UNIVERSITY OF CINCINNATI

February 18, 2005

Date:___________________

Mingfeng Li
I, _________________________________________________________,
hereby submit this work as part of the requirements for the degree of:
Doctorate of Philosophy
in:
Mechanical Engineering
It is entitled:
Active Vibration Control of A Gearbox System
With Emphasis On Gear Whine Reduction

This work and its defense approved by:

Dr. Teik C. Lim

Chair: _______________________________
Dr. Jay H. Kim
_______________________________
Dr. David F. Thompson
_______________________________
_______________________________
_______________________________
 Active Vibration Control Of A Gearbox System
With Emphasis On Gear Whine Reduction

A dissertation submitted to the

Division of Research and Advanced Studies
Of the University of Cincinnati

in partial fulfillment of the

requirements for the degree of

DOCTOR OF PHILOSOPHY

In the Department of Mechanical, Industrial and Nuclear Engineering

of the College of Engineering

2005

by

Mingfeng Li

B.S. in Acoustics, Nanjing University, Nanjing, China, 1994

M.S. in Acoustics, Inst. of Acoustics, Chinese Academy of Sciences, Beijing, China, 1999
M.S. in Mechanical Engineering, The University of Alabama, 2002

Committee chair: Dr. Teik. C. Lim

Members: Dr. Jay H. Kim
Dr. David F. Thompson
ii
 ABSTRACT

The gear whine problem that widely occurs in power transmission systems is typically

characterized by one or more high amplitude tonal acoustic signals. The whine response

originates from the vibration of the gear pair system caused by transmission error excitation that

arises from tooth profile errors, misalignment and tooth deflections. In very severe cases, the

gear vibrations can also reduce life and performance of the power transmission systems. The

fundamental gear whine problem has been of interest to many researchers for a very long time.

However, most of the work have focused on shaping the gear tooth profiles, adding vibration

isolation to the gearbox, and other passive control methods. Effort involving active gearbox

vibration control is still quite limited. In fact, other than those papers on the active control of the

vibration transmitted through support struts of gearboxes, only a few isolated studies that directly

deal with control of gearbox internal components have been seen. Therefore, addressing this

research gap is the primary aim of this dissertation study.

In this research, a dynamic finite element model is first developed for a power re-

circulating gear train system for use to examine the feasibility of applying an active shaft

transverse vibration control system near the gear pair in the effort to tackle the gear excitation

problem more directly. For this particular actuator setup and gearbox system, a complete set of

controller which is based on the FXLMS algorithm combined with improved frequency

estimation technique is designed. A series of computational studies are performed to refine the

algorithm design and to determine the most suitable parameters. Finally, a series of experimental

iii
studies are performed on an actual power re-circulating gearbox system that is equipped with the

designed active control system. In some cases, up to 14 dB of reduction in the vibration

response can be achieved. Also, the controller is capable of handling up to 6 harmonics

simultaneously. Furthermore, the vibrations at other locations on gear housing and gear whine

are also monitored and analyzed. The inadequacies of system are also quantified, and

refinements are proposed for future research.

iv
 ACKNOWLEDGEMENT

I would like to thank Dr. Teik C. Lim, who is serving as the chair of my academic

committee, advisor and research supervisor, for his support and instructions that is important to

this research and my graduate study. I would also like to thank Drs. Jay H. Kim and David F.

Thompson for serving as my academic committee members. Furthermore, I would like to thank

Dr. W. Steve Shepard, Dr. Y.H. Guan, Mr. Chris Moon and Mr. Jim Parker, who collaborated

with me on this same project, for their assistance, suggestions and insights.

The material is based upon work supported by the U.S. Army Research Laboratory and

the U.S. Army Research Office under contract/grant DAAD19-00-1-0158 (Project No. P40942

EGDPS). The ARO technical monitor is Dr. Gary Anderson.

And, finally my gratitude goes to my wife, Li Feng, and my parents for their patience and

supports.

v
 TABLE OF CONTENTS

ABSTRACT................................................................................................................................... iii

ACKNOWLEDGMENTS ...............................................................................................................v

TABLE OF CONTENTS............................................................................................................... vi

LIST OF TABLES......................................................................................................................... ix

LIST OF FIGURES .........................................................................................................................x

LIST OF SYMBOLS .................................................................................................................. xvii

CHAPTER Page

1. INTRODUCTION .................................................................................................................1

1.1 Introduction....................................................................................................................1

1.2 Available Active Gear Vibration Control Methods.......................................................2

1.3 Available Control Algorithms for Periodic Disturbance ...............................................4

1.4 Available Actuators .......................................................................................................5

1.5 Summary........................................................................................................................6

2. SMART MATERIALS AND ACTUATOR MODELING ...................................................9

2.1 Review of Smart Materials ............................................................................................9

2.1.1 Magnetostrictive Materials ...................................................................................9

2.1.2 Piezoelectric Element..........................................................................................10

2.1.3 Shape Memory Alloys (SMA) ............................................................................12

2.1.4 Electrorheological Fluids (ER) ...........................................................................13

vi
 2.1.5 Summary .............................................................................................................14

2.2 Model of Piezoelectric Stack Actuator ........................................................................14

3. GEAR PAIR SYSTEM MODELING AND ACTUATION SYSTEM...............................22

3.1 Review of Existing Models..........................................................................................22

3.2 FE Model of Gear Pair System ....................................................................................25

3.3 Dynamic Mesh Force and Stiffness .............................................................................32

3.4 Actuation System .........................................................................................................36

3.4.1 Concept 1: Inertial-Based Active Gear Actuator ................................................36

3.4.2 Concept 2: Semi-active Gear-shaft Coupling .....................................................38

3.4.3 Concept 3: Direct Active Bearing Vibration Control .........................................39

3.4.4 Concept 4: Active Shaft Transverse Vibration Control......................................40

3.4.5 Summary .............................................................................................................41

3.5 Electrical Power Requirement .....................................................................................42

4. ADAPTIVE ALGORITHM AND CONTROL SYSTEM DESIGN...................................46

4.1 Active Control Structure ..............................................................................................46

4.2 Adaptive Algorithm .....................................................................................................48

4.2.1 Least Mean Square (LMS) Algorithm ................................................................48

4.2.2 Filtered-x LMS Algorithm..................................................................................50

4.3 Effect On The Controller Due To Uncorrelated Periodic Signal ................................53

4.4 Secondary Path Modeling ............................................................................................58

4.5 Fundamental Frequency Estimation ............................................................................59

4.5.1 Existing Frequency Estimation Technique .........................................................59

4.5.2 Frequency Estimator and its Design Parameters ................................................62

vii
 4.5.3 Performance of Proposed Frequency Estimator .................................................65

4.5.4 Improvement of Frequency Estimator ................................................................71

4.5.5 Frequency Estimation Applied to Measured Vibration Data..............................73

5. PROPOSED ALGORITHM SIMULATION STUDY........................................................75

5.1 Simulation Study..........................................................................................................75

6. ACTIVE GEAR VIBRATION CONTROL SYSTEM AND EXPERIMENTAL STUDY91

6.1 Active Gear Control System ........................................................................................91

6.2 Adaptive Controller .....................................................................................................96

6.3 Actuation System Dynamics Analysis and Experimental Validation .......................100

6.4 Control Experimental Study ......................................................................................108

6.4.1 Single Gear Mesh Harmonic Control ...............................................................109

6.4.2 Vibration Control of Multiple Harmonics ........................................................117

6.4.3 High Torque Load Condition............................................................................121

6.4.4 Effect of Adaptive Linear Enhancer .................................................................125

6.4.5 Effect of Step Size ............................................................................................129

6.4.6 Effect of The Order of Secondary Path Model .................................................132

6.4.7 The Vibration At Other Locations And Gear Whine........................................136

7. CONCLUSIONS AND RECOMMENDATIONS ............................................................146

7.1 Conclusions................................................................................................................146

7.2 Recommendations for further Studies .......................................................................149

REFERENCES ......................................................................................................................151

viii
 LIST OF TABLES

Table Page

2.1: Specification of actuator parameters................................................................................. 21

3.1: Some main parameters of power re-circulating gear system ............................................ 32

4.1: Overall numerical process of the standard FXLMS algorithm......................................... 51

6.1: List of hardware for the active vibration control system .................................................. 95

6.2: Specification of power amplifier(Model E-420.00 by PI ) ............................................. 104

6.3: Vibration levels at four locations along gear mesh line-of-action (dB ref. 1g) .............. 137

6.4: Detailed Values of four cross-point transfer functions at four harmonics...................... 140

6.5: Average Phase difference between other locations vibrations and Accy ....................... 140

6.6: Control results when Accy is error signal....................................................................... 143

6.7: Control results when Acc2y is error signal..................................................................... 143

6.8: Control results when Acc3y is error signal..................................................................... 144

6.9: Control results when Acc4y is error signal..................................................................... 144

ix
 LIST OF FIGURES

Figure Page

2.1: Piezoelectric element and coordinate system ................................................................... 15

2.2: Piezoelectric stack actuator............................................................................................... 16

2.3 Effect of constant load on the maximum stroke of PZT stack actuator............................ 18

2.4 Effect of dynamic load on the maximum stroke of PZT stack actuator ........................... 19

2.5 Effect of structure load on the maximum stroke of PZT stack actuator ........................... 20

3.1: A two-degrees-of-freedom torsional dynamic model of a spur gear pair [24] ................. 24

3.2: The model of single stage geared rotor system................................................................. 26

3.3: A 2-noded beam element with transverse u z parallel to the gear mesh line-of-action,

bending rotation x and torsion y coordinates used to construct the input and output

shaft models ...................................................................................................................... 27

3.4: Used infinitesimal spring-damper gear mesh model ........................................................ 29

3.5: Dynamic mesh force spectrum of the test gear pair of interest due to 30m of

transmission error excitation............................................................................................. 34

3.6: Housing vibration due to 30m of transmission error excitation ..................................... 35

3.7: Smart gear with inertial actuators (concept 1) .................................................................. 37

3.8: Active control of gear-shaft torsional coupling (concept 2) ............................................. 39

3.9: Active control applied directly at the housing bearings (concept 3) ................................ 40

3.10: Active shaft transverse vibration control (concept 4) ....................................................... 41

x
3.11: Housing response due to a unit magnitude of actuation force .......................................... 42

3.12: Estimation result of required voltage fed into actuator..................................................... 45

3.13: Estimation result of required current fed into actuator ..................................................... 45

4.1: Proposed active vibration control system applied to a gearbox plant............................... 48

4.2: The adaptive linear combiner: (a) general form (b) transversal filter............................... 49

4.3: Structure of standard FXLMS algorithm .......................................................................... 51

4.4: Filtered LMS algorithm in Reference [71] ....................................................................... 52

4.5: Frequency domain of first element of FIR controller weights.......................................... 56

4.6: Frequency domain of output of controller ........................................................................ 56

4.7: Frequency domain of first element of FIR controller weights.......................................... 57

4.8: Frequency domain of controller output............................................................................. 57

4.9: The structure of the proposed FXLMS algorithm with online secondary path modeling

feature ............................................................................................................................... 59

4.10: The structure of the proposed frequency estimator........................................................... 63

4.11: Effect of initial frequency guess on convergence rates .................................................... 66

4.12: Comparison of results for different initial amplitude guess.............................................. 67

4.13: Result of frequency estimation for linearly time-varying frequency in the presence of

measurement signal noise ................................................................................................. 68

4.14: Result of frequency estimation for frequency jump phenomenon.................................... 69

4.15: Comparison of results predicted by the proposed frequency estimator ( ) and

Kims adaptive notch filter technique( ) .................................................................. 70

4.16: Comparison of results predicted by the proposed frequency estimator ( ) and

Kims adaptive notch filter technique( ) .................................................................. 71

xi
4.17: Comparison of estimated result for with filter and without filter case ............................. 72

4.18: Comparison of estimated result for with filter and without filter case ............................. 73

4.19: Result of frequency estimation applied to measured gearbox vibration data compared to

the true mesh frequency signal ......................................................................................... 74

5.1: MATLAB model with offline secondary path modeling feature...................................... 76

5.2: Control result using FXLMS algorithm with offline secondary path modeling............... 77

5.3: The finite impulse response of the offline identified secondary path............................... 78

5.4: The frequency response of the offline identified secondary path ..................................... 78

5.5: Control result using online secondary path modeling....................................................... 80

5.6: Amplitude and phase responses of estimated filter of secondary path ............................. 81

5.7: Control results of response with three sinusoidal components......................................... 82

5.8: Effect of amplitude ratios of the harmonic components in reference signal on the control

results ................................................................................................................................ 85

5.9: Control result of the frequency jump phenomenon jumping from 300Hz to 400Hz: (a) the

time series of the error signal, (b) estimated frequency.................................................... 86

5.10: Control result of the frequency jump phenomenon jumping from 300Hz to 320Hz: (a) the

time series of the error signal, (b) estimated frequency.................................................... 88

5.11: Control result of first two harmonics of measured data.................................................... 89

5.12: Measured finite impulse response of the secondary path ................................................. 90

6.1: Schematic diagram of the closed-loop, power recirculation gearbox setup equipped with

an active shaft transverse vibration control system .......................................................... 92

6.2: Photograph of experimental test system ........................................................................... 93

6.3: Closed-up view of: (a) actuation setup, and (b) PZT actuator.......................................... 94

xii
6.4: Enhanced FXLMS algorithm with secondary path identification and frequency estimation

modules ............................................................................................................................. 97

6.5: Designed GUI of gear active control system .................................................................... 99

6.6: Comparison of the frequency response functions of the secondary path for modal test

result combined with piezo-electric model ( ), and identification result applying

the developed control system ( ) ......................................................................... 107

6.7: Comparison of voltages at the actuator input: ( ), predicted; ( ), measured

......................................................................................................................................... 107

6.8: Frequency domain active control result of Accy (dB re. 9.8m/s2) with shaft rotational

speed of 200 rpm. (Keys: , control off; , 3rd mesh harmonic control;

, 7th mesh harmonic control)............................................................................... 109

6.9: Time domain error signal Accy amplitude at the 3rd ( ) and 7th ( ) mesh

harmonics for active control with shaft rotational speed of 200 rpm ............................. 110

6.10: Time domain response of controller output signal at the 3rd ( ) and ( ) 7th

mesh harmonics with shaft rotational speed of 200 rpm ................................................ 110

6.11: Identified secondary path transfer functions. (Keys: , 200 rpm; , 300 rpm)

......................................................................................................................................... 112

6.12: Frequency domain active control result of Accy (dB re. 9.8m/s2) with shaft rotational

speed of 300 rpm. (Keys: , control off; , 2nd mesh harmonic control;

, 4th mesh harmonic control)............................................................................... 113

6.13: Frequency domain active control result of Accy (dB re. 9.8m/s2) at the 2nd mesh

harmonic with shaft rotational speed of 700rpm (a) and 800 rpm (b). (Keys: ,

control off; , control on) ...................................................................................... 116

xiii
6.14: Frequency domain active control result of Accy (dB re. 9.8m/s2) with shaft rotational

speed of 250 rpm. Simultaneous control targets are the 4th thru 7th mesh harmonics with

amplitude ratios of targeted mesh harmonics in reference signal given by 5.5:1.04:0.5:1.

(Keys: , control off; , control on) .............................................................. 117

6.15: Frequency domain active control result of Accy (dB re. 9.8m/s2) with shaft rotational

speed of 250 rpm. Simultaneous control targets are 4th thru 7th mesh harmonics with the

amplitude ratios of targeted mesh harmonics in reference signal given by 1:1:1:1.

(Keys: , control off; , control on) .............................................................. 118

6.16: Frequency domain active control result of Accy (dB re. 9.8m/s2) with shaft rotational

speed of 230 rpm. Simultaneous control targets are 3rd thru 8th mesh harmonics.

(Keys: , control off; , control on) .............................................................. 119

6.17: Vibration response (dB re. 9.8m/s2) for simultaneous control of multiple harmonics: (a)

4th mesh harmonic; (b) 5th mesh harmonic; (c) 6th mesh harmonic; and (d) 7th mesh

harmonic. (Keys: , without control; , with control) ........................................ 120

6.18: Vibration Spectrum before control ( : low torque; : high torque) ............. 122

6.19: Identified secondary path response while system is operating at 250rpm harmonic

( : low torque; : high torque) ..................................................................... 123

6.20: Frequency domain control result for higher torque case and 3rd harmonic. ( ,

control off; , control on) ...................................................................................... 124

6.21: Frequency domain control result for higher torque case and 5th harmonic ( ,

control off; , control on) ...................................................................................... 124

xiv
6.22: Frequency domain active control result at the 4th mesh harmonic with shaft rotational

speed of 250 rpm. (Keys: , control off; , ALE off and control on; ,

ALE on and control on) .................................................................................................. 126

6.23: Frequency domain of first element of FIR controller weights (Keys: , ALE off;

, ALE on) ............................................................................................................. 127

6.24: Frequency spectrum of the controller output (dB re. 1 volt) for shaft rotational speed of

250 rpm. (Keys: , ALE off; , ALE on)...................................................... 127

6.25: Frequency domain active control result of Accy at the 5th mesh harmonic with shaft

rotational speed of 250 rpm. (Keys: , control off; , 4 order ALE; , 64

order ALE) ...................................................................................................................... 128

6.26: Frequency domain active control result of Accy at the 6th mesh harmonic for different

step sizes with shaft rotational speed of 250 rpm. (Keys: , control off; ,

=0.005; , =0.01; , =0.02; , =0.03) ............................................ 130

6.27: Time domain amplitude response of the targeted 6th mesh harmonic of housing vibration

for different step sizes . (Keys: , =0.005; , =0.01; ,

=0.02; , =0.03) ................................................................................................. 131

6.28: Mean value of FIR controller weights for different step sizes . (Keys: , =0.005;

, =0.01; , =0.02; , =0.03) .......................................................... 131

6.29: Identified secondary path impulse responses ( N=64; N=128).................. 133

6.30: Frequency domain of identified secondary path ( N=64; N=128) ............. 134

6.31: Frequency domain of control result for order 128 secondary path FIR model............... 134

6.32: Frequency domain of control result for order 64 secondary path FIR model................. 135

6.33: Frequency domain of control results( ,control off; , control on)............... 136

xv
6.34: Phase difference between Accy and Acc2y. ( ), 4th harmonic; ( ), 5th

harmonic; ( ), 6th harmonic; ( ), 7th harmonic....................................... 139

6.35: Cross-point transfer functions of the paths from actuator force output to Accy and Acc2y

......................................................................................................................................... 139

6.36: Cross-point transfer functions of the paths from actuator force output to Acc3y and

Acc4y .............................................................................................................................. 140

6.37: Spectrum of gear whine ( ) and gear housing vibration Accy ( ) ........... 142

xvi
 LIST OF SYMBOLS

e
K beam : beam element stiffness matrix

e
M beam : beam element mass matrix

K: system stiffness matrix

M: system mass matrix

C: system damping matrix

X: system response/displacement vector

D: system output coefficient matrix

Y: system output vector

Q: system modal coordinates

: system mass normalized eigenvector matrix

F: system force vector

I: identity matrix

Hjk: transfer function between k-th reference coordinate and j-th physical

respond coordinate

HhTE(z): transfer function between transmission error excitation and housing

vibration or primary path

Hhr(z): transfer function between actuator excitation and housing vibration

or secondary path

s: Laplace variable

xvii
 x j: displacement of j-th coordinate

Fk: k-th element of force vector

: element of eigenvector matrix

i: modal damping coefficient of i-th nature mode

i: i-th nature frequency

Fmesh: dynamic mesh force

Dmesh: dynamic mesh stiffness

S hr (z ) : identified secondary path

r(n): reference signal

r (n) : filtered reference signal

y(n): output of adaptive filter

wi: i-th weights of adaptive filter

: learning rate of FXLMS algorithm

LS1, LS2: driven and driving shaft length

LlS: large shaft length

RS: shaft radius

RlS: large shaft radius

W f: gear face width

Kbx: bearing stiffness

L: element length

I: area moment of inertia

Im: motor inertia

Id : load inertia

xviii
Ip, Ig: pinion and gear inertia

Ic: coupling inertia

J: polar moment of inertia

E: modulus of elasticity

G: shear modulus

m: beam element mass

R: beam element radius

uz: transverse direction coordinate of beam element

x: bending rotation coordinate of beam element

y: torsional coordinate of beam element

Km: gear mesh stiffness

Cm: gear mesh damping coefficient

Rp, Rg: pinion and gear pitch radii

e(t): transmission error excitation

yh,TE: housing displacement due to transmission error excitation

yh : housing total displacement

mp, mg: pinion and gear mass

mh: housing mass

mc: coupling mass

t: time

n: discrete time index

1, 2: estimated frequency and amplitude

Acc, Acc2: acceleration signals of four tri-axial accelerometers

xix
Subscripts

p: pinion

g: gear

h: housing

r: actuator

TE: transmission error

mesh: gear mesh

c: coupling

s: shaft

y: y-direction of tri-axial accelerometer

Superscripts

T: transpose vector

e: element

xx
 CHAPTER 1

INTRODUCTION

1.1 Introduction

Gears are essential parts of many precision power and torque transmitting machines.

However, they do generate tonal noise (whine) that is often considered highly undesirable and

annoying. In very severe cases, the gear vibrations can reduce life and performance of the power

transmission systems. Typical gearbox vibration and noise measurements contain several

narrowband tonal signals that are mixed in with a broadband response. The tonal signals are

basically the fundamental gear mesh frequency and its corresponding harmonics generated from

the perturbation in the gear meshing action, which are the primary cause of the annoyance

problem. On the other hand, the broadband signatures are background machine acoustic noise

that is usually not troublesome unless it becomes extremely high. Addressing the gear whine

concern is the main purpose of this research. Although the fundamental gear whine problem has

been of interest to many researchers for a very long time [1-4], the work on the active gearbox

vibration control are still quite limited. Hence, the intent here is to develop and test a new, more

effective method for reducing the annoying tonal gear mesh response by employing a unique

active vibration control approach that will be described further in latter part of this dissertation

proposal. In most active noise and vibration control applications, there are several major

approaches available, such as vibration transmission path control, excitation source control and

receiver treatment. The first two concepts have been of interest in gear vibration control

1
 2

applications. Since the vibration control at the source can potentially achieve global vibration

reduction, it is potentially the best choice to use among the above three methods. This

dissertation research will attempt to demonstrate the ability to control the gearbox vibration by

applying internal active actuation scheme directly at the vibration source. An advantage of the

source control approach compared to the previous attempts is the increased potential of reducing

both the structure-borne vibration and air-borne radiated sound pressure responses. The success

of this method largely depends significantly on gaining an understanding of the physics of the

gearbox excitation and vibration transmission mechanisms, and developing suitable actuators,

sensors and active control laws. Thus, a good system model including gear pair system and

actuator should be developed for analyzing the physics of gearbox dynamics. Furthermore, a

suitable actuation concept will be designed from the analysis of the dynamics of gear pair

system. As a result, the gear pair system model and control strategy such as adaptive algorithm

and actuation configuration should be proposed and analyzed in this research.

In the following subsections, the literature is reviewed.

1.2 Available Active Gear Vibration Control Methods

Although numerous applications of smart material systems for vibration control exist [5,

6, 7, 8], many of them only deal with controlling the response of individual modes that are often

less than a few hundred Hertz of simple structures, such as a beam or a plate. However, for

many machinery structures, the modal densities are often quite high especially in the mid and

high-frequency ranges. In these frequency ranges, the individual modes can be difficult to

identify and control. These difficulties lead to the lack of applications of vibration control in

practical industrial system. However, for a gearbox system, unlike the system described above,

the response spectrum is dominated by the mesh frequency (the product of the number of the
 3

teeth and shaft speed) and the well-separated harmonics that can span over a wide frequency

range in spite of the higher modal density characteristic. Furthermore, even though the gearbox

system can sometimes exhibit nonlinear and time-varying characteristics under light load

conditions [9], when the transmitted load is sufficiently high, the gear drives often behave quite

linearly in most applications. The discrete nature of the spectral peaks makes this complex

system a potentially suitable candidate for active control using smart material systems.

