Bala Murugan S FINITE ELEMENT ANALYSIS OF MULTI-DISK ROTOR-BEARING SYSTEM WITH TRANSVERSE CRACK

FINITE ELEMENT ANALYSIS OF MULTI-DISK
ROTOR-BEARING SYSTEM WITH TRANSVERSE
CRACK
A THESIS SUBMITTED IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
Master of Technology (Research)
in
Mechanical Engineering
by
Bala Murugan S
Roll No: 612ME310
Department of Mechanical Engineering
National Institute of Technology
Rourkela - 769008
March 2015
FINITE ELEMENT ANALYSIS OF MULTI-DISK
ROTOR-BEARING SYSTEM WITH TRANSVERSE
CRACK
A THESIS SUBMITTED IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
Master of Technology (Research)
in
Mechanical Engineering
by
Bala Murugan S
Roll No: 612ME310
Under the Guidance of
Dr. R.K. Behera
Department of Mechanical Engineering
National Institute of Technology
Rourkela - 769008
Declaration
I hereby declare that this submission is my own work and that, to the best of
my knowledge and belief, it contains no material previously published or written by
another person nor material which to a substantial extent has been accepted for
the award of any other degree or diploma of the university or other institute of
higher learning, except where due acknowledgement has been made in the text.
(Bala Murugan S)
Dr. Rabindra Kumar Behera
Department of Mechanical
Engineering
National Institute of Technology
Rourkela 769008
Odisha, India.
National Institute of Technology
Rourkela
Email:
[email protected]
[email protected]
Phone:
+91-661-246-2504 (O)
+91-661-246-3504 (R)
+91 8895444886 (M)
Certificate
This is to certify that the thesis entitled, “Finite Element Analysis of MultiDisk Rotor-bearing System with Transverse Crack” being submitted by Shri Bala
Murugan S is a bona fide research carried out by him under my supervision in
partial fulfilment of the requirements for the Degree of Master of Technology
(Research) in Machine Design and Analysis at Mechanical Engineering
Department, National Institute of Technology, Rourkela.
To the best of my knowledge, the matter embodied in the thesis has not
been submitted to any other University / Institute for the award of any Degree or
Diploma. The matter embodied in this thesis is original and has not been used for
the award of any other degree.
I wish him all success in his future endeavour.
Dr. R.K. Behera (Supervisor)
Associate Professor
Department of Mechanical Engineering
National Institute of Technology
Rourkela- 769 008
i
Acknowledgements
With deep regards and profound respect, I avail this opportunity to express my deep
sense of gratitude and obligation to Prof. Rabindra Kumar Behera, Associate Professor,
Department of Mechanical Engineering whose valuable suggestions, interests, patience and
inspiring guidance, constructive criticism throughout in this research work which made this
work a truly rewarding experience by several altitudes.
I am sincerely thankful to Prof. S.S. Mohapatra, Head, Department of Mechanical
Engineering, for his advice and providing necessary facility to carry this work.
I would like to thank Prof. S.K. Panda, Department of Mechanical Engineering, for
his advice and providing necessary suggestions to make forward this work.
I would also like to thank Prof. S. K. Sahu, Department of Civil Engineering, NIT
Rourkela, and Prof. Md. Equeenuddin, Department of Mining Engineering for their
talented advices.
I am also thankful to my friends and colleagues for standing by me during the past
difficult times. Particularly, I am indebted to Mr. Alok Ranjan Biswal and Mr. Sudhansu
Meher for their utterly selfless help.
I am greatly indebted to my grandfather Palaveasam Assary, Mother S. Velammal,
Father P. Subiramaniyan and Sister S. Amirthavalli for their loving support throughout.
Finally I am thankful to all my other family members and friends for their support in
completion of the present dissertation.
Bala Murugan S
ii
Table of Contents
Certificate ........................................................................................................................... i
Acknowledgements ............................................................................................................ii
Table of Contents.............................................................................................................. iii
Abstract ............................................................................................................................ vi
List of Tables ....................................................................................................................vii
List of Figures ................................................................................................................. viii
Nomenclature ..................................................................................................................... 1
1 Introduction ................................................................................................................... 4
1.1 Background and Significance .............................................................................4
1.2 Basic principles ................................................................................................. 5
1.3 History of Rotor Dynamics ............................................................................... 6
1.3.1 From Rankine to Jeffcott Rotor systems ............................................ 6
1.4 Research goals and Analysis Approach ............................................................ 7
2 Literature Review .......................................................................................................... 9
2.1 Introduction ....................................................................................................... 9
2.2 Dynamic analysis of rotor-bearing systems ...................................................... 9
2.3 Summary............................................................................................................24
3 Theoretical analysis ..................................................................................................... 25
3.1 System Equation of Motion without crack ...................................................... 25
3.1.1 Undamped flexible finite rotor shaft element ................................... 29
3.1.2 Energy equations .............................................................................. 31
3.1.2.1 Rigid Disc .......................................................................... 32
3.1.2.2 Bearings ..............................................................................34
3.1.3 Rotor element with variable cross section .........................................35
3.1.4 Undamped system equation of motion ..............................................36
iii
3.1.5 Damped flexible finite rotor shaft element ........................................37
3.1.6 Damped system equation of motion ..................................................39
3.1.7 System instability regions ................................................................. 39
3.1.8 Whirl speed analysis ..........................................................................40
3.2 Fault modelling in the rotor system ................................................................. 41
3.2.1 Linear mass unbalance in rotor ..........................................................41
3.2.1.1 Unbalance Response .......................................................... 41
3.2.2 Transverse crack modelling .............................................................. 42
3.2.2.1 Transverse crack element modelling ..................................43
3.2.2.2 Open crack ......................................................................... 43
3.2.2.3 Equation of motion of the system with transverse open
crack ................................................................................... 44
3.3.3 Lateral displacement responses of bearing using ANSYS ................ 45
4 Numerical analysis and Discussions ........................................................................... 47
4.1 Undamped rotor bearing system without crack ................................................ 50
4.1.1 Natural whirl frequencies and mode shapes ...................................... 50
4.1.2 Unbalance Response .......................................................................... 52
4.1.3 Natural whirl speeds .......................................................................... 53
4.1.4 System instability regions ................................................................. 54
4.2 Damped rotor bearing system without crack ................................................... 55
4.2.1 System with hysteretic damping ....................................................... 55
4.2.2 System with viscous damping ........................................................... 57
4.3 Undamped rotor bearing system with transverse crack ................................... 59
4.3.1 Natural whirl frequencies and mode shapes ..................................... 60
4.3.2 Unbalance response with transverse crack ....................................... 62
4.3.3 System natural whirl speeds with transverse crack .......................... 62
4.3.4 Effect of crack depths on natural whirl frequencies ..........................64
iv
4.3.5 System instability regions with transverse crack .............................. 66
4.4 Frequency domain and phase – plane diagrams .............................................. 67
4.4.1 Undamped system without transverse crack .................................... 67
4.4.2 Damped system without transverse crack ........................................ 70
4.4.2.1 Hysteretic damping without transverse crack ................... 70
4.4.2.2 Viscous damping without transverse crack ....................... 72
4.4.3 Undamped system with transverse crack ......................................... 74
4.5 Bearing reaction force (ANSYS® - v13) ......................................................... 76
4.6 Observations .................................................................................................... 78
5 Conclusions and Future scope .................................................................................... 79
5.1 Conclusions ..................................................................................................... 79
5.2 Future scope ..................................................................................................... 82
Bibliography ..................................................................................................................... 83
List of Publications .......................................................................................................... 91
v
Abstract
The vibration analysis of rotating systems is pronounced as a key function in all the
fields of engineering. The behavior of the rotor systems are mainly resulting from the
excitations from its rotating elements. There are several numerical methods present to
analyze the rotor-bearing systems. Finite element method is a key tool for dynamic analysis
of rotor bearing system. The current study describes a multi disk, variable cross section
rotor-bearing system with transverse crack on axisymmetric elements supported on
bearings in a fixed frame. The shaft in the rotor-bearing system is assumed to obey EulerBernoulli beam theory. The equation of motion of the rotor-bearing system is derived by
Lagrangian approach along with finite element method. Finite element model is used for
vibration analysis by including rotary inertia and gyroscopic moments with consistent
matrix approach. The rotor bearing system consists of two bearings and two rigid disks.
One disk is overhung and the other one is placed between the bearings. Internal damping of
the shaft and linear stiffness parameter of the bearings are taken into account to obtain the
response of the rotor-bearing system. The rotor has variable cross-section throughout the
configuration. The disks are modeled as rigid and have mass unbalance forces. The critical
speed, unbalance response and natural whirls are analyzed for the typical rotor-bearing
system with transverse crack. Analysis includes the effect of crack depths, crack location
and spin speed. The results are compared with the results obtained from finite element
analysis. The bearing configurations are undamped isotropic and orthotropic. The natural
whirl speeds are analyzed for the synchronous whirl for both the uncracked and cracked
rotor bearing system using Campbell diagrams. The effect of transverse crack over the
starting point of the system instability regions in the rotating speed axis with zero
asymmetric angle is examined. Further, Houbolt’s time integration scheme is used to
obtain the phase diagrams and frequency response for both the bearing cases to study the
stability threshold. Analyses are carried out by using numerical computing software.
Keywords: Finite Element Method, Rigid disk, Transverse crack, Unbalance response,
Whirl speeds.
vi
List of Tables
Sl. No.
Table caption
Page No.
1.
Table 4.1 Rotor element configuration data .............................................. 48
2.
Table 4.2 Physical and mechanical properties of shaft and disk ............... 49
3.
Table 4.3 Natural whirl frequencies of isotropic bearing .......................... 50
4.
Table 4.4 Natural whirl frequencies of orthotropic bearing ...................... 50
5.
Table 4.5 Comparison of critical speeds for isotropic and orthotropic
bearings ...................................................................................... 52
6.
Table 4.6 Natural whirl speeds for isotropic bearing ................................. 53
7.
Table 4.7 Natural whirl speeds for orthotropic bearing ............................. 54
8.
Table 4.8 Natural whirl speeds for isotropic bearing ( η H = 0.0002) .......... 56
9.
Table 4.9 Natural whirl speeds for orthotropic bearing ( η H = 0.0002) ...... 57
10.
Table 4.10 Natural whirl speeds for isotropic bearing ( ηV = 0.0002s) ........57
11.
Table 4.11 Natural whirl speeds for orthotropic bearing ( ηV = 0.0002s) ... 59
12.
Table 4.12 Natural whirl speeds for isotropic bearing ............................... 60
13.
Table 4.13 Natural whirl speeds for orthotropic bearing ........................... 60
14.
Table 4.14 Natural whirl speeds for λ = 0, ¼, ½ and 1 for isotropic
bearing ........................................................................................................ 63
15.
Table 4.15 Natural whirl speeds for λ = 0, ¼, ½ and 1 for orthotropic
bearing ........................................................................................................ 63
16.
Table 4.16 Natural whirl frequencies with µ = 0.1, 0.2 and 0.3 for isotropic
bearings ...................................................................................................... 65
17.
Table 4.17 Natural whirl frequencies with µ = 0.1, 0.2 and 0.3 for
orthotropic bearings ................................................................................... 66
vii
List of Figures
Sl. No.
Figure caption
Page No.
1.
Figure 1.1 Rotor model (a) Rankine model (b) Jeffcott rotor model ........... 6
2.
Figure 3.1 Typical Rotor-bearing-disk system configurations ................... 26
3.
Figure 3.2 Cross section rotation angles .................................................... 27
4
Figure 3.2.1 Relationship between slope and displacements.......................28
5.
Figure 3.3 Finite rotor element and coordinates ........................................ 29
6.
Figure 3.4 Displaced position of the shaft cross-section ........................... 30
7.
Figure 3.5 Sub-elements assemblage ......................................................... 35
8.
Figure 3.6 Relative positions of the shaft and transverse crack in
circumference ............................................................................................. 42
9.
Figure 3.7 Rotor-bearing system with SOLID273 axisymmetric elements 46
10.
Figure 4.1 Rotor elements with variable cross section ............................... 47
11.
Figure 4.1.1. Finite models of the system (a) Undamped system without
crack (b) Damped system without crack, (internal dampings η H = 0.0002 &
ηV = 0.0002s), (c) Undamped system with crack ....................................... 49
12.
Figure 4.2 Mode shapes at 0 and 30000 rpm (a) First mode shape (b)
Second mode shape (c) Third mode shape ................................................. 51
13.
Figure 4.3 Unbalance response of rotor with isotropic and orthotropic
bearings ...................................................................................................... 52
14.
Figure 4.4 Campbell plot for rotor-bearing system with both bearings...... 53
15.
Figure 4.5 The starting points of instability regions related to I and II FW
whirl modes .................................................................................................55
16.
Figure 4.6 Natural whirl frequency of rotor with hysteretic damping on
isotropic bearing ......................................................................................... 55
17.
Figure 4.7 Natural whirl frequency of rotor with hysteretic damping on
orthotropic bearings .................................................................................... 56
viii
Sl. No.
18.
Figure caption
Page No.
Figure 4.8 Natural whirl frequency of rotor with viscous damping on
isotropic bearings ....................................................................................... 58
19.
Figure 4.9 Natural whirl frequency of rotor with viscous damping on
orthotropic bearings..................................................................................... 58
20.
Figure 4.10 Rotor bearing system with variable cross sections and crack...59
21.
Figure 4.11 Mode shapes for spin speed 0 and 30000 rpm (a) first mode
shape (b) second mode shape (c) third mode shape ................................... 61
22.
Figure 4.12 Unbalance response of rotor with transverse crack (h/R = 0.3)
for isotropic and orthotropic bearings ........................................................ 62
23.
Figure 4.13 Campbell plot for rotor-bearing system with transverse crack
for isotropic and orthotropic bearings ........................................................ 63
24.
Figure 4.14 Campbell plot for rotor-bearing system with µ = 0.1, 0.2 and
0.3 on natural whirl frequencies for isotropic bearing ............................... 64
25.
Figure 4.14 (a). Magnified view of II FW and II BW whirls for rotorbearing system with µ = 0.1, 0.2 and 0.3 on natural whirl frequencies for
isotropic bearing ......................................................................................... 64
26.
Figure 4.15 Campbell plot for rotor-bearing system with µ = 0.1, 0.2 and
0.3 on natural whirl frequencies for orthotropic bearing ............................ 65
27.
Figure 4.15 (a). Magnified view of II FW and II BW whirls for rotorbearing system with µ = 0.1, 0.2 and 0.3 on natural whirl frequencies for
orthotropic bearing ..................................................................................... 65
28.
Figure 4.16 The starting points of instability regions related to I and II FW
whirl modes with non dimensional crack depth ......................................... 66
29.
Figure 4.17 Response and phase-plane diagrams of disk 1 and 2 for spin
speed 1000 rpm with eccentricity of 1mm for isotropic bearing ............... 68
30.
Figure 4.18 Response and phase-plane diagrams of disk 1 and 2 for spin
speed 5000 rpm with eccentricity of 1mm for isotropic bearing ............... 68
31.
Figure 4.19 Response and phase-plane diagrams of disk 1 and 2 for spin
speed 1000 rpm with eccentricity of 1mm for orthotropic bearing ............ 69
32.
Figure 4.20 Response and phase-plane diagrams of disk 1 and 2 for spin
speed 5000 rpm with eccentricity of 1mm for orthotropic bearing ............ 69
ix
Sl. No.
33.
Figure caption
Page No.
Figure 4.21 Response and phase-plane diagrams of disk 1 and 2 for spin
speed 1000 rpm with eccentricity of 1mm for isotropic bearing with
hysteretic damping ..................................................................................... 70
34.
Figure 4.22 Response and phase-plane diagrams of disk 1 and 2 for spin
speed 5000 rpm with eccentricity of 1mm for isotropic bearing with
hysteretic damping ..................................................................................... 71
35.
Figure 4.23 Response and phase-plane diagrams of disk 1 and 2 for spin
speed 1000 rpm with eccentricity of 1mm for orthotropic bearing with
hysteretic damping ..................................................................................... 71
36.
Figure 4.24 Response and phase-plane diagrams of disk 1 and 2 for spin
speed 5000 rpm with eccentricity of 1mm for orthotropic bearing with
hysteretic damping ..................................................................................... 72
37.
Figure 4.25 Response and phase-plane diagrams of disk 1 and 2 for spin
speed 1000 rpm with eccentricity of 1mm for isotropic bearing with viscous
damping ...................................................................................................... 72
38.
Figure 4.26 Response and phase-plane diagrams of disk 1 and 2 for spin
speed 5000 rpm with eccentricity of 1mm for isotropic bearing with viscous
damping ...................................................................................................... 73
39.
Figure 4.27 Response and phase-plane diagrams of disk 1 and 2 for spin
speed 1000 rpm with eccentricity of 1mm for orthotropic bearing with
