DYNAMIC ANALYSIS OF UNBALANCED RIGID ROTOR WITH

TWO OFFSET DISCS SUPPORTED BY FOIL BEARINGS

A PROJECT REPORT
Submitted in partial fulfilment
for
the award of the degree of

BACHELOR OF TECHNOLOGY
in
MECHANICAL ENGINEERING

Submitted by:
CHALLA DHANUSH SRI KARTHIK REDDY (20107007)
DESHAVATH PAVAN KUMAR (20107019)

Supervisor:
DR. PRABHAT KUMAR

DEPARTMENT OF MECHANICAL ENGINEERING

NATIONAL INSTITUTE OF TECHNOLOGY
MANIPUR IMPHAL, INDIA- 795004

MAY 2024

1
 BONAFIDE CERTIFICATE

This is to certify that the Thesis entitled “DYNAMIC ANALYSIS OF UNBALANCED

RIGID ROTOR WITH TWO OFFSET DISCS SUPPORTED BY FOIL BEARINGS”
submitted by CHALLA DHANUSH SRI KARTHIK REDDY (20107007) and
DESHAVATH PAVAN KUMAR (20107019) to National Institute of Technology Manipur,
Imphal, Manipur (India), in partial fulfilment for the award of the degree of BACHELOR OF
TECHNOLOGY (B.TECH.) in MECHANICAL ENGINEERING is a Bonafide record of research
work carried out under my supervision. The content of this thesis, in full or in parts, has not been
submitted to any other Institution or University for the award of any degree or diploma.

(DR. PRABHAT KUMAR)

SUPERVISOR
Assistant professor
Department of Mechanical Engineering
National Institute of Technology Manipur
Langol, Imphal West, Manipur, India- 795004

The B.Tech. Project work examination has been successfully conducted

on........................................................and the report has been accepted.

(D. SABINDRA KACHHAP) (Dr. ASHUTOSH KUMAR SINGH) (Dr. KH. NIMO SINGH)
EXAMINER I EXAMINER II EXAMINER III

(DR. ANIL KUMAR BIRRU)

Associate Professor & Head of the Department

Department of Mechanical Engineering

2
 National Institute of Technology Manipur, Imphal.

DECLARATION

I declare and certify that

 The work contained in this thesis is original and has been done by myself
and the general supervision of my supervisor.

 The work has not been submitted to any other institute for any degree or
diploma.

 Whenever I have used materials (data, theoretical analysis, results) from

other sources, I have given due credit to them by citing them in the text of the
thesis and giving their details in the references.

 Whenever I have quoted written materials from other sources, I have put
them under quotation marks and given due credit to the sources by citing them
and giving required details in the references.

Challa Dhanush Sri Karthik Reddy Deshavath Pavan Kumar

(20107007) (20107019)

Place: Imphal, India

Date: 20th May 202

3
 ACKNOWLEDGEMENT

We wish to express my profound gratitude and indebtedness to Dr. Prabhat Kumar,

Assistant professor, in the Department of Mechanical Engineering, NIT Manipur for
introducing the present topic and for his inspiring guidance, constructive criticism and
valuable suggestion throughout the project work.

We would like to thank Dr. Sabindra Kachhap, Dr. Ashutosh Kumar Singh and Dr. KH
Nimo Singh for giving us support and suggestions during the progress evaluation of the
project. This helps us to go more deep in the analysis of current research area.

We thank Dr. Anil Kumar Birru, Associate Professor & HOD (ME), for his continuous
support during the course of this project.

Special gratitude to our B. Tech Coordinator, Dr. Prabhat Kumar, Assistant Professor
(ME), for his valuable guidance during the course of the project.

We are very thankful to the Faculty and Staff of the Department of Mechanical
Engineering, NIT Manipur for providing us the opportunity to do project under such
innovative minds. We are also thankful to our parents for their love and support which enabled us to
complete the project in time.

Challa Dhanush Sri Karthik Reddy Deshavath Pavan kumar

(20107007) (20107019)

4
 ABSTRACT

To overcome several operational constraints of traditional bearings, such as the low reliability

and high maintenance of machines, friction losses due to lubrication, wear and tear of bearing

materials, and suitability at low speeds only, the project report titled “Dynamic Analysis of

Unbalanced Rigid Rotor with Two Offset Discs Supported by Foil Bearings” proposes foil

bearing technology to support the rotor system in high-speed machines. High speed rotating

machines are very useful in several industries and production plants or factories. The rotating

components in these machines are usually prone to multiple types of faults. These faults can

cause large and heavy amount of vibrations in the assembled rotor systems during its operation.

