', Ind "
.......Jd ""'fui..... mudl l> i~ ......" .. l><m "f""'Clrum 10 d 'm ;.... I" bt'!..,...." lit......
two fT""J...·ncin.
\...., !hi. frequen cy .JIlI......' nm br ....ry imporllnl. l 'rldn th .. r'!th' '~>l1di ·
' i.. n•. ru b c.n p.odu.... ",. 'Ily X " bral ion, Un lit.. nttw. h.and. Ou,d-indu <'td
i...,.bili,y ' yp ic..lly.. "" u.. " a ",bs~'nchronou5 f , m1 . lill l.. 1...1..", lei X . ...1
i. u nlikdv . 10 <><:cU• •., ",,,.d lv I;,X. I ""'.. rmi ni n;l ' hi minlllv . """II d iff.. nm n'
in f ncy ca n boo wry iml'.. n an llO propPrly d i"ll"'''''''ll ' """-'h i"" m. lfu.....
tiu r <XtHI dd ck-. rly ........ lodfod K<'YJ!h.a dUI,' ~ '" 1",.... "," X vibra' IOn
. nd MJoptuoM doh rn<Mnll ~inst I.... diJ l"Cti of ' .......-.i"n f..r \, bralioa . 1
olo«!>ln'...... I.",n 1iI.'(,

n... unr.........d orbil t ... .... u....t todH......' .... t ,-,w, jt. I : ""::> r#vtbra·
_ .. Uw X dDTd ..... """,,pufd to ' M frft(.......-y of 1 in ,.... ) ' d ilft'tion
CF....... 5'- I.i~

I. PIot. n unfiltnN ......t.

2. SoIart . 1 • ~-phuor mark Ml(( '"""'" .." ... nd I .... mbtt. Sot", """',
"",m' lX"'ili>'C' i .... ""'!I'll''''} !'<'u...... """'...nl......d ..'ith ......pt"C1
10 0.,.. of ,he Ir. nodlKTNo-

F'II"" S·lS F~""""" '01"'" from_ .. !uri 0I lh<-

~ doI_ 1OIrld lh<- _ .Count Ihr
......- '" _ pr OCC .. in 'No If "'KIO '
_ on .... '-.:ion In "'" ~ "'" If "...-.c..r
.... fWO _ PN"> lone • . . . - _ 1hr0lh0<\.
_ the , , , _ . . - _<JI'IIO, ..... _ PH"- Tlw
H"-Y.a<>o ii l:-I
 3. R.-p<"lII Ihi. p, OCl'du ....wilh ' .... ..-ct I" Ihe ..Ih", l ra n.d" n ',. The
.... ali..... frNJuency can I><> s talt'd • •

Ix ' / r ( :'>-3)

o r Ih.. ""..., ....

In Ih.. f,gu ..... Iwo po.. il i..... p"'a ks ar...."""unt .. re<! in Ih.. X d i tion. a nd o n..
I" " iti..... f'<'ak is ..n.:o"",..red in II><> )"d i. <'Ction. Th " s th.. X: r f ncy rat io io.
2:1. a nd Ih.. compo n..", . a ....2X a oo IX. (Th...... i" at.... I X rom potK'nt in Ih.. X
di r<'Ct io.,,,. 'h . ........ Ied by Ih.. di ff..", ,,1 hor ilOn iai position s o f th.. two pea ks. ) As
"....n i" t his ..u n,p' '' . Ih.. s hal'" of .n " n filt..,t'd o rbit ca n p rmi d... im po , ta nt
infor mal io n aboul th.. f""l" ency co"'..nt o f machin...ry ' ib'al ion.

•~ can ab.., p n Md.. cl" ... to t he p""""n"" of .."""""i..... , adiat loads that
ma l-... i"l. High rad ial lood. l..nd to p ush Ih.. .....o r I.. h ~ (.'CCt'ntricily ral ios in
a l1u " l-film hyd rody na mi.. l...." ing. Th .. "il film tha i ""ppo tt. I.... ......., I....od .....
''''' y h igh "J' ri"lt "tilTo.-.,; a l h i!d' ,""u-ntri<-ity ' a l ~ >S_ Th is "li fT"".. is high 1 i"
Ih.. rad ia l di , <'Ct'..n a od 1" ..,.1 io I.... tao!!"nli,,1 d i 1.",n. Fig u~ $- Ib . h " h'"
u nfill...-t'd omit" fn)fll I d ilT......" t "'ooam tu rbin with h igh .adial l..ad .. ~..I"
Iha t th .. I..... mad " " .." lu m in opposil.. di ......tion. (black . n ...... ). Both " th,ts
di.play .-..m.a'ka bly "imila ' . hal'<"" in fact th ..y a ", " ..a t ly idt>m, cal t>.." COf-
...........1f'" " "'a t i" n .1i"",1.;" ". Th .. an', ;oo ical.. Ih.. lOCal,O" o f Ih rinl! Wllli.
a nd th.. n od a rrow. intlical e th.. likely d i t;"" of I.... appl it-d rad ia l l" atl.
So fa ' ...... ha.... . h.....soc.(... mca .......m nt. thai can ht- p<'rf..rrnt'tl OIl . iogl..

fiogu <e ~ - 1 6 . Uf<'CI of ,ad..1

load '"' orbit ,t..opo_Orbt<.,.. ,---- . ~ . ~-~-- .
I
_from (lM)d~

" ...'" t\J<t:»nM - " ~

rotat"", d"... --.. lblad<
/
"""""l_lloth rnochrw. ....
~""""'IIhigh 'ad'" _
1M ,od _ ondrea '.. "'"
opprt»lirna,.. dUction of til<'
'ppIi<'d 'odiolload- n... .."
","'......... l~ ptCIb.IbI<' " ' _
..bOIl of _ bNn"'.l ...oll
orbot... ~'n. "",up ..m.h c_ ~~ i tho!- P""""'" of tho!- orbtt plot.
Multiplor .>rl>rt.s a1n br ......oIN "''Of >pr'O'd. "''''''lOft.
........ a c+t.ongi~ m..... ,1W pn>n"K ~ Tlw rloanfl" !hlII appnr in muh...
""'Of c~ ....

pW orbnA hrip iIlumi nat.. ~ -..prrts .. ..-hi.... bO'Iw\io<.

Mult.,vorlll:rs ........ op«d ..., ctt!'aIN lL.... . dol. from . .... IItO'tiUJftl'Iot
plano- in tn.. ...-iJ,IW. Ourinfla oUrt up or ohutdmo-n. at loppIoptial:.. AJ"'fll '"
t ,nw inlona • data .am"'" to t.knl tNt is IIW'd 10 CI'O'atO' orbc.- ~ ,..m
J*>tA can br combirwd 10 """'" ...... rotor bt-l\.,.'ior (JOn op<"O"d.
MultlpW orbi t. of th" lund can br ...-j to idn11 '~' • 1>10111...... TnOfIa~.
Fi~r .. .... 17 .00.... .. .... in of I X·fill... td. C'Ofllp<"" . . IN ..ro,h h-k..., duri"ll l....
ohut down of . 75 MW It....,n t um , . Ik' mnnb...-t""l lh .. " ..,-ph.,;o, dol .., .
I.... pmilion of t.... rot or h ilth "f. nt ..t..." t K~-phu<" nl ' l<"C1l.... A. loc
........hin.. pa. .... t hrotlll:h t ""'n..... th al 'o". h ip t>rt n l oc lm bala ......
loca tio" (th.. hrol'Y Of"" ) and I roto, = pon... U.... hiP. .".Il] "ha " !«,,, TI><'
hu ,)' . pot loca tion d"... n.>I " han ll" (it i. fIX"" in Ih.. . ha ll al i'" to I....
" .,-ph , no och). h ur.. dur' lIfla .hutduwn. Ih.. hiP' opollll p Ih "'"Y spot I......
a nd as ~ m"",hin.. i"U"'<'" throu¢' I I><' ......""'....... r .... orbil ..00.... Ih.....
~ F"tl'It. u . po-rd <I<uft.- . thO' high 'P'"loca liun (which ito ~· Ih ..
kypIw..... dot) lyp ICai ly mo".... amund 1M orbIl in Ih.. .am.. dllft1lO11 rota·
tIOn. At thO' .a..... II...... thO' om.l oiu incrN- . "ao.-h..o a maximum u Ill<'
mat:~ P*-""" Ih"'"'!(h ~ 1'f'IOI\aDCf', and <I<uft...... ~in .......... I.... rno-
........... ~ . oo !hlII ~ OfWfltatlon of dl<' orbit ..u'J- cha.1I(lf"I as tho- o.h.oft
I!iD"' ~ tho- ~. ...... to thO' aniIOIropi<" oI,If.... il .....t iff....." IS 110'I
thO' _ ill aD radi.al du,ect _ ~CbapI ... Il l al thO' Ay-o.t" nL It..- tff<'C1I ....
a ~ altho!- I'OI-Or I~~ "'" dynamic ~ to u nlloolan.-.. (_ l hap<:... I I ).

§/' , ..
• • o-~- .

""""S. l 7 ~ """" ...... 'llftdft<lm I ~ MW _

"" IIlht _ ~ft<Im tl~~<lIu""""""~ ~ ..... _
t......... _
--
....., . . _ ... pIot ·

---
_ ...........-PIOW>!I-..;I\I~~ .. _1120 _ _ ~
........ ".. "'"""""'" <l01I"" ".. ........ _ _ , of ..... I ~ ,., . - dioK -
92 Oau.PlolS

M,dl~ oroi/& 0 pm";"" ~ '" ~ ,t'd ..h..n I.... data i. ta l ..n from ...,,,,,,,1
m..as " """..nt p lan at Ih "'" Un, Bo>ca " ... th.. IInf,II...1<'d orb it 010",•.,. th..
path of th ft ,,'.. rlin m ulti l,l.. or bil. o f this k ioo !Ii'", lI S ~ Ih.........d im.."-
.ional pi<1 u of t m.. ' ~>n of Ih.. . otor alon!l th .. l..nlrth o f th moc h i"", t.ai o.
Th i. I""h " iqu "'. I>.--t ......." mac h i..... a... rig id ly rouplm: Il ibk- coupl i n~
,mil..... thc in llu..",... of mal·h i" ... 00 £'aCh 011>0>•.
r igu S. 18 • .......,. . """",, .. f 1X-IiII....-..d o rbil. from a 125 ;>O\ W M...a m lu rbi"...
g....... al or Th.. K"n ,ha , dot. ma. k t .... 10Ul ioll of Ih...ha ft in ..ach plan..
at , h.. inSla nt th.. K~l' ...."" 0<"<"''''.
"t Th..... d..l. ca n 1M> lillkt'd lo ob ta in an
....t ima l.. o f Ih .. lh ......·d im..n. ion al mod c. ..r d ..Ike' ion. ,-,hap" alon lllhc roto•.
:'001.. th ai. on a . impl..- orbi t pl..t . W<' c~n nOI d o t his for I.... ...""' in....' . ofl ....
o rbil . Sha ft mo tion a lo nll th... o rbit ' ,,-CUM- al d,ff......nt rat ... in dill ..... nt part&o f
Ih.. o rbil . Witholll add ilional tim i" l1 mark call llo l say for .....rta in ..-h...... th..
• haft i. a t a ny pa rt ic ula. t im... Th .. K"YP ,," n,a rk {li ,.... II. 11>0> t, mi ng info,-
ma lion for on.. particular point o n cach orbit. If ..... ran acco,.... Ih.. indi,;d"al
d igital ... m"l.. poillts on th .. orbit. t ....n " can co nn"'" , im lllta tlt'O'" point . on
d iff......nl o rbll ' and obtain mode .hap" p" inlm m alion rm tI,,,,,,,
point s.
~..,.. al on tha i th.. d.. Ilo>ct ion . ....p" o f t haft i. a" .-.t imal... Orh.... mor ..
compl ical m <haP"" a po.....' ibl... a nd it " ...,ld " i.... m...... pia""" ..f ",..a.ur..·
""'''I to .... . lI. .. o f t d..Il""I,o n <hap" at this . pt'I"l. Th i. is ........mca lly th ..

HMPr"' .....

.-C). , • -l..... .~
. , . -'- •
~~?
' '. " , II

0_ .1
-----
Fi!l"'e
------
S· la.MuK;pe ort>rt. ave< 1lO'fI.... Imm . ' 2 ~ ". "Non lurbine _310< >eI. A_ of

_..
l X ·~""'ed ort>rt. a' l ~ ....... ",ale """" ""ptu~ al thr """""nt '" lime. Fe< _
""" ~ don <I.....- thr Io<ation d I~ <hat! """- thr lI.<yphaoor ......-n 0«:..... The
~ don """ r.. ""~ \.... r """,! 10 obl .... a" fl' '''''''e of the - . . . " , .t\aQr of It-..
" Otbfl.
"hje<:ti.'e o f mod.. id..nh oca ti"" p. obes (..... Chal'l..r 12). II,"'........~ Ihi, infor·
mat ,on Can bt' ("Omh"..... " i th in fo.mal ,on from a /loud ro lm ~pl<'m mode l 10
,..,., fi,m the ro t" r .klkction >ha p"',
Kn_ ledllt' of the roto r dNl eel io n sha pe ca n be hell,ful for int..rpol..lillR the
rotm ,ib,al i" n ..t mids pa n points ",eU a",a~' from Ih.. mt>a........m..nt Iot'alio n...
Thi, in fo,mat it' n ca n help id" nl'fy PM, ibJ.. I""a tion.. ...tI high vibrat ion of Ihe
rol" , miflhl ,"Ontlict " "i lh limited d ....ran...... ln t.... mach in .
Mullipl.. rmnls 0..... cha nging Of"'raling conditio".' Can be .....a h.od "'il h d ala
from .. . inld" pla n.. IOl m ultipk pla n ).Chang.... in I" .... o. opt"mt inlt rond iti"".
"ft ,." prod uct' chanll.... in vib. a tion ha' ..." . Th.. infu rmal ion from mull'pk
" rbit« can hclp clarify t h.. ca lL of. mach ine p rob!<·m.
h !l-" "" 5- 19 , h",.'S a i of orb it. tha t " ....... obtai ned fro m . , inRl" pla n..
"n a It"" pipclin.. com p ""•. A. suct ion p'''''i Ut¥ " " . ~a.io'{l. . "ib mt io n beha."ior
c han ltf'd d ram a tica lly. from no rmal. m... lJ}· I X "'·ha';o. to n" id ' md ucc-d inota ·
b ililv. T1Ji5 in formation wao lLit'd 10 d i. gnu,... a nd d,-...-1op a ...ml'd)· for Ih.. p. ob-
' m.;o.1ultipJ.. o rbits aJ'(' importa nt I""",,,... th",:, of't."n up ot h..r d im..n, ion. , "och
a••peed . lim.., load , o. "" m.. " Ih" han!linfl paramct"~ T.... ad d it ional inf",-
mal ion ca n be ""'y h..lpful fu' ....1 1ing a nd diag no.inlt machi n..ry prohl"",,,.

,
Su<bor> _ ,
t I t
' U6 12cll ' 1-19

•
L: o . /01
-/~~ ~"",
' • .n ......

Fig_ S·19 Mu~OpIoP Otbots 0"'"' cl'>angong Opera ' ",!!c<>n6llon> Imm onr tINting 01a 9'"
~ ccm~ _ ..... mad""" ..... .. , . lOCI rp-n at a cons,..... t di!Ch"'l' "'''''............
cwbU . - th... nud--ind",o<l in,tabollt)' occ~ _ In. <uroon "'..."'... .....
""~
[)ata Plots

The o..bitlTimebase PIa l

The or h't / lnneb.a,;e 1'1 01 com bin.... l he o rbil with t he two ti"""lu... ploh Ihal
are u,.."j to c reate it a nd p.mide. th e ""me m o ,.... o f d isp lay ,-hat,K im.tics: fil -
te red. u nfilter ed. o. co mpen..ated. Timeba .... plol" are d i.'K·u'lSed in Cha pter 4.
RenUy Ne"ad a orbit / l imeb a... plot . a rt' CIl"IIled it h the u rb,t un the left
s ide nf the plot. The Yl imeb.a.se plot i. displ a!-m abo th ... X l imebast' 1'101 to
the . ighl of th.. nrhit (I'igu . e .i-Wj. The plot conta in. in format ion on l he d, rt'C"
tion u f .. ~ ..tinn, the pint ""ali ng used in Ihe orb,t li he ..am.. .. ~ th O' .....rt ical llCak
in Ih... l imeb.a", 1'1014 th.. ' !"""d- a nd tht> time ocaI.. on th .. timeba"" plot •.
111... fill" rt' i. ..n ... .1..,,1'1.- ..f ho'. ' In u... a n o rbil l time ba... plot to local... a ....r·
fa.... d.-f"d u n a .J>a ft. Th e plnl <h.-. s low . 011 dala from a boil..,. fl't"d pu mp
m.~0r.1 h is " rb it has a Ia'll" .. mn un l u f g1 ilc h. p roba bly due 10 ""IlK" .haft "" •.
fa..... . la mag.... Surfa,,,, d.-f<'Cts di sl ~a~"'" in orbit. I~l'icall)' . hO\<O· up a . 'P,kn l hat
point in l he !l"ne. a l di ra:linn o f the t .a "...du"". "- Th e l imeb.a... plols help cla n ·
I)' the l im i"ll ... latiOflships o f Ihe -,'pik... a nd ma ke it p"...... ible to det e nni n.. Ih..
a ngular local io n of th e .... rf d arna ll". TIw spike in th.. I' plot occ u rs abo ul
50 illS a lkr t he ",,,"phas,,. ,,' nl. TIt.- pt"riod of on .. n··..o lurion is the tim..
be t.......n s lJIT("...i,... "'''YPh;,,,,, ""n l-s a nd i. a boul ·IU I m. t aet u all~' 39'! m,,-

•
/
/ --
•
I , ,
O..l...olld'" Ill .....
•
fiqu.. S·lO An O<btI1......t>o", plot from ~ int><>lnl "".... "9 of a _ ~ pump
"""or.n", pint """'" """ ~""'"" ' _ roI data from probe'I mounted .of 90" t
and O".Tho ' rnri>o... clra<ly 'Ihow5 "'" ''''''''9 '"""" la' ge po<iti~ noM sP~.
"",""<15 """'" up on ~ sign'b from both ~ 'lO" 'pa"- -., t"ol.on "'e
r I.......
1>0", pIoI. "'" ""ke ,,,:CuI<; abco.lt 50 ms"'''' "'" Kq<pha5Qr~ thu>. ""'10<""", '"
"'" m,rl< on "'" wit <.... boo ,je,... m,ned. s..."'" tr., lor dMMls.
 Chapter 5 The Orbit 95

based on the speed of 153 rpm). So, once the Keyphasor event occurs, the shaft
turns approximately

50 ms )(360 / reV) = 45°

0

( 400 ms/rev

until the defect is under the Y probe. To find the damaged spot, we would align
the Keyphasor notch with the Keyphasor probe, and look 45° against rotation
from the Y probe.
Remember that positive peaks on the waveform represent rotor passage
next to the measuring probe and that the probe mounting locations are shown
on the orbit plot. The Keyphasor marks provide an important guide for estab-
lishing the location on the orbit or timebase plots. The Keyphasor mark repre-
sents the same instant in time on all the plots.
This combination of plots allows us to correlate events on the orbit with
their corresponding events on the timebase plots. The timebase plots act to pro-
vide a time scale that can help establish accurate timing of observed events on
the orbit.

Summary
The orbit shows the path of the shaft centerline. It combines the one-dimen-
sional timebase information from two, orthogonal, coplanar transducers into a
two-dimensional plot of the lateral motion of the rotor shaft. Orbits can be unfil-
tered or filtered.
A Keyphasor mark on an orbit shows the location of the shaft when the
once-per-turn Keyphasor mark on the shaft passes next to the Keyphasor trans-
ducer.
To remove slow roll runout from the orbit, unfiltered orbits can be waveform
compensated, and filtered orbits can be vector compensated. Orbits can also be
effectively notch filtered by compensating the unfiltered orbit with two synthe-
sized, filtered waveforms. The end result is a Not-nX orbit.
Orbits can be used to measure the peak-to-peak amplitude of filtered or
unfiltered vibration and the direction of precession of the rotor. Filtered orbits
can be used to estimate the nX amplitude and phase.
Orbits are useful for various kinds of frequency analysis. The relative fre-
quency content of X and Y signals can be determined. With the Keyphasor
mark, the orbit can be used to determine frequencies of vibration relative to run-
ning speed. The Keyphasor marks are superior to a spectrum plot for establish-
96 Data Plots

ing whether the frequency of vibration is an exact submultiple of running speed

or not.
Orbit shape can yield information on the direction and relative magnitude
of static radial loads. Also, different frequencies of vibration can produce char-
acteristic and recognizable shapes in orbits.
Multiple orbits can be created over speed, axial position, or change in load
or some other parameter. By adding another dimension, multiple orbits greatly
increase the information available for machinery diagnosis.
Orbit/timebase plots are created from the combination of an orbit with the
two XY timebase plots used to create it. The timebase plots are displayed to the
right of the orbit with the Y plot over the X plot. Orbit/timebase plots can be
used to establish timing relationships for features seen in the orbit.
 97

Chapter 6

Average Shaft Centerline Plots

THE ORBIT, TIM EBASE, AND ORBIT/TIMEBASE PLOTS PRESENT dynamic (rap-
idly changing) shaft position data, but they do not show changes in the average
shaft position, which is also an important characteristic of system response.
These changes are caused by changes in the static radial load or changes in the
stiffness characteristics of the rotor system. They routinely occur during start-
ups or shutdowns and during steady state operation of the rotor system, over
relatively short or long time spans. When a rotor system with fluid-film bearings
changes speed, there are changes in the stiffness characteristics of the bearing,
which cause a change in the average position of the shaft. Thus, changes in shaft
position can provide very important diagnostic information.
The average shaft centerline plot provides this information. This plot is
designed to show changes in the average position of the shaft; thus, the plot is
effectively low-pass filtered and does not display rapidly changing (dynamic)
data. However. when the information in the shaft centerline plot is combined
with other information, such as known clearances, orbit dynamic behavior, and
centerline plots from other bearings, we can obtain a more detailed picture of
the motion of the shaft, its relationship to available clearances, and the static
radial loads acting on the machine.
The average shaft centerline plot is most often used to display changes in
shaft position versus speed. but it is also used to display changes in shaft posi-
tion versus time, so the changes can be correlated to changing operating condi-
tions. Because some malfunctions (misalignment, rub , and fluid-induced insta-
bility, to name a few) can produce noticeable changes in the centerline behavior,
the shaft centerline plot is a very important tool for correlation with other plots
when performing diagnostics.
98 Data Plots

In this chapter. we will discuss the construction of the shaft centerline plot
and the addition of the bearing clearance circle. We will then define the position
angle and the attitude angle and show how to obtain them from the plot. Then.
we will show examples of how the shaft centerline typically behaves in machines
with internally pressurized (hydrodynamic). fluid-film bearings. Finally. we will
show how the combination of the shaft centerline plot. the bearing clearance
circle. and the orbit can produce a very powerful and detailed picture of the
dynamic response of the shaft.

The Construction of the Average ShaftCenterline Plot

In the Chapter 5. we saw how vibration signals from two orthogonal trans-
ducers are combined to produce an orbit. The orbit shows the dynamic motion
of the shaft centerline about an average position and is constructed from the ac
part of the vibration signals. The average shaft centerline plot shows changes in
the average position of the shaft centerline in two dimensions and is construct-
ed from the dc part of the vibration signals.
The de information is contained in the gap voltage of the transducers. When
the machine is operating. the voltage fluctuates continuously about an average
value. For Bently Nevada 5 mm and 8 mm transducers. this corresponds to a
physical gap in the range of 0.25 mm to 2.0 mm (10 mil to 90 mil). If the gap volt-
ages from two orthogonal transducers are low-pass filtered. the results represent
the average position of the shaft in the plane of the transducers.
This average position information from two. orthogonal, coplanar transduc-
ers is combined to produce a point on an XY (Cartesian) plot. The point repre-
sents the shaft. as if it were spinning without vibration. at a particular location
inside the bearing clearance. When a set of this data is collected versus time or
speed. the plot becomes an average shaft centerline plot.
An example of an average shaft centerline plot (Figure 6-1) shows the out-
board compressor bearing data from the shutdown of a steam turbine compres-
sor train. The plot has equal scaling in both horizontal and vertical directions
and is square in shape. The reference direction ("Up" in the figure) and the direc-
tion of rotation (curved black arrow) are shown to aid the viewer in orienting the
plot to the machine and interpreting the data. The data points (yellow) may be
labeled with the speed of the machine or the sample time. Note that. unlike the
orbit plot. the probe orientations (which in this example are 45° Land R from
vertical) are not shown on the plot.
If the dimensions of a nearby bearing or seal are known. then a clearance cir-
cle (green) can be added to the shaft centerline plot. This is a circle (or an ellipse)
that is the path traced out by the shaft centerline as the shaft walks around the
bearing or seal, and it represents the available clearance. The diameter of the
 ChaptN (; A..... ~. SNoIt c.ntenine PIou 99

dr....anu cirde is "'" W wnw .. Uv ~arinc di.alTWtO'r. 11 ;. <'qual to Uv ~ar­

~ ~ ......\1& tlw shaft journal <harnft.... u.. mq.,...vAi ~ ~
potoillOll 01 Ow shaft ..milO IN dNrana cirde ;. oft..., 0'1~ as ib 0'CC:ftI-
IricitY f'III>O 1_ Ow ~ ~
To 1M np oi 1M plat ~ iDw.tn.tion s ol It.. ....It powlion
~ at ~ ol3-Kl rpm.,w;t 6 100 rpm (dNo t.anns c
in'"
u.. !>tar-
IS gmotly "Uf!-
8"'"~ fordarily~ 1llt'~lom""~ """"'Ow powlion at llw ....... rolI
~ ol~rp<n.~ tlwWft mots<:&<- to dw boclom olttv~ Th..
top ~"' .no.... dw rn.p<>nW of 1M WIt to u... static: fumPli and stiff.......... &I
Of'I""ung <p:'ed. Whu" It.. orbil""" Of"""
not ~ c:lv.ngod ~.,.,. many
chlllflO$ in ota tic: pooihon which.,.,. importaol to do....."",nl '''!! th.. respoo... of
It.. 'Yo«'m.

•--, , i
;-
, .. --- • I
4 "' ~

,
- ,

l -- -
.---~

• /
I ••
,
•I l~
, ••
• I
•• ., , .
~
~_.

....
.,..... .. , _Aro _190 <holt ,...,_ pIoI "r • ......-. d. ""'" ....- com~"",
It... (oucDoord """opo_ bNN>;; -.... 'Pl'Pd Thoo...- _ fOII,.,....,;""""",_
lhr 1-0.... 1' 110 I""l cloomotff , ....._ . OIdo (1l'ft"'J _ l I w .. cd t .. _ _.. clNt'.
If"U ~"'" PIo<\cyIndnul ~ The _ lI'J'ICiarl. "'"'" ............ _ ~
d l"e plot Tho ~ '" "" ~ po:>olClCIn d "'"'",..,. ~ -.c..d ........
b cWltiI_ '" 1M tlNN'IJ _ b lIw C(>lf1tlf"9 _ _ .... - - .
100 Oal.. Plot .

If Ill... m..as u rem..nt pla n.. Ls d o...

to Ill... oca . ing o r ...al. th..n th .. c"'a r- ( <<_ ~ R ..oo
n-.. ,..... 1''''''';'''' 0( ......It " ott""
an..... c ircle. if it is accu. a t..ly d ra...11.
ca n , ho w ",h..n til... awrag.. s haft
d-;-rib«! "" "8 ,h. <'C«'n 'ncilJ _ <
(G....k t......- CU< "J'iilon~ Th. """'''id-
po. il ,o n i5 approa ch mg or ..xce<'tiinll
the limil o fl he o,'a ilahl.. d.-a . a n,.... lf
IJ "'00 ill do-finod M
'a""" _ u..
,h.
«"",... ntio> or u.. d.
0( Ihr w it and
the , hafl "'-'nte. Ji.... exCt'f'ds til... a""il- 'M CftJIor of ,ho bNri"!lo '. ' 0 . ... avaiWIIo
abl e c learan...... th e n it is a ,illn that ,a<hal d "",..."". co
...melh ing COtIJd .... .... iou.ly ...-ronft
in t .... machi ..... ' uc h as .. "...,..,.. oca r-
, --,, ( I>- II

ing " ' pc. "'''''. 'ho ..... ft "". "~ i ........., ,....
1I0 .. ·.. v..r. if t il... m..a -", rem..nl dt-an~ ri ,d . loqu,•• 1rnt ' 0 ,h It
p la ne is ""me d iu a n<'l' a ,,'a,· fro m Ill... ...... Lto<1Ong ,.... bNri"l! '" ~ ,h • ..,.,.n-

oca ring o . ..... 1. . haft dril....·lion can Iri<itr ..,., IS OIM'. Wbm con, ........ ..
make Ihe .haft J"<" it io n app"a r 10
in ,ho middto of ,h. ""' n« ri ",to
exce<'ti the lim it o f th.. clea. ..n"" c ir·
<"'I",...... , '0 ,ho It """'II ",rod in
.... bNri~ or ~' ..,.,.n'nrity ratio
c'" u nd.... normal opcral ing cond i· " u-ro. I h """"00'" ",'iD'" .....
lions. "'"'''' "urnbor _ n «'ro
Figu t¥ &-2 show5 ano t..... . 'ana · and 0 .
lio n of t lu> sha ft ....n t..r1in.. plo t fro m
the oUlhoa rd oca rinll o f a 125 M\\'
IJP/I P tuthin... 111.. mach ine i. ru n ning a t a consla nt "f..-ed of .'It-O KI rpm. The
journ al i, s hifted donvn and to Ill... right as til... appl iOO load on the ..>l o r c hanJl<"'"
Th is u n oc th uJt o frclat i....ly h igh p"""'u....on o ne side " f Ill... rot o r du .. lo
un hala n. "t'd ,,,I,,, u..ncltlll- It is a lso po •• il ~ .. thaI t h" larfll" c han!!" " 'as
ca u'<"<! ~." cha nge in al illnm..nt a s th .. mac hin a ,·hed thMlTlai "'luiJibri u m
a ll.., th.. ""'d cha ng... Note t haI th.. plot has l im.. la I...... thal lh .. data can oc
wm-Ia t"d ....i l h the o pe ral ,o nal chan~
~o t>ea. inft d earance d ala ...as nailable. "" no d ....ra n(·.. circle has M n
dra..." on Ihi. plot. T~l'icaJ m achi ..... o f this tyP'" h","". h" ft dia mel ..... o f a bout
3lKl m m (12 in). .'\ n a pl' ro:<imat.. rule is that. f" r p l" in q lind rical oc arinp.
I..... ring dia mel ral d ..ara n is ~l',call~' ..............n 0.1'0( to 0.1.5% of t he sha lt d iam ·
"'..r. Th Us. in th is ca " wo uld ex pec t a d iam.-t.a l d e.,... ""... of oct....""n 300
" m ( 12 m il) to 450 " m ( 18 m d). Th " ,;ha ft (,>:,,,te. line pk,t .holO~ t...-o distinct dus-
t..... of posit ions thaI a ..- ah..ut 20 m il ( 510 " m ) aJ"'rt. Thus. it is p.oba..... Iha t
II>.- s halt i. m"" ng fro m o n...id.. o f th.. oca .i nlt 10 th.. ot h..,. d u.ing t his loa d
t-ha nJl<".
~ .. ~ A 'iNI't ,,",_ 10.. pIcII""'" """
_ - . v a l .IZS _ _ ..... /
_.-,. -. 'ItIt......- .. ~ ••
, .. __ -.dal)e(lOrpm bod " ,- "
.. '-"J ~ n.. bo;"g ohIt-
.., _ _ lID tht ..".. dutInq ..... bod
... '-,
p<lOII:IIJ <:Iut 1O....--a<l _
~.

~ .. 10 ........... 9"""'!" _

_ _ ....'.. otT1w_ .. ~_­

... _ _ al'lI"l _ _ """'"
"1
, '="'"-".
" '~ " 07
-.aI ............ •
~' ..."'"j
.,.."... Soe""" .... b _ _

,-
oS ,, '
-XI ."
, II

• • •
",

Infor ma tion ( o nlained in l he Average Sha h (enterli~ Plot

"''''''I!'' ....
T.... ft cent" rlin.. plot e..n ..-d 10 in rer I 8«"..t deal a bout a
machin.'. co nd il lOn. II ca n be .. ..-d to me '" th.. ohln po!<ili..n .~ ... nd ....li·
mat.. th...haft .uit..de a ngl.. a nd k-J dl tl'rtion. an d it e.. n .... .. ..-d to mo nilor
..... . inll_ ~ ..11PCt""'lalic O"n)$io..... . nd th....mal ..IT"",,," TIl .. startup .nd .....,.
o:J<-n bmavior o f a.haft in a nu id-film bNring ca n .... co mf'"'l'd 10 tMol'l'l,nI
and hll torical """""""1": ,....
e""ng.- in ohafl pooition ";f h ""'J'l'CIto th.. el.... •
ana' cirn.. i. C-'O mmonly uM'([ l<l d.-duce w!ll't.l'>l'r th...... n pooiho n chang.- i.
appr oprial" for !1w....p""'rd opt'I'1Itinll cond,t ...... and ""'arin. l!ft>!J\l'lry. SlWt
poo.itioa c:an .... comf'"'l'd tIICl'OM o..llJlnp. couplmp.. a nd mach i..... to dl't«t
poll'1>lw ~i(i:n"""lor coo pli"8 ",lIl* ms. f inally. k.d or "l""'"aling ror>dJ.
lion ....... ott.n ft"IUll in >haft ,-.tion c~ I,,"", a m .... ....arru n.-d f.w
..... ol abnonnal ~ n... nftalIl' ft nontftl" ... plot io. • p""''l'I'ful tool
b madu.....". diagnootJ<1 ~ uw l<l alf~" <bta. from ~
plot IOnnau and procno data.

T1v ohaft OI'rlt...tirv plot aD .... UMd l<l ............. 11w .haft pog -Ptt.
Thio ... ltwanp Iwt••• " th.- Ii..... l h..,.... ltw ohaft and "-rinft: ttnl and In
arbouary n:tr.........<d orectMHL In horizontal ma-hi.- 1M refeA>ncedir<"ction to
102 Data Plots

almost always down. The angle is measured from the reference direction, in the
direction of rotation (Figure 6-3). To measure the shaft position angle,

1) Create a shaft centerline plot with an accurate clearance circle.

2) Verify that the starting point (reference gap voltage) is correct.

This will usually be the low-speed point for a startup and is typ-
ically at the bottom of the plot for a horizontal machine.

3) Draw a line from the center of the clearance circle to the bottom
center of the plot. This is the reference direction.

4) Draw a line from the center of the clearance circle through the
average shaft centerline position for the rpm or time of concern.

5) Measure the angle from the reference direction to the line

between centers, in the direction of rotation.

In th e figure , the reference direction is down, and the shaft position angle is 80 0
•

The attitude angle, tf/ (Greek upper case Psi), is the angle between the applied
load and the system's response to the load (Figure 6-4). The measurement of the
attitude angle is similar to that of the position angle: from a reference to the line
of centers, in the direction of rotation. The reference, though, is the direction of
the applied static radial load , which can vary with operating condition. The stat-
ic radial load is the vector sum of all radial loads, including gravity, that are act-
ing on the rotor, and it can be in any direction.
Historically, the dominant radial load on a horizontal machine was consid-
ered to be gravity, and the attitude angle was thought to be the same as the posi-
tion angle. However, we know today that large loads can act in other directions
(for example, partial steam admission loads in a steam turbine, fluid-handling
loads, gear mesh loads, or misalignment loads) and that they can be strong
enough to move and even lift the rotor. Thus, the static radial load will include
contributions from gravity and all other loads; the vector sum may point in a dif-
ferent direction than down. Figure 6-2 shows a clear example of how a massive
steam turbine rotor can be moved to an unexpected position by a major change
in load.
Unfortunately, it is difficult to know the exact direction or magnitude of the
static radial load vector acting on the rotor shaft. However, the shape of a start-
up or shutdown average shaft centerline plot, combined with our knowledge of
 - .....
;..... .. ... < ,
......... ' " ..... '" ••• d_1'<"'lion
""'JIe All "-"--oJ""" <i!' "" .. .. pcM><:o'I
"_~doil "_" _dNf.
__
-.Clldtb"'~ ~" Iz,",,,,,
~" _d"'<l _._
_1:I'waqo_.
do_ ......... bNoo:tr"q
1
"""" """"'"" The""'Jle" _
_ ' l< ...... <10
..... "'e'o.d ..........
_
""'"
....... POS'I....

"r 17
_.
~r:=~ .. !
,- -
••
~ "
l"-,

...
<,
'J
"'J ~" 6-4 "'_~ d ""'~
""'9"'- " ~ _ ............. ""flo .. pot

:I
........ ~ d o o ;

.... ~_ "J l!'E...- <:W«t>on d _

---.-- -
__ ~_dio<:l ... _ ..
ll:-_ _ -.g~ """"9" ....
•
fw:>m 10 _ _ ..

i
"' __.011-.
o .
~
••
.
•
,-~ "
104 Data Plot.

how the ~haft .<1wl.l1d n,,,,... for th.. ~l'" of bearing in~talJed. ca n g i,... u~ a d u.. a s
to Ih.. lo ad d i.ect ion. DOC<' Ih.. load di rKlion i~ ;,,~umed. Ih.. atl itu d....nfll.. can
be esll mated..
In p ract ice. a n abno rmal pos ition angle p.",ld..s a n indicat ion Ihal Ih.. ",•.Ii·
al loa d has a di ff..... nt m"'/lnirud.. or d i. ection than "x p'-"'tcd. o . l hat th.. sli ff......
o f l l>e sysl..m has cha nged. Wh ..n a d irect ion for th.. loa d is a"u mro. it ca n I""d
to id..nl ifying th.. po l..ntial so urC1' o f Ih.. cha nge in Ih.. load .
In a mac hin.. "'ilh inl ..mally p ....." " izl'd (h.,TIrod ynam ic ). fl" id ·film bea r·
i nfl-~ a nd a .. a tic rad Ia l loa d. c han/(t'. in ro tor sJ-<.l will u~lJ3l1y I' rod""" a
chang.. in .....·'..g.. . haft I" ...ili" n. Th i. happe"~ ht-ca" "' 11>e sti ffn oflh.. bear·
ing chan~ ,,; t h n ,to , spet"<!. Il,ff....." t hl!armg t~l"'s a nd d iff " 1 ~tati" loa d.
p •.,d Ut,.. di ff..... nl hl!ha,loNo o, , . peed TI>e ~ha ft C1'nt.. rli".. plol "..n be u.<.ed 10
cl\eck .,n lh ,s beha vio r .. nd to the c han/(t's in II>e mag nitud.. a nd d i....,tio n
o f lh .. I ).
f ig" t>-:'i shu "-s a'''''''Il'' sha ft "..ntet line .hutd"'o·n dala from th .. o utboa'd
beuinll o f a hoow nlal oompre~"". tra in. ". ~ .. Ihal th .. mach "'e ro t..l... in a Y
to X (dod ,,,;....) d in'ction. l h.. la 'fI" poo.il io n .10111.. .. nd low ..........nlric ily ' .11" , a t
ru nn in!! spe<"d a re nut normal fu. Ih is Iype o f b<-a .ing (plain ~'Jl ind ri".. l) in this
"'''; C1'. a od im ply Ihal Ih .. loa d may be po iol cd a~ ,J,"wn l rwl) an d ha,... I.......'
Ihan nu.mal magnil "..... ..\ s I.... mach in.. sl",o·. dow n. lh .. shaft Ct'nl...-l ,n.. 1Jlift.
to a lJO"il ion Ihal iodi calt'"S Ihal gnn; ly (blue) is Ih.. tlor n;n" n t load o n Ih.. sha lt.
l h.. shaft pn. it ion o..ar Ih.. .. nd of Ih.. . huld" w n i. quit<' no rmal r." a plain.
C)-i ind rical bl-arifll( (th .. hydrodyoam ic flUId ,,-cdg" l'n oduct'S a 10""-' thai m,,",,,
Ih.. rot o r up a nd to t he 1..1t o f C1'nt..r fo.,hi. rot a tioo d i....,tio n).

.•-
•
FOg..... 6-s, """"""'"" b6l.......
.. ~"" " " ' _ bNnng
• / /'-t -:.: - ---
7" ~-
•
Tho mactw-c is eq."PPO'd ""th
pIo;n cyWldti<aI _g, .nd
......,'" in • r <oX (doc' w;",J
direction. The............,;lload_-
lOfS lot hogh \peed (tedl .nd .O:>w
• I. /:- - /<
,. ,.
,
~-

.". ~IOPOO""'" """".d-a- j

m.", dw><Je in loadIng d the a ,~

bea""9- See the "'''' lot _ ts

:~~ /'
/
•~
, • , •
I"" nr!'HI= l'~ ftUl'Iq'!'''Ul· iii Itt t
~r !~~l.rnrf .h ! .l r!lH1 q lit1tH j
i l
!'t "i'i lh lt i l l i'I!!I!B:~"rt
~:i 11J'l!'~r l If,lltt!lrHHmil!Jil !t ,
. ; t"h'~' i [. t [~Ilq
'Ii'I
P •• tlttr d !
~a' maim
;,~'i t~·
HHHI·nrt!Hhilh!t~lHh
i,r,,!,L! .U, II
fl.i 1,;;,,, u"I' r i r p
't';,f, rt HI 'I.lr Irlr,ndtthrbll
't • ., l

HHHHhilHHliJ'
rr: . .rIJ~ • r~ 1
!r PftHHWBH
1•[,,,if till'
HU IU[
f
riHt~ tiH th !•
>

iIHW;lf1!
~'i;lih! If'['fl
lapi i rUt
It 1 I .1,lt'l!H1·f lit!
hUt ilh!~HI'rl~J·14
r '11 It t'.· ~t II "ll
,i
1;"j!!1
, ; A: 1 l j1
!
HH;~m Mitt n liltbrhw! I ~
1
i
•

-a
 •...1constanl 'J"""d. chan~ in loa d ",ill al", prod""" a chang.. in Ih.. attilud..
a ngll"a rK!. th .....fo.... th.. pos ilio n an~e. Th ,... c ha nllf'S in Ihe posi tio n a ngle in ~
sha ft u ,nt...-l ine p lot ind ica t.. cha fllti ng loa ds. Som e of th""" l08ds may 1>.- d u.. 10
pn ocns chan!!"So bu t m ioalij(nment ca n al", prod """ s ign il1ca nl I..a d c hanfl<"" in
a 1>.-a rinf(.

Th.. a>'er"t!" . haft Cl'nterli.... plot is ~ JlO"",rful tool for dt"t .-ct inll c:htlngt>< in
tl/ignmtml in mach ine tra in.. In ~ h..riron ta l m....hin.. wh .-re t he primary radi al
loa d i. p a' i ty "ilh no ~a,i"g load.. no rmally load...t. pla in cyIirK!ncal. h~"' ....
~"TIamic bearin gs ..ill ha..... a .,..., itio n . ngl.. o f bt-t.......n 30'" ~ nd 45·, and the shaft
will normaUyoperal" ~t a n eccf' nt , icity rat io of 0-6 to 0Ji f1orizo nlal mach ines
" i l h tilt ing pad bearings ",ill no rmally ha.... .. position ~ ""'e of b"'w.... n 5· a rK!

• •

\
" I
/ ,/ ...
,.
' / '" / ".I
~.
'>. y
"

) f71
"j) 1
,-'. ~
" "
j:
i
~~
"""
I ),
i
,
:" ,
" "'
i
},
I

F>g~ .. 6-6 Shaft po\Ot...., pllm lor 8 ........g...... 1I\8dI..., t,";n. Tho: plrm show tl><'
-89" ""It """""'" and pO"tlon 8"9'''' lor the fOur bNti "9~ ee... ing 1 i' low.
~ 1 i' ho;JIt..n<> till" ,I\0Il, .'"
rig dy wuplood. Thi, !lOrtic...... corrOn........ 01 P8"
""".rod oongulo' ml.. ~n_ pnodue _ tr.fl1kt from bNnng, 1_'"'_.
i"9' 1 . n<> 3."'_ thl" ,holt _ high <"<c""""'I1)1 '" ~ 1 i' w<y
lightly 1oodo<1.n<> tl><' 'h.>II "",noon " '" the ~ hoIf 01tho
V>ofl _bOn.."",. thl" coupling" ., "WOS"" Q'-.nts-
""9- _ _ til.
15· . nd <'Cl'enlooh' .ah o ,,\0- In U'ro. .\ I...-Ji!ttWd m.iI....... un 011...... . k-J
I ..n ~........ br.. n...... and ",... "" """'" "",,,Tinp wtll carry """"" than tJ>c"
""-"' of I.... ndiallo.d. 1'lIU " ~ t-nnp rNlr..ry W>Io>:adrd or. pt.'Mi-
bl,:.1o.Jc.d in <>J'PO"M dinoc.1 1F.,;un- &-6~
Who.... ...m miNli!tnmntl ..u.r... ......§ 1.. ..-.iU...:l~· 1M dIafi CM1 add to
or...hl""'" from tho- jln'1t~ l<>.d at ..ad! hraTi~ .nd 1M r-t..I l<>.d ' >':dOll ca n
c~ dnl matically in """fInitude- a nd d irm -'. Th..... all .no..." m thC' r""",.
tNo ..... ft poolhon .nPr 1II ''J''''fal i''ll "f"""d can br 'TI')' d,ff"",-I too- "'... ad .........1
brafl" , in a mi....JiIlr>oi"d m...h inr. Sn-~ ... 20 f",. "'0'" inf""",,hon.

01 ....... maJfu nction ~ can p n >dut"l' dra malic cha n~... in o.h.ift Cl'nk'rl inr pooi-
Imn. l\,>(-a u... . ru b ....h u . n..... t.....ri"!l in Ih .. ~y" ..m. a ",1," -I"..'II.to. ru hca n

--
,
....,.
~
..... '
~
•• •
•
/
- - 'j , ._- -,--'j /

, • .- --
• •• , ..
•• , • ,, . •
j, •, ,
u _ u _
• • •
f"~ 6-1 . _ d a 9"' ~ (-""",, _ a _ _ (~ ond •
.... Ih< pn. ""'" -...,. a ond S ... It>l' _ . . . . - _ . dt_ _ ...
... ."" Iio .. pmIbon. _ «lOll 'POl n.. ....... _ "'" (O<4IIng _ Ie> _
..... ..... - .n..« ])00 """ IS _ n... «><e*w,J ..
_ .. ' ~ 5 6.
108 Data Plots

result in a radial load transfer that can load or unload nearby bearings. depend-
ing on the orientation of the rub. This causes a change in position angle. which
can be detected with the shaft centerline plot.
Also, as we have seen, the journal in a normally loaded, hydrodynamic bear-
ing will operate at a high eccentricity ratio. Fluid-induced instability whirl or
whip can cause a change in average shaft position, at the source of the instabil-
ity, to a lower eccentricity ratio.
Thus. the average shaft centerline plot can reveal or help confirm serious
machine problems. Figure 6-7 shows an example of the shaft centerline response
due to a locked gear coupling between a low-pressure gas turbine and a com-
pressor. The shaft centerline plots of the two turbine bearings, 4 and 5, are
shown during shutdown of the machine. The high-speed operating position in
bearing 5 is abnormal. As the machine slows to about 4000 rpm, the coupling
unlocks, causing the shaft position to suddenly move to the right. The path of
the shaft centerline appears normal from about 2300 rpm on down to slow roll.
Note that bearing 4 also shows an abnormal path at the beginning of this shut-
down. The shaft position was so extreme that it caused a rub in the machine
until the coupling unlocked at about 4000 rpm.

The Complete Picture: Orbit Plus Average Shaft Centerline Position

A vibrating shaft has an average position and dynamic motion about that
average position. The average shaft centerline plot displays only the average
position; the orbit plot displays only the dynamic motion. Both pieces of infor-
mation are needed to completely define the dynamic position of the shaft rela-
tive to the bearing in the measurement plane.
Figure 6-8 shows a sequence of shaft centerline plots from the shutdown of
a steam turbine generator set. Each plot includes an orbit captured at the same
speed as the position data (yellow dot) and displayed at the same scale. The
clearance circle has been estimated.
The orbit begins a large amplitude, Y2X vibration at 2580 rpm. From the
shape of the orbit, it appears to contact the bearing boundary in two places: the
lower right quadrant and the upper left quadrant. This was a violent, 18 mil pp
(460 urn pp) vibration that was actually heard by nearby personnel. At 1727 rpm,
the orbit has returned to a flattened, mostly IX shape, which still shows some
evidence of rub in the lower right quadrant. The combination of the orbit plus
shaft centerline has produced a striking set of data that greatly clarifies what is
happening to the machine, and the reason why!
 Ch••pl.. , 6 A.... ~ ShaftC .....' ...lin. PIn" 109

"
--
'/ / -- ~

,
.
" , ....~" .
•
• I
t.
/

f;gu.o 6-.
QrboI, .... pon"'PO>O"! on .natt _itton plots ""'... du...... ~ >hUl- ' 01•
,_'" ,!J'b"", _.>lOr.n.. dola from lht HPIIPoutboootd bNnn<;J ,how> .. ....... nt l n X
libatIOn .. 1';/l() rp<T\ ..... d> ,""".." ",.,uoa ... (h t.... bNnng "' ~ _ri<Jht ancl UPl'"'
Itft qo.oa<jf........

Su mm ..' y
Th .. . , !(.. ~haft ,.....t...t in lol i. a n .\T " lot of th .. a",,"!t.. po. il ion of I....
•ha ft in the m..nt p la n n... d rd .. d rawn on t .... " lot r<"J'«'s'm t . t ....
dia mel 1d n"" .. f a o..arb)· a ring 0 ' ..'al. T.... combinat ion allo"" us to
' ••uali t a, rall" po'il inn of th.. s haft ,..l"t i,... to I.... a"a ilable d eara n"" d uro
in!( s t. rt up. s hutd,,,,-o. or "'..., tirnt'.
Th.. pmition a ol':l"00 huri ,..'"t..l mac hin..,, ;s m..as ured from .. , .. n ;"a l rM'
t'n"n.... (" . ..ally dO\\.... ) to Ih.. li.... betwt'<'n (fflt...... of th .. bea rinfl: and . haft. in I....
d i=tio n of rot a tion.
T a ttit ud.. an!tl.. is th.. an~ ....t" .....o I.... a....l ied load a nd t he .~.I ..m·•
......poo lo th.. load .nd i. a key ..harat:te' i.. tic Oflh .. o ....rall 0fl: coooit ion o f t ....
mach i"". T}l'i<.<lllv. th.. luad i. 001 Lo",,-n. a nd th.. atl llud.. a ng\<' is ~imated
a od u....-d to work bac k from th.. av..ra~ shaft l" ",;tioo to tb.. d irt'Cl io n a nd m8/l.·
oilUd.. of Ih.. load .
T a ,...,8/l." . ha ft ""nl ..rlin.. plot i. u....ful fu, dl"!t"t" ing a nd co nfirm ing th..
n i" ..n of m ao~· m..chi .... m alfu nct ions. ouch as ruh. Ou id-,nd u.--...:l in..abilily.
....a ' infl: wea, 0 ' ..,o..ion. and m i. a.liltnm..nt.
Th .. combination of t .... . haft nlerl in.. plot . t .... ht-..ring d "a , a o"" c i,ct...
a nd th.. o rb il ....10 prudlX"<' a ''''ry JI"", rf,, 1a nd {"(Im" ll"! .. "i<:tu .... tlr th .. st.. I ;C a od
dynamic .....po n... o f th.. . haft rdal i In th.. ..... ' ;ng.
 111

Chapter 7

Polar, Bode, and APHT Plots

IN CHAPTER 3 WE DISCUSSED THE CONCEPT of the vibration vector. This vec-

tor represents the amplitude and phase of a filtered vibration signal from a sin -
gle transducer (usually a shaft relative, displacement transducer), where the
phase is the absolute phase of the filtered signal measured relative to the
Keyphasor event.
A vibration vector will sometimes change with machine speed. Changes in
IX-filtered vectors occur because of the way rotor systems respond to unbalance
forces and changes in the Dynamic Stiffness. Harmonics of running speed (2X,
3X, etc.) can also change in response to changing conditions in the machine. As
we will see in later chapters, these changing vibration vectors, combined with
our knowledge of rotor dynamic response, are a major key to understanding the
root cause of many rotor system malfunctions.
Because of the importance of vibration vectors in machinery diagnostics,
several vector plot formats have been developed to present this information.
Two of these, the polar plot and the Bode (pronounced BO dee) plot are
designed for the presentation of startup and shutdown data. Another pair, the
Amplitude-PHase-Time (APHT) plots, are very similar to polar and Bode plots
and are designed for trending.
Polar and Bode plots present the same vector data. Because of the differ-
ences in format, the two plot types complement each other in ways that allow us
to see some information more precisely than would be seen with either plot
alone. The importance of vibration vectors to diagnosis, and the clarity of pres-
entation of these plots, make the polar and Bode plot formats powerful and
valuable tools for the machinery diagnostician.
112 Data Plots

Because of the similarity between the polar/Bode and APHT plot families,
they will all be treated in this chapter. We will start with a description of the con-
struction of polar and Bode plots. We will then discuss slow roll compensation,
a very important technique for clarifying the dynamic information in these
plots. Then we will discuss the large amount of information contained in these
plots, such as the slow roll speed range, the slow roll vector, resonances (both
rotor and structure) and resonance speeds, the Synchronous Amplification
Factor (SAF), the heavy spot location, and mode shapes from multiple polar
plots. Finally, we will discuss how to generate and use the APHT plots.
While polar and Bode plots can be created from velocity and acceleration
data, they are most commonly used to present shaft relative, displacement
vibration data. Consequently, this chapter will be limited to a discussion of dis-
placement plots and their interpretation. See Chapter 16 for a discussion of the
important differences in the appearance of polar plots created from velocity and
acceleration data.

The Structure of Polar and Bode Plots

Polar and Bode plots present data from the startup or shutdown of a
machine. During this transient process, vibration vectors (IX, 2X, 3X, etc.) are
collected at small speed increments between the startup speed and operating
speed of the machine. This can result in several hundred vibration vectors for
each measurement point.
In Chapter 3 we showed how a vibration vector is plotted in the transducer
response plane (Figure 7-1). The vector is drawn from the origin to the point (yel-
low dot) on the plane that represents the peak-to-peak amplitude of the filtered
signal and its absolute phase lag angle. This phase lag is measured relative to the
transducer measurement axis in the direction opposite to the direction of rotor
rotation.
The polar plot is a straightforward extension of this concept. The plane of
the polar plot is the same transducer response plane. In the polar plot, though,
the arrow is omitted, and just the set of points (the locus of the response) is plot-
ted, with a line connecting consecutive points (Figure 7-2).
For IX data, the vibration vector represents the approximate location of the
rotor high spot relative to the transducer when the Keyphasor event occurs.
(Because we are describing a physical spot on the shaft, we will use the term high
spot only for IX response.) This is exactly true for IX , circular orbits, and
approximately true for IX elliptical orbits. The Keyphasor acts like a strobe that
illuminates the location of the high spot at the instant of the Keyphasor event.
The IX polar and Bode plots show how the high spot angular location and the
vibration amplitude change with speed.
 (hap!"" 7 Po"',.~. and APHT PlotS 11 )

, X-li_ _ .."" "'lO>al

'"_ 7·' ... ", btaoon _ lOr pIon~ ,., the non,,,h,,,.. 'e1oPOA'" pia "", The IX
..-<tor ;s 90 II"" PI' .:220' (l -5 mOl PI' n1O"}. The no" <leg,..., ,-...ce .. .. the
.......sd ue.. . ng oN, kx.,,,,,,,...clthe ~ log """eo"", rn . <;t;,l!<tIorl "PP"",e
'" the JI '" Y«CW) "".toorl. I.e<- Chop''' .1

"'.-
Fi9<><e 7·2 ... ' ll.. _ornpen<otO'd p<>I.o, piCt
""""""II d.m hem ,1>0 ....." ''''''''' cf .
....yge--19 18. "eom '''''''"",The
l OMW.
IT ",.,k "
f,
' kgr><'d WIth the "onsdue.. k>uncn. .tId ,hr
,.....,., log . ngIeos ;I'IC' eo", " the d .-eclK>n
<lPPC"1e '" ""ot"""~,.,,,_1or
"""" <:J thr tJD'IIt'-

l m i pp tul .....
 Th.. pola r plot ~ g......' atl.'d wit h dala fro m a 5ingl.. transd oct'~ Tho> u ro
degr.... mark o n th.. p<'rim..t.., o f Ih.. plot is a1il(nl.'d with I......nlll"ar mounl inl(
locat ion of th .. Iransduco>,. a nd I.... d.. gn'E" inc........ in a d in.-ct io n oppo silr 10
rolat ion.
In th.. figu th...... ..ng l..,; art' ma rkl.'d at 90" in t" ,,·a l,., b ut a ny con""n i..nt
im""'al n m u"'-"<i SJl""'d la 15 a rt' ad<kd to ,;clo>cll.'d point. f,... Uowj. Th..
d itt'Ction o f ro t..l io n. filt.., ing u (u...aUy IX). ..nd full sea l.. ra"l(" fo r a mph·
Iud.. a re indud l.'d a~ ran of I.... docum..ntat ion.
Rt-ca " ... o r Ih ....-a~ rot or ..,,;po n typ ically cllan~ . ....... a I'5Ona nco>. mo1lt
o r th.. poi nl ~ nn a po .lar plot t..nd to c,""....... tog<'1 h.., d O"-' to Ih.. lo,," a nd
h igh -' I....-d .. nd~ of Iii<> pIn!. \ \ 1Iil.. th .. pola. plot ca n .... zooml.'d to gi.... mo,..
,-i,ihilil )' in Ih.. IlIw-,1""f'd tang", th .. Bod" l'lot, which includ M a fr"'l" ..ncy scal ..,
"a n .... a ben... cho ico> for u a min ing dal.. a,,"a~' fro m ' .....na nco>•.
11>.- RII. l.- plot (Filll'tt" 7-31 displa~" th m.. vib r..tion ....cto r dal.. a. Ih..
pola r p lnl : Iii.. ,ibrat ~ .n a mplil ud.. a nd plla a p loul.'d"1>8 rat..I)· o n t....o. !'1'C-
tanllul"t plot . ..i l h , pt't'll (or rn"l.....ncy) On th.. ho, ilOnlal ax.... T.... up po>' 1>101

120 l Xth .... '....~_'!I ••

i ''..
. -

"-
i -
! 0'
110 _

,. J/!
,

, " u_ I ............ I

•1 ' t
I
Jt - - --tl'/
,
• '.
-~-~-,~~.~ .
l mlpp lul,,*
I
•
00

Fig " ' . 7-) A tx. "~ «l ~ pIo1_ polo, pIoI1llowirolj "'"' '"""" d.ThJ !bkJr)_ Tho
~ pIo1;s "" tUllil)' two pIotUho ~ pia( i, . ~ the ..... p1_ pIot. AI dot. "from • m-
g... tr'''''''''''.'''''nted ... r:r
ohown by tho 1o<.l1..... on lho poIor p10l Tho ...-litl«ed pIilU ;,
Ulown in ~ on "'"' ~ plot. " ";r>g'" .,..... <I>c:N<n '" oed on "'"' polo' pIoI.;, ""' lh
oed 10....... on "'"' ~ plot
 Chapter 7 Polar, Bode, and APHT Plots 115

displays phase lag in degrees on the vertical axis, increasing from top to bottom
rather than bottom to top (see Appendix 1). The lower plot displays the vibra-
tion amplitude on the vertical axis. The red vertical line in the Bode plot identi-
fies the same values of amplitude and phase as the red point in the polar plot,
but also explicitly identifies the rpm.
The unfiltered vibration amplitude can be plotted on the amplitude part of
the Bode plot (green), which allows easy comparison between unfiltered and fil-
tered data (phase cannot, generally, be measured for unfiltered vibration).
Because the unfiltered data contains all frequencies in the vibration signal, the
difference between the IX amplitude and the unfiltered amplitude is the Not-IX
vibration component in the signal. A difference as large as is shown in the figure
warrants further investigation.

Slow Roll Compensation

As we discussed in Chapter 3, slow roll compensation removes slow roll
runout from a filtered vibration signal so that (presumably) only the dynamic
response remains. Compensation is most often applied to IX data because of
the importance of IX dynamic information for balancing. The slow roll vector is
a vibration vector that is measured in the slow roll speed range, a speed low
enough that no significant dynamic vibration is taking place. The data from
each transducer will have a different slow roll vector. Compensation is the
process of subtracting an appropriate slow roll vector from a transducer's entire
vector data set before plotting. The resulting, compensated vectors produce a
compensated polar or Bode plot (Figures 7-2 and 7-3 are uncompensated plots).
An unstated assumption with slow roll compensation is that the slow roll
vector remains valid for the entire set of startup or shutdown data and from one
transient event to the next. All assumptions, including this one, should be regu-
larly checked. See the sidebar in Chapter 3 for some reasons why slow roll vec-
tors can change.
Subtraction of a vector from itself produces a vector with zero amplitude.
Thus, compensated vector data should have zero amplitude at the speed of the
slow roll vector. On a polar plot, that point should plot on the origin. On a Bode
plot, the amplitude will go to zero at this speed.
This behavior allows us to easily cross-check a vector plot for correct com-
pensation. A slow roll comp ensated vector plot will have zero amplitude at the
chosen compensation sp eed. If, on a compensated plot, the amplitude does not
go to zero at the speed of the slow roll vector, then either the slow roll vector is
incorrect or it is not correctly subtracted.
How do you choose a slow roll vector? First, use an uncompensated, IX Bode
plot to identity the slow roll speed range. This range extends from the slowest
116 Da'a Plot.

ru n ninll: .......-d o f , he "",ch i"" up to t he speed .....ere d.m a mic efTect~ d ue 10

u nbala nce become apJMrent. Lheck bot h the a m plitu de a nd pha... pl..t f.. r Ihe
appeara nce of ~T1amic enoo~ whe... t he a mpli tude or p ha... be¢ n to ~igntfl­
ca ntl y rnanll:e.
Xed:. fro m ..;,hi n the ~peed r3"1le, ....ect a ...mp le , h.lt i' lyp ical o f t h.. ""a,·
by data. E..<p<'Cial l)· at low .. .....-d~and low a mp lim d.... the p ha... dat a can "I ~ ...ar
'''rr noi~)·. d ue'0 th e Ii mit at ion~ o f , he instrume nt a t..ion. Som.. ,;sual ~n.""t h ·
inll: ma~' tw nec-e. ... 'Y to id..nlify ~aJ,,<'(o Iha t ..... Iyp ,cai or alyplcal fu r , Ite , a nll:e.
T he ... a... time. " il e n th e ph.."., a nd ..m phtu d" in a R..d" plo t will ~ta rt
c ha "lli ng from I he 1" ..-.....1 ~peed in th e <Lota""..,. Wh"n this hal'l"'n. , th mav
be 1\" clearly detinP<l , 10... .. ~I ~peed ",n!t". a nd th" .." Iy rn.. i~... ma ~ he 10 1...1
th e ......__ '"JI""d in the d alaha"", thai ha ~ " ......hl.. da ta.
... •Iow roll "l.....-d r3"1le i~ iden tified on th e Il<ode I'k.t in FilC" '" ' -4. At 120
' pm.. th .. "jo..- roD ''''';10' is O.bl m il PI' L IIllr (IS I' m 1'1' L 11ltI"l. Hh i5 ~Iow roll
n~tol ~....m....·hat alJIPical. bul i~ no' a bad data po int A bette, choice m il4J.
ha,'e been o n" in th e 2I K) 10 -il K) ' pm ,a n ~".) Th.. ,I,,,,,' .. ~I vect o r IS . uht ract nl

•" -c
I "'0 ~

i'
J
I ,, ", ."
\ -"v II
_....
oro -

, ..- .... I
,
! , •,

J ,
••
,
J\
Fi.,... 7-.4 _ and pojM pkrn """"""'.I thr~ , 01 dow mil """'P'""""'''''-On thr _ pIo\.
Ihr , low "'" """tor (re<! <loU on _ ploO: and re<! """'" on pola< plot) i. taken from thr .. ncorn·
~"'f'd amplaudo.nd ~ dala (bluo,) .... !'Iin thr _ roI "I.....:l ,_,C~ 01.
poIaI plot i-....""""_ ... ""et,,,,,, """""'"
a thr dow "'" """tor. 1M amount 01,,,,"
""'ll>ll to 1Mampnudo of ~ ""'" ,001 """t<".
 Chapter 7 Polar, Bode,and APHT Plots 117

from every data sample of the uncompensated data (blue) to produce the com-
pensated data (green). The compensated phase plot is slightly noi sier near the
compensation speed, wh ere the amplitude is low. Note that the zero amplitude
point on the Bode amplitude plot (red circle) corresponds to the slow roll vector
speed.
On the polar plot, slow roll compensation is equivalent to shifting the entire
plot by the inverse of the slow roll vector (the small red arrow), or by moving the
origin over to the slow roll sample point. This will place the compensation speed
point at the origin of the polar plot, as seen in the figure. This technique allows
polar plots to be easily compensated by inspection. Notice that the shape of the
polar plot does not change. It is much more difficult to compensate a Bode plot
by inspection.