viscous damping ......................................................................................... 73
40.
Figure 4.28 Response and phase-plane diagrams of disk 1 and 2 for spin
speed 5000 rpm with eccentricity of 1mm for orthotropic bearing with
viscous damping ......................................................................................... 74
41.
Figure 4.29 Response and phase-plane diagrams of disk 1 and 2 for spin
speed 1000 rpm with eccentricity of 1mm for isotropic bearing with
transverse crack .......................................................................................... 74
x
Sl. No.
42.
Figure caption
Page No.
Figure 4.30 Response and phase-plane diagrams of disk 1 and 2 for spin
speed 5000 rpm with eccentricity of 1mm for isotropic bearing with
transverse crack .......................................................................................... 75
43.
Figure 4.31 Response and phase-plane diagrams of disk 1 and 2 for spin
speed 1000 rpm with eccentricity of 1mm for orthotropic bearing with
transverse crack .......................................................................................... 75
44.
Figure 4.32 Response and phase-plane diagrams of disk 1 and 2 for spin
speed 5000 rpm with eccentricity of 1mm for orthotropic bearing with
transverse crack .......................................................................................... 75
45.
Figure 4.33 Bearing reaction forces – Transient analysis .......................... 77
xi
Nomenclature
Overall cracked element cross-sectional area
C
Bearing damping coefficient, i, j = V, W
[C]
Bearing damping matrix
Young’s modulus of the shaft material
Centroid location of in y-axis
e
Element gyroscopic matrix
h
crack depth in the radial direction
Second moment of inertia of the shaft
Constant quantities during the rotation of the cracked shaft, i = 1, 2
[I]
Identity matrix
(t)
time-varying area moments of inertia of the cracked element about X axis
(t)
time-varying area moments of inertia of the cracked element about Y axis
Area moments of inertia of the overall cross sectional area of the cracked
element about ( ) X axis
Area moments of inertia of the overall cross sectional area of the cracked
element about ( ) Y axis
ℐ , ℐ
Diameter and polar mass moments of inertia of the shaft per unit length
Kinetic energy of the element
[K]
Stiffness matrix
k (t )
Time varying stiffness matrix of the open cracked element
oc
k ,k
o1
o2
secondary stiffness matrices due to the open crack
Bearing stiffness coefficient, i, j = V, W
l
Length of the shaft element in the finite element model
[M]
Mass matrix
1
Disk mass
(s)
One dimensional quadratic Lagrangian shape function
Translational shape function, i = 1, 2, 3, 4
Rotational shape function, i = 1, 2, 3, 4
Potential energy of the element
Nodal displacement vector
Unbalance response associated with cos (Ωt)
! Unbalance response associated with sin (Ω" # )
R
shaft radius
s
axial distance within an element
T
Kinetic energy of the disc
$%
Total kinetic energy of the system
t
time
&%
Total potential energy of the system
(V, W)
Translational displacements in Y and Z directions
XYZ
fixed reference frame
' External force vector relative to fixed reference frame
' Unbalance force respect to cos (Ωt)
'! Unbalance force respect to sin (Ω" # )
Greek
[Ψ ]
Matrix of translational displacement functions
[Φ]
Matrix of rotational displacement functions
φ
angle between the major axes of the crack and the shaft
ε
spinangle
2
Ω
ω
spinspeed=45
whirl speed
µ
Element mass per unit length
µ
Non-dimensional crack depth
78 79 Hysteretic damping coefficient of the shaft material
Viscous damping coefficient of the shaft material
(θ , ζ ) SmallanglerotationsaboutYandZaxes
γ
Proportionaldampingcoefficient
λ
whirl ratio
Superscripts
T
Transpose
.
Dot, differentiation with respect to time
′
prime, differentiation with respect to axial distance s
d,e,b,s
j
refers to disk, element, bearing, and system
cracked element
Subscripts
T, R, B
refers to translational, rotational and bending
3
Chapter-1
Introduction
The subject rotor dynamics is called an idiosyncratic branch of applied mechanics which
deals with the performance and detection of spinning structures. The predictions of the
system dynamic aspect are meticulously essential in the design of rotating structures.
Generally it analyzes the behaviour of rotating structures which ranges from fans, gear
trains to turbines and aircraft jet engines. Rotating systems generally develop instabilities
which are excited by unbalance and the internal makeup of the rotor system and must be
corrected. This is the prime area of interest for the design engineers who model the rotating
systems.
1.1. Background and Significance
From ISO definition, rotor can be defined as a body which is suspended through a set of
cylindrical rest or bearings that grants the system to rotate freely about an axis secured in
space. In the basic level of rotor dynamics, it is related with one or more mechanical
structures (rotors) supported by bearings that rotate around a unique axis. The non spinning
structure is called a stator. When the spin speed increases the amplitude of vibration
increases and is maximum at a speed called critical speed. This amplitude is often elevated
by unbalance forces from disk of the spinning system. When the system reaches excessive
amplitude of vibration at the critical speed, catastrophic failure occurs. Normally turbo
4
machineries frequently develop instabilities which are mainly due to the internal
configuration, and should be rectified.
Often rotating structures originates vibrations depending upon the complexity of
the mechanism involved in the process. Even a small misalignment in the machine can
increase or excite the vibration signatures. System vibration behaviour due to imbalance is
the main aspects of rotating machinery, and it must be measured in detail and reviewed
while designing. Every object including rotating structures shows natural frequency
depending on the complexity of the structure. The critical speed of these rotating structures
arises when the rotational speed meets with its natural frequency. The first critical speed
can be encountered at the lowest speed. However as the speed increases further critical
speeds can also be spotted. It is very essential to reduce the rotational unbalance and
excessive external forces to minimize the overall forces which actuate resonance. The
major concern of designing a rotating machine is, avoiding the vibration in resonance
which creates a destructive energy. Situations involving rotation of shaft near critical speed
must be avoided. When these aspects are ignored it might results in wear and tear of the
equipment, failure of the machinery, human injury and sometime cost of lives.
1.2. Basic principles
To model the actual dynamics of the machine theoretically is a cumbersome task. Based on
the simplified models, the calculations are made to simulate various structural components.
The dynamic system of equations have interesting feature, in which the off-diagonal
elements are stiffness, damping, and mass. These three elements can be called as, crosscoupled stiffness, cross-coupled damping, and cross-coupled mass. Although there is a
positive cross-coupled stiffness, a deflection will originate a reaction force opposite to the
direction of deflection, and also the reaction force can be in the direction of positive whirl.
5
When these forces are large compared to the direct damping and stiffness, the rotor will be
unstable. If a rotor is unstable it typically needs a prompt shutdown of the machine to
avoid breakdown.
1.3. History of Rotor Dynamics
Rotor dynamics has been steered further by its practice than by its theory. This remark is
particularly related to the initial history of rotor dynamics. History and research on rotor
dynamics spellss at least 14 decades.
1.3.1. From Rankine to Jeffcott Rotor systems
The history of rotordynamics begins with W. J. M. Rankine [1], who first performed an
analysis of a rotating
ing shaft (Figure 1.1a) in 1869. He concludes that
hat beyond a certain spin
speed, the shaft is appreciably bent and whirls around the bent axis.. He illustrates that, this
speed as the whirling speed of the spinning shaft.
a.
b.
Figure 1.1 Rotor model (a) Rankine model (b) Jeffcott rotor model
Whirling of the shaft refers to the shift of the disk’s centre of mass in the deflected
position from a plane perpendicular to the bearing axis
axis. Whirl frequency
f
‘v’ mainly
depends on the stiffness and damping of the rotor and the amplitude becomes a function of
6
the frequency ω, (excitation force) and magnitude. Critical speed, ωIJ ,of the system exists
when the excitation frequency coincides with a natural frequency, ωKL , and this can lead to
the enormous vibration amplitudes.
De Laval [1] (Swedish Engineer) in 1883 succeeds with a single-stage steam
impulse turbine for marine applications which operates at 0-42000 rpm. He first decided to
use a rigid rotor, but later he came with a flexible rotor and unveils that it was feasible to
operate above the critical speed by operating at a spin speed closely about seven times the
critical speed. A flexible shaft of negligible mass with a rigid disc at its midspan is called
as a Jeffcott [1] rotor (Figure 1.1b). The fundamental theory of rotor dynamics first
recorded which can be found in a masterly article of Jeffcott in 1919. However Jeffcott
confirmed Föppl's [1] prediction that a stable supercritical solution occurs and decided to
continue with Föppl's study. Föppl used an undamped model to show that an unbalanced
disc would whirl synchronously with the heavy side flying out when the rotation is
subcritical and with the heavy side flying in when the rotation is supercritical
1.4. Research Goals and Analysis Approach
By the beginning of the twentieth century, the developments in the field of rotor dynamics
was summarised in the classic book written by Stodola [1] in 1924. Then turbine
manufacturers around the world started to design, analyze and operate rotors at super
critical level. On the contrary Rankine’s model neglects the Coriolis [1] acceleration and
states that the system cannot be stable if it operated over the critical speed.
The proposed research study aims to find out the behaviour of the dynamic rotorbearing system supported on two rolling element bearings. There have been much
investigations related to the field of rotordynamics during the past few decades. By looking
in to all the previous published works, the part of literatures on rotor dynamics is most
7
concerned with diagnosis of imbalance response, critical speeds, natural whirl frequencies
and instability analysis. The presence of transverse crack on the rotor has been a keen
attention for researchers. Presence of crack in the rotating systems will lead to the timevarying stiffness (parametric inertia) and causes instability and gives severe vibration
signatures under specified operating limits. Few researchers speculated on the opportunity
of Rotor Internal Damping (RID) or Material Damping (MD).
The following analyses are carried out for the current research work;
1.
Dynamic analysis of an undamped rotor-bearing system without transverse crack.
2.
Dynamic analysis of a damped rotor-bearing system without transverse crack.
3.
Dynamic analysis of an undamped rotor-bearing system with transverse crack
All the above three analyses are carried out for two bearing cases; case (a) isotropic
and case (b) orthotropic for the speed range of 0-30000 rpm. Finite element method is used
to model the rotor-bearing system by incorporating internal damping with crack. By using
Lagrangian formulation the equation of motion is developed for the discrete elements and
shaft. The effect of transverse crack is studied over its stability region in the spin speed
axis. From the finite element stiffness matrix of the cracked element, it is observed that, the
cracked element stiffness matrix has time-periodic components with the frequency of 2Ω.
The analysis is presented in detail for the uncracked and cracked rotor-bearing system in
subsequent chapters.
The shaft is modelled as Euler-Bernoulli beam incorporating translational inertia,
rotary inertia, bending deformation, gyroscopic effects and internal damping. Natural
frequencies of the cracked rotor-bearing system also carried out. The results obtained for
the dynamic analysis of the uncracked rotor-bearing system by using numerical analysis.
8
Chapter-2
Literature Review
2.1. Introduction
The dynamic analysis of high-speed rotating machinery on rolling element bearings has
been always a challenging task. The most interesting part is the design of rotating
machineries. In general all the spinning structures show vibration signatures and need
frequent analysis to stretch their performances. The rotating machines have great
engineering applications when it’s come to industries. Accurate analysis and prediction of
dynamic characteristics is prime objective in the design of rotating structures.
2.2. Dynamic analysis of rotor-bearing systems
There have been several investigations related to the topic of rotor dynamics for the past
few decades. The flexible rotor-bearing systems have been analyzed with many
mathematical methods. The use of finite element methods in the dynamic analysis of rotorbearing systems provides better results. Several researchers have done their extensive
investigations on the spinning rotors since the last four decades. Here some of the
literatures were studied for their contribution on the field of rotor dynamics that concerns
with prediction of critical speeds, mass imbalance, natural whirl frequencies and stability
thresholds.
9
The author mainly discussed about the basic concepts and methods of rotordynamic
systems. The analysis of various types of rotors are discussed and presented by Genta [1].
An important modelling and analysis method to obtain exact solutions for multistepped rotor-bearing systems with distributed parameters was carried out by Hong and
Park [2]. In first example the system was compared with FEM model for validation. In
second example a parametric study was carried out for two shafts with different lengths
and diameters. In the final example, an unbalance response analysis was performed to
show the applicability of the proposed method.
A new method was introduced by Joshi and Dange [3] for calculating the critical
speeds of a general flexible rotor supported on flexible or rigid bearings which includes the
effect of distributed mass and inertia of the shaft along with the transverse shear effect.
Bearing mass, damping, coupling flexibilities and external loads are also considered to find
the effect of coupling flexibilities in critical speeds.
Modal analysis for continuous rotor systems with various boundary conditions was
presented by Lee and Jei [4]. The mode shapes, backward and forward whirl speeds of a
rotating shaft are presented as spin speed and boundary conditions vary. Boundary
conditions with the effects of asymmetry on the system dynamic characteristics are
investigated by them.
Rao et al. [5] used the finite element technique to obtain the eigenvalue and
stability analysis of rotors by considering the distributed bearing stiffness and damping.
Two models with uniform and parabolic distribution were analyzed. The stability limits
were studied for rotor supported on cylindrical, tilting pad and offset and three lobe journal
bearings.
10
A method of dynamic reduction procedure which does not require either sub
structuring or lumping was performed by Kim and Lee [6]. The method of modal
transformation was derived from the isotropic undamped static part of the rotor equation,
and a comprehensive study of the effects of varying the number of retained modes and the
bearing properties was elaborated.
Zhao et al. [7] have established for symmetrical single-disk flexible rotor-bearing
system. The motions of journal and disk have been simulated with fourth-rank RungeKutta method. Non-linear transient simulation and unbalanced responses are also
investigated.
Shih and Lee [8] presented a new method for estimating unbalance distributions of
flexible shafts and constant eccentricities of rigid disks based on the transfer matrix
method for analyzing the steady-state responses.
Tiwari and Chakravarthy [9] used an identification algorithm based estimation of
unbalances and dynamic parameters of bearings by using impulse response measurements
for flexible rotor–bearing systems. The identification algorithm has been tested with the
measured noise in the simulated response.
The effects of bearing support flexibility on the rotor dynamic analysis for the first
forward and a backward critical speed was studied numerically and experimentally by
Sinou et al. [10]. They discussed the measured FRF for various rotational speeds,
eigenfrequencies and the associated Campbell diagram from the numerical model and the
experimental results.
The dynamic modeling of rotor-bearing system with rigid disks and discrete
bearings were analyzed with finite element method by Nelson and Mc Vaugh [11].
11
The effect of rotatory inertia, gyroscopic moments, axial load and internal damping was
included by Nelson [12]. He has not included the shear deformation or axial torque in the
observation. He generalizes the present study with his previous published paper by
utilizing the Timoshenko beam theory for obtaining the shape functions. He has compared
the results obtained from FEA with the classical Timoshenko beam theory (closed form)
for both the rotating and nonrotating shaft systems.
The analysis of Jeffcott rotor-bearing model is presented by Greenhill and Cornejo
[13] to predict the critical speed produced by the unbalance excitation of a backward
resonance mode. Prediction of the critical speed was given with the test data. Their study
shows that the resonance occurs due to the backward mode with the recommendation to
avoid the unique critical speed situation.
Nandi and Neogy [14] studied the stability analysis of asymmetric rotors in a
rotating frame using finite element method. It shows the efficiency of their method which
indicates that, only the non-zero terms and their respective rows and columns positions of
all the related matrices can be stored and carried forward for the analysis.
The non-linear dynamic analysis of a horizontal rigid rotor with unbalance is
studied by Tiwari et al. [15]. The concept of higher order Poincare map and interpolation
technique were applied to find out the fixed point and stability of the system.
To identify the fault in a rotor bearing system, Sudhakar and Sekhar [16] used the
equivalent load minimization method. Two approaches namely equivalent loads
minimization and vibration minimization method are used to identify the fault. Finally the
unbalance fault was identified for only one location by measuring transverse vibrations.
12
Bachschmid et al. [17] presented a model-based method for multiple faults
identification. They had done this by least-square fitting approach in the frequency domain.
To validate the identification procedure they presented numerical applications for two
parallel faults and some experimental results which are obtained on a test-rig.
A method proposed by Sinha et al. [18] is to estimate the rotor unbalance and
misalignment from a single machine run-down. From the identification they assumed that,
the source of misalignment is at the couplings of the multi-rotor system. Finally they
demonstrated the method by using experimental data from a machine with two bearings.
Arun and Mohanty [19] described a model based method to analyze the rotor–
bearing system with misalignment and unbalance. The experimental results were obtained
by the residual generation technique. They also obtained the residual forces due to
presence of faults. Finally with the help of this model based technique, the condition of
fault and the fault locations were identified.
Vania and Pennacchi [20] suggested a model-based diagnostic technique that can
be used for the rotor health analysis. To identify the faults in rotating machines they
developed a method to measure the accuracy of the results that are obtained with modelbased techniques. By using both the machines responses simulated with mathematical
models and experimental data on a real machine, the authors tested the capabilities of these
methods.
By representing the equivalent force system, the effect of the faults is modelled by
Pennacchi et al. [21]. The model is fully assembled by the sub models of the rotor discrete
elements like bearings and foundation. Some identification techniques such as the least
squares identification are used in frequency domain to increase the accuracy.
13
Lees et al. [22] overviewed the recent evaluation in the field of rotor dynamics
which has a significant practical importance. The models assists the complex turbomachinery monitoring which includes rotor balancing, rotor bow, rotor misalignment, rotor
crack and bearing parameter estimations.
The goal of Isermann [23] is to generate the several symptoms indicating the
difference between nominal and faulty status in the model-based fault detection. He
determined the faults by applying inference methods based on the different symptoms in
the fault diagnosis procedures.
Complete theoretical models of a motor-flexible coupling-rotor system was given
by Xu and Marangoni [24]-[25] to understand the dynamic characteristics of the faults
such as, shaft misalignment and rotor unbalance. They derived the equation of motion from
component mode synthesis method and conducted an experimental study to verify the
theoretical results with a simple flexible coupling and a helical coupling.