So, there is a need for analyzing the rotor behavior in the presence of faults and their

identification. This project report discusses the dynamic behavior of a rigid rotor with two discs

at offset positions and mounted on air foil bearings at the ends. In this system, it is assumed that

the discrete unbalance fault is present at discs only. The stiffness and damping coefficients of

both foil bearings are consider to be different and anisotropic in nature. Considering the forces

due to foil bearings, inertia force, discs unbalance force, inertia moment, and gyroscopic couple

effect, the equations of motion of the rotor system have been derived in the two-dimensional

transverse directions (vertical and horizontal directions) using the moment equilibrium method.

Moments are considered by different forces about both the bearings one by one in the horizontal

and vertical planes, and then these are assembled together to form equations of motion in matrix

form. Further, these equations are solved using a model developed in SIMULINK TM platform in

the MATLAB software. The obtained solutions of the equations are the time domain rotor

displacement in the vertical and horizontal directions. It would be very interesting to investigate

and study the unbalance fault effect through time domain spectrum, orbit plots, steady state and

transient responses, as well as displacement responses for different values of angular

5
 frequencies.

List of Figures

Fig. No. Description Page No.

1.1 Section view of bump typed gas foil bearing 9
1.2 Jeffcott rotor with disc at middle supported by two rigid bearings at the end 11
position
1.3 Complex rotor displacement in Jeffcott rotor 12
1.4 Schematic diagram of Rigid rotor system 13
1.5 Schematic diagram of Flexible rotor system 14
1.6 Two bearing rotor system with flexible foundation 15
3.1 Rigid rotor with two discs supported by foil bearings at the end position 21
3.2 A side view of axial representation of rotor with Unbalance 22
3.3 Linear and angular displacement of shaft in X-axis 23
3.4 Linear and angular displacement of shaft in Y-axis 23
3.5 Linear and angular movement of the rotor in XZ & YZ plane 24
4.1 Developed Simulink model of the system 26
4.2 Description of Simulink model 27
4.3 The displacement response for the initial transient state in (a) x-direction (b) y- 28
direction
4.4 The steady state displacement response at the angular frequency of 35Hz a) x- 29
direction at FB1 b) y-direction at FB1 c) x-direction at FB2 d) y-direction at
FB2
4.5 Displacement orbits considering the initial transient state at the angular 30
frequency of 35Hz a) FB1 location b) FB2 location
4.6 The steady state combined displacement orbits at different angular frequencies 31
a) FB1 location b) FB2 location

List of Tables

Table No. Description Page No.

4.1 Assume parameters of rotor bearing system for the numerical 27
investigation
4.2 Peak value of displacement at different angular frequency of the 31
rotor

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 Table of Contents
Chapter 1. Introduction and Literature Review
1.1 Rotor Systems…………………………………………………………………….10
1.1.1 Jeffcott rotor………………………………………………………………….11
1.1.2 Rigid rotor……………………………………………………………………13
1.1.3 Flexible rotor…………………………………………………………………13
1.2 Unbalance Fault…………………………………………………………………..14
1.2.1 Definition……………………………………………………………………..14
1.2.2 Identification & Balancing Procedure………………………………………..14
1.3 Bearings…………………………………………………………………………...16
1.3.1 Definition……………………………………………………………………..16
1.3.2 Classification of Bearings…………………………………………………….16
1.3.3 Foil Bearings………………………………………………………………….17
1.4 Summary…………………………………………………………………………..18
Chapter 2. Motivation and Objectives of the present work……………………………19
2.1 Motivation of the present work……………………………………………………19
2.2 Objectives of the present work……………………………………………………20
2.3 Summary………………………………………………………………………….20
Chapter 3. System Configuration and Modelling
3.1 Rotor - Foil Bearing Configuration……………………………………………21
3.2 Mathematical Modelling of System…………………………………………...22
3.2.1 Unbalance Force Model………………………………………………...22
3.2.2 Foil Bearing Force Model………………………………………………22
3.2.3 Equations of Motion of the System……………………………………..23
3.3 Summary ………………………………………………………………………24
Chapter 4. Results and Discussion
4.1 Description of Simulink Model……………………………………………….25
4.2 Numerical Response Generation and discussion……………………………..27
5.3 Summary ……………………………………………………………………..30
Chapter 5. Conclusions & Future Work………………………………………………...31

Chapter 6. References…………………………………………………………………….32

7
 1. Introduction and Literature Review
In the recent times rotatory elements are playing an important role in manufacturing of essential

power generating equipment’s like gas turbines, wind turbines, etc. [1-3]. Rotating components in

these machines need bearings for their support while in operation or in a stationary condition.