In 1999, Rebbechi, Howard and Hansen [10] presented a method for performing active

control of a gearbox by actively isolating the vibration between the shaft and housing via a pair

of magnetostrictive actuators mounted at one of the support bearing locations. It was

demonstrated that the response of the fundamental gear mesh frequency and its first two

harmonics could be simultaneously reduced. A reduction of 20 to 28 dB was achieved at the

fundamental frequency, while reductions of only 2 to 10 dB were achieved at the harmonics.

More recently, Chen and Brennan [11] proposed a control scheme that used three

magnetostrictive actuators mounted directly on the gear to produce circumferential forces for

suppressing the torsional vibrations. The experimental results showed about 7 dB of reduction in

gear angular vibrations at the tooth meshing frequencies between 150 and 350 Hz. Other than

these studies, no other recent work on the active vibration control of internal gearbox

components is found in the public domain literature. There are several related studies that

concentrated on active vibration control applied to the structural support system outside of the

gearbox [13, 14, 15] such as the gearbox strut. From these few available studies, it can be

concluded that active gearbox vibration control has not been studied thoroughly. In fact, the

application is very much at its infancy. Seeking for improved understanding to this problem and

a more robust solution is the basis of this dissertation research.

 4

1.3 Available Control Algorithms for Periodic Disturbance

From the viewpoint of active vibration control laws for suppressing periodic disturbance

in gear pair systems, there are fundamentally two available control laws. One is the filtered-x

LMS algorithm that is also used by Rebbechi, Howard and Hansen [10], and the other is a

frequency domain adaptive harmonic controller employed earlier by Chen and Brennan [16]. The

filtered-x LMS algorithm is in fact one of the most widely used control algorithms in real world

applications, but in this context, it requires a reference signal that is highly coherent with the gear

mesh frequencies. As a matter of fact, Rebbechi, Howard and Hansen [10] were unable to control

the response at higher-order gear mesh harmonics due to this problem. Fundamentally, the

problem is mainly due to the presence of frequency estimation errors that increase as the order of

the harmonic rises. Hence, the correlations between the reference signals of the higher order

harmonics generated by the multiplier and the true mesh frequencies are quite low. These errors

tend to impede the ability of the controller to attenuate vibration efficiently. To address this

limitation, the authors suggested using a finer measurement scale in combination with a divider

to generate more accurate reference signals at the higher order harmonics. Nevertheless, their

approach still relies on producing highly precise reference signals of the fundamental gear mesh

frequencies.

Frequency domain adaptive harmonic controller that is especially designed for nonlinear

applications [16] was also used to suppress the vibration of a gear pair system [11, 17]. This

adaptive harmonic controller only works for constant mesh frequency cases and requires the

identification of plant sensitivity arrays at the critical mesh harmonics [16]. The algorithm also

relies on having accurate gear mesh frequency data like in the case of the filtered-x LMS

algorithm noted above, and may be further limited by the fact that the reference signal and
 5

sensitivity arrays are time-varying in practice due to either variation in gear speed and/or mesh

stiffness. Hence, it is not particularly suitable for the present application.

In spite of this lack of active gear vibration control investigations, there exist several

algorithms to control periodic disturbances for generic mechanical plants. Some of these

techniques include classical feedback control [18], LMS-based adaptive feed-forward control

[19], higher harmonic control [20], linear quadratic gaussian (LQG) and H? [21], and model

reference-based adaptive control (MRAC) [22]. However, these existing algorithms either

require a detailed model of the whole control system, or require exact information of the gear

rotation speed or the fundamental gear mesh frequency. A detailed discussion of the control

algorithms proposed for this research is given in Chapter 4.

1.4 Available Smart Actuators

Because many conventional actuators, such as electromagnetic and hydraulic shakers, are

massive and have a limited frequency bandwidth, actuators made from smart materials have

undergone rapid development recently [23]. With recent technological advances in material

science and control theory, many forms of smart materials have emerged for use as advanced

sensors and actuators. These materials are able to overcome some of these drawbacks of

conventional sensors and actuators, such as size, that are not convenient to implement. Typical

forms of smart materials include: piezoelectric elements, magnetostrictive materials, shape

memory alloys, electrorheological fluids, and fiber optics, which have been successfully applied

in a variety of fields.

Actuators made from piezoelectric elements and magnetostrictive materials typically

have a large frequency bandwidth and are able to generate a large dynamic force. As a result,

these two kinds of actuators are often applied in active vibration control of simple structure, such
 6

as beam and plates [5, 6, 7, 8]. As noted above, magnetostrictive actuators have been used in the

gear vibration control [10, 16]. However, magnetostrictive actuators are typically bulkier than

piezoelectric actuators although the former can generate larger forces. One the other hand, other

types of actuator made from shape memory alloys or electrorheological fluids have a relatively

low frequency bandwidth. Consequently, piezoelectric actuators are selected for this research. A

more detailed discussion of these types of actuators is presented in Chapter 2.

1.5 Summary

As mentioned above, the use of active vibration control in gearbox systems is very

limited. The lack of applications may be due to the facts that: (a) the physical mechanisms of the

gear mesh excitation and vibration transmission control are not well established [2, 24,]; (b) the

bandwidth of the gearbox system dynamics is large, up to several thousand Hertz; (c) it is not

easy to find an actuator setup which has enough force and bandwidth; (d) the model of the smart

material actuators is not well defined [25]; (e) the gear system dynamics can be a time varying

due to variation of rotation speed and gear mesh stiffness; and (f) the control algorithm has to

compensate between the control performances and DSP hardware computational limitations [19].

These current limitations provide a need for further development of an active gear vibration

control system. In the following chapters, a simple model of piezoelectric stack actuator, proper

actuation concepts, proper control laws and the proposed experimental validations are discussed.

Note that the research discussed in this proposal has potential applications in the rotorcraft

industry, as well as in aerospace, naval, and automotive vehicle systems.

The organization of the rest of this dissertation proposal is given as follows:

Chapter 2 presents a detailed overview and applications of smart materials, such as the

piezoelectric and magnetostrictive actuators. A simple lumped parameter model of a

 7

piezoelectric actuator is also presented in this chapter. This model will be used to guide the

selection of actuators and amplifiers.

Chapter 3 presents a dynamic finite element (FE) model of the gear pair system used to

understand the vibration generation and transmission mechanisms, and also for demonstrating

the impact of the proposed active vibration control method. The four potential actuation

concepts for the gearbox system will be introduced there. To select a proper actuator, the power

requirements of actuator and amplifier based on the proposed FE gear pair system model and

piezoelectric actuator dynamic model are predicted. Through this analysis, the active shaft

transverse vibration control method is adopted for further investigation in this dissertation

research.

Chapter 4 presents a filtered-x LMS based control law for the proposed active shaft

transverse vibration control. The filtered-x LMS algorithm is one of the most common control

algorithms in real applications. However, in this context, the identification of the secondary path

has to be sufficiently precise and reference signals that are coherent with the gear mesh

frequencies are readily available. To overcome these two drawbacks, two enhancements are

applied. The first enhancement uses an offline/online secondary path identification using

additive random noise method. The second one applies an online frequency estimator for

estimating the gear mesh frequency directly while adding a reference signal synthesizer to the

control algorithm.

In Chapter 5, a series of computer simulation studies are performed in order to guide the

experimental work.

Chapter 6 discusses the result of experimental study where the designed active control

system is applied to a high density, power recirculating gearbox system.

 8

Chapter 7 gives the conclusion and suggestion for future potential research work.
 CHAPTER 2

SMART MATERIALS and ACTUATOR MODELING

In this chapter, the existing smart materials and actuators that potentially can be used for

gearbox applications are reviewed and discussed. There are several typical smart materials, such

as: magnetostrictive materials, piezoelectric elements, shape memory alloys, electrorheological

fluids [26]. In the following sections, general properties as well as applications of relevant smart

material systems are briefly discussed. Then, one piezoelectric stack actuator is considered in

detail and modeled as a lumped mass - spring system for the mechanical part and a lumped

capacitor for the electrical part.

2.1 Reviews of Smart Materials

2.1.1 Magnetostrictive Materials

Magnetostrictive materials are the substances that can generate strains upon application

of magnetic field. Conversely, a magnetic field is generated when the material is stressed. This

field, however, is proportional to the materials rate of strain [27, 28]. Although

magnetostrictive materials have been extensively studied, few practical applications exist

because these materials do not generally produce large strains and forces like piezoelectric and

electrostrictive materials for the same level of power consumption. This fundamental

disadvantage has been recently changed with the development of the so-called giant

magnetostrictors, namely Terfenol-D. This material is an alloy of iron, and the rare earth

elements of Terbium and Dysprosium. These magnetostrictive materials are able to offer strains

9
 10

an order of magnitude more than conventional piezoceramics with comparable force outputs.

Therefore, one of the primary advantages of a magnetostrictive actuator is that it can generate

very large strains in comparison with those created by piezoelectric actuators. However, the

constitutive relationship between applied magnetic field strength and generated strains is highly

nonlinear and exhibits significant hysteresis. Furthermore, magnetostrictive actuators are larger

in size and more massive than the equivalent piezoelectric actuators.

Magnetostrictive devices have a wide range of applications include vibration suppression,

micropositioning, damage mitigation and shape control. Preliminary results show that simple

analog control systems consisting of Terfenol-D actuators can significantly reduce flexural

vibrations in a rotating machine tool shaft [29]. Recently, three magnetostrictive inertial

actuators were clamped to a helicopter gearbox support strut to minimize the kinetic energy of

vibration of the transmitting structure [14].

2.1.2 Piezoelectric Elements

The piezoelectric effect is that certain crystalline material produces an electrical charge

on its surface when it is subject to a stress field, which was first discovered in 1880 by Pierre and

Currie [30]. It was also demonstrated that the converse effect, when an electric field is applied to

a piezoelectric material, its shape and size change, is also true. Piezoelectric materials generally

used in smart structures come in two forms, namely piezoceramics and piezoelectric polymers.

The piezos ability to actuate the structure is a function of its stiffness, electromechanical

coupling coefficients, flexibility, and applied voltage limitations. Piezoceramics, such as Lead-

zirconate-titanate (PZT), have large stiffnesses and mechanical coupling coefficients. Thus, they

are used as actuators. And, polymer types are usually applied as sensors because of their high

sensitivity to strain characteristics. It is common to make a piezoelectric stack by bonding two

 11

or more pieces of piezoelectrics together to produce larger sensitivity. A piezoelectric actuator

(PZT) can generate a large force up to 10 KN, offers very good response time (microsecond time

constant), and consume relatively low power.

Piezo-actuators are applied as either distributed form or stacked form. For distributed

form, two configurations are basically used: surface bonded to the host structure, and embedded

in the structure. For both configurations, the position of the actuator, the orientation of the patch

polarization, and the polarity of the input determine whether the induced loading is pure bending

or axial extensional / compressional loading. As an example, if only one surface of the host

structure is bonded with a piezo-actuator (i.e. a unimorph configuration), then it will exhibit a

bending effect. The unimorph configuration has been used in speakers for noise production and

cancellation applications. If both sides of host surface are bonded using a pair of piezo-actuators

with opposing polarity (a bimorph configuration), then the structure will experience axial

extensional or compressional loading effect. On the other hand, embedding the actuator allows

the designer to place it exactly where it is needed to get a desired effect. However, this design

can limit the service temperature of the structure, interrupt the load path, and prevent service or

repair work. As for the piezo-stack, they contain multiple layers of piezoceramics, which are

coupled mechanically in series and connected electrically in parallel. The stack are capable of

producing kilo-Newtons of force with only microns of expansion when connected with an

inertial mass or contained in a stiff structure. Their response time is around ten microseconds.

Piezoelectric elements have been used as actuators and sensors in a wide range of

applications such as optics, photonics measuring technologies, disk drives, microelectronics,

precision mechanics, vibration and noise control, and even life science and biology [23]. They

have been successfully embedded [31] in composites, and surface mounted on both composites
 12

and other substrates [32]. Typical composite applications include vibration damping [33],

vibration isolation [34], and acoustic control [35] due to the adaptability and lightweight

characteristics. In addition, the ability to be easily integrated into structures makes the

piezoelectric elements very attractive for structural control since no moving parts are needed

unlike conventional actuators.

2.1.3 Shape Memory Alloys (SMA)

Shape memory alloys are a special class of adaptive materials that convert thermal energy

directly into mechanical work. A variety of alloys, when properly mixed, exhibit this conversion

effect by heat treatments. Such an effect is accomplished through a phase change and can impart

large strains. The heat may be applied by numerous means including electrical resistive heating.

However, the frequency bandwidth of the actuator is limited by the speed at which it can be

cooled between states.

Common SMA forms are wire, ribbon, sheets and springs. They can also be conditioned

to go between two preset shapes to produce two-way shape memory effect. The advantages of

SMAs are: a) high power density (ratio of actuator output to actuator volume); b) superelastic

property of Nitinol(the most common type of shape memory alloys), which allows recovery from

strains of up to 6 percent without being heated; and c) excellent cycling performances in repeated

loading. As the result, SMAs have been used for applications like eyeglass frames, coffeepot

thermostats, satellite release devices, electrical couplings, robot, position control and vibration

control. However, SMA-based actuators have some disadvantages including low frequency

bandwidth, large non-linear hysteresis, and difficulty in developing good dynamic models.
 13

2.1.4 Electrorheological Fluids (ER)

Electrorheological fluids, which were first investigated by Winslow in 1949 [23], exhibit

a coupling between their fluidic and electrical properties. Typical ER fluid consists of

suspension of fine semi-conducting particles in a dielectric fluid. Cellulose, cornstarch grain,

alumina powder and silica gel have all been used as particles in ER fluids. Without an applied

electric field, the ER fluid behaves like a Newtonian fluid. However, once a sufficient electric

field is applied, the fluid undergoes a dramatic change, inducing a Bingham fluid-like behavior.

The most significant change is associated with complex shear modulus of the material.

Therefore, ER fluids are used in vibration-suppression application where variable damping

characteristics are needed to effectively control the response.

Most ER material applications can be categorized into two classes [26]: controllable

devices and adaptive structures. Controllable devices applying ER studied include valves,

clutches, engine mounts, brakes, dampers or shock absorbers, and damping of helicopter blades

[25]. On the other hand, adaptive types possess tunable properties by incorporating controllable

ER material [36] or damper [37]. Yao [38] studied a disk type ER damper to actively suppress

the vibration of a rotor system theoretically and experimentally. He showed that the ER damper

can suppress large vibration amplitudes around the critical speed, dampen any sudden

unbalanced response, shorten the transient process, and reduce the amplitude of the steady-state

response. In spite of these successes, one of the main drawbacks of ER fluids, which keep them

from being used in many practical applications, is the need for very high voltage levels.

Furthermore, the ER response is temperature dependent as well as lacks long-term stability.

 14

2.1.5 Summary

Among the above four types of smart materials, the piezoelectric elements and

magnetostrictive materials have the fastest response time, and their effective frequency range can

reach up to 10 kHz. Furthermore, they can provide large static and dynamic forces (up to 10 kN).

In contrast, SMA possesses the slowest response time. Although ER fluids have reasonably high

response time, they are difficult to integrate into a gear system due to lack of structural strength.

Since gear vibrations are dominated by high frequency tones, piezoelectric elements and

magnetostrictive materials appear to be the most suitable candidates for active gear vibration

control application. Accordingly, the model of piezoelectric element is discussed further next.

2.2 Model of Piezoelectric Stack Actuator

Piezoelectric elements have been studied and applied for active control applications

extensively [29, 31, 32, 33, 34, 39]. However, there are relatively few papers addressing the

modeling of piezoelectric stack actuator. A simple lumped parameter linear model for the

piezoelectric stack actuator based on the impedance model from Flint and Liang [40] is

presented. It is known that if the applied electrical field density is small, then the piezoelectric

element can be treated as a linear system. However, if the electrical field density is too high,

piezoelectric elements may exhibit large non-linear hysteresis [23].

The relationship between the applied force and resultant response of a piezoelectric

material depends upon a number of parameters, such as shape, composition, properties, electrode

position, poling axis location (that requires a set of suitable coordinate system be defined), and

the applied field polarity. Figure 2.1 shows an element of piezoelectric material and a coordinate

system [23]. Three axes are used to identify the directions of the piezoelectric element, namely
 15

1, 2 and 3 corresponding to x, y and z coordinates, respectively. The z direction is assumed

parallel to the direction of polarization which is represented by vector p.

3(z)

Applied
+ Field E 2(y)
V p

1(x)

Figure 2.1 Piezoelectric element and coordinate system.

Under an applied electric field that is sufficiently small, the mechanical strain of the

piezoelectric element reacts proportionally to the electric field density to produce a constant

mechanical stress. This is characterized by the piezoelectric strain coefficient denoted by dkp

(see [41]), which is defined as the ratio of free strain to the applied electric field E. The first

subscript in the strain constant gives the direction of the electric field associated with the voltage

applied or the charge produced, while the second subscript gives the direction of the mechanical

strain. The constant dkp can also be viewed as a relationship between the charge on the electrode

and the applied mechanical stress. In particular, the strain constants d33, d31 and d32 are of

primary interest. In fact d31 = d32 = -d33, where v is Poisson ratio, since the piezoelectric

material can be assumed to behave identically in the 1 and 2 directions.

A piezoelectric stack actuator shown in Figure 2.2 is considered here. The actuator is

constructed of by n layers of piezoelectric elements in mechanical series but electrical parallel.

Each layer has a thickness of t and the total height of the actuator is L. The polarization direction

is parallel to direction 3.
 16

Figure 2.2 Piezoelectric stack actuator.

Due to the rod-type configuration of the actuator shown in Figure 2.2, only the

extensional stress along the length of rod, that is direction 3, exists. Thus, the constitutive

equations of this actuator are given by

S 33 = s33E T33 + d 33 E3 , (2.1a)

D3 = d 33T33 + 33
T
E3 , (2.1b)

where S33 is the strain along the rod, T33 is the stress, is the complex compliance at zero electric

field E3 , d33 is the piezoelectric constants, 33

T
is the complex dielectric constant at zero stress

and D3 is the electric displacement which is given by the electric charge q over cross area A.

But, this set of equations is used in the micro-crystal form and not easy to use directly in the

model. However, if the working frequency of the actuator is far less than the first resonant

frequency of the actuator [40, 42, 43], the strain of each PZT layer can be assumed to be same.

Applying equation (2.1a) and (2.1b) to each layer and summing them up can yield the macro

equation of the piezoelectric stack actuator. Thus, for externally applied force: F = AT3 , the

electrical charge: q = AD3 , externally applied voltage: V = EL , equivalent piezo-stiffness:

 17

k p = A / s33E L and equivalent piezo-capacitance: C ps = nA 33s / t , the above equations can be

transformed into

F = k p x d 33 k pV , (2.2a)

q = d 33 F + C ps V , (2.2b)

Note that the second term of right-hand-side of equation (2.2a) can be defined as block force of

the actuator and also can be approximated as the multiplication of its stiffness and maximum

stroke

Fb = d 33 k pV k p l , (2.3)

where l is the maximum stroke of actuator. Actually, the block force Fb is the maximum output

force that the actuator can generate when it acts on a rigid structure with infinite impedance.

After obtaining the above fundamental equations of piezoactuator system, it is also

valuable to view the effect of external loads on the maximum stroke of the actuator can achieve.

Three different external loads will be considered here.

1). Mass load

The simplest external load is applying a mass on top of the actuator. For this case,

solving equation (2.2a) will get the instantaneous displacement of the actuator

x = d 33V + F / k p , (2.4)

Note that the second term of right-hand-side F / k p is constant, so the maximum stroke of the

actuator, L0 , is the same as the zero-applied force case. Compared with the free zero-applied

force case, the instantaneous displacement x shifts by an offset = F / k p = L N , which is

demonstrated in Figure 2.3.

 18

Figure 2.3. Effect of constant load on the maximum stroke of PZT stack actuator.

2) Spring load

When the external load is not constant or a mass, such as a spring load, then the

externally applied force is related to the instantaneous displacement of the actuator. Assume a

spring element is connected to the actuator as shown in Figure 2.4, and the spring stiffness is

k ext . In this case, the externally applied force is

F = k ext x , (2.5)

Substitution of the above applied force into equation (2.2a) yields

kp
x = d 33V , (2.6)
k p + k ext

As seen in equation (2.6), the maximum stroke of the actuator will be reduced. For a

stiffer external spring, the maximum stroke of the actuator will be smaller. This effect is shown

in Figure 2.4. If the external spring is infinitely rigid, then, the actuator cannot present any

stroke at all and will output the block force to external structure. If actuator is free of end, it will

generate zero force.

 19

Figure 2.4 Effect of dynamic load on the maximum stroke of PZT stack actuator.

3) Complex structure load

For a more complex external load, such as when the actuator is acted on by a complex

host structure, the equivalent dynamic stiffness can be used. For example, when the actuator is

applied on a shaft structure, as shown in Figure 2.5, then the driving point stiffness can be used

as the equivalent dynamic stiffness. Note that the equivalent dynamic stiffness might be a

complex value, which means the displacement response will have a different phase relative to the

force excitation. The equivalent dynamic stiffness can be calculated from experimental results or

a finite element modeling. One can still use equation (2.5) to calculate the external applied force

to the actuator. Consequently equation (2.6) can be used to calculate the displacement response

of the actuator. Substitution of equation (2.6) into equation (2.5) yields the effective force

applied on the host structure

k ext k ext
F = k p d 33V = Fb , (2.7)
k ext + k p k ext + k p

where block force Fb is defined in equation (2.3). From the above equation, the effective force

applied on the host structure depends on the stiffness of the host structure and the effective

piezo-stack stiffness. The stiffer the host structure is, the more force the actuator can generate.
 20

Shaft structure

kext
kext
Piezo-
actuator x kp
Fb

Figure 2.5 Effect of structure load on the maximum stroke of PZT stack actuator.

Furthermore, the electrical charge applied to the actuator can be calculated by equation

(2.2b). Substitution of equation (2.7) into equation (2.2b) yields

k ext
q = C ps [1 k 2 ]V , (2.8)
k ext + k p

where k 2 = d 332 / 33
T E
s33 is the piezo coupling coefficient, which is always less than unity. From

equation (2.8), the effective capacitance of the piezo-actuator is defined as

k ext
C p = C ps [1 k 2 ], (2.9)
k ext + k p

Taking a time derivative of equation (2.8) and substituting equation (2.9), the electrical

current I applied to the actuator is found as

I = i C p V , (2.10)

where i is unit imaginary number and is the angular frequency of the applied voltage. It

should be known that piezoelectric elements can only bear compressive force and not tensile

force. Thus, in order to apply the piezo-stack actuators into a dynamic application, the actuator

must be preloaded.
 21

Table 2.1. Specification of actuator parameters

(Model PSt1000/16/40 VS 25 by Piezomechanik GmbH [44]).

Parameters Numerical Value (Unit)

Operating Voltage, V 0 1000 V

Maximum Stroke, l0 40 m
Actuator Length, L 53 mm
Actuator Diameter, D 25 mm
Capacitance, C0 360 nF
Stiffness, kp 200 N/m
Resonance, f0 30 kHz

This actuator model of complex structure load will be used to calculate the required

voltage and current for active gear housing vibration control in Chapter 3.
 CHAPTER 3

GEAR PAIR SYSTEM MODELING AND ACTUATION SYSTEM

This chapter discusses the development of the gear pair system model and the ensueing

dynamic analysis performed. The dynamic model is used to assist in the design of the

experimental setup, to gain a better understanding of the nature of gear response due to

transmission error excitation, and to help in the design of the control algorithm. The simulation

conducted is intended to determine the effectiveness of the proposed active control concept and

to compute the required actuator parameters. First, a brief review of existing gear dynamic

models is given. Second, a finite element (FE) model is presented. Third, the proposed actuator

configuration for the gearbox system is introduced. At last, the electrical power requirements

based on gear system model and piezo-stack actuator model are presented.

3.1 Review of Existing Models

Analytical work on the dynamics of gear pair systems is abundant. This includes a series

of mathematical dynamical models that have been developed for different purposes. Reference

[24] groups the existing models into 5 different classifications based on the specific purpose of

the models as well as the dynamic characteristics of the primary elements in the system. The

classifications are (a) simple dynamic factor models; (b) models only with tooth compliance; (c)

gear dynamics models that include the flexibilities of other elements as well as the tooth

compliance; (d) geared rotor dynamics models in which the transverse vibrations of gear-

carrying shafts are considered in two mutually perpendicular directions, and thus allowing the

22
 23

shaft to whirl; and (e) torsional vibrations models in which the flexibility of gear teeth is

neglected but the torsional flexibilities of the shafts are included.