Forced response of undamped rotating shaft with distributed parameter are analysed
by Lee et al. [26] by using modal analysis technique. The analysis includes various
boundary conditions to analyze the undamped gyroscopic systems with Galerkin's method
for the forced response. Numerical examples for both the methods are illustrated and the
results are compared and discussed by them.
The study of a misaligned rotor-ball bearing systems driven by a flexible coupling
is done by Lee et al. [27]. They carried out the experiments extensively to distinguish the
difference between experimental and theoretical results. From their observation, they found
that the natural frequency of the misaligned rotor system increases largely with the
misalignment direction.
14
Sakata et al. [28] carried out finite element analysis of a lightweight rotor system,
which has discrete elements like flexible disk with flexible blades and a flexible shaft with
rigid bearings. In order to reduce the dimension of matrices, they considered the shaft and
blades as beam elements, and the disk as annular elements. They conducted the test on a
model rotor to compare the results with the experimental data.
Khulief and Mohiuddin [29] developed a dynamic rotor-bearing system by using
finite element method. Their model includes the gyroscopic moments and anisotropic
bearings. They obtained the reduced order model by using modal truncation for dynamic
response analysis and presented with two types of modal truncations; (i) with planar
(undamped) modes and (ii) with complex (damped) modes.
A finite element model is presented by Ku [30] to study the whirl speeds and
stability of the rotor-bearing systems. He combined the effects of transverse shear
deformations; internal viscous dampings and hysteretic dampings in the formulation in
addition to the effects of translational and rotatory inertia and the gyroscopic moments. He
compared the results of whirl speeds and damped stability analysis with other previously
published works.
The dynamic stability of a rotating shaft is studied by Chen and Ku [31] with finite
element method. Bolotin’s method is used to obtain the dynamic instability diagrams for
the various rotating speeds for Timoshenko beam. They conclude that the sizes of these
regions increases as the spin speed of the system increases.
Flexible rotor having unbalance, and supported by ball bearings was studied by
Villa et al. [32] for non-linear dynamic analysis. The bearings are modelled as two degree
of freedom system by considering the kinematics of the rolling elements. They analyzed
15
the system stability in the frequency-domain by using a perturbation method which is
applied to known harmonic solutions in time domain.
Finite element model of Timoshenko beam supported on hydrodynamic bearings
including internal damping were studied by Kalita and Kakoty [33]. By using Campbell
diagrams they calculated the critical speeds for synchronous whirl in different operating
conditions. They observed that in addition to the natural whirl frequencies, another
whirling frequency appears for every spin speed, and they also found that, this happens
when the spin speed is half.
The study of the flexural dynamic behaviour of a general rotating system, based on
the use of complex co-ordinates and finite element method is elaborated by Genta [34].
The study includes the non-rotating parts of the machine and two types of dampings,
namely viscous and hysteretic.
Thomas et al. [35] presented the formulations of three degrees of freedom system at
each of two nodes of Timoshenko beam theory. They studied the convergence rates and
compared with the calculated natural frequencies of two cantilever beams.
A complex rotor-bearing-support system was taken for the general analysis by
Adams [36]. Proper handling of various non-linear effects is a main feature of his analysis.
The study presents the developments of the analysis, comparison with experiment study
and examples and its use in the industrial applications.
A finite element model of non-axisymmetric rotors on non-isotropic spring support
in a rotating frame is presented by Nandi [37]. In his work he shows that the proposed
reduction technique works well for a rotor supported on non-isotropic (orthotropic)
springs.
16
Murphy and Vance [38] describe the modelling procedures for the rotor-bearing
systems which includes the effects of damping, gyroscopic effects. In addition to the
characteristic polynomial, stability prediction and critical speeds are also estimated with
good accuracy without missing any modes.
Lee and Choi [39] proposed an optimum design approach to show the speed and
load dependent stiffness effect on the system dynamic behaviour to demonstrate the
effectiveness of a multi-stepped rotor-bearing system supported on two angular contact
ball bearings. Transfer matrix method is used to obtain eigenvalue of the system and as an
optimization technique the augmented Lagrange multiplier (ALM) method is used. From
the results, they show the effect of stiffness on the system dynamic behaviour.
Dynamic behaviour of a complex flexible rotor-bearing system is studied by
Wenhui et al. [40]. The unsteady oil-film force model was described by three functions. In
addition to that the bifurcation and chaos behaviours were found by calculating the
maximum Lyapunov exponent of the system. They carried the experimental analysis to
compare the calculated results.
Patel and Darpe [41] studied the use of forward or backward whirl (full spectra),
and showed the possibility of misalignment. The information yields an important tool to
unconnected faults that generate the same frequency spectra (e.g. crack and misalignment)
and lead to a more definite misalignment diagnosis. Finally full spectra and orbit plots
were efficiently used to reveal the unique nature of misalignments.
Zorzi and Nelson [42] analyzed rotor system with internal damping. The model
consisted of viscous as well as hysteretic damping. It is shown that the material damping in
the rotor shaft introduces rotary dissipative forces.
Rao [43] explained the theory of computational aspects and applications of
vibrations in a simple manner with computational techniques.
17
Papadopoulos [44] presented the theory of strain energy release rate combined
with linear fracture mechanics approach to study rotating shafts with cracks. The main goal
of his research is to give the engineer an early identification of the crack in the rotor.
Chasalevris and Papadopoulos [45] investigated a stationary shaft with two cracks
with coupled bending vibrations. They used Euler–Bernoulli beam theory to define the
equations for natural frequencies and coupled response of the shaft. Their study focused on
the horizontal and vertical planes of a cracked shaft. Finally they presented the
experimental analysis for coupled response and eigen frequencies measurements of the
corresponding planes.
Dimarogonas and Papadopoulos [46] investigated an open cracked de Lava1 rotor
with dissimilar moments of inertia. Furthermore, under the assumption of large static
deflections, the analytical solutions are obtained for the closing crack. They found the local
flexibility function by experiments and a solution is developed for the same.
Shudeifat et al. [47] investigated the effect of crack depth and verified with
experiments through a general harmonic balance technique of a rotor-bearing-disk system.
They considered the breathing and open crack models in their analysis. FEM and general
harmonic balance solutions were derived for the two types of cracks which are valid for
damped and undamped rotor systems.
Chen et al. [48] studied a cracked rotor system with asymmetrical viscoelastic
supports to develop nonlinear governing equations of motion. The effects of crack and
other system parameters on the dynamic stability of spinning rotor system were also
investigated by them.
Darpe et al. [49] studied a simple Jeffcott rotor with two transverse cracks. The
effect of two cracks on the breathing mechanism and unbalance response of the rotor
18
system were also studied by them. They studied in detail, the effect of orientation of the
breathing crack with respect to open crack on the dynamic response.
Fu et al. [50] studied a rotating shaft with a transverse crack for the nonlinear
dynamic stability. They constructed the deflections of the system with a crack by using the
equivalent line-spring model. They found the unstable regions using Runge–Kutta method
and Floquet theory. The effects of crack depth, crack position, disk position, disk thickness
and spinning speed on the principal unstable regions were also discussed by them.
Gasch [51] - [52] provided a comprehensive analysis for a cracked rotor system to
predict its stability behaviour and forced vibrations due to unbalance of the crack and the
disk. He mentioned that his study is restricted to the Laval rotor and established the early
crack detection on the rotor.
Sekhar [53] studied the finite element analysis of a rotor system for the flexural
vibrations by including two transverse open cracks. His study also carries the eigenvalue
analysis and stability study of the system including two open cracks. The eigenfrequencies
are calculated for the influence of one crack over the other and the mode shapes for the
threshold speed limits.
Sekhar and Dey [54] focused on the stability threshold of a rotor-bearing system
having a transverse crack by using finite element method, considering various crack
parameters, shaft internal damping (viscous and hysteretic) and geometric parameters.
They showed that the instability speed has reduced considerably with increase in crack
depth and influenced more with hysteretic damping compared to viscous damping.
Sekhar [55] summarized different kinds of application on double/multi-cracks to
note the influences and identification methods in vibration structures such as beams, rotors,
pipes, etc. He brings out the multiple cracks effect and their identification by the state of
his research.
19
Sinou [56] analyzed a rotor system for its stability with a transverse breathing crack
by considering the effects of crack depth, crack location and the shaft’s spin speed. The
harmonic balance method used by him is to calculate the periodic response of a non-linear
cracked rotor system. He also investigated the system for the effects of some other system
parameters on the dynamic stability of non-linear periodic response.
The free vibrational analysis of a multi-cracked rotor is studied by Tasi and Wang
[57]. The cracks are assumed to be in the first mode of fracture, i.e. the opening mode.
Based on the Timoshenko beam theory, the frequency equations are constructed and
assembled with each segment of the multi-step and multi-cracked rotor. The effects of both
relative distances of cracks are taken into account for free vibration analysis.
A boundary tracing method is given by Turhan [58] for the construction of stability
charts for non-canonical parametrically excited systems. This method is used as an
extension to cover the combination resonances of the well known Bolotin’s method. The
proposed method reduces the boundary tracing problem into an eigenvalue analysis
problem of some special matrices.
Darpe et al. [59] studied a rotating cracked shaft for the coupling between
longitudinal, lateral and torsional vibrations. The elemental stiffness matrix of a
Timoshenko beam got modified due to presence of crack. Two analyses were included in
their study (i) coupled torsional–longitudinal vibrations and (ii) coupled torsional–bending
vibrations with a breathing crack model.
A rotating shaft is analysed for the influences of transverse cracks by Sinou and
Lees [60]. Two main issues such as the changes in modal properties and the influence of
breathing cracks on dynamic response are addressed by them. The resulting orbits during
transient operations of a cracked rotor are also examined by the authors.
20
A simple rotor with a breathing crack is studied by Jun et al. [61]. The equations of
motion and the breathing crack model is further simplified to a switching crack model. The
conditions for crack opening and closings are derived by using the switching crack model
for crack identification. They observed that the vibration characteristics of a cracked rotor
can be identified from the second horizontal harmonic components measured near to the
second harmonic resonant speed.
Finite element analyses of a rotor-bearing system having a slant crack were studied
for flexural vibrations by Sekhar and Prasad [62]. A developed flexibility matrix and
stiffness matrix of a slant crack are used in the analysis subsequently to find frequency
spectrum and steady state response of the cracked rotor.
A model based method was proposed by Sekhar [63] for the on-line identification
of cracks in a rotor. He accounted the equivalent loads and fault-induced to the system in
his mathematical model. Identification of the cracks are carried for their depths and
locations on the shaft. Finally the nature and the symptoms of faults are found by fast
Fourier transform analysis.
Online identification of malfunctions in the rotor systems was discussed by Jain and
Kundra [64] with a model based technique. The fault model of the system is referred for
study by the mathematical representation of equivalent load. Identification of unbalance
response is validated with basic principle of the technique, through numerical simulations
as well as by experiments.
A rotating shaft with a breathing crack was studied by Georgantzinos and Anifantis
[65]. They considered the cracked area with the effect of friction and by applying the
energy principles a portions of crack surfaces were found. The direct and the cross-coupled
flexibility coefficients were also determined by them.
21
A new method was proposed by Binici [66] to obtain the eigenfrequencies and
mode shapes of beams for multiple cracks which are subjected to axial force. The effects of
crack and axial force levels on the eigenfrequencies of the beam were studied. He
considered two cases in his analysis (i) simply supported and (ii) cantilever beam. During
the investigation the author found that the eigenfrequencies are strongly affected by crack
locations. The conclusion from his study shows that, the proposed method can be used to
predict the critical load for damaged structures.
Non-linear behaviour of a rotor with a breathing crack was analysed by Sinou [67].
The relative orientation between the cracks and the unbalance of the rotor system were
studied. The study indicates the emerging of super-harmonic frequency components which
provides useful information on the presence of cracks.
A Jeffcott rotor with a crack was studied for critical speed and sub harmonic
resonances by Darpe et al. [68]. The study includes three crack models; (i) breathing crack,
(ii) switching crack and (iii) open crack. The experimental investigations were performed
for the peak response variations and change in orbit orientation. Their experimental
investigation shows that, the orbit orientation changes through the sub harmonic
resonances.
A rotating shaft with an open transverse surface crack was investigated for the
coupling of longitudinal and bending vibrations by Papadopoulos and Dimarogonas [69].
The effects of unbalance and gravity are also included in their study.
Vibration and stability analysis of cracked hollow-sectional beams were carried by
Zheng and Fan [70]. Their analysis includes influences of sectional cracks and deeper
penetration on the stability of the beam. Hamilton’s principle is used by them to derive the
governing equations.
22
Hwang and Kim [71] developed a method to detect the damages of structures by
using frequency response function. The method uses only a subset of vectors from the full
set of frequency response functions, and calculates the stiffness matrix and reductions in
explicit form. A simple helicopter rotor blade was numerically demonstrated for the
verification of the proposed method.
A cracked beam is analyzed for the natural frequencies and mode shapes by Zheng
and Kessissoglou [72] using finite element method. The study results are compared with
the analytical results obtained from local additional flexibility matrix.
Sekhar and Prabhu [73] studied the transient vibration of a cracked rotor which is
passing through the critical speed. The study confirmed that the model can be analyzed for
the further results like, time histories with harmonics, and frequency spectrum.
On the remark of “The determination of the compliance coefficients at the crack
section of a uniform beam with circular cross-section”, Abraham et al. [74] studied
analytically the double integrals which are commonly encountered when the crack depth
exceeds beyond the radius of the cracked section.
Analysis of a rotating cracked shaft to identify the crack depth and crack location
with the application of a new method was carried by Gounaris and Papadopoulos [75]. A
rotating Timoshenko beam is modelled as a shaft with the gyroscopic effects and axial
vibration. The method is used by the authors to find the axial vibration response.
A dynamic study of multi-beams with a transverse crack was carried by Saavedra
and CuitinNo [76]. Based on the LFM theory, the additional flexibility of the crack was
evaluated by using strain energy density function. The dynamic response of a cracked freefree beam and a U-frame was studied with a harmonic force.
Nayfeh and Mook [77] emphasized the physical aspects of non-linear systems in
detail.
23
2.3. Summary
With the help of web-based tool, research articles related to dynamics of rotor-bearing
systems are studied and found that most of them are focused on the rotation of shaft of
uniform cross-section with and without crack. The area involving rotations of shaft with
variable cross section are less explored. There is a wide scope of research in the area of
dynamics of rotor-bearing system with variable cross section with and without crack. The
study of the proposed system is carried with mathematical model to understand the system
behaviour. Convergence study is done to ensure the system natural whirl frequencies.
Besides, Houbolt’s implicit time integration scheme is used to study the effect of spin
speed with disk eccentricity. The objective of the present work is to obtain the critical
speed, unbalance response, natural whirls and stability thresholds for the multi disk,
variable cross section rotor-bearing system with transverse crack.
24
Chapter-3
Theoretical analysis
The dynamic analysis of rotating machineries is generally carried out with the help of
vibration measurements and through continuous monitoring of the system. However, some
issues are difficult to identify and evaluate purely through measurement based analysis.
There are several mathematical methods available to analyse the dynamic behaviour of the
rotor bearing system. Out of these methods finite element method plays an important role
for the rotating system analysis. The current chapter incorporates the modelling of a rotorbearing system with Finite Element Method.
Rotor system configuration and interrelate segments
By using Lagrangian formula the finite equation of motion of the rotor shaft element,
bearings and rigid disks are developed. Shape functions are derived by using EulerBernoulli beam theory. The mathematical expressions for the system with damping and
transverse crack are also presented in the following topics.
3.1
System Equation of Motion without crack
The flexible rotor-bearing system is analyzed by finite element method for a typical
configuration. The system consists of rotor and collection of discrete disks. Rotor sections
are represented with distributed mass and elasticity with discrete bearings. Fig. 3.1 shows
the system along with two reference frames (fixed and rotating) that are utilized to describe
the systems equation of motion.
25
Figure 3.1 Typical Rotor-bearing-disk system configurations.
configurations
The rotating frame of reference and fixed frame of references
reference are represented by
xyz and XYZ triad respectively. The undeformed rotor, which is represented by X and x
axes are collinear and coincident. The rotating frame of reference is defined to fixed frame
of reference by a single rotation ‘ωt’ about X with ω denoting the whirl speed.
A typical cross section of the rotor in a deformed state is defined relative to fixed
frame (XYZ) by the translations V (s, t) and W (s, t)) in the Y and Z directions respectively
to locate the elastic centreline. The small angle rotation θ (s, t) and ζ (s, t) about Y and Z
axis represents the position of plane of the cross
cross-section
section respectively. The triad abc is
attached with the axis ‘aa’ normal to the plane cross-section represented in Fig. 3.2.
The angular velocities related to the fixed reference frame XYZ can be given as
ω a   − sin θ
ωb  =  cos θ sin ε
ωc  cos θ cos ε
1
0  ζɺ 
0 cos ε  εɺ 
0 − sin ε  θɺ 
•
ζ about Z defines a" b" c"
•
θ about b" defines O′ P ′ Q ′
•
(3.1)
ε about O′ defines
defines a b c
26
Figure 3.2 Cross section rotation angles
For small deformations
deformations, the rotations θ, ζ are approximately collinear with the Y, Z
axes respectively. The spin angle
ε
is constant for a spin speed system with negligible
torsional deformation Ωt
Ωt, where, Ω denotes rotor spin speed. The displacements (V,
( W, θ,
ζ)) of a typical cross section relat
related to fixed reference frame are transformed to
corresponding displacements ((v, w, β, γ) related to rotating reference frame by the
orthogonal transformation
{q} = [R]{p}.
(3.2)
With
cos ωt
V 
v 
 sin ωt
W 
 w
{q} =  , {p} =  , [R ] = 
 0
θ 
β 