From old times researchers have been using conventional rolling element bearings for rotor

support in which the rotor remains in physical contact with the bearings [4, 5]. This may cause

several efficient losses and low speed operations [6]. In the pursuit of enhancing machinery

efficiency and minimizing friction-related losses, the integration of foil bearings into rigid

systems has emerged as a promising avenue [7-10]. Foil bearings (refer Fig. 1) have gained

prominence in various engineering applications due to their advantages, such as high-speed

capability, low friction, and minimal need for lubrication [11]. These bearings, however, are

susceptible to vibrations that can impact their performance and reliability. The study of vibrations

in foil bearings has become a critical area of research to ensure the robust operation of machinery

and systems relying on this technology. In the recent years, Larsen et al. [12] proposed two

different schemes for stability analysis of a rigid rotor mounted on foil bearings. The schemes

include nonlinear time domain simulation as the first one and the frequency domain method as

the second scheme. They have also shown that the bump structure compliance had a very strong

influence on the journal lateral stability. Martowicz et al. [13] utilized smart materials in gas foil

bearings to enhance its capability and performance in terms of its mechanical as well as thermal

characteristics. The working operation of gas foil bearings remained stable even for demanding

excitations and environmental conditions. Khamari et al. [14] presented a brief review on

mathematical modelling and stability analysis of a rotor system supported by gas foil bearing.

They have also described several analytical models used for simulating the performance of gas

foil bearing as well as their correlations with experimental data.

8
 Fig 1.1. Section view of bump-typed gas foil bearing.

Unbalance fault is one main complex fault in rotating machines [15-18]. The force due to

unbalance fault increases with slight increase in the speed. The force is equal to the product of

mass, rotor eccentricity and square of spin speed. Thus, several researchers are working on

dynamic analysis of faulty system under different conditions monitoring techniques for

identification of fault [19-27]. Around 80 years ago, Baker [28] introduced mathematical model

for calculating unbalanced corrections in machines, where it is difficult to identify using

balancing machines. They experimented this method for finding unbalanced corrections in an

internal combustion engine crank shaft. Gupta et al. [29] proposed a dual rotor test rig set up

calculated experimentally the unbalance response, critical speeds and mode shape of the system.

Both experimental and theoretical results were compared and found reliable and effective.

However they didn’t consider the gyroscopic couple effects and rotatory inertia. Thereafter, Shih

and Lee [30] determined vibrational signals of pedestal and found the discrete unbalance state of

the rotatory machine. But this method is not reliable for finding noisy signal. Zhou and Shi [31]

worked on different theoretical models for both rigid and flexible rotor balancing. They have

stated that by using active balancing techniques we can control the induced rotor displacements.

Tiwari [32] calculated the unbalance and fluid film bearing dynamic parameters under a flexible

rotor system using the

9
 responses under two conditioning of the rotation, clockwise and counter clock wise and

identification equation. Later the active control technique is used by a researcher for determining

the unbalance state in Jeffcott rotor system. De Castro et al. [33] proposed metaheuristic search

algorithm for determining the unbalance magnitude and phase and its location in the shaft

supported with hydrodynamic bearings. This analysis is observed to be difficult to find the

unbalance parameters and position along rotor axis. Researchers used inverse problem for

determining the estimation equation and solving the characteristics of rotor bearing systems

unbalance [34]. They also referred Tikhonov regularization method for getting stable results. In

the recent time, the unbalance in a rotor and misalignment in active magnetic bearing were

observed in the published papers [35, 36] by developing novel trial misalignment method. This

method was alike to trial unbalance technique used for purpose of balancing.

From studying various past and recent literature survey, it can be concluded that most of the

research is associated with the vibrational analysis and identification considering the rolling

element bearings or active magnetic bearings as the rotor support. However, there are very less

paper in the area of analyzing the unbalance faults vibrational nature in a rotating machine

supported on foil bearings. Therefore, an attempt has been made in this paper for the numerical

investigation of dynamic analysis of a rigid rotor with two discs supported on foil bearings under

the effect of unbalance fault and gyroscopic couple moment. The displacement response has been

also analyzed at different values of spin speed of the rotor through orbital plots.

1.1 Rotor Systems

There are different types of rotor system which is important and utilized in industries for

analyzation purpose. These are discussed in this subsection.

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1.1.1 Jeffcott Rotor

Fig 1.2. A Jeffcott rotor with a disc at the middle supported by two rigid bearings at the
end positions.