The classifications (a)-(d) are based on the work by Tuplin [45] in the 1950s, which are

of interest to the present study. In his work, Tuplin introduced a simple spring-mass gear

dynamic model that can be used to estimate dynamic factors at frequencies well below the gear

mesh resonance mode. His theory forms the basis for modern gear dynamic models used

extensively today. Figure 3.1 shows a typical model comprising of a gear mesh stiffness element

that forms the coupling between the 2 rigid gear bodies. Using this concept, Ozguven and

Houser [46] developed a torsional model to analyze the dynamic behavior of spur gear pairs due

to loaded static transmission error excitation. In addition, Amabili and Rivola [47] applied a

similar model with time-varying gear mesh stiffness characteristic to compute the steady-state

response and to examine the stability of a spur gear pair. The calculations are made assuming

viscous damping that is proportional to the mesh stiffness. In a subsequent paper, Amabili and

Fregolent [48] reported the use of this model to identify critical modal parameters and gear errors

from gear vibrations. Other more recent studies that applied the same class of models are by

Zhang and Rescigno [49] who used a four degrees-of-freedom dynamic model assuming rigid

bearings to estimate the impact load on gear tooth due to torque pulsations. Cricenti et al. [50]

used a six degrees-of-freedom model that included both translation and torsion coordinates to

perform both static and dynamic transmission error calculations, and Lim and Li [51] developed

a general multi-mesh counter-shaft transmission model for analyzing coupled torsional and

translational response. Using essentially the same type of models, a team led by Houser [52, 53]

examined the role of sliding friction in gear dynamics computationally.

 24

yg

Km
yp
Cm e(t)
~

Figure 3.1. A two-degrees-of-freedom torsional dynamic model of a spur gear pair [24].

Steyer and Lim [54] also formulated a theory to relate the dynamic mesh stiffness to the

driver and driven compliances at the mesh. The concept enables one to determine the controlling

factor and compute the sensitivity of mesh force to drivetrain system dynamics. Using this

formulation, Donley, Lim, and Steyer [55] developed a unique scheme to compute the effect of

transmission error excitation in the context of a general-purpose finite element program.

Kahraman et al. [56] also extended the basic gear modeling concept to devise a more complex

finite element model that considers the effect of rotary inertia of the shafts, bearing flexibility,

and gear mesh stiffness to study the dynamic coupling between torsional and transversal motions

of the gears. Other more sophisticated finite element models have been proposed by Parker et al.

[57], Yakhou et al. [58, 59] and a team of researchers led by Velex [60-63]. Some of these finite

element models utilized more sophisticated mesh models compared to the one shown in

Figure 3.1. For example, the one proposed by Parker et al. [57] actually simulates the detail gear

engagement process by combining finite element and contact mechanics. On the other hand, the

one used by Velex relies on a discretization of the mesh stiffness along the line of contact.

Another class of geared system model is generated, by applying the finite element

method. In this research, a finite element model with lumped parameters of gear mesh model is

developed to model the single-stage geared rotor system. The widely used gear mesh modeling
 25

concept, applying a unidirectional lumped stiffness element, is adopted. In this study, the finite

element approach will be employed. The formulation of the dynamical equations of motion is

presented next.

3.2 FE Model of Gear Pair System

A cross-section of the power re-circulating gear system considered in this study is shown

in Figure 3.2. This system consists of two pair of gears, a belt driving motor, a torque

transducer, 3 shaft-to-shaft couplings, two additional support bearings at two sides of flywheel

on driven shaft, and a series of driver and driven shaft segments. The main noise and vibration

source of interest to this study is the gear transmission error at the mesh of test gear set. It is

commonly known that the transmission error excitation interacts with the system dynamic

characteristics to generate a dynamic mesh force that in turn excites the rest of the gearbox

structure. The resulting vibrational energy is transmitted through the shafts and bearings and

into the housing where some of this energy is ultimately radiated as high-frequency gear whine

noise. Since the frequency of the gear tones are significantly higher than the shaft rotational

speed, they can be easily monitored and identified. This frequency identification ability is

essential for the active vibration control implementation considered here.

 26

Housing

Pinion Coupling Bearing (? 4) Pinion

Motor

Torque
transducer

flywheel Coupling Gear

Gear

Figure 3.2. The picture and model of power re-circulating gear system.

A dynamic FE model of the gear pair system is first developed for use in examining the

proposed actuation concept. The model, which is constructed from beam elements and lumped

masses and springs, provides an analytical representation of the critical gear rotational and

translational vibration characteristics. The shafts are assumed flexible in the transverse and

torsional directions. The longitudinal degree of freedom is neglected since no significant axial

excitation is expected from the spur gears and to avoid unnecessary complexity in the analysis

[64]. Moreover, the natural frequency associated with the longitudinal vibration is much higher

than the frequency range of interest. Accordingly, each nodal point on the discretized shaft is

only defined by three degrees-of-freedom (DOF) that include transverse, bending rotation, and

torsion coordinates as shown in Figure 3.3. The beam element stiffness matrix formulation can

be shown to be
 27

12 K 1 6 K1 L 0 12 K 1 0
6 K1 L
6K L 4 K1 L 2
0 6 K1 L 2 K1 L 0
2
1

0 0 Ks 0 0 Ks
e
K beam = , (3.1)
12 K 1 6 K1 L 0 12 K 1 6 K 1 L 0
6 K1 L 2 K 1 L2 0 6 K 1 L 4 K 1 L2 0

0 0 Ks 0 0 K s

3
where L is the element length, K s = GJ L , K 1 = EI L , E is the modulus of elasticity, I and J

are the area and polar moments of inertia, and G is the shear modulus. The corresponding mass

matrix is

156 22L 0 54 13L 0

22L 4 L2 0 13L 3L 2
0

m 0 0 70R 2 0 0 35R 2
e
M beam = , (3.2)
420 54 13L 0 156 22L 0
13L 3L2 0 22L 4 L2 0

0 0 35R 2 0 0 70R 2

where m is the element mass and R is the shaft radius. These matrices correspond to the

coordinate vector {u z1 , x1 , y1 , u z 2 , x 2 , y 2 }T .

z
uz
y
z y
2
uz
y x x
y
1

x x
Figure 3.3. A 2-noded beam element with transverse u z parallel to the gear mesh line-of-
action, bending rotation x and torsion y coordinates used to construct the input and output
shaft models. The nodes are labeled as points 1 and 2.
 28

The gear mesh kinematics is modeled using a concept originally proposed by Tuplin [46],

which has been widely used by many gear researchers [24, 46, 50]. The model consists of an

infinitesimal spring-damper element positioned in series with the transmission error excitation

e(t) at the mesh point as shown in Figure 3.1. The mesh model couples the translational

coordinates of the gear and pinion centroids along the line-of-action. Additionally, the bending

rotation and torsion motions of the gears are considered. Hence, each gear body is represented as

a lumped mass element whose instantaneous position is defined by one translational and two

rotational coordinates expressed as uz, x, and y. The proposed gear mesh model is shown in

Figure 3.4. For the combined gear-pinion system, the position vector is

{u zg , xg , yg , u zp , xp , yp } , where the subscripts g and p represent gear and pinion, respectively.

The corresponding mesh stiffness and gear mass matrices are

Km 0 K m Rg Km 0 K m Rp
0 0 0 0 0 0

K m Rg 0 K m Rg2 K m Rg 0 K m Rg R p
K e
= , (3.3)
Km
mesh
0 K m Rg Km 0 K m Rp
0 0 0 0 0 0

K m R p 0 K m R p Rg K m Rp 0 K m R p2

m g 0 0 0 0 0
0 I xg 0 0 0 0

0 0 I yg 0 0 0
M emesh = , (3.4)
0 0 0 mp 0 0
0 0 0 0 I xp 0

0 0 0 0 0 I yp

where Km is the gear mesh stiffness, and (Rg, Rp), (mg, mp), (Ixg , Ixp), and (Iyg , Iyp) are the radii,

masses, and mass moment of inertias of the gear and the pinion about the x and y directions,
 29

respectively. Note that Equations (3.3) and (3.4) are similar to the one proposed by Lim and

Singh [65]. Here, our model has been extended to include bending rotation.

yg

uzg xg Km
yp
Rg uzp
Cm e(t) xp
~
Rp
Figure 3.4. Used infinitesimal spring-damper gear mesh model.

The belt driving motor is modeled as lumped mass moments of inertia, while the bearings

are formulated using lumped spring elements having the ability to resist motion in the transverse

(parallel to the line-of-action) and bending rotation directions. The housing structure is

represented as a 1-DOF rigid body (lumped mass element) connected to the gear train at the 4

bearing locations via a set of bearing stiffnesses. This rigid body housing approximation does

not pose significant limitation to the simulation work, as the focus of this study is to control the

overall housing vibrations via the internal gear-shaft dynamics. In fact, this simplified housing

model eases simulation efforts. The assembled system model contains 51 finite elements

including 12 lumped masses, 10 lumped springs, 27 beam elements, and two gear mesh stiffness

elements, and correspondingly twenty-nine 3-DOF nodes and two 1-DOF nodes for the housing

of slave and test gear setups. This gives a total of N = 89 DOFs for the complete system

coordinate set.

The system mass M and stiffness K matrices, each of dimensions N? N, can then be used

to form the classical governing equation of dynamic motion expressed as

 30

&& + CX
MX & + KX = F , (3.5)

where X is the response vector, F is the forcing vector containing the transmission error term, C

is the damping matrix assumed to be equivalent to the typical 5% uniform modal damping

& and X
observed for gearing systems, and X && signify the first and second time derivatives of X.

Here, M and K are the summations of the expanded version of each element matrix derived

using a standard FE formulation [64]. Note that Equation (3.5) is linear and its coefficient

matrices are time-invariant. An output vector, Y = DX , is defined for a subset of the n response

coordinates of interest, where D is the output coefficient matrix of dimension n? N. Applying the

theory of normal modes to Equation (3.5), X = Q can be used to transform the

representation from physical coordinates X into modal coordinates Q = {q1 q2 L q N } .

T

Here, consists of the mass normalized mode shapes (eigenvectors) where T M = I

resulting in the identity matrix. The resulting dynamical equations in terms of these uncoupled

modal coordinates are

q&1 0 1 0 0 ... q1 0
q&& 2 2 1 1 0 0
... q&1 F
1 1
1
T
q& 2 = 0 0 0 1 ... q 2 + 0 , (3.6)
q&& 0 0 2
2 2 2

... q& 2 F
2 2
2
... ... ... ... ...
... ... ...

where Fj, (j=1,2N) is the element of the physical force vector F, i is the i-th natural

frequency, and i is the corresponding modal damping coefficient. In terms of the modal

coordinates, the output vector is

Y = D Q . (3.7)
 31

Solving Equation (3.6) for qi and transforming them back into the physical coordinates yields the

output response of interest. Accordingly, the transfer function between any k-th reference

coordinate (forcing) and any j-th physical response point is

xj N ji ki
H jk = = , (3.8)
Fk i =1 s 2 + 2 i i s + i2

where s is the Laplace variable. This FE model and the corresponding transfer function results

shown in Equation (3.8) are employed in the subsequent active vibration control simulations to

examine the true capabilities and limitations of the four proposed actuation concepts to minimize

the gear housing vibration response.

 32

3.3 Dynamic Mesh Force and Stiffness

For the power re-circulating gear system shown in Figure 3.2, the design parameters of

each component are listed in Table 3.1. Using these parameters, the above FE system model can

be obtained.

Table 3.1. Some main parameters of power re-circulating gear system.

Parameter Descriptions Numerical values (units)

Bearing stiffness, Kbx 3.2x107 N/m
Support bearing stiffness, Ksx 5.0x105 N/m
Gear face width of slave gear, Wf 6.76x10-2 m
Gear face width of test gear, Wf 2.54 x10-2 m
Coupling mass, mc 0.56 kg
Coupling inertia, Ic 7.09x10-4 kg?m2
Outside driving shaft length, Lds 0.8 m
Gear and pinion mass, mg, mp 0.53 kg
Gear and pinion radius, Rg, Rp 7.5x10-2 m
Gear mesh stiffness of slave gear, Km 1.05 x109 N/m
Gear mesh stiffness of test gear, Km 3.94 x108 N/m
Housing mass, mh 200 kg
Outside shaft radius, Rs 2.06x10-2 m
Inside driving shaft radius, Rs 3.18x10-2 m
Inside driven shaft radius, Rs 4.29x10-3 m

Since the dynamics of the gear pair system is mainly excited by the transmission error at

the gear meshing point, the resulting dynamic mesh force is normally used to describe the

severity of the dynamics of the geared rotor system. For this power re-circulating gear system,

the transmission error on the slave gear mesh point can be neglected, because its smaller than

the transmission error on the test gear mesh point. Hence the formulated force term is really a

function of the magnitude of transmission error, mesh stiffness, and the pitch radii. As shown in

Figure 3.4, the transmission error excitation leads to a pair of translational dynamic force and
 33

torque acting on the gear and pinion. This dynamic force vector F in Equation (3.6) is expressed

as

F = [0 L 0 K m 0 K m Rg 0 L 0 K m 0 K m R p 0 L 0 ]T e , (3.9)

u zg yg u zp yp

where e is transmission error represented mathematically as Ee i mt m, and u zg , u zp , yg , and

yp are the gear and pinion translation and rotation coordinates. Here, E=30 is assumed.

Omitting the damping term, the dynamic mesh force can be formulated as [45-46]

Fmesh = K m (u zg u zp Rg yg R p yp e) . (3.10)

The predicted dynamic mesh force for the power re-circulating gear system of interest (see Table

3.1 for actual design parameters) is shown in Figure 3.5. The trend shows a general increase in

Fmesh as frequency increases. And, there is a peak around 3 kHz. This is due to the contribution

from the fundamental mesh mode (out-of-phase rotation of the gear pair) at around 3 kHz for this

set of spur gears at test stage.

 34

9
10

Amplitude (N)

8
10

7
10

2 3 4
10 10 10
Frequency (Hz)

Figure 3.5. Dynamic mesh force spectrum of the test gear pair of interest due to 30m of
transmission error excitation.

The vibratory response of the gear housing due to the unit magnitude of the dynamic

force at the gear mesh coordinate, expressed by Equation (3.11), can be calculated using the

modal superposition-based transfer function theory given by Equation (3.8). By scaling the

equation linearly to reflect the total mesh force and summing the contributions from the resultant

dynamic forces and torques on the gear and pinion, the net displacement response of the housing

due to the specified transmission error excitation, yh,TE, is

4 N
hi 4
y h ,TE = H hk Fk = [ Fk ] , (3.11)
s + 2 i i s + i2
2 ki
k =1 i =1 k =1

where Fk, k=1 to 4, are the four non-zero forcing elements of Equation (3.9). By taking the

second time-derivative of displacement yh,TE, the acceleration response of the housing is

obtained, which is shown in Figure 3.6. From this figure, one can see that the responses higher
 35

than 300 Hz tend to be almost same except the valley around 350 Hz. Higher response means

that the mesh frequency can be easily identified and controlled.

1
10

0
10
Amplitude (m/s )
2

-1
10

-2
10

-3
10
0 200 400 600 800 1000
Frequency (Hz)

Figure 3.6. Housing vibration due to 30m of transmission error excitation.

The dynamic mesh stiffness is the frequency response function relating mesh force to

transmission error excitation. It can be given by

Fmesh (i )
Dmesh (i ) = , (3.12)
e (i )

where Fmesh (i ) and e(i ) are Fourier Transform of mesh force and transmission error,

respectively. From Equation (3.12), the dynamic mesh stiffness should be nearly the same shape

curve as the mesh force shown in Figure 3.5. The dynamic mesh stiffness is also used to analyze

the response of the geared rotor system. The larger dynamic mesh stiffness leads to higher

response of system.
 36

3.4 Actuation System

Since the actuator is one of the primary elements and its setup may significantly affect

the performance of the active vibration control system, it is highly desirable to be able to identify

an effective configuration. There are four candidate active control concepts available. They are:

(1) active inertial actuators positioned tangentially on the gear body, (2) semi-active gear-shaft

torsional isolation coupling, (3) direct active bearing vibration control, and (4) active shaft

transverse vibration control. These approaches are designed to either impede the transmission of

vibration energy into the housing or reduce the effect of the dynamic mesh force excitation.

Here, these concepts are analyzed to determine the levels of required actuation efforts and the

corresponding driver electrical power requirements. Note that while the overall goal is to reduce

the housing response, each concept attempts to control the vibrations at different structural points

inside the gearbox system. Therefore, their control requirements may be inherently different.

Details associated with each of these concepts are provided next.

3.4.1. Concept 1: Inertial-Based Active Gear Actuator

The first approach uses three inertial actuators, which can be made of magnetostrictive or

piezoelectric stacks, directly mounted tangentially onto the side of the gear body at equal angular

intervals, as shown in Figure 3.7. As the smart material expands and contracts at the selected

excitation frequency, the acceleration of the inertial mass mounted on one end of the actuator

will provide a reaction force that acts on the gear body. Due to the circumferential arrangement,

these actuators can produce a net torque fluctuation as well as a translation force parallel to the

mesh load line-of-action. The relative amount of these two actuation loads can be controlled by

appropriately phasing the drivers. The net dynamic torque and force can then be applied in a

controlled manner to reduce gear torsional and translational vibrations. Since this actuation
 37

concept directly acts near the excitation source, it is expected to be able to suppress vibrations

transmitted into the shaft and bearings, and also reduce dynamic mesh force production.

Additionally, the overall design includes three miniature accelerometers, located near the

actuators, to measure gear vibrations and provide feedback signals to the controller.

To keep the actuation design as simple as possible, only the driven gear (or pinion) is

treated. Since one of the purposes is to minimize the generation of dynamic mesh force, which

involves lowering dynamic transmission error dictated by the relative motion at the mesh, it is

only necessary to actuate one of the two gears. The disadvantage here is the lack of vibration

transmissibility control through the shaft of the untreated gear. Note that a related design was

used by Chen and Brennan [11] as indicated earlier. The fundamental difference is that the

control object of Reference [11] was the gear torsional vibration, while the goal in this work is to

suppress the gear housing vibration.

Mounting flange
Pinion Gear
Slip ring Fp

Shaft

Magnetostrictive or
piezoelectric actuator Accelerometer
Inertial mass (mounted on gear body)

Figure 3.7. Smart gear with inertial actuators (concept 1).

Other considerations include the design of the fundamental resonant frequency of the

actuator, which ideally needs to be near or less than the excitation frequency, and the fact that we

have a set of rotating actuators requiring the use of slip rings. In the former issue, either an
 38

amplification mechanism, such as the displacement amplification or a series stiffness

configuration may be needed to lower the first resonant frequency below the working frequency.

This is especially critical when a piezoelectric type actuator is selected since its resonant

frequency is inherently high. In the latter problem, suitable slip rings with proper power ratings

must be used to ensure sufficient supply of power to the rotating actuators. Of course, the slip

rings are also needed for the accelerometer signals.

3.4.2. Concept 2: Semi-active Gear-shaft Coupling

The second proposed concept is aimed at modifying the torsional vibration transmission

path between the gear and shaft. In this setup, the gear body and corresponding shaft are

connected via several piezoelectric actuators, as shown in Figure 3.8 for one set of gear-shaft

coupling setup. The piezoelectric actuators serve two purposes. They transmit the mean torque

in the gear-shaft load path and simultaneously generate reactive dynamic forces to minimize

transmitted perturbations. Like the first concept, slip rings are needed to provide input power to

the actuators. Note that the translation force of the gear is transmitted via a rolling element

bearing between the gear body and shaft.

This particular actuation concept is similar to the application of an active dynamic force

directly on the shaft (the receiver) with a reaction against a gear body (the mass), as described in

Reference [8]. Therefore, the dynamic force required to isolate the torsional vibration

transmitted from the gear into the shaft is inversely proportional to the excitation frequency.

Accordingly, at relatively high frequency, only a small dynamic force is needed. However, when

the frequency is lower, a large dynamic actuation force is required. Note that this scheme may

also provide some level of passive torsional vibration isolation in the lower frequency range due

to the semi-active nature of the structural design.

 39

Pinion mounting flange Pinion

Shaft mounting flange
Housing bearing F

F
Accelerometer (mounted on gear body)
F
Piezoelectric actuator

Figure 3.8. Active control of gear-shaft torsional coupling (concept 2).

3.4.3. Concept 3: Direct Active Bearing Vibration Control

Since most gear whine problems are the result of structure-borne vibration transmission

from the geared rotor system through the bearings and into the housing, a natural path control

scheme inside the gearbox is to setup choke points at the support bearings. One way to

accomplish this is to use two pairs of piezoelectric stack actuators placed in two orthogonal

radial directions between the bearing raceway and the housing support structure, as shown in

Figure 3.9. One pair of opposing actuators is oriented parallel to the gear mesh line-of-action,

while the second pair is oriented perpendicular to the line-of-action. Note that although only one

pair of actuators oriented parallel to tooth load line-of-action is needed theoretically, this double

set configuration is recommended in practice to achieve a more robust ability to control general

transverse plane motions of the shaft-bearing structure. This is especially useful when non-

negligible transverse vibration orthogonal to the load line-of-action caused by misalignment and

friction force excitation exists. In this work, the analyses of concepts 2 and 3 only incorporate

the idealized single pair actuator case.

 40

Gear
F pair
Housing
Pinion Gear bearing

F
Piezoelectric
actuators

Housing

Figure 3.9. Active control applied directly at the housing bearings (concept 3).

This specific actuation concept is designed to directly add active dynamic forces to the

housing support structure by reacting against the bearing raceway, which may effectively

suppress the housing vibration. Unlike the previous two concepts, the actuators here are

stationary. As a result, slip rings are not needed in this design. Nevertheless, the housing

supports must be modified substantially to accommodate the actuators, which may not always be

feasible.

3.4.4. Concept 4: Active Shaft Transverse Vibration Control

In this configuration, two pairs of actuators are attached to the rotating shaft using an

additional set of bearing as shown in Figure 3.10. The actuators are acting against an additional

redundant bearing along the line-of-action of gear transmission error excitation. Although the

dynamic force is still being introduced between the shaft and housing, the added bearing

component does not carry the required static load. Therefore, this bearing can accommodate a

large range of motion. A similar idea was examined by Rebbechi, Howard, and Hansen [10],

where magnetostrictive actuators were used. The proposed system here applies piezoelectric

stack actuators along with an improved control algorithm to be described later in this proposal.
 41

For the present analysis, it is assumed that the actuation position is located at one-third of the

length from the gear position to the housing support location solely based on packaging

considerations. Furthermore, due to the possible run-out and vibration deformation of the shaft,

the actuators may see some lateral motions that may be harmful to the actuators. Hence, the

setup must be carefully designed such that the actuators are not directly connected to the bearing

raceway. Instead a thin stinger rod as shown in Figure 3.10 is used. This stinger rod provides

some degree of lateral flexibility, thereby reducing the potentially harmful lateral forces in the

actuator.

Gear
Additional pair
bearing Housing
Pinion Gear bearing

F Stinger rod
Piezoelectric
actuators
Housing
Figure 3.10. Active shaft transverse vibration control (concept 4).

3.4.5. Summary

Based on the work in Reference [12], the most feasible active vibration control concept

for suppressing the housing response is the active shaft transverse vibration control one. For this

proposed actuation setup, the housing response due to actuation force can be easily calculated

using the above system model. The result of this analysis is shown in Figure 3.11. This figure

can be combined with Figure 3.6 to determine the force requirement for the actuator [12]. The
 42

force requirement for the actuator at each frequency can be determined by dividing the curve in

Figure 3.6 by the curve in Figure 3.8 at each frequency point. After having the force

requirement, the suitable model of actuator, which can provide enough force to suppress the

housing vibration of gearbox system, can be determined for the experimental setup. Actually,

the force requirement is not the only requirement. The electrical power requirement must be

considered as well as the force requirement in next subsection. Note that these results are for TE

excitation with amplitude of 30m. In fact Figure 3.11 is the amplitude response function of the

secondary path which will be discussed later.