 ζ 
 γ 
 0
− sin ωt
0
cos ωt
0
0
cos ωt
sin ωt
0

0 
.
− sin ωt 

cos ωt 
0
(3.3)
(3.3)) with respect to time can be
After the first and the second derivatives of the equation (3
obtained as,
27
{qɺ} = ω[S ]{ p} + [R]{ pɺ }
(a)
{qɺɺ} = [R]{{ɺpɺ} − ω 2 { p}} + 2ω[S ]{ pɺ }
(b)
With
(3.4)
− sin ωt
1 ɺ
[S ] = [R] =  cos0ωt

ω
 0
− cos ωt
− sin ωt
0
0
0
0
− sin ωt
cos ωt
0 
0 
− cos ωt 
− sin ωt 
(c)
The shaft is considered as an Euler-Bernoulli type of beam. Any transverse plane of the
beam before bending is assumed to remain plane after bending and remain normal
norm to
elastic axis. Therefore, the beam cross section has not only translation but also rotation.
The shaft is assumed to have uniformly distributed mass an
and
d elasticity. The nodal
displacements are the function of time q(t). Four boundary conditions are needed to obtain
the shape functions.
RST, "U = S"U
RSV, "U = W S"U
XST, "U = Y S"U
XSV, "U = Z S"U
Figure 3.2
3.2.1. Relationship between slope and displacements
28
3.1.1 Undamped flexible finite rotor shaft element
A typical finite rotor element is shown in Fig. 3.3. Here it is assumed that the nodal cross
sectional displacements (V, W, θ, ζ) are assumed to be time dependent and also the function
of position (s) throughout the element axis. The rotations (θ, ζ) are related with translations
(V, W).
Figure 3.3 Finite rotor element and coordinates
The relation between these two can be expressed by the equation as
∂W
∂s
∂V
ζ =
∂s
θ =−
(3.5)
.
The coordinates ( q\ , q\# … … . q\Z ) are time dependent end point displacements
(translations and rotations) which are shown in Fig. 3.3. A spatial shape function is used to
express the displacement of each node by using Euler-Bernoulli beam theory. The nodal
displacement is a function of time, q(t). The translation of a typical point internal to the
element is chosen to obey the relation
V (s,t)
{W(s,t)
} = [Ψ(s)]{q (t)}
e
(3.6)
29
Figure 3.4 illustrates the shaft cross section at displaced position.
position An infinitesimal
area of radial thickness ‘dr’ at a distance r (0 ≤ r ≤_` ) is considered which is subtending an
angle d(Ωt). Ω is the rotational speed of the system in rad/sec and ‘Ωt’ is the subtended
angle varies from 0 to 2π
2π.
Figure 3.4 Displaced position of the shaft cross-section
section
The
he spatial constraint matrix for translation can be written as [11]
[Ψ] = N01 N0 − 0N N02 N03 N0 − 0N N04 