A Jeffcott rotor consists of a flexible, massless, uniform shaft

mounted on two flexible bearings equidistant from a massive disk

rigidly attached to the shaft. The simplest form of the rotor

constrains the disk to a plane orthogonal to the axis of rotation.

This limits the rotor's response to lateral vibration only. If the disk

is perfectly balanced. The coordinate directions shown are used in

this work. Jeffcott published a paper confirming Foppl’s theory

and the simplest rotor model got the name the Jeffcott rotor. The

rotor consists of a flexible shaft, with zero mass, supported at its

ends (Fig.1.2). The supports are rigid and allow rotation around

the center axis of the shaft. The mass is concentrated in a disk,

fixed at the midpoint of the shaft. The rotor system is

geometrically symmetric with respect to its rotational axis, except

for a mass unbalance attached to the disk. When rotating, the mass

11
unbalance provides excitation to the system. Hence, the excitation

is sinusoidal, occurring only at the speed of rotation. In the

synchronous motion of the shaft, the orbital speed and its own spin

speed are equal. The sense of rotation of the shaft spin and the

whirling are also same. The unbalance force, in general leads to

synchronous whirl conditions; hence this motion is basically a

forced response. It is commonly used to analyze the behavior of

structures ranging from jet engines and steam turbines to auto

engines and computer disk storage.

An interesting feature of the rotor dynamic system of equations is

the off-diagonal terms of stiffness, damping, and mass. If this

force is large enough compared with the available direct damping

and stiffness, the rotor will be unstable. When a rotor is unstable, it

will typically require immediate shutdown of the machine to avoid

catastrophic failure. A complex variable can be adopted to

describe the displacement of the disk center in the Jeffcott rotor.

Yx=Re(r)

(1.1)

Yy=Im(r)

(1.2)

where r is the complex radial displacement of the disk in the XY

plane, as shown in Figure 1.3.

12
 Fig. 1.3. Complex rotor displacement in Jeffcott rotor.

In the complex form, the equation of motion of the Jeffcott rotor

(shown in Figure 1.3) at a constant speed of rotation is described

by

mr̈ +cṙ+kr=muru

(1.3)

where

‘m’ is the mass of the disk,

‘c’ is the damping constant representing an external source of damping,

‘k’ is the spring constant, equivalent to the stiffness of the shaft,

r̈ and ṙ are the second and first time derivatives of radial position,

mu is equal to the unbalancing mass on the Right hand side,

‘ru ’ is the distance of the unbalance from the geometrical center of the rotor,

‘ω’ is the rotational speed,

‘i’ is the imaginary unit,

and ‘t’ is the time.

1.1.2 Rigid Rotor

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 A rigid rotor system is a type of rotational mechanical system in

which the axis of rotation is fixed and does not change. It consists

of a central shaft or axis, around which a load or load component

is rotated. The shaft deflection is very small, the mass inertia

forces are small and the majority of centrifugal force energy is

counteracted by bearing deflection. Rigid rotor systems are

typically used in applications where precise control of rotation,

position, and movement is necessary, such as in aircraft engines or

industrial machinery.

Fig 1.4. Schematic diagram of Rigid Rotor System.

Reference:https://www.researchgate.net/figure/Rigid-rotor-free-body-
diagram_fig6_257623338

1.1.3 Flexible Rotor

When the components of the rotor are rotated at high speeds or above the critical operating

speeds, then it is called a flexible rotor. A flexible rotor machine that has significant bending

during operation. These conditions are found in components where the length-to-diameter ratio is

at its extreme and the component is running at critical/operating speed. Flexible rotors are

commonly found in pumps, generators, and other rotating machines where flexibility is

beneficial.

14
 Fig. 1.5. Schematic diagram of Flexible Rotor System.

1.2 Unbalance Fault

1.2.1 Definition

An unbalanced fault refers to a condition where the vibration pattern of a machine or system is

not symmetrical, meaning that it is not balanced about the origin of movement. This can be

caused by a variety of factors, such as misalignment of components, uneven loading, or damage

to bearings or seals. Unbalanced faults can cause vibration to be amplified, leading to increased

noise, vibration, and potential damage to the system.

1.2.2 Identification and Balancing Procedure

There are two methods for identification of unbalance faults i.e., equivalent loads minimization

and vibration minimization method. Equivalent loads minimization is a technique used to identify

unbalance faults in rotating systems by comparing the loads experienced due to a fault with the

loads predicted by a theoretical model. The goal is to minimize the difference between these two

sets of loads, thus pinpointing the location and severity of the fault. Vibration minimization

method is another approach for unbalance fault identification in rotating systems. Instead of

focusing on loads, this method looks at minimizing the vibrations caused by a fault.