Information Contained in Polar and Bode Plots

Polar and Bode plots are an important window to the dynamic behavior of
rotor systems. They are used primarily for two purposes, balancing and the iden-
tification and analysis of resonances. Thus, it is difficult to discuss the use of
polar and Bode plots without touching on some rotor dynamic theory, a large
and complex topic that will be presented later in this book. Because of space
limitations, the rotor dynamic discussion here will be very brief. Balancing,
which is a complex subset of rotor dynamic theory, is covered in Chapter 16.
Basic rotor dynamic theory predicts that, for synchronous (I X) behavior, at
speeds well below a resonance, the he avy spot (which represents concentrated
rotor unbalance) and the high spot will be in phase (0 phase lag). Thus, low-
0

speed vibration vectors (which represent the location of the high spot when the
Keyphasor event occurs) will point in the direction of the heavy spot. At the res-
onance, the amplitude will reach a maximum, and the high spot will lag th e
heavy spot by about 90 In other words, the vibration vector at resonance will
0
•

point about 90 (measured in a direction opposite rotation) from the direction

0

of the hea vy spot. At speeds well above a resonance, the amplitude will decline
0
to a constant value, and the high spot will lag the heavy spot by about 180 the ;

vibration vectors at high speed will point in the opposite direction from the
heavy spot. See Chapter 11 for a detailed development of these principles.
These basic rotor behavior characteristics are used to extract information
from polar and Bode plots. In th e most common application, balancing, the IX
compensated plots are used to identify resonances, determine the mode shapes
of the resonances, and identify the location ofthe heavy spot(s) for each mode.
IX, compensated polar and Bode plots can also be used to calculate the
Synchronous Amplification Factor (SAF), a measure of the Quadrature Dynamic
Stiffness of the rotor system.
118 Data Plots

Equally important, IX and 2X plots, when combined with historical startup

or shutdown data, can be used to detect changes in rotor system behavior that
signal the presence of several types of malfunctions. 2X and higher-order polar
and Bode plots can be used to identify the presence of, and changes in, reso-
nances above running speed.
Vibration in machines always involves a system, which includes the rotor
shaft, the bearing supports, the machine casing, any attached piping and auxil-
iary equipment, and the floor and foundation structure. All of the components
of the system will participate in every vibration mode of the system, but some
parts of the system will vibrate in some modes more than others.
Usually, we are interested in modes that involve large amounts of shaft
vibration relative to the machine casing. These rotor modes are usually the most
important for machinery management. Rotor resonances tend to produce large
amplitudes of measured shaft relative vibration and are easily identified on Bode
plots as a peak and on polar plots as a large loop (sometimes called a "polar cir-
cle"). However, it is important to remember that the machine casing, and conse-
quently the observing probes, may also participate in the vibration. This is espe-
cially true for machines that have support structures with relatively low stiff-
ness. Thus, the shaft relative vibration at resonances (and away from resonances,
for that matter) may include significant contributions from casing vibration.
This vibration can either add to or subtract from the shaft absolute vibration,
depending on the phase relationship.
Modes can be excited where most of the vibration occurs in the piping sys-
tem, other attachments, or the foundation and support structure. These modes
(sometimes called structural resonances) can produce a relatively low amplitude
vibration of the casing, and, by extension, the measurement probes. These
modes do not usually involve a significant amount of rotor motion, but, because
of casing motion relative to the rotor shaft, they appear as small, but occasion-
ally large, loops on polar plots. It can be difficult to tell the difference between a
mode where the rotor and casing are both significantly vibrating in phase and a
mode which produces a small vibration in the casing only. Both modes will pro-
duce small loops. Casing transducers, by revealing the amount of casing vibra-
tion, can help resolve the difference (see Chapter 12).

A balance resonance has the following characteristics on a Bode plot:

1. An amplitude peak, and

2. A significant increase in phase lag at the frequency of the peak.

 On a pobr plot. roto< ",,>oks wi.ll prod,..... Ia'll"' , Yin!l lo.""!'" or p;o.tiIII
10.-...... ()l lIn modn .. d l tnId toprOOUC1' ......u 1oops. ill IM opI it I>alana .....
.............. cau....d ~ an .ootI>,...- (u q ... U ... ppon ol,fJ...- t..... Ch.p4n IJ ~
~ n ....... ~ ..... oft ""'"' vil.iblr on • polar plot btoc.uw tho:-
.oft.... arrnr'" a dootinctiw .....u loop.
F'!lU"' 7· 5 ...,_ .• fI..d .. a nd polar plot. al. balaOCf' ncr and ..
atlwr maII ..........ncn. A quod ri>rd< ...,.,.... thar: It..- I X, comprn..t ....
,"'" On 1M Bode pIoI. IIw ,....mArY rotoI" (ba1anCf'I ............,..'" idnlt lfi.d "lo' In..
, k l hal iIJ'l"'MS .01 juot O'Ofl 211lXl rpm IIJ>d t h .. ,,,,-nuintl: plw.., ~ ..~th
i "1't"• • inlt "peed. T h..ol.o...... ...... ' M OCf' ~ ... at tt... "f"""d of max ,mu m
ampln ud...
1'h.. pola. plot . I>o.JWS Ih.. ....In.. data a. a la' ll" 1' ~ 1 '. , ..... max ,mu m a n,p' ;'
lud,- " f l .... loop k,cat.... 11>0.· ..•••ma...'" al 20"10 rpm. ~, ml>l.· "'10' modeli" l1 p....
did . lhal the plul... b ll 81 .....' nanCf.> will rnan ll" t.y .1>0:"" 'IIJ" f... m Ih.. I"",·

wr-""--- • ..
... "
t '•. .-
!
I ••

"'~
~I
,.. -"'- V
. '·'-
I
J ........... K ...

...
f Mju .. 7·S u..>g _ _ poloI pion. 10 _
.. _ ~ • P"'_ in ~
-,

' .............., On. ~ pIot, . _

"~

""""'""'"
an in< _ .. pII.tw ~On. poloI pIot, ....
_ " . . - . . . . . . . II:qI
~ ......d _ n . . . _ oIwIt _ _
...... poloIpIoC~...... ~ .. _ I o < l t ~ _
..
_~Iog~in._IIOn_I<>
~ _.

......
2OXI<;>mSmlll .......
pIots_

--~---~
120 o.ta Plou

'f""""l phaw ~ Thu", ~I"'" I..........umum ....plnudoo Of I 'lO' pha... chIo.

can ....." 10<:. .. 1"" .................. In I"" p..lar pIol: in I.... fl,CU , phaMo a t thl'......-
............. pnl< (aboul 16lT 1 bp I 'f""ftl. phaM' ( 110" 1 ~ 50': Th .. ..
oillmfun~ In.; 1h.o.n fl"d innJ ~ 1 ,imp'" rnodool buc 01111 p""...s..s • ....-ful
<n>«<Iwd<.
In I polar plot. I""" io. .o ........ luup .at I"" .....,.~ rnd 01 1.... dol.t and
~ Ioopo at It... ~opronJ rnd dthO' obI__ T' - ""JP- "'J'P"'"' ' ' pftaW'
and .. mpl,lude di... ~ Oft It... !lo ,k pIoI. ",10>0.1 w...ty. I""""' . ,.., oma.lI .~
r?'M............. IM An... brftI. f'U""i'lnJ by u... _ "r o~m u!lb.o.Lo....... ~.
~ 1M II>ou¢' . "'mplor rutur .... xkl p_he u a ')II" pt,.... riJ• • at
• bal. ....... ~ a nd a llll)" pt, m-ngt." thn~ a 1>001" """"" Ih..
d n..1 al...,..~. 0,,"," " in praet il Th .. id"" h fyi" g eh.lr...-I i<l ie o r a n ........
nalM i. a flt""k in am phlud.. ...-curring .01II>.- ... m.. Ilm.. a' a n '1KTt"a_ in pha...
I~ In pl"nl mac hi n<'ry. Ih.. 1,,1.01pha.... cha ~.. lh rou!lh .. ......"'""ct' may t...<""ll·
. i<f,.rahly mo... o r I Iha n IllO'". 1 hi. i. l rut' "".." fo. omall ,",l"a"...,.. which i.
"";d.. ,,1 if }"" u vi lh.. sn,aU . mali loop Of] Ih.. 1',,1.0. IJI". ... .. mi"iaru... pola.

-
pIoc

•!'~·:'·::'''':''7·----------
f,
I
.,.
.-
..•
,n -
--- -- ~~
I
•

-. -- - -, -
' , r: :0.-
,- , L
. -
--,
vo1 '. -

---
1 - '-
1

I /
 Chapter 7 Polar, Bode, and APHT Plots 121

The Synchronous Amplification Factor is a measure of how much IX vibra-

tion is amplified when the system passes through a resonance. Systems with
high effective damping tend to have a low SAF, and systems with low effective
damping have a high SAF.There are four methods to measure the SAF (Figure 7-
6), three using the Bode plot (Peak Ratio, Half-power Bandwidth, and Phase
Slope), and one using the polar plot (a variation of the Half-power Bandwidth
method).
The SAF is based on the dynamic response of rotor systems to a rotating
unbalance force. Because of this, all methods for measuring the SAF use l X,
compensated polar and Bode plots. Slow roll compensation removes the part of
the rotor response that is not due to unbalance. Because the Half-power
Bandwidth method is so widely recognized, it will be described in detail here.
See Appendix 4 for details of the other methods.
The Half-power Bandwidth method [1], endorsed by the American
Petroleum Institute, is also known in the USA as the API method. The term
"half-power" originated in ac electrical circuit theory; the half-power point cor-
responds to the -3 dB (0.707) point of the voltage response curve of an electri-
cal oscillator (because power is proportional to the square of the voltage, -6 dB
(0.5) power corresponds to -3 dB (0.707) voltage). Since simple electrical and
mechanical oscillators are mathematically equivalent, the method can be
applied to rotor systems and yield meaningful results.
The half-power points of the Bode plot, Dlol\' and D higlz in the figure, have a
vibration amplitude of about 70% of the maximum balance resonance ampli-
tude. Once these speeds have been identified, they define the half-power band-
width (the blue region). These two rotor speeds (or frequencies) are used togeth-
er with the speed of resonance, D res ' to calculate the SAF:

SAF=-----'-""--- (7-1)
(n high - Dlol\' )

Because the SAF is dimensionless, any consistent speed or frequency unit can be
used.
Figure 7-6 shows Bode and polar plots that have been generated by a rotor
model. Using the example in the figure, the SAF is found to be

SAF = 1230 = 2.5

(1550 -1050)
122 Data Plot.

" word of caution is approprial.. h....... ..... m ,· l h.~ ls f", meas..flo g SAF a ....
ba..-d on ideal. iiOt ropic rolor tw-hatior. Heal rol '... sY"te m, ';-a O (a nd u",al ly d o )
ha ........ m.. de" "'e o f a niwt ropy in th e . upport st ilTn.... tha t raUSf'S t he mea s·
ur.. "",nt o f <>A F .. <J "ll any met hod 10 he ... ns ili,... 10 Ihe o,ou ntin!! a n!ll.. of the
meaS......men t pr......... Be M'.... lo ..... <-1lapter 13 1>,·f",.. . PIlIying an), of the S.' F
m,"""",..n,en t met hod.s to data fro m your ma,·h ine.

I X. "'''''I...n''''t.... pola r a nd Rod e plols ,;-an he u......d to Iocof.. Ih.. di .-oc/inn <J/
Ih.. h",wy -'fHJf r,... bala nring PUl l""""" p robably their mo.st importa nl u..... A• ••,t"
hOI'''' ..i""',....... at sl......ts ",el l helow a ....sona ll«'. the h..,,,"y , pol a nti h .¢, spol
will J... in p M ..., the ~ihrat ion "Klor "ill poinl in I "" di ll"ctin n of the h...,,1' ' pol_
Th i. princip le ,;-an 1.... p" lit'd 10 eoc h ,';bra tion mod... of a mochin....
~IL'i" o f its f.. rmal. th.. polar p lot i. a lm...t alwa~.. eaSIer to u... for th IS
po rpo Mg" ... 1-i <h. ...... 1X. coml n...lro Ilo.>d... a nd pola r plot . from the sa"'"
. mall . t m tumine. wil h t he d' d '.. n of t he low·sp<'<'d ,......-tor l hea•.,.. 'J"" I
ioo icated in red. On the 1Io ~'" p l..t. t he d irection of t he low·spe<'<! vib ral ion "l"("-

J
"" ~W'
If
. '
, .,_ /... . \~ -,
,, ,I
~

f~'"
'.
1-1 Huv,- 'P<lC
,-........-,,,.., -
Ioc.'''''' .....ng _
,

.00 pola< plot,- ~ IINvy 'pol dttKtoon;, .'.x.>led

l'I' ,.... ,ed dol on It'>e polo' plOI. At ~ _ _ ...... ~1lU', ...... ph,o... on It'>e <~
",'ed pola<plot po,,'" tow.. d >')" <pol On ...... _ pIoO:. "'" pI'oioso'i"" - . . _ '<">0-
"""".. """"'I" iOOic" d. _I of t-..avy <pol The ~"" h"lh-<poed dota
.... ' " ...-.:I t8O" fr<m I >V)I 'PO' d._I for an idNI ,OlOt 'ySI""'.On poIa' pIol. 'he
b1.w Iir>es "'-' ...... ,~"" ....... h"lh-'P"ed <X<C1IOn> whoch <!no""..
from the idNI,
I .... " 1JIu.. hom .. ~ .."It"' ju... boof..... tlw pn.....~n.lO rn.~ d .... I " .....
0I>&I1U. Tb .. I'fO''1do.-o. a ...... . ,M,j" ('f~ 011 tho- polar pIooc in l~1Ofl-
\\nm .. BoooW pIoI ;. n,,11 .....w.bW. thf. 0". \llJ'. .. nd 1lll1" ........"""dupo o;:.on "'"
.......d 0<1 .. polar pIooc to In. ~ ..".....a.ft:tor d ,fO'Ction. In thf.~. tlw
bIUl' ....... ohow tlw pt..... '"" rnon.o.lKlP I2IJ.!tl rpm l a nd "" wft!.abo\". ........
nalKlP. ~ I ~ h oII.>uId "'" 90" a nd IIiO" frum I"'" hH\) 'P''' '
,' n,ootropic otiff,...,.. and aiM .maII rnon.o."""" I'ft"'o<'fII in tI".
mad"".. d l5llOft
I , , " , , " . n,bhonobipo I ,,", ClwpII~ 13~ £>0 if llw ""'-'} "PC'" b:..I _
U. d OC;>(M br !two I~ CUOl.II1lct""l II,," ~ l he- """" """'"" would he-
wnbi n ~ oL ~ .odcq....lt t .... u¢l f" .. b.o\.II.nci"", .'or
idral 'ot .... . ~-,;tnna
WIth ;""' ropir: "' ,ffn..... pnla' plot. from XYt .. nod MU .... 01< lik.- idf'Tll 1C&I
circlro . nd yicld id<-nl icaJ ~um.t ... o f lhe h } 'I"" luo:..tion. H.,........'.
a niOOl,..,p'" . , iff....... ..i ll ~.." I.... polar rlot~ I" I" ok . ifln, ftcantly d ill..... n.
IF'lIure 7·81 and r ,,,,lucr di rk n! in 'l'lied lo<:atio n. fu' th e hea,')' opt>!. Wocn
boIa ncinfl. when......... P' .... ibl... ( ml...... bol h pola' plot~ T........a.)' opol loc,, ·
IIU(U c;on Iw ;o.....r"!tf'd. ,, ' "'''"' ouph i.tkaled sillnal p"""". oing teo;h" iq....a ca n

. . <00 . ... ' _

"-r
-~
~.

.,,,•: ;~
- - r
tI

. --
,I!
•

~. • •
-
O' L ....

- -
•• •
•
,~.

111II' t / ""
- 0<$'
s ... "" ........

,".m llJrI>F.e _0\01'""""' ' ' p<obfo> ..

F ~ 7-.8. PoW pioI ' lhrru_ bU'I"og~" tO I,lW
WI ""'" ani>an<>pO: "dl
. . - <UIoo. ~ impl""" hMoy <pol !t'6I1oul
lloo:h plot> ... to lhr
(ted:. based 00" _-.:I
'*- ditK_ d HCh polar pIIa(._ """'" _ tit rhos ..,....... ~ <'M>
Ioul.... un bf' _agt<l CIt_ ~ >igr\lI ~ <rct>-
_unbf'~_C'-")_~ . Sl
124 0"." Plots

1... .. I'I ' Ii...:l. The... IKh lliqUh illclu<k Virmall'mbe R< ~ .. li,," 0'. ,·'... n bell "'. fo,-
""' rd ""'I"".... ' -..clor s (..... ( 'hap l'" 13 a nd Appt'"di~ SI-
Each ro lor """,,"a nCt'. 0 ' mod<-. ..,II ap P"'" on .. "'>de plol "s a n ..m phlud..
p"a "" ..; Ih a ll a s<ocialt>d pha... lall in.....,a..... On a P'~" r 1'1,01. t'8ch n" o<I.. ..;11
a w a r aA a 10<'1' III Ihe ..olar p lot . Som .. mod ,'s l.." d I" 1',, 0<1 "'-.... ..... ry ~maJl loop<
i" a polar plot. II " .. aft' d, scuss llll\ sill"ifi"'lIlt "",,' _ ..Ie. of r h.. ~~"'I .. m,
whk h uSWllly p mdu I... It.. Ioop< on a pola r plot.
PilluW 7-':1 , hows data from an t'Xpt'rim..n tal ma..hin.. " ' Ih d ....rly dpfint>d
fi",t (bl" .., a"d ",·<.'" ,d ( rt'd J ",,><I..... :'\01.. Ihal Ih.. pha.... lall i"'· """ in a d irt't;-
1'0" " ppos it.. t" ,oUlI,,)fl for ..a..h mod Th..... ,s a l..., a s mall m I.. a l jU..I ' I\'t'.
100 lO rpm Ilh.. "m,,11 It..,p o n Ih .. pola loI l-
l h.. .. ~ .,, _ ...... "I'lwaron Ihis pola. p lO! a. lwo la r!U' 1t...P'" Eat·h 10<'1' is a n
i"''''pt'n. k " t _ ><k ......h ha~ a low s peed. ......'" a rK"\'. a",1 hil\h' '' Ilt't'<i """'lio n
t haI has a " aPl'",.i""'l.. W, '.lU". a nd 1110" rela tion ship.. .....P''<1 i....ly. I" Ih.. h..,,,.,,
s pol for thut moo... Tht- r,tAl n,od.. h..a.'Y <pOI i. ma rkt>d wit h a blu.. . It,t. l h.......•

o "~~"""

f '. l ..
,
J
I
'.
~
]" •. , .-,
, • -,- ~- -
»e -
,
•
•
••
-...-.. -
" ./
,

"
I
.
j , -, ", .... pplul «Mo

••• '000 2000 lOOO eeoc SOOO 6tIOO 7000

.."""""""''' ' ' '

FOg..... 7-9 A two--rnoer """"""" from ...... ~ ......... I0 '....cho~. n...
fi"r ,bI",,: or><! .1>< "",end
1' <"<1) modo-> 'PPN' .. puks in rl>< _ pial or tfiSO.od St SO rpm .od as mulloploo Iool>< in tht'
poIor pial hch ~ .. M on .. soci.nt'<l p/la<l' .... of OW'''''.,'''''''''
tW,~ fif<I motIt
1><AY'f'l"" Ii ...., _ • _ dol....-d .......... ond modl' hNvy 'I"" , , _ witII" tt'<I dol.
T........ Ii ,,1<0. 'm p/>a>t' ch.tIQO' "<o<",t'd "' ~ n " ........ """"""'.. or >lKjtnly".... t OOJ rpm
 Chapter 7 Polar, Bode,and APHT Plots 125

ond mode appears to start at a point slightly away from the polar plot origin
(where the color changes to red) , where it adds to the residual response of the
first mode. The start point of the second mode can be mentally shifted to the
origin of the plot. When this is done, the second mode heavy spot at this trans-
ducer location is approximately at the red dot.

Polar plots from multiple planes of measurement can be used to estimate the
different mode shapes of a machine. Figure 7-10 shows IX, compensated polar
plots from vertical transducers in two planes. The measurement planes are both
inside the nodal points associated with the bearings.
In the polar plots, the first mode forms loops (blue) that have the same ori-
entation in both planes. This data corresponds to the approximate mode shape
in the top diagram. The shaft is bent into a simple curve, where each end of the
shaft passes next to the measurement transducers at approximately the same
time. Thus, this is called an in-phase mode.
The second mode loops (red) have the opposite orientation; the high spots
are 180 from each other. The shaft is bent into an "S" shape, where the shaft
0

pa sses a transducer at one end half a turn later than at the other end. This mode
shape is an example of an out-of-phase mode.
Mode shape information like this is very important to balancing, because
weight placement can influence several modes simultaneously or have little
influence on a mode if the weight is close to a node. Knowledge of the mode
shape is used to select the size of balancing weights and their axial distribution.
Measurement of modes can be tricky. The measured amplitude of vibration
is determined by the motion of the rotor and the motion of the measurement
probe. lt is possible for the rotor and casing to have large amplitude, in-phase
vibration in a particular mode, producing a relatively small shaft relative vibra-
tion signal that may look like a small system mode. Casing transducers can help
identify such modes. The phase relationship of measured modes will also
depend on the axial location of the measurement transducers relative to any
nodal points in the rotor. In the first bowed mode at the top of the figure , if
either transducer were moved to a location on the opposite side of the bearing,
the nodal point near the bearing would produce a phase inversion in the polar
plot. Knowledge of transducer axial location and the likely location of nodal
points is important to establishing the correct mode shape of the rotor (see
Chapter 12).
Specific machine speed points on polar plots from different axial positions
in the machine can be linked to produce a gross estimate of the mode shape of
the rotor. This method is similar in concept to linking Keyphasor dots on multi-
ple orbits to obtain the rotor deflection shape. However, unlike the orbit, polar
l Ui 0.1.. PIon

•..... - " - 1- ·...........

-I .. - -- ----IIl.+~-«l

,.'
. •.'
-
,...
~' V
'80'
" "
"om
f Og ~'" 7-10 Pl:OM pIols d''''''''nl mN.... ~m "'ll pa......"' •
m..ct......Two ell"""" """""..
~ vi"b":"'" r.,,""""' ;n bh.. .mel",..
"",on<! mo<!<' '" _ ~ ~ ""'" lot- NCh mo<!<'.", ,how"
!NqtMTIS oIt.... ~ ~ """""' b !hi, moe....... ~ <hown at
,.... ,op. se.-",.. ,~, "" • ,'""p...... d""",""',
 Chapter 7 Polar, Bode,and APHT Plots 127

plots only provide an approximation of the direction and amplitude of the

response. An unfiltered orbit displays the actual dynamic position of a rotor. A
polar plot, however, is created from filtered, no t unfiltered data, and it displays
the vibration vector as measured by only one transducer. This produces a rea-
sonably accurate estimate of the rotor position at the instant of the Keyphasor
event if and only if the orbit is circular and predominantly IX. However, most I X
orbits are elliptical, not circular. In the diagrams at the top of the figure, the pic -
tures represent an approximation to the shape of the rotor at the moment of the
Keyphasor event.
Finally, several possible mode shapes can have similar polar plots. We typ i-
cally assume the simplest mode shape. Here , knowledge of the machine con-
struction, together with knowledge of all lower modes, must be used to arrive at
the probable mode shape. It is possible that the mode shape could be much more
complicated. On a more complex machine, mode identification probes should
be used to obtain more information at different planes in the machine, or exist-
ing measurements can be correlated with a good model of the rotor system.
Polar and Bode plots are used to display startup and shutdown data. It is
very useful to maintain a history of these plots so that data from different start-
ups and shutdowns can be compared. Slow roll vectors, resonance frequencies ,
and the shape of the response should be checked for signs of change. Changes in
resonance frequencies imply changes in rotor system support stiffness. Any
change is evidence of a changing machine condition and should be investigated
for its significance.

APHTPlots
Amplitude-PHase-Time (APHT) plots are a variation of polar and Bode
plots. The vibration vectors are plotted versus time (trended) rather than versus
frequency. These plots are designed to be used for long term trending of vibra-
tion vectors while the machine runs at constant speed. Because of the possibil-
ity that the slow roll vector will change, APHT plots are not normally slow roll
compensated. APHT plots can display any harmonic of running speed.
Vibration is a ratio of the applied force to the Dynamic Stiffness of the rotor
system. Changes in vibration vectors mean that the applied force, the Dynamic
Stiffness, or both have changed. Thus, a change in a vibration vector (either
magnitude or direction) can provide early warning of a developing machine
problem.
Figure 7-11 shows a set of 2X APHT plots from a vertical reactor-coolant
pump. The data was trended while the pump was running at 1187 rpm. For the
first two months of operation, the 2X vibration did not change significantly.
Then, in late October, the 2X amplitude and phase lag began to change. The
am pl.lI\>& I"l'*d>rd a muilll<lm 011 12 ~"""""" and, ohortly .ltn" that. 1M
pMW l.o@: ,1IOCn'a-t d~_ Ao tIw Yiootioon d«noaood tIM' pu mp ....~ V,U I
dooo.-n ..... Wpecud. poMoihty I .... fi~ tilnC'. mach""" wu otoppftI ~ ci
_ _ ,>bration. It ..... """" rntal11Pd. but otoppo'd &pin • ,hart linN' IatrlO t:pon
1l"p"n...... 11w pump Wft _ found 10 b.. CI'aC'bd
In un. .,u mplf'. u... 2X ~ cha~ ~ t .... undrrly; ns mang., m
~mic Stdfnno. An m tnatin(l: a>pKt ci t ltis data is tbM t .... AJ>I-tT plot: Ioob
""'Y much hkr • da....: . ta rtu p or v,utdmm Bodo- plot: ci a madli... !P"II
thl'OUf(h.~ . y<i 1M ....mi... Op....tnl ll a ClIfIlUnl "J""""d durirlf!: thiI
"nl,", poriod In. t"ad of lho m..,hiM pa~5i"ll: Ih""""". ,""na""" Wllh <01>&"11'
ill,('1X'fd. A,.,...........,..,
ptJU«l r1lrtNgh twia! op«rU"'lJ .,-J .. h ~., th., rnac/tinoo
_ • • 1 con. tanl ' peN_Th.. im pl.... thlt .,ithn" lh.. rolo. m..... inO'O'. onI ill. ·
mal ical ly (u nlikely l or l he ro to r .~l.o'm . t lffn...... d ecn • ...! dr . matically. In fact•
.. th" ("rac k proP"ft'llni t hroullh lhe .h.ft. th.. .h. ft .... nding . t iffne.. dl'C.... ...-d.
lo.....ri"ll l .... na tural fro<j" ..ncy. A~ I naru ral fro<j"''Ilcy pa""'-'<l through ~
ru nninll ' f""nI. it ..... ... cilO'd by Ih ,m ml'l rJC . ha h . t iffn"•• d u.. 10 I"" ("flic k

."-
f w-
! ,~ .

1 :1
",= =====
'w. .-
1 ..
I .-
j ,-

'•...
F9n7·11 . ....
...... Tht _ _
ponoo:lThr!211.b
...... .-.1Sft
lS Soop "Oct

- 11_

dl X .PMT""'""d.-... _~

_b
'_
• Do<

<Il1mt._"'"_io......-.....
clr-..,....
' . - .............
_ _ ."" " . - .. """,
_~~

~_ttwougI'I_
__
...
1_ Ch.ptr< lil In n.......,...
Ii...Y'I..... .,...~ from . «>ndi1l0n tlv
_ ..raI fr-rqunx-r ..... ahoo... nne. ru tulInft: "P'""'d 10. cond ition It ....
........ t>on ru nnirut "f"l""d. Th .... non rutul~ 'f'"d .... in.tially J>do.. . rw·
..raI """I~. and n>dod " p ""'- It.. n.aI: rr..., i"'~ produrintt: Ihu m p ;-
l"O" and pn..... chItlj[n ~lC of ~.

""c.-punc:tr ~ Pion
A• • hlatonall dal........ ~ for • ....m,............... ""-'ba\'ior , ..n btr
drli""'<1 MI<ilIn... in,...od lv.lIh ohoukl hln ; bra! "", ,'f'<1on. OW.,.., ""J>O'III-
abIor for oim;lar cond itioN ,,f JPH'd .nd Ioaci 1 " bf ll •• n ,'0'<10,. 6houId ploc
in !tv l(l'1I<"ral ....... on I pol l r APII r plot . I nd boundl ';'" can btr dt-fined
th l l .. ndo o.lJCh ...~om.
SI..-h .. riot i. aloo ....11t-d In ~t"'ce rtgiotr pIol . A I.nc.. , ell:".n. "",,n·
ll in Ih... no.mal boo-h.,i", o r Iht· ,·ih,a l;"n '-....;10. ht-inll: m t<'d Th...... . .... " ....
..lIy IX o r 2X '"<'elor .. hut Ih..y Cl n .... filte wd to . ny o.ti.., of ru n ninll . pt"t'd.
ACC'l'pt ""l(ion. ca n .... d.. fint>d for di ff..rent loadI.. .1......Is. .. te.
Ftg" i -12 . h_.. I n ..u m pk of In ~ ..,.ion pIol. " ' Ih th.. "'tlion
(wdl of normal, hi.OI"'''' be""'''''' for Ih il machi at Ih il 1 poin t.
"P""'II ,ng "I'"""'l lnd <>p<'fJoI l"/l cunditiot1... the 1Cn'J'U "'~ .. dcofinnl
"'TO . 'I ~ of,ibratioc ampllt\Odf' f""" i .Ho 110 lOrn ",,1 ).010 4.3 mil "" lind
phaoor lag from 95° '0 ) fl(!". fbi. ~ rni~ ,ndudf' ~l'.caI bef>Iriof for IlfMl '
f'f1tIlor .at d ifft'l"nll lo.ttd... ~~ I of Ilv ."ibrltion W'C1,'" out ofllv ar:nptl~
..,c>on cun.. i1..t .... abnr................... and .........1d be i_ ~1Pd..

f ..... 1-'~
""""""
"" ""'~ ...,.... pQ n..., tl<l
~_ d .......... """-" ""
"- ,.
/!
~ •.
,... 1 ~ .,."'" _leo- .. _ ~ PO"'!- •
The ~ .. _ "" ,... '""9" d ......,."'"
~~ ........ ph llnPO"If' O<II_'hO O«"'P!-
'1'1(.. '''lIOn
tonstitIJI btlQffftOl ~ .....
. - bf' .......1'9"'«1
,.. •••
-
130 Data Plots

Summary
Polar, Bode, APHT, and acceptance region plots are designed to display vec-
tor data. Polar and Bode plots are used for startup and shutdown (transient)
data, APHT plots are used primarily for long term trending during steady state
conditions, and acceptance region plots are used to identify response outside of
normal, expected behavior.
Polar plots display the locus of a set of vibration vectors, in polar format. The
phase angle is measured relative to the transducer location, in a direction oppo-
site to rotation.
Bode plots display the same information in two, separate, rectangular plots:
phase lag versus frequency is plotted above, and vibration amplitude versus fre-
quency below.
APHT plots are similar to polar and Bode plots, except that the data is plot-
ted against time instead of speed.
Polar and Bode plots are usually slow roll compensated, using a slow roll
vector selected from the slow roll speed range. The uncompensated Bode plot is
used to identify the slow roll speed range and a suitable slow roll vector.
Polar and Bode plots can be used to identify resonances and resonance fre-
quencies. The Synchronous Amplification Factor (SAF) is most often measured
from the Bode plot, although a polar plot can be used for this purpose.
The IX polar plot is most often used to identify the location of the heavy
spot for balancing purposes. Multiple plots from different axial locations can be
used to identify the mode shapes for different resonances.
APHT plots are used for long term monitoring of nX vibration vectors. The
polar APHT plot can be used to define regions based on typical vibration vector
behavior. Such a plot is called an acceptance region plot and can be used with
software alarms to detect significant changes in vibration vectors.

References
1. American Petroleum Institute, Tutorial on the API Standard Paragraphs
Covering Rotor Dynamics and Balancing: An Introduction to Lateral Critical
and Train Torsional Analysis and Rotor Balancing, API Publication 684
(Washington, D.C.: American Petroleum Institute, 1996), p. 3.
 131

Chapter 8

Half and Full Spectrum Plots

IN PREVIOUS CHAPTERS, WE HAVE DISCUSSED the use of filtering to obtain

vibration vectors. These vectors are filtered to mul tiples of running speed, IX,
2X, 3X, etc., and provide us with information about the behavior of a machine at
one of those particular frequencies. This information is presented to us as the
amplitude and phase of the vibration at that frequency.
However, machines can vibrate at many different frequencies simultaneous-
ly. These frequencies can be related or unrelated to running speed and include
both sub synchronous and supersynchronous frequencies. Since these frequen-
cies are associated with the operating condition of the machine, the machinery
diagnostician must have some way to determine the frequency content of a
vibration signal in order to make an accurate diagnosis.
Vibration frequencies sometimes appear as a series of harmonics. The series
consists of the lowest frequency in the series, called the fundamental, and a
number of frequencies at integer multiples of the fundamental. In a typical
series , the amplitude of higher order frequencies will decline rapidly. To avoid
confusion , we will define a harmonic as any frequency that is an integer multi-
ple of the fundamental. The first harmonic is the fundamental, the second har-
monic has a frequency of twice the fundamental, the third harmonic has a fre-
quency of three times the fundamental, etc. Often, the term harmonics will be
used as a general term to indicate integer multiple frequencies that are above
the fundamental.
The fundamental vibration frequency of a serie s can be any vibration fre-
quency. Often, the fundamental is IX, but it can also be any subsynchronous or
supersynchronous frequency. For example, a series could be based on a funda-
mental at VlX and include IX, %X, 2X, %X, etc. In this serie s, the IhX is the fun-
132 Data Plots

damental or first harmonic. IX (= %X) is the second harmonic. % X is the third

harmonic, etc. Such a series can be generated by a Y2X rub. as we will see later in
this chapter.
While the timebase and orbit can be used to evaluate frequency informa-
tion. the most convenient plot for this purpose is the spectrum plot. The spec-
trum plot is created from the signal of a single transducer. It is the basic display
of a spectrum analyzer and has been a mainstay of machinery diagnostics for
many years.
In the past few years. an important new tool, thefull spectrum plot, has been
developed that uses the signals from a pair of orthogonal, shaft relative. vibra-
tion transducers. The full spectrum plot compares to a conventional spectrum
plot in the same way that the orbit plot compares to a timebase plot. The full
spectrum plot contains much more information than the spectrum plot. includ-
ing vibration precession direction and orbit ellipticity. It is so important for
machinery diagnostics that we now refer to the conventional spectrum plot as
the halfspectrum plot.
Spectrum plots are used to identity the frequency components that are pres-
ent in complex vibration signals and to trend changes in the amplitude of fre-
quency components. These frequencies include running speed. multiples of run-
ning speed. line frequency electrical noise, gear mesh frequencies, gear defect
frequencies, rolling element bearing frequencies, and vane and blade pass fre-
quencies. Rotor system natural frequencies that are excited will also show up on
the spectrum plot. Subsynchronous frequencies that are often associated with
fluid-induced instability, compressor rotating stall, compressor surge, or rub,
and supersynchronous frequencies that are often associated with rubs and shaft
cracks can also be identified.
In this chapter, we will start with a discussion of the complex vibration sig-
nal, its frequency content. and how that information is displayed on the half
spectrum plot. We will then discuss some technical aspects of spectrum signal
processing, followed by the meaning of and the enhanced information content
in the full spectrum plot.
After full spectrum, we will discuss plot formats th at present spectrum data
versus speed and versus time: the spectrum cascade plot for startup or shutdown
data and the spectrum waterfall plot, which is used primarily for steady state
trending. Both of these plots can be generated in half and full spectrum formats.
These plots are similar in structure, but have important differences in their
application.
The H.. 1f Spec tru m Plot
1.. 'l's ojart wil h a machin.. o p<'ral ing al .. ron~tant sf"'t'd_ Rt,.;au ,," o f a o "m-
binal ion of u nbal an<:.. a nd o lh e. effects, t he mach ine i, vibrating in" com pli -
ca led " ."y. Th... un filt....ed ,i bration ~ignal f,o m a t ,a nsd ucer o n t bi, mac hin...
will ....""a l all o f th.. complexity of the ma ch i.... ,ibral lOn Iha t fa lls wit bin th...
b<>ndw;"llh o f I .... d ata collect ion sy~lem. A filtera1 signal from Ihi, t ra nsd lK't"
will bo- a sine " .,,"" a l Ihe fill" , f"'"ll1<'ncy with son ,e a m plil ude a nd p h.....
l'iflUre 8-1 sh...... a rom plex. u m..b<>... ,-;br.. lio n sig nal (mI) " 00,, M" i.... o f
si.... wa'.....(bJu..) tha I add up to prod'Ke the t im""" 'l<' sig nal. U. ing the fo ur ,e,
t ransfor m. th.. f""l....ncy. am plitu de. a nd pha,.., o f th""" sin... " ."..... (called com -
f'O""nls ) ca n bo-compu led from a d igital ...ml'le of the o rigina l ti meba... signal.
The ph...... for ..""Ii sitlnal is meas u.-ed with lP1of"'Ct to th... Iri!'JI"" sillnal t hat

~ /.

- I
VV\N I,
V\J' f,
_.lW
-
• C,-""
- ". .
- . ., . .
'". ---

lLl~
s.
-
J,

"'\l"" &- 1.r_.oo ~e<lIJ<'O<Ydornu> ""ploy 01. compl.. ~gnol Tho Four... I,..wo.m
opplOd to. complt.. period ic. ~""'_ >ignol (' od) ptOduc~."" of ~ ,..., WOVO';,O<
J, J.

~ ""'" ~"'" componoont; ... 9''''''''''' at """"" rq,' by odd,,'11 tl-o<! ttlird di,.,..,sion,
frtoq-.cy '"'" pIOI is ""at,"" to ~ a !W<>-d,_ pk:o; 01 ~ __ 1r"l"""'Y. II<
tl-o<! ~. w 'P"'C!..... plot. ~ componoont; """"PPNI ... _ d --,,,oj ........ ""'"'" ~
~ ~ PNk-to-PNk _p1iMln d tl-o<! ~nol ~'" ~ tl-o<! tlrJ"oing (ph.t;<1
information is lost, (i, rIOll'O"'bI< to ''''''''''truCl ~ Ol'J9"YI-rorm from ~~"
din. 'P"'C!",m plot
. la rH II><> 'Ill mpl ing J""'-- a t IIn'e '" The Four i..r Im nsform OUtpu l i. "'I ui\1l'
1..,,1 In Ih.. o ulpul " r a ...,ri.... " r band -pa-"l fillers l hal ha,... bf't>n Sf'! 10 inl "g<>r
m ull ip l.... ..r lh .. 1.." ....1 f""lu..ncy ~ignaLJi. (In pmet ,.,... Ih.. Io", ... 1 di~pla~"f'd fr..·
'Iu,"".:y ..r a ~pet:l ru m p lot ...-ill w;ually be helo", Ih.. I0......1 m..a~u red 'ibrar ion
fre<I" "" ' 'Y·)
The comp",..." l . ;n.. ",'a\'... in Ih.. figu r.. a... pJol ~ o f d i~plac ..ment ,...<su,
l im... If ..... aho plot Ih..m w"'''~ fn-q" ..nry. " ... can e ...al.. a th ......ti"'..n.ionaJ
plot (u pper rillhl ). Wll h Ihi. pefSpet·t;~... "'.... '..n ""'" ""ch ""mpon..nt ·s f""lu..n-
cy. am plil ude. a nd pha se.
Thi~ plot i. ""m.....hal n 'pel il iou. alon g II><> ti me u is. M r all, " ... know
l hese a... . in.. ""a',,-,,", ",~....." Ii"".. "'p"a ling IIw sa me d.. ta . "' a nd m 't'r? If " ...
rOla t.. Ih .. plot so tlial Ih.. l im.. u i. ..i"'pJ'<'a ..... "'... ...l1J _ a 1",'<H.Ii meMionaJ
plot of a m plil ud.. '-""'''' fwq" y (bn Un m righl ). :"Inl.. tha t Ih.. comJ""",m sig-
na l. now ap pea r a. a ...ri... of rt iea l lin..,., ..ae h lin.. ... p."""nt. a si ngle f...•
qu......y a nd it ~ Ii..illbt is Ih.. am plil ude of Ih.. . ig" al. This is II half5p<'Clrum pM.
R~' liid ing Ih .. l im.. axi.. ...... h.n-..obta ined a ...I.. li ly . impl.. p lot formal t ha t
an....... u. to d .... rIy tli.. f""lu..nci"" a nd amp litud of II><> co mpon..n t ~i!l-
naJ•. l 'n fo n u na t..ly lian hidd..n a n impo rtant p i of info rmatio n. flu-
phase ofth.. comp''''''''1 " WIlli•. Bee..u", o f Ihi•. i. not J" _ ibl.. to .....,.,n"'ruc l
I.... o rillina l ...a....fo. m from Ih.. co mpon..nt f""l...." c;,'" ..nd a mplil ud.... Thi. L'
a d ra...-back o f a lllyp<' s of . peet mm plot •.
Th" amplil ud.. ""a l.. u n he " ith ... lin r or lo!tar illimic. I.Qlla. itlimic sc..bng
i~ use ful ,,-li..n tli is a need 10 compa ignal. "'i tli liol h ,...ry lar!fe a nd ...... ,.
' mall a mphrud Tlii. scaJing ...iU dearlydispla~' all sillnal compon..nt-sand th..
noi ... lloo. 1I"" r. ", h..n a ppl ied to rOla l i"!l mach in...-y r" . I~ rilh mic
""'li nll ma kes. il "'0 '" d ilfocuh to quickly diseri minate be"' n .illni fica nt a nd
insigni fican t ,ibral,on co mpon..nts, Lillt'ar ..,aJing has I.... ad,....I'4I" of show·
i"!l l h.. most signi f,ca nl compon..nl s; ......a k. in.igniflCa nl . a nd 10" ··1"'.... no ise
co mpon..nt. a... p ..ally red uced in sea l.. o r ..liminal ed. Beea " ....o f it. ad'·an t.. ll"
for mach in..ry w" rk. li"..a. scali ng " i ll he used for tli.. 'P"-'CIru m plots in this
boo..
Th.. fr"'l"..n'1' ..,al.. ca n b<> displa~'ed in ""'.... a l fr"'lu..n'1' unils. Mosl s J"-"C"'
lru m a na lyz.... d ..pl..~· Ih.. f""luenry in h... tz (liz) . Thi' is u",fu1 ,,-li..n co mpar·
in!,: ",ach ine vihral ion frequ..ncies 10 line fnoqu..neies. ~uch a s in ind uction
m" lot Of .. ..~ m lu rh i..... !':.." ..ralot diagoo.tiC1>. <'o m.....flwa... pack.og... ca n dis ·
pla~' "n il~ o f Ib .. CJ, m, o r ortt..", ur runn ing .J><"'d. Cp m i. a ",ry co m '..n i..nt unil
",Ii.. n " 'Irk ing wilh mac hin..ry.M a '-'-.... ii ,s ..a. y 10 com par.. a f""l,,,,n'1' in cpm
t.. Ih .. . " n n ing ' I....... of Ih.. m..chi ne in 'I' m. Spet:lrum plot.l t a r.. displa~·ed
in ord.. r< o f run ning s pt't'd ( IX. 2X. et c.) a... a lso eas~' 10 inr..rp l.
T"d",i<~ lulX"5
Wh..... on Ill" rf...... . "f""<'1rum ploI: .API-" oimpa. t o inlt"opl11. it ..
impoo;.t_1 10 und and Ib hmit.lion~ and 10 IW'OfIni:N' b.d Of q"..,.t..........
d.I&. n.... d i'lCWoOo..n ....... io .l>rirf ...m rnary 01. U~ ITItMoI ,mP"""U"t po;nl ..
Thor r......... t n~form (. nd it. cnmputahonal cou";n, lhor w t Four""f I ....n.-
ror-m. ,..- F FT ). ~ mn thal l hor unfiltnrd. , 'ibr.tion -.na1 hu aho~.. ~.nd
,,'ill ....'~.. hr. unclv.npod. ln 01 11... word<., thor ,.... \1hra1J011 Ol:~"""" not
rna",",.•nd thor oip>aI ...,....t. in nact!y tbr ..,. for"""- Th .... ~mption
"' an aJ...q""I.. appnn: i~ht'fl for mo<.( madl,,,... 1M """,,.1.. at a "'''ad:-' .tat..
opn'd or cha n~ ... ~ .Jt...t),
Th i" ••"" mpt ion u n b....k down badly to, madl' ...... Ih.ot ... ['<"Ii<'ncr . 00 '
tknly cluo n~"'jt , 'ibrat ion onnd" ion. (.och a . ' "lush..... o r ..",,,J <ilipp,.....) o r for
mu h",... Ih at a""..I..r. ' " (,nd uel ,o n mo , o ", ) nr d ...-..I.., a' .. ~ ..')- ra pi dl,·.
Sp..dm m p lots u n...', lh.....•c1 ",·um,lanc-e. un ha i~nir"an t ...-rt... in a mph ·
1" .1.. a nd freqUt'fl"Y' Sp<'<1 r. or rap tdly c hangin" t. can l'Xhibi, b ro.tdt'flrd
'J"'Cf ral hn... that . ... "!l",fic.ntl~' . hilled in frequ.. n.cy.
In pract ;e.,. lh.. r• • • l'oun..rl ..""f"rm (FFT) n lcuJ. I.... Ill...pt'ctru m fro m •
...mpk rraYd. ..-hi<h «'fIUin . ;I, "p«if", n umbrr 01. d~i1 a1 ..,.,...form samplM.
IIoaIUM tnl' "",,,pi.. ft'nInJ h.u ;I, f""to. Irnjllh. pali 01. In.. .lttorithm in..,l.....
n l..nd" '/I iu wngtll t,- ...........th- "TIppin!! In.. ~nal ...... nd .... it ...1f. rnIH.,
In.. nJmlbO'f ol ~ of In.. ..gnaI nactJ,- matdtn In.. lrnjph of I"" ..mpa.
.-..cnn:I1..1UdI ... hnpn>baotk\. Ihrr-..,.;Q b<'. di oconti nu,lr at tt.... juoct..... n. ..
d,ocnn tinuity introd~ _ l ~ 1into tt.... oprctru ........ iriI broalrns l hor
froqU<'nl;}' [;......, n-d....... IN calculal.... a mphludn. . nd illCl'<'. on tho- no .....
II"..
1 n'" pmbi<'m ill Il'duad ~ .~';",g..J, ..'in<!ow function appl it'd 10 l ho-
Yrnplr non>rd form tho-"'It"" tn .<TO.t t he rndpo;nt • . ~ in a ~ua1
and o.tnoofb "",nllft; n... hu Ibr .. fJrct of n iminalinilllbf' .tq>di!lCOntinuity in
IN ...t...........i!!na l. D.-prndinfl on I"" ""fht,.. .... OO'W'fal "1'"
functionn< aw ,,·ailal>le• ...m ,.;Ih il. "",. ••ham.lt... iOnd di... d.-..nl./I....
of " ,ndt...i n/l

[lIlf n' .. ; nd....'inl( h",.,lion< ..'ill ....." 11 in d ilf.....nt .·.I ,,~< of . m plit OO~ a nd
fn.tJ ncy ...hen aw li 10 Ih.... m.. da ta. A Ilan ninlt wind..w fu nction i. "," al·
Iy the be•••" mp rom i ror rolali njl, mach in..')· ...... I... I''''. itI in/i /lood a mplil ud~
and fn-q" ..n<'Y n-.olu li.. n,
Figu.... 8· 2 . 00 , 'ih'.li"n ..·.wform. <:on'ain illfl. • m l1lu ....of I X a nd ,"" X
fA'<!.......·ift. a nd "' ""m p[...,of hair.pt'ct llJm plot....- , h.. w p 'J><'CI rum plot
(I1M' middl.. plot in Ih.. f'/lurr). no windowi"fl t" ncto"n ,,- appIio-d 10 I m-
plr.O'<'Of1i :"ot .. lhal lhe 'i...cl rllm · Ii ......• arr "'.. <p>1I.. 1iDH; ,n ol<'1ki. ~ ...
..".... fm,I.. ...;dth and..-Klm . 1 IN boltom. A unaII noi... floor io ah;t> ''ioitk.
1'lli< io .n ..... mplr ol ........ d .... 10 In.. di<con ti nu ity .. tho- ..... pa. ft'COnl rnd ·
 ,.
,.
,.
, v,
F;o,Iu... a-J.A _ ", pIoI-...th r..o

IT-
.... ~ of I'IoIf <pKtTurn pIoI.. fO< .....
_ . plot..., wind_ function wo,
. pp1;..d to I mp. ' K O'd. n...
'PO'C. 'M
lnlm ' i"..-

!he bottom-
tIOl qUn., ~ ;"..eNd
!hoy ~"'" !I<lm<'OO.... _

n...
. <>:l - . .,
bottom plot """'" , ....
~ m colculot«l U,;"g • H.nn"":!
dow, n... l n X 'PO'C"oj ~ is ...... ,~.nd
""n-
""yl><'r.• <>:l , .... ,...."".1 '" 1I0oI...., w-
,
!'
•
,.
, , , , ,
.
.....1y<;j;..PP'!M«l- "

•
. . . 09_
F' t<lUOftCJ {'<om'

'm
8: 6-
!
!' .
},
..
" , , • , •
o..q_y okomJ
 Chapter 8 Half and Full Spectrum Plots 137

points and the limitations of digital sampling. The bottom plot shows the spec-
trum when a Hanning window is applied to the sample record. Note that the Y2X
spectral line is narrower and higher, and the residual noise floor has virtually
disappeared.
A digitally calculated spectrum consists of discrete frequency bins, or lines,
of finite width. The width of these lines, the resolution of the spectrum, is an
important consideration. The maximum resolution of a spectrum is determined
by the ratio of the spectrum span (the range of displayed frequencies) to the
number ofspectrum lines that are displayed:

. Span
Reso Iution = ----"~--- (8-1)
Number of Lines

The spectrum plot is a collection of these lines, arranged side by side. The
width of each line is equal to the resolution of the spectrum. For example, a 400
line spectrum with a span of zero to 200 Hz will have a resolution of

. 200 Hz
Resolution = 0.5 Hz/Line
400 Lines

Thus, each frequency line will, ideally, represent only the spectral energy in a 0.5
Hz (30 cpm) wide band from 0.25 Hz below to 0.25 Hz above the center frequen-
cy of the line. Accuracy in the displayed amplitude and frequency of a spectrum
line will depend on where the actual vibration frequency is with respect to the
center frequency and which window function is used.
The limited resolution of spectrum plots means that there is always an
uncertainty associated with any frequency we wish to measure. In the example
above, a frequency actually located at, for example, 99.75 Hz, is displayed at 100
Hz. A spectrum plot with poor resolution will have a corresponding large uncer-
tainty in the measured frequency. Even good resolution spectra may not be able
to discriminate between vibration frequencies of exactly Y2X and 0.49X, an
important distinction for malfunction diagnosis. Higher resolution (zoomed)
spectra can help, but orbits with Keyphasor dots can sometimes be superior to
spectrum plots for making this kind of discrimination (see Chapter 5).
Noise can be a problem in spectrum plots. The Fourier transform of a spike
is a series of spectrum lines extending to very high frequency. Thus, anything
that produces a sharp corner in the signal will produce a series of spectrum
lines. Sharp corners can result from shaft rebound at a rub contact point or from
138 Data Plots

an inadequate sampling frequency (causing a corner where a smooth transition

really exists), among other things. Spikes or steps in the signal can originate
from electrical noise problems or from scratches on the shaft. Spectrum plots
are calculated from uncompensated waveforms, which may contain significant
slow roll or glitch content. In general, the appearance of spectrum lines as a
series of harmonics should be viewed with caution. Use timebase, orbit, or cas-
cade plots (below) to validate the data.

The Full Spectrum

The half spectrum is a spectrum of a single timebase waveform. The full
spectrum is the spectrum ofan orbit. It is derived from the waveforms from two,
orthogonal, shaft relative transducers, combined with knowledge of the direc-
tion of rotation. The information from the two transducers provides timing
(phase) information that allows the full spectrum algorithm to determine the
direction of precession at each frequency. Because the timing information is
critical, the two waveforms must be sampled at the same time.
The full spectrum is calculated by performing an FFT on each transducer
waveform. The results are then subjected to another transform that converts the
data into two new spectra that represent frequencies of precession, one
spectrum for X to Y precession and one for Y to X precession. The last step uses
the direction of rotation information to determine which of the spectra repre-
sents forward and which represents reverse precession frequencies. When this
process is completed, the two spectra are combined into a single plot, the full
spectrum plot (Figure 8-3).
Figure 8-4 shows the relationships among timebase waveforms, half spectra,
the orbit, and the full spectrum. The Y and X timebase waveforms and their half
spectra are at the top. The two waveforms combine to produce the orbit at bot-
tom left. The data used to generate the half spectra are further processed to pro-
duce the full spectrum at bottom right. Note that you cannot generate the full
spectrum by combining the two half spectra.
In the full spectrum plot, the spectrum of forward precession frequencies is
on the positive horizontal axis and the spectrum of reverse precession frequen-
cies is on the negative horizontal axis. Thus, for each frequency, there are two
possible spectrum lines, one forward, and one reverse. The relative length of the
spectrum lines for each frequency indicates the shape and direction of preces-
sion of the orbit filtered to that frequency.
Figure 8-5 shows four, circular, IX orbits, with different directions of preces-
sion, indicated by the blank/dot sequence, and different directions of rotor rota-
tion, indicated by the arrow. To the right of each orbit is its full spectrum. Since
each orbit is circular, there is only one line, which is the peak-to-peak amplitude
 Chap1... 8 HaQa nd f ull Spe<1ru m ' lou 119

fu l~ T ,. n""" _

f;'u.. t.-l (iOkoJ.Jnon aI , .... fun 'P"CfrUtn."" FFT i> Pl'tfonno<I on ~ach ..._ m ftcwn.n ){Y
tr. ",dlX.. p.ow. Tho ~ 'PO'C"" tUf "'""" "'b!Kt'" to .......~ ".n,1o<m 110.. g<'n<'< "'"
,..., ' - spK1" Ihol , ..,.~ ~ ~ ~ aIp".. ".,,,,,,,, nor 'P"CfrUm for X to Y p~
_ nor for Y to x p"", ,,,,,,""- Tho d" ..rnon aI ",'anon i> u"" to detftmi.... wtlich aI "'" 'PtC1t.
repo..enl> forwa,d pm:o=ion _ _ h ~"" ~ p""""" "'" ~ .............
 ••
. - 0 - -- '- ",.- - -
1..
1• "
i'
• r
,
lQ
..
~._
, •

•
. -,-- - - 7'"""-
•
" ",--------"---
"
1..
I·. "
h•
• • ._- n• " • • , ,
,
'"'
•:--C--;'--,_;-'Lc--i ..
cr_ •

_IX _1t1X

., .
"'0'''"'
"
-.. 1 0 1
'_«I
_ "_ " - Y II..q>m/
4
I
0

'
... " ~Cd"~
r......- _ _ ...., ~~I'>r __
,tM... 1O """"",. tI'oo 0tt0I _Tho ~ It\OI ptOdu<n tt.. ~
on >PK"'""'.Tt..'-
~. io
ptQ<1I... d 10 P'Odu«' I'>r W..,..,...... .1IonoIn "9/'(. _ ......... ~ "' ..... _
 Ckapt.. , ""hnd Ful SpKtrum PicoU '4'

- r-- .
..
f.- ~·. .. ft

'''''''.l 111.no..orcua. """" _

BJ - - ,
...

~U
-.
f..-opKtt"- <pectrIIm Iiow ...........
on rht """'ord _ rJ rht i*ll b ",.

- -
' - _ _ ttIOI_ .. do: _ ..

--_..--..--
~

~......".. ~!iPOCl'\om Iiow ................

lho _ _ rJ rht i*ll b .... ' -
••

~ -..... -U
e- no.. _ ~ l rJ_ ..
_ <I",. ,. ...
* " , - " , - . . ..
_"'" W;o
-. ~.....,.., """_ rht IiiEC IiO I rJ

E -I -...
142 IH II Plot s

of 1M ,><bit Rty-rdlno of lhe- d ln"Ctiotl of roe.. tio n. wlwn w dut'(1 lOQ of prr-
~ i> to......... d iin thr d,r<"ctJon nr ,"l ation). thr Ii""' .. .... lhe- poo<itn... u "-
and wlwn lhe- dirmion of ~ .. ~"""'" l "i'P<-l!.~ t he- duff1:M.... of l'Utll-
lion ] Il ...... thr ....pI;.... uis.
11..- ~Ih of thnr ciraIlar ori>it. if, ~IN to,. • " "Ilk- , ot.. lin( ._lJw
.....-t' M' Io-ngIh ~ lM pnk plll of l.... orbII. lOllO"-h&lftlw I..... hnght ..t"m
"V'" .....l. pnk·to-pu k p.. I ~ It roe.. l....1 11v frnj....""'Y ofl Ii""' .nd in
IIv ..lIrmion of ~"'" iIMllOIN tr,.lM liM.."s lh .. W<1'M' ...e..l l he- tip of
I,,",...dot will traa> oul In.. p.l1hof.1w orbit . Tiv run 'fW<1.um 1;"", drfinn Ih i>
rn' al i.... fono-.nl ,'O'Ct0<, . nd. con..-q ......l ly. II... orbo t;\s WII h In.. half 'fW<1rum.
lh . .... is no pha".. in form..lion in Ih~ full ........-rmm p~~ . .. ,.1M'- t-:.,-pha._ doe
~ ''''I ir'n i. arbit rary.
f .. r lhe I X. dliplicaJ o m il in fll1U '~ 8-6. I.... d ir<'<:tion ..rp............ion is X 10 1".
lh. Mm.. u I.... roLaI,.. " d i 1i" n. "" I.... orbil i. "n, ~ ""r .. u mpl.. of fo. .... rd
I" ion. h . full ' I""-"'ru ", '• • IX. ro,,,·a.d p ""irm rom pone nt a nd a
.."'. U , I X. nega li " ....__'.. n com pon..nl. The .'t'Cl nd II .... d..fined hy
on.. · h.olf the ,-.I of llv I"'"'li." and negal;". lin r IIv fuU spectrum a nd
..." .e.. in oppo<JI.. 01"·••1 ;0,,.. 1 "" .um oC th.. ......1 nd II ~n ...aU' rhe poeh
of In. oro.l. &ocau..... llW'y .." .. I.. in oppooile di•.-.:1 "", wil l a!I HNlU'ly ;odd
lo .nd WbUOoC1 from....,h . >1 ..... ~ per ......... ucion. drfinmll tlw ""mll"aJOl
..nd " , m i m i " " , au-o of ."" ""ptical orbit. In... sum and d lfffftlX'l' 01."" full
~ ... m Ii,.... ~ . . .... pnk ·'o>-po-ak,~ .. nd .....leI dtofi,.... 1M """"" a nd
""nor urs~
Th..... . 1w run ~ ... m Ii,," ."1"' nlw ror-rd.nd 1'O'WI'V,'O'Ct..... ,n.t
dcfirw tt... orbit. :>d"'h.o'l lw orw-n tar of tJw orbrt ril ipw ........n In lhr ~
.. arbouary_ Thr full ~ ... m contaIn.. informatIOn aboo.lI_. ..alplrctty. a nd
d,Oft""'" of p«a'S....... but bn:.............. if, no pha.... in.......... l thrn:- io ItO
inf""",,lion .bout ~__ A.. ori>it ..ith 4Il)'"oOml.tiorl. bul p: ,"Il t""
..me rih pticil,-. >iu. a nd di r«1iun of J'f0U"\..;.,n. ..i ll h.o." Ibr ..me tuB -"JM"C'
l ru m _Th " ... thr full opr'<'1rum i.. ,ndrprnd<-nl of 11lIMd<>Cr'f ",,",la' M'!\. Th is i.
d ill..,...n' fru... half 'fM"ct"'.... ..-hil-h dt ..... d.,.....,d on I h~ u .nod...." , or ;"n•• I;"n.
 I
."

"
I~
- j
f' ,
'
Y ,,
~ .

,oV-\

""""'' ,,<1'' '

f "",e H Fo....", _ '''-'' _ ttw
'P"'<'n.n\ cI on O<bt ,"""" . I ll.. fo< w... d P' ""~ '""" com ~
,....,.., ~"""on con"4'O"rnl- ~.um 01_ Imgth. c1lhe two full WO'CI"""
.nd."""'. . .
cI & I X. 01101'10"" <wb<t. 1M luI
IX.

I...... " oqwll0 ~ Ioong1h ~. *P""' ) 01 ,.... m.jo<..." cI ,.... "hpt",t 0Ib<1
(\~ fht. d~~ e b<-l...,..., ,"" "'''9'''' 01m. two M 1.peCUUm Ii_ Os ~.I ' 0
!IlO' Iffi<.ItI> (pNk-t<;>-pNkl 01 !he m..... . x;> 01_ ofi pl Koi ""'" :1). Tho """'''
9"""""" by ,.... sum 01 "'O.<OJnl«J<>1.hng _ t<n W'ilh loongl.... opq.>II to h,oII
lhe 'PK'''''''' lirIe Imgth. ~ 01 ,"" lock d ~ """,mo,,,,,, !fl''''' fuH '<>K-
""m. t ~ Os nat ~ inIormatl<lt> to un~ .<'C",,,.t,,,et , .... ""9,,,,,1 orbrL
144 Data Plots

Figure 8-7 shows the progression of IX orbit shapes from forward circular
through line to reverse circular and the associated full spectra. The relative size
of the forward and reverse line heights correlate with the shape and precession
direction of the orbit:

1. A single component, whether forward or reverse, means a circular

orbit.

2. The largest component determines the direction of precession.

3. The smaller the difference between the components, the more

elliptical the orbit.

4. Equal components mean a line orbit.

Complicated orbits will have forward and reverse components at many fre-
quencies. Each pair of components represents a set of vectors that rotate in for-
ward and reverse directions at a specific frequency. The most complex orbit can
always be described by set of such vectors and full spectrum lines. The lines in
the full spectrum represent the precessional structure of the orbit. Each pair of
forward and reverse precession frequency components describes an orbital com-
ponent, a suborbit (circular. elliptical, or line) with a particular precession fre-
quency and direction. The entire orbit can be expressed as the sum of its orbital
components in the same way that a timebase waveform can be expressed as the
sum of its sine wave components.
Figure 8-8 shows a complex orbit from a steam turbine with a Y2X rub. The
orbit contains Y2X. IX, and some higher order vibration frequencies. The full
spectrum helps clarify the complexity. Note that the IX spectral line pair shows
that the IX component is largest. forward. and mildly elliptical. The ¥2X line pair
shows that this component is nearly a line orbit. Also, there is 2X vibration pres-
ent that is also a line orbit. Some small %X, the third harmonic of the YZX fun-
damental, is also visible.

At first glance, the full spectrum might seem abstract. What is significant
about pairs of vectors with forward and reverse precession? It lets us easily iden-
tify key orbit characteristics that might otherwise be obscured. Precession direc-
tion and ellipticity provide insight into the state of health of a machine. More
importantly. some rotor system malfunctions can have characteristic signatures
on a full spectrum plot that are not available on half spectrum plots. These char-
 I
- -
~"

~"

i' I
f;g u.. S·l .G'n Q , . nd d iPlOe.. IXorbots
with thooir lui

""",_IS-
'IlK" '. do"""""
n....
l""
''''~ d ,I>< oob<l i, indiU 'N by_
d """"",,, Iinr oI_furward . nd _ ""
01 i>""

pti<iIy i' M\prm;nod

- -
by II>< 'M..... "'" d ..... Iorw", d .nd " ."
'-..1 I
- -
"."" "" ,ompOn<'rrt, N""" !hat wIl<-n tl><
orb<ts .,~<ir<LN,. ,....., ;, onlvone WO'C-
'Juon I~. and when _orbit " . ~ .........
WO'<'rum , ompOn<'rrt, "' ~ <"ql>otl.
."
"I
- -
"

_. - I
."
1 ~6 Oa... Plol ~

acte ri8tics u n be u...-d 10 d iscri minate betwt><>n d iff...e nt malfunc tion5 thai
p rodu"" "h,a h o n wit h t imila r l'rl'<juenci.....
." n uamp l.. i, d i..criminat ion octwttn a Y..X ru b a nd nu id-ind lk'f'd iMUlbil·
iry. Fluid · induU'd i n~l abilil y al mo~1 alway. ap pear-. a. a p redo minan tly for " -a rd.
n..a rly d ~ular, ....bsy.1<"hrunou5 .,brah lm. u. ually a t a f..........ncy .....ow 'riX_
Co mpa'" the !>alf a nd full 5pectrum pion (Figu'" 8-9' of a I'.J X rub (1l"d datal 10
a nu id-ind uU'd in.tab d iry (blue d ala ). :";Ol.. the . imila rily in al'f'"" an"" of th e
half spt'<"l ru m p lOls (lo p). The l'rl'<juend.... of the sul:>s~·nch ronou5 lin.... li n
.. ,d m o f ma ch ine spe<>d ) a", w ry d """. II wo uld be ....ry d iffic ult lo d i=iminale
bel"'.....n Ih...... tv.... malfunctions giwn o nly half "pt'<"l ru m plol..
Co mJ"'",th" fuji s pec tru m plot. (middJ..). Th..... is a dN r diJT n<-.- in the
",Loli, ... 5ire o f the ror..·ard a nd r""er", . u....)lochm n.. u> component The plot
rOt Ihe rub (l rfl) sh",,·s Ih al t he .ub>ynchro oou.<comlN ' n..nl is r>: 1.-..mely ellip-
tica l. TIlt- full . pt'<"l ru m p lOl for the in>tab ility (ti ghl) . h""" tha i th.. SU~T1 '
"hruno us "bra tion is d ..arly fo rwa rd a nd n..a rly c1lt"ula•. The forwa rd. c i. cular
5ub.ynch rono u. be......; or i.. t)l'ica l .. r n uid -ind uU'd in51abiJiry a nd a ln lical o f
rub. The add illo nal info nn a tion .. n Ih.. full 5pr'e1ru m plot. wh ich may not be
im med iately otr.i OllS o n th.. .. rbil~ (butt om). d ea rly r...,-..al. 3. d iff....... U' in
bt-ha,; or t !>at i• •·a1uahle for diag ",.. tj~
FuU . pectrum i. a n..w tnol. On<-.- in ...lal ionsh ip 10 lhe orbit is und.. ,""ood,
ha lf spt'<"l ru m plot s ap pea r limit<'d b~· compa " ",n. The effort mad.. to maste r
Ihi. important n.. w fonna t will be "',,-a rd<'d by a n e nha nced u ooe..w nd ing o f
mach i....n """a,; or.