-2
10
Amplitude (m/s )
2

-3
10

0 200 400 600 800 1000

Frequency (Hz)

Figure 3.11. Housing response due to a unit magnitude of actuation force.

3.5 Electrical Power Requirements

When the pizeo-actuator model, gear system model and actuation are determined, it is

necessary to select proper actuator and its power amplifier for this gear vibration control system

by estimating the electrical power requirements of the system. The structure of gear system can
 43

be treated as an external stiffness as described in Chapter 2. Note that this external stiffness is

not a static stiffness but a dynamic one.

It has been showed that a piezoelectric stack actuator can be modeled as a spring element

of stiffness Kp when the working frequency of the actuator is far less than the first resonant

frequency of the actuator [40, 42, 43]. Thus the mechanical impedance of the piezoelectric stack

is Z p = K p / i . The block force Fb, defined as the maximum output force when the actuator is

acting on a rigid structure with infinite impedance, is shown in Equation (2.3). If the actuator is

acting against a flexible structure, such as the bearing housing, the maximum possible output

force Fr, shown in Equation (2.7), can be expressed further as

k ext Z ext
Fr = Fb = Fb , (3.13)
k ext + k p Z ext + Z p

where Zext is the external driving-point mechanical impedance, which can be modeled

analytically or measured experimentally using a force hammer. Notice that there might exists a

phase difference between impedances Zext and Zp. If these two impedances are in phase, then the

output force Fr will be less than block force Fb. Otherwise, Fr might be large than Fb depending

the relative phase between Zext and Zp. One advantage of the above analysis is that it doesnt

need to consider the existence of the actuator when estimating the required active force, because

the actuator dynamics has been considered in equation (3.13).

Assumed the gear system is linear and the loaded gear transmission error will not change

after the actuator is installed, it is relatively easy to determine the required active force for

completely canceling the gear housing vibration with the absence of the actuator. Using the

superposition principle, the nominal required active force Fn can be directly estimated by

Z ph
Fn = Ah , (3.14)
i
 44

where Ah is the gear housing acceleration and Zph is the cross-point impedance from the actuator

point to the gear housing measurement point. Notice that Zph can be directly measured from the

cross-point frequency response function (FRF) or estimated by the above FE model. Notice that

maximum possible output force Fr must be larger or equal to nominal required active force Fn.

Combining equations (2.3, 3.13-3.14), it is easy to determine the required driving voltage for a

particular piezoelectric stack actuator.

On the other hand, the piezoelectric stack actuator can also be modeled as an electrical

capacitor. Due to the inherent electrical-mechanical coupling, the dynamic electrical admittance

Y is given by

I k 2 k ext k 2 Z ext
Y= = iC 0 [1 ] = i C 0 [1 ], (3.15)
V k ext + k p Z ext + Z p

where k2 is the piezoelectric coupling coefficient, which is assumed to be 0.3, C0 is the zero-

stress electrical capacitance of the actuator, and I is the applied current [14]. It is assume here

that C0 is insensitive to temperature, in spite of the heat generated as the stack expands and

contracts. Hence, given the required driving voltage V, the operating current I can be calculated

using equation (3.15). Notice that this operating current I can not be large than the peak current

provided by the piezoelectric power amplifier.

Using the parameters of Piezomechanik actuator PSt1000/16/40 VS 25 shown in Table

2.1 and the estimated driving-point FRF and cross-point FRF at actuator installation location, the

resultant required current and voltage are shown in Figure 3.12 and 3.13. From the estimation

results shown in Figure 3.12 and 3.13, mesh frequency between 200 Hz and 400 Hz will be

control frequency range of interest, because in this area, the required current is less than 100mA,

which is the maximum output capacity of most power amplifier.

 45

90

80

70
Required Voltage (V)
60

50

40

30

20

10

0
0 200 400 600 800 1000
Frequency (Hz)

Figure 3.12 Estimation result of required voltage fed into actuator.

250

200
Required Current (mA)

150

100

50

0
0 200 400 600 800 1000
Frequency (Hz)

Figure 3.13. Estimation result of required current fed into actuator.

 CHAPTER 4

ADAPTIVE ALGORITHM AND CONTROL SYSTEM DESIGN

The adaptive controller is one of the most important components of the active control

system. This component can be implemented in a DSP board. It calculates the control force

(actuator input) from the dynamic signals picked by the sensors (reference and error sensors)

using a specially designed algorithm. Normally, a well-designed controller will be able to

provide sufficient reduction to achieve the desired response under certain conditions. In this

section, the proposed control structure and its corresponding filtered-x LMS (FXLMS) algorithm

will be given. Also, a frequency estimation technique is introduced and used to construct the

reference signal for the adaptive algorithm.

4.1 Active Control Structure

In the applications of active noise and vibration control [66], two major types of control

methods have been widely used: (i) feedforward control and (ii) feedback control. Some

researchers have also combined these two types of control methods. The first one that is the

feedforward control method requires both reference and error signals, and can be used to reduce

broadband error signal as well as narrowband ones. The measured reference signal that is

correlated to the primary disturbance (error signal without the effect of control input) is used to

derive the control input. The correlation between the reference and error signals due to the

primary excitation source is critical to the reduction in the performance of the controller. If the

correlation between the reference and error signals is perfect, it is theoretically possible to force

46
 47

the error signal to become zero, which can be a very attractive feature. However, if the

correlation is only partial, the proposed active vibration control system can only suppress the

primary response frequency that is correlated with the reference signal. Due to its inherently

stable and robust characteristics, the proposed approach has in fact been used in many practical

applications.

The second method mentioned earlier is the feedback control that requires only the error

signal. It is mostly used to attenuate narrowband or periodic response. Mechanical systems

posing difficulty in obtaining suitable references and active control systems characterized by

narrowband disturbances, typically employ the feedback control approach in order to avoid the

problems associated with obtaining a good reference signal for use in the feedforward

configuration. It is well known that a certain threshold level of vibration reduction can be

achieved within a limited bandwidth and the smaller the error signal is driven to, the higher the

control gain becomes. Hence, the less stable the system becomes as well.

As mentioned above, reasonably good performance of harmonic response reduction can

be attained using a feedforward control method only when a clean reference signal is available.

However, for the gearbox system, it is very difficult to acquire a suitable set of reference signal

for the feedforward FXLMS scheme. Accordingly, a proposed feedback controller will be used

in this study to reduce the gearbox housing response as shown in Figure 4.1. Unlike the standard

feedback controller, this one uses a signal generator equipped with frequency estimation feature

that synthesizes the reference signal. This effectively converts the feedback scheme into a

feedforward type. The resulting scheme is more stable than the standard feedback structure.
 48

Gearbox system
TE x(n) HhTE(z)
+

Hhr(z) +

error input

Adaptive Controller
output
Signal generator with
reference frequency estimation
input

Figure 4.1. Proposed active vibration control system applied to a gearbox plant.

In Figure 4.1, HhTE(z) and Hhr(z) represent the real transfer functions of the primary and

secondary paths, respectively. The primary path relates the primary excitation source to the error

sensor position where vibration is being controlled. On the other hand, the secondary path

connects the actuation force to the error sensor. In this analysis, the error sensor is mounted on

the gear housing as described in the earlier chapter. Thus, the housing vibration is the main

control aim and error signal.

4.2 Adaptive Algorithm

There exist many algorithms in adaptive system. In this section, two algorithms

associated with this study are introduced.

4.2.1 Least Mean Square (LMS) Algorithm

Although LMS algorithm is more restricted in its use than some other algorithms because

of using a special estimate of gradient is valid for the adaptive linear combiner shown as Figure
 49

4.2, its still important for its simplicity, ease of computation and not requiring off-line gradient

estimations or repetitions of data [67].

Input
x0k w0k dk

w1k Error
x1k k

w1k
x1k
(a)

Input -1 -1 -1
xk z z z

w0k w0k w0k

dk

Error
k

(b)

Figure 4.2. The adaptive linear combiner: (a) general form (b) transversal filter

For the controller shown in Figure 4.2, the error signal can be expressed as

k = d k X Tk Wk (4.1)

where dk is the desired signal which will be cancelled during adaptive process, X k is the vector

of input samples in either of two configurations in Figure 4.2, Wk is the weights vector. In LMS

algorithm, the gradient of performance surface, which expresses the expectation of square of

error signal = E [ k2 ] for different weights of control filter, can be derived as,
 50

k2 k
w w
0
= M = 2 M = 2 X
0
k 2 k k k (4.2)

k k
wL wL

With this simple estimate of gradient, the LMS algorithm to adapt the weights value is derived

as,

= W + 2 X
Wk +1 = Wk (4.3)
k k k k

4.2.2 Filtered-x LMS Algorithm

The above standard LMS algorithm has been widely used to adaptively update the

weights of the FIR filter in numerous applications. However, in many active noise and vibration

control applications, the standard LMS algorithm must be modified to improve stability. The

path between the adaptive controller output to the housing response point (error signal) is

generally known as the secondary path in contrast to the primary path that relates the external

excitation to the same response point. The existence of this secondary path transfer function in

the closed-loop system may cause the standard LMS algorithm used for active vibration control

applications to become unstable. This is because the error signal is not properly aligned in time

with the reference signal [19]. To address this problem, many solutions to compensate for the

effect of the secondary path have been proposed in the past [67-69]. One of the most effective

schemes is the well-known FXLMS algorithm as shown in Figure 4.3. The standard FXLMS

algorithm is summarized in Table 4.1.

 51

d(n) + yh(n)
x(n) HhTE(z)
+

r(n) y(n)
W(z) Hhr(z)

Shr(z)

LMS
r(n)
Controller

Figure 4.3. Structure of standard FXLMS algorithm.

Table 4.1 Overall Numerical Process of the Standard FXLMS Algorithm.

Description Algorithm
Initialization W (0) = 0
Update y ( n) = W ( n) r ( n)
(at each iteration) X (n) = [r (n) r (n 1) L r (n L + 1)]
X ' (n) = [r ' (n) r ' (n 1) L r ' (n L + 1)]
r ' (n) = S hr (n) r (n)
W (n + 1) = W (n) + 1 X ' (n) y h (n)

The FXLMS algorithm is well studied and has been used in many active vibration control

applications. In fact, both of its Wiener and non-Wiener behaviors have been studied by many

researchers [70-72]. In Reference [71], Bermudez and Bershad used the orthogonal subspace

approach to analyze the non-Weiner solution of filtered LMS algorithm for noiseless reference

sinewave signal. In their research, the proposed FLMS algorithm is shown in Figure 4.4.
 52

d(n) + e(n)

y (n)
X(n) y(n)
W(z) S2(z)

S1(z)
e(n)
LMS S3(z)
X (n)

Figure 4.4. Filtered LMS algorithm in Reference [71].

The difference between their filtered LMS algorithm with the proposed FXLMS

algorithm for gearbox vibration control is S3(z). Their non-Weiner solution for FLMS algorithm

is

Px Pd
wi (n) = Re{( 1 + 2 ) exp[ j[( x + d )n + 2 d i x ]]
M2 , (4.4)
+ (( 1* *2 ) exp[ j[( x d )n 2 d i x ]]}

where i=1,2,N (N is the length of adaptive FIR filter) is the weights index, Re{ } stands for the

real part of the complex number, and

= M 1 M 2 M 3 , (4.5)

1 = [cos[(1 + 1 ) x ]e j d cos( 1 x )] / T , (4.6)

2 = j[sin[(1 + 1 ) x ]e j d sin( 1 x )] / T , (4.7)

T = e j ( + 2 ) d 2 cos( x )e j ( +1) d + e j d
. (4.8)
+ NPx cos[(1 + 1 ) x ]e j d NPx cos( 1 x )

Here, Px and Pd are the powers of the sinusoidal signal x(n) and d(n), respectively. Furthermore,

x and d are the digital frequencies of x(n) and d(n), respectively. The transfer functions S1(z),
 53

S2(z) and S3(z) can be expressed by amplitudes M1, M2, M3, and phase delay 1, 2, 3 (where

=2+3), respectively, at frequency x or d. From Equation (4.4), it can be seen that weights

will oscillate at frequencies x+d and x-d if x? d. Only if x=d, the weights will converge

to fixed values.

Also, they gave the limitation of learning rate , which is the minimum value of

solutions of Equations (4.9) and (4.10)

c
sin( ) + sin[( + 1) ] = 0, 2k x , (4.9)
b

c
cos( ) + cos[( + 1) ]
b 1
= , (4.10)
a + 2 cos b

where a = 2 cos x , b = NPx cos[(1 + 1 ) x ] and c = NPx cos( 1 x ) .

4.3 Effect On The Controller Due To Uncorrelated Periodic Signal

From the discussion on filtered-x LMS algorithm in section 4.2, one know that the

weights of controller will oscillate at frequencies x+d and x-d if x? d. Only if x=d, the

weights will converge to a set of fixed values. To mimic the cases where only one harmonic is

being controlled with other uncorrelated harmonics in the unwanted vibration signals present,

one can assume that there are two harmonics 1 and 2 in unwanted signal d(n), and one

harmonic 1 in reference signal r(n). Thus, one can expect that the weights will oscillate at

frequencies 1+2 and 1-2 with a mean value because there is one frequency value contained

in d(n), which is at the same as the frequency of r(n). These weights can be expressed as

wi ( n ) = a0i + a1i ( e j [(1 +2 ) n +i ] + e j [(1 +2 ) n +i ] ) + a 2i ( e j [(1 2 ) n +i ] + e j [(1 2 ) n +i ] ) , (4.11)

 54

where i=0,1,N-1 (N is the length of adaptive FIR filter), and n is time index. The output of

controller y(n) can be expressed by

N 1
y ( n ) = W ( n ) r ( n ) = wi ( n ) X i ( n ) , (4.12)
i =0

where X i ( n ) = b( e j1 ( n i ) + e j1 ( n i ) ) . The above equation can be considered as three terms

separately. Note that the first weighting value is constant. Thus, the multiplier of the first term

and the reference signal is a sine wave with frequency 1. Based on the discussion in section

4.2, one can see that the amplitude a1i is independent of index i. That means all a1i are the same.

Similarly, all a2i are also the same. But their phases depend on index i ( i = i = i1 ). The

multiplier of second term and reference signal can be derived as

a1i ( e j [(1 +2 ) n +i ] + e j [(1 + 2 ) n +i ] ) b ( e j ( n i )1 + e j ( n i )1 )

(4.13)
= a1i b ( e j [( 2 + 21 ) n +i i1 ] + e j [(2 + 21 ) n +i i1 ] + e j [ n 2 +i +i1 ] + e j [ n 2 +i +i1 ] )

The sum then becomes

N 1

a
i =0
1i ( e j [(1 +2 ) n +i ] + e j [(1 +2 ) n +i ] ) b ( e j ( n i )1 + e j ( n i )1 )
N 1 N 1
= c1 ( e j ( 2 + 21 ) n e j (i i1 ) + e j (2 + 21 ) n e j (i i1 )
i =0 i =0
N 1 N 1
+ e jn2 e j (i +i1 ) + e jn 2 e j (i +i1 ) ) (4.14)
i =0 i =0
N 1 N 1 N 1 N 1
= c1 ( e j ( 2 + 21 ) n e j ( 2i1 ) + e j ( 2 + 21 ) n e j ( 2i1 ) + e jn 2 1 + e jn2 1)
i =0 i =0 i =0 i =0
j 2 N1 j 2 N1
1 e 1 e N ( N 1) jn 2
= c1 e j ( 2 + 21 ) n j 21
+ e j ( 2 + 21 ) n j 21
+ (e + e jn2 )
1 e 1 e 2

This term will generate two frequency components, 21+2 and 2 where 2 N1 = m . If

2 N1 m , only one frequency component with frequency 2 will be generated by the above

term. Similarly, the multiplier of third term and reference signal can be derived as
 55

a2i (e j[(12 )n+i ] + e j[(12 )n+i ] ) b (e j (ni )1 + e j (ni )1 )

(4.15)
= a2i b (e j[(212 )n+i i1 ] + e j[(212 )n+i i1 ] + e j[n2 +i +i1 ] + e j[n2 +i +i1 ] )

For this result, the sum is

N 1

a
i =0
2i ( e j [(1 2 ) n + i ] + e j [(1 2 ) n + i ] ) b ( e j ( n i )1 + e j ( n i )1 )
N 1 N 1
= c 2 ( e j ( 21 2 ) n e j (i i1 ) + e j ( 21 2 ) n e j (i i1 )
i =0 i =0
N 1 N 1
+ e jn 2 e j (i +i1 ) + e jn 2 e j (i +i1 ) ) (4.16)
i =0 i =0
N 1 N 1 N 1 N 1
= c 2 ( e j ( 21 2 ) n e j ( 2i1 ) + e j ( 21 2 ) n e j ( 2i1 ) + e jn 2 1 + e jn2 1)
i =0 i =0 i =0 i =0
j 2 N1 j 2 N1
1 e 1 e N ( N 1) jn 2
= c 2 e j ( 21 2 ) n j 21
+ e j ( 21 2 ) n j 21
+ (e + e jn2 )
1 e 1 e 2

Again, this term will also generate two frequency components, 21-2 and 2 where

2 N1 = m . If 2 N1 m , only one frequency component with frequency 2 will be

generated by above term. Hence, the controller output, y(n), includes 4 harmonics given by 1,

21+2, 21-2 and 2 if 2 N1 m , and only two harmonics at 1 and 2 if 2 N1 = m .

Figure 4.5 shows the frequency domain of first element of FIR controller. Figure 4.6

shows frequency domain of output of the controller. In this case, there are two frequencies that

exist in the unwanted signal d(n), namely 128Hz and 512Hz. The reference signal is a sine wave

with 128Hz, the sampling rate is 4096Hz, and the length of FIR controller is 48, which satisfies

the relation 2 N1 = m . Hence, the weights of FIR controller should include the mean value

and the two frequency components, 384Hz and 640Hz. And, in the output of controller, it should

contain two frequency components, 128Hz and 512Hz as shown in Figures 4.5 and 4.6. For the

case of two frequencies, 100Hz and 200Hz, in y(n), and reference signal at 100Hz, the results are

given in Figures 4.7 and 4.8. Furthermore, the length of FIR controller is 10. That implies that
 56

the weights of FIR controller should include the mean value and the two frequency components,

100Hz and 300Hz. Furthermore, the output of controller should contain the three frequency

components, namely 100Hz, 200Hz and 400Hz, because 21-2 = 0.

1
10
Mean Value

0
10

-1
10
384Hz
Magnitude

640Hz
-2
10

-3
10

-4
10

-5
10
0 500 1000 1500
Frequency (Hz)
Figure 4.5. Frequency domain of first element of FIR controller weights
4
10
128Hz
3
10

2
10
Magnitude

512Hz
1
10

0
10

-1
10

-2
10
0 500 1000 1500
Frequency (Hz)
Figure 4.6. Frequency domain of output of controller
 57

0.6

mean value
0.5

0.4
Magnitude

0.3

0.2

100Hz
0.1 300Hz

0
0 500 1000 1500
Frequency (Hz)

Figure 4.7. Frequency domain of first element of FIR controller weights

1.5

100Hz
1
Magnitude

0.5
200Hz

400Hz
0
0 500 1000 1500
Frequency (Hz)

Figure 4.8. Frequency domain of controller output

 58

4.4 Secondary Path Modeling

As stated above, the FXLMS algorithm requires knowledge of the transfer function of the

secondary path. If the system is time-invariant but unknown, an offline modeling technique can

be used to estimate it during the initial stage. After the initial stage, the estimated model of the

secondary path is essentially fixed during the operation of adaptive controller. However, the

offline technique has a disadvantage of not tracking the changes in the secondary path. Because

the transfer function of the secondary path may be continuously changing in a real system, the

online modeling technique is desirable. It is well known that as long as the phase modeling error

is within ? /2, the adaptive noise canceler will be stable [73]. For the case when the reference

signal is narrowband but the disturbance is broadband, numerical results show that the modeling

error with a phase error of 40? hardly affects the convergence speed of the algorithm [19].

Hence, in real life application, the secondary path is modeled in the offline mode. During the

operation of the controller, if there is significant change in the transfer function of the secondary

path tested, an online modeling technique is needed. The resultant algorithm is examined here to

determine its suitability in performing active vibration control of geared rotor systems.

Figure 4.9 is an illustration of the FXLMS algorithm with the online secondary path

modeling feature in Figure 4.1. The offline modeling technique used here is similar except that

the controlling filter does not update in real time, and the gearbox system is not running, i.e., y(n)

and x(n) are zeros. This online identification is defined as an additive white noise type since it

injects a small amount of broadband power into the system [70]. In this figure, HhTE(z) and

Hhr(z) represent the real transfer functions of the primary and secondary paths, respectively.

Furthermore, W(z) is the adaptive controller whose weights are updated by the FXLMS

algorithm, and S hr (z ) is the estimated secondary path. Here, S hr (z ) is identified online by

 59

feeding a white noise u(n) unrelated to y(n) into the actuator, and updated using the LMS

algorithm. Since the interest here is to control the dynamic response due to transmission error

excitation e(n), a frequency estimator scheme is employed to extract the required fundamental

mesh frequency information from the measured vibration signal and at the same time generate a

reference signal r(n) that is correlated with the primary excitation source e(n).

HhTE(z) d(n) +
x(n)
yh(n)
+

Sinewave r(n) y(n) +

W(z) Hhr(z)
generator +

Noise u(n) v(n) +

Shr(z)
Shr(z) generator
e(n)
Copy
Frequency LMS
estimator r(n)
LMS

Controller

Figure 4.9. The structure of the proposed FXLMS algorithm with online secondary path
modeling feature.

4.5 Fundamental Frequency Estimation

4.5.1 Existing Frequency Estimation Technique

Frequency estimation is a critical component of the proposed control system. From the

pervious sections, one can see that the control result depends on the reference signal of the

FXLMS algorithm. In this research work, the reference signal is generated through a frequency
 60

estimator and sinewave generator. Thus, the performance of the frequency estimator will

directly affect the control results and stability.

In applying the filtered-x LMS algorithm, the reference signal is one of the most

important parameters needed. For the gearbox system investigated here, the aim of controller is

to suppress a set of multi-tone vibration and ultimately the unwanted gear whine signals. The

dominant multi-tone frequency components in the housing vibration response spectrum typically

contain the fundamental mesh frequency and its higher order harmonics. Due to the

configuration of the gearbox dynamic system, it is quite hard to obtain the reference signal

accurately. Hence, the inaccuracy of the estimated frequency may affect stability and vibration

reduction performance. To deal with this problem, numerous frequency estimation approaches

have been previously developed. These include methods such as spectral estimation, adaptive

notch filter, phase-locked loop, etc.

In many signal-processing applications, not just in active vibration control

implementations, the problem of estimating the primary frequencies in a dynamic signal

containing multiple sinusoids is essential. Numerous techniques currently exist. One of them

uses a spectral estimation scheme. In 1994, Gough [74] presented an iterative spectrum

estimation technique based on the discrete Fast Fourier Transform (FFT) algorithm. In another

study, Kay and Shaw [75] proposed a principal component autoregressive spectral estimation

technique that avoids the computation of the eigenvalues and eigenvectors by filtering them out.

Similarly, to avoid the cost of computing the eigenvectors, Karhunen and Joutsensalo [76]

proposed an efficient Fourier transform-based approach for approximating the signal subspace

used to estimate the sinusoidal frequency of the signal. In all of these techniques cited, the

frequencies are found by locating the maximum or minimum values of the computed spectrum,
 61

which usually incur high computational cost even though they produce fairly high precision

results. Due to the high computational cost needed for these techniques, they are basically

impractical for use in real time dynamic signal processing.

An alternate technique developed for frequency estimation is the adaptive notch filter

(ANF), which is inherently stable, fast and efficiency. Classically, the ANF technique is

parameterized by the polynomial coefficients of its transfer function. These coefficients are a

function of the notch frequencies with zero or nearly zero gain. In this technique, the frequencies

are computed from the estimated transfer function coefficients. It has been a popular method for

estimating the fundamental frequency of a dynamic signal, and well studied by many researchers

in the last several decades. In 1991 Gegalia [77] proposed an improved lattice-based adaptive

infinite impulse response (IIR) notch filter. The proposed algorithm possesses good convergence

and an improved tracking ability compared to earlier designs. More recently, Li [78] proposed a

cascaded notch filter, which is purely parameterized by the conventional notch filter. This makes

the algorithm simpler, more efficient, and more robust. Kim and Park [66] applied the adaptive

notch filter to an active multi-tone noise control problem. In their approach, a reference signal

generator using frequency estimator is designed for the filtered-x LMS algorithm.