15
By measuring and analyzing vibrations, the vibration minimization method aims to identify and

quantify the presence of a fault in the system.

The balancing techniques have been described for rigid as well as flexible type rotors. Single-

plane balancing, and two-plane balancing, such as the conventional cradle balancing machine

method (off-site or off-field balancing) and the modern influence coefficient method (on-site or

field balancing), are different methods for rigid rotor balancing. The modal balancing method as

well as influence coefficient method are the two basic methods for balancing the flexible rotor. In

the area of flexible rotor balancing, Bishop and Gladwell [37] proposed the modal balancing

technique, which needed the correct values of the system’s modal parameters such as mode

shape, natural frequency, modal damping, etc. On the other hand, Drechslen [38] developed the

influence coefficient method, which exploited the amplitude and phase values of vibrational

responses to determine the amounts of balance correction mass. Therefore, the influence

coefficient method requires little information on the system modelling parameters as compared to

the modal balancing method in the balancing of a flexible rotor.

Figure 1.6. Two bearing rotor system with flexible foundation (Lees and Friswell [39]).

In continuation to this, Morton [40] also utilized modal balancing technique for balancing an

elastic shaft in the absence of trial masses and without knowing the values of the supported

bearing parameters.

16
Besides, the step force and deflection concepts were used at different nodes of shaft elements.

The balancing technique was observed to be very appropriate for balancing of flexible shafts in

the various bounds of the system’s critical speeds. However, one of the drawbacks of this work

was neglecting the gyroscopic couple effect of the shaft and its rotational damping during the

mathematical modelling of the flexible rotor system.

1.3 Bearings

1.3.1 Definition
Bearings are devices that enable relative rotation between two or more components while
restricting motion in undesired directions. They are commonly used in mechanical and
hydraulic systems to facilitate smooth movement while minimizing friction and wear.

The design of the bearing may, for example, provide for free linear movement of the moving
part or for free rotation around a fixed axis; or, it may prevent a motion by controlling
the vectors of normal forces that bear on the moving parts

1.3.2 Classification of Bearings

 Classification based on direction of force

i. Radial bearings:

Radial ball bearings are friction reduction devices that carry loads radially around its
axis. A subtype of ball bearings, they operate through the use of lubricated steel balls
placed between two grooved rings. They are frequently called deep-groove bearings or
Conrad bearings.

Example: single row bearings, double row bearings, internally self-aligning

Bearings, externally self-aligning bearings etc..

ii. Thrust bearings:

Thrust bearing is a mechanical component that allows a rotating object, such as an

17
 engine or pump, to generate thrust or force in a desired direction.
Example: hydrodynamic bearings, ball bearings, and rolling element bearings etc.

Classification based on friction

I. Sliding Contact bearing:

In this type of bearings, the surface of the shaft slides over the surface of the bush. To
prevent friction, both surfaces are separated by a thin of lubricating oil. Generally, Bush
is a made from bronze or white metal.
Examples: Plain bearing, journal bearing, sleeve bearing

II. Rolling Contact bearing:

Rolling contact bearings refers to the wide variety of bearings that use spherical balls or
some other type of roller between the stationary and the moving elements.
Examples: Ball bearing, Spherical bearing, Cylindrical bearing etc..

1.3.3 Foil Bearings

A foil bearing, also known as a foil-air bearing, is a type of air bearing. A shaft is supported by a
compliant, spring-loaded foil journal lining. Once the shaft is spinning fast enough, the
working fluid (usually air) pushes the foil away from the shaft so that no contact occurs.

The shaft and foil are separated by the air's high pressure, which is generated by the rotation that
pulls gas into the bearing via viscosity effects. The high speed of the shaft with respect to the foil
is required to initiate the air gap, and once this has been achieved, no wear occurs.

Unlike aerostatic or hydrostatic bearings, foil bearings require no external pressurisation system
for the working fluid, so the hydrodynamic bearing is self-starting.