...
,"
~
- - - - a

!
• i,e s
, , ,. _" _' _" ll'
" ."ll'
."
<
.• ,
, ..
_..
-
~.

•• • , • •
~ "--'<Yt k<pml

Fog..... H An Otblr • .-.:1 fuN 'P"CtnJm from 0 ,~ am " , _ ..."". If.!X rub. Thp " .
""",trurn ~ dorf>' thr coo.....' orbrt, _ ......... 0I1f.!"J.. I"J.. ond ."...... hq,-
'" orde< -.;briItIOn Thp t XC""""""",,, orbot ;, thf' tafQ<"Sr Iot_otatld mldly .... pn-
col. Thr If.!XA.-.:I 1X_ or.......,ly w.. _ " ....-..lllIlX ",mp"roer" i, aw .,.;,;-
!>Ie and i, ltw tho«! l\atrYO"li< 01 It.. IIlXfundamen,al_
 •• - _ h* I_ '.,
I.
I "
"
i: "
I•
"
•
.. ..-
• • • •
10 . ....
•
;_'==:
, •
,
I : I·~;-·;::'l ." IS - '7'"--- '
I
l' _ 1&
1
" 110
.,me . ."
t• ,.

-
....

_
·1 0 1

-~
•

....,.., """"'""'" 11.;. ...1

•

• ..
• •
•
•
• • • ""• 0_•
=1-" •• •
• .
..-
I

~a_ . .... _ ... ~ ~ d . tn •

....-., itIUI dotoI _1Ioe """" in
pIoo$ lI;IpI. ,.,. ~
_ro<e
pO;O:>dNtIJ -.01
...c.I ~ dotoI _ . ~

dtt-. two h.t ~

dI.t,,,..... "". t Il d .... .,...
....._ _ .. ...u.".lr
_ _ -,, <:>o<wMtn.. ... _ .......
,d al ~ . . . . . - , _..
b eoo,_ _ • fcr-
Spe<;t ru m CaKad.. p ion
During a slart up or ~h"ldown. ~pectra ca n "" la l<o.-n a t d iff..r.. nt sp<'<'ds.
The .... ' peclra ca n "" d ispla~l.'d ~. arldi ng a thini d irt"'ns io n to th.. s pect ru m
p ot . rot o r "......d. Th.. spect ra a .... po" ' tionnl in lhe md.... o f inc... a~ing ~p<'<'d
" 'ith th.. Iow"" t speed spectru m in fro nl. SU<'h a p lot i. cal lnl a s pect ru m ca~·
cad.. plot. or. mor .. ~, mpl~·. a ca"""d.. plot. A ca ......... plo t can he con~truclnl
" it h a ...' i.... o f half sll<'ctra or " i l h a "-'fie. o f full "J""d ra. Th.. full "f't"Cl ru m ca~·
cad.. p lol is p ....f..rrt"d ben , u", o f ils hogh.... info nnation oonl ..nt.
T he fo n 'p""'·t ru m u ' !lC>lde plot in Fipl re 8· 10 s!>om's the st a rt up o f a
mam in.. " i th a rub. As " i l h the fuJI sJ>ecl ru m p lol . l he hori7.ont a l .., is 1t"J'A>--
""' nls p......... .ion fll"l.]u..ncy. The ,'e rt ical ""ale to the rijrhl is for a mpil ... I... ,,'hie h
i' m....s" rt"d frum th.. Ita"..Jine o f eoc h specl ru m. T h.. ,'{"rt ica l ,,'a le to t he left is
for th.. l hini d im..ns i" n. n.tm ~I""...j. Eoch "J""drum is p aced o n th.. p l'''- .... t hat
its Ita... line {"(' rTf'Spondsl " th.. ...tor speed a t wh ich the m pl.. wa . ta ken.
Th lft' p rima ry ....Ial ion>hip' >h,,,, ld he e u minnl " n " C"i nll a ca"'ade
plot. Firsl is th.. ditl!1;Qfral .....at i" n.h ip: 'ihrat ion f.........ne i"" l hat c han!!.. ,,; t h. o r
Iradr.. ru nn ing sp'--ed. OOO" r lint which ...... dr awn d iallonally fro m th.. o r;"in
(, ..ro speed. U"ro freq uency). a th.. m" , t typical d iago nal ....latio nship (l h......
a ....no t a~ailable whe n I"" fll"l.]u" r>cy ax " i. in Oltk>....of ru nn ing speed). Th.. i I X
ord..rli ....s ..how the points " n th.. . I.....tra ", he r.. , he ,ibrat ion or p......... .ion f......
qu..ocy is ",!ual to run ning ' r<-ed. 111.- :t2X ",d..r lin"" iok ntify r.....ucn e i"" "",ua!
10 nor i ru nn in!!spo.-ed. the i 'f.&X lin.... id..ntify fr",!ue nc ie-s ",!ua l to o ne- ha lf ru n-
ning ~peet.l. e tc .
Ordo>r lin.... quick1~' ""ta bli. h impo rta nt I....rking r.........ncy relal ,o n. h ipo..
Th.. mos t olniou s a r.. t h.. IX o rd cr lin..... \ ',b not io n ..n t ",,",-, lin... is l}l'ical ly
ca used hy a comb inal io n o f ru noul••haft Ixn<.; a nd u nltala n..... ~ormally. th... IX
' i b ....t '''n co mpon..nt " . 11 It.- h igh er th an I"" - I X ro,np'-"wnt 00:>1,,'" unbal ·
a n"" UMlaily l' rod uc... ro r... ard o rb it •. Th.. fill" '" sh".... I" " p,obable ltalan....
... so nan ,,-hkh appear as p<'aks in a mpl it le. Rt",.." ..., o f the lack of pha...
in f" rn,at n o n Ih.. spect ru m p lot. s".pectnl ,.." na nC't"S ..hou ld ah.'a~·, he ron ·
firmt'tl ..n a Rt..... o r polar p lot .
R.......mh..r tha t ...." ,d machin.. accelerat ion. a nd d......l..rat 'o ns rna. ' ca u,...
. m..a rinll a nd off...,t of f, ,,,!,,..ne ;'" in th...pect ra. A 1l00d c hec k i~ I" ''''ily th ai
th.. I X >ib"'lio n" ..Iig....... ,..it h th.. I X o oo.... linc. An" s igni f,ca nt "ff....t impl,,.,,
Iha t all fn-qu" ",,·i... for th..t "!..-..tru m sa mpl.. ha ..... lw<-n . hi ftnl.
T he "'-"COnd key n'la tions hil' is ho rizontal . Ho rizo n ta l ....lat.io n.h ip. ..xi51 for
diff..rent fll"l.]uen c .... at th.. same sl.......d (in th .. sam.. s pectrum l. One u ..mp l.. o f
Ihi. ....Iati..n. hi" is lh.. ,nl ..g..r mult iple. (harmoniCT> 1o f foreinll fll"l.]" ..nd ('§ pro-
d uced by n"" lin..a'i' ..... a nd a.y m ..... trie. in rolor .~.,.tem n .. p rimary forcin g
freq u..ncy. IX d ,... to " nhalan..... ca n p rod uct" harmo ni a t IX, 2X. :lX...te.
Anoti>o'r namploo i. tM ... m.nd d ,tff'f'f"!K'O' fn..q......n.-. (~l produ«<!
wh<-n one ~lInai modulaln anot he...
Ila rmunin can aloo "" 11".....","'" by rum. Anyt h 'ng in a mach' ne lhal ce"....
" . ouddo:on chango: in d' lKtion ofl........ ft ","ill prudUoClP har....mlU in th.. op«-
n...
t ........ J!wpft the- tum. thC' rid>n- the' MrmonIC opp«nmt. Suboyudtl'Ul'lllW
nbroU>n frnju .ll<on. am aIM> pn>ducC' harmoo.ia: lOr eo:.unpk a " X rub can
prod.- harTnonoD at IX. h X. n . ""x. ftc I F~ 8-II ).!Kntd>n on .... rt.
...... ouddm m.n,:n In d..p.:..n...nl ~ !hat lleOOIIt In hanaonics. no.....
"'snaI' III.., tr.k runn,nll '('ftd and III.,. cun~,,"l in ampl'h>dC' "'". a wide
5pnod ranlt".
lJur irontai ..~al ,o h ip" r a n in.·ol ..... mo ,.., than harmo nirt.. The .. X vibra ·
lIOn on FijtUrt' 8·10 is ......t by • rub that ill thor _ "It o f high am plilude
~ in the' -...00 bala""" rt'SONlnn. T"" ho>ritonta! rrial,.,...."p n impor·
tant b.cauMo 01. 1"" mnftatlOll olthe ... X vibnh..O'l "'lth thC' ....-.ann•• nd it.
maIbd with thC' ~~ .
1M third k~ ~ ;. "l'ftical \"n1 lCa1 rri.honslup- C'IlW for!hi..-
thai happm •• the _ f""lL.llml"Y .,...... d iffO'R'nI o.p<'ftI. (m ult, pIe apt'("1 ra ).

•
-- /

10
..-
In I ~I"-
/

"

. C::J

~
.
....

-_...,pot_ -mK_
, -----.-
..
_

.,-og..... 10ftftlurn<Il.
-l. l .
. . . - fooquM<y rkqft!

• full _lfum"~ plot "'.

Kale IS10 _
• •
>tMlw. The horilonYi
nght. ~
-- rep.........
_loft
Oodi<o ....... ct... ~ ""'" _ oA;;oI.'' ' ' ' . ~ VK~ _ _

.. .
;':'":~:_: _ _ ........ '...... ID.....,;ng......sn.. _ _ -tOI...
d
,,, _
t d _ .... . . . . . - . _...... _ . _ _ .....wCI_~Sft.... Wo1ba ...
1SO o..Uo PloU

Ma ny ",tor ,~,u'm natu .a1 f' <'<Iue nc"" ll.'lTIa in const ant 0..... 'f'""d a nd apl ar
un c" ",a d plou a, a ""'ri5 olli ne, on d iff...... nt , pect ra, a rra ng"" ,...rt ically ~
" moc hin c hang<" sp<'<'d. if th I X forci nl( fnnction coinn ...... wit h (cro.s on
a .",,,,,,de plot) a nal ural f'<'<I" ncy of t he rotor. a r..-..ona nct' " i ll 'Il'Cl.l r. If tl><>",
a rt' h annonk~ II><> ha. mon ic " h rat ion is also a,-ai lable to n cit... a ny Il"'IOnanre_
Th" ... if 2X , ';I>,a tion m ind d " i th a nat ural fn.<j"""c;.-. th..n it wiD n cit... t hat
na tu ra l f""l....ncy as it l... ~ th rough. Th... R"SOna nef' " i Dshow on th.. 2.'1: o rd....
lin a, a pt'a k of a m plilode th at ..ill be ,,," ically "'i!/.ned " 'ith th... pt'a k o f ampli·
tn d on t l><> IX lin... 'SO'" Figu",:n--1 for a n .......m pl .
Simila rly. J X. 4X. etc., vibrali..n (for ...u m pk . t t p. od"ced by a bla.....· o r
'·an....."" 5.~ m<'C ha n ~.m ) can al.., n cil'" a ""'.o nanef'. Th ....... w " kal .... at io n,hip'
can be im porta nt f... ma lr" "",Oon diagnos is.
In Ihe fl~u",. Ih.. ...· l1i..&1 .. II iI'''' show, t he n...a rl ~· """ k al "-,,lat ion.h ip
bet", .,.,n t he rub vib rat ion f""'-lu..ncy a nd I"" fi.", ha la n..... . l"SOna n..... f"",,,,,ncy
of this mac h ine. Th ... o ff...t in [""" "'''''1' i' p .-...Jie-tabk. ""'-"a".... th ... rotor ~..t..m
ha • ......., "' iffen"" by Ill<' rub co ntact. mmi ng th .. rot". '~S1<'rtl nat" . a1 fr<'<l"... n·
cy h i!/.h...r.
Th .. full ' p<'Ct rum cascadc plot i. a good t,. ,1 10 .."" ,,-h..n a ,ugni flCa nt
a mou nt o f M t ·1X act i,it~· i. .....t.,.,t..... 1 h.. compon..nt. oft he ~ ..t--I X .illrat ion
ca n be q"ickly id... n lifi.... h~ n. ing this plot. ", hieh "ill " id in ide nt ifying the
sour..... o f th... 'ibrat ion.
l 'nbala nef' pon... " il l ".u.all~· prod lK" negli/t:ihl.. ,; lIra' ,on a t slow roll
speed•. Beea" ' p"d ru m plot. a"-" # n...." tOO f",m " nco m!"'''...t.... data "
la. g >low roll ",nou t ..ill p .oduct' . igniflCa nt .i brat io n a lonl( the IX o.d..,. line
do" to w . y 10" · . 1....... .. a nd it can be ea , iIy """'!':n iz..... rlll n"-,, 8 -10 sh s this
t>t> 'i..r. S o t... th at th e +I X ,ib.atio n i. d ....r1y 'i s ible a t >1,,,,- roll , thi.
machin... had a ,ign lflca" t ro to r how.
SlIafl .....ral ..h'" ..a n Im >due<' a ric h ....rmo nic s pt'Ctrum. Beca n'" th... p" ",nl
,,-aw fo ,m. a rt' uncomp"n ""t"". t he scra tc h ' p<'Ct ru m " i ll app"" " in all ..r th ..
' I.......ra on the ca...ad.. p IOl. Thi. beha,1o' make. it fairly ..a~y lo r",;ogni7 1r lh ..
sa "''' t "f rmo nic$ 'i.' ibk a t aU , peed.. th...n th' 'Y a... probahlv du 10 on ...
or m' ' fl ....a l£h Multipl ra t..h.., can a p!"'a. a. 8 mu tu ... of f"""a rd
a nd "" ha.m""ic~ " n a full " "'m. d "p"nd ing on th .. nu mb...-of seral,iJ ·
... and their ~poci nll on \ 10.. ,1I&1t.

Spect' u m Walerfa ll Plots

Wh il... , ,,,'el rum caM:a.... I'k>" a re d""ign...t to d i' play m ult ipl.. spt'Ct ra , n ·
""" s,......J, from t ra n,i..nt. , la'h' l' Of fohutd,wm data.•pectru m ,,·a te rfall plot~
are d...iltn"" to display m nll ip l " .....tra ,......... t Im.......ually duri"it COflsta nt
' '''''''' opt' rat 'on_\\·&I....rall pi l>sliIUU' tim.. fo . roto r , p<'ed on t he th i.d .......
 CNptH 8 ~If and ""II Sjl<!< tru m PlOIS lS I

(. thou¢! rut ot opt'<'d ~. rna~ L

TIw hu......... tal .nJ "oeal .... la ho n~ h;p' 01
tlw c• ..,upO<>U.~ lnt.. inN. but thoP dull'on.ol r uo... lupo. ....... typiocalIy.
d ,\I""<'<l ...... linn mn not br otrai¢!t.
\ \:.tnfall pMJtJ. ..... cuonmonly ...-J I .. .. urn'.... how machinr .-;bn1JOn
cho ...... wtl h .. thangoo in an opnal '~ .-ranwtn. "". run 'f"I"'tI"un> "lItmaU
ploI in FIftW't' 8-11.r..~ t1w......".,..... 01 . C'<>ml'"""_ ....t h . c......... ..,..) in~ ....
bddy pablo" ...\>. suction prn.o.w'r wd, ... brallIJI'"I br/wn...... ~ dnmat..
~ '""" I'I<>rm.OI, ~ IX br/wn 10> n"id ·tnductd ~;t" . nd t.ck
~ Orbi h for dJff.. lPnI suction p •. , ' ~ COPd'I........ Xoo:.. tlul llw
hdI opod .....m ~ th.at th.........yndo........... \1brAtion pn'domm.o.nl ·
Iy bwatd a1 that fl'r<l""~~ t/w, "'. '(""'lIl ,ndJc:ato<lhat tIw problorm ;,. . llllid·
indll<'rd ~y.

-
",. ..
••
•

...
, .., .,'.".
.......
'
,
"

•
........ • "_ 4, _
•--
,.....a-I I ~· w . . - - ' I I ................ IJIKt'._-'

.-,.------_
..- . Ihe _ Il>ff'd .. _ ClWOIUi"Of 01 ~ _OIerOt plot
b ,toa ' pol.'••• no....-.g Il>ff'dright 100~ thoo" d_on_
.....-- _..
<CO
Itlf' "9"C _ <11_ plot. _ _ ., •

""".""""'"
5 . . . . . . . _ . . -

"""Ioo¢. ~ II,if <pocln,IInO _ de<doe<l .... __ - .

..
152 Oau Plot ,

At limn, ITWChines ..inhr .c artni. run

Th" ;•• n idral.pp1 .... t "", for a ..·.I....a11 r'ot- 1
"If a ", II""". a nd I...... r.hlll l!<:MTI.
,....all pl..t ca n I"""w
ll" od .... ,hiJ ,ty .>f Ihnor Ih"", diff.......,1 "'lli<>f>& on 0flC' plot. F s- 12 .no..'S a
l'ull "f""M. llm ..'M....a1I oi t " lIp- oU'-n- ot.rO'• • nd oRLJ'Id T1 oi ... indu.r-
lion .....t , lh an O'iO'cIncal p",b" ,,~ In thO' ot.artUP "'llion (nodl thO'wry
h~ . mpl """ 01 11..... Ill. :t6O Hz .. Ii... f ~ _ . n... d X mol...
rotor bnt"", ~ on thO' dLa,l:Onal .. I.... motor ..... ,.t.....\\'hO'n tht
mol "' It.O' "'l""""'I"'ft: .,.-:I oi arou nd J5o'lO rpm. O'iO'cIlinoi p<>WIl'I" ..
rniu«d. ,"'" IX fUlor fO"SJ'O""'".nd Hl Hz Ii fnoquO'ncy combl... into
a 'l~ imO' 1P""" ..-,;>on l Wh.... I"'"'ft it cut a nd 11:29. t :tbO Hz ""......
cumponml d~.~ " .., nil I"" 21X coa<td<-n ribration tb ).
ThO' ." Pl IT plot: f...... It.lI mach ..... ~ • ,,",",odic npplO' in . mpln ......
ph.o... ( . .. ...! b,- 'hoe be.ol f~..........,. ~ t .... W III t J6lJl rpm) h.... noi>O' .od
' hoe I X rot'" rn-q"""J'.r 1'>90 cpm. Thit ....'mall plot pn......J thai thO' prublorm
in thO' data ..............-tl"bolft'l and I\<)( rotor -m.tfti.

a_
...... . l l ...IU&~_d .. .. ~
.......... , : I I.Tht-,tqI~lnn.riO"'''lhO' -.t
_ ~ ........ _lhO' -""<1I......-.::I
lS9l) ~ .. -......_"... , I boaIfooO'
<11 1._...-...._60 ... _ . - . . _ _ ......-.::111;29.....
60 Hllnndo!.aA>N' ~~"...,1t. I .....
 Chapter 8 Half and Full Spectrum Plots 153

Summary
The conventional, or half spectrum plot displays amplitude of vibration on
th e vertical axis versus frequency of vibration on the horizontal axis. It is con-
st ructed using the sampled timebase waveform from a single transducer.
Spectrum plots can be used to identity the frequencies of running sp eed ,
harmonics of running speed. sub- and supersynchronous vibration frequencies,
gear mesh frequencies, gear defect frequencies, rolling element bearing defect
frequencies. vane and blade pass frequencies, sidebands. glitch. and line fre-
quency noise.
The full spectrum uses the waveforms from an orthogonal pair of vibration
transducers (usually shaft relative). The full spectrum displays frequency and
dire ction of precession on the horizontal axis. Forward precession frequencies
are displayed to the right of the origin and reverse precession frequencie s are
displayed to the left of the origin.
The full spectru m is the spectrum of an orbit, and the forward and reverse
frequency component pairs represent orbit components (filtered orbits). The
ratio of the amplitudes of full spectrum component pairs gives information
about the ellipticity and direction of precession of the components, important
characteristics for malfunction diagnosis. However, there is no information
about the orientation of the orbit.
Spectrum cascade plots are sets of spectra that are collected during the
startup or shutdown of a machine. Cascade plots can be constructed from either
half spectra, or full spectra. Cascade plots have important information associat-
ed with vert ical, horizontal, and diagonal relationships.
Waterfall plots are collections of spectra obtained. usually, during steady
state operating conditions and plotted versus time. They also can use either half
spectra or full spectra.

The spectru m plot is a powerful tool when carefully applied. Because of its
wide availability, there is a temptation to use the spectrum plot to the exclusion
of other plot formats. But the spectrum, however powerful, is not a substitute
for the information that can only be obtained in other plots: the filtered ampli-
tude and phase in polar and Bode plots, the sh aft position information in aver-
age shaft centerline plots, the shape and frequency information in the orbit, and
the waveform information in the timebase plot. All of this information is need-
ed for comprehensive machinery management.
 155

Chapter 9

Trend and XY Plots

IN CHAPTER 7 WE DISCUSSED THE Amplitude-PHase-Time (APHT) plots, a

form of polar and Bode plots, which are used for trending vector data. Besides
vector data, there are many other parameters that we would like to trend: vibra-
tion levels, position data, process data, or any other parameter that can be use-
ful for machine condition monitoring. The trend plot shows changes in and the
rate of change of parameters that may signal a de veloping or impend ing prob-
lem in a machine. This information can be used to set limits or thresholds for
action.
We may also wish to examine how any of these variables change with respect
to others. Such correlation is the heart of a good diagnostics methodology. No
single variable or plot type can reveal everything about a machine. Data is usu-
ally correlated with many other pieces of information to arrive at a diagnosis.
This kind of correlation is often done with multiple trend plots, where several
variables are plotted against the same time scale. Correlation can also be done
with XY plots, where two parameters are plotted against each other.
This chapter will deal with the construction and uses of the trend plot and
provide several examples. We will briefly discuss problems that can arise when
the data sample rate is too low: Finally, we will discuss XYplots, a special type of
trend plot used for correlation of two parameters.

Trend Plots
The trend plot is a rectangular or polar plot on which the value of a meas-
ured parameter is plotted versus time. Trend plots can be used to display an y
kind of data versus time: direct vibration, nX amplitude, nX phase (the API-IT
plot is a trend plot that displays both), gap voltage (radial or thrust position),
156 Data Plots

rotor speed, and process variables, such as pressure, temperature, flow, or power.
Trend plots are used to detect changes in these important parameters. They are
used for both long and short term monitoring of machinery in all types of serv-
ice and are, typically, an example of a steady state (constant speed) plot.
The data for a trend plot can be collected by computer or by hand. Figure 9-
1 shows a trend plot of hand-logged gap voltage from a fluid-film bearing at the
discharge end of a refrigeration compressor. Due to improper grounding, elec-
trostatic discharge gradually eroded 280 J.lm (11 mil) of the bearing, allowing the
rotor shaft to slowly move into the babbitt. The trend plot alerted the operators
to the fact that something was wrong, and they scheduled a shutdown in time
to prevent serious damage. This is an example of how a very simple data set pro-
vided valuable information that saved the plant from an expensive failure.
Even though a trend plot may look like a continuous history, it is not.
Parameters are assumed to be slowly changing, so the data to be trended is sam-
pled at intervals that depend on the importance of the machinery and the data.
If a sudden change in behavior of the parameter occurs between samples, the
data will be missed.
Some data values may fluctuate periodically. For example, IX amplitude and
phase may change periodically due to a thermal rub. The period of change for
this kind of malfunction can be on the order of minutes to hours. Amplitude and
phase modulation can occur in induction motors, due to uneven air gap, at twice
the slip frequency; the period here is usually a fraction of a second. If the sam-
pling frequency is less than twice the frequency of interest (does not satisfy the
Nyquist criterion), then the frequency of the changes in the trend plot will be
incorrect, an effect known as aliasing.
The trend plot from an induction motor (Figure 9-2) looks like, at first
glance, a timebase plot. However, it is a trend of unfiltered, peak-to-peak vibra-
tion that is changing periodically. This motor, which drives a boiler feed pump,
has an uneven air gap problem. The vibration amplitude is modulated at a beat
frequency equal to twice the slip frequency of the motor.
The data in blue was sampled very rapidly, at about 10 samples per second.
This produced a high resolution trend plot. The data in red is a portion of anoth-
er trend plot from about two hours earlier, when the motor was experiencing the
same problem. The sample rate was one sample every 10 seconds, a factor of 100
slower. Note that the frequency of the change of the red is much lower, the result
of aliasing. This would not be obvious unless the data taken at the higher sam-
ple rate was available. The observed modulation frequency is much lower than
the true modulation frequency. Parameters which change periodically like this
are relatively rare, but this example demonstrates how the sample rate can pro-
duce a misleading picture of machine behavior. This effect can also happen
 Chap ler 9 Trend . nd XV Plots 157

,,.,.-.:1
f;gu.., 9-1 A 01......... ~ QOP
"""""" from • flud-lrlm br.,""'l .,
lil<' d,,·
<ho ~ on<! 01. ,,,,;g.,.-ation ,""""........
0....", imp<_ Q"",,ndo~~t""'"'Oc
d-.cN'Q< Q'dl'Uy eroded 180 ~m (11
mil 01,........ mg.~ lil<' ""'" "'-It

--
'" <lowly _ ;"' 0 tho babbott, no." l'ond
ploI p<ovodo<I ~ . tIy w"'""' ng 01,.... prOO-
lr<n_ Id<ont_ , ,.~ 01 chong<o.

,
,--
•-.ng • >d'o""'-'1<cI td"""'. Th;, 1'0'-
llCU"" machine probIoom woo."" <6,.
"">«I by [,_m.nn;" ,O'fto,,,,,,,~ [11
• 10 1~

fIqu ' . 9-2. '"""""'l rn. t~ pIaL Thl>i'

not . n~ plat! Thi, _ ~ pump
fTl<ll'" 1\0> on ......-. 0;' gap. The ..
'fYIJlI"""- " rnodu "' ~ ..._ ~""n­
<y ""ll>Ol '" ~ lil<' ''if> ~"""'Y 01,....
molOr. Thr d.ota in bIuo wo> ~ ..
tw.,,,,,,
,
• bout 10 """pies PO" second-no." dol. in
! '
'od is. pottoon 01.
..mplo<l obou'
,,<'I'd
ploo: IN' wo>
,wo houn. ... ~~.., ~

""mplr ~ 10 sec"""'- . foct<>' 01100

j.
_ .When tho two IIftI6> . ,., <omp;l~
~;, cINt 1hoI tho ,~ d " .I .....,.j;tho
~ ,.,. .... """ f """"Qh '" oo:;u-
"'01)' coP' '''. the <hongesin the _ation.
158 Dat a Plot<

...h"n. to inc rea ", the l rend in t .-a l o n th.. ho ri7v nla l u i... th.. databa... of ",m·
plt>d dala is dt'Cima l...:! ( mp l aw Ihn ,...n ",. t l.
:\Iull iple para "'''I ca n a l", lw displa~-..d on tilt- ... m .. plot. or on ",'-eraJ
plots with the ... me ti" ·a le. tn ~..r... lat" e ha n!l"" thaI <",..·ur in a mam ine.
rigu re 9-3 "''''''. a tr.. nd plot of " w n' J'"""""r " i l h a "",nl ..r ",aJ ins la bility pro!>-
I.. m. Ro th IX l hl....' a nti u nlil t.. ...-d (rt'd l ,i hral ion a mplil ude a ... . ho" n on Ihe
...me plot anti 10 tilt- same ..." Ie. Wh ile th.. ma l·h ine "'a . ru n ninji: a t full s peed.
t h"~Ul1inn p ......u ... "'a ' vari...:!. and a n uid · intluced ins ta h iJity a ppear...:! or di.·
a ppe a " .. 1d..pend inKo n Ih e I........ o f suaion p r...... u..., :\0I.. 1hal l h.. I X ,ib,al io"
am plitu d.. decrrn....J wh ..n Ih.. unfil l.-rnl ,i b03lion incrrn...-d. T h is can ha p pe n
,.·Ilt-n hiKh a mplit ud ... :\o' · IX \i b03l io n mo'.-"" lh.. roW r to a "'ltio n ",here the
l':on a mi" Sti ffn...... is d ilT" wn t c han ltinlt lh .. IX ... spon..... A plot of Ih.. ae-t u.d

fOg",. g. )
.,,1/1 • ' ·
tr pIoI of • ,om~lOI
1in".M,y 1'1_ . 80lh
t I t
1X :t>I ) _ urmh",..-d (red) \lib,,,,,,,,,
. mpitud... .... ...,..." on ~ ..rrc pIo(.'"
til ...."'" "all" """It" tho- """"' fUIl -
""'9'" lull ~ ,1'0<- ""'fun pr '''''' ViI' -
!<'d:. flud-;nd ucl"d in ""~ .pp.-atl"d"'"
di"PPN'O<I.~·ng on tho- - . of "", .
bon _ ... _ thaTt"" lX _ ",ion
d"'...... ~ ........, ,~um;~",.,j ,.;t>r",,,,,,
ll'-l6 IU l IU O
,~ i..... ""'" to< fImh<'f ... 0......""""1.
 "'" Pl", 9 Trend lind IV Plou 159

.........
P""'''''~

F
dali. ...-hioch ........ "".~, ... >uk! hiJlhli,dl1 tM

~
CUfrriallOQ

~ trn>d d.ot.t frool a 3OJ000 hp. "'~ ~lrc1:rio; IllOCllf

ci tro.

dm1nfl fIo,.- air compl ' '''0; llIr plot .1oooJon n bnhoQ .ampUtude- from a
d~ol pc-ot:.. l bl..... riplt ~l .nd .. ,"","il) tnon>d~ (P"""'-" lrit
""....).Th mol.... had .. ....u fourodal ...... ~" l"" Ihat olowly 10<_ ...... aJk".'lIfl-
1M .;bn!"... ampl itude- w ord ~ both lr. n'ld....-c-n.. to ~h ~.. ....
\ \ -...... 001. cI d>ock ~ iMtalIrd and tM found.chO<l bolt. rl"ti¢'t.......d
around JO .-\~o.t. In.. \;!mlt ..", d..,W ord .., ~. fol\oo<.nl ~ . ",IC hor.- .kI..- ri ..,.
s.-U ln.. rnd oft"'" rO'COl"d. t tl<' m...t.ill~ wu .hut d<Jwn.. and thr f"u lldation .... .
"''''' I*"u 'lr ""...m aulll'd. n... m,IC ur il....r .. .... found to ..... .. J"Oblc-m.... nd 1M
Ull il ...... rt'p1aCt'd..-\fi .., .... larl ill'" I vibral lOn r<1 umll'd I.. '...r~· lc ...· l.-vcl•.

" ----;;;- "

." ! •

f~"'" T....-.d dol. ""'" II -'0.000 lip. Yf"K1TorIous ~ 1Ktt>< molor ~.1f'9 atI ""..,
lIow• • compr..."". ~ pIol U>owo """.""" ~ from. do>Ploo<_ prot><-
(blue. ~'" ""'..: .nd • _oty ,..nod""t< (Q1~ Itft !UIfo~ Tho, moIor had. _
~ 'Y''''''' m.. """""t - . ~ .o-ilion _p1~<do ~ "" ...:1 on

__
b<:!t"otr..-..du<... .,..;I_)O~.menomc/:(ho<l ........ .-.. ...........
..-.:I ~<WI - .
_ .... _
<J>t,~_~_
,""""'fd Tho """'..".., ......... It-..p drop."""""" b1.
tho ..-.:I c/: tt.. """"'" IN .....,..... _..-.. _ _ ..... b..nGi-
160 Dau Plot.

XY Plou
Where Ih.. tre t>d p lO! di'PJa~ o ne o r mo re para .....te. . .......u. tim... th e XY
plot (n ol to be confu..-d ..ith Xl' ........ or XY t ra n.duee.. ) ca n be u..-d 10 di .play
a ny two pa ram.. t<> rul. each O! he~ Co ....lat ion. be t""""n t he pa ramet ..... will
. how a diagonal alio n.hip. ,\ complete laclr of corre lation will "'ow ei th er a
horizo nt al or .....rt ical . ...atio n"'ip.
Figu . e 9--5 .how. a n :x.... plot of vibra tion a mplit ude ,...... u. ga p ."Oltage from
a 125 :\1\\' oUea m tu thine gene r" to•• II P{lP u nit ru nn mg a t 36CX) 'I'm. The p~ .t
' '-'1'at I"" nl I ..t...n th .. mach i n~ u ndergoes • loa d ch a ng... Ilrl......... P.' ml$ I
a nd 2 (mil. th~ r iO! d ....rly . hnW1; a co. .......l io n be twee n ehanging . hafl .....iti..n
(measu. <-d by th .. gap ~nlt.~) a nd IX vih ",tion a mp litud... A. I.... s haft m. ....es
(hlue) to r " int 3 .b u inll th .. nU l t h~ hlllJ r'$. th.. ' i h'at ion dC'C l'P"-"" whil.. th e
gar voltage rem a in••p .....x;m.,..ly cunst.nl. It i. P'""ihl.. th.1 a gra .i ty b" ",
may be ""\Irking it....f OUI o. Ih.. m..a'U ....mffi t p robe tna~· be vi..,.;ng a d ill......nt
...ction " f Ih.. , haft as th .. mach in.. ...aeh n th ...-ma l "'Iuilihriu m. Th.. ""aft u k...
o nly about 33 minut... to mo'... from poinl 3 10 point 4 (~n ), I1lmt likely in
'0
....P" n... a no th l"T load ch a nll".

.. , - - - - - - - .,-
!l-S An )('I pial 0/ I~ .."'II""
f Og_ ,"'pIio
..-~' 9"P_ ~trorn. ,lS MW
Sl-.. nnt>r.e 9"....,"""... P/IP uM ""'nilWl
'" l6lXIlJ)m .The pial cINrIy - . ~ the
<~"., btol_ cha ngi"9 """" poWon
(-.nu,"" by the g.p """9") ...-.1 8.1'1
inc"'. ... in ! X ~t>r;mon MfIp1rroo. whon ,,,"
Iood ~A, tho ,t>ah ............ '0 p:JOn')
GUi"9the ~ _ ~the..b<II;on

~ wt>;'" the g.p -..,... ,"""'''''

approll""'''''' <:om!¥>!
Aft
(no < ~IalIonl-
that, the shaft I...... onIy .bout 3~ min·
, ,
ut to ~ to pot'oI..o Ig''''''''. ""'" h•...,.
in ''''IlOll''' '0 "-"" Iood cha"'J<'.
 eh.lple, 9 T.~ ..d .....t XV PIon 161

Su m..... ry
Th.. t.....nd plot i~ a ' <"C1a ngula r or po la r I ~nt o f II m..ao",n! pam m..te. ",,,u~
tim... T..... nd plol. can I><' u.ro to d i~pla~' " ny kind o f dat a U"rs,,~ time: di rect
' -;l>tati,,11, nX a m plitud... nX p gap ,'olta~. rotn. ~p<"t"d. a nd pn~ ,,,ri o
al....... "',... a, p re" " re. I.. mpetatu 1101" or 1'0" ...... Da ta ('an I><' ro. rclatn! by
plutlin!!..·",.,,1~ariabk. o n the ....nI.. t im.. """It-.
Tw ndnl da la i~ ,a mr! .-d al pe. iodic i"'""'alfi,, a nd il i~ a",umn! to I><' ~I ......,·
If cha..gins- If d ata "al",-", can c han ge I.... ~ ..li,·a lly. t he .....m pling fr....uency m ust
.... a t I"",t "'..... 1 f""l"ency of c hange II .... "~"l u i.1 c n te.ion). If ""t. th.. d al..
will b.- a lia.ro; I display.-d ht-ha,iOT " f Ih.. t:rend d ala will nol I><' rorf<"C1.
The Xl" plot c.. n I><' " <edl" di,p1ay a ny "''' pa m mete.~ ..gain"" ..ach olh.. ~
Correlat ion, I><'lw n I h.t' pa . a n,,·t ..r. " i ll sh"w II d ia!l<mal ...Iation >h ip..'\ rom-
p i..... lac k of co I..t;on ..ill >h" w .. ith... a l>" ri7"'''' lal o r '...rt ical ....l..t ions h ip.

s ereeeeees
I. Ei...n ma nn. Roht-rt C . Sr~ .rnd l'iw nma n n. ko l>ert C . Jr.. •t /achmn y
Malfu nction Diagnosi. aad l .f.JTTOClio'l (L:p pe. Saddle Riw l: p ..... nu ...... II..U.
l nc~ 19')8 ).PI'. i.~ I·i5S.
The Static and Dynamic Response
of Rotor Systems
 165

Chapter 10

The Rotor System Model

ROTOR SYSTEMS ARE SUBJECTED TO MANY KINDS of forces. Forces can act in
radial and axial directions, and torques and moments can act in angular direc-
tions. These forces can be static, or unchanging in direction and time, or th ey
can be dynamic, where they can change in magnitude or direction with time.
Static forces acting on the rotor system produce static deflections of rotor
system elements. For example, a static radial load applied to the midspan of a
rotor shaft will cause the shaft to deflect in a direction away from the applied
load. Or, when a torque is applied to the shaft of an operating machine, the shaft
will twist to some extent in response to the torque.
Dynamic forces acting on the rotor system produce vibration (Chapter 1).
Vibration can appear in the form of radial, axial , and torsional vibration. Usuall y,
we measure radial vibration in machinery because radial vibration is the most
common vibration problem. Axial vibration is less frequently encountered but
can produce machine problems. Torsional vibration is very difficult to measure
and tends to be overlooked. Both torsional and axial vibration can produce radi-
al vibration through cross-coupling mechanisms that exist in machinery.
Unbalance is the most common example of a dynamic force (the force direction
rapidly rotates) that produces radial vibration.
How do dynamic forces act on the rotor system to produce vibration?
Somehow, the rotor system acts as an energy conversion mechanism that
changes an applied force into observed vibration. The rotor system can be
viewed as a very complicated "black box" that takes dynamic force as an input
and produces vibration as an output (Figure 10-1). If we can understand the
nature of this black box, we should be able to understand how forces produce
vibration. We should also be able, by observing the vibration and knowing the
166 l1w SIal ic and Dyn.amic R.. ~ol ROlo . S ~'l .. tm

wo rkin!/." o f th .. black box. to dl.'d oce soml'1hin g ahout Ih.. fofCt'<> th ai p.odu,'t"
the , i bralion.
We ca.n t r,' 10 gu e•• th.. ront ..nls o f th .. black box by shaki ng it us ing a I('(:h-
n iqu.. ca llro penurbalwtr a nd obS<'"i ng th.. bdla, i o. o f III.- .,..t..m. Th i. is th..
""m... I""h niqu.. som.. p<'OpI.. uS<' wh..n " , in!! 10 g""'" I .... ront.. nl~ of a
"Tapped gift. Th..,. shako> th.. gifl a nd ""aluat .. th.. w..ight.. hala nce. a nd sound.
Wh..n appliro 10 rotor s,·st..ms . Ih hak ing a""lifos a known forc e tn th .. ro to r
.,"t..m. a nd Ih.. vihra tion .....pom to th.. fo. C('is m....s urPd.
W.. ca o also tr,. to ....timat e th.. w nten t. n f the bl....k bo.- by d e,...lopinj( a
mal ....ma tkal modP/ of th.. rolo • •,.."t..m..~ !l'uj m..d el " i ll a ll..w us 10 relat e
oo....r..-l.'d ,ib.al io" to th .. fo thai ac l " n lhe ,,·slem. TIl;" ..ill allow lUI to
d..tect.. id..ntify. a nd ro•......,l p" ntia] p",hl..m. in Ih.. ro l..r 'pt.-m.
A Iloo<! mod.. 1",i ll also gi,-e us th.. a hilily to prnliCl hnw cha ng in foret.
" i ll a ffect ,ib.al ion. Th i. i. Ih.. l<'y to "ff.-cth... bala nc ing I hn iqu Imalline
tr ~inll to bala """, a <:('"'1'1... ma.·h i"" by m......l,. gu..... ;n!! ho .- m uch l\'Pij(h. 10
add a nd wh I" 1'1....... it- Wil l>o"t a sysl..mat ic approoch ba.ro on knowledg..
o f roto r I a'.;ot. acrural.. bala ncing wo ,,1d bt- "rtua ll,. imposs ibl... A mod'"
gh..,,; u' th.. fnu ndat ion fo r a .y.t..-matic. efficirnt.. and d J""ti,... balancing tec h·
niqo...
In thiS chapt..r we "ill .........I..p a si mple mod d of a rot o r s,"t..m. Th .. pri ·
ma ry .....ull of th.. mod el. o,·tromic Slijfn... ,. is the solution 10 11K- mpt••) · ins i""
It... black hnx. fl i. a f"n<.Lom..nlal a nd important rona-pt fo r u nd...rs tandi nll
roto r "..ha\int. it p",vidn a p""... rful t....l for mal funct ion d iallnos is. a nd il is l he
k..,. to s lJC\-,.,., ful ba lancing.
In d....-..lop ing th is mod d . Ih..... is no "lIy to ",'oid th .. mal ....malic. of d if·
f..... ntial ""l"at ions a nd romp l num lx>rs. w.. ",ill ma ko> ....wy l'IJon 10 "'-"'I' tt...
m al h..mali... a. si mp!.. and d , a. J"OS'ibl... TIlOS<' who d o not h3\'" th.. ma th ·
....,...tica.l bad <gro" nd ,J,,,,, ld Ix> a hl.. to skip th .. malh and st ill und ..rs ta nd tt...
ronct"J'ts. 1 h.. math is 11'1 for 11'1 0><' who wond..r " il..r.. it al l rom.... from.
\\ .. " i ll foclls o n the d loprn.." t o f a roto r sy.l..rn rnod..1 ba«>d on rad ial
,';I:m.lion. Th.. bas ic con ls thai "'...",il l d ...,...lop ........can b.. ..a.i1~· ..xl<'fld<'d to
", ial a nd to rsi..nal ,i bration.

f/gl" .. 10. 1.T1'or ""'" 'Y"e", . ... 'tlIocl

box.'T1'or ""Ie'" OOI"vem dynamic;npo,n
for= 1O..... po,n \/b';o\lOn. -
 Chapter 10 The Rotor System Model 167

Introduction to Modeling
Everyone has seen scale models of aircraft. A model airplane mimics impor-
tant features of the real thing. When viewed from different directions, a well-
built model airplane can look very much like the real thing, but it includes only
certain features of the full-sized airplane. However, even though it looks the
same, a model airplane does not behave in all ways as a real airplane; if it does
not have a working engine, it cannot fly. In fact, all models are simplifications
designed to represent particular features, and they will not function properly
when pushed beyond the limits of their applicability.
Our rotor system model is a mathematical representation that is designed
to mimic certain important features of the real rotor system. The rotor model is
an attempt to describe the function of the black box that transforms dynamic
forces into vibration.
Because it is only a model, it will have limited applicability. The limits of the
model are stated in the assumptions used to derive the model. Assumptions
almost always involve simplifications that make the solution of the model easi-
er. Applying a model beyond the limits expressed in the assumptions will usual-
ly lead to error. Sometimes the error can be tolerated, sometimes not. Often the
amount of error is unknown.
Most real rotor systems are very complex machines. We cannot, given the
current state of computers and mathematics, construct a model that duplicates
the behavior of these machines in every detail. We can construct complicated
computer models that do a good job of mimicking some aspects of complex
behavior, but the results of such models tend to be very narrowly focused and are
difficult to generalize.
Simple equations can be more easily understood and interpreted. The bene-
fit of the simple approach is clarity and, we hope, an intuitive understanding of
basic rotor system behavior. Because of this, we will develop a model that will
yield some relatively simple algebraic equations. The price we pay for this sim-
plicity is that the model may not have the capability to accurately represent the
behavior of complex systems. We must always keep in mind that a simple model
will be limited in its application. In essence, we will trade detail for insight.
The model we are going to develop is a variation of an early rotor model,
often called thejeffcott rotor, developed by Henry jeffcott [1] in 1919. Our model
extends the Jeffcott model by including the effects of fluid circulation around
the rotor and by using complex number notation to simplify the mathematics.
It is necessary to include fluid circulation effects if we want our model to predict
the rotor response in machines with fluid-film bearings, seals, and other areas
where fluid is in circumferential motion. Our model will be a slight simplifica-
tion of th e model presented by Bently and Muszynska in reference [2].
168 The Static and Dynamic Response of Rotor Systems

Here are some basic definitions of terms that will be used in the derivation
and discussion of our model and throughout the book:
The rotor system includes all parts of the machine that are involved with
vibration. This includes the shaft with any attached disks, the bearings that sup-
port the shaft, the structures that support the bearings, the machine casing, the
foundation system, coupled machines, and attached piping systems or unse-
cured cabling. The rotor system can also include all of the plant equipment that
is involved in the process in which the machine is imbedded. When working
with simple models it is easy to forget that real rotor systems include all of these
components.
The rotor is the rotating shaft assembly that is supported by bearings. The
rotor may be rigidly coupled to other rotors in other machines, effectively form-
ing a large, extended rotor.
The stator is the stationary part of the machine that contains the rotor. The
rotor rotates in and is supported by bearings in the stator. The purpose of bear-
ings is to eliminate friction while preventing unwanted contact between the
rotor and the stator.
Forces that act on the machine will be divided into internal and external
forces. Internal forces are those that appear from the machine's interaction with
parts of itself. Support forces in bearings, forces resulting from shaft deflection,
and forces due to interaction of the rotor with the surrounding fluid are exam-
ples of internal forces. Externalforces are forces that are applied to the rotor sys-
tem and produce some sort of perturbation, or disturbance of the system, such
as impact forces due to rotor-stator contact, static radial loads, or deliberately
induced perturbation forces. Even though rotating unbalance is generated inter-
nally, it will be treated as an external force.
The term synchronous refers to anything that is rotating at the same fre-
quency as the rotor. Unbalance is an example of a synchronous rotating force.
The term IX is used to describe a synchronous frequency (-IX is also consid-
ered to be a synchronous frequency).
The term nonsynchronous refers to any frequency other than synchronous.
A nonsynchronous frequency may be either supersynchronous (higher than run-
ning speed) or subsynchronous (lower than running speed).
A rotor system parameter is a property of the system that affects system
response. Mass, stiffness, and damping are examples of rotor system parame-
ters.
The term isotropic describes the properties of a system that are radially sym-
metric. For example, isotropic stiffness means that the stiffness of the system is
the same in all radial directions (Figure 10-2). The term isotropic is distinct from
the term symmetric, which implies a geometric (shape) symmetry.
 Chapter laThe Rotor System Model 169

Isotropy Anisotropy

Figure 10-2. Isot ropic and anisot rop ic systems. A

system propert y is isotropic if it is the same in all
radial directions. A system property is anisotropic if
it has different values in different radial direct ions.

A system property is anisotropic if it has different values in different radial

directions. Fluid-film bearing stiffness is isotropic when journals operate at low
eccentricity ratios and anisotropic at high eccentricity ratios, where the stiffness
in the radial direction is typically much higher than the stiffness in the tangen-
tial direction. On the other hand, rolling element bearing stiffness is usually
isotropic.

Modeling of physical systems usually follows a structured process:

1. State the assumptions that will be used. These usually involve

simplifications that allow easier solution of the problem, but
limit the application of the model. These limits must be kept in
mind when applying the model to the real world.

2. Define a coordinate system. The rotor system moves in space,

and there must be a measurement system to describe the
motion.

3. Describe the forc es that act on the system. Forces ar e modeled as

physical elements which depend on displacements, velocities,
or accelerations. For example, a rolling element bearing support
force can be described as a spring element where the force is
proportional to the displacement of the spring.
170 The Static and Dynamic Response of Rotor Systems

4. Develop a free body diagram. This diagram contains the rotor

mass (or masses) and all of the forces acting on it (or them).

5. Derive the equation of motion. This is the differential equation

that combines the forces and mass elements with a physical law
that describes how the system must behave.

6. Solve the equation of motion. The result will be an expression

that describes the position of the system over time.

7. Compare the predicted behavior to the observed behavior of the

machine. Theoretical behavior is compared to the results of
experiments.

8. Adjust the model if the description is not adequate.

Assumptions
The assumptions define the limitations of our model. They make the model
easier to solve at the expense of detail in the final results.

1. The rotor system will have one degree offreedom in the complex
plane (l-CDOF). One degree of freedom implies that there is
one, independent, lateral position measurement variable (r,
which will be complex), no angular deflection, and one differen-
tial equation to describe the system. This will produce a model
capable of only one forward mode, or resonance.

2. The rotor system parameters will be isotropic. This will allow us

to use a more compact mathematical description for the model.
The effects of anisotropy will be discussed in Chapter 13.

3. Gyroscopiceffects will be ignored. Gyroscopic effects can cause a

speed-dependent shift in rotor system natural frequencies. This
can be very important for overhung rotor systems, but we can
ignore gyroscopic effects and still gain a good understanding of
basic behavior.
 Chapter 10 The Rotor System Model 171

4. The rotor system will have significant fluid interaction. All the
fluid interaction will be in an annular region; that is, a fluid-film
bearing, seal , impeller, or any other part of the rotor that is
equivalent to a cylinder rotating within a fluid-filled cylinder.

5. Damping will be viscous and due only to fluid interaction. There

will be no other source of damping in the system. This reduces
the number of parameters in the equation and simplifies the
mathematics.

6. The model will be linear. This is a difficult term to define briefly.

A useful definition is that a system is linear if a multiplication of
the input by a constant factor produces a multiplication of the
output by the same factor. For example, if we multiply the
unbalance by a factor of two, then the vibration of the machine
will increase by a factor of two. Also, if a linear system is sub-
jected to a dynamic input at a particular frequency, only that
frequency will appear as an output. While machines can and do
behave in nonlinear ways, nonlinear mathematics can be very
difficult to solve algebraically. Fortunately, most machinery
behavior is approximately linear, and the linear models approx-
imate real machine behavior well enough to be quite useful.
(That's good , because balancing techniques depend on linear
behavior.)

7. A fluid will completely surround the rotor. Fluid-film bearings

will befully lubricated (360°, or 2'Tr). While normal hydrodynam-
ic bearings operate in a partially lubricated (180°, or n) condi-
tion, misalignment can unload the bearing, resulting in a tran-
sition to full lubrication and fluid-induced instability. Also, seals
are designed to operate concentrically with the rotor; thus, they
operate, by design, in a fully "lubricated" condition. For these
reasons, the fully lubricated assumption is both realistic and
necessary to adequately describe fluid -induced instability prob-
lems.

8. A nonsynchronous, rotating, ex ternalforce will be applied to the

rotor system. We will see that rotating unbalance is a special
case of this general nonsynchronous force.
172 The Static and Dynamic Responseof Rotor Systems

The Coordinate System and Position Vector

Figure 10-3 shows a basic physical description of the rotor system. The rotor
can be described as a single, concentrated, perfectly balanced rotor mass, M,
located in the center of a fully lubricated, fluid-film bearing that is fixed in place.
A massless shaft is supported at the left end by an infinitely stiff bearing that
provides only lateral constraint (no angular constraint). Thus, all of the rotor
mass is concentrated in the disk and is supported by the bearing. The only stiff-
ness element in the system is associated with the fluid bearing (which can also
represent a seal). The rotor rotates at an angular speed, n, in rad/s, in a coun-
terclockwise (X to Y) direction, as shown in the section view. The bearing clear-
ance is greatly exaggerated for clarity.
Note that, even though this description implies that the rotor mass and
shaft can pivot in the small bearing, the rotor is assumed to be constrained to
move only in the plane of the bearing with no angular deflection. The most accu-
rate graphical description of the model would eliminate the shaft and small
bearing altogether, leaving only the rotor mass and the fluid-film bearing.
Figure 10-4 shows the coordinate system for the measurement of the lateral
motion of the rotor mass. The X and Yaxes represent the real and imaginary
axes of the complex plane. The terms real and imaginary come from the mathe-
matics of complex numbers. Both of these directions are quite real, and nothing
about the rotor position is imaginary. The origin of the coordinate system is the
equilibrium position of the rotor when no external forces are applied to the
rotor, and, in our model, it is located at the exact center of the fluid-film bear-
ing.
The rotor position vector, r, represents the position of the center of the rotor
relative to the equilibrium position. It is defined in the rectangular complex
plane as

r=x+ jy (10-1)

where x is the position of the rotor in the X direction, y is the position in the Y
direction, and

j=~ (10-2)

The length, or magnitude, of r is A, where

(10-3)
 Chap'.., 10 Thl! Rotor System Model 17)

--
figooH 10- ) Th<- b.o. ... pt.yso<.1 ""'" 'r<'~ n.. rotor i1 • ~ng~,conc""tr.t .
od ""'.~ -",
_
kw;.,,,,, in "'" (~, d • ftuod-filtrl t...ff'09. no.. WIt po"""'" roo
'100 tt... fOlo< m. -.; 011 of "'" "'... .. ....pported l"I' ...... bNr"'9- All of
...... " "",",,' Ifl ,"" .Y'I........ "' Ie ~ Ioc.,,,,,
in !he lIuld b..ring
(\Ntli<h u n ...., I<'pR'>ffit • "". l d d ..dl. ~ ,,,,,,, """' (""",.«Iock-
wi'" ~ to Yl . 1"" Otlgul¥ -..d fl in r.odl~ ., """"'" in , OK,""" ___

P"
jr

A
A
Jr

X X
• •

~ 10-4 £<><1 _ 01 fOtO< ¥>d iI. ,,,,,,d,,,,,'.. 'Y"orn- At left.t il<- rotor
.. _~ ~ ""' "'" « ·nof "'" .)'>,'""'_
n." ~.; 1M 1 right , t>ows
tho- ~ ITlOft' dNrIy.Tho X .>i,
. r><! It.. y . ", ar" , ,,,,,,d,,,,," . . ... d
"'" romp!,,, pl. .... ~!he tnt !of <lot.H..
174 The Static and Dynamic Response of Rotor Systems

Th e angul ar po sition of r is measured as a po siti ve angle in a counterclockwise

direction from the po sitive X axis. This angle, e(Greek lower case theta), is given
by

e= arctan ( ~ ) (10-4)

Also,

x = Acos e (10-5)
y = A sine

Notice the similarity between this position vector notation and the vibration
vector notation in Chapter 3. The two are very closely related; in fact, the solu-
tion of the model's equation of motion will yield vibration vectors.
Leonhard Euler (1707-1783) showed that the po sition vector can be
described using an exponential notation, which is very compact:

r = x+ jy=Ae jil (10-6)

where e = 2.71828... is th e base of natural logarithms.

The elements in front of the exponential function (in this case, A , but th ere
will be other elements) define the length, or amplitude (magnitude) of the vec-
tor, r. The exponential function defines the angle of r.
If r rotates around the ori gin with constant, nonsynchronous circular fre-
qu ency, w (in rad/s), then the angle, e, becomes a fun ction of time:

e =wt +o: (10-7)

where 0: (Greek lower cas e alpha) is the absolute phase angle at time t = 0, when
the Keyphasor event occurs, and r is located at an angle 0: with the horizontal
axis. The Keyphasor event acts like a strobe, momentarily illuminating the rotat -
ing vector at the angle 0: . If we substitute Equation 10-7 into Equation 10-6 we
obtain a general expression for r that will be very useful for our purposes:

r = Aej(wt+a ) (10-8)

Equation 10-8 des cr ibes a po sition vector that rot ates; the tip of the vector
and the center of the rotor precess about the origin in a circular orbit. We obtain
the velocity (the rate of cha nge of position) by differentiating the po sition with
 Chapter laThe Rotor System Model 175

respect to time, assuming constant amplitude, A, and constant angular velocity,

w:

v = dr = r = j wAe j(""t+n) (10-9)

dt

We differen tiate once more to obtain the acceleration,

a = dv = r = -w2 A eJ(wt+o ) (10-10)

dt

A few words about j are in order. Whenever j appears outside the exponen-
tial, it basically means "change phase by 90° in the leading direction:' In
Equation 10-9. j orients the velocity vector 90° ahead of the precessing position
vector. Thi s makes sense if you realize that, as r precesses in an X to Y direction,
the instantaneous velocity of the tip of r points (for circular motion) 90° from r
in the direction of precession.
Note also that in Equation 10-9, the amplitude of the velocity, wA, is propor-
tional to the circular frequency, w.
In the acceleration expression in Equation 10-10, the negative sign indicates
that the direction of acceleration is opposite to the direction of r. The negative
sign is the product of j .j (P = -1), so acceleration must lead displacement by
90° + 90° = 180°. The amplitude of the acceleration is proportional to w 2 •
Finally, note that the mathematical angle measurement convention is that
for positive w, r precesses in a counterclockwise (X to Y) direction, and the
measured angle is positive. This is opposite of the Bently Nevada instrumenta-
tion con vention. where phase lag is measured as a positive number in a direc-
tion opposite to precession (see Appendix 1). This difference is very important
when trying to relate the results of the model to measured vibration.

Lambda (.\): A Model of Fluid Circulation

Whenever a viscous fluid is contained in the annular region between two ,
concentric cylinders which are rotating at different angular velocities, the fluid
will be dragged into relative motion. Thi s motion can have a complicated behav-
ior. What we need is a simple way of quantifying this behavior. .\ (Greek lower
case lambda) is a model of fluid circulation that reduces this complexity to a sin-
gle parameter. Though our discussion of .\ will focus on fluid-film bearings, keep
in mir>d thai th...., con(' e pt~ can be a ppl iN to a ny M mi l~ r p h)-..icaJ _ituat ion,
such u seaJ~or pump ;mp...le.....
lmagi r>e ...."(>. infini te llal plain ...pa nott'd by a n u id-fiIkd II"P CFigu n: 10-5 ).
1 h.. UJ'J""T p la t.. m." wilh a con' la nl Iin....r ,...I" c ily. ". a nd th.. lo"'....- ..]at.. ...._
z..ro ,...Ioei" : fkcau o f fricl io n. Ihe lir>ea r ",locity " f Ih.. llu id r>e~t I" the ~u r-
fa.... of II>.- m","ing p la t.. "'il l be v. wh il.. lh.. wloci t~· of tl>.- n uid nn t to tl>.- ... r-
faet." of th.. . tati o na ry pl~te "'"ill be zero. lh.......Iociti e. in Ih.. n uid will form a lin-
ear velocity p rofik .. the ,...locit:< smoot hly rna nll'" from on.. ... rf,..... 10 th e
o ther. Th.. a'''''''!l.. hn..ar ,...Ioeit:< o f th.. llo id (rN) must be som..wh..r.. berwee n
ze ro a nd ". a nd . fo r this . it Udtio n. it i. O.5".
:'I:ow ima¢ne wrap p in!l th.. t....o plates into ...."(>. conc..nt ric, in fin i tel~' lonll
C)iind...... as , h....... at th.. bottom of th.. figu.... This i• • imilar toa roto roperat ·
inlt in.id.. a flu id · fiI m bearinll- The fluid i~ trap ped in IlK> a nnular region
be........ n m.. cyIir>d... . . t h.. in r>er cyIind... rot a te. at so me angular ,...Ioeity. fl.
a nd the o " t... cylinde r .e mai ns mot lonles•..-\ ,l h the n at pial .... t he n u id ne~t
to ( he .... rf.."... of m.. cyIind.... m,," ha,... th " rfa.... ,-..Iocit:< o f th.. C)'linde ...

f ,9"'" l ll-S Vi«<lus flutd lIow _n

n.... f1a1 pia.... ....., ;nlinrte <reu'" <yIrner..
Al1Op, the u~ '¥"t pial.. movn wilt1•
I"'oIi-

_ftotpLI,...
<"",tort! ine.. ...eIo<ity. r •• n<! ,..... """'"
pI.ote t\a< zero velocPly. 8«..".. of ""'''''''''
•
of the fluid, the ~_, '""""'''Y of thel'oJid
im.....oOMeIy ......... 1O, .........I"".. ofthe".."..
~ pIo'e ""Mb.- •. The "ud rext to the >U'-
fO«' 01 th...... llQn.tfy t>\Ol.. wi/lhaooo! "" 0
welo<ity The weIocit'! ('ed) of
<it""'''''' ;, 0-5., Now. ~·
!he I\uicI lot !h;,
. . . _.ppotlg".. . "'" plol... ""o twO <0<>-
centnc.infirntely long <yWIden,.<<>ntAon'"'.l
!he flutd in the .........., ""9"'" _ the
<y4inde<' (nghll- The in ..... <yindet 'o~.....
"""'" ""'JUItH "'"""'''Y. Q _ out...- <yItnd...-
' n. mooon with the not plat....
, flutd......... 10 1..,.. of the in.....
<~ "" Mhaooo! ongular velocity fl •.....,
!he I\uicI ......t ,,, the ..n..,e 01_ out...-
cylinder willhaooo! ",ro ong ular weIoc;,y.F",
_ inIinol<'ly lotoq <yJ;nden,. _ _oge

."9..... .......,..;,.,. """ be .Imost 05fl

 Chapte, 10 n.. R oto ' S Y'I ~mMode I 117

l he a,,!!-u lar .....loci h· of l he fluid next to t he inner q 1inder is fl. ..nd th" angu l..,
.'"loc ily of th.. fl" id n..xt to l he ~urfa ..... o f t he ou le , cylinde, i. ,,' ro. The fluid in
lhe an nula' It'gion ....ill han ' a n a..... "'ll" a "/lula , •·...ocity betw n ""ro a nd !J. Fu,
t....... mfi"""(~ {Ollg cyIind..... th.. a",rag<> a ngul ar ,...Ioci ty ill he a lmos t 0.5 f!.
R.... I jou' na l. a nd bea rinll" a ,.. not infinil ..ly long. In ,..al hearing.,< t .... fluid
is 1' ''' 1 tI " .. lo ..nd I""k"ll" a nd has to he re placed. Th i. i. u.ua lly accomp liAh.-d
m' ra";all~ injectinlt mak.-u p flu id int o th.. hea ring t h, ou gh on e or mote po, ts. lf
Ille po rt . a re radial Ih..n when t his flu id fi"'l e nt "O'$ the bea ' lng. it will ha'", zero
a"!lulu .....loc;ly. The new fluid p ad ually und..rg."'s a ngula ' an't'I",al ion du.. lo
lhe . h"" rinll act ion o f lhe moving n uid tha I is already in th.. bea ' ;n!!. Rut. al Ihe
"" me tim... bee a..... ofthe p......"'e d ilf......ntiaJ .............,..n th.. injm ion point a nd
Ille end of lh .. bearing. Ihe nuid . tart ' mm i ng axially. As a It. th.. n " id path
l race . o " t a s pi. al IFil{ure 10-6 ) a nd may he ejo'CI.-d lwfo.... it c lle. Ih.. a nllu,
lar ,...loci ty ......n in the infinil e cylinder. Fo. th is ...a'o n. t ile ..' a g<' flu id anllu ,
lar >....toe;ty ;n 1}1' icaJ. fully flood.-d. h~-d.ody oamir; beario!!, is Ivp iea lly ""'.....
than o.so.

.i I
. • I
J.