1
2
3
4
(3.7)
Where
N 1 = 1 − 3β
N 3 = 3β
2
2
+ 2β 3 ; N 2 = L(β − 2β
− 2 β 3 ; N 4 = L (− β
2
2
+ β 3)
+ β 3 ) and β =
s
.
l
By using equations (3.5)) & ((3.6) the rotations for the system can be expressed as,
{ζθ } = [Φ]{q }
e
(3.8)
With
 [ Φθ ]   0 − N1′ N 2′
=
0
 Φζ    N1′ 0
[Φ] = 
Where,
0
N 2′
0
N 3′
− N 3′ N 4′
0
0
1
( − 6 β + 6 β 2 ) ; N 2′ = 1 − 4 β + 3 β 2 ;
l
1
N 3′ = ( 6 β − 6 β 2 ) ; N 4′ = − 2 β + 3 β 2 ;
l
N 1′ =
30
0
N 4′ 
(3.9)
3.1.2 Energy equations
Lagrange’s equation provides a general formulation for the equation of motion of a
dynamical system. The Hamilton’s principle can be used to derive the Lagrange’s equation
in a set of generalized coordinates′ ′. The coordinates are selected as the variables that
determine the position of the system. The Lagrange equation for a system can be expressed
with kinetic and potential energies as
d  ∂ TS   ∂ TS   ∂ U S   ∂ D 

−
+
+
 = Qi
d t  ∂ qɺ i   ∂ q i   ∂ q i   ∂ qɺ i 
(3.10)
Where i=1, 2...n., ‘ TS ’ is the total kinetic energy, ‘ U S ’ is the total potential energy, D’ is
the Rayleigh’s dissipation function and ′' ′is the virtual work done on the system.
n
δW = ∑ Qiδ qi
(3.11)
i =1
The Lagrangian equation gives a group of second order ordinary differential equations.
These equations are non-linear and non-homogeneous. For the differential disk which is
located at the position (s) along the axial direction, the elastic bending and kinetic energy is
expressed as
dd
g
9
P.E: db = ce
dd f h
#
5 g
9
K.E: d = ce
5f j
#
T
T 9 dd
i c fds
e dd
and
μ T 95
5 g l ce5f ds+ 45 # ℐ mn+ cop5 f j #
#
T μ
T
(3.12)
T o5
l c fds - 45X5q ℐ mn.
p5
Using equations (3.6), (3.8) & (3.12) the potential and kinetic energies of the shaft
elements can be written as,
P.E: db = EI g Ψ g Ψ mn
#
and
K.E: d = µ5 g Ψ
g Ψ
5 mn + s5 # ℐ mn+
#
#
#
ℐ 5 Ф
g Ф
5 mn
g
−s5 ℐ 5 g vФp w Фo mn.
31
(3.13)
The total potential and kinetic energies of the complete element is obtained by integrating
the equations (3.13) over the length of the shaft element which is obtained as,
b + = g yb + 5 g (zg +z{ )5 + 45 # + 5 g 5 (3.14)
#
#
#
Where
}
(a)
}
(b)
zg = |` μΨ
g Ψ
ds
z{ = |` ℐ Ф
g Ф
ds
}
g
= |` ℐ vФp w Фo ds
(c) (3.15)
}
yb = |` Ψ′′
g Ψ′′
ds
(d)
= S − g U
(e)
The Lagrangian equation of motion for the finite rotor shaft element using the
equation (3.14) and the constant rotational speed restriction, 45 = Ω can be written as
Szg + z{ U~ − Ω 5 + Syb U = ' (3.16)
The force vector ' includes the unbalance mass, interconnection forces, and
other element external effects. For the shaft element with distributed mass center
eccentricity S7SnU, XSnUU , the equivalent unbalance force using the consistent matrix
approach can be expressed as [11]
' Z = ' cos (Ωt) + '! sin (Ωt)
3.1.2.1
(3.17)
Rigid Disk
The disk for the analysis is assumed to be axisymmetric and rigid with rotational and
translational motion. The total kinetic energy of the system is represented by the sum of the
rotational and translational kinetic energies. Using the Lagrange method the kinetic energy
is given by the following expression,
32
T
1 Vɺ  m
Td =    d
2 Wɺ   0
ω a 
0  Vɺ  1  
  + ωb 
md  Wɺ  2  
ωc 
T
I D
0

 0
0
ID
0
0  ω a 
 
0  ωb 
I D  ωc 
(3.18)
By using equation (3.1), equation (3.18) becomes,
1 Vɺ  m
Td =    d
2 Wɺ   0
g
T
0   Vɺ  1 θɺ   I D
 +  
md  Wɺ  2 ζɺ   0
0  θɺ 
  − εɺζɺθI p
I D  ζɺ 
g
− = z ~ – Ω 5 5
(3.19)
(3.20)
The Lagrangian equation of motion of the rigid disk for the constant spin speed
restriction ( εɺ = Ω ) equation (3.20) will become
Szg + z{ U~ − Ω 5 = ' (3.21)
The equation (3.21) is the equation of motion of the rigid disk referred to the fixed
frame of reference with the forcing term including mass unbalance, interconnection forces,
and other external effects on the disk. By using the equations (3.2-3.4) and pre-multiplying
by [R ] , equation (3.21) is transformed to the form of,
T
([M Td ] + [M Rd ]){ ɺpɺ d } + ω (2([Mˆ Td ] + [Mˆ Rd ]) − λ [G d ]){ pɺ d } − ω 2 (([M Td ] + [M Rd ]) + λ [Gˆ d ]){ p d } = {P d }
For the case of thin disc (I P = 2 I D ) equation (3.22) become,
(3.22)
([M Td ] + [M Rd ]){ ɺpɺ d } + ω(2[Mˆ Td ] + ( 1 − λ ) [G d ]){ pɺ d } − ω 2 ([M Td ] + ( 1 − 2λ )[M Rd ]){ p d } = {P d }
(3.23)
The equations (3.22) and (323) are the equation of motion of a rigid disc referred to
rotating frame with whirl ratio λ = Ω ω .
33
3.1.2.2
Bearings
The virtual work of the forces acting on the shaft can be written as,

δW =−99
RR − 9e
R − ee
− e9
R − Q99
R R–Q9e
5 R

−Qee
5 − Qe9
R5 (3.24)
The equation (3.24) can be expressed by the following relation,
δW = 9 9 + e e (3.25)
By neglecting the influence of slopes and bending moments the main characteristic link
forces will become

9 = −99
R − 9e
− Q99
R − Q9e
5
(3.26)

e = −ee
− e9
R − Qee
5 − Qe9
R5
(3.27)
Representing in matrix form, the equation (3.26) and (3.27) can be expressed as

FV 
 k VV


 F θ  = − 0
 b
F W 
k WV


 0
 F ζ 
b
0
k
0
0
0
b
VW
0
k
b
WW
0
0

0
0

0
V  cb
 θ   VV
  − 0
W   b
  c WV
 ζ   0
0
0
0
0
c
b
VW
0
c
b
WW
0
0

0
0

0 
 Vɺ 
 ɺɺ 
θ 
Wɺ 
 ɺ
 ζ 
(3.28)
The system is subjected to only one type of interconnecting component which is the
bearings. These bearings are linearized and the stiffness only considered in the analysis.
The equation of motion of the bearings as follows,

5 + = 5 + y (3.29)
5 + y = ' (3.30)
Where,

y = 99
e9

Q99
9e

and
=

ee
Qe9

Q9e
.
Qee
34
3.1.3 Rotor element with variable cross section
Figure 3.5 shows the variable cross sectional properties of a typical rotor shaft element. For
these elements the equation of motion can be obtained by either evaluating the integrals of
the equations (3.15) and (3.17) by using the variable properties of the system.
Figure 3.5 Sub-elements assemblage
Then the set of assembled sub elements possesses [4 x (total number of sub element
stations)] coordinates which can be reduced by following procedure.
The sub element equations in the assembled form, in fixed frame coordinate can be
represented as
{q e }a 
{ }a 



([M Te ] + [M Re ]) { }b  − Ω[G e ]{q e }b  + [K Be ] = {Q e }
 e 
{ }c 
{q }c 
 qɺɺe
 e
 qɺɺ
 e
 qɺɺ
(3.31)
The internal displacements {q e }b and the end point displacements of the element
{q }
e
a
and {q e }c are having the displacement dependency between them. Thus, this can be
adopted by considering the static, homogeneous case of the previous equation (3.31).
35
[ ] [K ] [K ]  {q } 


[ ] [K ] [K ]  {q }  = {0}

[ ] [K ] [K ]  {q } 
 Ke

 Ke

 K e
e
aa
ac
e
ba
a
e
e
bb
bc
e
ca
e
e
ab
e
e
cb
(3.32)
b
cc
c
From the second row of equation (3.32), the internal displacement vector {q e }b can be
written as
{q e }b
[ ] [K ] {q } − [K ] [K ] {q }
= − Ke
−1
bb
e
e −1
bb
e
ba
a
e
e
bc
(3.33)
c
The following constraint of equation can be written by using the equation (3.33)
[I ]
{q e }a  
 e  
e −1
e
{q }b  = − [K ]bb [K ]ba
{q e }c  
[0]
[0]
 e
{q } 
− [K ] [K ]bc   e a 
 {q }c 
[I ]

e −1
bb
e
(3.34)
From the equation (3.34), the elements in the columns of the constraint matrix
represents the static mode shapes. By applying the equation (3.34) to the equation (3.31) it
reduces the number of coordinates and associated components of force to eight, and this
provides the same element equation form as the equation (3.16). In this research work,
reduction of co-ordinates technique [11] is used to model the rotor elements having
variable cross section.
3.1.4 Undamped system equation of motion
The undamped system equation of motion in the assembled form which consisting of the
component equations form the equation (3.16), (3.21), and (3.30), is of the form,
[M ]{qɺɺ } − Ω[G ]{qɺ } + [K ]{q } = {Q }
S
S
S
S
S
S
S
4 n ×1
.
(3.35)
Where
[M ]= [M ] + [M ]; [G ] = [G ]+ [G ] ; [K ] = [K ]+ [K ] ; [M ] = [M ] + [M ] and
S
e
d
S
e
d
[M ] = [M ] + [M ]; {q } = {V W θ
d
d
T
d
R
S
S
i
i
i
e
B
b
e
e
T
ζ i Vi + 1 Wi +1 θ i +1 ζ i +1 ........... N + 1}T .
36
e
R
3.1.5 Damped flexible finite rotor shaft element
The previous study is extended to incorporate the internal damping in the finite element
formulation. In their finite element formulation Zorzi and Nelson [42] have considered the
combined effects of both viscous and hysteretic internal damping of the rotor-bearing
system. By using the both 79 and78 , which denotes the viscous damping coefficient and
the hysteretic loss factor of the shaft Material.
The potential energy U e and kinetic energy T e of the element, can be given by the
{ }
nodal displacement vector q e as respectively,
[ ]
1
{qe }T K Be {qe }
2
1
1
T
T
T e = {qɺ e } M Te + M Re {qɺ e } − Ω{qɺ e } N e {qe } + I P lΩ 2
2
2
Ue =
([ ] [ ])
(3.36)
[ ]
(3.37)
Here, the stress-strain relationship can be expressed as [42],
σ = E {ϑ + ηV ϑɺ}ϑ = −r cos[(Ω − ω )t ]
∂ 2 R0 ( x, t )
∂x 2
(a)
(3.38)
∂ 2 R0
∂
ɺ
ϑ = (Ω − ω )r sin [(Ω − ω )t ] 2 − r cos[(Ω − ω )t ]
∂t
∂x
 ∂ 2 R0

 ∂x 2





(b)
It is evident from the equation (3.38, b) that, when the system spin speed and the
whirl speed (synchronous state) matches (Ω = ω), and when the orbit is in circular shape,
the term (∂ ∂t )(∂ 2 R0 ∂x 2 ) = 0, besides the component of viscous damping which provides
no variation of the axial stress σ of equation (3.38, a). Hence, for the synchronous circular
orbits, the component of internal viscous damping can’t produce any out of phase loading
to reduce the critical speed orbit.
The bending moments at any instant about Y and Z -axes can be expressed as,
37
2 π r0
∫ ∫ − (V
MZ =
+ r cos (Ω t ))σ rdrd (Ω t )
(a)
0 0
(3.39)
2 π r0
∫ ∫ (W
MY =
+ r sin (Ω t ))σ rdrd (Ω t )
(b)
0 0
The above equations (3.39) for bending moment becomes by substituting the appropriate
values and by integrating on the limits,
ηV
 1 ηV Ω V ′′ 
M Z 
+
EI
=
EI


 


 M Y 
 0
ηV Ω − 1  W ′′
0  Vɺ ′′ 
 
− ηV  Wɺ ′′
(3.40)
It is observed that, the strain energy d and the dissipation function d for an
infinitesimal element, by neglecting the shear deformations which can be expressed in the
form of
θ ′ 
1 θ ′ 
dP = EI   [η ] ds
2 ζ ′
ζ ′
T
θɺ′  θɺ′ 
1
e
dD = η V EI    ds
2
ζɺ ′ ζɺ ′
T
e
 ηa
ηb =
(3.42)
ηb 
[η ] = 
ηa =
(3.41)