There is yet another class of frequency estimator, which uses the concept of the phase-

locked loop. In 1997 Bodson and Douglas [79] presented an algorithm in continuous time

domain form. The new formulation is shown to produce a good local convergence property.

This property is very helpful for some geared rotor system with an almost stationary rotation

speed of the transmission such as in helicopter systems. For this class of system, numerous peaks

exist in the spectrum. Having a good local convergence property eases tracking and locking

fundamental frequency. However, it needs an initial rough estimation of the frequency to initiate
 62

the calculation process. Later, Bodson [80] also implemented the algorithm in discrete time

domain form. He showed that the algorithm could be used to effectively suppress the sinusoidal

signal in the presence of high noise through simulation results. From his results, the estimation

error is not only determined by the signal noise present in the sinusoidal signal, but also decided

by the transfer function of the compensator. Therefore, a good choice of compensators will give

a small estimation error. This does help in the design of the algorithm. This particular algorithm

is employed in this research.

4.5.2 Frequency Estimator and its Design Parameters

Figure 4.1 shows the overall control scheme proposed for the gearbox plant of interest.

The adaptive filter shown is updated by the filtered-x LMS algorithm. The error input of

FXLMS algorithm is essentially the housing vibration and the reference input is generated by the

proposed frequency estimator. The subsequent output is used to drive the actuator for

suppressing the housing vibration. The primary path is denoted by HhTE that relates the

transmission error excitation to housing vibration, while the secondary path Hhr is from actuator

to housing vibration as well. In the figure, the frequency estimator is a critical part of the

controller.

The proposed frequency estimator is shown in Figure 4.10. This algorithm is similar to

Bodsons algorithm [80] as mentioned above. However, Bodsons algorithm contains an

additive G matrix that is used to compensate for the primary path. Here, the G matrix is defined

as the identity matrix. Thus, it is not shown in Figure 4.10. In the schematic shown, yh is the

input signal whose frequency must be estimated, and n is the additive signal noise. The signal r,

which is used as the reference signal for the filtered-x LMS algorithm shown in Figure 4.3, is the
 63

output of the estimator. Finally, C(z) is the compensator of the estimator designed to stabilize

the system, 2 is the estimated frequency, and 1 is the estimated amplitude.

+
Normalization
yh(n) +
u(n)
r(n)
2 y1
Shr(z)
?
? cos() C(z)
1 1 y2
?
y(n) z 1
-sin()

Figure 4.10. The structure of the proposed frequency estimator.

From Reference [80], for the proposed frequency estimator, the control law is given by

C1 ( z )
1 z 1 y1
= C ( z ) , (4.14)
2 2 y2
z 1

where C1(z) and C2(z) are the compensators of the two channels, as

C1 ( z ) = g1 , (4.15)

z za
C2 ( z) = g 2 . (4.16)
z zb

In the Equations (4.14)-(4.16), g1 , g 2 , z a , and z b are the design parameters. There are in fact

two simple methods available for the design of these parameters. The first uses

3(1 z d ) 2 z +2
g1 = 1 z d , g 2 = , za = d , z b = 3z d 2 , (4.17)
d1 3
 64

where z d is some desirable location to place the closed-loop poles, and d 1 is the roughly

estimated amplitude of the input signal yh. The other option employs

2(1 z d ) z +1
g2 = , za = d , zb = 0 , (4.18)
d1 2

in which the pole of the compensator for the second channel is placed at the origin, thus allowing

one to neglect its effect entirely. Note that the design parameters mentioned above are critical to

the frequency estimator. Poor selections of these parameters can make the system unstable

and/or ineffective.

From the parametric design formula, it can be seen that the parameter is quite sensitive to

the amplitude of the input signal. In real application, the error signal is reduced by the

adaptation of the controller. This will lead to the instability of the frequency estimator to track

the frequency. In order to address this problem, Bodson [81] also proposed a method, which

subtracts the effect of the control signal feeding into the actuator from the error signal. Using

this technique, the amplitude of signal feeding into the frequency estimator remains constant if

the external excitation is steady. This technique can be used if the secondary path is modeled.

This implies that the offline modeling technique should be used first. However, this kind of

method is applicable to the case when the secondary path remains constant. For the time-varying

system, the secondary path must be identified online. Thus, this particular method cannot be

applied. Also, the system response is different for different frequency point. Under this

condition, the parameters must be updated constantly, which is not very convenient. In this

research, the method is improved by dividing the input signal of frequency estimator by its RMS

value. Thus, the signal is almost normalized to a unit amplitude. This improved the

performance of the algorithm significantly.

 65

4.5.3 Performance of Proposed Frequency Estimator

The performance of the frequency estimator is studied next to provide guidance in the

future experimental work. The sensitivity of the performance of the frequency estimator toward

critical design parameters is studied. This includes the initial values of the integrators used in the

algorithm, tracking ability of the algorithm due to the changing frequency signal, and the signal

noise effect on the estimation result.

A) The effect of initial values of integrators

There are a total of four integrators in the algorithm. Two of these are very sensitive to

the initial values. They are amplitude and frequency integrators. Here, the amplitude integrator

uses its output as the estimated amplitude of input signal. Similarly, the frequency integrator

also uses its output as the estimated frequency of the input signal. The other two integrators

initial values are based on the estimated initial phase of the signal. For sinusoidal signal, the

phase is not very critical, and therefore can be set to zero.

Figure 4.11 shows the effect of various initial value of frequency integrator on the

performance of frequency estimation. In this case, the true frequency value is 500Hz. From the

figure, it can be seen that the initial guess of 510Hz needs about 0.1 second to converge to the

true value. When the initial guess is at 520Hz, about 0.14 second is needed, and for a 530 Hz

guess value, a much longer time (about 0.24 second) is needed. Hence, the less accurate the

initial frequency value is, the longer it takes to converge to the true value, as expected.
 66

550

540
Estimated Frequency (Hz)

530

520

510

500

490

480
0 0.05 0.1 0.15 0.2 0.25 0.3
Time (sec)

Figure 4.11. Effect of initial frequency guess on convergence rates. (Initial guess values
of: , 510Hz; , 520Hz; , 530Hz)

Figure 4.12 shows the effect of the various initial value of the amplitude integrator. Here, the

amplitude of input signal yh is 10. After normalization, the signals amplitude is almost 1. In this

figure, three cases are studied. The initial values of the amplitude integrator are 1 (solid line), 10

(dotted line), and 15 (dash line), respectively. From the figure, it can be seen that the algorithm

cannot converge to the true value when the initial value is 15. In general, the farther away its

initial amplitude guess is from the true amplitude, the worse the result becomes.
 67

200

150
Estimated Frequency (Hz)

100

50

-50

-100
0 0.05 0.1 0.15 0.2 0.25 0.3
Time (sec)
Figure 4.12. Comparison of results for different initial amplitude guess. (Initial guess values
of: , 1; , 10; , 15)

B) Tracking ability of frequency estimator

In practical applications, the frequency of the excitation source may be changing with

time. In order to further the study of the performance of the proposed frequency estimator,

example cases in which the signals frequency is changing with time are studied and results are

shown in the following plots.

In Figure 4.13, it is shown that the tracking ability of the frequency estimator in the case

of the frequency of signal varies linearly with time from 500Hz to 900Hz within 9 secs. In this

simulation, the effect of measurement error, in the form of white noise, is added into the input

signal. From this figure, it is observed that some initial time period is needed to converge to the

true value. Once the estimator reaches the true value, though, the estimator is then able to track

the time-varying frequency quite faithfully.

 68

950

900

Estimated Frequency (Hz) 850

800

750

700

650

600

550
0 1 2 3 4 5 6 7 8 9
Time (sec)
Figure 4.13. Result of frequency estimation for linearly time-varying frequency in the presence
of measurement signal noise. The initial frequency guess is 600 Hz and the true start frequency
is 500Hz.

Figure 4.14 shows the tracking ability of frequency estimator for the frequency jump

phenomenon. In this case, the frequency of signal jumps from 100Hz to 150Hz at 1 sec. From

the figure, it can be seen that the proposed frequency estimator shows a good tracking ability for

frequency jump case. For some real gearbox systems, the mesh frequency can be kept almost

constant. From the result, this frequency estimator can track any small change in frequency

values.
 69

200

180

Estimated Frequency (Hz)

160

140

120

100

80

60
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
Time (sec)
Figure 4.14. Result of frequency estimation for frequency jump phenomenon.

C) Comparison of proposed frequency estimator and Kims method

The performance of the proposed frequency estimator is further evaluated by comparing

its performance relative to the adaptive notch filter technique in Reference [66] by tracking

simulated signals with two strong mesh harmonics. Two cases are examined with both involving

a steady-state dynamic signal made up of a fundamental mesh frequency at 100 Hz and its

second harmonic at 200 Hz. In the first case, the amplitude of the fundamental mesh frequency

is twice that of the second harmonic, which is characteristic of geared rotor systems. The

frequency identification results of both the proposed and adaptive notch filter techniques are

shown in Figure 4.15. The comparison results show both techniques converge to the true

fundamental frequency. However, the proposed algorithm converges faster than the Kims

adaptive notch filter technique. Also, the transient signals of the proposed frequency estimation

technique are more bounded while the adaptive notch filter signals go through a larger variation.
 70

However, the proposed algorithm must rely on an initial guess as described earlier, while the

adaptive notch filter is not so sensitive to the initial value, which is actually easier to implement.

In a second case, the amplitude of the fundamental mesh is assumed to be half of the amplitude

of the second harmonic. The simulation processes are repeated again using both techniques and

results are shown in Figure 4.16. The proposed technique is still able to converge to the

fundamental mesh frequency solution as long as the initial guess is not too far off. On the other

hand, the adaptive notch filter zooms in on the second harmonic and completely missed the

fundamental frequency.

600

500

400
Frequency (Hz)

300

200

100

0
0 0.5 1 1.5 2
Time (sec)
Figure 4.15. Comparison of results predicted by the proposed frequency estimator ( )and
Kims adaptive notch filter technique( ).
 71

600

500

400
Frequency (Hz)

300

200

100

0
0 0.5 1 1.5 2
Time (sec)
Figure 4.16. Comparison of results predicted by the proposed frequency estimator( ) and
Kims adaptive notch filter technique( ).

4.5.4 Improvement of Frequency Estimator

From the above results, it can be seen that when signal noise is present, the estimated

result also contains signal noise. In order to get a more accurate result, a low-pass filter is added

to the algorithm to filter out the signal noise in the estimated result. In the following analysis,

the effect of the filter on the estimated results is examined. In this case, two low-pass

butterworth filters both with 6 orders are used. Their cut-off frequencies are 20Hz and 100Hz,

respectively (Sampling frequency is 20kHz). In Figure 4.17, the result for cut-off frequency of

100Hz is shown. From this figure, residual signal noise can be seen in the upper plot, while the

bottom plot is much cleaner. Also, in Figure 4.18, the bottom plot converges to the true value

(100Hz) better than the ones in Figure 4.17. This is because it has a lower cut-off frequency than

the first case. However, notice that the low-pass filter results do introduce a phase delay in the
 72

algorithm. Thus, the response without filter converges in 0.05 sec (upper plot in Figures 4.17

and 4.18). For the 100Hz cut-off frequency of low-pass filter, the response converges in 0.1 sec

(lower plot in Figure 4.17), and for the 20Hz cut-off frequency, it converges in 0.3 sec (lower

plot in Figure 4.18). Therefore, a suitable low-pass filter is required depending on the nature of

the overall system.

Estimated Frequency (Hz)

200
(a)
150

100

50

0
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
Estimated Frequency (Hz)

200

(b)
150

100

50

0
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
Time (sec)

Figure 4.17. Comparison of estimated result for with filter and without filter case. Filter cut-
off frequency is 100Hz: (a): without filter; (b): with filter.
 73

Estimated Frequency (Hz) Estimated Frequency (Hz)

200

(a)
150

100

50

0
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

200

(b)
150

100

50

0
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
Time (sec)

Figure 4.18. Comparison of estimated result for with filter and without filter case. Filter cut-
off frequency is 20Hz: (a): without filter; (b): with filter.

4.5.5 Frequency Estimation Applied to Measured Vibration Data

To further test the capability of the proposed frequency estimator technique, a set of

measured vibration data acquired from an actual gearbox setup is used. In this experiment, the

gearbox is continuously ramped up in speed and then ramped down. The estimated result of the

fundamental gear mesh frequency is shown in Figure 4.19 in comparison to the true gear mesh

frequency measured using an optical speed sensor. In general, the results obtained are fairly

good even though it took a brief time for the algorithm to converge. It can be expected that if the

shaft speed is almost constant, the frequency estimator can lock into the gear mesh frequency

much quicker.
 74

1000

900

800

700
Frequency (Hz)

600

500

400

300

200

100

0
0 20 40 60 80 100 120
Time (sec)

Figure 4.19. Result of frequency estimation applied to measured gearbox vibration data
compared to the true mesh frequency signal. ( , Estimated frequency; ,True
frequency)
 CHAPTER 5

PROPOSED ALGORITHM SIMULATION STUDY

In this section, the simulation study of active gear casing vibration control based on the

proposed algorithm, which is described in the previous chapter is performed. The simulation

results are used to guide the experimental design of the active vibration control system in the

next experimental work.

5.1 Simulation Study

In this subsection, simulation studies are performed on the active vibration control of the

gear pair system. The system model used in simulation is a state-space model whose state space

matrix is generated as described in Chapter 3. Figure 5.1 shows the MATLAB model with

offline secondary path modeling feature used in this simulation study. The model includes one

FIR adaptive filter with 64-order that is updated by the FXLMS algorithm. The Sw block shown

in Figure 5.1 is the FIR filter represented by a vector length of 1000. It represents the offline

identified secondary path. The bandpass filter used is an eight-order Butterworth filter with low

and high cut-off frequencies of 100 Hz and 4 kHz, respectively. The switch shown is used to turn

the ANC on and off. The time is controlled by a clock. The signal compensator is used to

compensate the effect of the control force on the error signal in order to maintain an almost

constant error signal, which is crucial in achieving a well working frequency estimator. The sine

wave generator is used to produce a reference signal for the FXLMS algorithm using the

estimated frequency. It can be easily controlled to generate a suitable signal with desired

75
 76

harmonics. The signal and spectrum displays are used to monitor the control results. The

sampling rate used is 20 kHz.

signal TE excitation butter

Housing Vibration
TE excitation Actuation force
Signal
Gear Pair System Bandpass Display
State-Space Model Filter
Error
In Out In Out Save Data
Err FXLMS
Sinewave Sw Taps
Generator
Control Filter Clock

0
Sw
Switch
Constant
Signal
Out In Out
Sw

Frequency Signal
Estimator Compensator

B-FFT
Spectrum
Dispaly

Figure 5.1. MATLAB model with offline secondary path modeling feature.

Figure 5.2 shows the simplest control case applying the proposed FXLMS algorithm with

an offline secondary path identification method. In this simulation, the sinusoidal TE excitation

occurs at 450 Hz with amplitude of 30m. The active controller is switched on at time equals to

3 sec. From the figure, it needs about 0.3 sec to converge. Note that the converging time is

dependent on the learning rate of the FXLMS algorithm. The larger the learning rate of the

FXLMS algorithm, the sooner it converges. However, it does have a limitation as described in
 77

Chapter 4. The finite impulse response of the offline identified secondary path is shown in Figure

5.3. Figure 5.4 illustrates the resultant frequency response function. The upper plot is the phase

response and the bottom one is the amplitude response. From this figure, the poles and zeros of

the system can be easily identified. The main pole of the system is around 420 Hz. Here, only the

response below 1kHz is shown, which is the main concern here.

2.5

1.5
Housing Vibration (m/s )
2

0.5

-0.5

-1

-1.5

-2

-2.5
2.9 2.95 3 3.05 3.1 3.15 3.2 3.25 3.3 3.35 3.4
Time (sec)

Figure 5.2. Control result using FXLMS algorithm with offline secondary path modeling
(450Hz).
 78

-4
x 10
4

Tap Weights of FIR Filter 2

-2

-4
0 100 200 300 400 500 600 700 800 900 1000
Tap Index (n)

Figure 5.3. The finite impulse response of the offline identified secondary path.
Phase (Degree)

100
0
-100

100 200 300 400 500 600 700 800 900 1000
Amplitude (m/s )

-2
2

10

-3
10
100 200 300 400 500 600 700 800 900 1000
Frequency (Hz)

Figure 5.4. The frequency response of the offline identified secondary path.
 79

Figure 5.5 shows the control result using online identification method. It shows the RMS

value of housing vibration. The active controller is switched on at time equals 0.3 sec. The

secondary path modeling begins at time equals 0.3 sec. The TE excitation signal is at 400Hz.

From this figure, it can be seen that a much longer time is needed to achieve convergence. Also,

in Figure 5.5, the amplitude of housing vibration behaves complicated. The reason is clearly

seen in Figure 5.6, which gives the plot of amplitude response functions of filter S (z ) and

illustrates the plot of phase response functions of filter S (z ) . The solid line is the offline

modeling result, which can be thought of as the true response of the system. The dashed line,

dotted line, and dashed-dotted line are online modeling results at time equals 1.2 sec, 1.8 sec, and

5 sec, respectively. From the phase response, we can see that at time equals 1.2 sec and 5 sec,

the phase of the estimated transfer function of the secondary path is around 0 at 400Hz, the true

phase is also around 0 radian from offline result. This is the reason that leads to the decrease in

housing vibration. At 1.8 sec, the phase of the estimated transfer function of the secondary path

is around 2 radians at 400Hz, and the phase error of the estimated filter is larger than /2 radians.

This shows that around 1.8 sec, the error signal increases. Also, from Figure 5.5, it can be seen

that the additive noise remains in the residual signal after 4 sec. From Figures 5.6(a), one

observed that the amplitude response of S (z ) is not good. It can be concluded that phase

response of S (z ) is more critical than amplitude response for the control system. In all of the

above control cases, the reference signal is the same as the TE excitation signal. This is done

intentionally to test the secondary path modeling method.

 80

8
RMS Value of Error Signal (m/s )
2

0
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
Time (sec)

Figure 5.5. Control result using online secondary path modeling.

 81

-1

Amplitude of Response of Secondary Path (m/s 2)

10
(a)
-2
10

-3
10

-4
10

-5
10

-6
10
0 200 400 600 800 1000
Frequency (Hz)

4
Phase of Response of Secondary Path (rad)

-1

-2

-3

-4
0 200 400 600 800 1000
Frequency (Hz)

Figure 5.6. Amplitude and phase responses of estimated filter of secondary path.
( , offline result; , online at 1.2 sec; , online at 1.8 sec; , online at
5 sec).
 82

Figure 5.7 examines the case of attempting to control three sinusoidal components, i.e.,

the fundamental one at 200Hz, the second and third harmonics. The solid line is the housing

vibration spectrum without control. The dotted line is the housing vibration spectrum with

control turned on. The amplitudes of these three sinusoidal components in TE excitation signal

are all 30m. Also, a small amount of white noise filtered by a low pass filter with cut-off

frequency of 2 kHz as background noise is fed into the system. The result shows excellent

performance by the controller after convergence. In this case, the three harmonic components in

the reference signal are generated with same amplitude.

1
10

0
10
Housing Vibration (m/s2)

-1
10

-2
10

-3
10

0 200 400 600 800 1000

Frequency (Hz)

Figure 5.7. Control results of response with three sinusoidal components. ( , without
control; , with control)

However, its far known that the converging rates at those different harmonics are

different and associated with the amplitude response of secondary path for this time domain

FXLMS algorithm. But for frequency domain FXLMS algorithm, the converging rate can be
 83

adjusted by setting different learning rate value to different frequency component. Here,

simulation results show that one can control the converging rate of each harmonics by setting the

amplitude ratio between the various harmonics in the reference signal.

In order to get a suitable amplitude ratio between the various harmonics in the reference

signal, several cases are studied next. Those results are shown in Figure 5.8. In all cases shown

in Figure 5.8, the amplitudes of three harmonics of TE excitation are 30m. In all cases, the

learning rates are the same, and the frequency response function of the error signals shown in

Figure 5.8 are taken at the same time. In plot (a), the amplitudes of the three harmonics in the

reference signals are 2.49, 1, and 4.48. These values are taken from the amplitude response of

the identified secondary path. The amplitude response function of the identified secondary path

at 230Hz, 460Hz, and 690Hz are 5.6? 10-3 m/s2, 1.4 ? 10-2 m/s2, and 3.1 ? 10-3 m/s2, respectively.

The amplitude response of 460Hz is about 2.49 times that of the 230Hz, and 4.48 times that of

the 690Hz. From this plot, it can be seen that all of the harmonics can be reduced by almost the

same amount simultaneously. In plot (b), the amplitudes of the three harmonics in the reference

signals are 1, 1, and 1. From this plot, it can be seen that the first and third harmonics cannot be

reduced by much simultaneously, but the second harmonic frequency response can be reduced

significantly. In plot (c), the amplitude ratios of the three harmonics in the reference signals are

square of the ones in first case. Here, the second harmonic cannot be reduced more than plot (b).

From these three cases, it can be seen that in order to reduce all harmonics simultaneously, the

ratio of the amplitude of all harmonics should be inversely proportional to the amplitude

response of the secondary path. To further demonstrate this effect, it is assumed the secondary

path is a pure delay, which means that the frequency response amplitude of this secondary path

at each harmonic is the same. Moreover, the amplitudes of the three harmonics in the reference
 84

signal are set to unity. The control result of this case is shown in plot (d), which gives the same

vibration reduction in all three harmonics. From the control law of the FXLMS algorithm shown

in Table 4.1, the updating of the weights of the control filter depends on multiplication of the

learning rate , filtered reference signal X ' (n) , and error signal y h (n) . Since the learning rate

for each frequency component and the amplitude of each frequency component of the error

signal are set to be same, the convergence rate of each frequency component will depend on the

amplitude of that frequency component in the reference signal that is filtered by the estimated

secondary path. If the amplitude ratio between the various harmonics in the reference signal is

set to be inversely proportional to the amplitude response of the identified secondary path, the

amplitudes of each harmonic in the filtered reference signal must be almost the same amplitude.

Thus, the convergence speed and ultimately the vibration reduction of each harmonic component

turn out to be almost the same. From the above analysis, it can be concluded that the amplitude

ratio of the various harmonics is closely related to the amplitude response of the secondary path.
 85

(a) (b)
0 0
10 10
Housing Vibration (m/s2)

Housing Vibration (m/s2)

-1 -1
10 10

-2 -2
10 10

0 200 400 600 800 1000 0 200 400 600 800 1000
Frequency (Hz) Frequency (Hz)

(c) (d)

0 0
10 10
Housing Vibration (m/s2)

Housing Vibration (m/s2)

-1 -1
10 10

-2 -2
10 10

0 200 400 600 800 1000 0 200 400 600 800 1000
Frequency (Hz) Frequency (Hz)

Figure 5.8. Effect of amplitude ratios of the harmonic components in reference signal on the
control results. ( , without control; , with control): (a) amplitude ratios between
three harmonics is 2.49:1:4.48; (b) amplitude ratios between three harmonics is 1:1:1; (c)
amplitude ratios between three harmonics is square of case (a); (d) amplitude ratios between
three harmonics is 1:1:1, and Hhr(z) is a pure delay.
 86

(a)
2

Housing Vibration (m/s )

2

-1

-2

-3
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
Time (sec)

550

(b)
500
Estimated Frequency (Hz)

450

400

350

300

250
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
Time (sec)

Figure 5.9. Control result of the frequency jump phenomenon jumping from 450Hz to 350Hz:
(a) the time series of the error signal, (b) estimated frequency.
 87

Figure 5.9 shows the control result of the frequency jump phenomenon. In this case, the

TE excitation frequency jumps from 450Hz to 350Hz at 0.3 sec. Plot (a) is the time series of the

error signal (housing vibration). The ANC is switched on at time equals 0.1 sec. From plot (a),

it can be seen that it takes about 0.1 sec for the controller to drive the housing vibration to zero

for the 450Hz signal. The controller cannot work well at 350Hz immediately when the

frequency of excitation signal jumps from 450Hz to 350Hz at 0.3 sec. It takes an additional 0.3

sec to converge again. This is because of the delay in the frequency estimator that cannot

converge to 350Hz immediately as seen from plot (b), which shows the estimated frequency.