F''9'''e 10 ·6 Ru<l flow .. . journoI be.lnog. FOlK! is in"",..-d mit> the beamg
!h<o.Jg~ one Ot _ POt'" _ th" ftuid filS' """ b'IA'ing. i1 has .....0 .og...
~ -..eloaty. Th" ftuid grodualy .........'90'" . og...... KC t:iondue to !he _og
K t' "" of ...... rnovino;llUd th.lt is "'Ndy .. !he _go EIec"",. 01 ..... ~""" gr;o-
dOe'" _ t..... iNe! POtIIl'ld !he """ 01 the beM;"g. the ftuCl " K'" ""' . 'P,.I
PIIt~ 10!he oM oI!he _,og. ~ n....,. not ",OdJ !he circu"'~"''''.¥1<I'JIa' ~Io< oty
n would ",OdJ on..., , nfi n ~ cylinde<.1h@ ....... "9" ftu<ld'9Jlor wlo< rty is V I
178 The Static and Dynamic Response of Rotor Systems

If the fluid average angular velocity is vavg ' then we define A as the ratio of
the average angular velocity to the angular velocity of rotor rotation:

(10-11)

Thus, the fluid circumferential average velocity is

V avg =Afl (10-12)

A is called the Fluid Circumferential Average Velocity Ratio. It is a dimen-

sionless measure of the fluid circulation around the rotor and is a powerful tool
for understanding rotor interaction with fluid-film bearings and seals. Typical
values of A for a fully flooded, hydrodynamic bearing with only radial injection of
fluid are between 0.35 and 0.49. Hydrostatic bearings, because of their higher
injection pressures and decreased exit time (less circumferential flow), can have
values of A less than 0.1.
If the injected fluid has a tangential angular velocity component, it will
affect the value of A. If the fluid enters the bearing or seal with an angular veloc-
ity component in the direction of rotor rotation (a condition called preswirl),
then A can have values considerably greater than 0.5. This can happen when
fluid is preswirled by a previous stage in the machine.
If fluid enters the bearing or seal with an angular velocity component oppo-
site to rotor rotation (antiswirl), then Acan have a value much less than 0.5, even
approaching zero. Fluid is sometimes deliberately injected tangentially against
rotation (antiswirl injection) in order to control a fluid-induced instability prob-
lem. This will be discussed in more detail in Chapter 22.
Bearing geometry can also affect A. Plain cylindrical bearings tend to have
the highest values of A: 0.43 < A < 0.49. Many bearing designs have been devel-
oped to break up circumferential flow and reduce A. Examples include tilting
pad, lemon bore, pressure dam, multi-lobe, and elliptical bearings. Eccentricity
ratio also affects A, and this will be discussed later in this chapter.
For the purposes of modeling, A will be assumed to be constant.
 ( twptl!'t 10 The ROlor S)'$le m Model 119

Fluio.film Bearing ForCIH ~nd StiffMsSof'i

VI. t-", di>.cuo.wd how a C)til>def rotallng irWde a l>O(iwl.lb>id-filIcod cyIi!t"
..... can ott the nWlt Into moIlOn. TIm io a ~ rn<Jdlol fof a pa.trI. cylindnal,
jo. bNringor waI.
A me the rotor .. mliOC ,,,,,1II ~ wIocir~ rim a counlerdod:wi8of (X
10 Yl directlOlL Whffl the jour nool io perffClly CffI ,",,", in the bnting (f'Cn'n o
trleu}' l'iOtio. € = 0 ). the onl}' ~ iOCl' ng Oft the ,hiOft '"' $bnnng ror.:n a...,..
o.,ord ..m. the flu id n...;oom.I KU bkr a pump. - " l l the- fluid around tile
annular clfoan.Q('f in the bnnng.
Who,,, the " " .... i. CC'IlIc«d on W ..... ring. the r;odial cIn"'...... i. lhe ...me
aU aruu nd lhe ci "",mfom-"ct o f Ihe bea ti ng. H""","",. If a ola l" Io8d i . applio'd
10 tile ' otor. th e ,..I.... io d i"f'la.:ood fr..rn tile ....nte~ ;.;,_ the moving flUId
enoo unl .... a tl"duction ". . ... He bl.. elWin."",," an d mu. 1&I..w do..·n. Th.. dect'k-r-
. Iion of lhio fluid ....,." Ito in . n inc........ in til.. local pr u... in th.. fl" id (Fi!!"...
1(07). Som.. of til.. fluid ...cap.-. u iall y. bul til.. fluid n r th.. axial Cffit.. r of th ..
btartngeannot. The flUId produ...... . ci""' mf........ tialpmuu... ....dg.. tIl. 1 pu.h-
"on the roto r.nd rn"",," il I.. l he . ide. The rot.... rn....... u m d th.. fora' pro-

""_ 10. 7. Tho 100<... on • ftuod·fim b"f\0"9 A "at.."""•.~. Os apphfollO fOt<lr.

_ II """rroq _ ""'9'U' ~ n. Tho mtor ..-s ""lJI"pOl.obQn r ,.....
b'OI produc«l bl' ..... _ on It.. of ~ .ltKII)I ........ 1he bte!hol ..

---.._"".. . . . __.. _--,.

- - . . to ..... _ boo.-....,
. . <#.-
The ~ ~ """'"" un
tt.<_dlhe-...g._ •• .,. _""......... _
rwo corropo--
... -..;._<_
dtt. ... " - . .. .... '" _ .d _ The ......d-...rwo
"""'"" .. ~ _ •• lID tnt - - . " bote_
180 The Static and Dynamic Response of Rotor Systems

duced by the pressure wed ge exactly opposes the force applied to the rotor. The
position vector, r, extends from the center of the bearing (the equilibrium po si-
tion and origin of the coordinate system) to the center of the rotor and is not
precessing for this static load example. Thi s pre ssure wed ge is the primary
means of rotor support in hydrodynamic journal bearings.
The force due to the pressure wedge can be resolved into two components
(Figure 10-7, right), a radial component that points in the opposite direction of
0
r toward the center of the be aring, and a tangential component that points 90
from r in the direction of rotation. Both forces are assumed to act through the
center of the rotor. Th e vec tor sum of these two forces is equal and opposite to
the applied force vector.
These two force components behave like forces due to springs. The radi al
force component, F B , can be modeled as

(10-13)

where KB is the bearing spring stiffness constant in N/m or lb/in, FB is propor-

tional to the displacement and the stiffness, and the minus sign indicates that it
points in the opposite direc tion of r. Such a force is also known as a restoring
force, because it always acts in the direction of the original po sition, attempting
to restore the system to equilibrium.
The tangential force component can be modeled as

(10-14)

where D is the damping const ant of the bearing. The j indicates that the dire c-
tion of F T is 90 leading relative to r (in the direction of rotation, D). In the fig-
0

ure , FT is rotated 90° from r in an X to Y direction (the rotation direction of the

rotor). If the rotor were rotating in a Y to X dire ction, th en FTwould point in the
opposite direction. The term jDAD is called the tangential stiffness. See the
Appendix for a discussion of the origin of this expression.
The tangential stiffness is proportional to th e fluid damping. More impor-
tantly, it is proportional to the average fluid angular velocity, AD. Thus, the
strength of the tangential force depends on both the rotor speed and A; it will
get stronger with increasing rotor speed and increasing A.

Other Sources of Spring Stiffness

Besides fluid bearings, many other elements of th e rotor syst em behave like
springs. The shaft acts like a beam that is supported at two points by bearings.
When a force is applied to th e shaft, it will defle ct, producing a restoring force
 (l\ii pl~' 10 The Rot", S,.I~m ModoN 181

d' l'ffted I",..a n! th~ appli<'d fol'C1'. Shatu. ca n be ""Iati"ely Ilexible••uch a. in

.......od.. ri~ative ga~ lu rnin..... o r very s tiff. Mlch as in ..I<'ct ric mo to ..... ll.-ca...... lhe
.ha ll acu like a ht-am. be-am defled ,on .... ualio n. un be u..-d to ....tima te ils
st iffne..,
Som.- be-aring. <'an act like a p we . pri ng. ..i lh lit tle or no ta ng..ntial "' iff·
ness. Rolling element bt'ari ng~ a nd low·.pe<'d bu <oh inll-' h ,lt"h a. o iH mp...-gnal·
ed hron ,.<- bear i ng~J a"" exam pi.... AI"". hyd wst"tie bea. infl,&. bt'<:a u... o f l hei.
high "Pring .t iffn.... a nd low A. ha ,... ""la ti,...ly 10,.. la ng..ntial . li ffness.
Seal. also conl , ibu t p. mg a M la ng..ntill! ' Iiffn<_ 10 Iii.. roto• •)...t..m.
SOmctim.... t hi' "' iffn u n be . igni flCant ,..hen .....1. a "" Iocat <'d d o.... to th e
mid. pan o f th .. roto •. A oal ing """ l, thaI becom.. locked can p rod uC<' d . a 'l ie
increa ...... in roto . "".Iem st iffn......
:\11 bea n nfl" a"" .upport<'d ,n .t ructu...s l lia t can defl<'ct un<kr load Th u-'<,
bt>a.i ng Mlp po rt p<>d tah and th e limnda tion .yst.. m also cont rib Ul.. "Pring
. ti ff........to Ih.. ' ol or ") t..m.
In o ur d.....e lop ment o f th .. "'tor ",.odeLall """ l'C1" o f ' p ring . tiffn.... in t he
rotor ,y.te m (incl.....ing th e bea ring , pri nll st iffn..... ) a "" comh ined lu p ru<luC<'
the ..ff<'ct ive sprmg !lliffn ..ss . K. Th i. consta nt i. fo rmed from tn.. ,;e" .... / pa "'llel
co mbi nalion of ti,e ~ari"u. d iff<"",n t 'Pring ' ti ffn........ We ,..ill u,... a n ",! ui"alent
"Pring fo rC<' .......or. Fr.. to de ... ribe Iii.. 'Pring force:

no- rs:
Like t lie bea. in!! ~pri ng foret', f j is proportiona l to K. and tn.. m in" . s ign ind io
cates l hat it a l,,-ap pu int. in t he opposite d irect ion o f r (Figu re l o-8j.

Fig u' . 10-3 '"'" 'P""9 """~_ Th<'.CII<Jf i. d,,-

IQ<...:I lO .n .ortli"I'Y P<J'OlOn, r , from ' be Otigi-
Nt """"""'- Th<' 'P""9 ""~ i>....Y' 0"''''...:1
b.d ' oward thi, ong;na1 poootiof\ _ i>"""",",-
llONI ,,, _ m. ~ 01 ..... di.place-
_ _ ...... . prr.g ~ K. Kcon ........ t
""""" UIIlroo>s conmt>.Jt>ons in tbe fOIOt ' Y'-
'"'" ,,,",,, bea<,ng ..,.ing "ofIne». be"ing ",p-
pot! ,,""""\ _ fourdonon """"",-
182 The Sl ~ Ii( ~nd Dynam ic Respo nse 01Rotor Sy>IINl'lS

Th.. Oam p ing For c..

All ro tor . y.t..... . run... int ..rnal fore thai ca u", th.. d i, s lpation ol ..n"<!tv.
Frict ion i. a n n a mpl.. o f a d is. ipati,'" fo FrictIo n fo ...... can ,...,.., It from con-
tac l u f t l><> ro tor ..i lh Ih.. . ta to Lt h.. mm l.'TTl('ni o f attacl1 ro pa ris o n tl><> ' 0 10 '. o.
fro m in t" m al frict ion in th .. roto r mal..rial il ...ll(al"" calkd h.'sl""tic d",npl1Jg ).
.~11 0111><>... frict ion fo ill be 1I1T1<'d to be . mall
Th" da mpin!! fo iA !!"""ra tro h..n a ' . ...0 11. fluid i•• h..a red be"'"l'E't1 two
su.fa~"l"S in tI,,,
I<ol i.... mo tjon o' ...h " n Ilu id pa...... Ihrou " h a . mall o ritic An
... a m p!<- ... t fore.- that is !!"o" rlliro in a n aUlo mo bil.- ,;hoc k a b.., T....
s l>ock a"so ""'" hit, a p is to n Iha t fo.e... Irapl'ro flu id th roll¢! a n n"f, con·
....-rttn!! m....' hit.lIcal .. nergy into h..at in th" .." rkin!! fluid.
A" ..tl>t-. .. u m l'le i. a he"' l bein!! pro.... lkd th rou gh th at.... If th.. ..nllin..
s top"- th.. vi"" "ity 01 th....."t" . ..il l d i;..'lip.at" Ih" kin"' ic ..n /lY of Ih.. boat, a nd
it ..ill rom.. tu a st up. T\t.o> foret'S acl ing h...... a .... a co mbinal ion of . h....rinll a nd
p.....,.'u.... d.ag_
Simila r da mpi n!: fo""," OC<"U ' i" a flu id-film I"'a ri"!! " h" o th.. rolor ...." .
........... in Ih.. bea ri"¢. Put .impl~; t da mpi,,!! (,."'.... ' l<'C'Jr th ",ugh a combina·
tion o f . h..arinll o f th.. flu id a nd p ,.... drafl. T\t.o> a m"u nt " f da mpIng 10"'"

Fi<Ju'" IG-9 Thodamping lofer. Tho rotOf h.o,

.n insWtt .neou> _ity vector. r _Tho d.1T4'"
ing 100:.. i' propottJOf\III to tho both t!w "*
""'" d.n'p<Ig con".o~ D. ond tho mogM~
01_ _ "Y. • nd n act> in • dO'KtlOO <>QI>:>-
"'.. to t!w w locity _ .
~all"d .. m.1..a to tho- \rioOI ~' of the JOUrn;d in th~ """'ring. .'l.ddilk>n.o.l
damp "lIt Can en"'" [n lm ro l." int..,-",1i'>fl .., th th~ _ ,king flu id tha t ... m lun.u
il. ~ute lluol r ......." .... d r"!t ..IT..eta M"" n..t, in a <trict ...n..... "<luiva!..nl to \1OC<>\l.
dl1lfl- l l....~. for umplic,ty........-.I]I" mp p~u ..... dr"fl rff"". inlu OUr dc-ocrip-
t...... 0( tn.. &oml"1lfII1<Jrn lh&1 .. ba...t Of\ \'Hcou' d,~
llwd<unpiRf! foono >TCtor. F,., it ".oJ k d by

n... da mpmlt fo.... i. p roponiu " a1 lo ,.... d amp ,"ll oonManL I J, <lnd ....h in a
d ir.-ctiu " op p."it.. to t .... i".t"n'..n ~"'''''ily ",,:t.... , ( fi jlU"" l o-9 ~
Th.. ",aO(J1 ,luoJoo of th.. da mpi ng r d.,...nd. (>fl tout h t.... da mJ""Il a'ldl h..
,.......lty: ,f tto.. vn.>cily io ....ry om.aIl t he n tn.. damJ"~ fof"" . -.I] a1.., t... .m.o.Il
Th;. en happm in . OfOf .;<'1................ nodal permt ;.Iocal..a i~ a fluld-
film t-rinlil- Tlv wnalI ,;bmion at tlut Io<.ation.-.I] prodUCO' a rn.ti>~ tmaI1
damJ"~ ror.,." C•...........dy. If a rnodf. ..... J""produ<-n ala"" a.......nl of ..;brali.."
inWdoo a fluid -film lwarin~ 'hO'n lhe da,npi nll fOf,,", pruduc...t . -.1] bl' !'1.'Lollmy
Ja~. Th..... "a"a, ion~ in d amp ''lIt fon,", art' .... all'd to th.. f'lllk,<"pt o f modal
daml'mN " 'hich w,ll be dis<:u,,,,od in CI'"I ~ ''' 12.

Th. Peortu,b.alion f ore:.

'\ l"'"urlJot_ IS a di\turt....... to a 'Y"'Mn. PO'nurlMtion f",,'n an' ....1....,.]
fo,n.. lhat d"turn IIv rotor from it. ftfUilibriwn !"",lion. n...... forcn c",n br
<t"'ltc. ... .....,. can bl' dp "" moc. SlMjc fW"Iurb.otion "~haw oonR.I.nl """fI:R1'
ludP and direction and prodUCO' ,u.tic d..Ila1ioru ~ in rotor p""'IlUR.
~' include It',,,"y Inad. in horirun. .. 1 mad "" fluid ........tion Inad. in
p um ...... un"'lua! .t<e" m ",Imi ",io" 10,,,Ja in . I..a m •",to"",.. ot h.', p"""''''' ......1"-
or a """,bi"al ~,n .. ft~. Mioa!'p' mf'fll can ca....... , u tk rad ",lload.o tu app..ar
bo.o..,... of load I f... am<>l>ll b.-....i~ Ca.,,,!! d<.f<ont1.lItioa and pol''''/! 'n
oon "",ft b.-anng ppoo1.... lt.IJng to ".,...'P'........ t and ,n.. .PP'.......,..
of ·
i<: rad>aI ......... Oy......ue fW"IWbd ..... ""'"" penodocally ~ '" magn iludo-
or d'A"Ction. Of t..th, ....... tb<'y produa- a dp....mic " " .... ~""' ..... """J'O"!'" lhaI
~ .. \;bnolion. Unba4noe ia an ~wmJ*.' of .. ,ut.. ting. ~-nd>ror>< .... pt'r-
tuTblttion fU" T.
.\ lany ""lOren uf IlOn<ylk-hru""u' 1","u rba' ~ln U ~.I in ,,,t..r ~~I ..nl'"
Kot.. t "" .1...1 i" enm...........'" C..n pmd ll<'<' a ""....ynm "'nuu~ n,tahnlt J""" ur -
boal~'" '><'-'0'. Kat uymrrwt.it'oI (ouch .. I'f'>'luad by a oIwfl a", H coupli"fl
J" ' ~'" Ill.. "'nd J"'"" can c",u...... prnynclt.......... prrturt.u ..... Imv-1-ing
...........-ution caliai-"" prod~ ~ ~ pnl1P'bltt...... oftto.._.,...-
,~
 •-
~.-nn
q
, ~ , s e sl p~ ~
;' 3 -:t' ;'''' "'2!:l -2,. i!~

~
_
t,....£
HiHWH "f
=,'
/" ' P"',·.lH
'! qH
- ~ '"'<!p_I" Ji "n
. 1 I, ~,
_.
l
"

UlHPH
widf! l
i i
,;: .... ·5-~" ~ 1".
: l ;r ~ ;"~i r 1I
" I'l't
l I~

!'f!'!FJi l,t1I' 'ir s !tS ; i ; ~

, . }. ~ ~ " 'ti''~ Ii ~,
u 'f r..
tjHt ~Hi [! i . ~i';:l ' n; ' 1sf!I 11'1 q-~'i. .;:
\. ~ f ~. "~~ '.l s~ lr •~ •! lj·i
i ! : ·l " i r·
! iI' , ! I ',l:
j1 f \~ '<'3,l ., ",
'" : ;; p, ! "q '~ !r ,~
.. h. I
.
!i! e
-\ '!:;!i' ~: ~P' P
' i hlB . i"l p
~ -i
'! !:"L~
• • 2. .3 1-~ '_ ·e:a.i.
' · 'r 1~ .. I
j ~ 5; ~ . T
"T' if ' ,j, .. _~.w;
'I~i!.! l"' II
",:;; I:~ '~.III t-=t
'~ ~_!
·1 i: ....
k '
' i
... . ~
:; 'i ' ".l
·'. l;::J ---_~ 3 _, l;"l _' . "
' ' ' 11- { ~!"
'" ,;! " ':r;~"3
.·l5.;0•~ --...e ,.-=- .-='-
.. 1; 0I. ;r ",.
~ , if
,~...
Th. F,.. Bod.,. DY<j.am
Sow I","" driintd tlor for<n act '''l': un I"" m10r ~~ ........ _ can con ·
.. ruct 1M r ~, d " p"'m Th ... i. a ~,mpMo d~ thai ~ 1M rotor mu.
and I'" ...,Ii ~......... offnrt'n act inll o n it.
r igu... \ ().11 Ahmo... th .. f..... ho<l}' d'''llram. Th .. rul ot ..,t h coOC<'nlr. ,M
rna......\ 1. ",lilt... ilt • Ap<'l'd. U. in ; In X 10 t·di ..'(1: ,.. n. a nd il ill d i&p ldITd from th.,
l'<lu~ihriurn pooUl " .n to " 1""';' ,,, n, , . At thai p... ition , the rotor io mO'.; "III a' . n
arbitl'ilry. ill-ola nl,,,,.·o... ,,,,I,,':lIy. r .
•-\11 ~><'T'O'O act ,,,!! on tlw .." u .... .......t to act Ihroujth It... n>nt.... ul .........
of W n il .... n... oprinll oI'ff ~ f~. 1'0 11>1$ t-io; tow.rd It.. ""l-'ih hrium
p<>a'UorL 11>0' .....l iaI.l ,ff_ f.....,... F l" p<-U '10" from r In tIw di,.,n"", of
roI.aUon. n.. dampl lljl. f<>rco'. f '& point. in. d ired ion ~t .. to tJw ir!aUnl..
_ .............y""'<1o<. r . n... n llM I"ft: J"'flu rbooc ..... ~. F,. lI .......n .to '''fI'l'
Iu po:>wtoon ~ lho- in.um d .. .....",na....-""'I '><"nl.....

,,-",10-11 n-..r.n, ~ ~..... '"- _ _ CO........ _ _ M .""'"

• • opH(I.O. .. .... x"" _ _ "..
~ ~~lIOa _ _
- - . ' _ . _ • ........, i- .<IIb< .. a<t>n;tOtl ".<1
"'«t-..". _ _ d _ d _ _ "- bm. F•. _
bo<* - . , . . . . , """I'"wI ""'*"'- Thol.-.,"'wlll_ farot # .. _ 'lII" _ ,
.. .,..., .."'..... d_~ ,n-..~b<:" #l>'_ ... doE<hD••
_"'...., ............... -.oy _ • '"-~ _ _ b<:"
F~ " .........'"' ""9UIar _OOOI~, _ ... . -
_ ~ ~<x<un.
'86 The Static and Dynamic Response of Rotor Systems

The Equation of Motion

The rotor must obey physical laws in response to the forces acting on it. We
apply Newton's Second Law: The sum ofthe forces acting on a body is equal to the
mass ofthe body times the acceleration ofthe body:

(10-18)

Substituting the expressions for the forces , and being careful to keep track of the
signs, we obtain

(10-19)

Rearranging, we obtain the equation of motion in the standard form of a differ-

ential equation,

(10-20)

This is a second order, linear, differential equation with constant coeffi-

cients. The appearance of this equation differs from a standard mechanical (or
electronic) oscillator equation because of the presence of the tangential stiffness
term,jD)..fl. This term is what makes rotor system behavior so interesting, and
it this term that is ultimately responsible for fluid-induced instability. If there is
no fluid circulation, ). is zero, and the tangential term disappears, making the
form of this expression identical to the equation for a simple oscillator. Note that
the rotor system internal elements appear on the left side of the equation, and
the applied, external force appears on the right.
If there were no forcing term on the right, the resulting equation (called the
homogeneous equation) would be equal to zero and would describe the free
(unforced) behavior of the system. For example, if we moved the rotor system to
some position away from the equilibrium position and suddenly released it, the
resulting rotor motion would follow the rules defined by the homogeneous
equation. Free vibration will be discussed in Chapter 14.
The equation ofmotion is a set ofrules that governs the behavior ofthe rotor
system. The acceleration, the velocity, and the displacement of this rotor system
must be related to the applied force by this equation. However, the equation of
motion tells us very little about the vibration behavior of the system. We would
like to know how this rotor system behaves over time, or what the amplitude and
phase relationships are between the force and response. We now need to convert
the set of rules into something more useful. This conversion process is called the
solution of the equation of motion.
 Chapter 10 The Rotor System Model 187

Solution of the Equation of Motion

To solve Equation 10-20, we must find a displacement function that, when
differentiated (converted to velocity and acceleration) and substituted into the
equation, makes the equation true. Since this is a linear system, perturbation at
a frequency w must produce a vibration at the same frequency w. We will assume
that the amplitude of vibration will have some nonzero value and that the
absolute phase of the vibration response will be different from the phase of the
perturbation force, 8.
Fortunately, the solution of this type of differential equation is well known.
We assume a solution of the form,

r = Aej(wt+o ) (10-21)

where A is the length of the rotating displacement response, or vibration vector

(the zero-to-peak amplitude of vibration), and ex is the phase, or angle, of the
response vector when the Keyphasor event occurs at t = 0.
We differentiate Equation 10-21 to find the velocity and acceleration. This
has already been done in Equations 10-9 and 10-10. The displacement, velocity,
and acceleration terms are substituted into Equation 10-20, and, after some
algebra, we obtain

(10-22)

In this form, the expression on the left side of the equation represents the
rotating response vector. The terms in the numerator of the expression on the
right represent the rotating force vector. Note that each exponential term has
been separated into a time term and a phase term. The time terms are responsi-
ble for the rotation of the vectors; the phase terms convey information about the
absolute phase of the force and response vectors. Because the time terms are
identical on both sides of the equation, we can eliminate them:

(10-23)

This equation describes the amplitudes and absolute phases of the vectors at
the moment that the Keyphasor event occurs (Figure 10-12). In the figure, the
response vector is shown lagging the force vector, but we will see below that the
188 The SI"Ii( ....,;l ()yni>mi ( 11<>""""", of ROlo, Syste m.

"""P"'n'" ' -("ClIO' u n lead Ih.. forCf' •....,10' und.., wm.... ifCum~I'1U1('''''' Th ..
d .."" m;"" to. of Eq ualio " 10-23 i. cal loo I.... no nsynchro>wu5I1yn(""ic St,jJnt'JJ,5.
In ito simpk.t fOl m. Equallo n 10·23 .Iat.... l h.. foll"", n!\,

FOIc..

\ , bm lion is Ih.. ral io o f t.... " 1'p1ioo f"....., 10 Ih.. f)yna mic St iffne••. When""e."~
mea.,,", .,b.a ho n (fo. elam ple u<ing a vibrallo n mo nilo . ). " ... a rt' act uaU,·
mea. " . inl\ th.. '-a il,,.. o f lhi~ ...l i,L Tho,,- a change in vibration ""nIH" cau SRtl by
..il"", a chang.- mjm__e or a chang.- in 1M D.vna mic SlifJneS!; or balh. This i. a n
important mac hin" 'l di~", ....tic. (0"""1'1. lll'ca U>e all of th ...., e l..menls a ,..
''-'Clo , quanl it ies. chaO!l'-' ca n appea r a< e i' hl'. a cha ng<> In a m plilud.. or pha se.

,
\

f_ e ltl-l l fOI<:.. _ .. br..,.,.., 'floPCI..... _ ·

'01>- The two _
ttn ........ h """'" .ngulor w.m
~~, .'" """'" ~n .. _ m<JOl<'IlI 01
_ ~<Of , The 100:.. vec10r .. """'~
Ir~ _ ~ _tor (II'>< " " ' _ '''9'
_ force). but , ..... ~ .. GO~ be ''''''_
 Chapter 10 The Rotor System Model 189

Nonsynchronous Dynamic Stiffness

The nonsynchronous Dynamic Stiffness, K N , is

K N =K -Mw 2 + jD(w->.fl) (10-24)

Dynamic Stiffness is a complex quantity that consists of two parts, the Direct
Dynamic Stiffness,

(10-25)

and the Quadrature Dynamic Stiffness,

K Q = jD(w->.fl) (10-26)

Direct Dynamic Stiffness acts in line with the applied force; Quadrature
Dynamic Stiffness, because of the i acts at 90 to the applied force.
0

Dynamic Stiffness is a very important result of the model. It is a function of

the perturbation frequency, w, and contains all of the rotor parameters in our
model, including the rotor speed, fl. Dynamic Stiffness is the black box that
transforms the dynamic input force to the output vibration ofthe rotor system. It
is a major key to understanding machine behavior, and it will be discussed in
detail in Chapter 11.

Amplitude and Phase of the Vibration Response

Equation 10-23 can be manipulated into expressions for the amplitude and
phase. First, let

(10-27)

Multiply and divide the right side of Equation 10-23 by the complex conjugate of
the denominator to eliminate the j terms in the denominator:

.
AeJO = -- - - [K
Fej6 D - jKQ ]
----"- (10-28)
K D + jKQ K D - jKQ
190 The Static and Dynamic Response of Rotor Systems

Now, multiply through and combine the exponential terms on the left,

(10-29)

We now have an expression that mixes exponential and rectangular notation.

The exponential form on the left already separates the amplitude and phase. The
amplitude of the rectangular part is found by taking the square root of the sum
of the squares of the direct and quadrature parts:

A
(10-30)
F

which reduces to

F
A=-,==== (10-31)
~Kb+K3
Dynamic Stiffness controls the difference in absolute phase (the relative
phase) between the force (the heavy spot) and the vibration response, {; - 0:.
This change is found by taking the arctangent of the ratio of the quadrature and
the direct parts of Equation 10-29:

(10-32)

Solving for 0:, the absolute phase angle of the vibration response,

0: = s- arctan [ ~~) (10-33)

Thus, 0: differs from the location of the heavy spot, {; (the phase angle of the forc-
ing function) by the effect of the Dynamic Stiffness. The negative sign indicates
that when K Q and K D are positive, the rotor vibration vector lags the heavy spot.
 Ch.o pc~ 10 The Rolor Syu.... MocHI 191

The Attrtuc» A"9Ie: Rotor Fl"9O"W to ;I St;lti( R;Jdi;lllNd

.\ ,.... toral oJ. ttwo ~ t. to .... bow thr rocOl" l'HJIO'odo to ttwo,opplicatJocl
of;l!Ut>c IWiw kl.d. \\" .....u ~ w rut;II l"fl un~1>« wtth ;l o.e.tIC loMl.
f -. W~ tiO' d"'t 1M ~ i. ;lJ>plied .wt>e:ally do..-nW1lrd .. tuk 1M _ i.
rot;lti"~ ;lt m~ I f"""d. Jl. F IS 1101 rotati ng. 10 th is ... !I.f""CW catoe of . non-
sync hronous p"Ml1rbat ion ru l'U' '" t h~ fJl"<[U<>" CY. w', i. , ..ro. Th"... th~ rom·
po".."t. of th.. J)~"II.mk St iffn bo>«>m~

)I;,c,u!hlt. k>r tlus _ ic "-:I catoe. thr Du'K1 D)-....mic Stiff...... to idmli<:al to
tlMo opnllfl.hlf_.nd tlMo Qaldratu ... 0, .., Stiffnroo it idmticll (nnpc
lOr thr loitlnl to the- lanfto>ntw ~ ,ffnno. I»t it lltill({ I,"","" ~Jo.mo>nu into
EqlJltion l6-ll givn ... th~ prp<hctlPd ~pon"" p hlw 1lIg1~.

" '"' ~ + u eta" KMII

[O (16-35)

wtwft _ h..~ pIIOM'd the- nrpt_1lipt Ih~ 1M ~ I ful\ction.

TIm rn<dt i. illuAtn1l'd 111 F...... 16- 13. T1w' pooiI- NfIJ1 ill Equation 16-35
,ndic:dn that thr ~ in roc.. JI'O'ltion ~ " lII-"o t...pooitnT. prodUCIll({
. eba ..... ill poIJt>Oll ill aJ>l- IetJddJ.f£<1ioo. Tluo is nacth· wlwl: _ rIpt'Ct r..

_r_.. . .
Fog"'" to-t ) llotor_fO • ...., ,...,...
lood "'-_ac1""9~ T ~.
~~ec: ...,
,",,- , ..... F "'" " ' - ongIt. ... d r ..
'-90' "'"" tho pt,.. ongIt. ~ d F '"""_.
i

....
enceWl ..,, _ _ ~ .. _ l .. . ....

~-
192 The Static and Dynamic Response of Rotor Systems

a rotating, statically loaded journal in a fluid-film bearing. The pressure wedge

forces the shaft to an equilibrium position, which is in the direction of rotation.
The amount of the phase lead will depend on the relative strength of the tan-
gential and spring stiffnesses.
Note that this situation is identical to the basic definition of the attitude
angle that was defined in Chapter 6. Thus, the attitude angle, ![t, is defined by the
model as

![t D>.D )
= arctan (K (10-36 )

Synchronous Rotor Response

A good definition of synchronous is at the frequency of rotor rotation. If we
lock our nonsynchronous perturbation force to the rotor, it will rotate at the
rotor speed and become a synchronous perturbation , where w = n. If we sub-
stitute D for w in Equation 10-23, we obtain an expression de scribing the syn-
chronous rotor response to unbalance,

.
Ae JO =
mr.
u
sr.» (10-37)
[K -Mf2 2 + jD (I->' )f2]

Because we are now modeling synch ro nous (IX) behavior, {; represents the loca-
tion of the heavy sp ot, the unbalance in the rotor; the vibration response
ab solute phase, o, represents the high spot of the rotor.
We can see that synchronous rotor response is actually a special case of gen -
eral, nonsynchronous rotor response; however, it is the most important from a
practical point of view. Ordinary, unbalance-induced IX rotor vibration is a syn-
chronous response. The rotor behavior described by Equation 10-37 will be thor-
oughly explored in the next chapter.

Synchronous Dynamic Stiffness

The denominator of Equation 10-37 is called the synchronous Dynamic
Stiffness:

K s =K -MD2 + jD (I->. )D (10-38)

 Chapter 10 The Rotor System Model 193

The synchronous Direct and Quadrature Dynamic Stiffnesses are given by

K D=K - M ft 2
(10-39)
K Q =D(l->.)D

The Direct Dynamic Stiffness is identical in form to th e nonsynchronous

case in Equation 10-25, but the Quadrature Dynamic Stiffne ss has a different
form than Equation 10-26. These differences will be discussed in the next chap-
ter.

Predicted Rotor Vibration

We have already examined the predicted behavior of the model to a static
load , and we have found that it produces a reasonable result. We will now exam-
ine the predicted rotor vibration response for both nonsynchronous and syn-
chronous perturbation over a wide range of perturbation frequencies.
First , we will look at rotor system behavior for a system with low (subcriti-
cal) damping. A system like this is also referred to as an underdamped system.
Low damping means that the Quadrature Dynamic Stiffness term, which
depends on damping, is also low; this is a typical condition for most rotating
machinery. The parameters used in the model are summarized in Table 10-1.

Table 10-1. Mo del pa ram et ers

Parameter Low KQ High K Q

D 52.7 x 10 3 N . s/ m (30 1 lb · s/i n) 1.58 x 10 6 N · s/rn (9.02 x 10 3 1b · s/i n)

n 900 rpm 1500 rpm
M 1000 kg (5.7 lb · s2/in) same
K 25.0 x 106 N/ m (143 x 10 3I b/ in) same
A 0,48 same
m
ru
0.0 1 kg ' m (13.9 oz · in) same
s 45° sam e
194 The Static and Dynamic Response of Rotor Systems

With two minor differences (zero-to-peak amplitude and the mathematical

phase convention), the model output is equivalent to a set of startup or shut-
down IX vibration vectors taken from a single transducer. In this simulation,
since all angles are measured relative to horizontal right, the vibration trans-
ducer is mounted at horizontal right. The rotating heavy spot of the perturbator
is located at 45° from the transducer, in the direction of rotation, when the
Keyphasor event occurs.
Figure 10-14 shows the Bode and polar plots of the predicted behavior of the
model. For the nonsynchronous perturbation, the rotor is operated at a constant
speed, n, of 900 rpm. The perturbation frequency, w, is swept from zero to 4000
cpm, and the model produces vibration vectors that are filtered to the perturba-
tion frequency.
For the synchronous case, the perturbation frequency is set equal to the
rotor speed (the unbalance heavy spot is now the perturbator), the rotor speed is
varied from zero to 4000 rpm, and the model produces response vectors that are
filtered to rotor speed. Thus, the horizontal axes represent cpm for nonsynchro-
nous perturbation and rpm for synchronous perturbation.
The phase produced by the model is in mathematical form, where phase
lead is positive and phase lag is negative. However, the phase in the plots in the
figure is presented using the instrumentation convention, where phase lag is
positive, increasing downward. Phase is measured relative to the positive X-axis
of the system (the location of the transducer).
The amplitude produced by the model is zero-to-peak. On the plots, the
amplitude has been doubled to produce peak-to-peak to conform to the instru-
mentation measurement convention for displacement vibration.
Note that, at frequencies near zero, the nonsynchronous response phase
leads the heavy spot location slightly. However, at zero speed the synchronous
response phase (the high spot) is equal to the heavy spot location. This is an
important finding with application to balancing.
A resonance amplitude peak is clearly visible and is accompanied by a sig-
nificant increase in phase lag. For this low damping case, it can be shown that
the nonsynchronous resonance occurs near

w=~ (10-40)

Equation 10-40 is also called the undamped naturalfrequency, or, less accurate-
ly, the natural frequency of the system. This is sometimes referred to as the
mechanical resonance.
 Chapt ... 10 Th,. Rotor Sf""" Model 19S

'. ----
.,'...
J:: il
••
"" ,..., ,Oi:
_ .'
_..,,,,-';~ ' _ ....t w _
".
,-
.... ..,_ 'lOll - .
_ ".._ ....

f ig u... ' O-l a. Bode MId p<>Lor plo'" of p'.,licl<'d 1Olo< vibraTion fol low Quad''''~
Oynornl[ "',.,.,...'- 80lh nonsynchron<o.lS (_ I • .-.:1 sytICIyonou> (9'''''') Ptrturba-
bon rcwIn . ... . - .. T~ hNvy 'POl d"""oon i> .no-. in ~ . To""'ke 1ht ....ulIS
consOU.." ""'h who1 wou ld r.. rnN>U ~ on ' ma<;hi~, 1ht omp;tud. olin. ""Odool
1>0, - . do<bled '" prod"". PN!<-to-PNi<. ond t~ ph. ~ Os <IIown in ,"" instnt-
"...""""'" ~ntion.W'he<e """SO' ..~ ~ _ .. d_P~ Os ""'''''''od<@i -
.,..., I<> the p<arI~ X-... i, of ,....ySte....
196 The Static and Dynamic Response of Rotor Systems

The synchronous resonance occurs near

n=Ji: (10-41)

A commonly used term in the industry for a synchronous balance resonance is

a critical, or critical speed, but the term balance resonance is preferred when
speaking of a synchronous resonance due to unbalance. At this frequency, the
phase of the response lags the heavy spot location by 90°.
When damping increases to a supercritical (overdamped) value, the predict-
ed vibration changes dramatically. Figure 10-15 shows plots for both nonsyn-
chronous (blue) and synchronous (green) perturbation. For the nonsynchronous
case, the rotor speed,n, was set to 1500 rpm.
At low speed, the nonsynchronous response phase leads the heavy spot by
about 80°. This is the attitude angle ofthe system. Using the parameters in Table
10-1 and Equation 10-36, it is calculated as 78°.
Because the system is overdamped, there is no synchronous resonance.
However, there is a nonsynchronous resonance near

w=>.. n (10-42)

This resonance is sometimes referred to as the fluid-induced resonance. At this

perturbation frequency, the phase of nonsynchronous vibration is equal to the
heavy spot location.
Both the mechanical resonance and the fluid-induced resonance are differ-
ent manifestations of the same thing. Recall that our model is only capable of
one resonance. The frequency of this resonance depends on whether the system
is underdamped or overdamped, and on whether the perturbation is nonsyn-
chronous or synchronous. The fluid-induced resonance is only visible when

1. The rotor system is overdamped, and

2. The rotor system is subjected to a nonsynchronous perturba-

tion.

Both conditions must be true. Because operating machines are typically under-
damped and subjected primarily to synchronous perturbation due to unbal-
ance, the fluid-induced resonance will never be visible under normal operation.
For a typical machine, the resonance will occur at the balance resonance speed
given by Equation 10-41.
 Chapll'f 10 Th RotCII' S~ .... Mod... 197

'"
," .

I"',.
I
--
\ ------------:-----'''-~
#- ---
I o· ,
•,
•
,.
.', , •• -.. - -'. ,
~r' ..'· .......
",,, """......""" """ """
•

•
I ,.
I ••,. •

--....._. - .... -
'O ......... ~ ...

•• "" "... "'" ,...

~ " "..

F;g.- 10- 15 Bodo . Cod POI.' pIoU <l P'e<lO:1od MOO' ";b<. lion few ~ j g h Qu.d.. , u,,"
!>l'n.mic St;tIn...~ 80th non"""hronou, (bI",,) ond ,ynchronou, (9'......:1 ~ ­
non mults .. ~ -'>own. Tn. ~ ,pot lout"", I••_ in ~ . Tn. ...,p1nuclo! of tho
modooI ..... _ n doublo<l ' 0 produ<~ I'Nk-to-pNl<. and tho pho", is "''''''''' in the
""tnl" - 'Wion ~lIOn""' ore pt>."'...., . .
PO>IIiYe <!<>wnword.P_ "' _ ·
u ~ f@l rolt>o!_ X-4'i.of tho ' y>l~ m . rn. 1ow .~ non,ynch'''''''''''
phno _ , ,...., 10 tho hN vy spot ,~o"ItS~ . no(Udo, ' ''9~ of ..... ' Y" e m.
198 Th ~ Slal;c a <><l Dynam ic Il ~ . pon ~ 01 RotOi Sy. tem.

Nonlinea ritie.
T",.., basic aM"mplions uoed in I.... d..ri' -al io n of l h.. m. odd ...... thaI Ih.. 'JOS-
le m isli n r a nd Ihat Ihe rotor sr"I ..m p<l ra mct..rs a....cr>nstant. Ro lto r shaft ..nd
bearinjt ppo rt .Ii ffn, ,'<n a..... for th.. m<>st p<l r1, ;nd..pend..nt 01roto r po.ilion.
Fluid· film bca rinjt a nd ".....] pa mmct...... a.... al'pro,im..td y wnstant a t 1_-
.....,.,nt ricit)· ratios; h"".... "l'I. Ih"y ..,m c hanlle rap id ly al h igh n trlciIJ· rat ;',.;.
Th.. Irp ical bc ha" or of ' prinjt . tiffne da mpi njl. a nd>. u. {'('C .. nt ricily
r..tio i. sho.. "O in Figu 16- 16. Sprinjt . riffn i. approximalely cr>n. ta nt at 10" -
..,c..n tricit)· a nd inc d ra ma tically nea r Ih.. bca ring . u rfac Ila mp ing al'"
incr..a.... n..ar t h.. bcaring ... rface. ", hil.. >. dec ....a..,•. 111.. d ec a.., in >. i. ...a·
..mable ", h..n ~..,u cr>n. id..r l hat llu id c i.culat ion is rMt ricled as th.. rot or
approac..... th .. limiu o f a,·a ilabl.. dea ra n..... .'\Iso. hyd rodyna mic bca rinll" .. -ill
usually l ransition to p<ln ial lubnc..tion at a hi¢' .....,.,nlricily ralio. d ra. tically
.-...Judng >..
I " spit.. " It he f.. ct thaI th...., I"' ra m..t .... ar.. not constant land Ih.. ..... ult iOjt
d iff......ntial .... uation ill no nlin....r). . ot m ht>ha'ior d not uSl1iIlly de " a t.. fa r
ftum Iin..ar ht>ha,'i..r." nd ou r simpl.. m..,it'l ....mains u. ful..'\ft... an. most ro tor
spt..ma ro n in nu id-film ......rinll-sal mod...atelyh igh .....,.,nt ridly ra tio•. and the
....ha.' .. r .. r most "J"'tem. i' p.-...Jict..ble en",, ¢, to a llow l>ala ndnjt using tec h·
n iq" ... de ri' -...:I ffOm a lin..... mod..l. Rot or ht>ha" o r i,approximalely Iin..ar .......n
.·i....-...:I in .. . m.. l1 ....ginn a rou nd th.. "'lu ilib riu n' p.... ition ("'hich can bc al h igh
t'('C:f'ntrid ly und..r st..l ic . ad tal load ~ L:....all~; 'ihrat ion in roto r . y.t..ms i. small
..no"gh to ""t i,~~ t his a l'p m>cima t ion.
.' \n nlin....rit ier. <an appear f...m o th..r ...u. r .... lIub conlac l n....r th e
mid ' p<l n o r a rotor i. a con" nrint thaI signi ficanlly inc.......... K. either m.. m..n-
tarily o r COfllinuou,ly. lo<:.... J><".. in support .~-"t ..m. ca" p rod" .... int...min ..nl
o r cont i",,,,,,. dec....a...... in K. Muc h of d ialtn,"" ic mt1 lw.odc,I' >g)' i n\'(~...,. C't" .....
la ting c han ll"" in ,,·.' ..m I:reha,i or " , th c hang,," in K. Th.. ""nlin..a'it ir's pr0-
duced ~' 80m.. ma lfunctiona f...-d throu!U> ""r li""", m"d '" in II ",a~' thaI .....can
u..-f"II~ inl.. rp w t
Vib.-ation ailtn..1" -a,...form d i' I" rt 'o n i. e,-;.... nce .. f tht> p•..,..,nee of n" " lin-
..arit i" Id..al.lilK"ar roto r spt..m ,-;bm lio n ......"',,,... 10 " nha lan ce ..il l ptnd ure
a .inttle.•inu",idal sillnal. NonliJ><"ariti... I'rOO,,('(' de-oi a tio n. from sin"."idal
,,·a,...form, l hat r,-"ull in more complicat ed orbits and harm n nic ""<1"" in 5P<'<"
l ro m...

The 8enelilS and limita tio ns Oll ht' Simple Model

Th.. mod..l lhat " ... ha.... p.......nted has ........ raJ majo r .... n..fit'" il i.....>l~abl ..
in an a nalyt ical fo rm thai ia r..lali,-e1y ..asJ· to und........nd. a nd IIw.- '"'l""tionslhat
com.. from th.. model pr",id.. (lood insillhl in to IIw.- OOSK" o r rot-<>. I:reha';"',_ W..
 Chapter 10 The Rotor System Model 199

Bearing
center

Q) I~
~ i' ~. Q)
ro , -, , - -- _ . ". -- ro
~
u • ~
:J .
' : :
:
I
1 A ........... ..
/ U
:J

Iz ' c J
1 .s 0
Eccent ricity ratlo. s
.s 1

Figure 10-16. Qualit ative plot s of flui d-fil m bearing

parameters versus journa l eccentricity ratio. Stiffness,
K. and damp ing, D, are minimum w hen the journa l is
at the center of t he bearing, and t hey are approxi-
mat ely constant for low eccentr icity ratios. As the jo ur-
nal nears th e bearing surface, stiff ness and damping
increase drarnaticallyA behaves in the opposite way.
When the jo urnal is at t he cente r of the bearing, ), is
maximum . As th e j ournal nears the wall, t he fl uid flow
is increasingly restricted, until ), nears zero at the wall.

will use this insight to examine synchronous rotor behavior in more detail in th e
next chapter.
As we have mentioned, vibration is a ratio, and changes in Dynamic Stiffness
produce changes in vibration. Dynamic Stiffness is a function of the rotor
parameters of mass, stiffness, damping, lambda, and rotor speed. By relating
vibration behavior to changes in rotor system parameters, the model provides a
conceptual link between observed vibration behavior and root cause malfunc-
tions. Thi s is a major advantage of this modeling approach when compared to
matrix coefficient methods. The basic relationships between simple rotor
parameters and malfunctions will be exploited throughout this book to solve
practical machinery problems.
The model provides a n excellent description of the lowest mode of a rotor
system and can provide some information about higher system modes. However,
accurate treatment of multiple modes requires a mo re complicated model with
additional degrees of freedom.
200 The Static and Dynamic Response of Rotor Systems

Extending the Simple Model

The simple, isotropic model provides an excellent description of the lowest
mode of a rotor system. However, it does not adequately describe the behavior of
rotor systems with anisotropic stiffness or with multiple modes. In this section,
we will present examples of a single mass, anisotropic rotor model with two reaL
degrees offreedom (2-RDOF), and a two mass, two-mode, isotropic model with
two compLex degrees of freedom (2-CDOF). Readers can skip this section with
no loss of continuity.
The anisotropic rotor model is similar to the l-COOF model that we have
been discussing in this chapter; the rotor is modeled as a lumped mass with sig-
nificant fluid interaction. There are two primary differences:

1. The displacement is measured using two, independent, reaL

variables, x and y (two degrees of freedom). The complex plane
used for the simple model is not used (although complex nota-
tion will be used to simplify the mathematics of the solution
process).

2. The rotor parameters are different (anisotropic) in the X and Y

directions. In general, any parameter can be anisotropic, but in
this discussion, we will only treat the spring stiffness (Kx and K)
as anisotropic; all other parameters will be assumed to be
isotropic.

The rotor free body diagram of the anisotropic system (Figure 10-17) is sim-
ilar to that for the isotropic system (Figure 10-11), with two exceptions: the force
components are now shown aligned with the measurement axes and the tan-
gential stiffness terms appear without the j that is used in the complex plane.
The tangential stiffness terms cause a response at right angles to the dis-
placement. Imagine that the rotor, which is rotating here in an X to Y direction,
is deflected a distance x in the positive X direction. Because of the fluid circu-
lating around the rotor, a pressure wedge will form that will push the rotor up,
in the positive Y direction. Thus, the tangential force in the Y direction is
+D>"[2x.
Similarly, if the rotor deflects a distance y in the positive Y direction, the
pressure wedge will try to push the rotor to the left, in the negative X direction.
Thus, the tangential force in the X direction is - D>"ily .
The tangential force terms cross coupLe the X and Y responses. As long as the
tangential force term is nonzero, any deflection in one direction will create a
force that produces a response in the other direction.
 - 0. - ~r

,.

,-~ _10-11 ~rr...bodyd"· •

9'_ "" a 2~ _""I'ic
_ _,Tho bU' """-"
"""" _ "'-'" aIoryood *""
II-.
- - . . - ~ Tho tangIt<"O-aI
.. 11-.

J ..... _ _ " ... ~ ... +OAflx

.... U>, . . . --.r. N._ ill
,.,. -~""'''''''''-'9
_ ""'""'" }( -._Iorte....
-~""""'
.... ...- 0ft9I0 t. tu. ... . -
""'" "'" r .... _ k>rr;e ....
Ilhaw 0ft9I0 ~ - ~ 2

x
202 The Static and Dynamic Response of Rotor Systems

The perturbation force is also expressed in terms of X and Y components:

F; = mru w 2 cos (wt + 8)

Fy = mru w 2 sin (wt + 8)
(10-43)

However, even though we are modeling the system with real numbers, it is math-
ematically simpler to use complex notation and take the real part of the result.
Then, the perturbation force can be expressed as

F
x
= mr:u w 2 e j (wt+8)
j[wt+8-~] (10-44)
Fy =mru w2 e 2

where the 7r/2 is the angular difference between the two coordinate system axes.
These two expressions identify the same rotating unbalance vector, which is ref-
erenced to each coordinate axis (see the figure).
The 2-RDOF system requires two differential equations in x andy:

Mx + Dx + K x x + D>'[2y = mru w 2 e j (wt+8)

j[wt+8-~] (10-45)
My+Dy+K yy-D>'[2x=mru w2e 2

We assume two solutions of the form:

x = Aej(wt+a)
y = Bej(wt+.3) (10-46)

where A and B are the amplitudes of the rotating response vectors, and a and (3
are the phases. The solutions will provide a set of rotating response vectors, each
of which is measured relative to its own axis. The instantaneous physical posi-
tion of the rotor is formed from the combination of the real part of these vec-
tors:

x(t) = Re[ Aej(wt+a)]

(10-47)
y (t) = Re[ Bej(wt+(3)]
Solution of the system of equations 10-45 leads to
 Chapter 10 The Rotor System Model 203

2
Aejn = mr w2e j8 K y -Mw + jD(w + >..n)
u (K x -Mw2 + jDw)(fS, -McJ + jDw)+(D>..n/

K x -Mw 2 + jD(w+>..n)
oc, -Mw2 + jDw)(Ky -McJ + jDw)+(D>..n)2
(10-48)

For each vector, the amplitude is found by taking the absolute value of the
expression; the phase of the response is the arctangent of the ratio of the imag-
inary part to the real part,

Im(Ae jn)
a = 8 + arctan . (10-49)
Re(Ae P )

The 2-RDOF, anisotropic model (in scalar form) can be converted to the
simple, isotropic model quite easily, a procedure that validates the anisotropic
modeling of the tangential force. We make the system isotropic by setting
K x = Ky = K. Equations 10-45 are modified to use the perturbation forces of
Equations 10-43, and the y equation is multiplied by j:

Mx + Dx + Kx + D>..ny = mruw 2cos(wt +8)

(10-50)
j(Mj + Djr+ Ky- D>..nx) = j(mruw 2 sin(wt +8))

When the equations are added, we obtain

M(x+ jj)+D(x+ jjr)+K(x+ jy)- jD>..n(x+ jy)

(10-51)
= mruw 2 [cos(wt +8)+ jsin(wt +8)]

This reduces to

My + Dr + (K - jD>..n)r = mruw 2e j (wt+8) (10-52)

which is identical to Equation 10-20, the equation of motion for the simple,
isotropic model.
204 The Static and Dynamic Responseof Rotor Systems

In the two-mode, isotropic system, the rotor is modeled with a complex dis-
placement vector in each of two, axially separated, complex planes (two complex
degrees of freedom, or 2-CDOF). There are many ways a system like this can be
modeled; what follows here is only one possibility.
The rotor is separated into two, lumped masses, M I and M 2 (Figure 10-18,
top). A midspan mass, M I , is connected through a shaft spring element, K 1, to a
stiff bearing at left. The mass experiences some damping, D 1• The mass is also
connected through a shaft spring element, K 2 , to a journal mass, M 2, at right.
The journal operates in a fluid-film bearing with damping, DB' bearing stiffness,
KB , and A. The resulting free body diagrams are shown at the bottom.
As in the anisotropic model, a two degree of freedom system requires two
differential equations, this time in two, independent, complex displacement
vectors, r i- and r 2 :

M1rl + D1rl +(K1 +K2 )rl -K2 r2 =m1rU1w2e}(wt+/\)

M 2 r2 +DB r2 +(K2 +KB - jD BA!?)r2 -K2rl =m 2 ru 2 w 2e}(wt+IJ,)

(10-53)

Note that there are two, independent unbalance masses, each with its own
mass, radius, and phase angle. We assume a solution of the form

r
l
= A1e}(wt+o\)
r = A 2e}(wt+o
2 )
(l0-54)
2

The solution is, again, two expressions:

(10-55)
 ehdple, 10 n..,Rotor S)'S' ..... M ~ lOS

> •
K , K ,
D,

m , " '~. 1~A J(. ' +~ )

,
-D,',

- ", (',- ',)

F"og_ 16-18 2-axJF ' olOt ~ .....:I h~ body <iogrom.. Tho fQ\o< i, "'PO'~'e<l
m"""""",,",..
" ' 0 rwo. ""'pe<l ","""" .\I, .....:I .ll ~ Tn. M, . ~ eonn KlO'd {h,0UtJI'l
~ ",.ft 'Pfin'l ....... nt. K ,. '0.
",1'1 bNnng . , loIt-Tho """" ..'" e~"'~ """"
damping. D,_Tho """, ;, ."" ~ tt>rough ~ >h.Jft 'P""'9 .............. K" 10.
jou",al ma,.. .>r.. .n hght,Thojournal Oll""~t~ in a1lu;,j -film bNring ",,",h damp-
.....V • . beanng"~ K...od .I.. n.. h~ body d<a9''''''' """" the be... II<'Oog
on the rol or ma,,,,," ineh.olOog rwo. independ"", unbal.onc:e fon:....
206 The Static and Dynamic Responseof Rotor Systems

As with the ani sotropic example, the amplitude is found by taking the
absolute value of the expressions. The phase is found using th ese expressions :

(l0-56)

Summary
Lambda (A), the Fluid Circumferential Average Velocity Ratio, is a nondi-
mensional number that represents the average angular velocity of the circulat-
ing fluid as a fraction of the angular velocity of the rotor.
Using assumptions of a single, complex degree of freedom; isotropic param-
eters; no gyroscopic effects; significant fluid interaction; and linear behavior, a
set of forces were defined that act on the rotor system. The se forces are the
spring force, the tangential force due to a pressure wedge in fluid-film bearings
and seals, the damping force , and an external perturbation force.
The forces were combined in a free body diagram, and used with Newton's
Second Law to obtain the differential equation of motion.
The solution of the equation of motion provided the rotor system Dynamic
Stiffness, an important result. Dynamic Stiffness is the "black box" that relates
input force to output vibration. The general, nonsynchronous Dynamic Stiffn ess
was found to be

K N = K -Mw2 + jD (w-.\ fl)

The response of the rotor to a static radial load led to an expression for the
attitude angle of the rotor in terms of rotor parameters. The atti tude angle was
found to be equal to the arctangent of the tangential stiffness divided by the
spring stiffness.
By se tting the non synchronous perturbation frequency, w, equal to the rotor
sp eed , fl, an expression for synchronous rotor vibration resp on se was obtained.
Synchronous rotor response, which is the mo st commonly observed mode of
 Chapter laThe Rotor System Model 207

operation of machinery, wa s found to be a special ca se of the general, nonsyn-

chronous model.
The model behavi or over frequency or speed was explored. Th e model clear-
ly shows a resonance, coupled with a 180 ph ase change in th e lagging direction.
0

The frequency of the resonance depends on the spring stiffness and mass of the
system, th e type of perturbation used (nonsynchronous or synchronous), and,
for nonsynchronous response, on the Quadrature Dynamic Stiffness of the sys-
tem. For underdamped rotor systems, the resonance occurs near the undamped
natural frequency,

w=Jf;
References
1. jeffcott, H. H., "The Lateral Vibration of Loaded Shafts in the
Neighbourhood of a Whirling Speed.-The Effect of Want of Balance;'
Philosophi cal Magazin e 6, 37 (1919): pp . 304-314.
2. Muszynska, A, "One Lateral Mode Isotropic Rotor Respon se to
Nonsynchronous Excitation;' Proceedings ofthe Course on Rotor Dynamics
and Vibration in Turbomachinery, von Karman Institute for Fluid
Dynamics, Belgium (September 1992): pp. 21-25.
 209

Chapter' ,

Dynamic Stiffness and Rotor Behavior

TH E CON C E P T OF DYNAMIC STIFFNESS IS an important result of the rotor

mod el that was developed in the last chapter. Vibration was found to be the ratio
of the applied dynamic force to the Dynamic Stiffness of th e rotor system. Thus,
a change in vibration is cau sed by either a change in the applied force or a
change in the Dynamic Stiffness. By understanding how Dynamic Stiffness
affects vibration, we can understand why rotor systems behave the way they do .
This understanding will lay the foundation for balancing and the malfunction
diagnosis of rotating machinery.
In this chapter we will explore Dynamic Stiffness in more detail. We will
sta rt with a discussion of the physical meaning of the components of Dynamic
Stiffness. Then, we will show how the rotor parameters of mass, spring stiffness,
damping, and lambda (A) can be extracted from plots of Dynamic Stiffness ver-
sus frequency. Concentrating on synchro nous rotor behavior, we will show how
Dynamic Stiffness controls rotor response over the entire speed range of a
machine and how it is responsible for the phenomenon of resonance. Finally, we
will show how changes in Dynamic Stiffness produce changes in vibration in
rotating machinery.

What Is Dynamic Stiffness?

Physically, Dynamic Stiffness combines the static effects of spring and tan-
gential stiffnesses with the dynamic effects of mass and damping. We will dis-
cu ss the physical meaning of this shortly. First, recall th at the nonsynchronous
frequency (of both perturbation and vibration), w, is completely independent of
th e rotative speed, n. Th e equation for the generalized, nonsynchronous
Dynamic Stiffness is
210 The Static and Dynamic Response of Rotor Systems

KN = K -Mw 2 + jD (w->..f! ) (11-1)

Thi s can be broken into the nonsynchronous Direct Dynamic Stiffness,

(11-2)

and the nonsynchronous Quadrature Dynamic Stiffness,

(11-3)

The Direct Dynamic Stiffness acts along the line of the applied static or dynam-
ic force. The j term indicates that the Quadrature Dynamic Stiffness acts at 90°
(in quadrature) to the instantaneous direction of the applied force.
When Equation 11-1 is written as

K N = K - M w 2 + jDw - jD>" f! (11-4)

the stiffness terms can be eas ily related to the internal and external rotor system
forces.
K, the spring stiffnes s of the rotor system, is a combination of the shaft stiff-
ness, the fluid-film bearing stiffness, the bearing support stiffness, and the foun -
dation stiffness. This term behaves like the stiffness of a simple spring; when a
force is applied to the rotor system, the rotor deflects and the spring is com-
pressed, producing an opposing force (Figure 11-1, middle). Simple spring stiff-
ness always acts in a direction opposite to the direction of the applied force;
thus, the K in the Dynamic Stiffness is positive, showing that it will oppose the
applied force. Positive springs are stabilizing in the sense that the force pro-
duced by the spring pushes in a direction back toward the original position.
The second term, -Mw 2 , is the mass stiffness. It is a dynamic term that
appears because of the inertia of the rotor. Imagine a mass that is vibrating back
and forth about the equilibrium position in a simple system (Figure 11-1, bot-
tom). Whenever the mass moves beyond the equilibrium position, the spring
force acts to decelerate the mass. However, the inertia of the mass creates an
effective force that acts in the direction of motion of the mass, opposite to the
spring stiffness force. Thus, the mass stiffness is negative, and acts to reduce the
spring stiffness of the system. Negative springs are potentially destabilizing in
the sense that the force tends to push an object farther away from the equilibri-
um position.
 Chapt('f 11 Oynam K StilfM" and Roto r ~avio. 211

."

". ."
"",

f~ 11-' . Spnrq ond "'"" Sl Th<' "",pi<- 'P"ngf......,

"Y""'" is shown ....th no apph 1oK,,-. "ati< 1oK,,-_ •
~ ~. W1lrn. >Lilli< 10«. ~ .~ IrrOdd"~ "'" 1'0'0-
I... 'P""'.l "ofIn~.. ';. proO"cfi. 10«:. pr Oportl<>rlal to "'" di,·
p lacftl'O'nt thol GPPO""' "'" ~ bc•. _ • <ty""",,,
10K.;, appJ;Nl (b<m<lml. ttl< inm.. et the ...... is ,;;"",..,.
_ ;oy Item "'" ""9,,...; I'O'tt.,.,. inc fNSing "'" d,'I'Ia<..........
"'"' , ""'."'oct;, oquMlenllO • "O'Q<>t~ " .......... . - lh.
"""''' _.11..._· ,
 T..... th ird trnn.J0 "" onJi...l.... in t IM' damping ruroe
and is calIo<i 1M aa""..
"'if ft<Jfrw's.s.. Thor j i.......,..... that tM daml""ll ltiff_ lI('\ f " 'lill" t.o I1M' dlR'C'-
tion of u.., appI...... f<Ifn', 1hP f&1: that it ~ pmUM lnd ic:al.... that tM da mJ'l""
!It;ff_ a;'I~ 10 oI.t>iliu tlw roIor ~y"'em.liU I1M' oI,lf....-. I.... darnplng
!It,!f......~ ' a ~mic oI,ffno-. and ,t only ~ lhe roI or ",",1m"... ha.
1'IOnun> >TIocih.
Thor lao! Ierm. - jD>.!l. ~ from the WlV"ttiaJ fonT of t .... fluid-film
1"""'8"", ..~ in I.... ~"" or...al. The j indiclot.... t hai Ihi. ttvzgmtitIJ " ""
ItrU t....." aJ.., ...-t . a! 9tI' w t d il't'O;tic>n of!be .pphed forw >'t'CtOl: RecalI lhaI
t .... taflll:"nti.ol . tiff in Ihe a...", chapter (E<j "" IlOII t o- Iol) wu pouti.....
In t he D).'nam ic Suffn Ih.. U "I!"" tia! .tiff_ term b.rom.... " ..gatn... ind,·
....t mll l ....t it ac t_ like . n"\lAti, "J'ri"ll- II..... I.... ....Ilo'l ive- . pri" l1 act _ lo rrJ"""
the 01 _bilizing damp mll . Iiff The I"ng" "l i. 1 .tiff ..... . l...--m i. _ fuoct ion or
th.. fl" id circ" mf....." ti.1 a, ra,w . "!lul" , ~ ..I..., ity. AU. A_ Ih.. ,o to, • .,...'t1
IO<Tt'. ..... fl" id cirelll"" o.. ino:n- nd th.. 1"" .... U. lly d.."' a bilil inlll " "Il"I'''''' )
t. ng..ntial ""lfn..... IlPl. 1. ,f{O'f. Wh Ih.. rotor . pt..-.! i. hitth ...... ~ th i~ I.., m
ran " "'lfali".. In.. tbtmp lOlt "' iffn and I, illll'" fl",d ·ind" Cftl ,n"' .blllt"
lO>al't ... 22 ).