− η b η a 
1 + ηH
(a)
(b) (3.43)
1 + η 2H
ηH
1 + η 2H
+ ΩηV
(c)
By integrating the equations (3.41) and (3.42) over the whole length of the element,
the strain energy, ‘ P e ’ and the dissipation function, ‘ D e ’ with internal viscous and
hysteretic damping of the shaft element, gives the following set of equations
{ }T [K Be ]{q e } + 12 ηb {q e }T [K De ]{q e }
Pe =
1
ηa q e
2
(3.44)
De =
1
T
ηV {qɺ e } [K Be ]{qɺ e } .
2
(3.45)
38
Where
l
[ ] = ∫ EI [Ψ′′] [Ψ][Ψ ′′]ds
T
K De
0
3.1.6 Damped system equation of motion
Through the use of Hamilton’s principle, the Lagrangian equation of motion obtained for
the damped finite rotating shaft element in the following matrix form as,
([M Te ] + [M Re ]){qɺɺe } + (ηV [K Be ] − Ω[G e ]){qɺ e } + (η a [K Be ] − ηb [K ce ]){q e } = {Q e }
(3.46)
From the equation (3.46) all the matrices are symmetric, except the gyroscopic
[ ]
[ ]
matrix G e and the circulation matrix K ce which are skew symmetric. Here the instabilities
[ ]
resulting from internal dampings are characterized by this circulation matrix K ce . The
material damping which is in the form of viscous, contributes to the circulation effects and
[ ]{ }
also providing a dissipation term ηV K Be qɺ e . Due to this nature, it can provide a rotor
system in the stable condition, providing that this dissipation term dominates. This form
can be achieved, when the rotor system with the undamped isotropic supports and the spin
speed is less than the first forward critical speed.
Hence for the damped system equation of motion can be expressed as,
([M ] + [M ]){qɺɺ } + (η [K ] − Ω([G ] + [G ])){qɺ } + (η [K ] − η [K ]){q } = {Q }
e
d
S
V
e
B
e
d
S
a
e
B
b
e
c
S
S
(3.47)
3.1.7 System instability regions
Instability regions can be divided into two types: first one is primary instability regions
(PIRs) and the second is combination instability regions (CIRs). The starting points of
these instability regions for the periodically time-varying system could be expressed in the
spinning (rotating speed) axis as [77]
39
2
 p
ωi ,
2
Ω
=
n

n
,

2
p
2Ω = (ω + ω )
i
j
 m
m
n = 1, 2,...
for PIRs
(3.48)
i ≠ j, m = 1, 2,... for CIRs
Where ωi and ω j are the whirling frequencies of ith and j frequencies of the system.
The results from the literatures show that the ωi and ω j in the equation (3.48) includes
only the forward whirling frequencies. Here n = 1 and m = 1 has taken for the PIR and CIR
respectively. The results for these instability regions are computed for the first and second
forward whirling frequencies. The system rotating speed lines are plotted with
ωi = Ω, ωi = 2Ω − ωb1 and ω i = 2Ω − ω f 1 for the both uncracked and the cracked rotor
systems which are presented in the results chapter to figure out the instability regions
related to the first two forward natural whirl modes.
3.1.8 Whirl speed analysis
Generally when the rotor shaft is in rotation, the shaft enters into transverse oscillations.
The centrifugal force due to the shaft unbalance is responsible for vibration. If the shaft
speed matches with the natural frequency of the transverse oscillations, the system
vibration behaviour raises and indicates the whirling of the shaft. This shaft whirling will
damage the rotating systems. So, it is essential to balance the system very carefully to
reduce this effect and to design the system natural frequency for the different spinning
speeds. For the computational purpose of the system equation of motion, the eigenvalues
can be obtained from the following equation,
[0]


−1
− [K S ] [M S ]
[I ]
(Ω[K ]
S −1

1
{
h0 } = {h0 }

α
[G S ] 
)
(3.49)
The equation (3.49) represents the conjugative pairs of the pure imaginary with the
magnitude for the orthotropic bearing which is equal to the system natural whirl speeds.
40
3.2
Fault modelling in the rotor system
In this research study, two types of faults are considered. First one is unbalance in the rotor
rigid disk and the other is transverse crack at the rotor shaft element j. The asymmetric
angle s, non-dimensional crack depths µ and location of crack are investigated to identify
the effects on the instability regions of the system. The modelling of fault is carried as the
same way the rotor system was modelled in the previous sessions. This is done by using
Lagrangian method.
3.2.1 Linear mass unbalance in rotor
Unbalance in rotor system is unavoidable and it cannot be completely eliminated. It
happens when the mass centre of the shaft is misaligned with the rotation centre or bearing
centre axis. This makes the rotor to the wobbling motion and major source of vibration. To
correct these unbalance, first it is essential to determine the unbalance. The presence of
unbalance changes the dynamic behaviour of the rotor system. Linear mass unbalance
distribution can be expressed by using the mass center eccentricity.
3.2.1.1
Unbalance Response
When the speed of the rotor bearing system increases, the amplitude is commonly excited
by the unbalance forces presents in the system. These vibration amplitudes frequently
passes through the maximum speed is called critical speed. When a constant speed is
considered the unbalance force for equation (3.39) in the fixed frame coordinates, the
equation for the system unbalance force can be given in the relation as [11]
{Q S } = {QcS }cos(Ωt ) + {QsS }sin(Ωt )
(3.50)
The steady state form of the solution will be,
{q S } = {qcS }cos(Ωt ) + {qsS }sin(Ωt )
(3.51)
41
By substituting the above equations (3.50) and (3.51) in the equation (3.35) yields,
−1
 {QcS }
{ } ([K S ] − Ω 2 [M S ])
− Ω 2 [G S ]
 

=
{ }  Ω 2 [G S ]
[([K S ] − Ω 2 [M S ])] {QsS }
 qce
 e
 q s
(3.52)
From the equation (3.52), the solution can be obtained by the back substitution to the
equation (3.51), which gives the undamped rotor system unbalance response.
3.2.2 Transverse crack modelling
The present study proposes the vibration analysis of the rotor-bearing system with
transverse crack based on the finite element approach. The dynamic behaviour of the rotorbearing system with periodically time varying stiffness and various crack depths are
investigated. The effect of transverse crack on the starting point of instability regions of the
rotor-bearing system is also carried out in the analysis. The following section elaborates
the finite element equations of motion of the rotor-bearing system with transverse crack.
The transverse crack appears at the shaft element j (Fig. 4.10). The relative position
of the crack in the circumference is illustrated in the Fig. 3.6.
Figure 3.6. Relative positions of the shaft and transverse crack in circumference
42
The angle between the major axes of the crack and the shaft is shown by ‘s ’. ‘s’
is called as the asymmetric angle, which is equal tos Ss = sU. The asymmetric angle ,
is an important factor for parametric instability of rotor system. The element matrices for
the transverse crack are introduced and derived for the assemblies of the FEM model of the
rotor system. In addition to this, a case of transverse crack: an open crack is taken in the
derivation. The crack is assumed to be present at an angle of s relative to the fixed
negative Z-axis at t = 0, as shown in Figure 3.6. As the shaft starts to rotate the crack angle
with respect to the negative Z-axis changes with time to (s+ Ωt).
3.2.2.1
Transverse crack element modelling
For an open crack case, the stiffness matrix of the cracked element in a generalized form
similar to that of the asymmetric rod can be written as [47]
− 6lIY (t ) − 12IY (t )
− 6lIY (t )
0
0
0
0
 12IY (t )


− 12I Z (t ) 6lIZ (t )
12I Z (t ) 6lI Z (t )
0
0
0 
 0


− 6lIZ (t ) 2l 2 I Z (t )
6lIZ (t ) 4l 2 I Z (t )
0
0
0 
 0


0
0
4l 2 IY (t ) 6lIY (t )
0
0
2l 2 IY (t ) 
E  − 6lIY (t )
j
(t) = 3 
 (3.53)
ce
0
0
6lIY (t ) 12IY (t )
0
0
6lIY (t ) 
l − 12IY (t )