One can see that it takes almost 0.25sec to converge to the true frequency 350Hz after the

frequency jump point.

Figure 5.10 shows the control result of a second frequency jump case. In this case, the

TE excitation frequency jumps from 450Hz to 400Hz at 0.3 sec. Plot (a) is the time series of the

error signal. The ANC is also switched on at time equals 0.1 sec. From plot (a), it can be seen

that it takes about 0.1 sec for the controller to attenuate the housing vibration to nearly zero.

When the frequency of the excitation signal jumps from 450Hz to 400Hz at 0.3 sec, the

controller needs to adapt again to the new frequency. Note that it takes almost 0.15 sec to

converge again. This is because the frequency estimator cannot converge to 400Hz immediately.

This can be easily seen from plot (b) depicting the estimation result of frequency estimator.

Here, one can see that it takes almost 0.05 sec to converge to the true frequency of 320Hz after

the jump point. From these two cases, it can be seen that the smaller the frequency change

occurs, the sooner the proposed controller converges after the jump point. Thus, for a relative

stable system, this proposed frequency estimator is satisfied.

 88

3 (a)

Housing Vibration (m/s ) 2

2

-1

-2

-3

-4
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
Time (sec)

550

(b)
Estimated Frequency (Hz)

500

450

400

350
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
Time (sec)

Figure 5.10. Control result of the frequency jump phenomenon jumping from 450Hz to 400Hz:
(a) the time series of the error signal, (b) estimated frequency.
 89

Figure 5.11 shows the control case in which the excitation signal is based on measured

data from a real gearbox system. In this simulation, the error signal due to TE excitation is

replaced by the measured data. This new error signal will be added into the error signal due to

the actuation force yielding a total error signal as the control error signal of the FXLMS

algorithm. The secondary path (from actuator excitation to housing response) is a measured one,

which is shown in Figure 5.12. In this case, the first two harmonics are attenuated greatly.

-20

-25
Power Spectrum Magnitude (dB)

-30

-35

-40

-45

-50
0 100 200 300 400 500 600 700 800
Frequency (Hz)

Figure 5.11 Control result of first two harmonics of measured data.

( , without control; , with control).

 90

0.03

Impulse Response of Secondary Path

0.02

0.01

-0.01

-0.02

-0.03

-0.04
0 20 40 60 80 100 120
Tap Index (n)
Figure 5.12 Measured finite impulse response of the secondary path.
 CHAPTER 6

ACTIVE GEAR VIBRATION CONTROL SYSTEM AND EXPERIMENTAL STUDY

As discussed in previous sections, the proposed active gear control system with

developed algorithm which is enhanced filtered-x LMS algorithm with frequency estimation

feature is tested in simulation study. In this section, the propose control system is implemented

on a power re-circulating gear system. And experimental studies are performed to test effect of

the proposed control system on gear mesh suppression.

6.1 Active Gear Control System

The cross-sectional view illustrating the inner details of the closed-loop, power re-

circulating, gearbox setup is shown in Figure 6.1. And, the real system is shown in Figure 6.2.

Figure 6.2(a) shows the gear system and used equipments for active control study. Figure 6.2(b)

shows detailed information of the target gearbox system. The basic experimental setup consists

of a belt driver, a torque transducer, a speed sensor, a clutch, a flywheel, one slave gearbox and

one test gearbox. The test gearbox, which is being treated with the active vibration control

system, contains a spur gear pair with a 1:1 tooth ratio. Each gear has identically 35 teeth, 25.4

mm (one-inch) face width and 152.4 mm (6-inch) pitch diameter. The pressure angle is about 20

degrees. The torque transducer is used to record the operating load. The speed sensor provides a

rotation pulse train that consists of 60 pulses per shaft revolution for computing mesh

frequencies. To simulate the high power density without the need for a high power driver, the

91
 92

gears can be preloaded by adding torque to one of the shafts coupling the two gearboxes. Of

course, this will also create a matching static torque in the other shaft. The flywheel and

couplings are used to isolate vibrations in the slave gearbox from the test gearbox.

Added Needle
Bearing

Belt Driver
Acc Acc4
Pinion Coupling Housing
Motor
Bearing
Speed Sensor

Torque Load Cell

Transducer Acc2 Acc3 Mic

Stinger Rod
Actuator

Flywheel & Coupling Pinion

Gear
Clutch

Slave Gearbox Test Gearbox

Figure 6.1 Schematic diagram of the closed-loop, power recirculation gearbox setup equipped
with an active shaft transverse vibration control system.
 93

Power Amplifier &

Signal Conditioner

Drive Motor

Power Recirculation Shafts

PC & dSPACE System

Actuator Support

Test Gearbox

(a)

5 6
10 11
8

7
3
Accy
Accx
9

4
2

1
(b)
Detail information of the active gearbox vibration control system: (1) test gearbox; (2) slave
gearbox; (3) belt driver; (4) clutch and flywheel; (5) torque transducer; (6) speed sensor; (7)
piezoelectric stack actuator; (8) adjustable preload nut; (9) accelerometer; (10) power amplifier.

Figure 6.2 Photograph of experimental test system.

 94

20mm

(a) (b)

Figure 6.3 Closed-up view of: (a) actuation setup, and (b) PZT actuator.

In order to monitor the vibration levels and used as error signal to be controlled in the

experiment, one tri-axial accelerometer is attached to the gearbox housing just above the driven

shaft bearing location facing the slave gearbox (labeled as Acc). The x and y axes of the

accelerometer are perpendicular and parallel to the gear mesh line-of-action, respectively. The

angle between the y direction and vertical is adjusted to 20 degrees, which is identical to the

pressure angle. The y-axis signal from the accelerometer, denoted as Accy, is used as the error

signal for the controller. Other than this tri-axial accelerometer, to monitor the vibrations at

other locations on gearbox housing, three similar tri-axial accelerometers are attached to gearbox

housing right above other three support bearings as shown in Figure 6.1. Among these three

accelerometers, the tri-axial accelerometer labeled as Acc2 is at same side housing panel with

Acc. And, other two accelerometers, Acc3 and Acc4, are located at another side panel. The

acquired vibration signals are processed by a PowerPC board located in a host PC. The actuation

method includes a piezoelectric stack actuator, which is shown in Figure 6.3, connected to the

driven shaft via a needle bearing. The actuator is driven by a high voltage amplifier that receives

command signals from the controller. Note that the actuation direction is parallel to the gear
 95

mesh line-of-action, such that the active control force is better able to disrupt mesh force

generation and vibration transmissibility. Also, in the mounting of the actuator, a proper preload

level is set to avoid problems associated with bearing clearances as well as gear eccentricity.

Furthermore, to compensate for possible run-out and excessive shaft deflection that can generate

harmful lateral forces to the actuator, a thin stinger rod that is stiff in the longitudinal direction

and relatively flexible in the transverse direction is used. The other end of the force sensor is

mounted directly to the secondary bearing outer raceway.

Table 6.1 lists all hardware and equipment used in the experimental study.

Table 6.1 List of hardware for the active vibration control system.

Component Model/Type Manufacturer

Personal Computer Pentium 2.0G Hz Dell

Actuator PSt1000/16/40 VS 25 Piezomechanic GmbH

Power amplifier E-420.00 PI

Accelerometer 356A11 PCB

Signal conditioner Model 481 PCB

Force sensor 208C03 PCB

Oscilloscope 4CH-Digital Tektronix

DSP board ACE Kit 1104 250MHz dSPACE

Digital recorder LX 10 TEAC

Sound level meter NA-27 RION

Magnetic speed sensor Key Transducers, Inc.

Torque Sensor Key Transducers, Inc.

 96

6.2 Adaptive Controller

As noted above, the proposed control scheme involves a modified filtered-x least mean

square algorithm with frequency estimation. The filtered-x least mean square (FXLMS) control

algorithm, which is an extended version of the LMS algorithm typically used for dynamic

systems with phase delay secondary paths, is well studied and has been widely applied to many

active vibration and noise control applications. This controller requires detailed information

about the secondary path transfer function. For the gearbox system being studied here, the

secondary path transfer function describes the part of the system from the output of the adaptive

controller to the measured housing response (referred to earlier as the error signal). The

controller also requires a clean reference signal that is coherent with the primary disturbance.

The first issue, concerning the need for a secondary path transfer function, can be handled by

applying a system identification method. Because of the periodic nature of the gear vibration

signal, a predictive error filter (PEF) can be used to increase the accuracy of the secondary path

modeling. The PEF filter, which was proposed by Kuo and his coworker [82], reduces the effect

of a periodic primary signal on the results of the secondary path identification. This approach is

illustrated in the overall control model illustrated in Figure 6.4. Although the overall approach

has already been mentioned, the diagram will be described in more detail below.

The coherence between the reference signal and gear transmission error excitation at the

target mesh frequencies greatly affects the achievable vibration attenuation performance. To

ensure an adequate coherence level between the reference and excitation signals, the exact

fundamental gear mesh frequency must be obtained in advance. For most geared rotor systems,

the shaft rotational speeds vary to a certain degree even for steady-state operating condition,

which makes it difficult to obtain the instantaneous shaft rotation speed precisely and thus the
 97

gear mesh frequency as well. Furthermore, the estimation error tends to increase at higher

harmonics due to the multiplying nature of the calculation. The estimation errors encountered at

the higher harmonics can deteriorate the vibration reduction performance especially for orders

above the third mesh harmonic, as discovered in the previous study [10]. To alleviate this

problem, a speed sensor is located inside the torque transducer. This sensor has a higher tooth

count (60) than the test gear (35) employed in the experiments. The acquired pulses from the

speed sensor are used in a frequency estimator that uses the phase-lock loop concept [79] to

estimate in real time the instantaneous shaft rotation frequency and subsequently the fundamental

gear mesh frequency and its harmonics. The performance of the proposed frequency estimator in

terms of tracking the signal frequency change was discussed in detail and tested in the previous

sections. Using this proposed technique, the reference signal r(n) at the target mesh frequencies

can be accurately synthesized.

TE Excitation d(n)
HhTE(z) +
Rotation te(n) yh(n)
Pulse Train Gearbox System +

Frequency r(n) u(n) +

W(z) H2(z)
Estimator
+ Predictive
Error Filter
Noise v(n) - +
Generator h2 ( z )
h2 ( z )
LMS2
LMS1
Controller

Figure 6.4. Enhanced FXLMS algorithm with secondary path identification and frequency
estimation modules.
 98

In the control schematic diagram shown in Figure 6.4, HhTE(z) represents the transfer

function between the gear transmission error (TE) signal te(n) and the untreated gear housing

vibration response d(n). The secondary path transfer function is given by H2(z), which relates the

controller output to the gearbox housing vibration at the control point. The housing vibration

response along the gear line-of-action, Accy, is used as the error signal yh(n) in the filtered-x

LMS algorithm for adjusting the adaptive controller W(z). The estimated secondary path transfer

function h2 ( z ) is identified by injecting a small amount of white noise v(n) through the actuator.

By using a secondary disturbance that is uncorrelated with the controller output signal u(n), the

secondary path can be updated using the standard LMS algorithm. Note that the overall

performance of the control system was simulated and refined using Matlab/Simulink software

package in an earlier simulation study in Chapter 5.

The active control system described above has been implemented on a dSPACE DS1104

PowerPC board. The processing chip is a Motorola PowerPC 603e model that operates at the

speed of 250 MHz. The Simulink real-time workshop software can be used to generate the

required C codes and also to transfer the resultant objective codes to the processor. Although

this step is very convenient, some of the auto-generated C-codes are not efficient enough to allow

for real-time execution of the control algorithm. This problem is more acute when both the

controller and secondary path transfer function require the use of high order FIR models, such as

those larger than 64. In order to improve the computational efficiency of the algorithm, several

C-coded S-functions were programmed manually to implement some of the critical controller

blocks, including the control law updates and the secondary path identification part.

Furthermore, in order to communicate directly with the PowerPC processor, such as for adjusting

critical control parameters, and monitoring and downloading important signals, a graphical user
 99

interface control desktop was designed in the Matlab environment using MLib, which is part of

the dSPACE system. The designed UGI of gear active control system is shown in Figure 6.5.

As shown in Figure 6.5, some control parameters of algorithm such as step size, order of FIR

filter, probe signal strength, etc., can be adjusted manually in the designed GUI. The identified

secondary path transfer function can also be shown in the graph.

Monitored time domain signal

Identified secondary path response Monitored frequency domain error signal

Figure 6.5 Designed GUI of gear active control system

Now that the test system hardware and the active controller have been described in detail,

experimental results associated with measuring the performance of the active vibration control

system will be described in the next sections. The following sections also discuss the range of

experiments performed to evaluate the ability of the proposed active system to control various

mesh harmonics over a range of operating speeds.

 100

6.3 Actuation System Dynamics Analysis and Experimental Validation

After the actuation system is set up, its system dynamics can be varied by comparing the

experimental study and analytical result. The system dynamics analysis including piezoelectric

actuator model is introduced in previous chapters. At that time, the system dynamics is based on

the FEA model. In this section, only measured system transfer functions by modal test are used

for analysis.

In the analysis, the gearbox system is assumed to be a linear time-invariant plant and the

transmission error excitation is the only excitation source to be controlled. To suppress the gear

housing response resulting from the gear mesh excitation, an actuation force, Fr, exerted by the

piezoelectric actuator is applied to the gear shaft using the hardware configuration described

above. The net housing response along the gear mesh line-of-action direction, yh, is

y h = y h ,TE + H hr Fr . (6.1)

Here, yh,TE is the housing vibration due to transmission error excitation and Hhr is the transfer

function relating the actuator force to the housing response, which is the cross-point frequency

response function measured using a standard modal testing technique. In order to completely

suppress the housing vibration, set yh = 0 and solve for the required actuation force as

Fr = y h ,TE / H hr . (6.2)

Notice that the needed active force in equation (6.2) is an ideal control force for a perfectly

modeled actuator and gear system. When the control force is generated by a real actuator, the

dynamics of the actuator must be considered as part of the analysis as well. Furthermore, before

predicting the required electrical requirement for suppressing gear housing mesh response, an

analytical description of the stack actuator is needed. A model developed for those purpose is

now discussed.
 101

From the chapter 2, the block force, i.e. the maximum possible output force that the

actuator can generate when it acts on a rigid structure with an infinite impedance, generated by

the actuator is shown as

Fb = d 33 k pV
. (6.3)

The maximum block force is actually equal to kp*lmax , where lmax is the maximum stroke

of actuator. The maximum free stroke of the actuator is equal to d 33Vmax , where Vmax is the

largest operating voltage. When an external load is applied to the actuator, then the maximum

stroke of the actuator will decrease. Thus, the effective output force will also reduce.

When the actuator acts against a complex host structure, such as the bearing supported

shaft structure in our system, the effective force applied by the actuator on the host structure can

be derived as

k ext k ext V k ext

F = k p d 33V = Fb = ( k p l max ) , (6.4)
k p + k ext k p + k ext Vmax k p + k ext

where Fb is block force, kp is the effective piezo-stack stiffness and kext is the equivalent

stiffness of that host structure. The driving point stiffness can be used as the equivalent stiffness

of the host structure along with equation (6.4) to calculate the applied force. Note that the

equivalent stiffness might be a complex value (i.e. dynamic stiffness), which means that the

displacement response will have a phase difference relative to the force excitation. From the

above equation, it can be seen that the effective force applied on the host structure depends

strongly on the stiffness of the host structure and the effective piezo-stack stiffness. The same

actuator can generate a higher dynamic force when acting on a stiffer host structure. Note that

since kext may vary with frequency, the force applied to the structure may also vary with

frequency, not only with applied voltage. When selecting a proper actuator, the maximum
 102

applied force determined using equation (6.4) should be larger than the required control force

determined using equation (6.2). Once the actual vibration level yh,TE , transfer function Hhr and

kext have been identified, equations (6.2) and (6.4) can be used to guide the selection of a suitable

piezoelectric actuator that is typically rated by the operating voltage range and maximum free

stroke.

The transfer function relating the actuator supplied voltage to the actuator force output

can also be derived from equation (6.4), where

F k ext l k ext
Hf = = k p d 33 = k p max
V k p + k ext Vmax k p + k ext
. (6.5)

By using this transfer function, the secondary path transfer function from the active

controller digital to analog converter (D/A) output to the gear housing vibration, can be

synthesized by

H 2 = g * H f H hr H o
, (6.6)

where H hr , H o and g are the measured cross-point frequency response function, the low-

pass filter transfer function between the D/A output and the piezoelectric amplifier, and the gain

of the piezoelectric amplifier, respectively.

As stated in Chapter 2, to determine the required actuator current, one can first determine

the electrical charge needed to drive the actuator by

k ext
q = C ps [1 k 2 ]V = C pV
k ext + k p
, (6.7)

where k 2 = d 332 / 33
T E
s33 is the piezo coupling coefficient, which is always less than unity,

and Cp is the effective capacitance of the entire piezo-actuator stack. By assuming a harmonic

time dependence, e jt , the electrical current I applied to the actuator can be found to be
 103

I = i C p V
, (6.8)

where i is the unit imaginary number and is the driving angular frequency of the voltage.

Since the required electrical current found using equation (6.8) should be less than the maximum

output current of the piezoelectric amplifier, equation (6.8) can be used in selecting an amplifier

after the piezoelectric actuator has been identified. It should be noted that heat generated by

continued operation of the actuator will increase the actuator electrical capacitance, Cp, as well as

the amplifier requirements. As a result, a slightly oversized amplifier should be selected. For

example, if the calculated maximum current value from equation (6.8) is 80 mA, an amplifier

with larger output, such as 100 mA, should be considered for the system.

With the analytical tools described thus far, it is now possible to select components for

the experimental active gearbox vibration system described earlier. The following section will

discuss the components selected for the particular experimental effort as well as provide a

discussion of specific measurement results.

Before the results are analyzed, though, one must first determine the effective stiffness of

the structure. To obtain the effective stiffness of the host structure as noted above, the driving-

point frequency response function is needed. Furthermore, to estimate the required actuation

force, the cross-point frequency response function is needed. These two frequency response

functions can be determined by performing a standard modal test without the actuator connected

to the host structure. The hammer impact location is the point where the actuator connects to the

structure. For the cross point frequency response function measurement, the accelerometer is

placed on the side panel of the gearbox housing right above the support bearing, as denoted by

Acc in Figure 6.1. Based on the measured frequency response functions and the operating

gearbox vibration spectrum, the analyses presented above were applied to select a commercially
 104

available piezoelectric stack actuator. Detailed specifications for this actuator are provided in

Table 2.1.

For this specific experimental gearbox system, the housing vibration is determined to be

at most 0.2 m/s2 at the mesh frequency between 550 Hz to 1,400 Hz. To suppress this level of

vibration using the actuator listed in Table 1, the power amplifier should be able to supply about

70 mA and a little less than 60 volts. In fact, the acceleration is actually much lower in some

cases. Based on this requirement, the most suitable amplifier is the E-420.00 model by PI (see

Table 6.2 for detail specifications). This amplifier also matches the stack actuator rating.

Table 6.2 Specification of power amplifier(Model E-420.00 by PI ).

Parameters Numerical Value (Unit)

Maximum Output Voltage 1000 V
Input Range -5 5 V
Output Range 0 1000 V
Gain 100
Maximum Average Output Current 100 mA
Average Output Power 100 W
DC offset 500 V

As described above, the actual force applied on the structure depends on both the

effective stiffness of the host structure as well as the actuator. Of course, when an actuator is

installed in the gearbox system, additional hardware including a stinger rod and support structure

is also needed. Here, only the stiffness of the stinger rod is considered because the other support

parts are significantly more rigid when compared to the gear structure. Therefore, the effective

stiffness of the host structure other than the actuator is

k sr k gs
k ext = , (6.9)
k sr + k gs
 105

where ksr is the effective stiffness of the stinger rod along its axial direction, which is 43 N/m

for a 0.32 cm (1/8 inch) in diameter and 3.81 cm (1.5 inches) in length stinger rod. The effective

stiffness of the gear structure is kgs and can be estimated from

F 2 2
k gs = = = , (6.10)
x &x& / F H rr

where Hrr is the measured driving point frequency response function.

There are two basic approaches to finding the secondary path transfer function, H2. The

first approach involves using equations (6.6) and (6.9-10) to calculate the H2 before the actuator

is installed. If the actuator has already been installed, then the alternate control system approach

described earlier is utilized. The secondary path transfer functions obtained using both of these

approaches are shown in Figure 6.6. Overall, the two results are found to be quite close to each

other. The small differences may be due to the fact that the modal test setup contains no actuator

preload, unlike the identification technique using the control system where the installed actuator

has a preload that acts along the gear mesh line-of action direction. Furthermore, when the

modal testing was conducted, the stationary torque in the system is held constant at 74 Nm.

When the system is running during the system identification process, the torque level may be

different and even fluctuate. This change in torque may also have contributed to the differences

between the results for both approaches.

To further verify the models, the predicted voltages for powering the actuator can be

compared to measurements taken while the closed-loop control is on. Under real operating

conditions, the predicted voltages for the actuator required can be calculated from equation (6.4),

measured cross-point FRF and the measured gear housing vibration level. The predicted voltage

and measured results for the piezoelectric stack actuator needed are shown in Figure 6.7. The

comparison demonstrates that the predicted values are in general close to the measured ones.
 106

Some differences can be seen as expected. From Figure 6.6, one can see that identified H2

response is lower than predicted result below 800 Hz or so. Thus, one can expect that the

experimental voltages for the actuator below 800 Hz are larger than the predicted values. This

effect is evident in Figure 6.7. Similar conclusion for the relation between measured and

predicted values can be obtained for other frequency points. Another reason can be the non-

exactness of the driving point Hrr and cross-point H hr frequency response functions. Those two

frequency response functions, measured using a standard modal testing setup, are obtained while

the geared rotor system is stationary with a certain level of locked torque. The actual system

response at a specific running speed may be different from those of the stationary geared rotor

system, since the overall alignment may be different. In addition, the sensitivity of the driving

point transfer function Hrr to accelerometer location may also affect the predicted force result.

Unfortunately, the electrical current provided by the amplifier cannot be measured due to

hardware limitations. Therefore, equation (6.8) cannot be directly verified. In spite of these

limitations, the results are still quite reliable.

 107

0
10

Frequency Response (g/volt)

-1
10

-2
10

-3
10

-4
10
0 500 1000 1500
Frequency (Hz)
Figure 6.6 Comparison of the frequency response functions of the secondary path for modal test
result combined with piezo-electric model ( ), and identification result applying the
developed control system ( ).
2
10

1
10
Voltage (Volts)

0
10

-1
10
500 600 700 800 900 1000 1100 1200 1300
Frequency (Hz)
Figure 6.7 Comparison of voltages at the actuator input: ( ), predicted; ( ),
measured.
 108

6.4 Control Experimental Study

After the implementation of the control algorithm and the design of the Matlab-based

GUI control desktop, a series of experiments were performed to verify the effectiveness of the

new control system. In this section, general control results for single harmonic control and

multi-harmonic control will be described at first. Then, detail results discussion will be

performed for different control system parameters.

The purpose of these experiments was to gauge the performance of the active control

system for a range of mesh harmonic targets and operating conditions in a high power density

gearbox system. Prior to describing the results obtained, there are a few system parameters that

should be defined first. During the experimental study, the FIR length of the controller W(z) can

be chosen from four options: 4, 16, 32 and 64. All of these options can be set in the control

desktop software interface described earlier. The length of the identified secondary path impulse

response h2 ( z) was set to 128. The controller parameter update rate was set to 4,096 Hz since the

control frequency of interest is below 1,200 Hz. All signals fed into the control board were

filtered by a low-pass filter that has a 1,400 Hz cut-off frequency. As described earlier, the

signal Accy along the gear mesh line-of-action (y-axis of accelerometer Acc) was taken as the

error signal. All vibration results shown below are given in decibels (dB) referenced to a 9.8

m/s2 (1g) acceleration amplitude. Also, it may be noted that in these experiments, fluctuations in

the shaft rotational speeds were observed. These speed fluctuations increase the difficulty of

achieving satisfactory active vibration control results. In spite of this difficulty, it will be seen

that the control results are quite reasonable. First the results for controlling a single mesh

harmonic will be presented. Then, the effect of two important control system features on the
 109

ability to control the vibrations will be discussed. Finally, the ability of the system to

simultaneously control the response at multiple mesh harmonics will be evaluated and discussed.