~ 1 1 ,2
~ _
_'i, o:I.oo .... ~
cloogwam. The 00Kt_
d""'~_I1 ... __-
_oI<t>1r~Ioo<: .. _ _ """
~ .......... ,,\Ill' ... the o-.ct ...
n... Jour ~5rJfInesi _ ......
~I~--' .... the~
.......
~ _ _..,. Il.• The
~ " " " - ' rho-Ipploed 100<.. _ the
Dyrwmi< 5,,,,,,,,,,,- _100
Ii the ...........
,....."'1<' be,,,,,,, ,..... 'OW"'ll b~
_ the ";brltlon teiPO".... _1OI

_ s __
 Chapt..r 11 D ~m i< Slifftw« a r>d ROlo< Ileha vior 21l

The Quaob a tu Dynamic Stiffn...... i< .....pon<ible fur th.. , mooth «'SO.... n""
tra n,il ion o f r ha lag valu..... \x>tween ()" and Illll". Thi, ca n be ..... n in th.. ,....,.
tor d iagr am of lh e nu n, ynchronou5 Dynam ic StiffnPM (Figu ... 11· 2). Th .. D....c1
axi , of th.. D}'nam ic Stiffn.,.,.,; i5 align oo " i th the a rrl ioo furet' ,...-ctor. a nd lh e
four D~Tlamic St iffn"", term, add u p to th.. no n.ync hron o u, Dynamic St iffn....
,'-'Ctor. The angle between the di rec tio n of the applioo force vecto r a nd lh e
Dynamic Stiffn.,.,.,; ,...-ctur i. lh .. ....me a ngr., '" th.. d iff"r..nC1' beh......n t he rotat.
ing fo rce ' tor and the ,i bra tio n .... pon,... '.....lo r. (T he a ng!.... a pl'..a r in th e
or l'u"it n OCcam.> the Dynam ic St iffn...... i. in th.. denomi na tor o f th.. rutnr
.....1""' eq uation. Equation 10-23. See a oo F-<jua tion 3·8.) If both quad ra tu re
" ,ffn"", t.. rms ,,~ ... zero. o nly the d irect . ti ff...." t.. rm. would ,.-rn" in, t he
Dynami" Stiffn.,.,.,; w etor a ngle wnuld he ()" or 180" (d..pend ing o n wh ich I.. rm
Wa' larger ), a nd the rotor vibra tion '.....1Or wu uld be alway. be aligned " i th or
"I......il.. lo the a pplied fOK e ",etor.
When a <.ta tic rad ial load is appl i...J tu the ro tat ing rot o r. l he f""rt urb<o tion
freq uency. ~'. i. , .. m, a nd the nonsyn ch ronou5 O)Tlamic St ,ffn...... r.,.J"~...' to only
Ih.. fir<.t an d las t te rms. K-jlH.U.

Fig_ 1' -) Sync:hronous Dyr\om<:

s,,,,"",,, _ d ""l'om. The QuaodooMU'"
Dynom" ~ conW< of on" """
'''''' ond """,""" OS".... gle ~_. like
the <'CJlSytI( """""" -'or d""'J'om, 'ho
"'91e af the 'Y'""hrcroou. Dynamic
SU'!....., Y@CIor .. the ...measthe""9'"
_ the "e.wr'poI d".rnon ....., the
. b o' "", ~""".... W'Ctor(higl't <pot).

, i. --,
214 The Static and Dynamic Response of Rotor Systems

The synchronous Dynamic Stiffness is obtained from the nonsynchronous

Dynamic Stiffness by setting w = D:

K s = K -MDz + jD(l-,\)n (11-5)

The synchronous Dynamic Stiffness is most important for everyday machine

applications, because machines vibrate primarily in response to rotating unbal-
ance, a form of synchronous perturbation.
The physical meaning of the synchronous Direct Dynamic Stiffness is simi-
lar to the meaning of the nonsynchronous Direct Dynamic Stiffness. However,
the Quadrature Dynamic Stiffness term in Equation 11-5 is different. Because D
is common to both quadrature terms, it is factored out.
Figure 11-3 shows a typical synchronous Dynamic Stiffness vector diagram.
Note that the Quadrature Dynamic Stiffness now consists of only one term and
appears as a single vector. Like the nonsynchronous vector diagram, the angle of
the synchronous Dynamic Stiffness vector is the same as the difference between
the heavy spot direction and the vibration response vector (high spot).
We will discuss the behavior of K s and its relationship to rotor behavior in
detail in a later section.

Rotor Parameters and Dynamic Stiffness

Both nonsynchronous and synchronous Dynamic Stiffness can be plotted as
functions offrequency (Figure 11-4). Dynamic Stiffness is plotted in two sepa-
rate plots, with direct stiffness above and quadrature stiffness below. Both non-
synchronous (N) and synchronous (S) data are shown. The horizontal axis rep-
resents perturbation frequency, w, in cpm for the nonsynchronous case, and
rotor speed, D, in rpm for the synchronous case. Rotor speed for the nonsyn-
chronous perturbation is 900 rpm. In the Direct Dynamic Stiffness plot, the non-
synchronous and synchronous cases plot on the same line. Below the Dynamic
Stiffness plots are Bode plots of the corresponding rotor responses.
The Dynamic Stiffness plots can be used to obtain the rotor parameters. The
figure was created using the same model and rotor parameters as in Chapter 10
(Table 10-1), and the key points are marked in red.
The Direct Dynamic Stiffness plot (top) is a parabola. At zero frequency, the
mass stiffness term is zero, and the Direct Dynamic Stiffness is equal to the
spring stiffness, K. The frequency at which the Direct Dynamic Stiffness is zero
yields the resonance speed (this will be discussed below).
The Quadrature Dynamic Stiffness plots (second from top) are different for
nonsynchronous and synchronous perturbation. Both plots are straight lines,
but with different slopes and different Y intercepts.
 ~ S<o'I'fTw\. plct,

.<-'''_'1
S K - .\ff P

. ,'- '.:-'-'
~

-.
_
- - -
...1
-- _

~I I "' _ _ D,nomoc

~ P'<O:'l "" .,.... 1."''''''''

-I .
-
6l _ """"1""<"""'" iN,
_ 000e<10 , . - So""""" ,, I D(I - 1}!J
- '-'•,-- ....
".,. ""_ ·rod."n"".
""' """'''' ...... .-..
- . . "II"'d lor ~.......,....
<hoonou> 1>0'1"'-""' ;" \100
'prn.SHT_ 10.1 lor Hom-
~ "" Cf/''''OIIW .....
•
-._
. -,-- .--- .• - •

Tho Dyn,o..... SO".- pIol c.n

til! ""'" lO ubi ~ 'otOl

-
p.or Cf/ ,.". bi<oc _
:so... t IIor. lui ~np.

I\ .I
, --
216 The Static and Dynamic Response of RotorSystems

The nonsynchronous Quadrature Dynamic Stiffness (blue) has a negative Y

intercept. The absolute value of this intercept is the tangential stiffness, DA[2, of
the rotor system. The stiffness then increases and becomes zero when the per-
turbation frequency, w, is equal to AD. The slope of the line is equal to the damp-
ing, D. There is enough information in this plot to obtain D and A. Thus, the
direct and quadrature components of the nonsynchronous Dynamic Stiffness
can provide all of the rotor parameters of our model: K, M, D, A, and D.
The nonsynchronous plots also define the Margin ofStability, the frequency
range between the zero values of the Direct Dynamic Stiffness and the nonsyn-
chronous Quadrature Dynamic Stiffness. Rotor stability will be discussed in
detail in Chapters 14 and 22. For now, we state that, if both the Direct and
Quadrature Dynamic Stiffnesses become zero at the same nonsynchronous fre-
quency, w, then the Dynamic Stiffness of the rotor system will vanish. This zero
in the denominator of the rotor response equation would result in a (theoreti-
cally) infinite response amplitude, a condition called instability.
The Y intercept of the synchronous Quadrature Dynamic Stiffness is at the
origin (0,0), and the slope is damping modified by fluid circulation. This is less
than the slope of the nonsynchronous stiffness, which is D. Thus, we can define
the effective damping (or observed damping), DE' of the synchronous rotor sys-
tem as

(11-6)

Because A is usually a positive number less than 0.5, the effective damping
for synchronous behavior is usually less than the actual damping constant, D.
We have already mentioned that the tangential stiffness term is negative, which
acts to oppose the stabilizing damping stiffness term. One effect of this negative
stiffness is to reduce the effective damping of the system.
This makes sense if we imagine a rotor in a forward, circular orbit.
Physically, the damping force acts to remove energy from the system. Force
times velocity equals power, and the damping force produces negative power
because the direction of the damping force is opposite the centerline velocity. At
the same time, the tangential force acts in the same direction as rotor motion
and pushes on the rotor. This is positive power, and this power input to the sys-
tem partially cancels out the power loss due to the damping force. Thus, because
of fluid circulation and the pressure wedge, the effective damping of the system
is lower.

PT. EMOMI
DOC. ROOM
Synchr...-J$ RotQf s.t>..w:..
n... prima.-y 'OOUnT of nhratlOn in _or ~~ nn. ;. dow 10 unbaLo.........
U~ ~ .o synduonour. ( I .'( ) rnponw in .n rol atin, madl""",:,, a nd
ill thor _ 001I1""""" ~ ~ f1K.ou... of lib i mporU~ ..~ will
c...oc.. , lnI.. on STTldIJorIOUIl _ bo-N,'ior;
~ ..... rolor""'f'Ol""'"' <'an br d j,ldrd into th -...tM.
fd ~o.: .....u
boobo- '""""...~ (..t. id1 ..~ ..,II call >pfftI-~ /ftORlI ~•
.oncl-U .o1x.. ~ ................ (-h~ "J"""'d.~ In NdI ~ .o d JI.......1 1nm of I....
I)ncilnmo.... 1»nomic Stiff """I rois th .. l'ftoJI'On ... of thO' r ,)-"",,_ In ow
d, ~m. ..~ .. i ll ..... lIMo ")· fOIl0U5 r"' 01 "",dd IN I d~ in 1....
t...1 p!<'•. in " t.ich I Ix> IX pon... ~edor. ,_ i, ......,n.d 1».

( 11-7)
:\) 0

........... "n..n 11>00 "..,-pha>t........... l OC'CU"'- th.. rotah nll unboo4nt:., d.......mic f,,~
II.........,.. <pOl'~ hal. m~nlludr -.0' a nd if; louI«i at angu1o.r p'",ition t . and
Ih.. ""'f""'''' >1'rtOf tth../fifJI.,...1 hal. ..... <>-I..-prak amphl..... A and io Ioc..l fd
at a~ o ..-\ ..."pArt form of thIoo npnov."," a n boo modoo by ouI::tIotitl1Ii"ll Ih..
. . -...uon IC1wpl~ 3 ) for Ilw 000000000000tial lUnct""""

O Hl)

Synchronous 8eha Yior 8@olow Rnonanu '

FijIu... 11-5 """""~ plolo of ap .dlJo nou. Oi""" l .ond Q......dr..tu O)nanl K-
Sl l l f _ t!Opland Ilod .. and pc>la. rloh of th.. IX rOlor V1hral ~m p iet ed by
11>0' rnod<-L .'\1 th.. . iJlhI of th .. d~-n. mic . ti ff 1'1"11 ..... th .. s)Tlch ron .. w.
Dyn..m ;c Sl ,ffn~s ''{'("(o, dl&ftla m. fo r Iow·speed na ...,... a nd h ijlh·.p'"<'d
.-ond llion... Th.. o .wnl..t;on o r th.. llil't'ct Dyna mic Sh IT n i. cor... s pond. 10
th....... ,.,. spot loca l IOn. 3 15".
In t.... Iow·"J""""d •• "tt"' ( ll"""n ~ ........&1 Ihin!'l-" . ... "ppa... nl :

1. n... rotor h ip> opot: (vibrat IOn nospon.... ' ''':'0' ) is in the .....
d o.....,""' .. , l hO' ''''''''' "f'Ol. TIliI io appr<lXOmalf'ly 11'UO' , Ix>
..- .'f"""d r....... . nd ;. ... _~. tlUl' in ,n..limil of UfO rotoI
"'P""""i n... hIM)" opot and hilth 'i"'l ...... 1d 10 hf. ... ~
218 The Static and Dynamic Response of Rotor Systems

2. The vibration amplitude increases as the square of the rotor

speed.

3. The synchronous Quadrature Dynamic Stiffness is close to zero.

4. The Direct Dynamic Stiffness equals K at zero speed.

The high spot/heavy spot relationship is an important key to balancing. Part

of the balancing problem involves determining the direction of the rotor heavy
spot. Either the Bode or the polar plot can be used to do this, although the polar
plot is much easier to interpret.
At very low speeds, D is small, and the Dynamic Stiffness is dominated by
the spring stiffness, K. Because of this, the mass stiffness and quadrature stiff-
ness terms can be neglected, and Equation 11-8 becomes

Below resonance: r=A L:a = mruD2 L:6 (11-9)

K

Because the mass and quadrature stiffness terms are neglected, there is no
phase change, and the response is in the same direction as the applied force ,
a = 6. Thus, the high spot and the heavy spot are in the same direction.

Synchronous BehaviorAt The Balance Resonance

Several aspects of the balance resonance region (red) are important:

1. The amplitude of vibration reaches a peak, and, at the same

time,

2. The phase of the response lags the heavy spot by 90°. This occurs
in the Bode plot where the phase slope is steepest and in the
polar plot close to the maximum amplitude of the polar loop.

3. The Direct Dynamic Stiffness becomes zero. This can be seen on

the direct stiffness plot and in the Dynamic Stiffness vector dia-
gram.

4. The Quadrature Dynamic Stiffness is the only stiffness element

available to restrain the rotor.
 1>;1-1<11

-- - '.----,.......
~
.... ---
.-1

- ---- - --
•• ,
'. ,'~',.;.-~;;::;;~~==::J.
..--J
'
.~
••• . ~

I ~I
I

f:
o.