Advantages and Disadvantages of Foil bearings

 Foil bearings have low friction torque at high rotational speeds and good dynamic
properties over a wide range of speed.
 Foil bearings have reduced wear and tear on the shaft and bearing surface due to the
presence of the foil, smooth and quiet operation, and the ability to separate the shaft
from the bearing surface if necessary
 Higher load capacity, Improved damping and Improved coatings

18
  The disadvantages of foil bearings also need to be considered, such as the fact that they
may require more maintenance and replacement than other types of bearings, and the
potential for issues if not properly installed or adjusted
 Lower capacity than roller or oil bearings and Wear during start-up and stopping
 High speed required for operation

1.4 Summary

 At the end of lesson we can able to identify the type of rotor system and it’s functions
and their applications
 We can identify the faults and its balancing procedure
 Bearings and it’s types and about foil bearings

2. Motivation and objective of the present work

19
2.1 Motivation of the present work

From studying various past and recent literature survey, it can be concluded that most

of the research is associated with the vibrational analysis and identification considering

the rolling element bearings or active magnetic bearings as the rotor support. However,

there are very less paper in the area of analyzing the unbalance faults vibrational nature

in a rotating machine supported on foil bearings. Therefore, an attempt has been made

in this paper for the numerical investigation of dynamic analysis of a rigid rotor with

two discs supported on foil bearings under the effect of unbalance fault and gyroscopic

couple moment. The displacement response has been also analyzed at different values

of spin speed of the rotor through orbital plots.

2.2 Objective of the present work

• To develop mathematical model of the unbalanced rigid rotor-foil bearing system

20
 • To develop equations of motion of the rotor system using moment equilibrium method

• To obtain the time domain displacement responses of the system at foil bearing locations

• To analyze the system responses based on changing various parameters

2.3 Summary

In this chapter we have discussed about our main motive towards selecting this topic for our

research discussed about our objectives in this project

21
 3. Mathematical model of the rotor-bearing system
3.1 Rotor-Foil bearing system configuration and assumptions:

Fig.3.1: A Rigid rotor with two discs supported by foil bearings at the end positions

Assumptions:

• It is an unbalanced rigid rotor with two offset discs mounted on two foil bearings.

• The shaft has been assumed to be rigid and massless based on lumped mass parameter
model.

• The translational degrees of freedom are considered to be in vertical (x-axis) and

horizontal (y-axis) directions.

• Based on lumped mass parameter model, shaft is compared to be massless as compared

To disc mass.

• The system having two degree of freedom (i.e. vertical and horizontal).

• There is no coupling of motion between x and y direction.

• For developing equations of motion, moment equilibrium method will be used.

Here l1, l2, r1, r2 are considered to be of different lengths.

22
 3.2 Mathematical modelling of system

O y
y
Disc
x Shaft

C
e
· G

ωt
β
x
Fig.3.2: A Side view axial representation of rotor with unbalance

Force due to unbalance can be written in x and y directions as

2
f unbx=me ω cos ( ωt+ β ) (3.1)
2
f unby=me ω sin ( ωt + β ) (3.2)

Where, m is disc mass, e is disc eccentricity, ω is the spin speed of the rotor, β is the phase of
unbalance.

The force due to FBs is also bidirectional, the x- and y- components can be written as
f F B =¿ K xx x + K xy y +C xx ẋ +C xy ẏ
x
(3.3)

f FBy =¿ K yx x + K yy y +C yx ẋ +C yy ẏ (3.4)

where K is stiffness parameters of both FBs. C is damping parameters of both FBs

Considering the moment of unbalance force, Inertia force, gyroscopic effect in z-x plane and z-y
plane about the Centre of FB2 is,

[
M x =−I t φ̈ y −I p ω φ̇ x −( m ẍ ) L2− m 1 e1 ω ⅇ
1
2 j (wt +ϕ1 )
( L2 + R1 ) +m2 e2 C xx ( L2−R2 ) ] −f bx 1 L=0(3.5)
1

I t φ̈ y + I p ω φ̇ x +(m ẍ) L2+¿

Similarly for My₁,

I t φ̈ x −I p ω φ̇ y +(m ÿ )L2 +¿ (3.7)

m ẍ
23
 φy
x=x 1 +( x 2−x 1
L ) L
L ( ) ( )
L
L1 x= 2 x 1+ 1 x 2
L
… (3.8)
x1 x x2

Fig 3.3: Linear and angular displacement of shaft in x axis

m ÿ

φx
y= y 1+ ( L )
y 2− y 1
( ) ( )
L
L
L
L1 , y= 2 y 1 + 1 y 2
L
…. (3.9)

y1 y y2

Fig 3.4: Linear and angular displacement of shaft in y axis

24
Substituting the value of foil bearing force f bx 1 and re-arranging the terms we get,

( m l̄ 22−it ) ẍ 1 + ( m l̄ 1 l̄ 2 +it ) ẍ 2+ ip ω ẏ 2−i p ω ẏ 1 +{K xx 1 x 1 + K xy 1 y 1 +C xx 1 ẋ 1 +C xy 1 ẏ1 }=−¿