− 12I Z (t ) − 6lIZ (t )
0
0
12I Z (t ) − 6lIZ (t )
0 
 0


− 6lIZ (t ) 4l 2 I Z (t )
6lIZ (t ) 2l 2 I Z (t )
0
0
0 
 0


0
0
2l 2 IY (t ) 6lIY (t )
0
0
4l 2 IY (t ) 
 − 6lIY (t )
k
From equation (3.53), l represents the element length, E is the elastic modulus. The
expressions for the time-varying quantities (t) and (t) are given in the following
consequent sections.
3.2.2.2
Open crack
The factors (t) and (t) are put to the time-varying quantities of the open crack. The
expressions for the time-varying quantities (t) and (t) are considered as [47],
43
I Y (t ) = I 1 + I 2 cos (2(Ωt + φ ))
(a)
I Z (t ) = I 1 − I 2 cos (2(Ωt + φ )) .
(3.54)
(b)
Where I1 = S + − # U and I 2 = S − − # U. These variables are constant
#
#
quantities throughout the time of the shaft rotation. By considering the profile of the
cracked element cross-section, the following quantities can be obtained by deploying the
non-dimensional crack depth (µ =ℎ ), and the shaft radius (R) as,
π R4
(
4
2
+ R (1 − µ ) ( 2 µ − 4 µ + 1) γ + sin − 1 (1 − µ )
8
4
π R4 R4
−
(1 − µ ) ( 2 µ 2 − 4 µ − 3 ) γ + 3sin −1 ( γ )
IZ =
4
12
−1
2
A1 = R ( π − cos (1 − µ ) + (1 − µ ) γ )
IY =
(
)
)
(3.55)
(3.56)
(3.57)
3
e=
2R 3
γ
3 A1
(3.58)
µ (2 − µ ) . Hence the finite element stiffness matrix for the jth element with
Where γ =
open crack can be given as,
k oc (t ) = k o1 + k o2 cos(2(Ωt + φ ))
j
j
j
(3.59)
The equation (3.59) represents the time-periodic stiffness matrix with frequency of 2Ω.
3.2.2.3 Equation of motion of the system with transverse open
crack
The global equation of motion for the rotor-bearing system with the transverse open crack
can be written in fixed frame coordinates by neglecting the unbalance force as
[M ]{qɺɺ (t )} − Ω[G ]{qɺ (t )} + ([K ] + [K~ (t )]){q (t )} = 0
S
S
S
S
S
S
(3.60)
[ ]
Where, M S = global mass matrix
[G ] = global gyroscopic matrix
S
[K ] = global stiffness matrix of the un-cracked rotor-bearing system equal to ko1j
S
44
[K~ (t )] = [K~o ] cos 2(Ωt + φ )
[K~o ] = global stiffness matrix of the cracked element, equal to ko2j .
{q S (t )} = {q1e .......... .qie ,........q Ne +1 }T is the global displacement vector.
[ ][ ][ ]
[
]
~
The matrices M S , G S , K S and K (t ) are having the dimensions of 4(N+1) x
4(N+1). The equation of motion of the system is a second order differential equation with
frequency of 2Ω for the open crack. This system is periodically time-varying.
3.3.3 Lateral displacement responses of bearing using ANSYS
The variable cross-section rotor-bearing system is modelled with axisymmetric elements
(SOLID273) to determine the bearing response. These elements possess the variable cross
sections of rotor sections with impulse excitations along the X-axis at a node situated in the
left overhung part of the rotor. Translational and rotational DOFs about the axis of rotation
at the bearing locations are constrained. Fixed support conditions are applied to the nodes
of the bearing elements. The rotor-bearing system of axisymmetric elements is given in
Figure 3.7 which is developed in ANSYS-v13. The analysis was carried out using the
commercial ANSYS software package. The axisymmetric rotor was modelled as a
configuration of eight master plane nodes. Two undamped linear bearings were located at
nodes nine and fifteen respectively as shown in Figure 4.1.1 (c). Modal analysis is
performed on rotor bearing system with multiple load steps to determine the natural
frequencies and mode shapes.
SOLID273 is used to model axisymmetric solid structures. The element has
quadratic displacement behaviour on the master plane and is well suited to modelling
irregular meshes on the master plane. The plane on which quadrilaterals or triangles are
45
defined is called the master plane. It is defined by eight nodes on the master plane, and
nodes created automatically in the circumferential direction based on the eight master
plane nodes. The element has plasticity, hyper elasticity, stress stiffening, large deflection,
and large strain capabilities. The proposed system under considered for analysis is
overhung.
Figure 3.7 Rotor-bearing system with SOLID273 axisymmetric elements
46
Chapter-4
Numerical Analysis and Discussions
For numerical analysis a rotor bearing system is considered which is represented in Fig.
4.1. The system has variable cross sections along longitudinal direction and is modelled as
seven stations which have eighteen sub elements. The rotating shaft is supported by two
linear identical bearings which are located at stations four and six respectively. Two disks
are placed at stations three and five for analysis.
Ω
X
STN 7
Y
STN 6
STN 5
STN 4
STN 3
STN 2
Z
STN 1
Figure 4.1 Rotor elements with variable cross section
The natural whirl frequencies, mode shapes, unbalance response, critical speeds,
natural whirl speeds and frequency responses with phase-plane diagrams of the system are
analysed by using finite element method. The effects of hysteresis damping and viscous
damping on the above parameters are also discussed. The design variables for the various
47
cross sections of rotor elements are listed in Table. 4.1. The total length of the typical rotor
bearing system is taken as 353 mm. The relative positions of disk-1 and disk-2 are 0.28 and
0.66 respectively from the left end of the system.
Table 4.1 Rotor element configuration data
Station No.
1
2
3
4
5
6
Node No.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
Axial Dist
(cm)
0.00
1.27
5.08
7.62
8.89
10.16
10.67
11.43
12.70
13.46
16.51
19.05
22.86
23.62
27.43
29.46
31.24
32.26
35.30
Inner Dia.
(cm)
1.52
1.78
1.52
1.52
Outer Dia.
(cm)
0.51
1.02
0.76
2.03
2.03
3.30
3.30
2.54
2.54
1.27
1.27
1.52
3.30
1.52
1.27
1.27
3.81
2.03
2.03
The physical and mechanical properties of the shaft and the disk are represented in
Table 4.2. The bearings are modelled as linear springs. Two types of bearings are
considered for the analysis. First one is isotropic bearing for which the stiffness values of
the bearings in Y and Z axes are KVV = KWW = 4.378e7 N/m and the second one is
orthotropic bearing for which the stiffness values are KVV = KWW = 3.503e7 N/m, KVW =
KWW = - 8.756e7 N/m. Isotropic bearings have physical properties such as stiffness same in
all directions, whereas orthotropic bearings have physical properties such as stiffness
independent in three mutually perpendicular directions [11].
48
Table 4.2 Physical and mechanical properties of shaft and disk [11]
Notation
Description
ρ
Density
E
Modulus of elasticity
μ}
Ω
I
I
m = m# =
Value
Element mass per unit length
Spin speed
7860kgm
2.1 x 1T N/m#
0.3
0-30000 rpm
Diametric inertia of disk
Polar inertia of disk
Disk mass
0.0136 kg-m#
0.0020 kgm
1.401 kg
The analysis is carried out by considering the following three cases of the rotor bearing
system.
Case 1: Undamped rotor bearing system without crack.
Case 2: Damped rotor bearing system without crack.
Case 3: Undamped rotor bearing system with crack.
a.
b.
c.
Figure 4.1.1. Finite models of the system (a) Undamped system without crack (b) Damped
system without crack, (internal dampings η H = 0.0002 & ηV = 0.0002s), (c) Undamped
system with crack.
49
4.1
Undamped rotor bearing system without crack
The undamped rotor bearing system without crack is analyzed for natural whirl
frequencies, mode shapes, unbalance response and critical speeds. The starting point of the
instability regions related to first two forward whirling modes are determined. Two types
of bearings i.e. isotropic and orthotropic are considered for the analysis.
4.1.1 Natural whirl frequencies and mode shapes
The natural whirl frequencies for the first three modes of the uncracked rotor bearing
system with variable cross sections supported on isotropic and orthotropic bearings for the
speed range of 0-30000 rpm in the fixed frame coordinates are found for the 18 elements in
first trial. Due to the complexity of the system configuration, convergence study is done
with three sets of elements which are 18, 25, and 30 respectively. It is observed that the
frequencies of the first three mode shapes are converged with 30 numbers of elements. The
convergence results for natural whirl frequencies of isotropic and orthotropic bearings are
listed in Table 4.3 and 4.4 respectively.
Table 4.3 Natural whirl frequencies of isotropic bearing.
Modes
I
II
III
Natural whirl frequencies in (Hz)
(No. of elements)
(18)
(25)
(30)
61.70
61.39
61.35
247.36
247.18
247.01
402.27
402.01
401.9
Table 4.4 Natural whirl frequencies of orthotropic bearing.
Natural whirl frequencies in (Hz)
(No. of elements)
(18)
(25)
(30)
Modes
BW
FW
BW
FW
BW
FW
I
60.78
61.70
60.55
61.5
60.42
61.48
II
241.82
247.36
241.63
247.22
241.45
247.15
III
373.64
402.28
373.38
402.15
373.25
402.04
50
The first three mode shapes are obtained for spin speed of 0 and 30000 rpm and are
represented in the Figs. 4.2 (a), (b) and (c).
a.
x 10
-9
1.5
1
1
0.5
Y (m )
Y (m )
0.5
0
0
-0.5
-0.5
-1
-1.5
1
-1
1
0.5
0.5
x 10
0
0
-8
Z (m)
-0.5
-1
5
0
10
15
x 10
-10
Z (m)
-0.5
10
15
5
0
X (Nodes)
X (Nodes)
b.
x 10
x 10
-8
1.5
3
1
1
0.5
Y (m )
2
0
Y (m )
-9
0
-1
-0.5
-2
-1
-3
-1.5
2
-4
1
1
0.5
0
-0.5
Z (m)
-1
0
5
10
15
x 10
0
-8
-1
Z (m)
-2
X (Nodes)
10
5
0
15
X (Nodes)
c.
x 10
-9
3
1
2
1
Y (m )
Y (m )
0.5
0
0
-1
-0.5
-2
-3
2
-1
2
1
1
-9
0
x 10
-1
Z (m)
-2
0
5
10
15
x 10
-9
0
-1
Z (m)
X (Nodes)
-2
0
5
10
15
X (Nodes)
Figure 4.2. Mode shapes at 0 and 30000 rpm (a) First mode shape (b) Second mode shape
(c) Third mode shape.
51
4.1.2 Unbalance response
The element with linear mass unbalance distribution is considered for unbalance response
analysis. The disk with mass center eccentricity of 0.001m with (η L , ζ L ) and (η R , ζ R ) are
calculated and plotted for the two cases of bearing stiffness. The undamped system
unbalance response for the fixed frame co-ordinates from the equation (3.52) for the rotor
speed range of 0 – 30000 rpm is plotted in Figure 4.3 with isotropic and orthotropic
bearings respectively.
Figure 4.3 Unbalance response of rotor with isotropic and orthotropic bearings
The results are validated with the single disc system [11] for the speed range of 0 – 30000
rpm. The differences for the present study with single and multi disc are listed in the Table
4.5.
Table 4.5 Comparison of critical speeds for isotropic and orthotropic bearings
Types
Isotropic
bearing
Orthotropic
bearing
Speed range
(rpm)
Ref. [11]
(rpm)
Present work
(rpm)
Difference
0 – 30000
1.71 x 1TY
1.688 x 1TY
2.2 %
0 – 30000
1.65 x 1TY
52
1.632 x 1TY
1.80%
4.1.3 Natural whirl speeds
The undamped natural whirl speeds associated with the eigenvalue problem of the equation
(3.49) for the spin speed of 0-30000 rpm are computed and plotted for the natural whirl
ratio of 0, ±1/4, ±1/2 and ±1 in Fig. 4.4 for isotropic and orthotropic bearings respectively.
Figure 4.4 Campbell plot for rotor-bearing system with both bearings
The first three natural whirl speeds for each whirl ratio are listed in Table 4.6 for isotropic
bearings.
Table 4.6 Natural whirl speeds for isotropic bearing.
Natural whirl ratio
1
1/2
1/4
0
Natural whirl speeds (RPM)
Positive
Negative
3855
3558
16164
13974
27396
21828
3783
3643
15456
14358
25626
22920
3743
3672
15126
14610
24858
23514
3702
14841
24136
53
The natural whirl speeds of the undamped system by using equation (3.49) for three
sets of spin speeds 10000, 20000 and 30000 rpm are obtained for orthotropic bearing. The
first three natural whirl speeds for forward and backward cases are presented in table 4.7. It
is observed that as the spin speed increases there is an increase in forward speeds and
decrease in backward speeds for all modes of vibration.
Table 4.7 Natural whirl speeds for orthotropic bearing.
Natural whirl Speeds (rpm)
Spin speed (rpm)
10000
20000
30000
Forward
Backward
4065
3303
15469
13994
24707
21902
4416
2962
16356
13441
25764
21009
4851
2656
17371
12970
26955
20095
4.1.4 System instability regions
The starting points of these instability regions for the periodically time-varying system
could be expressed in the spinning (rotating speed) axis by using equation (3.48) related to
the first two forward whirl modes. In order to find the system instability regions, the
rotating speed lines are plotted which is shown in Fig. 4.5. The initial points of instability
regions which are related to the first two forward whirl modes are determined. These
instability regions can be given as Ω p = 3799.8 rpm for PIR and Ω c = 9690 rpm for CIR.
54
Figure 4.5 The starting points of instability regions related to I and II FW whirl modes
.
4.2
Damped rotor bearing system without crack
4.2.1 System with hysteretic damping
The multi disk rotor bearing system which is supported on the two identical linear bearings
at stations four and six are analyzed with hysteretic damping loss factor η H = 0.0002.
Figure 4.6 Natural whirl frequency of rotor with hysteretic damping on isotropic bearing.
55
For the synchronous natural whirl, the frequencies of the first three modes are
plotted with the speed range of 0 – 30000 rpm for isotropic bearing and shown in Fig. 4.6.
Table 4.8 represents the comparison of natural whirl speeds for damped and undamped
isotropic bearings for first three modes. From the table it is observed that due to hysteretic
damping the natural whirl speed decreases as compared to undamped system.
Table 4.8 Natural whirl speeds for isotropic bearing.
MODES
Damped (rpm)
( η H = 0.0002)
Undamped
(rpm)
I FW
3822
3855
I BW
3536
3558
II FW
16116
16164
II BW
13932
13974
III FW
27360
27396
III BW
21798
21828
Similarly the frequencies of first three modes with hysteretic damping are plotted
for orthotropic bearing in Fig. 4.7. The comparison between the hysteric damped and
undamped system for orthotropic bearing are represented in Table 4.9.
Figure 4.7 Natural whirl frequency of rotor with hysteretic damping on orthotropic bearings.
56
Table 4.9 Natural whirl speeds for orthotropic bearing
MODES
Damped (rpm)
( η H = 0.0002)
Undamped
(rpm)
I FW
3795
3831
I BW
3508
3544
II FW
15966
15990
II BW
13752
13788
III FW
26502
26538
III BW
20898
20922
Figure 4.9 shows the natural whirl frequencies of the uncracked rotor bearing
system supported on orthotropic bearing with damping coefficient η H = 0.0002. Form
tables 4.8 and 4.9 it is observed that due to hysteretic damping the natural whirl speed is
decreased for both isotropic and orthotropic bearings.
4.2.2 System with viscous damping
The effect of viscous damping with damping coefficient ηV = 0.0002s on natural whirl
speeds for isotropic bearing is analysed. From Table 4.10 the first three natural whirl
speeds are obtained and is compared with the undamped case. It is observed the
frequencies decreased due to the viscous damping in all cases.
Table 4.10 Natural whirl speeds for isotropic bearing
MODES
Damped (rpm)
( η V = 0.0002s)
Undamped
(rpm)
I FW
3822
3855
I BW
3536
3558
II FW
22128
16164
II BW
13920
13974
III FW
27486
27396
III BW
21738
21828
57
Figure 4.8 shows the variation of rotation speeds with first three natural whirl frequencies
for isotropic bearing.
Figure 4.8 Natural whirl frequency of rotor with viscous damping on isotropic bearings.
Similarly for orthotropic bearing the natural whirl frequencies for the first three
modes is shown in Table 4.11 and is compared with undamped natural whirl frequencies.
The Fig. 4.9 represents the variation of rotational speed with natural whirl frequencies for