•
,. I ....... ' ..-

,..... 11·5
--------
"
10>,,<1 ; --
........~-
~~~ ",,,,,,,," PIct> oI ' ~
<0......... 001 ,,'
...... 11 _-","w.'tlon<Jorl)
__"CO' ..
~

1oooo_~_ ...
0,.-.-........ ~"""''''''id.. _

_SH~_bllul~
POI¥Pct>oI_
dlq_ "IJI'r: _ _ b _
220 The Static and Dynamic Response of Rotor Systems

The zero crossing of th e direct stiffness is near the speed of the resonance
peak. This is typical for a machine with a moderate Synch ronous Amplifi cation
Factor. The onl y way the dire ct stiffness can become zero is for the spring stiff-
ness and the mass stiffnes s to can cel each other. Let fl res be the speed at the zero
cro ssing of the direct stiffness . Then, for this condition,

K -Mfl;:es = 0 (11-10)

wh ich leads to this important relationship:

(11-11)

This expression is referred to as the rotor system balan ce resonance speed, reso-
nance speed, or critical spee d. Because resonance occurs when the perturbation
frequency is equal to th e rotor system natural frequency, this expression is also
called the natural fr equency. More accurately, it is the undamped natural fr e-
quency , which ignores the effects of damping. For most rotor systems, damping
is relatively small, and the damped natural frequ en cy (the natural frequen cy) is
close to and a little below the undamped natural frequency.
This expression is one ofthe most powerful tools in rotating machinery diag-
nostics. It shows that the balance resonance speed is determined by the spring
st iffness and mass of the rotor system. Changes in the balance resonance can be
ca used by many rotor system malfunctions. Changes in spring st iffness are usu-
ally responsible for significa nt changes in the resonance speed (mass doe s not
usually change). For example, spring stiffness can increase because of a rub or
severe misalignment, or it can decrease because of a weakening foundation or a
de veloping shaft cra ck.
This expression can be used to solve problems such as piping resonance. For
exa mple, say a machine has high amplitude piping vibration when the machine
is at running speed. One solution to the problem is to move the resonance of the
piping away from running speed. This could be done two ways: add mass to the
piping system, which would lower the resonance, or add stiffness (bracing), to
increase the resonance frequency to above running speed. The last choice is
probably the best one, because the resonance will be moved completely away
from the operating speed range of the machine.
Note that, in the Bode plot, the actual peak of th e resonance occurs at a
slightly higher speed th an th at given by Equation 11-11. This is because the
rotating unbalance force increases with the squa re of th e rotor speed. As th e
 Chapter 11 Dynamic Stiffness and Rotor Behavior 221

rotor passes D res' the force temporarily increases more rapidly than the
Dynamic Stiffness, and so overcomes the natural tendency for the vibration
amplitude to decrease. The higher the quadrature stiffness of the system, which
flattens the response and slows its decline, the more the peak is shifted to the
right. Because this effect is usually small and not important to rotating machin-
ery malfunction diagnosis, we will ignore it.
At the resonance, the Direct Dynamic Stiffness is zero , and the rotor
response equation becomes

At resonance: r= A LO = mruDz LD (11-12)

jD(l-)")D

The only remaining stiffness term at resonance is the Quadrature Dynamic

Stiffness, and a major component is the effective damping, D(l-)..). Thej in the
denominator is equivalent to subtracting 90 thus, at resonance, the high spot
0
;

0
lags the heavy spot by 90 •

The magnitude of the quadrature stiffness at resonance determines the

magnitude of the vibration response. Because the quadrature stiffness is in the
denominator, a smaller value results in a higher amplitude peak, and vice versa
(low quadrature stiffness produces a high, narrow peak and high quadrature
stiffness produces a low, broad peak). The quadrature stiffness expression shows
that damping, fluid circulation, and rotor speed all playa part in the behavior of
the rotor at resonance.
The effective damping applies only to synchronous rotor response. For the
machinery operator, a primary concern during startup and shutdown is whether
or not the machine can get through a resonance without an internal rub, and the
effective damping controls the peak vibration amplitude.
When a machine is running at steady state, away from a resonance, non-
synchronous dynamic forces may exist in the machine that act to excite the nat-
ural frequency associated with the resonance. In this case, the nonsynchronous
rules apply, and the full damping is available to limit machine response.

Synchronous Behavior Above Resonance

The rotor behavior at high speed (blue) shows two important relationships:

1. The amplitude of vibration approaches a constant, nonzero

value.

0
2. The high spot lags the heavy spot by 180 •
222 The Static and Dynamic Response of Rotor Systems

At high speed, the mass stiffness term dominates the Dynamic Stiffness. The
[22 term becomes so large that the other stiffness te rm s ca n be neglected. The n,
th e rotor response equat ion becomes

mr. [22 L 8
r = A L O' = _ =ll_ _- (11-13)
- M [2 2

The spe ed terms can cel, and th e equat ion reduces to

mr
Above resonance: r=A LO'= _ _ll L 8 (11-14)
M

Thus, at speeds well above th e balance resonance, the amplitude of vibration is

constant and independent of rotor speed. It can be shown that the zero-to-peak
amplitude in this equation is equal to the distance from the geometric center of
the system to the mass cente r of the system. Thus, above resonance the rotor
syste m rotates about its mass center.
Th e minus sign indicates that the high spot d irection, 0', is opposite th e
heavy spot direction, 8. In othe r words, the high spot lags th e hea vy spot by 180 °.

How Changes In Dynamic Stiffness Affect Vibration

When the Dynamic Stiffness of a machine cha nges, the vibration of that
machine will change. Dynamic Stiffness contains all of the rotor parameters, M,
D, K, .\ and [2, and each of th ese parameters may affect th e vibration, depend-
ing on what region th e machine is operating in. For a machine operating at
steady state, the most likely parameter to change is the spring stiffness, K. This
is because K is a combination of so many stiffness elements: shaft stiffness,
fluid-film bearing or sea l stiffness, bearing support st iffness, and foundation
stiffness. Changes in any of these components can ch ange K, and change the
vibration in the low-speed or resonance regions. The change can be in ampli-
tude, phase, or both. Interestingly enough, an increa se in K does not always
cause vibration to de creas e, or vice versa. What actually happens will depend on
wh ere the machine is operating relative to the balance resonance.
The Bode plots of Figure 11-6 show the vibration response of a rotor syst em
to changes in K. In all plots, th e original rotor response cur ves are green. On the
left, K has increased, shifting th e resonance to a high er speed (red curves). On
th e right, K has decreased, shift ing the resonance to a lower speed. In each plot,
 (hapl ~r 11 Oytl,mk Sllliness,nd Rot or Bet>.wio' 22)

r"",."....' lInli "f"""do ;Uld u..- ....,.."'..to'd dloonfl'!' In ..brat"", ..t t h._ .pttd. .....
""- Fur u...,,~. inagi...... ",..,1Un.. th.J1 "'-,. ",,",",'~ ~ "i""f'd flt- M ln ....-
naI moot "" 01 "I""'"ation. ,1>00 found.r.l.ion ~O'Iiono-INI. ,O'd"..,in~ tlw .,"em
OflI LI\fl .. ,ff.......... K. •• ohoooll in IN lind.. plot on IN n,:bI..-Y " drcn-.>onI. u..-
,"""",ncr ~ ~.;,«J. """"nll lhr ~ pt'ak to lhr Irft. Thi.
caw.r.. I"" .omplitudr and ph.a-eo 10 dloo"fl'!' from thr in fl loll val...... ( I I to u..- mod ·
,fll'd ~aJW.O In Th _ I"" rrduction In .. iff....... prodUC'N ' ~ In .'ibnllon
arnpIi ludr . nd ... inoo:'rN... in ph.w \a¢.

•• •• ,
I.
il \, .\.
~~==
n. f'. II, '1, fl , I.,.~a~.:a;.===:
Hr · .- ,J::
~-
I

i-
--_en. ..
I ,
n. n,
-- 11, 11, 0, 11, II, U.

~ 1 1-4 ~ .. >p>ng " " " ' - _ _ cf\onges '" ~

_ I!'>r"""J""I1'OI:Ot .....,.,.,... C>o"'Oft In 9"ft'\ Qp, _ If't I: hi!. ~
<-'9 tI'>r ~~ 10 >hot: "' • ..,.... -.d _ r..-.! Qp, ..... "'7"t I:
hi!. dI< .G..... """"""9 _ ~ 10.. _ -.d-" N<tI pig(. four OP'O'.
-.g""",,, _ _ n,. ~<.do....-d

. ...
~_~._~conti<>..oedlO_ lhOI-.d
 ,,- chang<' in .t iffnf'% can al"", produ".. a cha ng.. in nX ,ib ration (Figu re II -
7). Th.. 2X APHT p lot ( r..ft j from a w rt ica l reoc tOl-coola nt p um p sttm..~ dala
that was tre ndrd " il ilo- th.. p ump was ru nn ing al 1187 rpm. Beeau... o r a 8haf1
crack. t .... 2X ,ib m!ion brga n ' flcha ng<' d ramatically. produci ng ,,-hat loo ks like
a rewna nct' on 12 :-;"'....mbrr.
The right half of tJI.. fil!" re shows a ...1o f Bode pints gen..ratl.'d by I rotor
mode L " i th a re..,nanC<' fn.qUl'TlCY nn r !wict , .... ru nning spre<l o r t pu mp.
Th e plots show th....!fret of dec reas ing spring s tiffn.......K.u n rhe ",,,, ,na nC<'. Th..
initia l o pe ral ing co nd ition is sho...... in grttn. a nd th.. final. low s tilTn"", operal -

. lX 1181rpm
0

I• I
••• r
I .. I
.... ....
.
~

-------,
,
.o-
" "- ,."•
I
J
f «r.
0

! • y". ""-
0

. • .. - - ,

- ~

Fogute 11·7 2XAI'Hf pIoO: . nc! _

o~ n o>

"
••
"
rotor mpon"'. n.. 2X. AI'Hf pIol "" m..
I<'ftis from • ~ ' _ '...--cool.nt pump. n.. <lat. ...... nn<Iod _ "' !hf" pump
.... rurvling at 1181 rpm. 1M non'y<'Ch''''''.,." Ilod .. plot ",,!hf" fight gone<. ted
.,""'Iot
by , .... rotor moc\ooI, tI>e rotor .... PO""' .......... _ rotor 'PH<1 f l . ;. t 181
rpm. _oily. it "" nc.. pum p sr:-d ( g ,","~ M _ rlg
"Ilfnrss. K.docrN= _ ,,,,,,,,,.nc"''''lurn<y """' to. - . fr"'l .-c)'. _ ,, "'"
lhrougt> _ 2Xfr"""oncy.n.. moI,low "''''''''. Of>!"lting <orld~oon; . shown in
'l!d.The 2X P~tv<I .. . nd plio", « <>lore<! dais! <h.o"9" IS_ .Y',. . . ......
'01 fie.
q""""y pas tht wgh iWO:.. QPef.nng .PO'l!d
 Chapter 11 Dynamic Stiffnessand Rotor Behavior 225

ing condition is shown in red . Th e IX and 2X pump frequencies are shown. In

the amplitude plot, the 2X amplitude changes (colored dots) as the system res-
onance frequency passes through twice operating speed. The amplitude peaks
and then declines, ju st as the APHT data shows. The Bode phase plot is not as
good a fit to the APHT plot data, but it does predict that the 2X phase lag will
increase as the rotor system natural frequency decreases.

Summary
Dynamic Stiffness consists of th e static spring and tangential st iffnesses of
the rotor system combined with the dynamic effects of mass and damping.
Two general types of Dynamic Stiffness exist. Nonsynchronous Dynamic
Stiffness, the most general form, controls the rotor response to an applied
dynamic force at any frequency, independent of rotor speed. Synchronous
Dynamic Stiffness, a special case of nonsynchronous Dynamic Stiffness, con-
trols the rotor response to a synchronous (IX) forc e, such as unbalance.
Dynamic Stiffness can be separated into Direct Dynamic Stiffness, which
acts along the line of the applied force vector, and Quadrature Dynamic
0
Stiffness, which acts along a line oriented at 90 to the applied force vector.
The comp onents of Dynamic Stiffness are related to the forces that act on
the rotor and the inertia of the rotor itself. Dyn amic Stiffness contains spring
stiffness, ma ss stiffness, damping stiffness, and tangential stiffness.
Synchronous Dynamic Stiffness terms are associated with the rotor
response in three speed ranges: below, at, and above a balance resonance.

1. At speeds well below a resonance, spring stiffness dominates,

and the rotor high spot is in phase with the heavy sp ot.
Vibration amplitude increases as the square of the rotor speed.

2. At the resonance speed, Dire ct Dynamic Stiffness goes to zero,

a nd only Quadrature Dynamic Stiffness remains. Rotor ampli-
0
tude peaks, and the high sp ot lags the he avy spot by 90 •

3. At speeds well above a resonance, mass stiffness dominates, the

vibration amplitude becomes constant, and the high sp ot lags
0
th e heavy spot by 180 The rotor system turns about its mass
•

center.

Changes in Dynamic Stiffness produce changes in the amplitude or phase

(or both) of vibration. How vibration changes depends on where the rotor sys-
tem is operating relative to a resonance.
 227

Chapter 12

Modes of Vibration

TH E ROTOR MOD E L WE DEVELOPED IN CHAPTER 10 provides a good descrip-

tion of basic rotor beh avior. In Chapter 11, we used the model to understand the
basic pr inciples of synchronous rotor behavior below, at , and above a resonance.
Because the model has onl y one mass , it is limited t o describing a system with
only one , lateral, natural frequency, one forward resonance, and no gyro scopic
effects.
While our model has single, lump ed parameters of mass, st iffness, damping,
and lambda ().), real machines have continuous distributions of parameters (and
often several sources of )., from different bearings and seals) , and larger and
higher-speed machines often exhibit several resonances during startup and
shutdown. These distributed systems are theoretically capable of an infinite
number of resonances. In practice, we are primarily interested in onl y the low-
est few resonances that the machine will encounter on the way up to or down
from operating speed or that exist at some integer multiple of running speed.
When a rotor system encounters a resonance, the system vibration will be
amplified. For large , distributed systems, the vibration amplitude and phase will
be different at different axial positions along the rotor. Also, the machine casing
will participate in the vibration in some complicated way. The total system
vibration will affect rotor-to-stator clearances along the rotor, possibly leading
to internal rubs on seals or blade tips.
Th is complicated, vibration deflection shape of the rotor system is com -
monly called the mode shape of the system. The mode shape describes the axial
distribution of vibration amplitude and phase along the rotor system, and it
changes with rotor speed. It is a function of the ma ss, stiffness, damping, and ).
228 The Static and Dynamic Response of Rotor Systems

distribution along the roto r, combined with the distribution of unbalance along
the rotor.
In this chapter we will discuss of the concept of natural frequencies and free
vibration mode shapes and show that the forced mode shape is the sum of sev-
eral free vibration mode shapes, each of which is excited to varying degrees by
th e unbalance distribution of the system. We will show how rotor system mode
shapes are influenced by the relative stiffness of the shaft and bearings, and we
will introduce the concept of modal parameters. Finally, we will discuss some
different techniques for estimating the mode shape of a rotor system using
vibration data.

Mode Shapes
All mechanical systems have natural frequenci es of vibration. These natural
frequencies can be excited by momentarily disturbing th e system from its equi-
librium position. If the system is underdamped, it will vibrate at one or more
natural frequencies until the initial input energy decays away. Because the sys-
tem is not forced continuously, this kind of vibration is called free vibration.
Our rotor model is capa ble of only one natural frequency, wll ' which, for low
damping, is approximately

(12-1)

As the number of masses in the system increases, so do the number of nat-

ural frequencies. A natural frequency is also called a mode, or natural mode of
the system. Each natural frequency is associated with a particular vibration pat-
tern, called a mode shape , and each mode shape is independent of all the other
mode shapes of the system.
Figure 12-1 shows an example of the mode shapes of a simple, two mass.Iin-
ear system. Because the system has two lumped masses, it is capable of two inde-
pendent free vibration modes. The lowest mode of this system is an in-pha se
mode (left, blue), where both masses move in the same direction at the same
time. Note that the amplitudes of the motion are different. The second mode
(red), occurs at a higher frequency and is an out-ofphase mode, where the ma ss-
es move in opposite directions. The frequencies of the modes and the relative
amplitudes of vibration of the masses depend on the valu es of the mass and stiff-
ness elements of the system.
Our l -Complex-Degree-of-Freedom (l -CDOF) rotor model ha s only one
moving element, the rotor. We can extend that concept to view the rotor and
 Chap,.... 12 Mod,... ofV;b'~' ion 229

-- - , - ---
..........
K ,
..... d lfto...

K ,
[I] ~ .
IT: n JI:

Fi<Juo' ~ 12 -1 .The modo >hopes of ~ .. m~. ""'> ...... "",.",. llo!uu", _ .~ t"'O ............ the
'Y'1<"" i«opo~ of two, indoponcloonl. ftft .;tIM "" rnodO's. The ~, modo of !hi< 'Y""'" ..
•n "1>_ modo (_1 _ ~ both d,,,..,,,,,,
"""""'"' ........... it ...... .."'" """'_ _
............. impI~""'" of ..... mot"'" d""'",,,,- The _Of>d modr (..,.;II occu" i t . h'9hrf h...
~...-.I i'..., out-d"1l_ rn:>do. ~ ............... """"'" in _~ dl'''''"",,-
230 Th. 5.a . ;<and Dynam ic R."pon.. of Rotor 5,.,.....

F1gu,", 12-2 [ n,h ".... 01.. ""or

""'''''' w lh !WO rno<l<'S- n.., ,,.-
,..... is su~ to 1><'_ ~ k .. two,
lumP<'<! ,..... Ttl.. '"'''''l is
,hawn ;" light rN.nd II><' rotor;"
lig'tt bIu... ar<l ~r ,O'<u.... ortMto,
...........rod ;" tl><' ;.-...tJ.I Iro", .. '<'P"
r......... <'d by tl><' bLKl< .,;, I........ ..,..
, I>owfl;" rod . nd blU<'.-n-., four>.
dar.., i, .,"'-""t<! to I\av<' .........
comp!ion,.. 1M ..- - . """,on 01
th<'<_.In tl><' I<'Itcolumn,lh..
rotor . nd , .... "9 """""' m <l"I>\J in
..n "PPfO',m.t",>, ;,,-ph -
1"""IIop (lOp to bcttom).ln I....
I<'<end coIu....., I<'<Ond mod .. i.
""'"'" ,."..... , rotor . nd .... "9
'fIO'>'@ ., .n.PP'".;.......,.. tyOUl-of-
pr"".. "-,gd(,,""',, tI1.~ "" , .... 1<'< -
and _ . tI><' rotor ", tMatoon
..... plnude ;, ........... tho n /of 11><'
fi", modo. W lh .. ohm ",I,oti. ..
mohon of 11><' '0(0< i, "'9<"
 Chapt ... 11 Mod" H .1 Vib.atiotl 1H

«I~"'g a s .. 2-('flOI ~~~t"m. "·h,,r.. th.. roW, a nd «I~"'ll a rt' 1rt'..It"d a~ lu mJ"od
ma ~ Ul..e th.. nsl ..m in I'ill" .... 12·1. th is t" ... maM 'y""'" i~ ab... ....pal.... o f
I" " fn.e 'ih",li" n m"d ..... lIul. in.l..ad of b<''''ll constr..int"d to "" >I",,, in a line.
bo th pa.ts a rt' ....pahle o f I" -d im..n. io n.. l. independ.. tlI. pliOna. mol ,,,n Iha l. ro ,
an i....I" 1"" 'pl..m, in",I ,",eircula r orbit,;.
Figu. e 12-2 ,.I""". a " d .." d 'w,,' of th is ,otor .pt<'fYI. Th.. ea~i "g is . """-" '"
ligh t a "d th.. rol or ill lii/.ht blu... Th.. blaek lin.... repn-«'nt th.. as,.,. of th..
ine rt ial r,·... ".'l" rra ID<'. The foundalio n i.. a..umt"d 10 "II" ... ",..tion of t he ea. ·
ing_ In t l.·(\ eolu m n. t he rotor and ea. ing moo-.. in a n " 1'I' ""im..t"'v in ·p.......
.... ..t i" ns h ip. IIo .th Ih.. rotor a nd «Ising moo-.. in eircul.., orbit. t....1fo , Ih" "a. ·
in~ bt.... fo , Ih" rolor }a bout Ih.. ine rt ,a1 "..nl.., of th.. s....t..m. l'o , Ihi. mode. th..
">I,,, ..nd easin!! a... d elkctt'<l from th .. in..rt ial ....nl..,- in appro.i matd~· th..
"'" tnt' d ir-..ct".n al t he Mml' tim... producing a n tn-ph""" mot"'. In the "'-'COnd
ool",n" . a st"COnd mooe . hal'" i~ s ho...n .......... Ih.. rot". a nd ..... in!/. mm'l" in a n
appro. imately ou/--o/-phase mode. :\o t.. that. f" r Ihi, "'t'COnd m..d". Ih" s ha rt ,..I·
al'.... m' >l i",. o r the roto, i' I" . g..'.
R..al rot o , s~~t ..m, ar.. COfumuvus s~"t ..m", I" . t..ad of lu mped ma..""s ...,d
. p rinlU-o lh...... ~'.I ..m. ha,.. "->fiti""",•• dlStrib"t io". of rna.. a nd st iffn""" a nd
po. .... . ma ny na tu . a l r.equ..ncie•. Figun" 12-3."""." t he f,rst th r... mod...hap....
o f the ' implest ...a mpl.. o f a co n tinu" us .~-,;tem. a . t rinlt " 'hieh i. cla mp..d at
bo oth e nds. Th.. d iagram~ . how only th.. ... I...m.. J'O'it ions that th t ring ....ach ·
... d urinll it. , -;b "'t;on. Wh ..n ,-ib.at ing a l a natural f""lu" ney in f ,i b. a tio n.
the st rinll can , ib",t.. only in a p.<.t k u lar_na tura l moo e . hal.... Th" top mOO..
s....pe in the ligur.. cO" I" "'d. lo th.. It",- , na t" ra l frequ..ncv. a nd Ih.. m idd l..
a nd bot to m mod...haJl <:or.....J"-lIId t.. s u' iwl~· h illh" r ....l u. a1 freq lJenc i~._

Hg " ' e 11·) Tho- fnt th.~ ~ wpe> 01

o frftly .. b<O"ng .t'"'9 ."lh <iornoed er-.h
~ ..;bto''''g 0' 0 """',01 fr"'l.-.;y in
<::::::::_-->
'''"'9
_lIb-oliorl. ttw c,"" ,,;btot~ <I<VI< in
o pa rtie..... """"•• .,..,.;., VlOp"_ Tho-~", ,
........ Vl 0p" (top) in lhr figu~ corr<"ll""d.
to ,t,r, """",,, n....... ~y, tt,r, "'odd"
0 00 bottom 'T>Otlt Vl0p'" con"""""" '0
succ...iYo'/y .. g..... frO!QU_ ...... ,
232 The Static and Dynamic Response of Rotor Systems

Rotor systems combine the characteristics of a continuous system (the rotor

itself) and a lumped system, where the rotor and casing can behave like large
lumped masses. In addition, lightweight, relatively flexible machine casings can
deform in their own set of mode shapes.
Because of this complexity, the term mode shape can be somewhat confus-
ing. It is fairly easy to imagine a rotating, flexible rotor that deflects in some
complicated, three-dimensional shape like a piece of wet spaghetti. We will refer
to this as a rotor mode shape. However, vibration modes involve the simultane-
ous vibration of all coupled components in the system: rotor, casing, foundation,
attached piping, etc. The rotor and casing vibrate at the same time because the
rotor and casing are coupled to each other through the bearing stiffness and
damping. The casing can transmit vibration to the rest of the extended system,
and it can transmit vibration to the rotor. The term mode shape can refer to the
complicated pattern of rotor, casing, foundation, and piping system vibration.
We will refer to this as the system mode shape. The rotor mode shape is part of
the overall system mode shape, and each rotor and casing mode shape is associ-
ated with a natural frequency that includes the entire system.
When a rotor vibrates, it moves (precesses) in an orbit. The precession of the
rotor produces a set of rotating reaction forces in the bearings. These forces are
transmitted to the casing, and the casing responds dynamically to the rotating
forces. Thus, the casing can move in an orbit in response to the rotor vibration,
and rotor and casing mode shapes are a complicated, axially distributed set of
rotor and casing orbits.
Rotor mode shapes are strongly influenced by the ratio of the shaft bending
stiffness to the combination of bearing, casing mass, and casing support stiff-
nesses. If the stiffness ratio is low (relatively high support stiffness), then the
bearings and casing will strongly constrain the rotor motion at the bearings, and
most of the motion of vibration will occur through bending of the shaft. Rotors
that experience bending modes, such as those in aeroderivative gas turbines and
boiler feed pumps, are called flexible rotors. On the other hand, if the stiffness
ratio is high (relatively high rotor stiffness), then the rotor is likely to exhibit
rigid behavior, and the rotor natural modes will be rigid body modes. In this case,
bending of the rotor will be relatively small compared to the motion of the rotor
in the bearings (although there will be some rotor bending). A large electric
motor rotor supported in fluid-film bearings is an example of a rigid rotor.
Figure 12-4 shows typical lowest rotor mode shapes for cases of high and low
stiffness ratios. A high ratio of rotor stiffness to support stiffness tends to pro-
duce rigid body modes (left), while a low ratio tends to produce flexible rotor
bending modes (right). The first modes (top) are essentially in phase, as shown
by the Keyphasor dots on the orbit, while the second modes are out of phase.
 (t>.aple' 12 Mod ... of V ib'~tion 2n

~ ot e th e p ita... in...·"'ion acro"s Ihe nodal po,m (t he po"" ..

f minimum ~ih,a-
I,o n amp litu de). Mach i i th nu id -film t.....rings can Ita.., rigid h. ody t...1ta,i o ,

......
in the '""f). l..........t mod a nd tra nsition to flexible roto r helta,ior at hi¢'e ,

Rotor mode ohap<'s a re al", in nue nced by Ihe d i<lribul io n of "" ffn.... a nd
rna •• a lo ng t he rotor shaft. :\Ia ny wlo , , haf," d iIT...e nl Sof'CIion d ia m"'e ",
or .....U thickn........... al d iffe. ..nt ax ial loca l io Th i. p rod UCt'S ~arial ion. in th..
I..cal hend ing ..iff"""" "f lh.. sha ft. P IS"" a nd irnp<'I1 i th d iff.."'nl rna"""",a .t>
f>l a ~-ed al d, ff.....n l a xia l l.",a ti..n", "'''ull, ng in a n un.." ax i..1rna " dislrihul ion
alon g the oIla ft.
Th"", fact n", cumhin.. In aff",-I t .... " ,t. " mod e "'a~ Se<:tions ..f t .... ,, ~ '"
..i th low 1........1 be nd ing ...,ffn ' ",i ll ""nd Inn ... tha n Ih.. M"Ci in ns wit h high..,
st iffn<"<.$. At th E' I. """"t mod ma SS cuncen tra tin ns I..nd 10 p rodu"'" lio'l!'"
d..n"",;.",. n..a r lhe ma •• !wcau.", it is ha rtl.., for Ih E' "'a ft to co ns t. a in t .... in.., -
lia of the ma ,"_ Al h igh.... m. id..... th .. ine rlja ..f Ih.. mu . pT< od ,, ~.... h igh local
rna.... sl iffn...... a ...1can "",ult in ' '''''- klW , ';I,,-a l;on l...... ~. d ..".. I.. t hoe mass <vn-
cnll rat ion

G8{FG 8
..... ,
-~

-- ,,

"",gu' ,, 12-<1 TypUl lownt !'NO roto< modi' , " - ' lot COSM of h;gh """ 10..- st,W....,
'anos." IItgh ,auo 01 totor ,,,I!....... to "'t>POI1 std!ness,~ '0 pmdu<. og .. txxIy modes
(lrfrl. ......... low <tritno:s, ,..no trods to prod"", """ ""or brnd,"9modes log ht)_ Botn
fim modes ltol>J. '" in-ph. .. ""","-as """"'" by t!la orb<! ~ <lots. """,It>
bo1tl
_end modes ... out-<i-pha>OmocIes. ... acI'io.... -th t1u.. -1im !>N''''9' con " - fig"
body ~ in t!la W'Y _ , modes . nd """"""'" to 1Ie_ 'oto< _ atl'Ogheo'
mocIes. " """'..... .., '" """.,,"" ;,. ukd • nod<. or ood.tl pOInt. """",,urn '" ...1><"""" is
",,11rd an ann"""". ""t. thol phase in _ that ocnn""lOS<!he nod.o l poin~
2J4 1lw Stalic and Dynamic R<KpO<l ~ 0/ ROlor Sy.t~....

Fig u... U -S Typol ""'or rno<Io'

"'.pn for th_ com""", ""'or con·
figvtal Oonl. For Nth <onfigu"''''''
th~ ~ .... ~ ohown wMn..~,

. , "'P- The 'OIl 9"'"J p """"" " god . cod

bonding mo<t... fo< typeal .. ng io
-" ""'!l '0l0tl.. W"-n ,""" "'".......
" m.t~ hq,. ,t><- ngod rno<Io' wiI
' PP<'M_"'O~ ~ ...ibio ""'on .,;11 st-ow

~
th ~ fV>" and _ord _ n g """""" _ ""11

Thr dO"bIo ~ conf.,.,ot"""

c. pabio of """ rigod bod)' mo<t...
H.. ~_ ..... <)<de, of 'PPN,.<'lte of , ....
"9<1
<JIftdri<"
:..- ~C
rigOd mode, <!f'I><'nd, on ' .... ~ ~

.... •
" " !!
st""",,'h of n.. <y<~m .ngu'" ' otor "9<1 ~- B
suPport "rII....... to ,h~ ,. ,...1 ~" ""l
•
"dIror... fh " " ,.~ determine<! by
"'" beMing ,pacing System, wM ....
' _ 1)" close<y ,paced ~ . , w.;,";11
...Pf'<ioon<e. ~ ., moder... _
-~, .
.-•..- -.
«-• •
bNo"lngs .... e w""''>' 'P.1Ce<!. .ng"'''
""","", i, h"l l\ . nd n.. fim rigod
bod)' mode wiR ~ cylindric "
-~, ...-
e-
_ .-
-~

"'9dl
•
a- ... . ~~

..
The most common '0I0t <o"'g"n-
' '''''. _~ most of ,"" "'tor """"' "
l>etwee ~ the ~'''''9 center<. The fi",
" -B- 1 · ,!••-
....
,.,ode i, . cylInd rical or pWot.1 t9d
bod)' mode; w!'O<h om .... r.." "': 11
•• " ,I
~pen d on ,"" bN"ng """'ng. .ng...
"'1Ia'e'al .. rtf....., "be>. . nd ..... rn."
..... ' - &-.... •
II ,• 2
m_
d"tribulion. r <st . nd second bend,ng
modes folkM- ""~ or both pM:ital

• i
.....' -<.~
 Chapter 12 Modes of Vibration 235

Figure 12-5 shows several rotor mode shapes for three common machine
configurations. For each group, the modes are shown with the lowest mode at
top. The first group shows rigid and bending modes for typical, single overhung
rotors, such as a single-stage pump. The rigid and first bending mode are very
similar; which form will appear depends on the ratio of shaft stiffness to bearing
stiffness. When shaft stiffness is relatively high, the rigid mode will appear. More
flexible rotors will show the first and second bending modes. If the rotor disk
has a relatively high polar moment of inertia, gyroscopic effects at higher speeds
will tend to resist bending in the area of the disk, forcing the disk to the near ver-
tical orientation shown in the figure.
The double overhung configuration is capable of two rigid body modes.
Here, the order of appearance of the rigid modes depends on the relative
strength of the system angular rotor support stiffness versus the lateral bearing
stiffness. This is largely determined by the bearing spacing. Systems with rela-
tively closely spaced bearings will have relatively low angular stiffness and will
experience a pivotal mode first. When bearings are widely spaced, angular stiff-
ness is high, and the first rigid body mode will be cylindrical.
The last group shows typical mode shapes for the most common rotor con-
figuration, where most of the rotor mass is concentrated between the bearing
centers. The first mode is a cylindrical or pivotal rigid body mode; again, which
occurs first will depend on the bearing spacing, angular/lateral stiffness ratio,
and the mass distribution. First and second bending modes follow one or both
pivotal modes.
Because of the relatively large rotor vibration amplitude in bearings during
rigid body modes, damping forces can be very high, and these modes may not be
visible on a polar or Bode plot during startup or shutdown.
Some points along the rotor mode shape have relatively high vibration, while
others have little or no vibration. A location with no significant vibration is
called a node, or nodal point (see also Figure 12-4). Locations where the vibra-
tion amplitude is maximum are called antinodes.
Nodal points are important because the vibration on either side of a nodal
point will have a large phase difference, often 180°. If we fail to detect a nodal
point, our perception of the mode shape of the system may be incorrect.
Antinodes are important because they are regions of high vibration ampli-
tude. Because, for flexible rotor modes, bearings usually constrain nearby rotor
vibration, relatively high vibration tends to occur near the midspan of the rotor.
This has the potential to produce rubs that can damage seals, blade tips, or
impellers. For these reasons, it is desirable to know the forced mode shape of the
rotor
236 The Static and Dynamic Response of Rotor Systems

Forced Mode Shapes and Multimode Response

Up to this point we've discussed free vibration. However, operating rotor sys-
tems are subjected to forc ed vibration. Subsynchronous forcing can be caused by
aerodynamic instabilities in compressors (rotating stall) or by fluid-induced
instabilities associated with fluid-film bearings or seals. Supersynchronous forc-
ing can be caused by compressor or turbine blades passing a close clearance, or
pump vanes passing a cutwater. Rotors with cross-section asymmetries have a
bending stiffness that depends on angular orientation. Asymmetry can be by
design, such as in generators and motors, or the result of a shaft crack. When
such a rotor is subjected to a static radial load when rotating, the asymmetry
can produce supersynchronous forcing, most often 2X. Coupling problems can
also produce 2X forcing.
Rotor-to-stator rub can produce impacting, a special situation where a peri-
odic impulsive force (the rub contact) produces free vibration that decays until
the next contact. Rub impacting can produce subsynchronous vibration, but,
more often, a rub will produce a mild, once-per-turn impact that modifies the
rotor stiffness in the area of the rub and the IX unbalance response of the sys-
tem.
The most common form of excitation of the rotor is rotating unbalance. The
axial distribution of rotating unbalance produces an axially distributed force
system that is available to excite natural frequencies. This force system has a
shape (magnitude and angular orientation as a function of axial position) that
is similar in concept to a mode shape. The degree of excitation of any natural
mode depends on how well the unbalance distribution fits the particular natu-
ral mode shape. A good unbalance fit will produce a relatively large excitation of
a natural mode, a balance resonance. A poor unbalance fit will result in little or
no excitation of a mode (Figure 12-6), and little or no resonance. This charac-
teristic is often deliberately exploited to balance one mode while not changing
the unbalance state of another mode (Chapter 16).
The concept of an unbalance force distribution that matches a natural mode
can be extended to include all of the excitation sources we have discussed. Any
forcing function in the rotor system is available to excite system natural fre-
quencies when the frequencies coincide. As with unbalance, the degree of exci-
tation of a system mode will depend on the magnitude, orientation, and distri-
bution of the source. The discussion that follows, while oriented toward unbal-
ance excitation, applies to all sources of excitation in the system.
When the rotor speed equals a natural frequency of the rotor and the unbal-
ance distribution approximately fits that natural mode shape, then that mode
will be strongly excited, producing a balance resonance. The forced mode shape
of the system will be dominated by that natural mode shape. As the rotor speed
 (haplE'f 12 ~s o'Vib<abo n 231

inC«'a scs a bm... Ih.........." a n..... ~p'..-d, th.. am plil u' ''' tlf rihra titln ,,~II ded irtf'
to"a .d a ......itIuaJ I,-,",'d. If th .. unbala",... tli-t rih " l io n i_ a 11•• ><1 fit t.o Iii<- n..xt
hillIK-r natu ral mot'" shap'·. th ..n. as th.. mad ' in.. a. u -l..ra le ", II>.- n..xl rn,><I.. ", ill
0Cc01TK' rno..... . t ronflly <,xcitt'<!.
At a typ i<"a l o pt'ral inll . pt't'<1 a " ·ay ' rom a .......Il/O n..... l h.. furn-d m'od<> s hap..
o f th.. rolor ,,~II co nt ain si mu lta nt·ous om tt ib"l io n. ' rom ........ral u ' Ih.. na!l",.1
mod...... Th......Iua l fo ...ro roto r rnt..... sha pt' ,,~II d....."'1tin Ih.. lil a",1 magni-
Iud.. of tl>.- unbala nC1' d is tribul ion 10 ......b mod... a nd on Ih.. am plif...a l io n o f
......h mod.. al 11K- pan i<"ula. spc<'d. Th.. compl.·x inl..rplay oc t><o.....,-, unbala nC1'
d is tribul ion shapt'. natural mode shapt'. a nd .....sona n"" a mpli fica l io n p rod " "".
a fo l'O"t:1 ro tor mod.. shap,- lhat ca n conl inua lly cha n!!" ",i lh spt..-d.

~_ L__ L E

--
-
Fogu", 12-6 \II'IboIo",,, ",,,'obu!,,,, .nd _ _ aOloon. Tht' 'OCo< ....,....
t\os two mt>tJ<o.:,., pha", ..-.I out '" 1lN"'_Tht' ~ unbolon<" dl"'~
t>tmon {\t'ftl"odI'" ~ fW<I modo. but. .. ~ a do<'< not fit _....:ond
_ . a do<'< not "001,, ~ second _ _ Tht' OVI-d-pha", unbolon<..
dow,but.... (right) fots _ _ ond _ .M not _ fW<I
 Co«>oo...... inboord ....._ _ b<.III
•"
I,: r
t
,
".
I,.

,
IJ I
_ •
'\
•
:......1 • ..
, •, , ,
!
, • I

J, \
•• • ..

f;gu,.. 12 -7 Two Iorc<'d modi! <h'pn 01• "Nm 1Ubo~ 9"".... ,0< "" ~
dict«l l>v . >OP!>;"""lod 'OlIo< m~ progr'M. 'Jh<o foo, "'Iua ' unbolaro<; ~
~.... _

1tI" _ _.tOt
pla<od .. rr ... <11 ""P«t to tho ~ tr.n><lIc"'l)<PIIow dol"
plot """""' ..... I X ",br..""" resp:>n ~" ..... ;nboo rd t.....
'ng. ~ rotor mod .. sh.pn I.... tho 1X o<biI ~ dot lo<.ot""" Of 1500
rpm :9' '''') _ l600 rpm l' O!dj ,
 Chapter 12 Modes of Vibration 239

Figur e 12-7 shows the pred icted behavior of a steam turbine generator set.
Th e unbalance was the same ma gnitude at all four locations (yellow dots), and
0
all ma sses were placed at 0 relative to the me asurement transducer. The Bode
plot , generated by a sophisticated rotor modeling program, shows the synchro-
nou s vibration respon se to th is unbalance distribution. The rotor mode shapes
at 1500 rpm (green) and 3600 rpm (red ) ar e det ermined by the IX orbit
Keyph asor dots at each location .
It is interesting that this complicated syst em st ill follows the behavior pre-
dicted by our simple rotor model. At 1500 rpm, the machine operates below the
first balance resonance, and the phase of the gen erator response (top) is close to
the location of the heavy spots . The steam turbine and generator rotor mode
shape (green) deflects toward the heavy spots.
At 3600 rpm, the ma chine operates between the first and second modes,
wh ere the phase of the generator response lags the he avy spot by about 180°.The
rotor mode shape (red) shows that both the turbine and generator rotors are
deflected away from the heavy spot s.
Mode shapes are often three-dimensional; these mode shapes, though, are
almost completely in the plane of the paper.

Modal Parameters
Rotor system behavior involving multiple modes is quite complex. The
mathematical expressions necessary to accurately de scribe such behavior are
well beyond the scope of thi s book. Instead, we would like to develop a more
intuitive approach, which will allow us to extend the simple concepts we have
already de veloped to the more complicated multimode rotor behavior we
obs erve.
We have stated th at the natural frequency of a on e mode system is approxi-
mately given by Equation 12-1:

wh ere K is the combination of shaft spring stiffness, fluid -film bearing spring
stiffness, and support spring st iffness, and M is the rotor mass.
We want to apply this simple expression to the natural frequencies of high-
er modes of the system. In th e development of the sim ple model in Chapter 10,
we ass umed that the rotor parameters of mass, stiffness, damping, and A were
con stant. Obviously, to obtain a higher natural frequ ency from this equation,
240 The Static and Dynamic Response of Rotor Systems

either K must become larger or M must become smaller. We will show how both
of these things happen.
In a fluid-film bearing, the spring stiffness, damping, and>. are nonlinear
functions of eccentricity ratio. Thus, static radial load (which affects the average
eccentricity ratio) and rotor speed (which affects the amplitude of vibration and
the dynamic eccentricity ratio) produce changes in K and D, and in >., which
change the effective damping of the system.
More important than these eccentricity-related effects, though, is the mode
shape of the shaft, which directly influences the effective stiffness, damping, and
mass of the rotor itself.
A simple, vibrating, mechanical system involves the continuous cycling of
energy between the potential energy of a spring and the kinetic energy of a mov-
ing mass. When the velocity of the mass is zero, all the energy of the system is
stored in the compressed spring in the form of potential energy. When the veloc-
ity of the mass is maximum, at the equilibrium point, all the energy of the sys-
tem is stored as kinetic energy of the mass, and the potential energy of the
spring is zero. It is the ratio of these energy storage elements, K and M, that
determines the natural frequency of the system.
When viewed from the side, a rotor can be viewed as a more complicated,
vibrating, mechanical system. The energy in the system is traded between the
potential energy of shaft deflection (the spring) and the kinetic energy of shaft
motion. The mode shape of the shaft influences how much deflection is avail-
able for energy storage.
From the side, a deflected rotor shaft looks very much like a simple beam.
At the top of Figure 12-8, the rotor behaves like a beam that is supported at the
ends, which corresponds to a typical, first bending mode. According to beam
theory, a beam that is supported in this way will have an effective stiffness in
response to a static deflection force applied at midspan that is inversely propor-
tional to the length cubed.
In the second case, the beam has the same shape as a typical, s-shaped,
bending mode. This is equivalent to the beam having a pinned joint at the
midspan nodal point, which prevents any deflection there. The beam is now
similar to two beams with one-half the total length. A force applied to the one-
quarter point will produce a deflection, but the perceived stiffness of the beam
will be much larger than for the first mode shape. Thus, the effective stiffness of
the rotor is much higher in the second mode than in the first mode. We use the
term modal stiffness to describe the effective stiffness of a system in dynamic
motion. The modal stiffness of a rotor will be different for each rotor mode.
The only rotor mass that is available to store kinetic energy is mass that is
available to vibrate. The rotor mass near the center of the beam span (top) can
 ( hapt... 12 ModM of Vib, alion 24 1

"""...Ih.. rno", . /I;..a...r Ih.. ..1M.... I.... a moun l of motio n d<'c... a " m il th.. llea rn
i~ conSlra inl.'d a l III<> ..ndp..inl"- AI th.. I>"l lo rn. tl>... rolor rna near III<> Ct!nt.. r
nodal poim i. aJ.... " nahle'.. m,,,..,, TI>... lotal rno'"in~ rna.. for I hi~ mode m ust
be 1...... l han Ih.. mo,; ng rna", fo r Ihe fi.~t m..d Th us. Ih... clf""l i.... rna", oft h..
rotor ;" 1_ .." for th .. SE'COnd mod.. l han f.. r I fi~1 mod W.. u"" Ih.. I..rrn
modal ma... ' 0 d....,rih.. Ih., effPct" dyna mic rna... of Ih ~·'I ..m. Th.. moda l
rna.. ..."iIl be d iff... en l for ....cIt mod a nd xC<'pt wh n a n ..nl ,,,, .otor "".to''''
riltidJy (a rill,d body I ran~I ..liona l modd . th mod al rna of th.. rolo . will hto Ito,.,
tha " Ih.. stalle w..ij(hl o f Ih... ro lo r.
In 0 '" ..u mrle, Ih ... M'COlld mode modal . ti ffn is high'" lhan Ihto fi"'l
mod.. modal "',ff"..... a " d Ih ... second mod.. moda l rna i. Io....., Iha n Ih... fi ~1
mod.. modal rna,... Thu. I.... ral io o f II. to M mu,1 be hi¢,<,' for tb.. ....,.,nd m...Ie.
a nd Equa lio n 12-1 will yiot'ld a h i!!..... nal ural freq u..nc y. ......ieh i' u ac Uy ",ha t " ...
OW" ....

I'
t

f igu ... n ·s "'odal stJffnoo>., of • fOIOr. AI top. ~ ro<or

~ like. beam "'"' ;, ..... p/y "'ppOO1od .. ~
end, WIth.. ,t.>lI< d<1Iec_ , !of<;ropplit<l at mid,pan;
lh" .hope < ~ 10 0 '*" fi",
bend""l mo<le
AI bottom, lhe '0100' ~ I..... beam hoYm<.l iHl
oddn"""'l pir'oIed jOint .. "'" rnlChp1n ""'" 0 fon;"
applird to "'" one-<l... n.. POint- 1hi> se<ond mode i,
o typw;". Htwopod. bendi~ mode,The beam flOW
~ ~ r- tIN_ wnh one--l'>alf "'" tot.ollen<Jth
.r>d .. pe", ~ ,,"""'" mudllo<g« lhiHl for lhelim
mode ..... "". The ~"'" stIff...., oIthe ""or
i' """"h
hill'« in the _and mode thon In lhe 6,,1 ""Ode
 In.......t _ Jr!'l the- pnmary ........... of damp'''fl ;' the- fluid- film bear·
i"fl- n,.. doom~ tun. dc-pc'Qd~ OIl ~ ~I~ atlhr """'" CftIU'rl,,,.. In tM
lu t>rxah nj!: fluid. If a , BI ,.. h.... mode ..... J"l' IhM prodll,:... . Ia.'~e .mpl nudr at
VlM.I..." . A. in a be.,,,nl\- lhen I"" ' ibtalion ""loC1~ 'nIpl;1u.k. ~ = A11. " i ll be
h i!dl dod Ihe d a ml>i ng r'm,..., Ih' = OAf! ",jJ " I"" be . el. 1,wly l. rge. Th is i~ I....
......... I."
rillid I>o ody " " od"", Ih.. . d al i,...Jy ~l ifT roto r 1w.. 1a.~ ..mp lilu d.. vib.al ion
;nlM bearinp lHg"n- 12-9). "".. t his ........... . ~ body modf.!. ofle n ha, hi¢!
Qu..o<!,..tu,", Dy.......i.- Stiffnna and k- Srndu Ampl;fo tion FlICl In
ma~' CI.'In. lho- ''"Y ~ rigid ~ rnodrol -...niamprd and . ~
..........na don n"l1 appnr at aI1.

...... -

~ 'l., _ ~ .., ,,,,,,, ~ _ al "'"

d ..,."...,g fot<e ~ '" 1\.001_ -.ng...-.:I., _ _
_ to _ tm<.. -.cy of _ fClIlor '" _ bNt ·
~.I\I

~,on lhol ~r. ", .~ "~ to ' - _ ........ dirnpoo--q

con"......
~ _ .....,.--.g; 1or<:e..1lIndoo<"J """"""' _to
D IlIgoclllcdy ft'ICIdeS podur:llItgI ~ '" I....
ptO-

_1tl
dtIIll ................. " ' _ ~ ~
Ior<e.. _ "'""-"l ell.......
*"'.
tho! ofII<_ dar, ..........
.,_~_"_
 Chapter 12 Modes of Vibration 243

On the other hand, flexible rotor modes tend to have nodal points that are
located relatively close to bearings. This results in lower vibration amplitudes in
the bearings, which produces low vibration velocity and a low damping force.
Such modes tend to have lower Quadrature Dynamic Stiffness and higher
Synchronous Amplification Factors, with higher vibration at the antinodes.
We use the term modal damping to describe the actual damping force avail-
able to the system. The modal damping depends on the actual damping of the
bearings and seals combined with the mode shape.
This qualitative discussion shows how modal stiffness, modal mass, and
modal damping depend on mode shape. The rotor modal parameters combine
with bearing parameters, which change with eccentricity ratio, to produce over-
all modal parameters. Thus, each mode of a rotor system can be viewed as hav-
ing a different set of modal parameters that are associated with each natural fre-
quency. The variations in modal damping will produce a different Synchronous
Amplification Factor for each mode.

The Measurement of Mode Shape

Measurement of rotor mode shape involves determining the three-dimen-
sional, dynamic deflection shape of the rotor at any point in time. A complete
description of instantaneous rotor position would require unfiltered orbits and
average shaft centerline position data. However, for the purpose of defining
rotor dynamic behavior, we are usually concerned with defining only IX rotor
behavior versus speed. For that reason, the discussion that follows is concen-
trated on the measurement of IX mode shape.
In principle, a set of axially spaced, shaft relative transducers can be used to
determine the rotor mode shape over the entire operating speed range of the
machine. This can be combined with data from casing transducers to establish
the system mode shapes. This can done fairly easily with IX polar plots or more
accurately with IX orbits.
IX polar plots provide the easiest approach to estimating the mode shape.
This technique is commonly used in balancing and works best when orbits are
circular. However, when the system has a significant degree of anisotropy, which
produces elliptical orbits (Chapter 13), polar plots can produce misleading
results.
The polar plot technique uses data from a single transducer per mea sure-
ment plane to display the startup or shutdown data at points along the length
of the rotor. The plots have the same full scale range and are positioned in a way
that represents the axial location on the machine. Points at the same rotor speed
are linked by a curve, which estimates the mode shape.
244 The Sialic. and Dyn amic. RMponse of RolO1' SystlOm'

'o- 'o-
ro.... "" lui .... ro...."" ruI .....
" "
{ ...... _ _ I1 . .... ..-J ~
~ - - - -- - - - ~

t !

s-< I " TO """I

f igu.. 12-10 ~ poIaI pIot.,o .-I'malP ,lop '0101' modP ,twopp,..,, !'NO
bNn"'.l
Thp IX
r<>1or
pol., pkm _two
h., >I\iIIt ,....., "'" p<obeo moun,p,j ius, tIboat<:I oI,hp bNnngs.
bak.oncp ''''''''"''''nat 16SO 'pm and Sl 10 'P""<
in thP 'totluP dol ... Thp "" ;malo<I <TlQ<lP •.....,... ~ """"" . , thP l>o1toon-
 n... polar plou In r ....... 1-10 .........· IX Mar1up dal. f""" . ton:> bnrillfl
........ with Wit ......1Ln probn molintni juol inboud of the bnrinp.- BaIartno
1'f"IOII&fICft ..... ot-rwd at l b.";!' rpm .nd 51 ill rpm. n... "tiftUllrd lIlOdc- oNoprt
..... ohown . 1 th.o bol:tom.
If lhor rotor om .u . ... cunda~ lhn> u... point on lhor polar pM>t riI -:a.flIti'4:o
.'1". ....... 11v loc al ion of thor rotor (.. twn .tjuolrd for IIv pN-k-lo-pl'.ok .mpli-
I " of lhor polar plot ) .. h .... IIv Mypka.............t occu n... ~rd modt
~cw'" -...ld b.. rqui...w.n, to linlu"ll llv ~ doh lllv location of
thor rotor wf>o>fI thor ~-rtoa- nt oocurs) of. onv.. of cwcular omit...
,f thor orb,t> .. r. _ ci.nd but ftIipticallhn> thr .m plilw.. . nd pha.......-
un o n thor poIa. plul may not Jdo,nlJfy lhe corfl'ct local i" n oflh. 1i:~-ph.'K>I' dots
on u e orbit... Figu," 12· 11 . h"",.. 1\00", pola. plot. fn lm I'" ... m. mt'."".... mt'nl
plane. This slu m I,n h,". Il..n alor has hil(hJy >1 1 .. rbl1 ~ . nd l he 1\00",
Jl'~" l'lols 100" ,...ry t1ifT..ft°nt-Th is a la rge d i""ll' n nl ... 10 the loc. tion of
1M tutur het .....,..,n Ih.. l" probe da~ (le ft I a nd the X p~ da la.

".II•
r

- .' x"
"
•
-~
•
.

• 110· .•
••
~.,

,1\0.",,'9 _ .fht

__.
f ~ ~ r . 12 ·1 1 Po&. plou _apo< poIM-
plou •• ""'" J(I' probn mounted in !he some~ . The pCllor
plou.r- . duo lO ~ ~

_ ~ ........ cjOll\(u11 10.......__ ....... " " _ *

a - I l b .. ~~_c......
246 The Static and Dynamic Response of Rotor Systems

The mode shape is more accur ately determined using IX orbits (Figure 12-
12), which require two tr ansducers at each measurement plane. Filtered orbits
are constructed for each measurement plane, the orbits are plotted at the same
scale , and the Keyphasor dots are connected.
If the orbits are digitally sampled, and all the waveform samples are syn-
chronized, then individual mode shapes can be constructed for each sample
time (individual point) in the sampled waveforms. For example, if 128 X and Y
waveform samples were taken for each revolution, then X and Y sample 99
defines a point on the orbit. This point can be linked to sample 99 points on
other orbits, defining the rotor mode shape. Th e collection of the mode shapes
for all the samples defines the three-dimensional envelope formed by the orbits.

Mode Identification Probes

Accurate determination of the rotor and casing mode shapes would require
a large set of XY shaft relative and casing transducers spaced along the axis of
the machine. In practice, such a large set of shaft relative transducers cannot be
installed because of physical limitations. It is not physically possible (and not
economically acceptable) to install large numbers of shaft relative transducers
in extremely high -pressure or high-temperature regions, or where transducers
would interfere with process fluid flow paths. Because of these considerations,
shaft relative transducers are usually mounted near bearings, where access, tem -
perature, and interference with the process are not a factor.
Thus, mode shapes must be interpolated between a small set of measure-
ment points. Usually, this interpolation includes places far away from the actu-
al measurement points. This is unfortunate, because often our primary objective
is to determine the clearance between the rotor and stator at the midspan of the
rotor, exactly the area of highest uncertainty.
On large , critical machinery, modern management practice dictates the
installation of XY shaft relative transducers at each fluid-film bearing.
Unfortunately, this set of transducers will not always provide enough informa-
tion to measure complicated, higher-order mode shapes when nodal points
exist in the interior of a machine; often, more than one possible mode shape can
fit the data. Additional probes, called mode identification probes, can be
installed on both sides of each bearing, to provide more information.
The problem often occurs when nodal points exist at a location outside the
bearing. Figure 12-13 (top) shows a machine with transducers installed on the
inboard sides of the two bearings; the actual mode shape is shown in black. In
this situation, a nodal point occurs inboard of the left probe. More than one pos-
sible mode shape (middle) could fit the observed data from these probes. When
additional probes are installed outboard of the bearings, the additional infor-
 ,
• • • '
,
. ,
•
•
•• •
•• •I •
•
C) - o ~ ~y/'
,-..... 1} _1 J [-..,., d _ - * """",,,....-;I _.po
-.. ftoot ,~ ""' _""""'_
plontd .. _ ....... ICM -no". ~ _ "'" ,. lOW-
_d 1ho~_0« .....
~ __
~ ... _...-e_Iot<""'Il. __ "-

r.... U ·I) _ dtft,~­

' ''''' poobn ....... -~

-,,-,~.....-
If...o..r- _ _ .... IN
.... -,, ~- - - -

--- -
_-.d''--'"''

--,......--
n.e..-_"-,,
-"'-"qJI.-
.....
1~~.Il'It .......
..x ; _ --...
- ""'" -

....u.....
-_~
poobn.
..-..." Gl"'" -.g.lbct-
~
• --•
-..-
...". htIIIo............ o<twI
...,.".
_ ",-Otld_.-
___d
248 The Static and Dynamic Response of Rotor Systems

mation helps identify the actual mode shape. For very complex mode shapes,
even this probe configuration may not provide enough information to unam-
biguously define the mode shape.
The best technique for mode shape estimation is to use advanced rotor
modeling software in combination with vibration measurements. A model of the
rotor system is constructed using accurate shaft and rotor disk dimensions and
material properties, and the program calculates the theoretical mode shape
based on known physical laws. When combined with actual measurement data,
the software can automatically optimize bearing parameters to provide an accu-
rate mode shape of the rotor. The time is coming when such software will be able
to use real time vibration measurements to provide accurate rotor mode shape
information.

Summary
A rotor mode shape is the rotor's three-dimensional, dynamic deflection
shape, which changes with axial position.
A system mode shape includes information about the relative motion of the
rotor, casing, piping system, and any other part of the coupled system.
The rotor system can exhibit free vibration at one or many natural frequen -
cies. Each natural frequency, or mode, has its own characteristic mode shape
that is different from the mode shapes at other natural frequencies.
A free vibration mode can be forced, or excited, by the distributed unbal-
ance of the system. An unbalance distribution will have its own characteristic
three-dimensional shape, with both amplitude and phase as a function of axial
position. If the unbalance distribution shape is a good fit to the rotor mode
shape, then that mode will be strongly excited, producing a balance resonance
when the rotor speed is near the natural frequency.
Rotor system forced vibration includes contributions from many free vibra-
tion modes. Each mode is excited to some extent by the unbalance distribution,
and the resulting rotor response is the sum of the contributions of the individ-
ual forced modes.
Each mode can be characterized by a set of modal parameters. The square
root of the ratio of modal mass to modal stiffness determines the natural fre-
quency for that mode. The modal damping determines the Synchronous
Amplification Factor for the mode. Modal mass, stiffness, and damping derive
from the mode shape of the rotor.
Mode shape can be estimated using polar plots from different axial loca-
tions or from orbits at these locations. Nodal points can make determination of
the mode shape difficult. Mode identification probes are used to provide more
information about nodal points near bearings.
 249

Chapter 13

Anisotropic Stiffness

THE ROTOR MODEL THAT WE DEVELOPED IN CHAPTER 10 is isotropic, pro-

duces only circular, IX orbits, and predicts one forward balance resonance. The
model also serves as a basis for defining the Synchronous Amplification Factor.
An important result of the synchronous rotor model is that, at low speed,
the heavy spot and high spot are approximately in phase. This allows us to use
polar and Bode plots to identity the angular location of the heavy spot for bal-
ancing. This capability depends on the assumptions used in the model, which
include single mode behavior and isotropic parameters of mass, stiffness, damp-
ing, and lambda.
However, in real operating machinery, multi mode behavior, in combination
with fluid-film bearings, can sometimes produce a larger than expected phase
lag near a bearing. This effect will be discussed in Chapter 16.
A more common problem is that IX orbits are elliptical, and the orientation
of the ellipse usually changes with speed. Elliptical orbits produce shaft center-
line velocity variations that affect the interpretation of phase, leading to a break-
down of the assumed heavy/high spot relationship used in balancing. Also,
because of this ellipticity, measurement of vibration amplitude depends on the
orientation of the orbit relative to the measurement probe.
Many machines that produce highly elliptical orbits have closely spaced res-
onances that have a similar mode shape, called split resonances. In between
these split resonances, over a short speed range, the rotor may even travel in a
reverse, unbalance-driven, IX orbit.
These effects are a result of anisotropic stiffness in rotor systems. In this
chapter, we will discuss how anisotropic stiffness influences rotor system behav-
ior, with an emphasis on measured vibration amplitude and heavy spot location.
250 The Static and Dynamic Response of Rotor Systems

We will start with a discussion of the meaning of anisotropic stiffness and the
physical reasons why it is common in machinery. We will then discuss how
anisotropic stiffness manifests itself in rotor behavior, show how, for ani sotrop-
ic systems, measured vibration amplitude and phase depend on probe mounting
orientation, and how thi s behavior can lead to ambiguity as to the location of
the heavy spot.
Finally, we will present two signal processing techniques that improve the
vibration measurements of anisotropic systems: virtual probe rotation and for-
ward and reverse vector transformation.

Anisotropic Stiffness
For the purposes of this discussion, a parameter, such as mass, stiffness,
damping, or A (lambda), is isotropic if it has the same value when measured in
all radial directions. A parameter is anisotropic if it has different values when
measured in different radial directions (Figure 13-1). Because mass distribu-
tions, shafts, bearings, and support structures are not perfectly symmetric, all
rotor parameters exhibit some degree of anisotropy.
Uneven mass distributions on rotor casings and support structures con-
tribute to anisotropic modal mass in rotor systems. External piping can cause
different observed modal casing mass along the axis of the piping compared to
directions perpendicular to the piping. Rotor mass (and stiffness) can also be
anisotropic due to shape asymmetry, which is common in electric motors, wind
turbines, and generators. However, these asymmetries, because of rotor rotation,
typically manifest themselves as higher-order excitation of the rotor system,
particularly in the presence of a side load. A stationary observer sees an average
value of rotating rotor mass, so we will assume all rotor parameters to be sta-
tionary in this sense.
Most rotor systems have relatively low Quadrature Dynamic Stiffness. Thus,
damping and A anisotropy will be assumed to have a relatively small effect on
rotor response, and the tangential stiffness, DAn, will be assumed to be rela-
tively isotropic.
However, anisotropic spring stiffness is common in rotating machinery and
has a strong effect on rotor system response. Figure 13-2 shows an end view of a
typical, horizontally split machine and the stiffness contributions of various
components, including the piping, fluid film, and support structure. Note that
the XY vibration measurement probes are mounted at ±45° from the vertical, to
avoid the split line. This is a common mounting orientation that has important
implications for vibration measurement of anisotropic systems.
The method of mounting the casing to the foundation can also produce
anisotropic stiffness characteristics. Angular stiffness about the long axis of the
 - ' " 4 .

N,u" l J-I . ItOt'OP< _ ....""''OPt..,...

,......"'Y''''''' propo<tJ i<~ ~ ~ ~ IS
....... in ~. rod'" do'"" ........", ,)'1', pmp-
«11 i. -.op.c ~ n "'"' <lilio< ~ ,n
,,"""..., r od-.l ""f'<1""",-

-,!, ._
..... 1).-1 Sc:u-...,.d_ .._
.. _

_A
~ X T ! t w f :

..........-. _ _ ..........., S· K

_
~
- .I0_ __
frooo> ....
--...-....,.""'
.......
~
bA:io:>on._ PlIlr'9_'"
~.~""""""
eIlocn. ~ 1'Ud- ~'" _ng IS "fOI'9lr
~ 01 "";J~ ""~tricoly .. ,10>; ....
/OUI'
.

""" . 1'IIg~ ""''''9 <loflrooM on ,.....

'td "'~ Wong .... """ con_'"'lI
tINnI'lg _ ~ <"" 0<>1 , ....... on tl>r un-
9'"''"' do~"",,- The.....,.,. (_,Mo'"
ore . -to ~ 10 prcduc. """"",.
UoIt\t - one! ....,,,""'" "-'9 ""'-on.
--
252 The Static and Dynamic Response of Rotor Systems

machine can be significantly lower than vertical stiffness; thi s can appear as a
relatively weak horizontal stiffness .
The stiffness of the casing and support is influenced by the stiffness of the
piping system and it s attachments, and, typically, it will be different in the hori-
zontal and vertical directions.
Most importantly, a typical rotor is supported in fluid-film bearings.
Remember that the rotor model was developed with the assumption that the
rotor was operating, fully lubricated, in the center of the bearing. In this region
of the bearing, the spring stiffness is essentially isotropic. Lightly loaded, plain
cylindrical fluid -film bearings operate at low eccentricity ratios and can have
large attitude angles, sometimes reaching 90° or more. Also, externally pressur-
ized (hydrostatic) bearings normally operate in a fully lubricated condition at
very low eccentricity ratios; these bearings are essentially isotropic in behavior.
However, normally loaded, internally pressurized (hydrodynamic), plain
cylindrical bearings operate in a partially lubricated condition at moderately
high eccentricity ratios. At high eccentricity ratios, because of the action of th e
hydrodynamic fluid wedge, the journal sees anisotropic spring stiffness.
(Imagine it much smaller th an the bearing; ifit were sitting in the bottom of the
bearing, it couldn't move as freely down as it could move left to right.) The
anisotropic stiffness resolves itself into a strong and a weak axis. The strong axis
is approximately at the position angle of the rotor, acting in a radial direction,
and the weak axis is at 90° in the tangential direction. (Be careful here. We are
talking about variations in the sp ring stiffness in the radial and tangential direc-
tions, not about tangential stiffness, which we assume to be isotropic.) In a hor-
izontal machine, a properly aligned, gravity-loaded rotor with plain, cylindrical,
fluid -film bearings, will operate in the bearing at an attitude angle of a few tens
of degrees. For tilting pad bearings, this is typically less than fifteen degrees.
Thus, the orientation of the st rong spring stiffness in the radial direction and
weak spring stiffness in the tangential direction will be approximately vert ical
and horizontal, and the horizontal spring stiffness will be lower than the verti-
cal spring stiffness.
In an anisotropic system, the radial spring stiffness distribution can be a
complicated function of angle. In this chapter, we will assume that the spring
stiffness distribution has an elliptical shape (like that in Figure 13-1) and can be
resol ved into strong and weak stiffness axes that are perpendicular to each other.
Th e orientation of these axes can be in any direction, but because of th e
machine characteristics we have discussed, we will assume they are approxi-
mately horizontal (weak) and vertical (strong). We will call the weak spring stiff-
ness K weak and the strong spring stiffness K strong'
 Chapter 13 Anisotropic Stiffness 253

During the following discussion, remember th at this stiffness orientation is

an assumption based on gravity loading of the rotor. Some machines may have a
radial load vector that points in some other direction (for example, a gearbox),
and the rotor may operate with a large position angle. In this case, the stiffness
axes may be oriented in some other direction than horizontal and vertical.
As we discussed in the last chapter, different modes of vibration have differ-
ent modal parameters . By extension, the degree of a nisot ropy may change from
mode to mode. Thus, a rotor system with a relatively low degree of anisotropic
stiffness in the first mode may have a higher degree in the second mode, or vice
versa. Anisotropy may also change with axial position. We will primarily discuss
rotor response from the perspective of a single mode.

Split Resonances
A split resonance consists of two balance resonances that have a similar
mode sh ape, but are separated in frequency. Split resonances are a direct result
of anisotropic spring st iffness. Recall that, for low damping, a balance resonance
will occur when the rotor speed, fl, is equal to th e natural frequency of the rotor
system,

(13-1)

where K is the spring stiffness of the rotor system and M is the mass of the rotor.
In our system with a nisotrop ic spring stiffness, two values of spring stiffness
exist, K Weak in the horizontal direction, and K stroTlg in the vertical di rection. Thus,
the system can produce two resonances, one associat ed with the horizontal
spring and one as sociated with the vertical sp ring. The first resonance, as soci-
ated with the weak spring, will occur near

(13-2)

and the second, associated with the strong spring, will occur ne ar

(13-3)
M
2S4 ~ Stal i( and Dynamic Re'PO<"" 01 Roto r Sy,t"rm

If t h.. <.t,ffn"". ax". a. c o riented I..." izon ta lly and ' ''rt icalJ)·. and Ihe
Qu.ad 'a llJ'" I)~. nam ,e Stiffn'.... o f the 5\-,;tem
. is low. Ih.." tl,.. rotor mo tion in l he
Ii",t "",,,,,a ,,.... ...i ll ha...' a la.fI" I>orironta l co mpo" ..nt. and i" the ><'Co nd ...._
nan."'. a lalfl" \-...tical romp" ncn!.
n ... ""'po" .... d ue 10 a t:ol'ical 5plit ........"a n.·.. "' . h....... in Fig" ... 13· 3. This
data. f'm n a rotUl s~· .. cm mod..J"il h a ni""" rop ic sllff" """ i ho. ilO"taJ w..ak and
....l li,"'1 >trong )••h" ....,; l he rotor ' ''''po n... a. it would he ",eas ured by a p.obE>
mount.-d al -l~' I~ I'< ,te thai Iwo. d isl IlIel """'na n"e I"""h ca n be ......n in th..
IIo 000e p lo t. The p ha", lag i"'....a""" l hwugh th e ho. i",,,la l split "'''>lla '''.... Ilu>n
<1...." ",.." a s Ih.. s~.,.t ..'" al.p roac h... Ih.. w . tica l . plit "",."a n.",. IMJ t IlIC.."'....
a!,\" i" a5 th e ' :,"'1.." , g...... Ih rough th.. w rtical "PHI.. 0 " th.. 1• •la' 1,1, >t . th..... is a
c1ea. int emal 10.,1', ",h i,'" eorr""pond5 to t he am plilu' J., itnd I"'a...., .' hang"",,
bet"'....n II,.. split ",,,,,,,a n...... O rbi.. a... 5hoo.mll for k,'Yfl'''' I'0 inls.

-'..
~ c-:~£-(J
-- .",. .- '''''
-o,

.............-

f Og" '" 1]-3 A typic;tl 'I'In .....,..,....". p,;, dato .. fr<:m . rotor 'YS""" model _h
....""fOl>O:: " ofI-no1,s (hofiron' a1 --. ~ and ...... "".I."ongl.nd ,"""" tt>" ",,,.. oe<pon....
'" n would tie "'.",."... b'I • probe moun'<d '" as' L T_ d.'lm ,....,........,,, pe.>ks U "
tie ...." In , I>e _ " plot.The pIwse 109 in<......... ''"rough , he fi", d
,.... ,.....,.....,.....
,...." -...a"" ., ,.... ')'S'''''' .pp""""...
the _and ,,,,,,,,al1C"_
PI>a.., 109''''''' .nt'......-
... _ '" !h'O"'lh "'" =ond ,"""""'," On t..... poi¥ pIo1.." in,,,malloop ,, ... bl" !ha '
cor'espond' '0 "'" ampln.-.ond ph. .... ,~ bt'h " • t.... 'p1~ "" on.I1C"'-
 !-1 ' '''Ui
' ~ i :-
11.1' 1" 1< ~ i Jr ..
~ Ii ~i'"~3 ; -+
1 ~r ' ~n"i
u
-w ~
"l' ~8<i ,~ !": !l" ll 'q
n .,, ~1
s !t' J , l .c .. a s
i n'.H = :I :I ,. ~
Jtl lhi~ni!f
I','!'II - _
ilJ!"l.
ghiihH i i C
d'
·hl!
~i 'linn l~,]lth
l"l - -
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~ :I 1c.p~n
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U I

' ~ i I~HI
'!\I "" - - - '~H ,." ",- l,
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- ~ I ?~Hil" ~l'il;
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h~ Hi - ~Ihl HH {
'---.) . 8'
• }j<
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"·§l'ii'~ ~_ ~,, ,...:!
ll.~'~ 3 <~~q <:" ~ ~13"' ;.< I~
...
; , l li' "q' '' t i
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IlHHfll .if
;"' . '=> !';.~fl
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!Ill",l =
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II, Is'
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"•
256 The Static and Dynamic Responseof Rotor Systems

peak of the split. At 4000 rpm, the system is well above the split resonance, and
the mass st iffness in the model dominates the response. Becau se mass stiffness
is isotropic, the orbit becomes nearly circular. (If the next rotor system mode has
anisotropic spring stiffness, then orbits will become elliptical when the second
mode begins to dominate the rotor response.)
The appearance of split resonances can vary from a slightly broadened, sin-
gle resonance peak to clearly separated resonance peaks. The separation will
depend on the degree of anisotropy in th e spring stiffnesses, the amount of
Quadrature Dynamic Stiffne ss, and the viewpoint, or angular orientation, of the
probe. Low quadrature stiffness will tend to create clearly separated peaks; high
quadrature stiffness will tend to smooth the peaks together. The small polar plot
loop will also change appearance; clearly separated peaks produce a well defined
small loop, while , on a high quadrature stiffness machine, the small loop may
only appear as a small bump. As we will see shortly, probes misaligned from the
stiffness axes will tend to highlight split resonances, and those aligned with the
stiffness axes will tend to obscure the split.

Measured Rotor Behavior and Anisotropic Stiffness

There are significant differences in the observed behavior of systems with
ani sotropic spring stiffness compared to isotropic systems : elliptical orbits, dis-
agreement between high spot and heavy spot location at low speeds, and the
presence of split resonances. Also, in anisotropic systems, data becomes
dependent on the orientation of the measurement probes: there are differences
in the appearance of Bode and polar plots, in the observed Synchronous
Amplification Factors, and in the measured vibration amplitudes. Probe mount-
ing orientation becomes an important factor for anisot ropic systems.
Transducers (probes) are usually mounted in mutually perpendicular, XY
pairs in each measurement plane of a critical machine. The probes can be
mounted at 45° Land 45° R (typical for horizontal machines), 0° and 90° R, or
any other desired orientation. These probes generate timebase signals that are
combined to produce an orbit (Chapter 5). Each transducer signal can be used
to create a separate nX Bode or polar plot of a machine startup or shutdown.
There is a tendency to use the vector data from only one probe for analysis.
For isotropic behavior, this is acceptable, but isotropic behavior is rare in
machinery with fluid-film bearings. For anisotropic behavior, the use of data
from only one probe can lead to a serious misunderstanding of machine behav-
ior.
Imagine a rotor system that is producing circular, IX orbits. Because of the
symmetry, the I X vector from a single probe could be used to reconstruct the
original orbit. However, if a rotor system is producing ellipt ical orb its , it is not
 Chapter 13 Anisotropic Stiffness 257

possible to reconstruct the orbit with vector data from a single probe. Vibration
vectors are required from an XY pair to reconstruct a IX, elliptical orbit.
Similarly, a complete picture of anisotropic, nX rotor behavior versus speed
requires data from two probes: two polar or two Bode plots. A single plot will not
convey an adequate description of the vibration behavior of an anisotropic
machine.
Table 13-1 summarizes the important differences in observed behavior
between isotropic and anisotropic rotor systems.

Table 13-1. Isotropic versus anisotropic rotor behavior.

Isotropic Anisotropic

Circular 1X orb its Elliptical 1X orb its, possib ly reverse between

splits
No split resonances Resonances can be split
Measured vibration not viewpoint dependent Measured response changes with probe orien-
tation
1X polar plots look circular Polar plots may not be circular, and a small
loop may appear
X and Y polar plots look the same X and Y po lar plots are different
Low-speed heavy / high spot are aligned Low-speed heavy/high spot are aligned only
under special circumstances
Single mode SAF matches theory SAFis viewpoint dependent
258 The Static and Dynamic Responseof Rotor Systems

Because orbits are circular in an isotropic system, measured vibration

amplitude and phase behavior will not depend on the probe mounting orienta-
tion, and, consequently, polar plots from XY transducers will look identical
(Figure 13-5). In isotropic systems, the polar plot of the rotor response through
a balance resonance will have a circular shape.
Isotropic, single mode rotor system behavior matches the ideal behavior
used to define the Synchronous Amplification Factor (SAF). Thus, the SAF
measured on a single mode isotropic system will be a good match to theory and
will also be independent of probe mounting orientation. However, SAF meas-
urement can become troublesome even for isotropic systems when closely
spaced modes can interact and distort the SAF measurement.
In anisotropic systems, IX orbits can vary from nearly circular, for mildly
anisotropic stiffness, to extremely elliptical, line orbits. In systems with low
quadrature stiffness, orbits can also exhibit reverse precession between split res-
onances.
At speeds well below a resonance, rotor response is quasi-static, and the
Dynamic Stiffness is dominated by spring stiffness (Chapter 11). The relatively
weak spring of the anisotropic stiffness allows more rotor deflection in the
direction of the weak stiffness axis; thus, the major axis of the low-speed, ellip-
tical orbit will be approximately aligned with the weak stiffness axis.
Anisotropic spring stiffness can produce split resonances. The visibility of
the split will depend on the degree of anisotropy of the spring stiffness (the sep-
aration of the two resonances), the Quadrature Dynamic Stiffness of the system,
and the orientation of the viewing probes. Relatively high quadrature stiffness
will tend to broaden resonance amplitude peaks and blend them together, espe-
cially if the degree of anisotropy (and separation) is low. For this reason, split res-
onance peaks may not be clearly separated on polar and Bode plots.
If a measurement probe is not aligned with the axis of the elliptical orbit,
then a phase measurement anomaly will exist, producing an error in the inferred
heavy spot location. The data from an XYpair will disagree on the implied loca-
tion of the heavy spot. This will be discussed in more detail below.
Anisotropic stiffness can produce significant differences in measured IX
vibration response (both amplitude and phase) between X and Yprobes in the
same plane. This difference is largest when orbits are highly elliptical and disap-
pears when orbits are circular. Figure 13-6 shows the effect of probe mounting
angle (viewpoint) on the measurement of vibration amplitude of an elliptical
orbit. In the elliptical orbit at left, the Y probe sees a much smaller vibration
amplitude than the X probe. In the circular orbit, both probes measure the same
amplitude.
 -
. ......-
'
. - ........ -
F '9~ " ll-S Y_ XpglOI pIol. 01.... '''''>opt 'Y"....... Tho ~ "POI do"':I'on l~
doO i>up for bot~ plQO:,The _~'on<~ cI th r pie<' """ "Ol ~tId on Il'~ oh-
..,t.ot>o<l; on. plot.... 1 _• ._ oil 0'tMt> . .. <;,,;..... , In e ocr. t lse. tt..low-
!PHd """""'... -.ot'I po;..l """Old tho htovy 'POL Tho _ 'PN'd G ' b l _
"'. .... h9" """ ~_"" _ ; S in .... _ _ <100"";00' M .... ~ 'l"'t-

--_..-
r-. ... I J- 6 _ ... po
_..."...,..,... '*"'"""' ~_ompIIIuOe""""-~""~~N

_~
.....,r:oo _
__
CIfbo: .........-
............. _0Ja8'
f.ot
-.-
260 TIM Static and Dynamic Rnpon$@ of Rotor Sy'lt......

Bffa u of th" d"pt'nd"n~... "f ,~h.at ion ,""a,u ",m"nt on p robl" a nd " rbil
ax i' a1ill:n nl. po la r a nd Bod.. p1ol~ of riat" from X a nd r p robl"a can ha,... a
\'err d iff" ",nl ,;~o.aI app'·ara "' A81he or i..nta l ion of a n O'l bptica l o m it changn
Ih. ou gh "'ilOna ........ m..a "" vib l'lliio n a mplil ud.. a nd pha.... ",ill ~iJa ng.. in a
\'err di rr..... nt ma nnn lhan f"t t ho> circ ular orbit p red ictl"d Iry Iii.. . imp l...
i""trop'c "" x lo>l.
Figu.... \3. 7 ~ttm.... I X polar plots (If X1'll'a nS<luC1't ,.....-to r d ala p l'Oll" .,...11ry
a n a ni<olropic . ot o , ' p t..m mod t'!. Th.. or "'nla rion oJ t .... stiff ax..s is s hown
in red. along "'itli a 10",·, P"""d o rbil. Th.. Yp lot , " h ich has l he me .caJ.. .0..1....
X p lot . ha' a , mall..r maxim " m a mp litud.. a nd conlain, a ~malI loop lhat looks
lilt- a <.t ructu,a1 """"na nc... ,,-bile Iii.. X plot 61\,,,,,, a m uc h lar!!.., a mplil u<k a t
tli.. ."""na n"" a nd no <;gn. o f a loop.
Tho> loop in tli.. 1· plot j~ ,,01 dIU' to Q Mrw;fuml,..<m,a lrU. Th .. modd "sed
to g..n..",I.. th""" plot s i' not ca pa hle o f p rod uci ng a at.mclu, a1 """'na n"". Th ..
loo p is pu....ly a n ar tifact of Ih.. a n isotropic ro to r<y>ot..n, ""'po"'" a nd the p ,obe
,,,,,,-poin t. Thi. impl ~ t hat man~· or tb.. . ma UI""ps .....n on pol a ' p lots a not
st ruct u ",1 so ne aces a t a ll but an> du e 10 a nisotrop ic stilTn"",- Lal..... " ,,~11
....., ..xa m pl o f m"",hin.. data that show ....arly idf'fltica l bl"havi" ,.

... _. . .. -
- , <WI........ . . .

Rg..... U -7.lX poIat pl",n from a rot", "",om modrI w nl\ Mlisotmpic <pring n~...Tho "';oo-
t.>tlOO '" the 1SOt<OPiC <t;flne,., .on ~ _ r> r<d im dr .o. k:w-'PH<! 0ttJ;t Tn. y plot.
which 1\0\ _ >eat.. ", the ~ plot. _ • _ .......,. ..... " rna """"", arnplolude """, . ......
loop !hat _ like. nrocnnl .............,.._ tho !I plot _ . """'" lIrg<'I' . rrc>lmm .. the
.....,.,.,.,.. 1I'Id..., oig'" cI.o. loop_ Tho'\tnx"""I' loop ... 1..... Y plot;,. not . structural ....,.......
sy\''''''
at all:; ~;,. po lf'ly .... a<t~acl cl l ..... ...""ropic; rotor _nt
r""""'... .o.f\d the probe ........
 (hap.", 13 An isotropic St iffr>e.. 26 1

_ Cauw. ~ ....,...,... O<t> ~ . r

RoI"' n~ unbalonc r produ<w •
purfty forward p""urbahon of •
""or '~"<'rIL _ <an • . . . . -
orbit bo g....... ,..1 by fonnnl 1""'.
,out>.lion.
Thor koy ill , ..... pb. ","""on"".
•
".
---!il
-,
o
,"
_
....ich <an bo thoaof;h.of... po,<.,r

- ~
_ ............ o<cur in P"P"ndJ-

.'", ..
---
.....r d;...,.,-... AoIIu,,", <hat , hro<o
<IIrrdlon . ore hon..,Rlal eX. "'r
<II...,.,,,,,, of oil. _~ k .tdf. - I ...d
..." ical ( Y. 'h. di r..-tion of tho
It"",. .. 1Ifnorss~ t hat tho .,..."'" h
-
) ,"
•• •
- ,-_.-
-- - --
,-
"'-....
.-~ by"'" prtJl>r> """,..t..l !>or-
,.

f\ ~
,..,."an, ond wtticolly.And thot th o

Dun"ll ... ••••

...,.." ......tn ," . n X to Yd 'R eh",...

••• .....,.....<'<'.
pho... 1ag i"""'..... from IT '" about
,80" R I. " ,,", to th o h""", . pot .
" """' • opIM .....,......., i. -u _
•!• ) /
- - ,------
-, -
arrtt.... ond thr quadr.tu"" IttfI'nt.a
of .....~t ..... ,,1‫ס‬0o th .... " iI ........ ••
.... for .... horizon.aI r -...""" to
....... th rooWt mooI of ,h. phasr

-.
<tuI~ _ .... 1''''''' lag of thr
_ d ............"" .ho"ll'" "pill._
A~
In thr figu",. tho X R .po..... -X ~X
W ,,,.-
(po<'tIl charl g<o pho "" and...",...
cur ... ( b1u . ~ T ....
....,...,..,.., po;n' prod....... . .,...
.,
o.-bot. ' n .... "Jl"'I'd ran go ..-t>r", thr
X pIut.. ill boIow th r r 1''''''' """'~.... Ihadi"ll~ "'" orbi. h """""'. A. th o r p-Iag
itt<-.... with th.-..nd ~ "'" Ypho "' ......... ' '''' '' .... tho X ph..... prodocl .. g.
b... _ t .. .... """";"!l 1J"'O'!. At his""" Jp«do. thr orbit is 0....... oIlip"cal fcno-.nl-
" "ho"'M _ """",IN . t • tp<rtI in thr )'<UOW ...... ' .... loon..,..... coo'"I" ." '" of ,i"""
bo.. wdIbo ~ ppro:<im.t "' out ofpha.. ...n h tho .....,. ....... »hile- th.wnbl rompoo....t .....
bo appt<lJl imatd)o in phasr. Th".,..... """tionolup prod"""" ~ ........... orbit. 1.... .-1 of tho
_ - ""Il ;" on X to Yd;r...-tion Ibonom ~ thr ""Of DOW m....... III ~ - X to Ydi t<o;tinn.
_ it- "I"n-aJoont to Yto X.
Wb.n tho ""'or.pood inc...... furt h.... th• . - . _ _ ' h """" t .... .......... """'''"''''"'.
whidt ....... .... pha.. lagoftM Y.._on to chongo by 180". nil produ.... ~ - X to _Y
do.... ""'n of 1'. ............ wh..h h "I"n-aJoon. to X to Y. or _<d.
.'lor . thot . ...... 1'''"'''" oo....darin only .pply -.iI........ 1""1- , R "'glOO'd WI'" thr
.ni_""*,,,,lfn<>or au> of ............In.
P' o be Mou nti ng O rie n latio n and Measu re d Re~nse
In i.....tropi.. . y,lem•. at ,peed, .......n ""low a .~""nanee, 1M- h~a,"y spot a nd
hi4J spol are in pha .... T h is bdla\;or allo,,-. t h~ h"", 'Y 'f'01 to be "a. i1)' locat<'d
on t h~ J"~ar plot. Ilo"'..,.... c a s "'... h.a,.~ di sc us...d, in a ni""t.o pic s~·M"m. the I"",·
s ",'l"d ",,"po nM' or X a nd , . pola , p lots can be . iltn ifica ntly di ff" "'nt , mak ing
uniqu.. id..nti fICal ion of th.. hea, ) ' spot di rect ion mo,.. diffic ult.
I h.. xr Bod.. a nd pola , plols in Figu' " 13...8 ........" !lg n...al<-d fro m a , ulOf
modd wilh a ni tropic .lI ffn........ ",he. .. I h~ a k .Ii ffn~ .. ax i. i' h" ri r"nla l a nd
l he . Irong . tifl n ""i. is ' '''1'1 '''&. Th.. "'lor po nw i. Ih.. ", me fm all 1'101..
on l~' l he p .ob<> o . i..mal ion. a . .. d Hf..... m. [n lhe Itlp p [ot . , th.. 1"'''-< a .... m is ·
a lign <-d f.o m Ih.. Miffn" .... ax .." in Ih.. bo n om 1'[0 1.. Ilwy are ..hgOf'd ....; Ih tile
stiffn..... ax .... T h.. SoA r i.. m..a ,<'d Iwo .... ay. for .. ach ....... na n"" p<'..k o n the
Bod.. p lo t: Ih.. lI alf· """.... Band ' d lh m<'lhud ..nd Ihe P»a k Ral " . melhod (i n
pa...nth...... '.
T h.. to p conl'i gu ...l io n is a 1)'p 'C& . tiffn...... ,>tit'nt ..ti"n a ",1 p, ot... mou nli njt
local io n fot ho, i"'ntal mach i..... with flu id -m m ht-ari ngs T ile plot.,;how I ha t
l he h ilth '1>Ot a l low 'I>e<'d i. not a [il!nt"d ....; Ih I h......·tual he..' )" .pol loca l ion (the
red d o!). T he l"(' i. a la rge d iIT.'ren,... in t he 5.\ F \,dun fo, the '·a , io". p<'al. a nd
fot th.. dilT..... nt m<'l h" .h. ..od Ih..re app<'''' 10 be thllT d i.l i",·1 ... ..,na nee lre-
qu..oc i....

figo,l' e 1 J-f; Probe <It.."""",,, "'-" " " "dI,.." • ...,..A11

of me piau .-:/le ",t<,. ...-.pOn><' hlm , ......"'"
a""""" pt< , V"em. wile", tho--..k 'bffo..., is oriented
,n ' l>orilonta' (j,I!Ction, and , .... ,vong " .......... is
_ ., the top plot. the probe, .. e """"med .. ~S ·
L...-.:lOS' R."d ......,...,ed Ito-n ,he ,,""'.......... The
_ """,.how '''''' me r-->:peed h'9'> 'POI is not
tho-....... to< NCh plOOe. The SAF is "'own fu' NCh ' ....
ononce peakon the Elodt plot '" two wOY'o t.... Hoo·
pa-. """d...mh rret"Od . nd 'Ile Peak Itnio method
On I"'en""""'l.Ther.. " a '"''/'-' d'\C'"'P""c)" between
the SM ~at ...... mea""ed for 'Ile ""iouI. pe.><' and by
dolfe<enl method. The bonorn piau """" m.... me
rr«h"'" """"" but "",..... ,ed by an XYpmbe 1""
. t""ed w m wtf...... """'- The ,p«<! ""PO"><'
pha... .... ~. wn~,.,., 10<.0,-"" of the hea..,. !POl.
 ( Mpl~ t 13 Anisot, opic: Stiffn,... 16)

---
,- _.- - -
' ''''' >OlIO ,.,.
,

•
•• •
•.!-..,_-::':_: "'" """ - .... ..... ....,
- ,.-
- ,---- - - ---,-,
(
t"
I • -, •
I
1
,. - -
,

I
-.-
• ""'
'- - - - - t'~t" ,'..
.... ,"" ""'-
X

! •
! •
•
••
- . ',,-- -_ .... ,,.. -
264 The Static and Dynamic Response of Rotor Systems

The bottom plots show the same machine response, but measured by a n XY
probe pair mounted at 0° a nd 90° R; thus, the probes are no w aligned with the
stiffness axes. The onl y difference between the upper a nd the lower plots is the
orientation of the probes, yet the two sets of plots are very different. When the
probes are aligned with the stiffness axes, the response (high spot) phase at low
speed agrees with the location of the heavy spot. The pol ar loops are more sim-
ilar to each other, and the sm all loop has disappeared. The calculated SAFs are
still different from each other, but within the range of values obtained when the
probes were at ±45°. The different resonance frequencies have condensed into
two, closer to what we would expect from the strong/weak anisotropic stiffnes s
model.
The heavy spot/high spot anomaly is related to the ellipticity of the orbit
and the mounting ori entation of the probes. In I X circular orbits, both the rotor
rotation and the orbital precession of the shaft centerline (high spot) have con-
stant angular velocity; thus, the high spot maintains a constant angular rela -
tionship with the heavy spot. However, in elliptical orbits the centerline velocity
ch anges, and the relationship is not constant. At low speed, the velocity varia-
tions in elliptical orbits can cause the high spot to go in and out of sync with the
heavy spot. They are in sync onl y at the locations of the major and minor axes .
If the probes are located at these points, then the phase of the vibration not only
identifies the high spot, but also, at low speed, the heavy spot (Franklin and
Bently [1]).
At low speed the orbit major axis will be approximately aligned with the
weak stiffness axis. Thus, if the probes are aligned with the stiffness axes (0° and
90° R in this example), they will also be aligned with the orbit major or minor
axis at low speed, and the inferred heavy spot location for each probe will be the
same and will be correct.
However, if the probes are mounted at some other angle (for example, 45° L
and 45° R), then a phase measurement anomaly will exist, which is evident by
the fact that the phase measurement for each probe will not locate the high spot
in the same location; this would incorrectly indicate that there are two he avy
spots
When making phase measurements with a single probe, we tend to make
the unconscious assumption that the high/heavy spot relationship is constant,
but this is only true of a isotropic system with circular orbits. Imagine th e
machinery diagnostician who views only the polar plot for the Yprobe at the top
of Figure 13-8. The conclusion might be that this ma chine's vibration wa s not
high enough to worry ab out. A very different perspective appears when both th e
X and Y plots are viewed at the same time, and when the probes are aligned with
the stiffness axes !
 Chapter 13 Anisotropic Stiffness 265

Thus, we arrive at some important findings for systems with anisotropic

stiffness:

1. At low speed, the high spot direction will point toward the heavy
spot only if the measurement probes are aligned with the low-
speed orbit axes (which, at low speed, are aligned with the
spring stiffness axes).

2. Measured vibration amplitudes will seldom equal the major axis

of the orbit, because the orbit, typically, is not aligned with the
measurement axes.

3. SAF measurements will be different, depending on the degree of

anisotropy of the system and the probe orientation, and results
using different calculation methods will differ from each other.

4. When XY probe data is available, polar and Bode plots should

always be viewed in pairs.

Anisotropic stiffness is common in machinery. Because of the way many

horizontal machines with fluid-film bearings are constructed, stiffness axes
tend to be near vertical (strong) and horizontal (weak). Since probes are often
mounted at ±45° to avoid split lines , the amplitude and phase measurement
anomalies we have discussed are common.
While it is often not physically possible to mount probes at 0° and 90° R,
other factors in the machine (such as process loads or misalignment) may cause
the shaft position angle to be different from what we expect, causing the stiff-
ness axes to be oriented at some other angle. Ideally, we would like to adjust our
view of the rotor response to any angle we choose and to have another method
for accurate determination of the heavy spot location. There are two methods
we will discuss: Virtual Probe Rotation, and transformation to Forward and
Reverse components.

Virtual Probe Rotation

In an ideal world, we would like to be able to install a set of probes at any
arbitrary angle. While we cannot always do that physically (or economically), we
can take a pair of XY probes and rotate them mathematically to any angle we
choose. We do this by modifying the original data set as a function of the angle
we want to rotate the probes, creating data from a set of virtual probes. For each
sample speed in a database, the original pair of vibration vectors are trans-
266 The Static and Oynam ic R",pon•• of 1!o1(M S,.",....

fo nTH'd to produC<' a ..,t of "h'at iun \....:1 u", "'lui,·al..nl to what would bo> .....n
by prot- mo untoo ..t lhe n..w .,.....n t..tioo. s.... ."pP'-"nd ix:5 for mort' d..taik
Th.. tn:h niq u.. can t... UM'd to c al.. a M"t u f "rt ual p robe.. that a rt' ..Iignro
", th th.. ax.... of Ih.. lo,,··s J-"i, IX Ilipt it·a l <>Ttlil_Wh "n Ih is cood ilion is ""lis-
firo. th.. h igh ~t a t low . 11<"l"d " i ll aligoro w;1h Ih.. hn,"y spot. a nd th.. X and
I'plot. "ill agr<'<' as to lhe dirt'll'un ..fth.. l>.-a'"y ...... 1.
Th.. pola r plot. in Figu.... l 3-<f W'-"" Ih.. .-ff.-ct o f 'in ual p robo> roLali..n o n 1X
. hut down dal.. fro m a ~mal1. 10 .\ lW. st..am tu m in.. [l.. n..ratoo ..-t. Th .. d ala i.
from Ih.. inh" .. rd bt'aring o f th.. . I....m lurbin...This is .. ho rizontal nuochin.. "ith
plain. C)'lindric..1Ouid-film bt>a rinll-" and " i th probo>. mo unt ro at ~ 45' fm m ....' -
tical. It i. lik.-Iy Ihe t th.. primary radi..l lood on th.. rolo r i.p3\lI)'.
Th.. origioal dat.. (t"p) """s ta k..n from th.. ph~·.ical. XY. "b tion p t .
Th.. l" plot (I .. lij has .igni flCantly Iow.-r ,ibr..tion ..m plitu d.. 0 th nt i...
•J""""d r.. n~... od Ih i. a larg.. inl ..mal loop. Com pa rt' th...... plot. to Ih.. th ....
obta ined fro m th nisot rop'c ro to r mod..l in Figun' 13·11. Th.. "'...., aU pau ..m i.
wry .imilar.."1." not.. thai. in FilO'r.. 13·9. I.... indic..t.... h..a,"y spot lont ion.
I.....dot., disag .
Tb.. 10" , t. of plou . bo,,· Ihe d al a afte r it h... b....o l ra n. formro 30"
d ock",;,...10 a ..-t o f " rrnal probo>. locat al IS" L I Y,,) and 75' R (X,, ). Th i. rot a '
l i" o ..ngl.. " 11' r h.""n 10 p",,-ide agr m..nt bo>rw....n th.. indicatro h....,"Y ~I
I""a l iu n. " f each rot alOO probe.
Th.. low.."pe<>d. 600 rpm orbil sho.... l hat lhe o m ll n,..jn. n ~ i. or icnlro
<lightly mort' tha n 15" from l he hori zontal. At this sJ""""d. th.. u rbit i...pp"",i-
m..I"'y ..Iig...... " i l b Ib .. w..ak .. nd strong .pr"'g.li ffn.... a X.... Gi'...n th.. proba .
ble ntdial load d irt'ltion (down). Ih.. rola ti"" di "'Ction (X 10 l1. a nd I.... flu id ·
film brartnl[. it is lih h' th ai th.. mino' ..xi. o f Ihe orbil i. 0 ,;..,,1.... close to Ih..
po.il ion a n!d.. land al1ilU"," a ngl..) of th .. roto, in II>.- bt'..rinll-Th i. SUAAhlS th ai
th.. d om ina nl souret' o f ..nisot ropic .tiff...... is Ih.. flu id -film hyd rodynam ic
bt'..rinl[.
OOCt' I.... 'irtual vibr..tion ~ector " a rt' fo und. it m..y br """........ry 10 ","ter·
m i.... t .... p h)-.ica l loca tion o n th.. rotor th.at COf. ... p..nd. to a 'i rt ual 'ibr atio n
wetor. On polar plots. ~lIi,... phB .... lag is al",a~.,. ",..... " ....t ......1'.... 10 th.. J"".. .
tion o f th.. m..a...rt'm..,,1 tr..ll>dlkl;'r l real nr \irtu..l) in a d irt'ct io n opp.,,,it.. lo
rotat ion. To ph.vsicaJ(v Ioc.. te .. ,-..clor on" marn in... adjust th.. 'in ual p ha... by
I.... amo.mt o f I"" ,i rl u..1 m la tio n a n!(l... but in Ih .. op pmit.. direclion. For
..""m p"'. if you ro ta l.. th.. prub.-ot <Jl:<'mst th.. d irt'ct ion o f rotal lOn (......ich would
reduu Ih.. p hast' lag ..t " gi~e n ph~-,;i<,..l loca tion ). add th .. rot ..tion a ngl.. to th..
" rtu..1p hast' to o bta in th.. lncat ion .... at"... lo th.. phy.ical probe.
 Chap!.... 13 Mioo1ropic S,iffne.. 267

//7
,
- -
- ---
\ '
.!.

-- ,
,

~~ .• ~,

Fig<>r~ 13 " V'fI",I!>'ObP fOlabo', Th~ POlo' pIo1> _ IX "''''dowt1 oa,.

hom. """'llO '-'W. ' .... m ruobtn< Qe 'al<lt ""- ~ is. I'Iot1lOtl1al .......03"... _h
p1a... q\lndrX:.l ~uod-film beat ,"'J' _ lh 1"00." moun,ed aI u S' h <>m V<'t -
lie._ Tho lop oe1 d poIat plot, """'" ~ ~ da'a a ....'... from 'ho physi-
cal, IIY. ,.;bt", ,,., ~ Th~ O>d".."" lIN>'\' <POl 10<",,,,.,, I"' " ""',l ,j;1oag_.
Tho bottom w 01 pIoB """'" tho '"'''''' dar. afl.. " h. , - . rr.",Iotm"" XI'
<Io<k.._ to . w 01"" ,,,,1~ x. y.. lo<;.l,,,,,ar 15' l. . nd '5' R Aflt!. 1OtI-
~on. ~~ is good ag",<,rnenl .. ' 0 ' ho """ a''''' 10<""" cf ,.... "oavy 'POl.
1hfo1ow-'P"<'d 600 'P'" otbit ;, . pproz; ,oIy .~nod ";!h lho -.... on<!
"'''''9 ",","9 -..... "".. . 0><1 is .109""<::1 '''9"''
h tho ""...... pl ot..
268 The Static and Dynamic Response of Rotor Systems

Figure 13-10 is an example from a gas turbine generator set, showing IX

shutdown data from the outboard bearing on the generator. This machine is also
horizontal, with fluid-film bearings, and has probes mounted at ±45° from the
vertical. The physical XY polar plots (top) display an overall pattern very similar
to the steam turbine data in Figure 13-9. The virtual probes (bottom) have been
0
rotated 28 against rotation from the physical probes. The heavy spot locations
now show good agreement, and the axes of the low-speed, IX, elliptical orbit are
in line with the polar plot axes, similar to the previous example. The virtual
probes are now aligned with the spring stiffness axes, which are controlled by
the fluid-film bearing.
When XY polar plots agree on the location of the heavy spot, the probes are
aligned with the major and minor axes of the low-speed orbit, and the polar plot
and orbit axes are aligned with the anisotropic spring stiffness axes of the
machine. These two examples, combined with the predicted behavior shown in
Figure 13-8, provide strong evidence that the primary source of anisotropic stiff-
ness in a typical machine is the fluid-film bearing.
When balancing is what you need to accomplish, there is a much more effi-
cient method of locating the heavy spot: the forward response vector.

Forward and Reverse Vectors

In Chapter 8 we showed that filtered orbits can be represented as the sum of
a forward and reverse rotating vector and that the amplitudes of these vectors
appear as frequency lines in a full spectrum plot. The full spectrum is created
using a transform of X and Y data that preserves the phase information. When
applied to startup or shutdown data, forward and reverse Bode and polar plots
can be created.
Research [1] has shown that the forward response maintains the correct
heavy spot/high spot relationship with the dynamic unbalanceforce acting on the
rotor. This means that, at low speed, the forward response vector will identify
the direction of the heavy spot.
When the system spring stiffness is completely isotropic, the reverse
response disappears. This agrees with our understanding of orbit shape and full
spectrum. A forward, IX circular orbit will have a full spectrum consisting of
only a single, forward frequency component at the IX frequency. The shape of
the orbit can be described by this rotating vector.
Anisotropic rotor systems produce elliptical orbits and, by extension, for-
ward and reverse vibration vectors that are the components of the elliptical
orbits. The magnitude of the reverse component is related to the degree of ellip-
ticity of the orbit.
 Cham ... I I A";.ou o pic Sl;iffM.. 269

f ..... 11- 10 .V,rtu.ol probr mliIlIOn d dat.> from • g.t> rurb.... gm-
" " or ~. n,;. is"so . honront.~ 1\uid-~l m ~"'l m K~ with
p<0000< mounted ... H S· from ,.... ~ . """ p/¥iut~XY_poIo'
plat.,., ""-' at top._ m ~ <7Wft" """'1
po " .... '0
"tt'olo, t ....
"Nlt\ tul'l;M.... dat. in Figu.. 13 -9_At tho bottottt "; rtu.ol pItlbeo
~ - . « _ ed 2lr do<'wi~ ' rom , ~ .. p<obfoI. """
hNvy '!><"- Io:. ....... now """'" g<><>:I O9, nd , k>w-
"P""d I ll ~h pt o;al 0Jbit" . bi;lned W>th m ~ polo , plol """ Lor

_ .tiftnMs
to tho~ m ~_ Tho 1Irt.... probn ' ~ . hgno<l with tho
_ h ,., <ont_ by ..... ft ud ~ 1rn bNting.
270 The Static and Dynamic Response of Rotor Systems

In Figure 13-11, the data used to generate the polar plots (top) has been
transformed to show the forward and reverse response polar plots (bottom). The
bottom plots are labeled with direction of precession, not rotation. The phase
markings on each plot are in degrees lag relative to the direction of precession
(forward or reverse), using a coordinate system based on the X probe. Because
the forward and reverse responses are derived from the data from two probes,
they are completely independent of the mounting orientation of the physical
probes.
The low-speed part of the forward plot (left) points toward the heavy spot,
and this direction agrees with that found using virtual probe rotation in Figure
13-10. For any speed, the sum of the forward and reverse vectors, which contain
both amplitude and phase information, will completely reconstruct the original
orb it.

Summary
Anisotropic spring stiffness results when the support stiffness seen by the
rotor mass is not the same in all radial directions.
It is common in horizontal machines with fluid-film bearings. While bear-
ing support, casing, and foundation asymmetries can contribute to anisotropic
stiffness, the primary source appears to be the unequal fluid-film spring stiff-
ness in the radial and tangential directions when the rotor operates off center in
the bearing.
Isotropic systems produce circular, IX-filtered orbits; anisotropic stiffness
produces elliptical, IX orbits. Because of orbit ellipticity, measured IX vibration
amplitude will depend on the angular mounting location of the measurement
probe. Also, the high spot, as it moves along the orbit, does not maintain a fixed
relationship with the heavy spot; the phase doesn't always accurately locate the
heavy spot.
Vector data from an XY probe pair observing an anisotropic system pro-
duces pairs of IX polar and Bode plots that do not look alike. There are differ-
ences in measured amplitude, SAF, and phase when the probes are not aligned
with the stiffness axes. The polar plots may show small loops that are not due to
structural resonances.
Virtual Probe Rotation transforms the IX vector data from a physical pair of
XY probes to what it would look like from a set of virtual probes mounted at any
desired angle. If the virtual probes are aligned with the anisotropic stiffness axes
of the system, then the low-speed phase data will point toward the heavy spot.
Forward vibration vectors from XY probe data eliminate the phase meas-
urement anomaly produced by elliptical orbits. Therefore, the low-speed part of
a forward polar plot will point toward the heavy spot, a benefit when balancing.
 (ha pl~ t 13 Anism,opic Sl iffnMS 27 1

---
r'- y ~'"
~ _____, ~
H~'.· '''''~ _
, ~" " L......

_
. .~

\lO~ O'

':: -3- '

~-.

I ,_
\ j

'1IO'~ '

f ioJu.e 1J-l l Fo<w,,",and ~ " - " 'm "",or plot. The pol .. plots
(bon<mJ , how IN! """"'d :J<ofI: and ..,"",'" :"'lht: r~ ... of m.. da.. "'
Fogu'. B-l 0 in acCl()ftin... ')1"'''' b.>«l on the X p_. n-., plo<> .r.
laboolrd nn dofO'<toOn 01 p~e>siol\ not """ lion,n", ~ "",rt ;"9' on , ....
pIot< in d<og'ft"l lag ""'''''' to , ... d.~ d I're"'. ..."". n.. Iovv-~
~ 0/ thr forw.rd plot peon" lOW¥<! _ hnv)o wot. . f'ld ," , d"<'C,",,,
09·...,. w'1l1 th o! found"''''9 y.,.... ~ ",..
t<>n. n.. ~"" ""1'O'l'e
. _ """ .mplnude .'lCl 1lI'. ... of ~ ,_ YKIor for eoctl ~ Fo< any
loPH'd. """ ,..., of """ forw.. d and ' -=to" wiN ,..;o'"lrua m.. ""'.1'_
naI orb ... """'. .lng in 0I~""00fl~ , pooood ph. ... 01 th.. ,_ pol.,
""" 00.-. f'IOl. tyoO::" 'V. po.>' In _ d ".." w of the ~ ,pot
272 The Static and Dynamic Response of Rotor Systems

This is a more efficient and accurate method than probe rotation for determin-
ing heavy spot location. The forward vectors are related to the isotropic stiffness
behavior of the system and are calculated using the same transform used for a
full spectrum.

References
1. Franklin, w., and Bently, D. E., "Balancing Nonsymmetrically Supported
Rotors Using Complex Variable Filtering;' Proceedings ofthe Twenty-First
Annual Meeting, Vibration Institute, Willowbrook, Illinois (June 1997): pp .
67-72.
 273

Chapter 14

Rotor Stability Analysis: The Root Locus

UNTIL NOW, WE HAVE PRIMARILY DISCUSSED various aspects offorced vibra-

tion in rotor systems. The model we developed in Chapter 10 was solved to
determine the steady state behavior of a rotor system. Steady state behavior
describes how a rotor system responds to a continuous perturbation over a long
period of time.
However, all vibrating systems also exhibit transient behavior, which
describes how they respond to brief disturbances over relatively short time
spans. An example of transient behavior is the motion of a pendulum after being
displaced from its rest po sition. Transient vibration always involves the free
vibration of a system at one or more natural frequencies. This differs from steady
state vibration, which depends on the presence of a continuous forcing function
and (for linear systems) takes place at a frequency equal to the frequency of the
perturbation; for example, IX vibration response due to unbalance. At any time,
a complete description of the vibration of a system will include the sum of both
steady state and transient vibration.
Transient disturbances in rotor systems are usually small, but they can occa-
sionally become significant. Examples of small disturbances are the periodic
impulses caused by blade passage across a small gap, the forces due to meshing
gear teeth, or disturbances due to turbulent fluid flow. Rub impact is an exam-
ple of a larger disturbance. All disturbances excite the free vibration of a rotor at
one or more natural frequencies.
In stable rotor systems, transient vibration dies out over time as the damp-
ing force gradually removes the energy associated with the free vibration of the
system. However, an unstable rotor system can respond to a disturbance with a
dramatic increase in vibration, causing vibration levels to exceed allowable lim-
274 The Static and Dynamic Response of Rotor Systems

its. Thus, the analysis of the stability of rotor systems involves the analysis of
transient vibration.
When an instability does appear, it is important to be able to recognize it
and to know how to eliminate it. In this chapter we will develop a powerful ana-
lytical tool, root locus, that can be used to reveal many general aspects of rotor
behavior and help analyze rotor stability problems in particular. This chapter
will present some basic analytical tools and concentrate on the data presenta-
tion of the root locus plot. See Chapter 22 for a discussion of the underlying
physical causes and the diagnostic symptoms of fluid-induced instability.
We will use the simple rotor model we developed in Chapter 10 to explore
the transient behavior of rotor systems. Our rotor model has a tangential stiff-
ness term that mimics the effect of rotor interaction with a surrounding, circu-
lating fluid. This fluid circulation can trigger instability in the rotor system.
Though our discussion will concentrate on this fluid-induced instability, the
basic analytical principles can be extended to any other type of instability.
We will start with a discussion of stability of both linear and nonlinear sys-
tems, followed by a transient analysis of our linear model. We will obtain results
from this model, called roots or eigenvalues, that describe the free vibration of
the system versus time, and we will show how these results can be used to deter-
mine the speed at which a rotor system goes unstable.
We will show how the free vibration behavior of a rotor system changes with
rotor speed and how the information can be displayed in a convenient form on
a special plot, called a root locus plot. We will show how to extract a large
amount of useful information from this plot.
We will show how the root locus plot is related to (and superior to) the log-
arithmic decrement, which is commonly used to express the results of stability
analysis. We will also compare the root locus plot to the Campbell diagram,
which is used to show natural frequency relationships in rotor systems.
Finally, we will show how to use the root locus plot to perform stability
analysis of rotating machinery.

What is Stability?
Stability is a broad term that can be interpreted in different ways. A good,
general definition of stability is that a mechanical system is stable if, when it is
disturbed from its equilibrium condition, it eventually returns to that equilibrium
condition. A system is unstable if, when it is disturbed, it tends to move away from
the original equilibrium condition.
We can think of a stable system as one that is easy to control and behaves in
a predictable manner. A stable system mayor may not vibrate; either way it
behaves as we expect it to. An unstable system will behave in a way that is unpre-
dKUblor and n'ty d ,lf......lt to euotrol and 1ll"'Y"'~ !to com~dy ""I 0( OOIll rul
A "mp/o' namplor 0( a n unotllbW S~"'I"'" is a btuom"'-d b.IsncN W't'titally 011
,.,..., hand It pow.iblir 10 ~ It. tul il d,ffiodl and ' oq&illM coni""'''....
.sdju....,..."h 0( ~'OW hand powtion. If ~_ stop ~11~!""" hand. !fw.
broorluoId .... l!:OouI 0( """" roi and faD "'_
Whm d i" u rbni a ot.aob&t s~ ..... wtth .....l..I n'l'iy h'ld' <bm~ .."O .Jc.wfy
m um 10 llor t'<fUilibrium powlion.. A staIW .~~C'm ,hsI ..... rtUu.\ "'Y lmo' dsmp-
'''ll will O!oCiIllItC' t", t>mC" a round th.- oquilibnum poo.,boo for _ time. .. !fw.
.ihrat ion~· ill ......i~· di uil"'tN Th i. \i bratioo ..;o sI..:sys OOXUr III a rulla -
rsl fn"'llH."n<)· lor frC'quC'n n...10( 11M- ")..., ...... (l hi. it the' 'ICe of 1.... 1""'" _ .
uroJ.frw1"t'-'J<"Y. T.... d i. lurhPd .y. l if Iefl slo..... \itorat al Ilti. f""".... "<1.1
A .... 1I . t th.. botlom r>f. ro n"' 1f,lIn! wil h • •'iOCOl>s fl uid I F ijl:u ~ 14--
1 .I ~ft .rid {"C'n t.. , ~ is a n C'X . mpIO' .,f . "Ny . imp le. •t. hle .~...l..rn. The bloll. d ue 10
I.... ro ..... of j!Ia ' 'ity. i. in .I alic <'< Iuilih riu m at I.... h"!lorn of 110......,....1. Ir il i.
"""ro From tit.. bou om and .......0..-.:1. I.... fo ne.. o r !Ua'il~ ..'ill rn'''-'' it t--k
I.""aro , h.. <'<ju ilibri um po..I''''''.. .h""T1 in Ih.. tirrwba P~ II~ Jr thO' fluid has
low ,Ucu!. i[)l (air ~ tlw h&U will """'C' rapidly b«k to It... uilib-riu m .,..,.il io'"
O\'C'I.hoot, climb I"'n .... y up the' oIl1ft' sidl'. a oo """iIL I.. wil h decr.... in ~ a mpli ·
I..... u ntil it COI\'lft te> rnI III tIN' bot tom. Ifl t... fluid .... hiP> vUoros,[}·ilhid. oil).

.........
lhe' ball ...i ll " -'C' oIowly t -k t o> I.... bottom .nd n>mC' Ie> fnl. ...-Tlll",,1 "'......

---
- ... -
':c=
. J"cd n
f;g.- ' 4-1
I
I f
!ot.IbIr ¥oil umW>lC'
_ n. '"""""'. ftuod. i>. _
I( '"
'Y"_.... IC'Ir.. NIl ... ,.... bonom 01. Con',.... ""'.. _
' 1'\le'T'. l'oIlon ,...."... .. ~ from ,.... tIQn<lm ..-.I _~ ,....
~

flooc.. 01 gr--,o .... _

_
. . . __
to .......... ~ ba<k 10 !he ~ """'.,....!he!'<ad "" ..... ,," '
«My (air!, thC' boot ... """'" ~ bac k to !he bottonI.""""'-'..-.I _ _ bacIt _ lor....
...... d C ' ( ~ ~....... ~._ IO
ail thC' ....... """"" - . . , t>ocIi to _ """"'"
lOghI."bolI.~.Ihr_d • .....-. .............
'" _
rf .... ' " " ....... ~
- . . -.nq. ....
, ~ ,,_
.... DIfI ... ItIC'----.m _
276 The Static and Dynamic Response of Rotor Systems

A ball in equilibrium at the top of an convex surface (Figure 14-1, right), is

an example of a nonvibrating, unstable system. If it is disturbed, the ball will
move away from the equilibrium position and, because of the shape of the sur-
face, will never return.
Many mechanical systems vibrate when disturbed from equilibrium. A sta-
ble, linear (in the sense of our linear rotor model) system with low damping,
when disturbed from equilibrium, will vibrate with decreasing amplitude. The
vibration amplitude of an unstable, linear system will increase forever. This is
impossible in a real, physical system. In real systems, either nonlinear effects
come into play that prevent the system vibration from exceeding a certain level,
or the system destroys itself. Because of these effects, unstable, nonlinear sys-
tems may eventually reach a new, stable, operating condition that is different
from the original one.
Rotor systems usually vibrate in a steady state condition (for example, a cir-
cular, IX orbit) about a static equilibrium position. While moving in the orbit, a
stable machine is in dynamic equilibrium; if it is subjected to some temporary
disturbing force, it will eventually return to the original, dynamic equilibrium
position.
However, when a rotor system with a surrounding, circulating fluid (for
example, in a fluid-film bearing or seal) is at or above a particular speed, any dis-
turbance will trigger a fluid-induced instability. This speed is called the
Threshold ofInstability. The machine will temporarily become unstable, and a
subsynchronous vibration (at a rotor system natural frequency) will begin and
rapidly increase in amplitude. As the vibration amplitude begins to increase, the
machine will still behave in a linear way (that is, the spring stiffness is approxi-
mately constant). However, as the dynamic rotor position nears the surface of a
fluid-film bearing or seal, the spring stiffness will increase dramatically (a non-
linearity), producing a stabilizing restraining force. The rotor will evolve into a
new, stable operating condition, characterized by higher amplitude, forward,
subsynchronous vibration (Figure 14-2). According to our general definition of
stability, the machine is stable in this new orbit.
The unstable orbit in the figure shows a mixture of subsynchronous and IX
vibration; the measurement probe was located near the midspan of the rotor
some distance from the source of the instability, a fluid-film bearing. High
amplitude vibration is very undesirable and potentially damaging; thus, even
though, technically, the new operating condition is stable, the machine may be
unstable from a practical point of view. Thus, a practical definition of instabili-
ty is an undesireable level ofsubsynchronous vibration.
Fluid-induced instability is an example of a self-excited vibration, where the
energy of the circulating fluid is converted into the energy of vibration. Like all
 Chapter 14 Rotor Stability Analysis:The Root Locus 277

Figure 14-2. Stab ility and pract ical instability in a

rotor syst em. Rotor systems usually operate in
dynam ic equilibrium about a st at ic equilibrium
positio n (for example, a , X orbit, left).When dis-
turbe d, a st able machine w ill eventually return to Stable Unstable
the original, dynami c equ ilibrium condition . Above
t he Threshold of Instability, any disturbance will

o
G
trigger what is called fluid -induced instability: the
rotor will evolve into a new, st able dynam ic equ ili b-
rium , dominated by high er ampl itude, subsynchro-
nous vibration (right ).The orbit show s a mixtu re of
, X and 0.48X vibration. According to ou r gen eral
definition of st ability, the machine is stable in this
new orb it, but the higher amplitude vibration may
be undesirable and potentially damaging; thus, the
machine is unstable from a practical point of view.

self-excited vibration, the frequency of the instability vibration occurs at a nat-

ural frequency of the system, with a mode shape associated with that natural fre-
quency. The amplitude of the mea sured vibration is affected by three conditions:
how mu ch the vibration increases before the system restabilizes, the mode
shape of the vibration, and where the measurement probe is in relationship to
the source and the mode shape. The highest amplitudes of the subsynchronous
instability vibration will be physically located at the antinodes for the mode
shape associated with the instability natural frequency. This could be, and often
is, some distance away from the source of the instability.
Accurate stability analysis of rotor systems requires models based on non-
linear mathematics, but these models are difficult to solve analytically. This is
why our rotor model was developed using a line ar model. However, rotor behav-
ior below, at , and immediately aft er crossing the Threshold of Instability is
essentially linear in behavior, and we can use the linear model to understand the
basic mechanism that leads to instability. We will see that adjustment of the lin-
ear model can explain the nonlinear effects that prevent the vibration from
growing forever.
278 The Static and Dynamic Response of Rotor Systems

Stability and Dynamic Stiffness

Im agine a rotor th at has been disturbed fro m a static eq uilibrium po sit ion.
The spring force , -Kr, tries to push the rotor back toward t he equilibrium po si-
tio n (see Chapter 10). However, the tangential force, +j DAS?r, pushes the rotor in
a direction 90° from the di spl acement and preven ts t he roto r from returning
directly to equilibrium. The tangential force is proportional to the rotor speed,
S?, becoming st ro nger as rotor speed increases, and it ac ts ag ainst the stabiliz-
in g d amping force; ultim at ely it can destabilize the ro tor system. Becau se the
tan gential force is the effect of the fluid circulating aro und th e rotor, we call this
form of instability fluid -induced instability.
This destabilizing effec t is related to a loss of Dyn amic Stiffness in the rotor
system . We have shown t ha t rotor vibration is th e ra tio of the applied forc e to
the Dynamic Stiffness of t he syste m . The Dynamic Stiffness acts to restrain the
motion in a way that is sim ilar to a spring (Chapter 11). If the Dynamic Stiffn ess
were to disappear, there would be no constraint to rotor motion, and, when dis-
turbed , the rotor would move away from the equilibrium position forever. Thi s
m eets the general defini tion of in stability. Thus, we ca n see that when th e
Dyn amic Stiffness becom es zero, t he rotor system becom es un stable.
Th e nonsynchronous Dyn amic Stiffness is

(14-1)

Dyn amic Stiffness is a complex quantity, contain ing both direct and quadrature
parts. For the Dynamic Stiffness to equal zero, both th e direct and quadrature
parts must be zero, simultaneously. Thus, the Dir ect Dyn amic Stiffness is zero ,

K -Mw2=0 (14-2)

and the Quadrature Dynamic Stiffness is zero:

;D(W-AS?)=O (14-3)
 Chapter 14 Rotor Stability Analysis: The Root Locus 279

For this last expression to be t rue , th e te rm in parentheses must be to equa l zer o:

w ->'S2= 0 (14-4)

Because Equations 14-2 and 14-4 are equal to zero, th ey are equa l to each
other. We can find th e rotor speed that satisfies this system of equations.
Eliminating w, and solving for [2 , we find the Threshold ofInstability, nt/I' to be

Threshold of Instab ility (14-5)

This is the rotor spee d at or abo ve whi ch the rotor system will be un stable.
This exp ression is a very powerful diagnostic tool, and it is the key to under-
standing how to prevent and curefluid-indu ced instability problem s in rotor sys-
tems.
This expression combines the Fluid Circumferential Average Velocity Rati o,
>., with th e undamped natural frequency of th e rotor system,

(14-6)

For plain , cylindrical, hydrodynamic bearings (internally pressurized bear-

ings) that become fully lubricated, >. is typi cally less than 0.5. Thus, th e recipro-
cal of >. in Equation 14-5 is typ ically greater than 2. This tells us that a typical
rot or system mu st op erate at more than twice a natural freque ncy to trigger
fluid-induced instability. For systems with multiple modes, the rotor syste m will
encounter the lowest natural frequency first. Instability is almost always associ-
ated with the lowest mode of a rotor system.
To ensure rotor stability, all that is required is to rais e nth above the highest
opera tin g speed of th e rotor. This can be done by reducing fluid circulation,
whi ch decreases >., or by increasin g the spring sti ffness , K. Externally pressur-
ized (hydrostatic) bearings, tilting pad bearings, and other bear ings of noncir-
cular geometry accomplish on e of both of these objectives. This will be dis-
cus sed in more detail in Chapter 22.
For now, we will move on to another form of stability analysis th at can be
extended to more complex rotor syste ms.
280 The Static and Dynamic Response of Rotor Systems

Stability Analysis
Stability analysis requires the development of a mathematical model of the
rotor system. The level of required detail in the model depends on the complex-
ity of the machine being considered and the likely instability mechanism.
Models can range from simple, lumped mass systems, to more complicated,
multiple lumped mass systems, to very complicated, finite element models.
Here, for clarity of understanding, we will use the same rotor model that was
developed in Chapter 10.
We start with the equation of motion of the rotor model,

Mr+Dr+(K - jD>'[l)r=O (14-7)

where the perturbation force on the right side is zero, because we want to con-
centrate on the free vibration behavior of the rotor. This form of the equation of
motion is called the homogeneous equation of the system.
We will assume a solution to this equation that is similar to the one we used
in Chapter 10:

r=Re st (14-8)

where R is an arbitrary, constant displacement vector. We know that the posi-

tion will change in some way with time, t, but, since we don't know how it will
change, we will use a general variable, s.
To solve Equation 14-7 we need expressions for the velocity and accelera-
tion:

r= slie"
(14-9)
r=s2Re st

Substituting these expressions into 14-7, and collecting terms, we obtain

[MS2 +Ds+(K - jD>'[l)]Rest =0 (14-10)

For this expression to be true, either the term in square brackets must be
zero or the initial displacement, R, must be zero. If R is zero, then the system is
 Chapter 14 Rotor Stability Analysis:The Root Locus 281

resting at equilibrium, which is a valid, but not very interesting case. We want to
examine the case where R is not zero, which requires that

Ms 2 +Ds+K - jD>'fl=O (14-11)

This is an important relationship known as the characteristic equation of the

system. It is a quadratic (second order) polynomial in s. The values of s that sat-
isfy this equation are called the roots of the equation. When both >. and fl are
nonzero, solution of this equation will yield two complex roots of the form

s] = 1'] + j Wd
(14-12)
S2 = 1'2 - jWd

where 1'] (Greek lower case gamma), 1'2' and wd are complicated functions of M,
D, K, >., and fl (see Appendix 6). I' is called the growth/decay rate and has units
of lis; w d is the damped natural frequency and has units ofrad/s. The meaning
of these terms will be discussed shortly.
The roots are also known as the characteristic values, eigenvalues, and, in
control theory, poles of the system. If we substitute these two solutions into
Equation 14-8, we obtain two expressions:

r] = (R]el,t )e jwdt
(14-13)
r2 = (R2el ,t )e- jwdt

where the complex arguments of the exponential function have been separated
into amplitude and frequency components. R] and R 2 are constant vectors that
depend on the conditions at the beginning of free vibration.
The complete, free vibration response of the rotor is given by the sum of r ]
and r»

(14-14)

where r] and r 2 are a pair of forward and reverse rotating vectors whose fre-
quency of rotation is the damped natural frequency, w d ' Because the frequency
of r I is positive, it represents forward precession. The frequency of r 2 is negative
and represents reverse precession at the same frequency.
The amplitudes of r] and r 2 are given by the expressions in parentheses. At
time to' the initial amplitudes, as a result of a disturbance, are R I and R 2 • Once
 •
.m
.~,;-.. . , !.r,;'1"'1
~~
u
-
-~r~ HH'I u i! <~'1 ~ qn· il"
,.j-,
... ~i~~~ill~ loot. , "'a~'~~
~
~
s s a t .-i" ,
--'~
\,I,I~,fff~ ~ ~ ~
!"'"

.2 '" a '" ~, :l .~ ";:, t, " 5" ~! a. 8 :- ~~ ..'-""

~, 5 eo ~ ~ .. ,''
~" ~ ~, "~' -!!! [
~S~lll~;t~~r~'·~ "i,~~~,~ r~&e-i ~
.~I·.~s~~~t~JJ!~~ i,~-~ ·tll ~.-~~s,"~f ~
i.~i· l~ .. ~.,, ~;r~
f!Ji.J ' i~f~~ '~!·Jt!i .!·i~~~fJ~lii~ i
l~~ t '- ~.t-"O~! ..
f
l l ~ l ., ~ l i ~ f ~ 1, ! l i ~i.;.
!O. -1fe.. ;l~' if p;'i:~
~~H ~5-[ ~~~-,!~~~ L,~ ~t;l~~fJ
• is Ii ~,!l i'~ '
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~~~~ 1~~'
': s,'q
·,~ ~"i._E
i q '
f
s,
~· t ~ f ~~f~~fi~~.~ ~ tb' 'hU.~!~&.
i
• -I"t P ~ ~ I I ~', "".."ii' I'1'.'!
~, t,' ·"'" a. ~ ~!; i....i ;" ' U
~ !... ~.E..
,~ e,
;~~ ~ ~l lllii~.f ~~~~~f al " !.I f
.!~rfi'Ngi~.
il l~"" !l.*f"~ '~i~lf
l

~ ~ llS~~ftt
"il''''~;;;' a-, .. ~~ !I=~~!i 1
;l.n- ;-iCl.l~·~-~3 ~.j~l; ~~l~~ -ip;i
:p;lhH~in!";Hl~llh.! ~~
~~I;:!~a~~~i i fl~~~~i3~~ ~I ~ ~
fILt~rtlf~laf~ rt(~J~!l'~~ &l~~
~~~hI33Blr~~'-~l ~n i~tqil~ ~ti~
z.'rfH·t ~g .. .jH'~L ~~~ ~ftt~M ~113
,i" ., ~_i~'''S-~3.;[Z. ! l ~ ~S'l~ ~ a • Eo}
'/ iLi.,H ~J li..f~i.. d ii'; ~' § ~i ~ f.~ ~ I
 Ch,opt... 14 Rotor SUbdlly M~ yo j s: The Roollocu. lU