Similarly for equation (2),(3),(4),we can get after simplifying

( m l̄ 22−it ) ÿ 1 + ( m l̄ 1 l̄ 2 +it ) ÿ 2 +i p ω ẋ 1−i p ω ẋ 2 +{K yx 1 x 1+ K yy 1 y 1 +C yx 1 ẋ 1 +C yy 1 ẏ 1=− j [ F unb 1 y ( l̄ 2 + r̄ 1 ) + F unb2 y ( l̄ 2−¯

( m l̄ 1 l̄ 2−it ) ẍ 1 + ( m l̄ 21 +it ) ẍ 2 +ip ω ( ẏ 1) −i p ω ( ẏ 2 ) +{K xx 2 x 2 + K xy 2 y 2 +C xx 2 ẋ 2 +C xy 2 ẏ 2 }=−¿

( m l̄ 1 l̄ 2−it ) ÿ 1+ ( m l̄ 21 +it ) ÿ2 −ip ω ( ẋ 1 ) +i p ω ( ẋ 2 ) + { K yx 2 x 2+ K yy 2 y 2 +C yx 2 ẋ 2 +C yy 2 ẏ 2 }=− j [ Funb 1 y ( l̄ 1 −r̄ 1 )+ Funb 2 y ( l̄ 1

(3.13)

Fig. 3.5: Linear and angular movements of the rotor in (a) x-z plane (b) y-z plane

With,

[ ]
2
m l̄ 2−i t 0 m l̄ 1 l̄ 2 +i t 0
0 m l̄ 2−i t 0 m l̄ 1 l̄ 2+i t
[ M ]= 2
m l̄ 1 l̄ 2−i t 0 ml 1+i t 0
0 m l̄ 1 l̄ 2−i t 0 m l̄ 21+ it

25
 [ ][ ][ ]
1 1 1 1
0 −i p 0 ip K xx K xy 0 0 C xx C xy 0 0
1 1 1 1
0 −i p 0 K K 0 0 C C 0 0
[ G ]= i p [ K ]= yx yy
2 2
[C ]= yx yy
2 2
0 ip 0 −i p 0 0 K xx K xy 0 0 C xx C xy
−i p 0 ip 0 0 0 K
2
yx K
2
yy 0 0 C
2
yx
2
C yy

{ }
( l 2 +r 1 )
− j ( l 2+ r 1 )
{ f unb }=m1 e1 ω2 e (
j ωt +φ 1)
+ m2 e 2 ω 2 e (
j ωt+ φ )
¿ 2

( 1 1)
l +r
− j ( l+r 1 )

Equation of motion for the rigid rotor system employed with FB can be written as,
[ M ] {η̈ }+ {−ω [ G ] + [ c ] } { η̇ }+ [ K ] { η }= {f unb ( t ) } (3.15)

3.3 Summary
 Objective: To develop a mathematical model for the dynamic analysis of a rigid rotor
system, focusing on its vibrations, stability, and overall behavior under operational
conditions.

 Applications: Used in mechanical and aerospace engineering for machinery like turbines,
engines, and compressors.

 Rigid Rotor: Assumes the rotor does not deform under operational speeds, simplifying
the dynamic analysis.

 Dynamic Analysis: Involves studying the system's response to various forces, particularly
unbalance forces leading to vibrations.
Mathematical modeling of a rigid rotor system's dynamic analysis is crucial for understanding
and optimizing its performance. The model relies on fundamental principles of mechanics, linear
assumptions for simplicity, and advanced numerical techniques for accurate predictions. This
analysis is instrumental in various engineering applications, ensuring the reliability and efficiency
of rotating machinery.

26
 4. Results and Discussion

Fig. 4.1: Developed SIMULINK model of the system.

4.1 Description of Simulink model

In this section, the displacement responses at foil bearing positions has been generated by solving

the equations with the help of SIMULINK model (refer Fig. 4.1) in MATLAB (Version R2023b)

software. The Runge-Kutta 4th order differential equation solver with a constant time step size of

0.0001 s was used for the solution purpose.

The assumed values of shaft, discs and foil bearings as well as unbalance fault parameters taken

for solving the equations of motion is given in Table 4.1. The air gap clearance between the rotor

and top foil of the foil bearing is considered as 80 μm. The generated displacement responses are

lesser than the clearance value to avoid any kind of collision of the rotor with the bearing.