6.4.1 Single Gear Mesh Harmonic Control

In the first series of experiments, active vibration control of a single gear mesh harmonic

up to the 7th order for various operating shaft rotational speeds was studied. The results were

observed to be quite successful for the 2nd harmonic and higher cases. However, for the

fundamental gear mesh harmonic, the proposed active control system did not perform as well.

Because the actuator used is not capable of generating a force large enough to control the

gearbox system dynamics at the fundamental frequency, significant control could not be

achieved at this frequency. Therefore, results for this case are not presented. In the discussion

below, experimental results for controlling the 2nd and higher harmonics are presented.

-35
7 x mesh
Target frequencies
-40
3 x mesh

-45
Acc y (dB)

-50

-55

-60

-65
300 400 500 600 700 800 900 1000
Frequency (Hz)

Figure 6.8 Frequency domain active control result of Accy (dB re. 9.8m/s2) with shaft rotational
speed of 200 rpm. (Keys: , control off; , 3rd mesh harmonic control; , 7th
mesh harmonic control)
 110

-3
x 10
10

9
Control On

Housing Vibration (g) 8

3
2 4 6 8 10 12 14
Time (sec)

Figure 6.9 Time domain error signal Accy amplitude at the 3rd ( ) and 7th ( ) mesh
harmonics for active control with shaft rotational speed of 200 rpm.

0.4

0.35

0.3
Output of Controller (v)

0.25

0.2

0.15

0.1

Control On
0.05

0
2 4 6 8 10 12 14
Time (sec)

Figure 6.10 Time domain response of controller output signal at the 3rd ( ) and ( )
7th mesh harmonics with shaft rotational speed of 200 rpm.
 111

Figure 6.8 shows the typical active control results for the 3rd and 7th gear mesh harmonic

control cases with 26.0 N-m (230 in-lbf) of nominal pre-load torque and a 200 rpm shaft

rotational speed. Due to the dynamics of the system, the transmitted torque actually fluctuated

between 23.7 N-m (210 in-lbf) and 28.3 N-m (250 in-lbf). For discussion sake, the mean value of

26.0 N-m (230 in-lbf) is used as a reference. In the figure, the treated spectrum of gear housing

vibration at the 3rd harmonic is shown by the dash-dotted line, while the 7th harmonic case is

given by the dotted line. The comparisons of the controlled versus baseline response functions

reveal about 10 dB of reduction in the gear housing vibration at these two mesh frequencies. The

time series of the amplitudes for both of these harmonics are shown in Figure 6.9. Here, only the

error signals (i.e. housing vibration response along the line-of-action at the denoted frequency)

are shown. From the plots, it is clear that the response amplitudes at the target frequencies show

substantial reduction going from before to after control is activated. For the case of the 3rd mesh

harmonic, the gearbox housing vibration signal at this frequency decreased sharply from about

0.0617m/s2 (0.0063 g) to 0.0392m/s2 (0.004 g), which is about a four dB drop, and reached a

relatively stable magnitude after only about 1 second into the control mode. Similarly, the 7th

mesh harmonic of gearbox housing vibration dropped from about 0.0813m/s2 (0.0083 g) to

0.0333m/s2 (0.0034 g), which is approximately an eight dB reduction. Although not shown,

similar performances were also observed for the other gear mesh harmonics. The associated

controller output voltages for the piezoelectric amplifier for these two mesh harmonics are

shown in Figure 6.10. The required control voltages were found to be about 0.35 and 0.02 volts

for the 3rd and 7th mesh harmonics, respectively. These voltage strengths will later be compared

to levels computed using the secondary path transfer function and the uncontrolled response

levels at the respective frequencies.

 112

200

Phase (deg)
0

-200
0 200 400 600 800 1000 1200
0
10
Magnitude (g/v)

-1
10

-2
10

-3
10
0 200 400 600 800 1000 1200
Frequency (Hz)

Figure 6.11 Identified secondary path transfer functions. (Keys: , 200 rpm; , 300
rpm)

Figure 6.11 shows the identified secondary path transfer function, which describes the

system from the controller output to the gear housing vibration. The two identified results are

for 200 rpm and 300 rpm shaft speeds, respectively. From the results, the identified path

functions for the two speeds are seen to be quite close to each other. The similarity occurs

because the system setup was essentially the same, the pre-loading torque was set to a similar

value, and the shaft speeds did not vary significantly (i.e. only 100 rpm deviation). Although not

shown, the identified secondary path transfer functions for much higher speeds, such as at 800

rpm, or at substantially different pre-loading torque levels, are different from the functions

shown in Figure 6.11. By using the transfer function shown in the figure for the current test

conditions, the voltages needed to completely suppress the desired harmonics can be estimated as

follows. From the solid line curve in Figure 6.11, the vibration response levels due to 1 volt of
 113

input at the 3rd and 7th mesh harmonics for 200 rpm shaft rotational speed are approximately

0.049m/s2 (0.005g) and 3.332m/s2 (0.34g), respectively. By comparing these levels to the peak

amplitudes for the uncontrolled case shown in Figure 6.8, the controller output voltages for the

piezoelectric amplifier needed to suppress the vibration response at the 3rd and 7th harmonics

completely can be computed to be 1.26 volts and 0.024 volts, respectively. The actual controller

output voltage for the 7th mesh harmonic, as shown previously in Figure 6.10, is quite close to

the 0.0024 volts predicted from the secondary path transfer function. However, for the 3rd mesh

harmonic, the estimated and actual controller output voltages are quite different. The reason for

this voltage discrepancy is that the identified secondary path in the low frequency region is not as

accurate as in the high frequency region. Several factors actually contributed to this inaccuracy.

These factors include very low response sensitivity levels in the low frequency range, large

dynamic range of actuation system, as well as possible nonlinearities in the system.

-30

Target frequencies 4 x mesh

-35
2 x mesh 5 x mesh

-40
Acc y (dB)

-45

-50

-55

-60
300 400 500 600 700 800 900 1000 1100
Frequency (Hz)
Figure 6.12. Frequency domain active control result of Accy (dB re. 9.8m/s2) with shaft
rotational speed of 300 rpm. (Keys: , control off; , 2nd mesh harmonic control;
th
, 4 mesh harmonic control).
 114

To further demonstrate the performance of the proposed active control system, Figure

6.12 shows individual control results at the 2nd (dash-dotted line) and 4th (dotted line) gear mesh

harmonics. In these two cases, the preload torque in the system was again set to about 26 N-m

(230 in-lbf) but the shaft rotational speed was increased to a nominal value of 300 rpm. The

results show about 5 and 10 dB of reduction in the gear housing vibrations at the corresponding

2nd and 4th mesh harmonics, respectively. Although this figure demonstrates reductions at the

target harmonics, the impact at the other harmonics due to this control is also worth examining.

From the two cases shown in Figures 6.8 and 6.12, it is evident that the 3rd mesh

harmonic for the 200 rpm case and the 2nd mesh harmonic for the 300 rpm case are each reduced

when they are the control object. However, the response levels at some of the higher harmonics

for those two cases actually increase. These unexpected increases, which are often referred to as

out-of-band overshoot, can be traced back to the wide dynamic range of the secondary path

transfer function shown in Figure 6.11. The magnitudes of the secondary path transfer function

between 800 Hz and 1,100 Hz are much larger than those at frequencies below 500 Hz. Due to

the non-ideal filter response of the controller, even a small controller output component between

800 and 1,100 Hz will be greatly amplified by the secondary path transfer function. This

amplification can lead to large responses in these other frequency ranges. Furthermore, to

control the housing vibration below 500 Hz, a large controller output for the amplifier is

required. This large control signal increases the possibility of the housing vibration

amplification between 800 and 1,100 Hz, as shown by dash-dotted line in Figures 6.8 and 6.12.

On the other hand, when controlling the housing vibration between 800 and 1,100 Hz, a much

smaller controller output is needed. Hence, the housing response below 500 Hz is not as easily
 115

excited, and therefore, the unexpected increases do not occur in the results shown by dotted line

in Figures 6.8 and 6.12 at these frequencies.

Apart from large controller output requirements and the out-of-band overshoot problem,

the lack of accuracy in the identified secondary path transfer function below 500 Hz will

deteriorate the potential active control performance in that frequency range. As a result, while

more than 10 dB of vibration reduction can be easily achieved in the 800 Hz to 1,100 Hz

frequency range, it is more difficult to obtain similar performance below 500 Hz. Work is still in

progress to improve the estimation of the dynamics in the secondary path at these lower

frequencies.

Since the control cases shown in Figures 6.8 and 6.12 were for relatively low shaft

rotational speeds of 200 rpm and 300 rpm, it is insightful to consider higher rotational speeds.

Figure 6.13 shows the active vibration control results at the 2nd mesh harmonic for a shaft

rotational speed of 700rpm and 800 rpm. In this case, the preloading torque of the system

remained at 26 N-m (230 in-lbf). For this case, about 12dB and 8 dB of reduction in the gear

housing vibration at the 2nd mesh harmonic was achieved for 700rpm and 800rpm, respectively.

Note that there is a very slight increase in some of the amplitudes at other frequencies very near

the 2nd mesh harmonic.

Furthermore, higher speeds up to 1200rpm are also examined in the experiment.

However, no reductions on mesh response can be obtained. Its found that the reason is the

preload nut lost its tight during high speed running. The loosing of tight resulted in the response

of secondary path becoming much lower than normal one. This means that system needs much

higher voltage output from controller to control same level vibrations. To overcome this

problem, the preload nut should be modified to prevent the loosing of tight.
 116

-15

-20
(a) Target frequency

-25

-30
Acc y (dB)

-35

-40

-45

-50

-55
200 400 600 800 1000 1200 1400
Frequency (Hz)

-10
Target frequency
-15 (b)

-20

-25
Acc y (dB)

-30

-35

-40

-45

-50
200 400 600 800 1000 1200 1400
Frequency (Hz)
Figure 6.13. Frequency domain active control result of Accy (dB re. 9.8m/s2) at the 2nd mesh
harmonic with shaft rotational speed of 700rpm (a) and 800 rpm (b). (Keys: , control off;
, control on)
 117

6.4.2. Vibration Control of Multiple Harmonics

In the previous simulation study, it was concluded that the amplitude ratio of the

reference signal at each gear mesh harmonic should be inversely proportional to the amplitude

response of the secondary path at those harmonics. To experimentally verify this conclusion, the

case of a 250 rpm shaft speed was tested. Here, the control goal is to simultaneously suppress

the vibrations at the 4th, 5th, 6th and 7th gear mesh harmonics. The amplitude ratios for this case

were set to 5.5:1.04:0.5:1, which is inversely proportional to amplitude response of the secondary

path at these four harmonics with the 7th harmonic level assumed to be unity. The results in

Figure 6.14 show that all of these four harmonics are simultaneously reduced by at least 6 dB.

For the 5th mesh harmonic, the reduction is actually closer to 11 dB.

-30

7 x mesh
5 x mesh
-35
6 x mesh
7 dB

-40
11 dB

10 dB
Acc y (dB)

-45 4 x mesh
6 dB

-50

-55

-60

-65
200 400 600 800 1000 1200 1400
Frequency (Hz)
Figure 6.14. Frequency domain active control result of Accy (dB re. 9.8m/s2) with shaft
rotational speed of 250 rpm. Simultaneous control targets are the 4th thru 7th mesh harmonics
with amplitude ratios of targeted mesh harmonics in reference signal given by 5.5:1.04:0.5:1.
(Keys: , control off; , control on)
 118

The active control result when the amplitude ratios were set to 1:1:1:1 is shown in Figure

6.15. While response at the 6th mesh harmonic was reduced by almost 14 dB, no apparent

reduction is seen for the response at the 4th mesh harmonic. Besides demonstrating

simultaneous control, this experiment also confirms the theory reported in Chapter 5. That is, the

amplitude ratio should be inversely proportional to the amplitude response of the secondary path.

-30

-35 7 x mesh
5 x mesh
6 x mesh

-40

9 dB
8 dB

14 dB
4 x mesh
Acc y (dB)

-45
~ 0 dB

-50

-55

-60

-65
200 400 600 800 1000 1200 1400
Frequency (Hz)

Figure 6.15. Frequency domain active control result of Accy (dB re. 9.8m/s2) with shaft
rotational speed of 250 rpm. Simultaneous control targets are 4th thru 7th mesh harmonics with
the amplitude ratios of targeted mesh harmonics in reference signal given by 1:1:1:1.
(Keys: , control off; , control on)

Figure 6.16 shows another active control result targeting the vibrations at the 3rd thru 8th

mesh harmonics when the gearbox ran at 230 rpm with the same torque used earlier. Using the

same reference amplitude ratio strategy at each harmonic indicated above (the ratios applied

were 26:49:18:5.5:0.8:1 for the 3rd to 8th mesh harmonic), reasonably good active control results

were again achieved at all target harmonics. Specifically, about 9, 7, 4, 6, 4 and 2 dB reductions
 119

were obtained for the 3rd to 8th mesh harmonics, respectively. From these results, it is clear that

this active control system can handle up to 6 mesh harmonics simultaneously for a constant

operating speed. Of course, the number of simultaneously controlled harmonics depends on the

system dynamics as well as the actuation ability. It is also of interest to consider the

simultaneous control of multiple harmonics over a range of operating speeds as discussed next.

-35
1 x mesh
Controlled Harmonics

-40
3 x mesh

4 dB
-45

6 dB
7 dB

8 x mesh
4 dB
9 dB
Acc y (dB)

2 dB
-50

-55

-60

-65
200 400 600 800 1000 1200 1400
Frequency (Hz)
Figure 6.16. Frequency domain active control result of Accy (dB re. 9.8m/s2) with shaft
rotational speed of 230 rpm. Simultaneous control targets are 3rd thru 8th mesh harmonics.
(Keys: , control off; , control on)

The simultaneous active vibration control of the steady-state housing response at the 4th

thru 7th mesh harmonics were then tested for a speed range of 160 rpm to 260 rpm at 10 rpm

increments. Because the sampling rate was set to 4,096 Hz, the highest frequency that can be

controlled is limited to below 1,200 Hz. As a result, the highest shaft rotational speed is limited

to 260 rpm. Figure 6.17 shows the active vibration control result for the Accy signal.
 120

Experimental results for the 4th thru 7th mesh harmonics indicate that an average of 5 dB

reduction with the largest reduction found to be 14 dB. The reductions that can be attained, to

some degree, depend on the secondary path and how large the target harmonic amplitudes are

when compared to background noise levels. Also, as shown in Figure 6.17, the housing

vibration response at most of the mesh harmonics did not improved much at the lower shaft

rotational speeds. In fact, most of the vibration reductions are seen in higher speeds between 230

rpm and 250 rpm.

-30
(a)
4 x mesh

-40
-50
-60
-30
(b)
5 x mesh

-40
-50
Acc y (dB)

-60
-30
(c)
6 x mesh

-40
-50
-60
-30
7 x mesh

-40 (d)

-50
-60
160 180 200 220 240 260
Shaft Speed (RPM)
Figure 6.17. Vibration response (dB re. 9.8m/s2) for simultaneous control of multiple harmonics:
(a) 4th mesh harmonic; (b) 5th mesh harmonic; (c) 6th mesh harmonic; and (d) 7th mesh harmonic.
(Keys: , without control; , with control)
 121

6.4.3 High Torque Load Condition

In all above cases, the torque loads are set to 26 N-m (230 in-lbf). This value is not so

high. To test the performance of the active control system working under relative high torque

preload condition, in the following set of cases, the torque load is set to 180 N-m (1600 in-lbf).

In Figure 6.18, the vibration spectrums of Accy before the control system is activated for

low torque and high torque preloads are shown. From the figure, the vibration spectrum of

higher torque (solid line) is somehow different from the low torque case (dotted line). Most of

harmonics for large torque case are higher than small torque case. Especially for 3rd harmonic, it

is hardly to be seen for low torque case which can be observed from the solid line in Figure 6.18.

However, its strong in when large torque (dotted line) is applied to system. The difference of

vibration spectrum results from the changes of dynamics of gear system due to different torque

preloads. In the figure, the frequencies of corresponding mesh harmonics for large torque case

are a little low than small torque case. The reason is that the speed of gear system might not be

very stable although the speed is set to 250rpm through control box. There exists small

perturbation for shaft speed during the system running.

 122

-20

-30

-40
Acc y (dB)

-50

-60

-70
200 400 600 800 1000 1200 1400
Frequency (Hz)

Figure 6.18 Vibration Spectrum before control ( : low torque; : high torque)

Because of the change of system dynamics resulting from different torque load, the

response of secondary path should also be different from each other. In Figure 6.19, the

responses of secondary path for different torque applications are shown. From the result, one

can see that the response of secondary path for larger torque case is lower than small torque case.

This means that system needs more voltage to cancel same level of vibration.
 123

200

Phase (deg)
100

-100

-200
0 500 1000 1500
0
10
Magnitude (g/volt)

-1
10

-2
10

-3
10
0 500 1000 1500
Frequency (Hz)

Figure 6.19 Identified secondary path response while system is operating at 250rpm harmonic
( : low torque; : high torque)

Figure 6.20 and 6.21 shows the control results for the 3rd and 5th gear mesh harmonic

when gear system is operating at 250rpm, respectively. From Figure 6.20, one can see that the

target mesh frequency, 3rd harmonic, is reduced from -36.2 dB (0.016 g) to -42.2 dB (0.0078 g).

About 6 dB reduction is obtained. For the case shown in Figure 6.21, the 5th harmonic is

reduced from -29.4 dB (0.034 g) to -39.6dB (0.011 g). About 10.2 dB reduction is achieved. For

other harmonics, such as 6th, 7th harmonics, they are also tried in the experiment. Similar

reductions are obtained for other harmonics except 1st and 2nd harmonics because of systems

ability.
 124

-20

-30

3rd

-40
Acc y (dB)

-50

-60

-70
200 400 600 800 1000 1200 1400
Frequency (Hz)

Figure 6.20 Frequency domain control result for higher torque case and 3rd harmonic. ( ,
control off; , control on)

-20

5th
-30

-40
Acc y (dB)

-50

-60

-70
200 400 600 800 1000 1200 1400
Frequency (Hz)

Figure 6.21 Frequency domain control result for higher torque case and 5th harmonic ( ,
control off; , control on)
 125

6.4.4. Effect of Adaptive Linear Enhancer

As we can see in the previous section, when the harmonic is controlled, the higher

harmonic response may be increased by a little bit. The reason might be discussed in section 4.3.

In that section, some simulation works and theoretical formula derivation are performed to prove

the increases at other frequencies when there exists uncorrelated harmonics.

The increase in the response at the higher-order mesh harmonics when a lower-order

mesh harmonic is controlled can be attenuated by adding an adaptive linear enhancer to the

active control system. The adaptive linear enhancer (ALE), when used on the error signal before

supplying it to the filtered-x LMS algorithm, can keep the magnitude and phase information of

the error signal at the target frequency while filter out all other frequency components. As a

result, the ALE actually improves the signal-noise-ratio in the error signal for a wide frequency

range. The unexpected increase in gearbox housing vibration can therefore be alleviated as

demonstrated in Figures 6.22-6.24 where the 4th harmonic is the target frequency for a shaft

rotational speed of 250 rpm. Figure 6.22 shows the spectrum of gear housing vibration Accy

before and after the controller is activated. The control results both with and without the ALE

are compared to the results without control. Although the 4th mesh harmonic is the control

target, one can clearly see that the 6th and 7th mesh harmonics are increased by about 2 or 3 dB

when the adaptive linear enhancer is not used. However, the increase at these non-targeted

frequencies is not seen when the adaptive linear enhancer is applied. The reason for this

behavior can be understood by examining Figure 6.23 and 6.24, which shows a comparison of

the spectra of first element of FIR filter weights and controller output spectra with and without

the adaptive linear enhancer. As shown in Figure 6.23, the harmonic components in the filter

weights are reduced a lot when ALE is used. From the discussion at section 4.3, this means that
 126

unexpected frequencies in controller output should be reduced, which is shown in Figure 6.24.

In Figure 6.24, there are clearly many harmonics components besides those at the target

frequency (in this case, the 4th mesh harmonic). The unwanted harmonics at the 5th, 6th and 7th

mesh harmonics can potentially induce large vibration response when the resultant housing

vibration is in phase with the original response. This behavior occurs because the response of the

secondary path transfer function is quite large around 800 Hz to 1,200 Hz, as previously shown

in Figure 6.11. With the introduction of the ALE, those unwanted harmonics, especially at the

6th and 7th mesh harmonic, are reduced by more than 13 dB. Thus, even though they are

magnified by the secondary path dynamics, these components are not as dominant in the gear

housing vibration response spectrum when the adaptive linear enhancer is applied.

-30
6 x mesh 7 x mesh

-35

Target frequency
-40
Acc y (dB)

-45 4 x mesh

-50

-55

-60
600 700 800 900 1000 1100 1200
Frequency (Hz)

Figure 6.22. Frequency domain active control result at the 4th mesh harmonic with shaft
rotational speed of 250 rpm. (Keys: , control off; , ALE off and control on;
, ALE on and control on)
 127

-3
10

-4
10
Magnitude

-5
10

-6
10

-7
10
0 200 400 600 800 1000 1200 1400
Frequency (Hz)
Figure 6.23 Frequency domain of first element of FIR controller weights (Keys: , ALE off;
, ALE on)

-10
4 x mesh

-20
6 x mesh
7 x mesh
-30
u (dB)

-40

-50

-60

-70
200 400 600 800 1000 1200 1400
Frequency (Hz)

Figure 6.24. Frequency spectrum of the controller output (dB re. 1 volt) for shaft rotational
speed of 250 rpm. (Keys: , ALE off; , ALE on)

Although not mentioned earlier, the order of the adaptive linear enhancer may also affect

the performance of controller. This effect is demonstrated in Figure 6.25, where only the
 128

spectral content between 700 Hz and 1,000 Hz is plotted to highlight the effect more clearly. In

this figure, three cases are compared: no control, control with 4 order ALE, and control with 64

order ALE. Actually, four different orders (4, 16, 32 and 64 order) are tested in the experiment.

For brevity, only two different orders are shown here because the results for the other orders

follow the same trend as shown in the figure. The results show that the order of the adaptive

linear enhancer should be set as small as possible, since a large order will increase the phase

delay of the error signal and thus reduce the possible vibration reduction performance.

-40
Target frequency 5 x mesh

-45

-50
Acc y (dB)

-55

-60

-65
700 750 800 850 900 950 1000
Frequency (Hz)
Figure 6.25. Frequency domain active control result of Accy at the 5th mesh harmonic with shaft
rotational speed of 250 rpm. (Keys: , control off; , 4 order ALE; , 64 order
ALE)
 129

6.4.5. Effect of Step Size

Besides considering the effect of the ALE order size on the active control system

performance, the effect of different step size of the controller adaptation on the active control

results is also of interest. In Figure 6.26, the frequency domain active control results for several

cases, each with a different step size, are shown. In these cases, all of the other parameters were

kept the same, and the gearbox was running at 250 rpm. Only the 6th mesh harmonic, which is at

990 Hz for shaft rotational speed of 250 rpm, was being controlled. The different reductions

achieved for the spectrum of Accy at the 6th mesh harmonic using different step sizes are clearly

shown here. The corresponding time domain response functions for the amplitude of the 6th

harmonic are shown in Figure 6.27. Note that the 6th harmonic amplitude in the original gear

housing vibration response before control was almost the same for all different step sizes. The

adaptive controller was then activated at about the 10 seconds point. All results, even for

different step sizes, converged to a different stable value in less than 1 second from the time

when the controller was activated. However, they did not all converge to the same global

minimum value. As described in section 4.3, the weights of adaptive controller will converge to

non-Weiner solution expressed as Equation (4.4). As shown in Equation (4.4), one knows that

the solution also depends on the step size when the step size is small. Within certain limitations,

a larger step size generally improves the vibration control performance (i.e. smaller residual

housing vibration). The differences in the active control results due to the different step size

used can be understood more in-depth by examining the steady-state FIR controller weights

shown in Figure 6.28. It is noted that the phase of the adaptive controller for different step sizes

is almost the same, but the amplitude is quite different. The controller output signal is a result of

the convolution of the reference signal and the above FIR filter. The controller output signal is
 130

itself convoluted with the same secondary path transfer function, and the resultant housing

vibration due to the controller output signal will have to be out-of-phase with the original

housing response in order to achieve a reasonable response reduction. As a result, it is natural

that all the above FIR controller weights will be in-phase with each other such that they can all

attenuate the original housing vibration, although with different attenuation levels as shown in

Figures 6.26 and 6.27.