d'.I...rbrd. 1"" rotor hrpn. I... 1"l'n"J' in .n orbl1. .ond I" am pl,ludM. chan~p
,,"h 11nw- .. funct""" of ~ , and , .. If a ~ ~ ...-p1'...·. lhm I " npont...l .... ..iD
I" ' h'" omalItT ...l1h Innt. • nd I" .mpliludt 01 .""alion ..-;n drcrn.'If': I"
mo..... ~i>.... tho f. ...... t. . ~ in .." p1it ude . If a ' is f'O"'iln lhm tht ....po-
...... tiaI function ....n bot ,",'" IarJrf l llfUW) ith I I...... .nd I m pl iludt of
.'ibral~'" ...'ill incn-. ... IOrt>n; For tt. ...u ~ tht tr-1h/tkalr role,
bot-au.. ltv l1UVlLludt .nd "'It" COT'llrnl I..", tht . 'ihr.fion arnpf ilud<>
i docn>• ...,. li.......
I~ 1 1 mo.. l tv IV""""fh/doc.o.~· •• te mnl",l. I..... ;ampl il udoo OWl'
lilllf. T,Jntba plc~ I" fnot ,;bration bt-h........ of. rotor .~·SlPll1 .~ il
.......k! boP m~ rtd by a ., ntV< I ranod... n'!. Al k1i.. th.. ' Y" PII1 i, d l"P loc...t 10 a "
Inilial pmilion. R. ;and Ih"" (II~ ... . R ;al'. ....." h . ...·a1ar a mpl ilud.-.)
Th... y....m hqt".. IO .,hral I I damped nal ura l r,,"'1 """')". ~-d' "'Ih a I.....i,....
or 2'"/"'d' Bee........ , <: 0. I m plitud.. dPCaY' al • ,.. t.. ue l<'. m i....... by th,-
.., .>on..ntiaJ fun ct IOn, th" dm phtud.. ""''''''1''' l lt'd j. Th .. middJ.. p1o1 ;.h , .... .
....h• •'io. at tltt' Th•...h"ld uf In. la bility........... -. _ O. F", Ihi. C"a.... th.. np'"
....." 1,./ funclion it al.....~ e'<luall u 011<'. ' " It>.. . m ph lud.. doo-s nOl. chan",... Al
n¢'1. ~ > O. and tht am pliludr grow...,Ih tilllf.

_<Ii"" n.
,I
R,
- ...
~_ O

~.,---,
/
. ..

~,

f""'" 14- 3 How tOW 9'OWIhIdtu y ,a ~ "",t"" I~ IlI'IQirluclt 01 ",bra"on .

TIII'ItbI... pion """" lhP h.... vb....,... ~ <IiI rolO' ..... ,..m .....,...,..~ by
I ""'9'<- t r - . ..}lt ltft.lhf ............ d1>p1aot<l to .... IMlII. ~ pmrtoon. lI.
ond ttINstd The.......... bf9n. to ,"",a(e.-the <I¥nptd ........... ~
CJ,..J,; ·tII I ~ of 1. / _,; Iltau5P ~ < ll.the ompfIIudt ~ (t<'d . . -
Ioptl'" I.-dim . .. d b,thelapoooeC.r.-- The _ pIcc-'
_"' Thod_oI--.,...- ,_
Iun<l>on il
'" _ .......,.. '" "9'C-
_._~
0. ...._ _ ...--.....
~,.ll.

_the..,pcudot _
284 n.. Stat ic and Dv-mic ~ of Rot or Syste m,

T hus, 1 IIi'..... us a math"mll1ll:al de f,nit ion of Ma b dity:

I. ~ < 0: t .... . pt"m is Mable. T he ,~b.at ion a"""'ia t.... " i lh a dis-
t u.b;.nCl" 10 th.. 8)'51" '" will die out a nd t he .~st"'" " i ll ... t" ", to
t h.. o ri!!ina l .-qu ilih. ium ",...,OOn.

2. ., = 0: the s~tem is ' 'J''''ral ing al Ihe Th .....ho ld o f Insla bility. If

th e s~'slem is d istu. b.-d. t ....n th .. ' i h. a lion " i ll be a Sleady , ta te
,i b. at ion that ,...ith...r inc:wa.... no. dP(:l¥a ..... " i l h lime.

3. -. > 0: t he s~"8I ..m is ullstabi,,_ Or><-'t' d iu urbt.-d. if allY "l > O.

th e n Ihe ,i bration will. t .......... lica Uv. im;:"'"... fore....,..

The two ro ta t ing ,-..cto..... r J a nd r 1" o f F,.qoa lio n ]4-14 co mb... e to ocsc,ibe
the free vibrat, o n be ha,i o , o f t he roto r ( rillu", 1 ~ -4) a~ a fun ct ion o f 1. , tarting
fnom so ",e initial d i,placern...nt ([Ut'l"n dot ). The ,", t of OIb' l$ o n the left r"P"'·
"",nt " -pIt,,,1 h,," avior for a Slable ' ys t...m. Bolh tli.. f",w art! (lor) and ,. r....
( m i. kll..) orb It d"",,~' in oppm.it .. d irections in circ ula,. "pitals. hu t the -er...
dt"Ca~" faste r ht"Ca u..... : l is more n..,.ali'·" t han ~ t. IW-o:a,,'" the ... ,'t'....... ""lIOn....
d i"" "'.. mo ... q uickly. t he tot al rotor ... sponM" (hot to m) ~ho"" an initial ellipti ·
c..1",b it ....·ol,·ing to a circ ula r. fo rward . d l"<"a}i nIl SP,ral.
In t he midd l... the "'" of re..pon for a ~y .....m operating at tlie
Th """,. ~d of Instab,lity. " il...... ~ , = a ln Ihis 1J < II. an d th e ... ' 't'. ... o rbi t
d"",,~... q u ickly. If...ving only th e fo rward o rbit. Th... am !~it ude of lhe fo rwa rd
orbil is u ll....t a nt a nd . ta ys tha t " " y for.......,..
...1 righ t, a n " n", ..hl.. . y....e m is s hown. "il f{' 1 , > Oa nd 12 < II. The ... ,..."'"
..... po n d l"Ca )" q uickly. ",h ile the forward pon... gc."'~ On,'" th e "" 't',,,,
rn;pon d i"'PI.......... o nly the u ns ta bl... for" 'a rd o rbit ... ma inr.. If no th ing
' -hanll in th.. 'y"""" t he orb it a mplit<lde would in.....a oe f" ... ,'t' ,. It i.~ inti'. ...t -
ing to n"t.. Ih at il ta ke. time for theorbil to increa ... in oi"",. The ""aU.., th.. JIO""
iii"" '·al of 1. th.. mo... slo",ly the o rbit grow>..
In pp<'nd ix b. " .... >h" t ha t if " ....... t 1 = 0 a nd s"l..... for th .. R,t". "!I<'t'd. n.
" .... obta in t .... ... m.. H p ~ion for the T h. eshold ofl nstabihlv. flt/l. as Eq uation
]4 ·.'i Th " .. "t...n ~ = 0 t IM> ' Olor is o p<'.aling at Ihe Th ... . h,~d " r Instahilily: a ny
d i, t urban..... " i ll ..."'.... tli... m to , ,·'h' a ting in a c ircula , orbit.
Each pa ir of roo ls " i ll al"·,,ys ha ,... the "" "'.. da m p<'<! nat ural fTt'<Ju" n,-y. w'd.
How e...... for no nl-<'ro " ~ atj\-..' "I .......". ..""h pair o f roots " i ll h;" ... d ilT".......1 ,'al-
ue. of ~. "'h...... Iyp ica lly. "'t > "'1" Fo. subcnlically damp..d lundl'roo mp<'d) .y'-
tern.. "il" n ro ta t i.... . jIt"t"d is U'f(,. : , = ~l' a nd t he rool ..., a nd 51" a'" oomplex
ronjulta t.... ln Ihis (·a.... r , and r 2 " i ll al"'a~"8 ha .... .-qllal a mplil ud<"$ a nd dt"Ca~·
 Chapl~ r 14 RcKo. SCabi ~ry Analysi$:The Roo1 l.0UIS 285

' '. - ~CIf """"'"

'. n
~

!,1 @f' 0 ~
"

• • •
I '. "!l< ol
'. '. "l, <0 "l, <0
1

!• ~ <0 ;,
J
" " "
,,
, , "
,
e

1 iI ~
,

f iogoure 1...... [ ~ , of .... 9 'o,o," "dKdy r , '~•.."on tho fr... o;bo'atJon

b<+Iavior oIlh~ rot",.T)'pic" ~ lot a 'Ubloo 'r>I~m " ~ on
I.... lth coUnn. The Iot><.-.Id [lOP)and ' ............ (mdd ~l orI"" """
from ;nitia! d"l'l«~ (g 'ft"l"l dotl . nd dKdy ;n a coowlar sp;ral.
bull ~~ orbt de<ay> f",«<!han tho forward orbot t>o<_ ." "
" ' - ~""' !han ',. Tho- tQtaI rotor ~ (bo ttom) " .., t'lIipDuof
orbo1lhor ~ '0 • eireul... fono-.. d do><ayot'l<} sp;ral.tn ...... _ ~
<:oIum....ll....Tht..- oIlnstaDkty, ~ , _ 0 a nd .., < 0. A90'n,......
_ ~ orbt de<ay> Quoddl',IN",'"".! onl)l , fono-.rd o<bof. Tho- plO-
'u<le of , forw.rd <>IbiI ;, conSl. nt at>d orbot conbn~ for...-
""'" cons' l .mpI~u<le . In I.... rigt>t coknJn ".n uns,.tblo: ......"'"
w .... ,., ~ , > Oat>d ~, < o.nrr ~..-spon~do><a)lSQuickl)l._
.... 1orw.1d "'P""~ grow>-The ..,..,..... p-o<oM<l' ;"\O . n unst.tbl ~.
9"'W'''9- forw.rd ,pif" orbit.K nolhOnQ ch.tI<}e<I "' tho ..,..1..... !II<'
orbit . mpldlXle wwId ""r... ~ tor_
0' grow at lh.. ...m.. ral... plOdllci n~ .. d...." ying (or ,<no.., n~ llin .. orbit Thus.
when a ~ubc.ilicaJl~· dam ped noto r ~~...t..m i. ~topped. lhe syst ..m ,,·ilI lh eor.. li·
call~' ""-haV<' like a ~i mple linear mcilla w •. (I ma~ine di~placin!! lh.. ~topped rot o r
by ha nd a nd .....ea~i "ll. it. Wil ho lll rolat ion·dri",n Ilu id c irculatio n. Ih..,.. is no
la n!lm l ia l fo.ce. a nd l he spring folC1' ..ill l ry 10 ,.. tum lh.. rotor direclly toward
l he "'luilib,ium 1",.;lio n. ·f h.. ,...ult i" Iha t th.. . ot or "ill .ibrat .. lik.. a ~tri n!lo in
a lin.. o rb il. For machine.,; ..it h h~'d rod~na mic bea rinp . this i. a n ",...,.i mplifi·
calio n. beea u..... al ...10 ' I...ro. lh ...... is t>O hyd rod~·na mic bea , ing . I,ffn",.. )

State·Space Fonnulalion of Ihe Eigenvalue Problem

The c haraderi<lic "'luatio n ,,'e haw de....l0l'..d for Ih.. si mpl.. mod '" i.
qu rat ic a nd "ol,·abl .. us ing alg.-b.aic l<"(h niqu.." (all ho ugh th.. complex stiff-
n t... m c. ..at ,..; con"Id... abl .. d ilf,c lllty). Mo .... complicalo>d .pl..01' i ll pro-
d uce h ijl:h.., --o. d..r rna raet ..riMic "'luat io n. thai m l1.<1 be ...1.-..<1 n llmo>ri a Uy. For
..""m pl.., a 2-l)tlf modd ...ill g..lI..ral.. a rna ra......r;'t", "'luat ion t hat i. a ~ t h ·
oni poIJ·"",,,ia!. ...-hid , ", iIl l",><1ucc ro ... ..;g.,"'·alu.... I" ~n ... aL th.. o , d o r
t h h..'act..,ist k "'lu..tio" ..nd Ih.. numJ-...r " r ..il\" ".·al",,. ",ill h.. two I"" th ..
" u mh... of deg. ..... " f f.-....d" m " ...,) in the m,><1"'.
Wh.." " o rne, i"..1 m,-lh, ><1" mu ,", be uM'<!. .. .<Ial.. -·YHJU maJriJ. "1'1"",,,'h Iu
th.. prohl..m i. fa" t... a ..... ....' i... 10 fo,m ul..t... T h.......t..... ."'.... "'....hod ..... , ts
wilh rh.. ho n"ljo\""" ' NJ" "'l"..l i,,"(. ) o f mol ,o n of Ih.. 'Y'lem. To ill".trat .. th ..
forn,,,lat i,,,, m..thod. ..... will u... Eq"at ~,n H -i a . " ur ...art ing I', ,,nl:

.\1; _ D; _( I.: - jm.n) . = 0 ( 14-, )

1 h.. fi..1 st "p i. 10 p" rfo rm a .educl"", of ",de' from .. ..",,,,'od ",. k-, di ff.., -
..nt;al ""I ";ol io n 10 ..0 ' ''I" i.·" I, -nl syslem o f r;..1 ..rd, ,, ""I"al i. m•. To d.. Ihi... .....
will m..k., a .·a,~.h'" ,,"b.1illllioo lI"iolt a du m m~' ~""'I'>r ,-..riahl ... u 1...1

( 14-F'l

1 h.. ,·..riabl .. ,"bslil"l;on etl<' OI':'''' " " ' ..i n~l ",'("ond ..rde , . liffc...nhal "'lila·
t ion into a ~~...t.. m of I...... firsl o , de, "'l"at ion...

», = u,
( 14· 16)
"'Ii, + I"', '" (A: - JOM})u, =0
 D (' jfl-V l ) (1 4-111
" •,=- -".-
.\ 1' .\1 ",

(1 ....18 1

......... tlw oqu.a«' rnotm .. t .... olGI~~ _ru.( )on- "J""rifor lIlIm.ncal .'aI·
..... . ... k-Itd. rnodt.-rn «J<nJ"'t.Z><m ,.,(1'><0.."" \tuch ., :\U f L \ 8"1a n OJ"P'fat..
d "....-u, 011 1.... ><I.lf-~ ....I' ...nd numrrically,obca,n bot h rt!l"m 'aJuP<o a nd
~ ....1 u" l" fuch at. ~'Ond IIw ocopo!' (l( Illi. d iw:w... iofl). In tlU. ca.... Ilk'
In.- ,,,•.....neal ,lll ue u... """ rijtrm.. l..... of Eq ual "' n 14-12 " "Ollid boo
obt.,nN
n h...d i1 .... il~· n temkd 10 morE'rompk• • y......,•.T hO' h~n .." ...
for m of th n lJt>I , " p l(: .}~ ..m of l'.<ju.Jlioo Io-H ~

.Ui _ Dx"' '' ,x - !H f!.r .. 0

.\/y -.- Of - K .y - IJ>.1lA .. 0 (1 4· 191

W.. ,<'d uu · lh.. " ,d .., o r thi, .v. tcm ...inll &du mmy ••:.Ia• •'anable. u.

" , _ Of
":z _ i
11....20)
11) =1
II, =.,

"'.\l1i,=..,- Da,'" /(.... - D"I./u, . O

(!-l- 2 1)
II) = u•
.'1"0- Dtl. - K ,II, - 0 .\1111, = 0
288 The Static and Dynamic Response of Rotor Systems

or, in matrix form ,

0 1 0 0
til -K_
_ x -D -DAn ul
- 0
u2 M M M u2
= (14-22)
ti3 0 0 0 1 u3
ti4 DAn -K -D u4
- - 0 - -y -
M M M

Again, the square matrix is the state-space matrix, which in this case will pro-
duce four, complex eigenvalues.
The 2-CDOF model of Equation 10-53 can be converted to state-space form
in the same way. First, we start with the homogeneous form of the equation of
motion,
MlTl +Dil + (K l + K 2 )rl -K2r2 =0
(14-23)
M 2T2 + DB ;'2 + (K 2 + K B - jD BA n )r2 -K2rl =0

Then we reduce the order, using the set of dummy vectors,

Ul =rl
U 2 =;'1
(14-24)
U3 =r2
U 4 =;'2

Th is leads to the state-space formulation,

o 1 o 0
-(Kl +K2 ) -Dl K2 ul
0
Ml Ml Ml u2
o 0 o 1 u3
-(K2 +KB - jDBAn) - DB u 4
o
M2 M2
(14-25)

This matrix will also produce a set of four, complex eigen values.
 Chapter 14 Rotor Stability Analysis: The Root Locus 289

The Root Locus Plot

Simple spring/mass systems have roots. or eigenvalues. that do not cha nge
with speed. However. in rotor systems. the two components of the eigenvalues.
, and Wd' are both functions of the rotor speed. n. This means that the vibration
behavior of the rotor system changes with speed; the rotor system is a compli-
cated oscillator with speed-dependent properties. Because of this complexity.
we need a way to present the roots in a way that is clear and concise.
The best way is the root locus plot. This plot was originally developed by
Walter Evans [1] for use in automatic control stability analysis. where the roots
are also called the poles of the system. (The terms roots. characteristic values.
eigenvalues. and poles are different names for the same thing.) Several excellent
modern texts [2.3,4] discuss the root locus from a control theory perspective.
but this approach is oriented primarily to the needs of control system designers.
We wish to examine stability and the control of stability problems from a rotor
dynamics point of view. such as was pioneered by Lund [5].
The root locus plot is an XY plot; the coordinates of the plot represent the
components" and wd' of the roots, s, ofthe characteristic equation. Because of
this. the XY plane is typically called the s-p lane. In the root locus plot (Figure 14-
5), the horizontal axis is the growth/decay rate, , (L's), and the vertical axis is
the damped natural frequency. wd (rad/s), with forward precession (+ wd) above
and reverse precession (-wd) below. The roots (yellow dots) correspond to r 1 at
coordinates v, and wd' and r 2 at coordinates -y, and - wd'
The root locus plot is divided into two half planes. Response in the left half
plane (yellow). where , < O. corresponds to stable behavior; disturbances pro-
duce vibration that decays over time. The farther to the left the root is in the left
half plane (r more negative). the faster free vibration will decay.
Response in the right half plane (red). where , > O. corresponds to unstable
behavior; disturbances cause vibration that increases over time. No real machine
can operatefor more than a short time in the right halfplane. Either the machine
will self-destruct or a nonlinear effect will act to stabilize the system. return the
root to the stable half plane. and limit vibration. The right half plane is a forbid-
den region of operation.
Moving farther away from the horizontal axis will increase the natural fre-
quency. wd' of the free vibration. Positive values correspond to forward preces-
sion; negative values correspond to reverse precession. A point on the horizon-
tal axis has zero frequency of vibration and corresponds to a case of either crit-
ical or supercritical damping (see Appendix 6).
The forward and reverse precession frequencies have the same meaning as
the forward and reverse precession frequencies associated with the full spec-
trum (Chapter 8). The full spectrum displays the actual frequencies (with in the
 .
'•

--
"' ·+"-4 )

,,",,0)' 9_

fig~ ~ a -s nw fOOtlo<"'plaLTh< ~ . I ..." 0$1( 1/>1.

w '" ".bI< <>i><'fot..,., to tho loft of an.....ond un....,.e to
"'e n9h1· lho ~mc" ..;,. i> !he c1a"'~ n.lur'" tr<q",,"C)<
.J~ (r.od!.)..."" ",......0:1 ~ ...,..", _ OM. - P'<'-
c... ,.", _ ,~ ""'" ' e '9"'""' ~J (.(lrre>pOn(l,.... to.,
i,
pion"" .. <;<lQItl.... ~ ~I . nd ~·.. . nd " Of <oord"'. ,... "
and - "".'''' tho IofI twt plorw ~Uowt diuurt\¥l<... produ<.
";br.,.,., th.n do<:ays. ln tho rig'" II.tf pI_ ( ~. d · " u , ­
bone ... Clu ,," Yb,wOll_ 9'''''''' I/o _ _ e,,"~ ­

o~.Io< """"' rt>aIl "short""", ., lW rifI/1' II<J/f ~. _ ng

f........ , aw~ fmm , .... honzonl.l ... i, "NiI inc~_ tho ~
M..o. of tt>e ~H vibt.lJOI'l n.ttut.llroq""",y, _.... F'o<n.....
"""'... ~d ,,, forward pr<'<fl"""- _ "" •• ,u to
"'~ P~.,';on_ A PO't't on the hofilon1.I.l<i. "" refO
~ofllibr .hon.nd~ to. ,_of . n"",
Q';~C'" or w PO"Citoc,", e-p;ng,
- -....:::.;?- '/,-J..
.-
i
B:'O .;...,•

•
· ,
•
·
· .~

~-,.
--
- - - - - - - - '. - ~"l' , .... ,(11l1
•

FOg..... 14-6 ~oot 10<.... plot 01 ~ ''''1'':.~ ,or", _ . Tho

rotor .PH<! .. v.fIO'd ~om 0 ,.0<11. to 1000 radls-AI ""'0 'ot",
~ (1M poiM< rn¥i«d w 'th an ·X"! th~ two rool< ... ~ com-
pie , con jU9""" !the 'Y'1...... is "'be";"::.Uy d .~. whr", ~l
_ " _ - ll ~ I\ . nd If'>ty pIoI d, ""tIy. ~ a.-.d boolow e"Ch
_ .....,PH<! ina.......... ""' ''''''' """"" in ",""""e d'''''I''''~
. nd both ~ .f>:l_'~d>. "'J<' WIth """"" "PHd Tho rOOl w;1h 10<-
w. ,d p'<'C~""", m""."
tow..... til. un'lOb" """ plane. will e
_ ,.......,., pr«...""" rool """'"" 'W6y [_""'Jr.at aI>OVl
811J ,..t!.:8300 'pml,t"" <0<01 >p«'<! IN< .... '''0 ThrO'<hold 01
Ins1.lbdoty.......,.. "'r _0<1 root rNtl>t. t'w ... ltOe.1 a';, of "'e
plot._ thil; ~ tho """ """'"" to ."" rig'" II.oij plan~.
_ ~ I "p<>'l~""''''''; "'nY''''''''' un>"bl ~_At_
Th<W>old alln ,~. , .... fnoqU<'flC)' of: . iIJf;>tion is.l th~ n.. u-
,.01 f~""""'Y of th e rot", "l"'''''''_" _
4 10 ,.odI<,O.• ;rx
292 The Static and Dynamic Response of Rotor Systems

instrumentation bandwidth) that comprise the rotor steady state vibration

behavior. Each pair of full spect ru m frequency lines represents a des cription of
th e steady state orbit shape that corresponds to that frequency.
In contrast, the root locu s plot displays the theoretical forward and reverse
natural frequencies, ± wd' that, together with the associated growth/ decay rates,
"h and "h. describe the predicted transient behavior of the system . Each pair of
roots represents a description of the transient orbit that corresponds to th at
natural frequency. The transient orbit is the sum of the forward and reverse
rotating vectors at that frequency. The shape and decay time of the free vibra-
tion orbit is determined by the relative magnitude of the two values of "l associ-
ated with the roots.
In control system applications, a mathematical model of the electro-
mechanical system is created. A set of roots is calculated for the existing system
parameters. Then, the gain of a feedback control component is changed slight-
ly, and a new set of roots is calculated. This process is repeated until all the
desired sets of roots are created and plotted on the s-plane, creating a locus of
roots: the root locus plot. One objective is to determine how much feedback con-
trol the system can tol erate without going unstable.
In a typical rotor stability analysis, a rotor model is generated. and the roots
are calculated for a group of rotor parameters (M, D, K. >.•.0). Because the behav-
ior and stability of a rotor system are related to speed, the mo st interesting
parameter to change, at least at the beginning, is the rotor speed, fl . Later in the
analysis, a parameter such as the spring stiffness. K, or the Fluid Circumferential
Average Velocity Ratio , >., may be changed.
Fluid interaction in our rotor model causes both the growth/decay factors,
"11 and "12' and the damped natural frequency. wd' to change with rotor speed. For
syst ems with significant fluid interaction or gyroscopic effects, changes in rotor
speed modify the mechanical behavior of the rotor system. At any particular
speed, the system will have a free vibration behavi or determined by the eigen-
values, which will be different than at another rotor speed. Because of this. a nat-
ural frequency responsible for a balance resonance at a lower speed may shift to
a significantly different frequency at running spe ed.
In rotor systems with no fluid interaction (>' = 0), such as machines with
rolling element bearings, no working fluid, no gas or liquid seals, and no gyro-
scopic effects, the speed-related changes in eigenvalues may be very small or
even zero. An electric motor with rolling element bearings is unlikely to have
eigenvalues that change with rotor speed.
In the root locus plot of our simple rotor model (Figure 14-6), the rotor
speed has been varied from 0 rad/s to 1000 rad/s, (We could use rpm, but rad/s
makes comparison with the natural frequency axi s ea sier. and it is a natural
 Chapter 14 Rotor Stability Analysis:The Root Locus 293

input to the model.)

Our rotor model has one complex degree-of-freedom (l-CDOF), so it pro-
duces one pair of roots. For the set of parameters used in the figure, the model
is subcritically damped. Thus, at zero speed, the two roots are complex conju-
gates, where '"h = 12 = -325 Is, and they plot directly above and below each
other. With this value of I ' the amplitude of free vibration will decline, in one
millisecond, to

or 72% of the amplitude at time t = O.

As speed increases, the roots move in opposite directions. The root with for-
ward precession (upper root) moves to the right toward the unstable half plane,
while the reverse precession root moves to the left.
Eventually, at about 870 rad/s (8300 rpm), the rotor speed reaches the
Threshold of Instability, and the forward root reaches the vertical axis of the
plot, where I I = O. Above this speed, the root moves to the right half plane,
where '"h is positive and the system is unstable.
At the Threshold ofInstability, the natural frequency of the rotor system, W d'
is 410 rad/s, considerably less than running speed. From Equation 14-4,
W = w d = Aft. In the figure, ft = ftth = 870 rad/s and w d = 410 rad/s, yielding
A = 0.47. (In fact, this was the value of A used in the model.) Thus, at the
Threshold of Instability, the frequency of vibration is subsynchronous at 0.47X.
At rotor speeds higher than ftth' the root locus moves into the unstable
region of the plot. This is purely an artifact of the linear model and cannot
reflect actual machine behavior in practice.
In reality, nonlinear effects come into play at this point (Figure 14-7) . As
rotor speed increases past the Threshold of Instability, the root crosses into the
right half plane. The system is now unstable, but because the root is still close to
the vertical axis, I I is a small positive number, and the vibration increases rela-
tively slowly. The orbit diameter increases, and, if the source of instability is a
fluid-film bearing, the increasing rotor dynamic eccentricity ratio causes an
increase in the bearing spring stiffness, which is part of the overall system spring
stiffness, K. If the source of instability is an internal seal, then the vibration
amplitude might increase until a rub takes place, which will also increase the K
of the system. Either way, the spring stiffness of the system will increase;
Equation 14-5 predicts that an increase in K will produce an increase in ftth '
At this point, an amazing thing happens. If the increase in K raises nth above
running speed, then the system eigenvalue will shift to the left half plane (with a
corresponding higher value of wd ), and the vibration amplitude ofthe now sta-
 I

- •

--==~-

.'--- -- - - t --
- -- - - - - - . - - ~-. · 1 1lSl
I

~ 1'" _ """"" . .... n, .~'-d~

... . lMI\qtnclo>..... . 1 film bHrrq .... _ _ PO"' "
.... Th • ..-.okl d ~h . - uouon "'7'l ftI//
pIone. e.e.c-
....... 1IOS'I.... _
.... fOOl is uti cbIoo '" "'" -..eM
, _ .... _ _ . I(> I
~,
m..-'!'
II •

-.,.n.. _ _ ""'_ _ .... ""' ~ ......

~«X-..ory, ~ .. _ .. .... ~
'P'io'9 Ol"""'L P¥1 d
_...........,-r-d
~ ............ 'P""9 ,,-.. K. •

--"9
.... ircrN~'" K r,; f /.. rt-. thr
•.,.,.... ~ ;tIm to .... 1OIt "", pIo ..... . nd ' ''0
o;br. non .... p1~_ wi ~ , n 10 <10<,_.
produe;"g • • m .l~
a..
_n._
. ' f(C... tricl' y'.. oo.dKr ..... ng Ii fl.. M f. ..
~ opff'~ng loPH'd. lt>en
"9h1 t-.If pIoono
"'"'""1""'_ ..."'. to ....
,.;bt.,,,,,, ,,¥t, to ""......, l'>o.J'-
...... _.....,. """'-""1'-4' c
<:tot""" 01 ~....p.t_ I<.ob>,."""",,o _41.K."..
'" COf>-
 Chapter 14 Rotor Stability Analysis:The Root Locus 295

ble system will decrease. This produces a smaller eccentricity ratio, decreasing
K, and lowering nth' If nth falls below operating speed, then the eigenvalue shifts
to the right half plane again, and vibration increases. The result of all this activ-
ity is that the system self-stabilizes through changes in K in a condition of large
amplitude, subsynchronous vibration. The eigenvalue will be located on the ver-
tical axis, and the system will be operating in fluid -induced instability whirl or
whip.
Further increases in rotor speed will push the eigenvalue into the right half
plane again, and this nonlinear stabilizing process will repeat, resulting in a larg-
er diameter orbit than before.

The Root Locus and Amplification Factors

Even though the eigenvalues, or roots, tell us much about the free vibration
behavior of the system, they also have something to say about the forced
response of the system. Whenever the frequency of a perturbation is equal to a
natural frequency of a system, then a resonance will occur. The root locus can
provide us with information about the nonsynchronous and synchronous
amplification factors at resonance. To understand this, we will first examine the
behavior of a simple, spring/mass system (a simple harmonic oscillator). In our
rotor model, when n = 0, the rotor system behaves like a simple oscillator, so
this provides a good starting point.
For a simple oscillator, the damped natural frequency, wd' for a sub critically
damped system is lower than the undamped natural frequency, wI!' because the
presence of the damping force acts to slow down the velocity of the system
slightly. The damped natural frequency is

wd =wl!~l-(2 (14-26)

where

(14-27)

and «Greek lower case zeta) is the dampingfactor, given by

D
(= 2JKM (14-28)
296 The Static and Dynamic Response of Rotor Systems

The damping factor (der ived in Appendix 6) is a nondimensional number

th at defines the decay behavior of an vibrating me chanical system. If ( < I , the
system is underdamped (subcritically damped), and a freely vibrating system will
vibrate with decreasing am plitude. The larger (closer to 1) the damping factor,
the faster the vibration will die away. As the damping factor approaches zero , w d
becomes w n (Equation 14-26).
When ( = 1, the system is critically damped, and the system will not vibrate
but will return to the equilibrium position in the shortest possible time without
overshooting.
Systems with ( > 1 are called overdamped (supe rcritically damped) and will
return relatively slowly to the equilibrium position without any overshoot or
oscillation.
Also, because there is no fluid interaction in a simple oscillator, the two
growth I decay rates are always equal. They are related to the damping factor:

(14-29)

Equations 14-26 and 14-29 describe the components of the eigenvalues of a sim-
ple oscillator. The po sitive eigenvalue and its coordinates are shown in Figure
14-8. Trigonometry shows that the distance from the origin to the eigenvalue is
wI!' Then,

. (w
sme=__ n = ( (14-30)
wn

Thus, for the simple oscillator, radial lines from the origin of the root locus plot
describe lines ofconstant dampingfactor.
For the simple oscillator, it can be shown that the amplificat ion factor at res-
onance, which we will call Q, is related to both the damping factor and the angle
on the root locus plot:

Q ~~=_l_ (14-31)
2( 2sin e

e,
Thus the angle, on th e root locus plot defines both the damping factor and the
amplification factor, Q. For exa mple, if 'Y = -100 Is, and w d = 800 rad / s, th en
 C~ 14 Rotor St.odify An.I)1;S:The Root l.oc:ut 197

11 = <iIKt4Ifl
1-'1=..-etanI ",>i. I• .
~
~~ (I(llJ ,ad '.
7.1

Q ..
I
2~ -
I
2(0.12)
,
A~ we Ita,... "" ill. w .... n ~'t .... r A '" 0 or J? = O. . .... roto , 8Y"t~m modcl lX'ha" ~'
na.1J,. hkt-a .impl~ o"d llator. (l l~r(' ..... illno", th.. " rnbl,·m of flu id · fIl m lX'a rinjil
ol,ffnt"''' a t ........ "p<"t'd. l>u t Ihi. fA c<"Ll.i nl~ apph..abl~ to r.. lh n~ ~I~"",n l lX'a rinjil
rn.... hin.....l Hm-......,r. ,,"'~n bulh A and 11...... n"n u-r.~ , h....Ltlio Mh ip inm h'njli
t h.. dompinll factor and nal ural f"'lU<"ncy i. m uch morT complica ll'd and
in...." ..... both fluid ci ",uIallt ..... ff..cu a nd tIM.- rotor oprt'd I""' ."q>p>ndix 6~

7
j
_.-
~.- •
<-.

i -- 1
~.I 4-' Coor-...dltla POI"
~~ ~ "'t_ r:J _
•
~toc "". , . __
~ ........ IIr"'lUft'CY.•• •
1ht' rod'" ..... frQm Itla Ofigin ~
_u ..1na aI ,,,,,,,... t ~P' ''9
'M;tOI' Sft ,.... ",,' Jo< _ .....
I
I -'. _"oJI- C)

,
•
'.- - - - - • - -
~ .. r_ , It""
298 The Static and Dynamic Response of Rotor Systems

While the roots of a simple oscillator do not change with forcing frequency,
rotor system eigenvalues are a function of rotor speed and move in the root
locus plot. When they move , both components of the eigenvalues, I and wd'
change, changing the angle e. Even if M, D, K, and A are not changed (meaning
that (, as defined by Equation 14-28, is not changed), a change in rotor speed
produces a change in e. We can see that the effective damping factor of the rotor
system has changed. The fluid circulation around the rotor, Aft, is solely respon-
sible for the change in root position.
Equation 14-30 provides a good estimation of the effective damping factor
of the rotor system, and we can use Equation 14-31 to estimate the steady state
amplification factor, Q, of the rotor system when the rotor system is subjected to
a periodic force. When the rotor system is operating at a constant speed, the
eigenvalues will plot at a particular position on the root locus plot. The Q in
Equation 14-31 represents the response of the rotor to a nonsynchronous per-
turbation while the rotor operates at a constant speed.
As a rotor system root approaches the vertical axis and the Threshold of
e,
Instability, the angle of the root, approaches zero, and the amplification fac-
tor, Q, of the system increases. At the Threshold of Instability, the Quadrature
Dynamic Stiffness is zero, the effective damping factor of the system is zero, and
Q is infinite.
Figure 14-9 shows three positive eigenvalues and their associated amplifica-
tion factors for nonsynchronous perturbation. Lines of constant Q are drawn for
Q = 2.5, 5, and 10. API 684 [6] defines a critical as any balance resonance with a
Q > 2.5, so the left line on the plot marks a boundary that satisfies this criteri-
on. Any eigenvalue left of this boundary would not be considered to be a critical
by the API's definition. API also defines any resonance with a Q < 5 as a low
amplification factor and any resonance with a Q > 10 as a high amplification
factor. The Bode plots show the non synchronous response to a rotating unbal-
ance for each case.
Synchronous amplification factors are important during startup and shut-
down. Nonsynchronous amplification factors are important because distur-
bances other than unbalance are capable of exciting rotor system natural fre-
quencies. If the amplification factor of a particular natural frequency is high (e
on the root locus plot is small) then that frequency may show up as a significant
vibration component of overall rotor response.
An example is aerodynamic noise in compressors. This kind of noise is often
broad band, similar to white noise, and it will excite any natural frequency in the
noise band. If the amplification factor of a natural frequency is high, then the
noise will be amplified at that frequency, producing a noticeable line in a spec-
trum plot.
-.f -- - -+- ------j
......
, -
fig..... 14- 9 f or<;<'<! ""'1'0"'" for th "'" PO';~~
..g .,,.~ ........ The . mplilic.hon f.octo" for nomJ"'l-
"""""", P!"lum••bon • .., """"" .."h I...... 01
'on><on, Q how Q ~ 2.5. S. and 10._ pIot"how
m. ornpIItu<l<o (..1 : and ~ 101 oIlh" r ~ ...
to • """'yn<h''''''''''' rot'''ng " " bolo",. lot _h
c. ....A, til . oNJ""...... ne¥'< ~ ~,... ...... 1""
'fYIIllifi<",,,,,, ''''''0< """ ","" ply.
 ..
,"_192.00'>
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• •

Fojw. I4-' O E" ...... <d .......lYonouIompW;cmon IOClOr.Q,

II t/'II, bo.....,~ ~. ""- "'"" 'PH'd iO ~ to .
"""" 'Y'~ ~ tlw<\. --..... ;, htO'Iy l<I
O<n>"." " - n_ tlw<\ tlw .,......., _ ~ t10w 10 .....

.....,.,.,... PN*.-.d .....- . - o l e s.,.o-"",

~ fOClOr ,, _ ~ £_ "" '.. .11.... ..... hMn-

_.,...._...... """""""" .
pIio. .. ..... ~ """"" d 11D11u11 91 1~ lin O 'P'>I.'
• 4r . "'" <iimping 1..-:10<. ' " 0.17. _ """ _
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 Chapter 14 Rotor Stability Analysis: The Root Locus 301

So far we have been discussing nonsynchronous behavior at a particular nat-

ural frequency unrelated to running speed. However, the root locus plot can be
used to identify a synchronous balance resonance speed and estimate the
Synchronous Amplification Factor.
A root locus plot can be generated by varying rotor speed. When rotor speed
is equal to a rotor system natural frequency, then a resonance is likely to occur.
Figure 14-10 displays the positive root of the rotor model. When n = Wd» then
the system must be close to the resonance peak, and the approximate
Synchronous Amplification Factor is given by Equation 14-31. In this example,
at the resonance speed of about 192 rad/s (1830 rpm), () is 9.8°, the damping fac-
tor, (, is 0.17, and the estimated synchronous Q is 3.
302 The Static and Dynamic Responseof Rotor Systems

Parameter Variation and the Root Locus

The root locus plot of Figure 14-6 was created by calculating pairs of roots
for a series of rotor speeds. It is also possible to create a family of root locus plots
which demonstrate how the basic plot of Figure 14-6 changes when a parameter
is varied.
The plot family (Figure 14-11) was generated by selecting value s for the
rotor parameters (shown below the figure) . calculating the roots for a range of
speeds. then changing the value of K and repeating the process.
The plot has several interesting features. For K = 31 250 Nzrn, the combina-
tion of stiffness, mass, and damping results in an overdamped (supercritically
damped) system. This produced the pair of root loci (orange) with their zero
speed points on the horizontal axis of the plot and separated horizontally. This
is typical for a supercritically damped system. At zero rotor speed, th e system
has a damped natural frequency of zero; thus, the system will not oscillate.
However, once the rotor speed becomes nonzero, the roots leave the horizontal
axis, and the natural frequency ha s a nonzero value.
For K = 62 500 N/m, the system is critically damped. Both root loci (blue)
start at the same zero speed point on the horizontal axis and move away from
each other on a straight line.
For the remaining two values of K, the system is underdamped. The zero
speed points are located vertically above and below the critically damped zero
speed point. For these cases, higher stiffness results in a higher natural frequen -
cy. as expected.
The red arrow shows that, at an operating speed of 400 rad/s, the system
moves from the unstable region up and to the left into the stable region as the
stiffness increases.
Figure 14-12 shows a similar set of root loci that was created by holding K
constant and varying the damping, D, from 354 N . s/m to 1060 N . s/rn. M and
A are the same as for the previous figure, and K is held at 125 000 N/m (714
lb/in). The zero speed points for the root loci have the same characteristics as
in the previous example.
Note that the positive roots pivot around the same Threshold of Instability
speed because, for the simple model, this speed does not depend on damping.
This speed is the same as shown in Equation 14-5. Also, the pivot point occurs
at the same precession frequ ency given by Equation 14-6.
Note that for the underdamped cases (green and violet) the natural fre-
quency decreases as the damping increases. Increasing damping tends to slow
down the system and lower the natural frequency.
..... 14- 11 .t.noIJd
'-1o<M pIgQ proclou<ed
._... .......
~

"om ".I<OOf <00><

_t>,-,...,._ _.
.. ""'""~ Tl'n pIol ..... ~ .'.
~","" by ,,-,:,""11 . ,....... 'i
/of ~ wr,n,;..~ffn...~ K.
ul<u" ''''9 ,.... '00110''; ""
• '""9" de ~~!hen
j' '•
';'

<~!"e •
'------:t,:-:;!<~-7~t'----_j
..... d: K
I •
..
-
_ -.n<)l!w ~ I / '
*",_~-_d , /~. »--
II" -
.. _
'"""""'" I' : ' , . ~
'9'tlrd..'"
I ..
eodI_oI K.tt>o.......
~ .... _ .... _o
,oci'- '" 1001) ~ Thr
""'" P"'~_
""""" _ t .... plot \.Ho
"'" t~" 101 drt~~
-- - - . . .. 10kg
"".,,
~ oy ._ - l l l t1

(S.7 • llt'lb ,1 M )
.. -
"", ~ '" "... 1357 • Ht ' .. - ""'l

• '"
....
) 1.l5O ...... 1 1 79 ~

,,,,..-,. ,m "",
6J3OlI'l.'1ft
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1 11 11(1 ~

hlJu.. 14- 1l AfMnl)o of

,
rtW>Iloc\n plou pooch" ",,

_..
by _ _ tile .....'_
dofo,pon; M _ \ _ II..
_in~

11.--' K _ l lS OOO~'"
11' . ~ _
l .­
. ••7.: 1
-"'"-""
) \.4 '" """ O-OJ .. , ""'l
o09!i~ a.'1b ""'l
101'" 00., .......
1or.o .. 11m ~ Ib - """

•
304 The Static and Dynamic Responseof Rotor Systems

The Root Locus of Anisotropic and Multimode Systems

A mechanical system will have one pair of eigenvalues for each degree of
freedom in the system. Our rotor model is isotropic, has only one mass, and was
modeled as a 1-CDOF system that produces one pair of roots.
More complicated rotor models will produce more information in the root
locus plot. These can range from multi-degree-of-freedom models that can han-
dle many modes, to finite element models with very large numbers of degrees-
of-freedom, which can produce a bewildering array of eigenvalues.
Because rotor systems are most likely to go unstable in the lowest mode of
vibration, a large set of eigenvalues which describe large numbers of modes is
not really necessary for stability analysis. However, a set of damped natural fre-
quencies for the first several modes can be useful for detecting situations where
a natural frequency may exist at some integer multiple of running speed.
Excitation of such higher frequencies is possible if a machine has nonlinearities
or vane and blade pass frequencies that coincide with these frequencies at run-
ning speed. The Campbell diagram was developed for analyzing this situation
and will be discussed later.
In this section we will present an example of a root locus plot for a multiple
mode system and for a system with anisotropic stiffness. Figure 14-13 shows
root loci of a typical 2-CDOF (2 axial planes) rotor model that was generated
using the state-space matrix of Equation 14-25. All roots were created by chang-
ing the rotor speed from 0 rad/s to 1000 rad/s. The first mode (blue) starts on
the horizontal axis with damped natural frequency, w d' of zero. The zero speed
root pairs are separated horizontally rather than vertically because the first
mode is supercritically damped. If the system was critically damped, the zero
speed root pairs would start at the same point on the horizontal axis. (As the
damping factor increases above one, the zero speed roots separate farther on the
horizontal axis.) Inspection of the root locus plot will show that, for the super-
critically damped first mode, the rotor speed can never equal the damped natu-
ral frequency; thus, no syn chronous resonance is po ssible. This mode is equiva-
lent to a rigid body mode with high modal damping. The forward branch cross-
es into instability at a rotor speed a little over 300 rad/s. This linear model pre-
dicts that the forward branch will cross back into the st able half plane at a much
higher speed. This does not reflect actual rotor behavior because, once unstable,
system nonlinearities will come into play and change the system behavior.
The second mode (green) zero speed points form a conjugate pair; thi s
mode is subcritically damped. There is a synchronous resonance near 240 rad/s,
where the rotor speed equals the damped natural frequency.This is best seen on
the forward branch. Both branches of the second mode move toward the unsta-
ble half plane, but the reverse branch will not cro ss the threshold. The forward
 -- _ _

2
-
~
i_
r \
!'.
<
_. • ~--------

J _.
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_.
-_=-c_=--c_=-:_=,_=-c_=-c_:;--:.•=~.:-,_=--c!_
~ ,.... ,1 11'1l

..,
•, 1.1 1<1J
1.1 "9
{6 .3 .1tY't> " lin )
(6.3 • l tY' t> - ,"Ii ~)

", 5l N ->lm (OlOIb - .m)

".
«,
~lO N ·>l m

l ~ kWm
U .o t>- <l",1

""'~~
<, l~ kW m (200 Ibf.,)
<
,• l~ kW m (2OO lbI.,)

""
F¥- I ~ I J. T)'Pic. '00110<; 01. rwo-pI.o..... ""'' 'Pi<. HDOF """" tr>O<!o'I .....ng ,....
"" ..-sp,><e ""'''" 01 Equo,,,,,, 1 ~ 2S . nd _ po,amet<n . 1l<Mo._ ,peed i' v.... .,j
from 0 ' adil ,e l tnl 'adI, The fi", m<><Io (blue) " ..,. en ,.... I>coilOn,.1 ..., WIth .
d.mped nat ur. ' froquency 01 _ 0. The ~ ~ "'"" po"' .... "'P".,.,j hotilonl.ly.
ind<;at;ng that _ fitJ' m<><Io .. ",,,...-<""c.o'" <l.omped k .uoe cl:th..._ t01<:w ~
...1 ~ oqual the d.mped nat",. ,Iroquent)\ ond no Iyntlvcncu< 'e>On.Itl[e Iof ' ''IiI
m<><Io ,, ""'_ .The fim m<><Io ~ branc h t:")I~"'''' ""..bih')';M• ""<:W 'P!"O<l •
•,11et:Nef :lOO 'adio. The _cwv:l m<><Io I!l' ~ "",lx:mic . Uy d. mped wM> . Synth' cncu<
.....,...tl[e ne.II 200 'adl.. bes' Sft'Il en _ Iofwotd bt..-v;h_The fcrw.. d bi'....:h <<os....
" 'e inst. bility at • ""'" 'P<"'d of obou! 6S0 ,adIo. _ ._ tcco< 'Y""""
~ 9"
"","bIe .. ~ mt mode at :tOO ,adt.. .., , .... _cwv:l mod e i ~,tab""'" would net be.....".
306 The Static and Dynamic Response of Rotor Systems

branch crosses at a rotor speed of about 650 rad/ s. However, the rotor system
would go unstable in the first mode at 300 rad/s, so the second mode instability
would not be seen.
The eigenvalue behavior versus speed is very easy to interpret when complex
numbers are used in the rotor model. Forward precession roots move different-
ly than reverse precession roots, and they are easily identified. That is not true
for our next example.
Figure 14-14 shows the roots of an anisotropic, single plane rotor system.
This two-real-degree-of-freedom model treats the rotor system as a single plane
mass (isotropic), but with anisotropic horizontal (weak) and vertical (strong)
spring stiffnesses. Compare this plot to Figure 14-6, which shows both roots of
an isotropic, 1-CDOF system.
There are several important differences in the appearance of this plot. Each
degree of freedom produces a pair of roots above and below the horizontal axis.
Concentrating on the forward set of roots, we can see that at zero speed, the
natural frequencies are well separated. As speed increases, the two roots move
vertically toward each other. The first of the synchronous split resonances is
encountered on the green branch at approximately 160 rad/s: this branch is
associated with the weak, horizontal direction. The second split is encountered
on the blue branch near 220 rad/s; this branch is associated with the strong, ver-
tical direction. At a rotor speed of about 260 rad/s, the two branches meet at an
wd of about 190 rad/s, Then, the horizontal branch moves in the direction of
instability, and the vertical branch moves toward higher stability.
The reverse set of roots are mirror images of the forward roots. The stabili-
ty crossing of the reverse horizontal root occurs at exactly the same speed as for
the forward root, a little over 500 rad/ s. It appears that the weak, horizontal stiff-
ness is responsible for triggering the fluid-induced instability.
WilHFpU PI",.,,,,,,,, "_ y..'. ,'_01
W;~lhli! g l~ ~ i
, ii' .. 1
-------;-,,
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.. i_ .. ij !r. .. ~ ~ 3l! "" 1 ggqq .I
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•
308 The Static and Dynamic Response of Rotor Systems

The Root Locus and the Logarithmic Decrement

The logarithmic decrement (log dec, 8) is a stability indicator that is based
on the free vibration behavior of a system. In its simplest form, it is defined as
the natural logarithm of the ratio of the transient vibration amplitude of two,
successive peaks (Figure 14-15):

8=ln[~]
An+!
(14-32)

where In is the natural logarithm, An is the amplitude at time tn' and A n+ 1 is the
amplitude at the time one cycle of vibration later, t n +1.
Because the log dec is defined by the decay rate per cycle of vibration, it is
related to the eigenvalues of the system. The free vibration amplitude is a func-
tion of time and is controlled by the real part of the eigenvalue, r:

(14-33)

where A o is the amplitude at time t = 0, and t n is the time at which the vibration
amplitude is An' One cycle of vibration later, the amplitude is

(14-34)

Taking the natural logarithm of both expressions, subtracting the equations,

and collecting terms, we obtain

(14-35)

On the right side, the difference in time is the period, T, of one cycle of vibration,

(14-36)
 ,
t= - 2 ".-'- = - 2 .. ta n 61 0 4 ·37 }
-,
whe", IJ is t he a ngle from t h" o. igin ..r th e rool Iocu~ plot to t he .. ~nvalu .. (......,
FillU'" 14·8 ). Th i~ up......~'''n s"'~ t ha I it i' poo5ibl.. 10 calruJal<! th.. ~ d C'(:
fro m th.. compone nt . o r a n e igenvalue.
Th.. log d"" i. ofte n calcul.. t.... from "" p<'nm.. " tal data. Thu~ if ...it. Ihto
damped natural fre<Juency uf ,;bral ion. is m... . .,r...J. (in ,ad / . j. then l' ..a n he
obtai ocd from th e log d..., by F.qua liu" 14--37.
Be<:a " ... oh h.. ""gati", lIi gn in F"l" a tion \4 ·37. the st ability ..ril..rion for Ih..
lot! d<'C i. opposit e t haI fur t he m e" 1. lCus. In th.. roo t locus plot. if ,. > O. the n
th .. system i. tlnstable. 11... 0J'p u.ite i. l ru... for l he 10fI: dee; ins ta b ility '''';1I",
........." 1: <;0.
Equa tion 14-37 ....,.... 1. th.. ma jor dr . ..-back of th.. log d ec. Be<:au... l hoe log
dec p...... nu the rutio of l he tw<I .. i~n,·a1 ..e compo nt • • il /0.. ... injnTmatiOfl.
The rool I....,..... p lot i•• upol"Tior 10 th.. 1otl dec bee il dj.pla~", bul h compo-
" ents of th.. e igenvalue. II aDo.... us 10 ..... hO'o'o· ra pidly a I"' l1kula r ..illt"n'·alu..
a pp. oa ch th .. ins la b ility th s hold "ilh ch ang in ..ith .., roln . spt"t"d Or olh..,
pa ra m..t All ho.- ... me tim il all....·• u. to ho... the nat ural f""Ju..llCie. o f
th.. s)-..te m a re d llor'llinf\.

fi9U~ 14- 15 CoIculol''''l tl\e ~ .

""""it d<'C~ Fe<. ,,.It>l.. 'l'>-
t""" I k .... .. bt",,,,,, ..~
dl"<ql exponl"I1!Wly. Th.. petiod of
vibtall<K1un ~ USl"d to McIIt>l"
rNlion<hp to tl\e """ 10<", l"I<}<'O-
....... s.... ,..... ~"b <ll"1ail>-

'. I~"
The Root Locus ;lnd t:n. CMnpb.l1 Di~';lm
no.. C.mpbofl d ~ is .. plot of rotor ~~otnn 1Mt ..... ~ ..........
~llCi'"I''''' f~ IFtplR' l ..... \ fo). Tho- pkJt ..n.....,..~ ...... to <W'lM .........
",It_ runninil 'f"""'d.If ito ...... monictl .. r~ d<>W 10 .. lMt .. ra1 f'"fI"M'CY oftM ,,...
' ..m. n...di.!/'.nal li..... ft1""W"11 u... nmn,nlopt'O'd.nd .... ~ ..nl".l ha ,mon... IT.-
qu.. nt·;"' l ha l w uJd Iw pt<J<l .....,d 1»' lhr n>I' " ~y" ..m. an d Ih.....,n i<:al linr i' l l>o>
. ct .... 1 run n ing .pt'<'<!. Th.. h" . ,mnl..1linn ... I......·nl nalu ral f."'lu..nn ....... . ...,.
rxi~l in til.. ma,:hi,... Mol" ,,, I~ ",acIu" r U "I Of'"IlIing 'f'Hd.
Th ill pi"....... i!t- ll..-f\d. "'P........ nh a n .,.,..".,mplitic. llOfI of rotor .... pnn~.
. u..... ""'... Ioft'<l. m..nv dromprd n..tu"d f~ies in a roCt.If .y..~m do d ••ul!l'"
with rotof "P<"""i ~ ..... rno:>dr. ~ thn d Inl rraction ..... ~
spri"fl ot,fJl">r'U rontrol rotof syAnn '""'J"'IIM'. Thr ~ diapam ~I'
ooJy~ infonnalion. a nd ....ac f.....urnrr infonnat M'"at . ha t: d -.uId I>r
drl!inblr.o I>r.blr to .........bili.y ' nfon nat iocl .... . M "'nw plot.
1M «001 lo...-uo plot 1"""'<""10 dMa !NIl '"i"t t. bilth o.l..tMl'I~'
.nd f""l"':ncy ;nfonnaIM.n,.nd it sho. nm..' n. lU tIll ~ 1\tir!. du.nll" . , lh
rot o r . 1........ ,\ dd itiona lly. I..... r i, inf<>r m. ' M>A ..00..1 am plifICation facl"" .1
.-...ona nCf'; .adi al lin... of enn.tant Q mak r AI' I b" .. nda" n fa~~ 10 ;d.-nl ily.

~ 1. ·1. A ~ ~ <Ngr"'"
..,.,. p l o l _ .. ~ ",-.co ""-
"""" ...",, d..-.." ""'"" c-.:to _ ..
"""""'Ir"'l"O'nt~ <I ~ ''''.''1
/_J
duct<! by tht roto< ,,.t.... '"
Tht cIolqo-
NIt Iron 'tP'........ , "" lroqutnC .... pro-
I,," "PHd • /
_ .....bt,

r
, '"'" _ ~ 'Ptt<1 Thr

"" -plol '."'"""~

- < l l ' - - "'",",,_"' __ ....... I •I Y
_ ......_ ; , .. __,,<IWd.
_<1_
....,....".,...,- p l o l _ .. ~­
~ bot\OWO
i ,
, I.

u._ •• ~ ... <._

 '"" 1...... _ _ -

j•
;. -,
-~

J:~
,
I
-~
• •
fMJouro 14-11'" """ ............... ....,
~.IlllOJl...... ~I$._

from 0 'ocLI1t<> 1000 rodl~ ""lh . "

"'......fOlor l'O'amo1<'t1 _ t0r>-
n-. n. ~oI_al*y"
I
]
-- I
_.lOoadI.......... _ ~

_ . . . . . , .. _ <JllftSUnl •

--1
~

8'0 radI~ _ n.. """" ..............

o..,,_d""'" ' COl IIWI ll""'
<IotNl _
_ 1 0 7OO
to -l(XXJ _ 11 ~
.t<Iml Thoo_'" .~
--
"' I'0OI

_ infO,.,...-.... W ~ . ... ,.,.