27
Table 4.1: Assumed parameters of rotor bearing system for the numerical investigation

Fig: 4.2. The displacement response for the initial transient state in (a) x-direction (b) y-

direction
28
4.2 Numerical Response Generation and Discussion

The simulation was run for 5s and the time domain displacements at FB1 position (ux1 and uy1)

for the first 0.5s has been shown in Fig. 5 for the initial transient response. The maximum

displacement in the vertical and horizontal directions at FB1 in the transient condition are

3.03×10-5 m and 2.50×10-5 m, respectively. The steady state response for the last one second

(4s-5s) is also represented in Fig. 6. Figure 6 (a) and (b) depict the x- and y-directional

displacements at FB1 and Figure 6 (c) and (d) show the x- and y-directional displacements at

FB2. The response of the system (in Fig. 5 and 6) has been obtained by running the simulation at

the angular frequency of 35 Hz. This frequency is chosen below the first critical speed of the

system to have rigid behavior of the rotor during its operation. The peak values of steady state

displacement in the x and y directions at FB1 are found to be 2.46×10-5 m and 2.42×10-5 m,

where the peak values at FB2 are 3.89×10-5 m and 3.75×10-5 m. The displacement orbit

responses for the transient and steady state conditions are, respectively, given in Figures 7 and 8.

Fig 4.3: The steady state displacement responses at the angular frequency of 35 Hz (a) x-

direction at FB1 (b) y-direction at FB1 (c) x-direction at FB2 (d) y-direction at FB2.

29
Fig. 4.4. Displacement orbits considering the initial transient state at the angular frequency

of 35 Hz (a) FB1 location (b) FB2 location.

The shape of orbits for the steady state are observed to be circular but little irregular. This is due

to different values of mass, eccentricities and phases of disc1 and disc2 as well as anisotropic and

different nature of both foil bearings.

Further, the displacement orbits have been plotted (refer Fig. 9) in the combined way at different

angular frequencies of the rotor (i.e., 20 Hz, 25 Hz, 30 Hz, 35 Hz and 40 Hz). This has been done

to show the effect of spin speed on the displacement response. Table 2 shows the maximum value

of displacements at FB1 and FB2 in the x- and y-directions.

The orbit in red color is at 20 Hz speed, the orbit in green color is at 25 Hz speed, the orbit in

blue color is at 30 Hz speed, the orbit in black color is at 35 Hz speed, and the orbit in magenta

color is at 40 Hz speed.

30
It is noticed that the size of orbit increases with the increment in the speed, following the concept

of increment in the unbalance force proportionally with the square of the spin speed.

The percentage increase in the x-displacement response (at FB1 position) at 40 Hz frequency with

respect to 35 Hz is 56.91%, whereas the increment is 146.86% relative to 30 Hz. Similarly, at

FB2 location, the percentage increase in the x- displacement response at 40 Hz frequency with

respect to 35 Hz is 53.47%, whereas the increment is 135.97% relative to 30 Hz. In the y-

direction at FB1, the percentage increase in the displacement at 40 Hz frequency with respect to

35 Hz is 58.26%, whereas the increment is 150.33% relative to 30 Hz.

Similarly, at FB2 location, the percentage increase in the y- displacement response at 40 Hz

frequency with respect to 35 Hz is 54.13%, whereas the increment is 138.84% relative to 30 Hz.

Fig. 4.5 Displacement orbits considering the steady state at the angular frequency of 35 Hz

(a) FB1 location (b) FB2 location.

31
Fig.4.6The steady state combined displacement orbits at different angular frequencies (a)

FB1 location (b) FB2 location.

Table:2 Peak value of displacement at different angular frequency of the rotor

4.3 Summary

This chapter concludes with the results and discussion for the considered rotor-bearing system

for obtaining the solution of equation of motion. Simulink model has been successfully built

and run. The displacement response, full spectrum and orbit plots for the displacement

responses has been successfully drawn in the figure. The response due to crack fault is

dominating at lower spin speed and unbalance dominates at higher spin speed. It is also shown

on how the crack fault increase its response.

32
 5. Conclusions and future work

In the present article, a rigid rotor system with two offset discs supported by two foil bearings has

been considered under the influence of unbalance fault. The mathematical modeling and

equations of motion of the system has been developed using moment equilibrium technique.

The Simulink model in MATLAB is used to obtain the displacement responses of the system. The

responses in the time domain and orbit plots are analyzed for initial transient state and steady

state conditions.

Considering the different values of spin speed of the rotor, the displacement responses are also

obtained and found that the orbit size is increasing with the speed increment under the effect of

unbalance force.

The orbits are also observed to be circular but little irregular in shape due to anisotropic nature of

foil bearings and distinct values of mass, eccentricities and phases of both discs. The analysis of

displacement response can also be performed for a flexible rotor system with multiple discs

supported on foil bearings as a scope of future work.

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