-30
6 x mesh
Target frequency

-40
Acc y (dB)

-50

-60
800 850 900 950 1000 1050 1100 1150 1200
Frequency (Hz)

Figure 6.26. Frequency domain active control result of Accy at the 6th mesh harmonic for
different step sizes with shaft rotational speed of 250 rpm. (Keys: , control off; ,
=0.005; , =0.01; , =0.02; , =0.03)
 131

0.025

0.02

= 0.03
0.015
= 0.02
Acc y (g)

0.01 = 0.01
= 0.005

0.005

0
2 4 6 8 10 12 14
Time (sec)
Figure 6.27. Time domain amplitude response of the targeted 6th mesh harmonic of housing
vibration for different step sizes . (Keys: , =0.005; , =0.01; ,
=0.02; , =0.03)
-4
x 10
3

= 0.01 = 0.03
2

1
Mean Weight

-1

-2
= 0.005 = 0.02
-3
5 10 15 20 25 30
Tap Delay (n)
Figure 6.28. Mean value of FIR controller weights for different step sizes . (Keys: ,
=0.005; , =0.01; , =0.02; , =0.03)
 132

6.4.6 Effect of The Order of Secondary Path Model

In specifying the control system parameters, the order of the estimated secondary path

FIR order N and controller FIR order are studied. Since the control frequency of interest is

below 1400 Hz and the controller parameter update rate is set to 4.096 kHz, it is natural to select

a 128-order FIR type controller. The offline identified secondary path (from controller output to

the gear housing vibration) impulse response is shown as dotted line in Figure 6.29. Apparently,

the identified impulse response weights are rather small for tap number greater than 65. The

offline identification process was performed again for N=64 case, as shown as solid line in

Figure 6.29. It is clear that the two identified impulse responses are almost the same before 64-

tap. A little difference is from the different system condition, such as torque preload setting.

The frequency domain responses of the two identified impulse responses are given in Figure

6.30. The two frequency performances are also similar to each other. Especially for phase part,

they are very close for N=64 and N=128 cases for the frequency range from 400Hz to 1400 Hz.

As we discussed in previous chapter, the phase information is very important for the control

system stability. From the identified result of secondary path response, one can conclude that 64

order of FIR filter for modeling secondary path is enough. The effect of large and small FIR

order N on the control performance is shown in Figures 6.31 and 6.32, respectively. Figure 6.31

shows the steady-state frequency domain gear housing vibration parallel to the gear line-of-

action (denoted as Accy) when the control is turned off and on (the step size is 0.001 and the

leakage coefficient is 0.99) for the N=128 case. Figure 6.32 shows a similar performance for

the N=64 case (the step size is 0.001 and the leakage coefficient is 0.99). Comparing these

two figures, it is seen that the vibration reductions at controlled harmonics for the N=128 case

are almost as good as the N=64 case. Since using 64 order FIR filter to model secondary path
 133

response will reduce the computation burden of DSP board, it has advantage over using 128

order FIR filter for some cases in which the computation efficiency is highly desired. In this

study, for most of control case, 128 order FIR filter is used because DSP board has enough

computation capability. In this experimental study, up to 6 harmonics for 230rpm shaft speed as

shown in Figure 6.16 are controlled simultaneously. Its found that the DSP will exceed its

ability while controlling more than 7 harmonics simultaneously and using 128 order FIR

secondary path model. If using 64 order secondary path model, more harmonics can be

controlled at same time.

-3
x 10
5
Estimated Impulse Response of Secondary Path

-1

-2

-3

-4

-5
0 20 40 60 80 100 120
Taps (n)
Figure 6.29 Identified secondary path impulse responses ( N=64; N=128)
 134

200

100

Phase (deg)
0

-100

-200
0 500 1000 1500

0.4
Magnitude (g/volt)

128
0.3 64

0.2

0.1

0
0 500 1000 1500
Frequency (Hz)
Figure 6.30 Frequency domain of identified secondary path ( N=64; N=128)

-30

-35

-40
Acc y (dB)

-45

-50

-55

-60

-65
200 400 600 800 1000 1200 1400
Frequency (Hz)
Figure 6.31 Frequency domain of control result for order 128 secondary path FIR model
 135

-30

-35

Acc y (dB) -40

-45

-50

-55

-60

-65
200 400 600 800 1000 1200 1400
Frequency (Hz)

Figure 6.32 Frequency domain of control result for order 64 secondary path FIR model
 136

6.4.7 The Vibration At Other Locations And Gear Whine

In the designed control system, only the vibration along gear mesh line-of action at one

location is controlled. Vibrations at other locations on gear housing are also important for gear

whine generation. To monitor the vibration at other locations, three more tri-axial

accelerometers are mounted on the face of gear housing at right above other three supporting

bearings as shown in Figure 6.1. In this section, the vibrations of other locations are discussed.

-30 -30

-35
-35

-40
-40

Acc2 y (dB)
Acc y (dB)

-45
-45
-50

-50
-55

-55
-60

-65 -60
200 400 600 800 1000 1200 1400 200 400 600 800 1000 1200 1400
Frequency (Hz) Frequency (Hz)

-20 -20

-25 -25

-30 -30

-35 -35
Acc3 y (dB)

Acc4 y (dB)

-40 -40

-45 -45

-50 -50

-55 -55

-60 -60
200 400 600 800 1000 1200 1400 200 400 600 800 1000 1200 1400
Frequency (Hz) Frequency (Hz)

Figure 6.33 Frequency domain of control results( ,control off; , control on).

Figure 6.33 show the frequency domain of control result. In this case, the system is

running at 250rpm. And from 4th thru 7th harmonics are controlled simultaneously. The error

signal is the vibration along the gear mesh line-of-action, Accy. Also, the vibrations at other

locations, Acc2y, Acc3y and Acc4y, are monitored. The results are shown in Figure 6.33 (a)-(d),
 137

respectively. From the results, one can see that Accy are reduced a lot at the four target

frequencies. This is because Accy is treated as error signal. The aim of controller is to reduce

error signal as possible as it can. Furthermore, Acc2y can also show some reductions, especially

at 5th harmonic. However, for Acc3y and Acc4y, all target frequencies only show a little bit of

reductions. The detailed vibrations levels before and after the actuator is activated are listed in

Table 6.3.

Table 6.3 Vibration levels at four locations along gear mesh line-of-action (dB ref. 1g)

Accy (dB) Acc2y (dB) Acc3y (dB) Acc4y (dB)

n Frequency
(Hz) Before After Before After Before After Before After

4 660 -43.8 -50.1 -37.0 -38.1 -31.0 -34.6 -28.5 -27.9

5 822 -33.5 -43.6 -31.1 -40.6 -21.8 -25.0 -24.2 -25.8

6 990 -35.8 -43.6 -37.0 -38.3 -25.6 -26.8 -29.4 -30.3

7 1152 -32.0 -38.2 -31.7 -34.6 -24.7 -26.6 -30.8 -32.2

The reason can be seen in Figures 6.34-6.36. Figure 6.34 shows the phase difference

between Accy and Acc2y caused by gear transmission error excitation, i.e., the controller is off.

From that figure, one can see that phase differences of 5th, 6th and 7th harmonics are almost

stable, however, the phase difference of 4th harmonic is changing a lot with time. The reason is

that the 4th harmonic is not so significant. The phase information at that harmonic might not be

accurate because of the mask effect of broadband noise. Figure 6.35 shows the cross-point

transfer functions of two paths which are from actuator force output to housing vibration Accy

and to housing vibration Acc2y. Figure 6.36 shows the cross-point transfer functions of two paths

which are from actuator force output to housing vibration Acc3y and to housing vibration Acc4y.
 138

Their detailed phase and magnitude information at four harmonics are listed in Table 6.4. From

the results shown in Figure 6.35 and Table 6.4, one can see that the phases of two transfer

functions are very close. This means that housing vibrations Accy and Acc2y due to the control

force are almost in same phase. However, the total resultant vibration also depends on the gear

transmission error excitation. Even Accy and Acc2y due to control force are in same phase,

Acc2y can not be cancelled if Acc2y due to gear transmission error is out of phase with Accy due

to gear transmission error because Accy should be cancelled finally. From Figure 6.34, the phase

difference of the shown 4 harmonics of housing vibrations Accy and Acc2y due to gear

transmission error excitation are within 90 degrees, except 4th harmonic. In average, the phase

difference of 4th harmonic also can be looked as within 90 degree. That means that if Accy can

be reduced, then Acc2y also can be reduced at these four harmonics. Furthermore, from the

Figure 6.34 and Table 6.5, the phase difference of 5th harmonic is smallest among four

harmonics and magnitudes are almost same. And, from Table 6.3, the vibrations levels for Accy

and Acc2y at 5th harmonic are -33.5 dB and -31.1 dB before the controller is switched on. Thus,

its reasonable to get 9.5 dB reductions at 5th harmonic for Acc2y when about 10.1 dB reductions

for Accy are obtained. This explanation can also explain why the housing vibration at other two

locations cannot be reduced like Accy.

 139

180

90
Phase (deg)

-90

-180
0 5 10 15 20
Time (sec)

Figure 6.34 Phase difference between Accy and Acc2y. ( ), 4th harmonic; ( ), 5th
harmonic; ( ), 6th harmonic; ( ), 7th harmonic.

200
Phase (Deg)

-200
200 400 600 800 1000 1200 1400
Frequency Response (g/N)

-3
10

-4
10
200 400 600 800 1000 1200 1400
Frequency (Hz)

Figure 6.35 Cross-point transfer functions of the paths from actuator force output to Accy and
Acc2y
 140

200

Phase (Deg)
0

-200
200 400 600 800 1000 1200 1400
Frequency Response (g/N)

-3
10

-4
10
200 400 600 800 1000 1200 1400
Frequency (Hz)
Figure 6.36 Cross-point transfer functions of the paths from actuator force output to Acc3y and
Acc4y.

Table 6.4 Detailed Values of four cross-point transfer functions at four harmonics
Accy Acc2y Acc3y Acc4y
n Frequency Phase Amplitude Phase Amplitude Phase Amplitude Phase Amplitude
(Hz) (deg) (10-3g/N) (deg) (10-3g/N) (deg) (10-3g/N) (deg) (10-3g/N)

4 660 -163 0.193 -144 0.186 -174 0.304 -173 0.146

5 822 -174 0.332 173 0.329 -162 0.171 -159 0.264

6 990 134 0.625 128 0.202 118 0.806 108 0.453

7 1152 -176 1.294 -158 1.260 30.6 0.907 27 0.318

Table 6.5 Average Phase difference between other locations vibrations and Accy

n Acc2y Acc3y Acc4y

Frequency (Hz)
4 660 -56.06 0.82 -99.77
5 822 -23.86 80.62 -109.86
6 990 -67.06 -126.07 -158.22
7 1152 -29.24 -120.30 -99.77
 141

Furthermore, we know that gear whine is from the vibrations of the whole gear system.

From the above discussion, we also know that the gear housing cannot be treated as a rigid body

in real application. Hence, the vibrations of one location at gear housing cannot represent whole

gear housing vibrations. In this section, experimental studies are performed to decide which

vibration from four bearing locations is more suitable for the sake of gear whine reduction

consideration.

To study the effect on gear whine when the vibrations at other locations are treated as

error signal, several cases are studied and both vibrations and sound pressure level are monitored

in the tests. An example of both gear housing mesh response and gear whine response are

plotted in one plot for comparison as shown in Figure 6.37. In Figure 6.37, the solid line gives

the spectrum of the gear noise radiated from running gear system which is operated at 250rpm.

The dotted-line is the spectrum of gear housing vibration Accy. From the figure, one can see that

the frequencies of some significant harmonics below 600Hz are not matched between gear whine

and gear vibration. The reason is that the monitored gear noise is not only from test gearbox, but

also from slave test gear and motor. To screen the effects of other potential noise sources, the

gear system should be enclosed in the further study. Due to the limitation of experimental setup,

in this study, the sound meter is placed nearby the test gearbox to minimize the effects of other

sources.
 142

75 -30

70 5 x mesh -35

Sound Pressure Level (dB)

65 -40

Acc y (dB)
60 4 x mesh -45

55 -50

50 -55

45 -60

40 -65
200 400 600 800 1000 1200 1400
Frequency (Hz)

Figure 6.37 Spectrum of gear whine ( ) and gear housing vibration Accy ( ).

Tables 6.6-6.9 list the reductions of gear noise and housing vibrations for different error

signal. Table 6.6 shows the housing vibration reductions at four locations and sound pressure

level reductions when housing vibration Accy is treated as error signal. Tables 6.7-6.9 show the

reductions of vibration and gear whine when Acc2y, Acc3y and Acc4y are treated as error signal,

respectively. From the results shown above, we can see that the phases of cross-point transfer

functions from control force to housing vibration Accy and Acc2y are close, and the phases cross-

point transfer functions from control force to housing vibration Acc3y and Acc4y are close, but

they are different for different housing panels. However, their magnitudes are different. The

control results should be analyzed for particular cases. For example, when Acc3y is the error

signal and 5th harmonic is target frequency, as Table 6.8 shown, Acc3y got 5.6 dB reductions at

5th harmonic. However, Acc4y did not show any reductions. When Acc4y is treated as error

signal, Acc4y got 5.8 dB reductions and Acc3y got 14.8 dB increases. The reasons can be found
 143

in Table 6.4 and 6.5. From Table 6.5, we can see that the phase of Acc3y due to TE is almost out

of phase of Acc4y due to TE. And, the phase of Acc3y due to control force is almost in phase

with Acc4y due to control force. Thus, while controlling Acc3y, Acc4y suppose to increase.

However, the magnitude of Acc3y due to unit control force is much larger than the magnitude of

Acc4y due to unit control force. Hence, Acc4y due to control force might not affect total

vibration too much. However, if Acc4y is error signal, the control force will make big impact on

the resultant vibration just as the shown results. Note that the measured transfer functions might

be different from the real transfer functions because system might change when actuator is

installed and preload is added on the system. However, the explanation can also be made in such

way.

Table 6.6 Control results when Accy is error signal

Accy Acc2y Acc3y Acc4y SPL

(dB) (dB) (dB) (dB) (dB)
2 4.1 1.1 -1.9 -0.3 -0.3
4 6.8 -0.2 5.6 -0.3 -2.6

5 6.8 6.5 6.0 1.0 4.7

6 10.9 1.9 -3.7 1.5 2.3

7 12.9 5.3 -2.9 0.7 -1.7

Table 6.7 Control results when Acc2y is error signal

Accy Acc2y Acc3y Acc4y SPL

(dB) (dB) (dB) (dB) (dB)
2 3.0 2.2 -1.1 0.5 -0.5
4 3.0 3.7 5.3 -2.3 2.7
5 6.5 8.9 -2.5 1.4 3.1
6 -1.1 7.9 -1.4 -0.7 -2.5
7 5.7 7.8 -1.8 -0.2 -1.6
 144

Table 6.8 Control results when Acc3y is error signal

Accy Acc2y Acc3y Acc4y SPL

(dB) (dB) (dB) (dB) (dB)
2 -0.2 0.3 7.8 2.2 -2
4 0.9 1.8 4.5 -1.0 2.6
5 -0.6 0.2 5.6 -0.1 1.0
6 -7.2 -5.8 9.0 -1.0 1.3
7 -5.5 -6.2 3.2 -1.0 -3.1

Table 6.9 Control results when Acc4y is error signal

Accy Acc2y Acc3y Acc4y SPL

(dB) (dB) (dB) (dB) (dB)
2 0.7 1.3 3.4 3.3 -2.1
4 -3.3 -4.0 -6.9 7.0 -4.0
5 -6.9 -2.6 -14.8 5.8 -4.6
6 -11.8 -10.6 -9.0 4.0 -6.7
7 -10.4 -12.6 -5.4 1.4 -7.9

From the above control results listed in Table 6.6-6.9, one can see that the best error

signal choice from four bearing location vibrations is Accy. The worst choice is Acc4y. From

Table 6.9, we can see that gear whine at all four harmonics are increased. And, the vibrations at

other locations are increased for from 4th thru 7th harmonics. All treated harmonics in gear whine

signal are increased compared with SPL before actuator is activated. When Accy is the error

signal, most of harmonic vibrations at other locations are decreased. However, for the gear

whine, only 5th and 6th harmonics show some reductions. Of course, Accy might not represent

the global gear housing vibration because multi-modes existed in the system. To find a suitable

point which represents whole gear housing vibrations, a more detailed system model should be
 145

analyzed. In that model, gear housing cannot be treated as a rigid body which is used in the

research. Alternatively, to control the global vibration of gear housing and then control gear

whine, several actuators with Multi-Input Multi-Output (MIMO) control system should be used.
 CHAPTER 7

CONCLUSIONS AND RECOMMENDATIONS

7.1 Conclusions

The focus of this dissertation research project is to tackle the gear whine problem through

an active vibration control means. Gear whine is assumed due to the transmission error

excitation at the mesh arising from tooth profile errors, misalignment and tooth deflections. This

frequently high level of tonal noise and vibration can be quite annoying, and also can induce

structural fatigue. Furthermore, rotorcraft interior noise and vibration requirement often limits

the achievable power density and efficiency of the propulsion system. However, current passive

and active techniques have not been able to reduce the gear noise and vibrations to the desired

level. The existence of this gap is the focus of this research work.

To address the desired goal, a finite element model is first developed for a power re-

circulating gear system for use to examine the feasibility of applying an active shaft transverse

vibration control method to tackle the classical gear whine problem. For this particular actuator

setup and gearbox system, a complete set of controller is designed. The proposed control system

is based on the FXLMS algorithm combined with improved frequency estimation technique.

Using the frequency estimation technique, the feedback control structure is naturally converted

into a feedforward structure that is more efficient and robust. From the simulation results, the

active shaft transverse vibration control system is shown to work well in attenuating the gear

vibration response at the fundamental mesh frequency and its harmonics. Also, using the

146
 147

flexibility of the sine wave generator, a suitable reference signal can be obtained that enables

most of the harmonics to be controlled simultaneously. The value of the initial guess is critical

to the frequency estimator. The results show the closer initial guesses to the true frequency and

true amplitude of the signal lead to quicker convergence of frequency estimator. The farther

away its initial guesses from the true values, the worse the result becomes. Also, the results

illustrate the proposed frequency estimator can track any acceptable change in frequency values.

Furthermore, when signal noise is present, a low-pass filter can give a much cleaner result.

However, this filter introduces a phase delay in the algorithm. In Chapter 5, the computer

simulation study results show the performance of the FXLMS algorithm with frequency

estimator applied to the gearbox system vibration control. The effects of offline and online

secondary path modeling on the control results are compared in this study. The results show that

offline modeling gives a more satisfied control result. However, the offline modeling can only

be used in time-invariant cases. Furthermore, the simulations on the comparison of different

amplitude ratios between the various harmonics in the reference signals illustrates that these

ratios depend on the amplitude response function of identified secondary path. All simulation

results show that the proposed active control system is feasible for the reduction of gear vibration

and whine noise. Its results also yielded insights that were applied to guidance the design of

experimental study.

Other than the above algorithm design and implementation, the actuator and power

amplifier selection is also critical to the approach. The actuator and amplifier should be chosen

to have enough capacity to provide enough dynamic force to reduce the gear housing vibration.

In order to select suitable actuator and power amplifier, a PZT stack actuator model under

structural load is proposed. This model combined with measured or predicted frequency
 148

response functions (FRF) can be used to calculate the requirement of actuator and amplifier. The

results can be used to select suitable actuator. The experimental work shows good match with

the predicted results. In this study, a simplified linear piezoelectric stack actuator model is

developed for use in selecting the actuator for an active gearbox vibration control system. This

model enables the selection of a suitable piezoelectric stack actuator and its corresponding power

amplifier before the active control system is constructed. The secondary path transfer function

synthesized from the model matches quite well with the directly measured one using the control

systems online identification feature. The proposed methodology will ensure that the selected

actuator and amplifier can meet the requirements of the practical active control system without

the usual costly trial and error process. In the Chapter 6, this methodology is verified through

experimental study.

In Chapter 6, experimental studies are performed for the target power re-circulating

gearbox system which is equipped with the designed active control system. The housing

vibrations along the gear mesh line-of-action were suppressed at gear mesh harmonics for a

range of gear rotational speeds. In some cases, up to 14 dB of reduction in the vibration response

can be achieved. Furthermore, it is shown that the use of adaptive line enhancer with the proper

order can reduce the out-of-band overshoot and improve the system control performance. The

experiments also reveal some limitations in the active control system due to the inherent dynamic

characteristics of the gearbox system. Furthermore, the controller adaptation step size can affect

the active control results significantly. For the cases considered here, a larger step size yields

better reductions. However, the performance gains achieved by increasing the step size do have

an upper limit because a very large step size can cause the controller to become unstable. The

experimental work also reveals similar performance of the designed control system on different
 149

system loading condition. Other location vibrations are also monitored and examined. It was

observed that only one actuator and one sensor (Single-Input Single-Output) can not achieve

global gear housing vibration control. And then, the gear whine can not be controlled as error

signal does. To achieve global vibration and gear whine control, Multi-Input Multi-Output

system should be used. In spite of the inadequacies in the prototype system, the results yielded

by the current active vibration control system are quite promising, and the underlying concept is

believed to have the potential for practical use.

7.2 Recommendations for further Studies

Even though the active gear vibration control system has been developed and

experimentally verified, there are other fundamental issues that should ultimately be addressed in

the future.

(a) The selection of cost function and error signal is important to whole control

system. Since the gear dynamics at different frequencies and different locations

play an important factor in actuation system design for global control goal, a more

detail gear housing dynamic analysis is needed to guide the selection of more

suitable error signal location and cost function to meet global control goals.

(b) More actuators might be considered for better control the gear housing vibrations.

Currently, only one actuator has been considered. Theoretically, one actuator has

only one degree of freedom to control one vibration response. To achieve gear

housing global vibration control with only one actuator is likely an over-

determine problem. There is no global optimal solution for such case. However,

as it has been shown in Chapter 6, at most frequencies, the four housing vibrations

along the gear line-of-action are not all independent each other. A detail modal
 150

analysis of the gear housing might be needed in order to pick up the independent

gear housing vibration modes. After the number of independent vibration modes

within the frequency range of interest has been selected, the same number of

actuators might be needed in order to control the gear housing vibrations globally

and ultimately the radiated gear whine.

(c) Eccentricity and misalignment of gear system reduce the efficiency of actuator

system. For better performance, the gearbox setup should be assembled with care

to reduce the system misalignment. A study to quantify the effect of

misalignment on the performance of the actuator is also recommended.

(d) The actuator mounting method and mounting position are also important to the

control system operation. Due to hardware limitations, the current actuator

mounting position cannot be changed easily and the gear system cannot operate at

much higher speeds. In fact, the load transmitted through actuator may

deteriorate for speeds above 900 rpm. Thus the actuator will fail to generate the

required dynamic force for some cases. Alternate mounting configurations should

be considered to address this structural limitation.

(e) The nonlinear effects of gear system and hysteresis of the piezoelectric stack

actuator might need to be considered for cases where the required force is larger

and the gear rotates at higher speeds.

(f) To monitor and study the gear whine only from test gearbox, the test gear housing

should be enclosed in order to screen the effects of other potential noise sources,

such as belt driver and slave gearbox sources, leaving only the test gear housing

noise.
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