"""om plot. rotor 'J"'<'d 1$_ tor>-
...ant ., 81'0 r a<l/~_ l i' ~ oJU'd
from 0 . ' ~ dol...... "-110 ().lQ,
_n'W:Mt19 ...... .,....... fOOII _
_ ~ IUllir f\aIf ""'.'Ttooo .. an
1
~ G
........ rJ _ _ t.acc.om-
phN<I !>l' ~.- ...... to . '"
.-...g or wa

..
)(.----- . '
'
••

'-.,_•.•._.._._•..•..
_ ;-,.:-,.;-~.
~_ ' I''''
Roo llocus Arwolysis of MK hin. Sta bility Problfoms
In ordrr toobtain ~ ....... ...w Sf"""'d . . .... fllo<>drl "" .... <'Oft-
Stru<1..! of I'" machi"" "ndot" udy. This can ~ from pW. Iurnpnl
..... 1IIOdorl1o ..-.ph..11nI10"d finn ~ ~"",.. nl rnodf.l
'Thr KCUracy ofl rnults will doopend 011 W ~ of W mociool C"'n1
W nlfTMIl ~I~ of t modoofi"fl Mt.1hoono ..... manr .. nn"ItlIl nl~ .......... mod -
d l"fl rotor .~-stnn bo-ba.u ,,",ri"fl or...-al stilfneu. damP"'Il. .nd Lombd.o ...
an funct....ns of Kn'nIricily ........ 111'"- ">II ,,," ~..... "*' '"J'OfI""'~. br ot""'ll-
Iy MpooOOm! on Sl4l ic radial!ooldi"ll. ..-bid> in<lin"dly .. lfrrt~ tlww piI••mrlns.
In m.n, machi...... Sl"Iit" ...Ii.d lc-li"fl rna , d>oo"!l"' ..-,u, opo.... lin!! ClM1d ,tions:
fnr uampl~ . ..ilh nouw >ortllnp in st." "... tumin.... 1'h<-rmal jlrOW! h du rin!
>t••lu p ca n at", .ff<'CI. aI'!tn m"" l . nd ht-a ri ng rttnll ' K"ity ra lio. Bcca u of
I ...•... unn>rt.inll.... il iA unlikdy I .... ' . ny mud ..J-ha ..~1 .U1hilily . n.... y. i ill
I' rrd ict I.... Th .... h.~d of In'lah,lily ~ " ' lh h igh ... ",ura~y_
Ru u l locus an'll)-r.is iA suil"! for ""'noil ;'ily .1I" h to ...1.... a stahilily
pn>blt-m. (; ,.....n • ntJK"hi ,Ih a .Iahihty p. uhll'm. h n n t he probl~m Ill'
d ,mi.... l..! in l h~ ~ ~ O'ffm ;'", m. ""....? .\Ia ll~· pos.;I~~ ..ngin....n nll .. ~u ·
tions ""'Y Ju.... 10 be ....-luat<'d bu l moot. d irt"ctly ,.,. ind .. rrt l,. inn",.. .-han~ ...
In stiff.-.or ambda. 1 tw n>t>lIoo:u' C11ll hrip "'M1 IIi......p. t.... JIO""ibi.lit ..... nd
find I t..-.r; .aut,,"'_
F" 14- 17 shawo" an t'umplor 0(. roo:>l Jonu, .~...;... AI lop. tM "P""f"I ir.
d>oo"fll"d from 0 radls 10 IOI () radii ..i1h aU <>e rotor par1Imffi'n hdd rnn-
>tanL Thfo ~ ul illfolablllty is aboul 870 , ... In thO' middW plot. I....
n Jloe" "f-.:l jot hrid .......anl at 870 radI.... nd I ip l"fl .. ,fI........ IS ~
from l(JI(I lb / in (tht' dorfauJl ' -... . .J lo -lIUllblin (17S 1..' /m 10 700 kX/m). TJw
i~ in ..d'r_ """,," II.. rotor S)-r.tnn ...... Imt> 1M otablr half ......... .'n
inu in st iff...... iiI<. Iii.. could to. acrompli~ by ..'l!in .......L ..1I",h -...Id
in<:n'Ol'" t ht' kwod on • f'""'"1OtL~. lipltiy IoMlftl bra.. n~ nr by ilKl1"U' ''tl !two
p............, in . ~t1""" .ti<" "".n~ From thio ...-w p ...ilion in IIw It>t>t Wus plot.
II wou ld rt'</u i.., a "'n .itlrrahW inc......... in . 01•.,. . 1,.....) 10 ..... l" b ilizo> th.. rutor
sY"tt'm.
In tli.. bollom p l' JI. fOtn r op<'<'d is ht'ld con.lanl a l 1170 .ltd/ and A i•
•l"dun -d fro m OA7 lI .... ol,·r,m ll v. lu..) 10 0 .20. Th i. is an t'u mplt- " f "' t ... otJ ld
br a("("()mplioOCd~· USIll~ an li.wi.1 in,..." ion in a br.rin~ or ,,·al.
Both mcth"ds otahiliuo , .... n ,tor ' y<l"'" I..... F.q""'linn \ -f-S). b ul anl i.. will
injt"Cl"'" 1fIO\"n Ih .. " ~ ,,r ro<Jl I. n ...... away fro m th .. Th.....h"ld of I n.tabilil~.

Su m ~ ry
., JUUd- ~ dofinll"'" ,,( SlAbil lty ir. thai .. .....m..n..-.J oy>l U ~ if.
"-#"',, It u distlUbrdfr-r u. ~ ~ it "'"" ....t1)· WlU teo /hat
 Chapter 14 Rotor Stability Analysis: The Root Locus 313

equilibrium condition. A system is unstable if, when it is disturbed, it moves away

from the original equilibrium position.
Rotor systems that have significant fluid interaction have the potential to
become unstable (a condition called fluid-induced instability) because of the
tangential force caused by fluid circulating in bearings, seals, or around
impellers. The rotor model predicts the Threshold of Instability given by
Equation 14-5:

This expression is a very powerful diagnostic tool, and it is the key to under-
standing how to prevent and curefluid-induced instability problems in rotor sys-
tems.
The simple rotor model was shown to have two roots, or eigenvalues. These
complex numbers are of the general form 1 1,2 ± j wd: where 1 controls the rate of
vibration growth or decay over time, and w d is the damped natural frequency of
precession. The growth/decay rate, I ' is a useful stability criterion: 1 > 0 implies
growth of free vibration amplitude and an unstable system.
Rotor systems with significant fluid interaction have eigenvalues that
change with rotor speed. A set of eigenvalues can be calculated from a mathe-
matical model over a speed range and can be plotted on a root locus plot.
Radial lines from the origin of the root locus plot describe lines of constant
damping factor, ( , and constant amplification factor, Q. The Q at resonance is
related to the damping factor and the angle on the root locus plot. The reso-
nance is located approximately at the point on a forward root branch where the
rotor speed equals the damped natural frequency.
The logarithmic decrement can be calculated from the two components of
an eigenvalue, but it loses information because the components are in a ratio.
The log dec concentrates on stability characteristics at the expense of frequen-
cy information.
The Campbell diagram does the opposite. It displays the natural frequencies
of a system at the expense of stability information. The Campbell diagram fre-
quencies are based on running speed only; the plot does not show how natural
frequencies change with rotor speed.
The root locus plot is superior to both the log dec and the Campbell dia -
grams because of its ability to display both stability and frequency information.
314 The Static and Dynamic Response of Rotor Systems

References
1. Evans, Walter R., Control-Sy stem Dynamics (New York: McGraw-Hill, 1954).
2. Nise, Norman S., Control Sy stem s Engin eering, Third Ed., (New York:
Addison-Wesley Publishing Company, 2000).
3. Ogata, Katsuhiko, Modern Control Engin eering, (Upper Saddle River:
Prentice Hall, Inc., 1990).
4. Kuo, Benjamin C, A utom atic Control Syst ems, Fifth Ed., (Englewood Cliffs:
Prentice-Hall, Inc., 1987).
5. Lund, J. w., "Stability and Damped Critical Speeds of a Flexible Rotor in
Fluid-Film Bea rings:' ASME Journal ofEngin eeringfor Industry (May 1974):
pp. 509-517.
6. American Petroleum Institute, Tutorial on the API Standard Paragraph s
Covering Rotor Dynamics and Balancing: A n Introduction to Lateral Critical
and Train Torsional Analysis and Rotor Balancing, API Publication 684,
First Ed., (Washington, D.C.: American Petroleum Institute, 1996), p. 3.
7. Bently, D. E., Hatch, C. T., "Root Locus and the Analysis of Rotor Stability
Problems," Orbit, v. 14, No.4, (Minden, t\TV: Bently Nevad a Corporation,
December 1993).
 315

Chapter 15

Torsional and Axial Vibration

~DIAL. ALSO CALLED L AT E RA L. VIBRATION is the most commonly meas-

ured type of vibration in machinery. It involves the lateral, oscillatory mo vement
of machine components in planes perpendicular to the long axi s of the machine
in a combination of rigid body and bending motion. Measurement and control
of radial vibration is important because of the potential for unwanted rotor to
stator contact during machine operation and because the cyclic bending stress-
es associated with vibration can lead to fatigue failure of machine components.
While we cannot measure radial vibration in all places in a machine. it can be
relatively ea sily measured with currently available transducers.
However. lateral motion in the XY plane does not co mpletely describe all the
possible directions of motion of machine components. The rotor can also move
relative to the machine casing in the Z direction (ax ial vibration), and it can
statically and dynamically twist about its rotational axis (torsional vibration).
Torsional and axial vibration do not get the attention that they should. Since
torsional vibration is more difficult to measure than radial vibration. it is often
ignored. due to an "out of sight, out of mind" att itude. In fact, torsional vibration
can be quite severe and is capable of producing damaging cyclic stresses that
can cause a fatigue failure.
Axial vibration is relatively easy to measure using thrust position probes.
Unfortunatel y, the usual application of thrust probe measurements is for static
position measurements, and the dynamic information from these probes is
often ignored.
Both radial and axial vibration can act through the spring-like elements in
bearings to directly produce vibration in the casing and other attached machin-
ery components. Torsional vibration cannot act directly through the bearings,
316 The Static and Dynamic Response of Rotor Systems

because they have no spring-like element in the rotational direction, and they
are designed to minimize power losses due to friction. Thus, few direct me cha -
nisms exist for coupling torsional vibration to other parts of the machine.
However, both torsional and axial vibration can indirectly cross couple into
radial vibration and vice versa. Thus, it is po ssible to see some of the effects of
torsional or axial vibration in a radial vibration signal and not recognize them.
Axial vibration problems in machinery are relatively rare; torsional vibration
problems, while more likely to occur, are harder to detect. Because of the relative
importance of torsional vibration, it will be the primary emphasis of this chap-
ter. We will describe what torsional vibration is and how it behaves. Then we will
show how the dynamic parameters of torsional vibration are similar in concept
to the mass, damping, and stiffness parameters of radial vibration, which leads
to similarities in Dynamic Stiffness and vibration behavior.
We will discuss the sources of static and dynamic tors ional loading in rotor
systems and how the rotor responds to these loads. Then, we will discuss how
radial vibration can produce torsional excitation and the reverse. Finally, we will
present a summary of torsional vibration measurement te chniques.
We will conclude the chapter with a brief discussion ofaxial vibration, start-
ing with static and dynamic axial forces. We will follow with a discussion of
axial/radial vibration cross coupling and axial vibration measurement.

The Torsional View of the Rotor

All rotating machines are involved in the delivery or transformation of
power, and power delivery requires transmission of torque. The applied torque
causes the shaft to twist about its rotational axis. Thus, instead of measuring
displacement relative to an equilibrium position, torsion requires measuring the
angle of twist of the shaft at different axial positions. Figure 15-1 shows th e twist
e
angles at several locations along a shaft. At each location, the angle represents
the angular deflection of the shaft relative to the untwisted, equilibrium posi-
tion represented by the black line .
Unfortunately, because the shaft must rotate to deliver power, we cannot
directly observe particular spots on the shaft and calculate the angle of twist;
the angle of twist ofthe shaft is added to (or subtracted from) the continuously
increasing angle caused by the spinning shaft. Torsional deflection and vibration
involve only the amount of twist of the shaft , but the spinning shaft is what
makes the twist so difficult to measure.
Torsional defl ection is measured as an angle, in degrees, which is analogous
to the displacement measured in lateral deflection; analysis of torsional vibra -
tion also requires angular velocity and acceleration. Torsional vibration ampli-
tude is usually measured in deg pp , angular velocity in deg/s pk, and ang ular
 ( " ap t... , S Tonional ...t Ax;aIVibral;on 317

acr.-In-alio<l in ,1..• / 01 pk. TIw.>w . 'a1un ar c ...." ••U. nl". ...n nl to ra<!i.o n.o .. f1O'n
uW<l'n ~""'".......<> hrnrrfuJ to kny> I . ..k ci t.... ",,,I ..
w.. ha> rrn>drird radial ro( or 'Y'! rm " I>r.lio<l u 'ling n>t or p.uamrt.... cI
fotw damptl1g. olIff"...,.. a nd lambda. Tonoo......J vihrollion a~.q • .-..
I.... ...,.w.u orqeu, ....... t.o cl tM fir-.t four p.o.ra mrlrno 1'-'lL>C'. mommt cI inMtia.
ton>onaI damptl1g. and Ioroional oI,ffnno.. In u...tot....""" ....,.Jd. Ihrw- .. 1M>
""",naJmt 10 Ia"ll""'hal.tilf lam bda » IX>t uoN .
" "'"I- (al to eallN a 'IX"' t offo'rr~ T prodUCf'd wt.m a f~ io
appIwd at ........ d .<lana. caIIord 1M .....-.., from I rot.alio<l ~ Whm
, - ..... a A, c heh 10 l '!dtl.... a bolt. you 1IJ'PIy . f, to I nod of I.... " Tf11Ch.
Ttw '''''It! h of t wrftK"Il II 1M INNTWI1t . n n,. and I ro.-c. prod"",,", . lorquor
IhaI .......... I holl to ro(.r~ T""'l.... ha.t lU1il .offort-.. tll" .... di "'......,... X · m
li b . ft ). In L:5. .-u,;t0fTl0an" u nit .. i.nchn .........r,........
ub!.c,wlm for furL
r......nocall)<. t. orqut' and a,.war d'''J'la<-1. ,'f'Iocity. a nd aced a"'on ....
Ih........J,nw n "ft."tOr quamilin. BK hm llir>jt our di '..io<IlO
...... du"m "l\nr aDo of l m.d" llh no ~-n>OCOJ'1C O'll.cu illt. ...1
aU of I acaJar <jUAnl ir.....
n... ftJUn'a1C'nl ol rna.. io In.. ( _l momntl of.,...,.,.... I. Ttw
...... mo.·nl of inen ia hao. tlle un il of m aM In" ... r~,.... "'Ill&n'd. or k!f • m '

F.,.... I,._' ThlN_,....

_ ~of

_-"'-
rigdr ......... "' ......

---
ltcauwoJ~Ior<

........
_lIoc_.
..... At-'ttcatiof\' _
_

--
_ IN . . . . oJ

-"..,.-"'-~
318 The Static and Dynamic Responseof Rotor Systems

(Ib- ft . S2). As the units suggest, the moment of inertia depends on the amount
of mass and the square of its distance from the spin axis . Rotor disks with a large
amount of mass located relatively far from the spin axis (for example, large
diameter pump impellers) will have a large moment of inertia. Gyroscopic forces
in rotor systems are related to the moment of inertia and can modify the radial
vibration characteristics of a machine.
Torsional damping, Dp corresponds to the radial damping, D, the force per
unit of velocity. The torsional analog is torque per unit of angular twist velocity
(in deg/s), so DT has units of N . m . s/deg (lb . ft . s/deg).
The damping associated with radial vibration is caused by the movement
through a viscous fluid and produces a force that is proportional to the lateral
velocity of the rotor centerline. However, the primary source of torsional damp-
ing is the internal frictional damping of the material. It produces a torque that
is proportional to the angular velocity of twisting of the shaft (the rate of strain
of the material). This weak damping torque can be supplemented by magnetic
field effects in motors and generators, or by impeller vane and blade interactions
with the working fluid in fluid-handling machines. It is usually assumed that
these additional torsional damping effects are small; thus, for most machines,
torsional damping is assumed to be very low.
Because there is no tangential force equivalent in torsional vibration (no
lambda), the torsional Quadrature Dynamic Stiffness consists of only the tor-
sional damping stiffness. The low quadrature stiffness results in very high ampli-
fication factors at torsional resonances. For this reason, special torsional
dampers are sometimes installed in machinery. These dampers can take the
form of fluid dampers or special couplings that contain rubber blocks. The rub-
ber material provides some energy absorption that increases the system tor-
sional damping. The rubber blocks also act like soft, torsional springs, reducing
the torsional stiffness of the system. Elastomers can deteriorate and harden over
time and lose their ability to absorb energy.
Torsional stiffness is defined by the angular deflection of the rotor in
response to an applied torque. The unit of radial vibration stiffness is force per
unit displacement; the unit of torsional stiffness is torque per unit of angular
displacement: N . m (lb . ft) when radians are used and N . m/deg (lb . ft/deg)
when degrees are used.
For a uniform, circular shaft, the torsional stiffness, K p is given by

(15-1)
 Chapter 15 Torsional and Axial Vibration 319

e
where is the angl e of twist along a shaft section oflength L, G is the shear mod-
ulus, or modulus of rigidi ty, and] is the polar moment of inertia, a cross-section
property. For a hollow sha ft with inside diameter d, and outside diameter do'

J = .!!...(d 4 - d 4 ) (15-2)
32 0 I

Thi s expression can be used for solid shaft s by setti ng d, = O.

Equation 15-1 shows that the longer the shaft, the lower th e torsional stiff-
ness . Stiffer materials will have a higher shear modulus and produce a torsional-
ly stiffer shaft. For example, the shear modulus of steel is about three times that
of aluminum; thus, a steel shaft will have three times the torsional stiffness of an
identical shaft made out of aluminum. Also, because] is related to the 4th power
of the diameter, large diameter shafts are torsionally much stiffer than small
diameter shafts.
Since most rotors are not uniform in diameter over their entire length, the
torsional stiffness of a typical rotor shaft will var y with axial position. The total
shaft st iffness will be equivalent to a number of short shaft segments in series,
each with its own torsional stiffness. These segment s behave like springs in
series, so that the total torsional stiffness will be less than the stiffness of the
weakest segment. For a shaft consisting of n segments,

-1= -1 + -1+ ... +1- (15-3)

KT Kn K T2 K Tll

The power delivered by a shaft is the product of the torque and the angular
velocity of the shaft,

P=Tfl (15-4)

where P is the power in watts, Q is the rotor speed in rad/s, and T is the torque
in N . m. If the power and speed are known, the torque can be found by invert-
ing Equation 15-4. If the torque and torsional stiffness are known, the angle of
e,
twist, of the shaft can be found from Equation 15-1. The angle of twist (windup
or wrap-up) of most rotating machinery is quite small; a large steam turbine
generator set may have a wrap-up of only one to two degrees from one end of the
machine to the other.
320 The Static and Dynamic Responseof Rotor Systems

The twis t produces a material strain whic h is largest at the oute r surface of
the shaft. This strain produces a tors iona l shear stress, T (Greek lower case ta u),
at the outer surface that is given by

Tr
T = - (15-5)
]

whe re r is the rad ius of the ou ter surface of t he shaft.

A co mpariso n of the var iables and typi cal units of mea surement for rad ial
and to rsiona l param eters is presented in Table 15-1.

Table 15-1 . Radial and torsional vibration parameters. For torsional vib ration, rad ians are usually
subst it uted for degrees in analyt ic calcu lations. See Appendix 7 fo r convers ion factors .

Parameter Radial vibration Torsional vibration

Displacem ent r IJm pp (m il pp) B deg pp

Velocity r rnrn/ s (in/s) j deg / s
Accele ratio n r 9 1 deq/s?
ForcefTorque F N (lb) T N 'm (lb · ft )
Mass/Mom ent of Inert ia 1v1 kg (lb· s2/ in) I kg ' m 2 (Ib . ft . S2)
Dam pi ng D N 's/m (lb vs/ in) DT N . m . s/deg (lb ft . s/deg)
Stiffness K N/m (lb/in) KT N · m/deg (lb ft /deg )
 Chapter 15 Torsional and Axial Vibration 321

Static and Dynamic Torsional Response

A machine that is operating at a steady state spe ed is subjected to several
torques that are in static balance. The driving torque originates either in a sep-
arat e prime mover and is delivered through a coupling, or, if the machine is a
prime mover, internally from multiple steam or gas turbine stages or magnetic
fields. In a prime mover, this torque appears as an axially distributed set of
torques, where some fraction of the tot al driving torque is input to the rotor at
different axial positions. The driving torque is balanced by the load torque on
th e system. These loads originate in aerodynamic forces in com pressors and
fans, fluid loads in pumps, or in magnetic field interactions in generators.
As long as the driving torque is equal and opposite to the load torque, the
rotor speed will remain constant, and the rotor shaft will be twisted slightly at a
constant angular deflection. The amount of twist at each section will be deter-
mined by the local torque at each shaft section and by the requirements of
Equation 15-1.
Any variation in driver torque or load torque will produce a change in the
angular deflection of the shaft. Torque variations can be periodic in nature or
impulsive. Steady state, periodically changing torque will produce a steady st ate,
torsional vibration of the rotor system. Impulsive events will disturb the syst em
and cause a decaying, free torsional vibration.
Sources of periodic torsional excitation include flow oscillation in fluid -han-
dling machines, misaligned couplings (IX and 2X), rubs (subharmonics, IX , and
harmonics), gearbox tooth profile errors (harmonics or nonharmonics of run-
ning speed), reciprocating machine crank forces (IX and harmonics) and cross
coupling of radial and axial vibration.
Impulsive torques can cause torsional ringing of the system. Impulsive
torques can be caused by bad synchronization or sudden electrical distribution
load changes in generators, or by rub impacts, faulty rolling element bearings ,
slipping poles in synchronous motors and generators, and older variable fre-
quency drives.
Variable frequency drives use a synthesized electrical waveform, which is
made up of a series of closely spaced voltage steps. Older drives have fewer and
larger voltage steps and produce large torque steps that excite torsional ringing
of the rotor system. Newer variable frequency drives have improved the smooth-
ness of the voltage waveform and reduced the torsional ringing.

As in radial vibrat ion, periodic torque variations produce a dynamic tor-

sional response of the rotor. The torsional modeling of a rotor system is per-
formed in a way similar to the modeling of radial vibration (Chapter 10). We will
$um mari Ih... mO'>l im portanl a'Jl'l"C1 . of 11M> lo ..iona l mod..l lo ~hnw t IM> . im-
ilarity bt-t>< n lo ....io nal and ,adial " b ,ation dyna mic "<Jual io "," a nd ........'i nr.
Fig" ... 1:;' 2 s!l0\\" Ih.. 5impl....1 lor sion al mod..1 of a rotating sys"'m. Tht'...
a... two rot n , mom...n ls ofin..rt ia. II a nd I?" ....hic h "'P'...... nl a mo tor ronn <'Ctt'd
10 a dri ...·n mach i"". Th.. mac hin.... an> ronn"':Il.'d b~ a ~haft with to....io ""l <.tiff-
nt'M. Kt » a nd lorsional da mp ing. Dr. Each mom..nt o f i""n ia is ca""hle o f mov-
ing ind..pt'nd..nlly (subjecl 10 sliff"".. and dampi ng constrai nlS ~ '" Ih..... a...
two anp d a r dis pla m..nl ,·ari...blh, {it and (iT T W<l inokpt'nd..nl ln"I".... T , a nd
T?" ...... a ppl il.'d .0.1 a ch ...nd ofl ht'1JJaft.
To ..iona l ,; bral ion in,·ol~ .... pt'. iodic. . t'lal i,... a ngula. d i..plac...m nt o f t h...
Iwo in rt ia. and "'i s li njli of Ih.. sha ft. nOl " "'8d).' .ra t... ro ta tion o f th nti n" ~ .
1<'1TI. R m...m be•• t ho u¢'. Iha t any ,;bral,o n ".gu lar d i' l'lac...m..nl i5 addl.'d to
..·hal"' conslant an~ . .. o f slatic twist al...ad~. ex i$IO du.. to th.. 8'·.... 811..
. 10001
being t .a ns milt l.'d t h. ou¢' th... sha ft.

f ;g u.. I S·2 Somplo....."... t

~t at • Jt>lOr '1'''''''.Two
rotor "'O'Ile"" 01"'e<l""1 ..,d
1,.'''1''..... '1\. m>1o< ,Ot'O'\Kl.
." by" ,haft _ <;oypI"".lIO'
d""''' moc,"" ".The \hafr 110.
tOt\Oon:ot ,,"""'" Kr . nd 1<lI .
.......... da""""'9 D~Eoch
"..,...."of,,,,,,,,. n capable 01
m<M"'.! ~n'lo·nlll' l\ubjoKI
to """""• ..,d damping co<>·
",a...." t .,;t/l ""'.!utor d;'~
........19 , .nd 9~ Independoon]
tOfq..'!'S. T, .nd r,. a'" appiOed
a, .. oc ~...-.d 01...... ,l\afc
 Chapter 15 Torsional and Axial Vibration 323

We will define the dynamic relative angular displacement (angle of twist ) as

¢ (Greek lower case ph i),

(15-6)

If Ii = 12 = I, we can develop a relatively simple equation of to rsional motion of

the system:

(15-7)

Here, the difference in torques is expressed as T, which is assumed to change

periodically at a nonsynchronous frequency, w, and have an initial phase angle
of 8. Solution of this equation leads to

<Pe jo. = - - - - - - - - - (I5-8)

(2K T - Iw 2 + j2DT w)

where <P (Greek upper case Phi) is the amplitude of torsional vibration, and 0: is
the phase of the response. Here, we use the mathematically convenient expo-
nential form even though we are not modeling the system in the complex plane.
The physical result is the real part of the solution, equivalent to a cosine func-
tion.
The w in Equation 15-8 represents any nonsynchronous vibration frequency.
n
A synchronous expression can be created by substituting for w in the equa-
tion. The result will be identical in form to Equation 15-8, so it does not matter
which form is used; the behavior will be the same.
The term in the denominator is the torsional Dynamic Stiffness, which, like
the radi al vibration Dynamic Stiffne ss, has a direct and quadrature part. The
direct part has a torsional spring stiffness , related to the twisting of the shaft,
and an inertial stiffness. Unlike radial vibration with fluid interaction, the quad-
rature part consists of only the torsional damping stiffness, which is typically
small; there is no torsional equivalent for tangential stiffness.
The behavior of torsional vibration for this simple system is very much like
the radial vibration behavior presented in Chapter II, with one important dif-
ference . In torsional vibration, there is no analog for rotating unbalance, so the
periodic forcing torque is assumed to have constant amplitude. This changes
the low- and high-speed behaviors of the torsional system when compared to a
rotating-unbalance-driven , radial vibration.
 Figu '" 15-3 s tMm's Itw "'.pan"" of thf' . im ple to ....' o nal mod f'1. :\ t low fw.
quf'nc ;.... well bdow """na nct". l he t" ....ion'" ' -ibr"lion will be a pp. oximatdy in
p ha -ith th f' fo ,...inl! t<>rqu.., . od," ""to f.l"<Juf'ocy. thf' a mp lit ud e bernm.,.. a
sl y s la l.. 1",'i. l l"<JuaI T12K r'
Th<- d efm ihon of ph a ... in lor . ional '-ibrat ion ,..,..w; "",m.. e xpla nal ion. For
10" iu" ,,1,1I>ra l io n, the - lii!fh . poI. i. Ih.. maxim um of the to ..ional vib ration
l imeba... "'a,...fo. m, "'li ich, for a " ll" la r d isp l""...ment .wt a. "'pl1"Sl""t s a max i-
m um a nlll" o f t...-ist. lik.. rad ia l " bra lio", t .... ab60lute p hase is de fined as Iii"
n umb..r o f d~. of Iii.. ,i bral ,o n cycle from Ihe KeH ' hasor ......nt 10 t .... fi..1
positi' peak of Ihe to rsio"al vib ... l io n ", a,...fo. m.
:\ 1 '"a tli.. ampl it ud e o f Ih.. ' ibral io n will J"-"" k sharp lv Pffa ,,,'e of Ih..
10>-' quad .at u Iiffn. %. a nd tli.. pha.... of lh e response ilIlall lli.. ph a.... of l h..
f" ... i" ll to rqu" by about 'l(r. To ..,o nal "-"SOna "O' pca h usually ....r:-. narTO'o'o',
...i t li a m plifica tion fact o .. f" ... to te n time. Ihme for . ad ial ,ib ration. TIi "er~'
liillh a mplificat ion fact or. can p rod uct" hi!fh a lt. ",alt nj(s....ar st Ie ls ,,-li..n
Ihe rotor sr'te m '-'X peri ..nces a ..."'na ....... T....... shea . >I s ar.. , u\,,-,rim ·
posed on t .... . ta tic to..ional load st.........,; and any . ad ial " b ra tion bend inj(
" ...s.... Ttw combi nal ion can product"d a mag; "{l le,...ls o f sI ....s an d can I..ad to
fatill"'-' crack iniliat ion. Ru n ning a mac hin.. al a s Jl'<"-"'i l"<Jual to . Ior . ional . '-"IO-
na nO' or at • sJl'<"-"'i l"<JuaI to a s ub mullipl.. o f a lor . ional na nO' is a 'l'f)' bad
id.... Fortu nal..Jy, tli.. narrown...... of to .. iona! "'-'SOna n mak.... il ..... Jik.ly
t hat tlilS " . ll ....pp"n by acrid..nt .

•
. ~

Figu,,, 1. ·J r""""",1 t1!'SPO"S'O 0/ a I

...m ~ rot", model.At Jrequon< ... ! . ~

t<;w"""'"

- -_- - - -
__ ~ ..,W
",b'a'''''' ....11 be .ppro.. """ .... in
l .. ~
ph. ... _h '~forc"" ''''Q''''. T. • .-.I
••
the _ _.. "",. be eQUal to Til~ f
• ..... ~-

..."-
.t '""0 treq""ncy, Amplotudoe """
ph. .........,""""'1""'""" ,p<'e<l a, e
"md.o' to th<»e for r.,ba l"';brat"",

'-
Be<......... 0/ low '''''"''''''' d.fY19ing.
.mpkfw::at;or, 'acto<> at a 'oosionol
"""""nee can be Qune hilt'.
,-
f...-
.-• Tj2K j

• • •
--.• ......• - •• ••
1\01.. w",mdarity 10 1M ~ ~ for nod... VIbnloon (£q\l&l lOn 10-
-10). n,.. fr"'l""'nq oIill in>'Oh"" ttw ratio el oIl1f....... 10 rn&u.1"'J'<~ntfti Iw..
br ttw rnotnc'nI el ir>n1ia.
For this lin/d.. modo n olnn..t fn,q~..
fuM of'VJralw" ...JI ~ um.•nd tht
riI.""',. rHO""""".
pot>
WtunpU·
pha .....ill lag,.... forn ng
phaM- b,· aboul 180". 111.... with oJ igh l d ill ba.ic tor sion al vib ,at ion
Iw.......'" lJ, wry s im,la, to th . t of radial , ib' .lio n.
A. " i th radial >i b,.I;oo. mune cm opl... ma<·h ".... ... ca l"'bJ.. o f .....·....1
mod.... o f tor . ional vib rat,on, with d ,ff.....nl _ >tit ..... J1""L fiW'... Is-! , how. t.."
t~l'icaI lomonal mod.. wp"" fUI • madl ;n i lh lh r mom..nts o f i..... tia
(a no th.... modr shiP"" nor shown. iJ, . .... g..n t in -p mod<> th l! is ",!"i>..·
Ioont to rot.l ion el l},., ft). Each mod<- shiP"' .. pr.....nl. d ,ff... ""n1 ,,,i.I"ljI
rombinalions in 1M .....-h, TIw fim modo- ohapt has "'... ad)lCftll m."".,.."
of inon" """'-m,; in phaw ilh l hor th in! out of pba ... . TM sr«>nd rnudt lo!wpl'
I...."'"" 1M middIt momml el iof'l"t", """.... out el .,.,..... _h ttw hn> .-nd
_ 1 $ el iJwnia. .<\mIaI modo- ~ ..iII dtpmd 011 1M ~udt and d.....
,nbuIion 01 diF........,t inonias and 011 tht tor<ioflll oIUJ- dau n buloon ... tM
... ~

"""'" 1H r...".,..., . . - ~ T-..o

"'-."' ---:----'\
--
_ ...,.,.. _ """"" /tIr • rnoct>orwo
.....,lIV... mom«~.d_l ~
_ "'"'Pt."" \I'IowI\ ~ a ~'""'. ---l
~_ ,.... "' .-qu"-" "l ""a·
I"", of tht lhoftJ 'n tht fin' """'" I/wIpt
(topl. two odja<"'" mom« ." 01 onM..
_ in ~....., tl'>t 'M" our 01 pt>aw,
'" lhe _
_
pI\aw_
......
"""'" " - !t><:lnOfI'llhe
•• o .... d _ ......... OulId
N~Actuol_" -
""tht~ __
01 ... "0 ....... 01_ _ 001 .... _
_ _ 01..,.,.,... ~ '" \!'or"""'-
-
Torsio ""lfRa di a l (ross (oup lin9
In a si mrl~. m.alh.. m~I IOlI n ,to , rnod~l , adial a nd oo...,o nal \,brali on a ....
compl.-t.-ly ind..J""n....n t. Th is 'iI U~It..n is only tru~ ",h~n Ih.. rot o r ,;haft i.
" brati ng in a circul a r. I X .. rbil ""n«'. .... o n Ih.. axis o f I ~·.I ..m. In m. lily.... ~ •
..raI lhi np ta ke plitl'" in .. ph~'skal rOior .~..te m l ha t cau radial " bration 10
prod oce a J""riod ic to rqu e a nd \,.... '·e........ a p h..no me non ..ai led CroI<S coup/mg.
Ellipti cal o rb it. prod" .,., cha ng.-. in Ihe t>.-Ildinll of th~ rotor .md cause a
period ic torqu~ to app'-"a ~ whkh can ca " ... lor siona! ,-ibrat ion. Figu ... l.i·5
.00..... a roto r . ha ft , u'J""nd ed b.>N..... n tWQ b.-.rinp. ;\1 I..ft. th.. ",t ,,,. ,;ha ft is
. traight and spins a t aRj{Ul.... , -docity fl. lh~ d ri,..,' of th.. machi ne ...... a . haft
" ,o",.. 1ll o f in..rt ia. /. eq ua l to I.... mom..nt. o f i....rti a o f Iii.- , ha ft a nd d isk. At

right. IX , ad ial vibrallo n has d..n'-"C1ro I.... ,;haft so l hat th...haft C('nl..r1 in.. IS
,..m.. d istan.,., from th .. axis I red) oft h.. rot or s~-,;t ..m. " ..w Ihe disk. ",nt "r al so
J're«s",s arotJ .... th.. . pin axis ",il h a"\lUl......Iocitv f!. ll.-<:a ...... / i. I''''p',,''o n-
al lo ma •• ti~ radi"s "Iuar ed. lh.. lot a l mo m.. 1ll of iO<'rtia is no" " /, = / + M,.l.
wh...... ,11 i. th .. m.... of lh .. di.k..
In Ih.. a b...n.,., of changing tu rqu..... th .. angular mom ..ntu m of the rot or .ys-
t.. m.I,U. m" ,1 ... ma in I.... same. A. th .. , hafl d..n'-"C1 ion ioc...a...... the incrt'a ...
in I, req u i.... a redoctiOll in . ha ft . ...nt ..rline angular ,-..locity. U. a nd Ih.. d ri .
....... a to rque tha t l rln lo s lm" d..wn Iii.- .y.t..m. O n t h.. ot h... ha nd. wl>cn. haft
d..nection decn"a th .. dri•.."....." a t<lfqu"lhat Ifi to . pe.-d up the .~..t..m_
Recause of t h d f'-"C1 s. a oy urh ill ha t produ a c ha nl!" in pmitioo Ia -
ti.... 00 th .. .,.,Ill.. rlin.. o f th.. row r . y.....m ..1U p rodu.,., a ,·anabl.. torqo~. l h ",,0-
ta tio n ..a n b.- IX"r mo .... rompl u. an" it will a pp..ar b lo .. ional " br ation of th~
rot o r .... t.. m.
Throu gh t l>.- sa me m....hani.m. a ny tor , ional vihration ..1U c ross couple inlo
rad ial ,~brallon. M a llse oorqu.. ,·,,,iations " ,11 a!l e n' p t to t ran. f.., rotation al
..n..rll." illlo radial . 'ihrat ion enerlt.~.
•<\ d iff.....nt c ross·coopling m<'C han;,; m ' ICCUn. wh..n a rotati"lt ,;haft has an
.....~m "' ..t ric c ross seelion. suc h as might "",oil fro m a . haft c rack.. a nd i••ub-
j....t.... to a .ta tic radiall..., d . Th .. a'Jmm("! ,~' m..a n.l hat t he b.-ndin[l. sliffn..... of
II>.- .haft i. not t h.. sam.. in aU rad ial " i"",1 i..n«.Thu,,-wh..n . ubj t to a st a t·
ic rad ia l ]""d. th .. . ha ft ,, 'iIl.... nt'Ct m",....-h" n th.......ak ,;haft Miffn axi. i~ ori·
..nt .......il h th e applied load a nd I..... wh"" the st roo ll .. iffn..... ax is i.ori..nt:ed
" i t h II>c 1,,,,01 (FigtJO' 15-6). Th.. most inte....li ng c..... i~ wh..n t h.. ,;hall i.
b.-tw....o th..,... t"n u",d ilio".. Then. ••ta tic rad ial loa d wm p",d UU' a d..n •
tioo ,,"h a rornponMlt at righ t a ~ lo th .. l""d. 8eea ll-.... lh.. shall is d..I1....I....
....lOy from th .. ",to r '}-st..m ....",..rlin Ihi. P"'J'I"floJi<'ula r Ct,rnl" .""nt act. as a
cra nk.. " i.-..ctlJ p rod ucing a tOrqll.. in th .. rotor. T h~ ....1i<Jn " ..-cors rapidly. pro-
d uci"lt a . napping act io n a. th.. ,(long Sliffn"". ax;' mm 'e. aer"". t h.. load ,
 --
f_ IS-~ f;1IW;td ~ dt'.. I10.. ..., ... " .. ' .... 01 .......... loti,
_ _ orWIft .. "".,;gtIc ...,. - . . . -.q.MI ~ rl ~.-.- at ....
.....-_.............."......"'_t....
_ '. . . <1#_ "' '9'C. , . ,~ AI
lOlhe ...... d..-.e
...,., .....
do",Oi!d !he ~ ""
_ .... _ "'""
01 _ _ """",,, _
. -....:e.
- <frot.,.....,..,
_

.... ..,U.TM_"OCI' " ' _ ..

"""'" '-'9"" II . Mr' ._
_~~_""'"' 01_.,_.
Ill .. _ ...... oIlt>r ' " " ' - ..........

"-_ 'H T.-- ~ bo, . ~ _ ,

-.._....,. .. .,-y." _ _ .. ftY"""ft-
~
. ....,_....."
"" . . - _ _ oca... cr.... _ ~ " " ~ •..."..
__ Allti\ - . . .. ~_ .... <lI".DO" .. "9'-- ~
.... - .
_ .. ...-.When _ _ . _ CO
_ ._ _ <lIolIKb .......
._ rwocoo_ . .
"""'9 iW< .. . . .... - . "'" lc>od. _ _ """",,.

_ro.
l t f t ~

IliOIqwr.oboul ........ _ ... 8fauw......,.....,....-

"''fPH!'o-r _ ~ _. . boo. 2Il_"...;.
__ .~O<tion .. _ - . .......... _ _
f~ "-9' _ ..... d 0C!CJft d"..,_
Ileca use l he ~han asJm m.-try Il"J""'IS Iwi"" p"" ",...,I " tion. Ihi. snappi"ll a.1i..n
",il l p rod " "" a 2X lo",io nal ~~cil ..I...n in th .. mlo~ Th.. " o nli","a r " ..I" .... of lh i..'
m<"Cha " i.m "a" a1"" R.." .. "'t ~ h igh.., han " "nie•• ..,ch a. 3X. 4X. ~tc.
G..a , oo... ...a ct io" load. a'" a nOlh.., . tro ng SOUI<"t' of " " 1M co upling.
( .....rbo~"" I'an. f..... 10«\.... Ih rough Ih.. ocl ion o f Ih.. fore<' al th .. /I...., m...h. This
fore<' .mo ach a s an (id..a1ly ) sla tic. radial load o n th .. rolo ,. Any varia lion in
torq .... ",i ll a pp"'ar a~ a ~arialion in th i5 radial load a"d ",iU a1"" app"'a , a. a ",ri-
al 'on o f Ih.. , adial load on Ih.. s haft. c...., profilE HroT<. Ih..,..fo,... ",ill show up
as hnl h radi al a nd tor . "",al \' b ,ation. In fact. p " , fll<' .., rors (..ith.., pilch rad iu~
or toolh spaci" I(.., ro...) a", " , ua lly d..tect Ih rough f""[....ncr a naly. i. of .ilt·
na l. from g..a rbox· mo unl.... acrt>...romel "" h ieh a", "",,,s it i,... to , ad ial. bul
not to rroi..nal. ,i hral"'''.
To rs io nal/ rad ial ,,""'. cou pli"S ill gea ,bo"' .... "..n p rodu"" la 'ge. radial \ib,a -
l ion fOfl-"n. Figu... I;i.7 ~"".• a h..lf spect , um ".."",..de plot of startup d al.. from
.. mol .. ,-dri~..n cornp..·."',. A 6-poI.. "J""chron' ..'. mo tor (1200 ' pm ) i. con·
nl"Cled through a d iap h' ..gm couplinlt 10 a spn"d-irl('rea.i" S lt....rbo:<. which
d , iws a 5614 , pm, 5500 hI' (4100 kW I romp"'....'. Th~ 1'1"1 ~ows .halt relal i,...
,i brahon d al.. fro m thE' h igh -~ pinion .
Th E' 6- poI ~ .ync hlOl\O'" moto, has a Ma rt up to rqu .. .....iII ..I"' n th a i occu r;
at fiX slip /rf'qup ncy. Th i<. tor;ional ..x" ,la l ,on f""lu"llCy (d iagonal red line)
'·an..s fro m i ll lO 'l'm. whE'n Ih.. moto, i~ slopP"'l to 0 <1'm. when il i. a t full
~. Th i. to rqu.. <><.cillalio n H c it... torsional ,i b.-.. ti.. n. which strongly CTOM
roupl.... inl" th E' ob<.t'r~ ed rad ial ,ibrat ion at cri tical f""lu"""i...... Th.. lorsional
"""'na n<...... calcul.. ted at 127;' " pm a nd 3260 " pm. aT>' ..sci t..d in ......·..r".. o rde,
d lUlRg l he slartup.

To,s io n..1Vibr-.. tio<l Menu, e"",nl

Earli.. ~ "'... defmed lor . iona l VIbrat ion a. th.. d)' nam ie twisti" l1 o f IhE' . haft.
Th.. a ngl.. of N i.1 i. th .. diffr....n~ bel"....,n l he a nlluiar d i"..la<"<'TT\f'nl. or angu -
la, posil ion. at two. asially separated poinu (Equal ion 15-6).ld....lly. 10 m ur..
this twist. " wou ld m~asU"" rh.. a ngul ar ru",I..{"("mcnl a t I",.. p.. IOU: bo"' '.
m.....ur..m t of a ngul.., di<.place m..nt i~ co mplica ted by Ihe rotali..n of Ih..
•baft . Th a n a ngul.. ......ocity. n "a " .... a conl lRuou~ in.......a"" in .."gular d i...
pla.........nl. (J = Ot. o n top of wlla t..,-..rlorsio nal " b,al ion may be p",...nl. Thu$.
fI i. nol a u<oeful m.....""'mcn l for lor . iona l ,·ihral ion.
TI\<> cha ng.. in a"lt'l la r ,...Iooty i. a h<"f l.., m......ur..menl c ho i...,. ...1 any a~ illl
. -iLion. to15io nal \'b ration will aJ'P"'a ' as a period ie varial ion in the a " gular
~ ..Iocity ",ound Ih.. m....n "'I..... U. \ "lrtLldlly al l lo rsionai ,ib,alion m....su"'m..nt
sys.l..ms "'....." ... ,·",illtio n. in .." gu l..r \l'locity; beell" se of th i.<. Ihis m.-thud i.
a l"" called a n F.\1 {f Tcqumcy M"duluUan } Iltc....m ..nttll ll"Chniqu~. 1 1\<> .....ulling
 ,

-::- - '
, , ,
'.
-- •
f ' O<l_. lktponl
•

f igute 1S-7 T""""".V,ad'" ,,.,.. """4lling d.-;ng ".nup of .

'Y"'<"""""" motof _ ~"" "6-pole (llOO rpm) ')'11 "
ct.""""", molo< drivo's. 561, 'P'" ~ th"""Qh I
dOopl>'ogm.~ <""'flIIng_ _ ~ inc "'~ gN<ba<. ~
I\oIf lPK""'" ""~ plot 01 the .lO<tup _ <N it ......,"'"
d'>PIoce......,' "",~ <lot. ~ 1t'91>-~ pn<)"- n.,
from
motOf h. .. ......, ..., to<qUO'
osollo1.on that <XC"" at 6J( .lip Ire-
1~
""0!fI(jI. ThO. .."'" f<~""ncy ldi.o</<><WI ff'd Woe)
..riM Itom 1100 <p/'I'I (twi<;e Ii.... ~""Y) w~ "'" "'ot", i$
""PP<'d, to 0 <pm W't>en the mol'" i$ I I Iu~ ~ Th e~ ...
O!d\afoon ..at... ,"""""", _ _ ""'"'" <lrongl)' ero" COo>-
pin into ~ _ ,od lal "'branon f\NI' the ,.leu..,.... tor-
_ , ~... 011 17~ <pm ond J260(pm.n.,~

_ It> "".... 7 ~50 <pm .. e ~ _Md twmoni<: of !lit' 117S <pm

"""""",,e. NoO:e lhOl ,"" '"""",""", .'" _ fIe<I WI ' - ""'..
during !lit' ".""p. Dma ~yafC/lmln ./0<1_ [1].
330 The Static and Dynamic Response of Rotor Systems

torsional vibration signal can be integrated to angular displacement and

expressed in degrees.
Various types of noncontacting, torsional vibration transducers have been
developed that measure changes in angular velocity. One type uses an eddy cur-
rent probe to observe the teeth of a gear or toothed wheel (Figure 15-8) that is
installed on the shaft. The timing of the pulses produced as the teeth pass next
to the probe provides a measurement of the angular velocity.
The wheel rotates at an average speed, n, but torsional vibration causes this
speed to vary periodically. A transducer observing the teeth provides timing
pulses. The time between pulses, t, is constant when no torsional (or radial)
vibration is present (top series of pulses). When torsional vibration is present,
the time between pulses changes, and this information can be used to calculate
the torsional vibration frequency and amplitude.
The signals from two probes are used to cancel out the effects of radial
vibration. When using a single probe, radial vibration in a direction perpendicu-
lar to the probe axis produces a timing variation that looks like torsional vibra-
tion. The signal from the second probe on the opposite side of the shaft is com-
pared to the first signal to cancel out the radial vibration component.
This measurement method has the advantage of robustness in an industri-
al environment; the system is relatively immune to dirt or oil contamination of
the observed surface. The disadvantage is the need to install the toothed wheel
on the shaft.
A variation on this principle substitutes an optical probe and a special tape
that is attached to the surface of the shaft. The tape is printed with alternating
light and dark bars, which generate the timing pulses in the probe output. The
tape is easier to install, but is vulnerable to contamination, tape adhesion, and
temperature problems.
With either the toothed wheel or tape. the torsional frequency response is
limited by the number of timing pulses per second. The Nyquist criterion
applies: the highest torsional frequency that can be measured is less than ':h the
pulse rate. Thus, high frequency response requires many, closely spaced teeth or
marks on the tape.
Lasers are now being developed for torsional vibration measurement.
Typically, the laser beam is split and directed at the shaft in two, coplanar loca-
tions. The surface velocity of the shaft causes any reflected light to be shifted in
frequency (Doppler effect). The laser light reflected back toward the sensor is
recombined, and analysis determines the changes in velocity due to torsional
vibration while negating the effects of radial vibration. Laser methods, though,
can be sensitive to contamination problems in an industrial environment.
 ( " ap t.., 1S TonionAl and AlIial V-1b,aliOft )J 1

,~nochcT ~ ..... ..............." u ..... Sfra i" P!l" 10 1T>O'tiUl~ t he driorm..•

lio<l olta>U of l.... shaft .. ~ it ~ if! ~ to aD ~>pl1O"d torq ..... I'Iw .ua in
cN"It"'" lhr "",i'-ann- of lho' "rain PfIn. ........ can br _ wlPd .rn~ a "",, 0.1 ,
eeoe 110. _ n...
map- 11'dlni<.al J"ubIotm ""...n_fIltll,"llw... rain ~ .......
t.d to , ou lsidir ..nr\d from tho- rotati"!l 'haft.
On« I ill tra n.tC'fTC'd t....... rai n. ohafl: (I:I'O"l<'I ~ a nd WfI main'"
pl.".. .,,... an' u...d 10 cak""".. It... Iorqur l ha t .. t......" dd.....rNi lh"""fth Ilw
o/v.ft at 1M p;>inl. k _... otr.un ~ ~_mic l ra ....n.c...r.. m.,. can br
u....t I" .............. dynamIC Ootraln .nd lorq .
Variou, rncth"d. h".... I..- u....d 10 tra n,r... In,. .1rain !t"tt" .~n.aI-.n_ I....
mlalln! ~p. u.....• f"""d " I'f'l .....t io" . h.a,... II..-d •.hp r'''1l a"d b ru.h h1~
"hi,... call .... noi..·. Ot h... ",,·thud. " '" ind llCl i" n ,, ' a n ~' M n .. " . min ....

--
~­

Fig_
--
. ..... _ _nJln,-,'iru'i:J "1JULI-
I So, _ , _ d ~ vibrotO'l _ • t<lt>!N'd ......... Tho _
II ......, _ ~ fl .but ,lIl'$iorI.Ol_.lIO<'l cau,"" n... - ' " ' 0
«>lain
d'I_ po<iocI;caIly A
"_ocer ~ lI>r _ h P"""dn nming lIU"'"- n... bn... bot 00 , pu...... ~"
"""'.- (top! who .. "" torlooORaI .. 'odoa! """Moon .. P'O'Wftl_ .."....... _ ...... ..
JI"1""I.I'"
l!w
btot", 0 , .,........ ~ _ """...,."..,..,...., boI_ to
__ ~ _ ""'I*tudo_Tho ,;gnall1Iom both P'<Ibt'I _ _
c-.
'" 01""" _ ofItorn d . - """'J' ~ Il'<Ibt; ..........
• _ _, .............. 10_ ptobo . .. """""","" """"9 _ .... _1oI<r
_ _ _ Tht'"l"'fl : __ d __
~
"
"
_
'
"
.., l:'C' ...... ;: d 10 _lIroI '"l"'fl 1O~ ............... _ _ "'"'........
 332 The Static and Dynamic Response of Rotor Systems

Strain gages provide a measurement of dynamic torsional strain, stress, and

torque at a single axial position. This measurement is most useful for calculat-
ing dynamic power by multiplying torque by rotor speed using Equation 15-4.
When power is measured at the coupling between two machines, it provides
information for efficiency calculations.
Torsional vibration signals can be processed and displayed much the same
as radial vibration signals. Timebase plots can display the filtered or unfiltered
torsional vibration signal. Filtering can provide nX amplitude and phase (using
a standard Keyphasor transducer as a reference), Bode and polar plots can dis-
play this data, and spectrum plots can provide frequency analysis.
Note that any torsional vibration measurement should be made at a location
well away from a torsional vibration node of the system. Couplings are often a
poor location because torsional nodes tend to be located nearby. It is a good idea
to perform a preliminary lumped-model torsional analysis of the system to esti-
mate the locations of both nodes and antinodes before deciding where to install
torsional vibration transducers.

Axial Vibration
Axial vibration is another commonly overlooked behavior in rotating
machinery. Most axial (thrust) measurement applications are concerned with
the static position of the rotor, because of clearance issues during startup and
operation, and the importance of monitoring thrust bearing health. Usually, lit-
tle attention is paid to axial vibration; in fact, thrust probe installations often
ignore the potential for vibration measurement and filter out dynamic informa-
tion in the monitor. This is unfortunate, because malfunctions such as mis-
alignment and surge generate axial vibration, and both radial and torsional
vibration can cross couple into axial vibration.
Axial deflections of the rotor, like radial and torsional deflections, can be
either static or dynamic. Static axial forces are caused by gravity in vertical
machines, axial components of gear loads, and differential pressure effects in
fluid handling machines, such as steam turbines, gas turbines, compressors, and
pumps. Electric motors and generators do not normally produce significant
static thrust loads, because magnetic forces maintain the position of the rotor.
Static axial forces are balanced by thrust bearings, balance pistons, dual
(opposed) flow configurations of stages, or by combinations of these.
Dynamic axial loads can appear because of flow irregularities in compres-
sors and pumps, cocked wheels, coupling problems, or from cross-coupled tor-
sional or radial vibration. The most dramatic example of dynamic axial vibration
is surge in a compressor, where vibration amplitude can be large enough to cause
violent rubs and machine destruction.

L
 lltooIu,,", of llk' ma ny ..... " ..... of u ia l forcin~ u ial ~, bra' ion m..asur..meot .
ca n be a n importa nt ""UIl,.l ' of inf" """,lion for cor....la' i..n ....... n ' '), nlt 10 sol~l'
a ma ch inery I"" hl..rn.
D~'narn ically. u ia l ~ihrati"o i~ rno~1 like II.... " b,alio n of a ~i rn pl ...
sp n nlllma"" .~· .te ol. The m lo r rep ....... nt s a o ' n,...n tra ted rna •• l,,·hich is act ual·
Iy "'Iual lo II><' ...eilthl o f II", rotn. ill Ihi. ca...l. The act i"" ~d.. of the ' hrus' bea r-
i" lt prm 1d... the pri ma ry .., ppo tt s tiff"""" in II.... lUial d ir<."CI io" .
\\1',, 0 f"'rlorm i"l/. a " a naly.is of lUial "bra lio" . i\ is irnporta ot 10 iocl ude
I.... ..1Jt'(1. of o luplinll . ' ilJ"'....... Couplin!\S a lJl'Cl bot h , I... rna .. a nd stilr..... ~
lwh a,"''' " f II... s~..t..m. lIillid co upJi"l(S pfO\1de \'Cry h ilth ax ial sl iff......... pro-
d .....inlt wh al i. ..1Jt'(1 i'·"' y 0 .... lar~ rotor. O n a long. rilt idly coupled mach 'ne
tra in, th.. n ~ ", m",,' wiU .... ual lh.. ' 0(.11 mas. o f , I... n ltidl y cou pled rolor .. a nd
II><' SlllJn.... in I.... syst..m will be supplied by th e th rust bea n np.
C , " uplin!!s. b<=.- of Ih..ir ax ial freedom of mol ion. p rm , de ""10 "" i,,1
. lilTn u"d,.,. no rma l cond il ion . a nd acl '0 ,solal" roto .. from each "dM" This
..1Jl'Cl iu-lv dl'COUpl..... ax ial ~, bra" on bet .."..n machines.
lJiaph rallm a nd d isk cou plings act hk.. ax ial ~pnng.. a nd tra " , m i' 'ihral io n
fo"",," from o n.. ro lo r 10 a no lher. T hus, a ma ch ine will. , ....... 'VJln of co" pli ng.'
act. lik.. a multip le ·d"W""-of· fll'edom """Wa lor ......k h is c" ""hI.. " f m" Jlipk
mo,res of ax ial " b, al ion.
A si mpl.. model fo r lUial , 'ib rat;on is >!low n ,n Figu re 15-'l. Th .. rot,,, rna", is
rtl< "Jded as a . inl(k.l um ped rna . s. ,I/ . a nd Ihe ~I, ff" ........ K. a nd dampi" s. 0 . a ...
p n wided by , he Ih rust h<-ari n~ A . i"ltl.. ~·a"ahl ... z. mea "" re. II... d i.pla<"t'fTl<'n'
in t he lUlial d ir.-ct io" relal i.... lo Ihe Ih rusl l...aring ,uI ~ " 'rt. 11><'... is no lanll"'"
' ia l .. ilJn.... "'Iui"a lent for a s ial vibra tion.

f~. H·9 II .... pIo ITIOdooI for..... t

",brabOn- ~ rotee mas. i' . . . - asa 'l K D
" ~.\JmP"d mass M, and m. , ~ ru"

<lo_
"""...,,'9 ~!h<' ,,1fInr!;\ K. oOO
M ,

--
V.Tho ~ .............oN:l
In Ihr .... do~ lion. .. '<'Ion...... '" m.
••
lhtusl bNrong "'flIlO't. Tho'" I> " " tongrn-
I""-"~ for ..... . ibr olion. /
n""" - " 9
334 The Static and Dynamic Response of Rotor Systems

The model leads to the equation of motion,

Mi + Dz + Kz = Fej(~t+lJ) (15-10)

where z is the axial position of the rotor mass, M, D is the axial damping (pro-
vided by the thrust bearing or working fluid around an impeller), and K is the
axial stiffness. The dynamic axial force is assumed to have constant magnitude,
F (no rotating unbalance here), frequency, w, and phase, 8, when the Keyphasor
event occurs. As in the torsional model presented above, the exponential form
is used for convenience, and only the real (cosine) part of the solution represents
the measured vibration.
Solution of this equation leads to a form very much like that for radial vibra-
tion:

FejlJ
ze» = - - - -2 - - - (15-11)
(K -Mw + jDw)

where Z is the amplitude of axial vibration and a is the phase. The denominator
is the axial Dynamic Stiffness, which has a direct and quadrature part. The
behavior of the system is similar to that for torsional vibration, with a natural
frequency and resonance near

(15-12)

In Equation 15-11, the nonsynchronous frequency, w, is used, but n can be sub-

stituted to obtain the synchronous response. In large, rigidly coupled machines,
the natural frequency of axial vibration is likely to be low, because of the large
effective mass of the rotor.
When multiple machines are coupled with diaphragm or disk couplings, the
machine train can possess multiple modes of axial vibration, one mode for each
major mass of the system. As for radial vibration (see Chapter 12), each mode
will have its own characteristic natural frequency and associated mode shape.
Axial mode shapes consist of longitudinal patterns of axial motion, with differ-
ent phase relationships. As with other types of vibration, the most likely first
mode shape is an in-phase motion of all coupled rotors (see Figure 12-1).
Radial vibration can cross couple into axial vibration. One such coupling
mechanism is the bending of the rotor shaft during radial vibration. Bending
ohonf1U l boo IrngIh 0( I.... oh.t11 .uItIIlJ~ ;u . &h.olt .. ndcofl!Ut"" p"'i..dic rna"!l"
,n Wft .,...,I..n".. J'<>"il..", durin!! rood;,! ... b...h p U 0( In.t ""-It ",iU u ndcof-
!!U a J"'riodtc ~ ' n .... 1III ""'"I....... Ttm """'I chanJtr. 1"ItC1hn- ..,Ih 1M
.....,.,.01 tho- l'OlO<. . nd <UlJ pllnl: and booa.-'''f! oI. ,IJn, _llf"'"iOI'" pt'riodioc a~""
r..m... lhat can .._ .... 1iOI >. ,,!ion.
\ "anablr I hruOl kJtId."ll from ~~ hl'i.lCi01 !/"'iOn it . ........... of r..d a nd lor'
oioonaI Cf""" (IlUpli.... 1<> ...... "bnoti<KO- .'\ ~. pr'riodioc rna. in r..d "-l or
tOfqlM' at tfte ~ar mnh ..-ill "PJ"'iOr .. . dl.an!(i "l! Ih ru olhtd. Thi. ",..n.. bot h
"'-.yo: ... ia1 >i bnohon can ..... pIc- in lll radial a nd h ..-oioonaI vibr . 1ion Ihrot.l~ Itw
......... mrri> .ni. m .
Turbi n.. hWd ("On r1 I.... ...... .10· 0( u w ly IDO\i "ll Ou id 10 10 'q" e, h ,l..
com p........ bI"" ("0"' 1 It... I'",!,,,, 10 " oc ' !lY in lh.. nll id. ." . flllid fl .
a round. 11Ia,I.-. It... fo",.. p.. " luC"O'd al ....d , b1ad.. ha. hol h la n".cnlial (wh ich 1'" ,.
d llu'" It... IOIll".. ) . nd u ial e..mp<",.. nl~ .i\ny flo..· ir"1/.lIla rili... in Ih...... 'l'"
1.. 1lU wiD -.h..w "I''' . imll ll. ....... " . e ha ....... in lorqu.. .... I . . ial forc... Any n" w
... rial i<",~ ("lin <'XCi I.. frn- Of forcnl lo rs ional a nd ax ial, ,h ratio n.
~1 ....h.",io'.1 a nd d r in l run.... U call .....k lik.. u i.ol vih..tion. A.....I po>i·
In it. " .,lally ..,..a .u by Ih ru <l p""ilion prot.... I...' ............ . Ihr ' ('()I1a. or
1M ..nd 01 lhor W it . .-\ a d •..d or "'"a\)" t hrw.l: .~.n.r ca n prod t-'c..hc
cha"'V" in IranodlXft ""'ruL Kunou! can be •• much at- 2.5 to 50 "'" pp (l ln 2
mil ppL .....1 mi. K ..,...• • nd it "' not u<ually _ . " obloon •. Runoul can tw a
morT ..-.iou. probIrm l boo l .. lIod.....-n <Iboorn'nl: l boo .. nd of . n Uh pn!'"
parN ....ft. Kunout ~ion ol ....ial " briOl ion dala can "'" po--rtor........ if
.................
Summ ary
Ton.innaf \\bn .tion i. u..~.... mic ""-Kt,,,!! o( lhor rnee..-....fl . boul i10 ..... .
lional ax TIlIo 1"'~linl! t, •..Lalrd 10 IIw dJlf..wnct' in an !fUlar dl"fk<1io n
!wi"', iaUy ..,......I ...' ..........,....... nl pia .......
l two dynam ic ..q"al i" n, . nd "'Iur pa.~ m l"l "", . "" ~ mll ~ . 10 lh ...... for ..d ,..1
, i h •• tio n. To ....i.. ""I . lj/fl"l<'&S i Ia t....l 10 I.... mal..rial l' '''p''. h ... a nd en . M'<:.
t" ,""l !!....m"'. y o f Ih.. ~hafl. T i..n..1d a ml, in!! i.. pro.i<....d I,rirnaril~' b~· ma t..·
rial int.-rnal friet i"n in lh.. . h" ft . nd j, """aU,· "" ry '"w.T h.. l",.;' .nal '"'lu i.·al....1
o f rn i& ltwo (rna..) ",o m.. nt of irwrlia. ..i1i.-h i• • "'.... u,... of I.... rad ial rna",
d iatr i l i"" of I.... ",I< .r.
80Ih ....lie and .n'na m il" "''''nTlI of I....ia nal k h .." e . i>t in "".1'.."
rnchin .-ry-..'" Wllh radial vih.-.l i<KO- to nional ..,br.b Ih.. rallO 01 ~'nam lC
torq.... 10 InnionaI dvuamOc , ,,IT n... brhavior of l ><>n.al vibralion "'" .
&po<r<l it. \imila. lo thal 0( . ad.... ..,bnt"",. ancl llw ~. it..-..lat .
nl to ...... "'lUiOR' root 0( IIw ral io of toroional Miff 10 ,no-rtia ! EqlWion 1 ~'1~
336 The Static and Dynamic Response of Rotor Systems

Because of usually low torsional damping, torsional resonances tend to be nar-

row, with very high amplification factors.
Torsional vibration can cross couple to radial vibration and vice versa.
Causes include deflected-rotor crank effects and gearbox reaction loads. Cross
coupling of radial vibration can produce IX torsional vibration. 2X torsional
vibration can also be excited by rotor shaft asymmetries acting with a static
radial load.
Measurement of torsional vibration usually involves the detection of varia-
tions in the angular velocity of the rotor shaft. Strain can be measured by strain
gages; the strain, shaft geometry, and shaft material properties are used to cal-
culate the dynamic torque in the shaft.
Axial vibration is the periodic axial movement of the rotor shaft. The pri-
mary sources of axial stiffness are thrust bearings and diaphragm or disk cou-
plings. Axial vibration can be measured using thrust position transducers.

References
1. Jackson, Charles, and Leader, Malcolm E., "Design, Testing, and
Commissioning of a Synchronous Motor-Gear-Axial Compressor:'
Proceedings ofthe 12th Turbomachinery Symposium, Texas A&M
University, Texas (November 1983): pp. 97-111.