orthotropic bearing with viscous damping ηV = 0.0002s. From Table 4.10 and 4.11 it is
observed that by considering the viscous damping in the system the whirl speed decreased
for orthotropic bearing.
Figure 4.9 Natural whirl frequency of rotor with viscous damping on orthotropic bearings
58
Table 4.11 Natural whirl speeds for orthotropic bearing.
4.3
MODES
Damped (rpm)
( η V = 0.0002s)
Undamped
(rpm)
I FW
3795
3831
I BW
3509
3544
II FW
15972
15990
II BW
13752
13788
III FW
26592
26538
III BW
20838
20922
Undamped rotor bearing system with transverse crack
The undamped rotor bearing system with transverse crack is analyzed for the system
shown in Fig 4.10. The transverse crack is assumed to present with relative position of 0.73
and non dimensional crack depth of 0.3. Crack is assumed at an angle of s which is related
to the negative fixed Z axis with the time t = 0.
Y
Z
X
Figure 4.10 Rotor bearing system with variable cross sections and crack
When the rotor shaft rotates the transverse crack angle keep changing with time s
+ Ωt with respect to the negative Z axis. The system with crack is analysed for the natural
whirl frequencies, mode shapes, unbalance response and critical speeds for the timeperiodic stiffness matrix with frequency of 2Ω.. The starting point of the instability regions,
effect of the various non
non-dimensional crack depth (µ) on the system natural whirl
59
frequencies, frequency response and phase-plane diagrams are plotted for the system with
transverse crack for isotropic bearings and orthotropic bearings.
4.3.1 Natural whirl frequencies and mode shapes
The natural whirl speeds for the first three modes of the cracked rotor bearing system with
variable cross sections supported on isotropic and orthotropic bearings for the speed range
of 0-30000 rpm in the fixed frame coordinates with the non-dimensional crack depth µ =
0.3 are computed and listed in Tables 4.12 and 4.13 for isotropic and orthotropic bearings
respectively for the three sets of spin speeds.
Table 4.12 Natural whirl speeds for
isotropic bearing.
Table 4.13 Natural whirl speeds for
orthotropic bearing.
Spin speed Natural whirl speeds (rpm)
(rpm)
Forward
Backward
Spin speed Natural whirl speeds (rpm)
(rpm)
Forward
Backward
3979
3243
3833
3085
15862
13894
15447
13395
24734
23265
24661
23190
4366
2919
4204
2779
16968
13063
16533
12591
25630
22676
25552
22605
4752
2629
4579
2502
18096
12316
17712
11862
26641
22236
26559
22108
10000
20000
30000
10000
20000
30000
The first three mode shapes of the cracked system are computed at the spin speed of 0 and
30000 rpm and are shown in the Figs. 4.11 (a), (b) and (c).
60
a.
-4
x 10
1
4
3
2
1
Y (m )
Y (m )
0.5
0
0
-1
-0.5
-2
-3
4
-1
4
2
2
0
-5
x 10
-2
Z (m)
-4
0
10
5
15
x 10
0
-4
-2
Z (m)
X (Nodes)
-4
0
-4
0
10
5
15
X (Nodes)
b.
x 10
-5
x 10
3
4
2
3
2
1
1
Y (m )
Y (m )
-4
0
0
-1
-1
-2
-2
-3
1
-3
4
2
0.5
0
-0.5
Z (m)
-1
0
10
5
15
x 10
0
-4
-2
Z (m)
X (Nodes)
15
10
5
X (Nodes)
c.
-6
x 10
3
1.5
2
1
1
0.5
Y (m )
Y (m )
x 10
0
-5
0
-1
-0.5
-2
-1
-3
1
-1.5
2
1
0.5
0
-0.5
Z (m)
-1
0
5
10
15
x 10
-5
0
Z (m)
X (Nodes)
-1
-2
0
5
10
15
X (Nodes)
Figure 4.11 Mode shapes for spin speed 0 and 30000 rpm (a) first mode shape (b) second
mode shape (c) third mode shape.
61
4.3.2 Unbalance response with transverse crack
The element with linear mass unbalance distribution is considered for the unbalance
response analysis. The disk with mass center eccentricity of 0.001m with (η L , ζ L ) and
(η R , ζ R ) are
calculated and plotted for isotropic and orthotropic bearing stiffness. The
unbalance response of the undamped rotor bearing system with a transverse crack in fixed
frame co-ordinates from the equation (3.52) for the rotor speed range of 0 – 30000 rpm
with the non-dimensional crack depth µ = 0.3 is computed and plotted in Fig. 4.13.
Figure 4.12 Unbalance response of rotor with transverse crack (h/R = 0.3) for isotropic and
orthotropic bearings.
4.3.3 System natural whirl speeds with transverse crack
The undamped natural whirl speeds associated with the eigenvalue problem for spin speeds
of 0-30000 rpm are computed by using equation (3.49). The variation of rotational speeds
with natural whirl frequencies for whirl ratio of 0, ±1/4, ±1/2 and ±1 are shown in Fig. 4.13
for isotropic and orthotropic bearings. The non-dimensional crack depth of 0.3 is
considered for the analysis.
62
Figure 4.13 Campbell plot for rotor-bearing system with transverse crack for isotropic and
orthotropic bearings.
Tables 4.14 and 4.15 represents the natural whirl speeds of the cracked rotor system
for the natural whirl ratio of 0, ±1/4, ±1/2 and ±1. It is observed from table that, as the
natural whirl speed ratio decreases, there is decrease in whirl speeds.
Table 4.14 Natural whirl speeds for
isotropic bearing.
Natural
whirl ratio
1
1/2
1/4
0
Table 4.15 Natural whirl speeds for
orthotropic bearing.
Natural whirl speeds
(rpm)
Positive
Negative
3742
3470
Natural
whirl ratio
1
Natural whirl speeds
(rpm)
Positive
Negative
3673
3352
16326
13332
16584
13584
26256
22542
26190
22482
3679
3530
3626
3389
15642
14154
15414
13878
24948
23172
24882
23106
3647
3554
3610
3403
15240
14460
15054
14148
24420
23538
24354
23466
3633
3567
3605
3407
14976
14689
14859
14304
23950
23945
23947
23815
1/2
1/4
0
63
4.3.4 Effect of crack depths on natural whirl frequencies
The transverse crack with the asymmetric angle s = 0 for three non-dimensional crack
depth, µ = 0.1, 0.2 and 0.3 on the rotor are investigated with the isotropic and orthotropic
bearings. The corresponding frequencies for the crack depths for isotropic and orthotropic
bearings are shown in Figs. 4.14 and 4.15.
Figure 4.14 Campbell plot for rotor-bearing system with µ = 0.1, 0.2 and 0.3 on natural
whirl frequencies for isotropic bearing.
Figure 4.14 (a). Magnified view of II FW and II BW whirls for rotor-bearing system with
µ = 0.1, 0.2 and 0.3 on natural whirl frequencies for isotropic bearing.
64
Similarly for orthotropic bearing the Campbell plot for rotor bearing system of non
dimensional crack depths 0.1, 0.2 and 0.3 are plotted in Fig. 4.15.
Figure 4.15 Campbell plot for rotor-bearing system with µ = 0.1, 0.2 and 0.3 on natural
whirl frequencies for orthotropic bearing.
Figure 4.15 (a). Magnified view of II FW and II BW whirls for rotor-bearing system with
µ = 0.1, 0.2 and 0.3 on natural whirl frequencies for orthotropic bearing.
Table 4.16 Natural whirl frequencies with µ = 0.1, 0.2 and 0.3 for isotropic bearings.
Modes
Natural whirl Frequency in (Hz)
µ = 0.1
µ = 0.2
µ = 0.3
FW
BW
FW
BW
FW
BW
I
60.67
60.45
60.63
60.03
60.55
59.45
II
250.18
249.16
249.99
247.28
249.60
244.82
III
399.18
399.16
399.17
399.12
399.17
399.09
65
Table 4.17 Natural whirl frequencies with µ = 0.1, 0.2 and 0.3 for orthotropic bearings.
Modes
Natural whirl Frequency in (Hz)
µ = 0.1
µ = 0.2
µ = 0.3
FW
BW
FW
BW
FW
BW
I
60.57
57.35
60.36
57.13
60.08
56.78
II
249.69
240.54
248.77
239.71
247.65
238.41
III
399.17
397.03
399.16
396.99
399.13
396.93
From the Tables 4.16 and 4.17 it is observed that as the crack depth increases there
is decrease in natural whirl frequencies for both forward and backward conditions for
isotropic and orthotropic bearings. Again for orthotropic bearing the natural whirl
frequency is less as compared to isotropic bearing for a given non dimensional crack depth.
4.3.5 System instability regions with transverse crack
For computing the starting points of instability regions related to the first two forward whirl
modes equation (3.49) is used. In order to find the system instability regions, the rotating
speed lines are plotted which is shown in Fig. 4.16 with non-dimensional crack depth 0.3.
These instability regions can be given as Ω p = 3742 rpm for PIR and Ω c = 9774 rpm for
CIR.
Figure 4.16 The starting points of instability regions related to I and II FW whirl modes with
non dimensional crack depth.
66
4.4
Frequency domain and phase - plane diagrams
The frequency domain and phase – plane diagrams are obtained by using the Houbolt
method. This method is an implicit integration scheme, which gives the solutions to
coupled second order differential equations. In this, the standard finite difference equations
are used to approximate the acceleration and velocity components. This will be in terms of
displacement components. This method helps to avoid the critical time step limit, and the
time step ∆t can be generally used as large as the value given for central difference method
[43]. The method utilized here to study the effect of the system spin speed at 1000 rpm and
5000 rpm with disk eccentricity of 0.001m respectively. The responses are analyzed at
various nodes in the system.
The following configurations of the systems are analyzed with both the bearing
cases to obtain the frequency domain and phase – plane diagrams for the above mentioned
parameters,
1. Undamped system without transverse crack
2. Damped system without transverse crack
3. Undamped system with transverse crack
4.4.1 Undamped system without transverse crack
The responses for undamped rotor system with isotropic and orthotropic bearing without
transverse crack are shown in Figs. 4.17 - 4.20. The analysis is carried out for spin speeds
of 1000 and 5000 rpm with disk eccentricity 0.001m.
67
Figure 4.17 Response and phase-plane diagrams of disk 1 and 2 for spin speed 1000 rpm
with eccentricity of 1mm for isotropic bearing.
Figure 4.18 Response and phase-plane diagrams of disk 1 and 2 for spin speed 5000 rpm
with eccentricity of 1mm for isotropic bearing.
For undamped rotor bearing system without crack at 1000 rpm for isotropic
bearing, the amplitude is 1.10e-5 dB for disk-1. Whereas for the same condition, for
orthotropic bearing the amplitude is 1.65e-5 dB. The difference in amplitude is due to
variation in stiffness of the bearing. When the speed is increased to five times the
amplitude for isotropic bearing is 1.13e-5 dB. For orthotropic bearing when the speed
increases to 5000rpm, the amplitude is 1.56e-5 dB.
68
Figure 4.19 Response and phase-plane diagrams of disk 1 and 2 for spin speed 1000 rpm
with eccentricity of 1mm for orthotropic bearing.
Figure 4.20 Response and phase-plane diagrams of disk 1 and 2 for spin speed 5000 rpm
with eccentricity of 1mm for orthotropic bearing.
For disk-2, at 1000rpm for isotropic bearing the amplitude is 6.62e-5 dB. For
orthotropic bearing at same speed, the amplitude increased to 19.6e-5 dB. When the speed
increases to 5000rpm, the amplitude changes from 6.69e-5 dB for isotropic to 20.45e-5 dB
for orthotropic bearing.
69
4.4.2 Damped system without transverse crack
The effects of damping on frequency response without considering the transverse crack of
isotropic and orthotropic bearings are discussed. Two types of dampings are considered for
the analysis. First is hysteretic damping where the energy dissipated is independent of
frequency of oscillation. Second one is viscous damping in which energy dissipated per
cycle depends linearly on frequency of oscillation. The frequency responses along with
phase-plane diagrams are shown in Figs. 4.21 – 4.28.
4.4.2.1
Hysteretic Damping without transverse crack
For hysteretic damping the response for frequency with isotropic and orthotropic bearings
are shown in Figs. 4.21 – 4.24. Two spin speeds i.e. 1000 rpm and 5000 rpm are considered
for the analysis. The disk eccentricity is assumed to be constant as 0.001 m.
Figure 4.21 Response and phase-plane diagrams of disk 1 and 2 for spin speed 1000 rpm
with eccentricity of 1mm for isotropic bearing with hysteretic damping.
70
Figure 4.22 Response and phase-plane diagrams of disk 1 and 2 for spin speed 5000 rpm
with eccentricity of 1mm for isotropic bearing with hysteretic damping.
For hysteretic damped rotor bearing system without crack at 1000 rpm with
isotropic bearing, the amplitude is 1.79e-5 dB. Whereas for orthotropic bearing the
amplitude is 2.64e-5 dB at same speed. When speed increased to five times the amplitude
for isotropic is 1.86 e-5 dB and that for orthotropic is 2.61e-5 dB for disk-1. Similar
characteristics are obtained for isotropic and orthotropic bearings with hysteretic damping
for disk-2.
Figure 4.23 Response and phase-plane diagrams of disk 1 and 2 for spin speed 1000 rpm
with eccentricity of 1mm for orthotropic bearing with hysteretic damping.
71
Figure 4.24 Response and phase-plane diagrams of disk 1 and 2 for spin speed 5000 rpm
with eccentricity of 1mm for orthotropic bearing with hysteretic damping.
4.4.2.2
Viscous Damping without transverse crack
Figures 4.25 - 4.28 represents the effect of viscous damping on frequency response for
isotropic and orthotropic bearings without consideration of transverse crack.
Figure 4.25 Response and phase-plane diagrams of disk 1 and 2 for spin speed 1000 rpm
with eccentricity of 1mm for isotropic bearing with viscous damping.
72
Figure 4.26 Response and phase-plane diagrams of disk 1 and 2 for spin speed 5000 rpm
with eccentricity of 1mm for isotropic bearing with viscous damping.
For viscous damped rotor bearing system the amplitude for isotropic bearing is
1.40e-5 dB whereas for orthotropic bearing the amplitude is 3.24e-5 dB for disk-1. When
the spin speed increases to 5000 rpm the amplitude is 1.39e-5 dB for isotropic bearing and
3.27e-5 dB for orthotropic bearing. It is observed that the difference in amplitude is
negligible when the spin speed increases for both isotropic and orthotropic viscous
damping. This is due to very small change in frequency for isotropic and orthotropic
bearings.
Figure 4.27 Response and phase-plane diagrams of disk 1 and 2 for spin speed 1000 rpm
with eccentricity of 1mm for orthotropic bearing with viscous damping.
73
Figure 4.28 Response and phase-plane diagrams of disk 1 and 2 for spin speed 5000 rpm
with eccentricity of 1mm for orthotropic bearing with viscous damping.
4.4.3 Undamped system with transverse crack
The responses for undamped rotor system with isotropic and orthotropic bearing with
transverse crack are shown in Figs. 4.29 - 4.32. The analysis is carried out for spin speeds
of 1000 and 5000 rpm with disk eccentricity 0.001m.
Figure 4.29 Response and phase-plane diagrams of disk 1 and 2 for spin speed 1000 rpm
with eccentricity of 1mm for isotropic bearing with transverse crack.
74
Figure 4.30 Response and phase-plane diagrams of disk 1 and 2 for spin speed 5000 rpm
with eccentricity of 1mm for isotropic bearing with transverse crack.
Figure 4.31 Response and phase-plane diagrams of disk 1 and 2 for spin speed 1000 rpm
with eccentricity of 1mm for orthotropic bearing with transverse crack.
Figure 4.32 Response and phase-plane diagrams of disk 1 and 2 for spin speed 5000 rpm
with eccentricity of 1mm for orthotropic bearing with transverse crack.
75
Similar characteristics are observed for isotropic and orthotropic bearings for
undamped cracked rotor (h/R = 0.3, x/L = 0.73) for 1000 and 5000 rpm. For a cracked rotor
with isotropic bearings the amplitude is more compared to uncracked rotor. However for
orthotropic bearings
between cracked and uncracked rotor the change in amplitude is
negligible. This is due to very small change in natural frequency between cracked and
uncracked rotor.
4.5
Bearing reaction force
The shaft is modelled with SOLID273 axisymmetric elements by using ANSYS® - v13
software. The element has quadratic displacement behaviour on the master plane and is
well suited for modelling irregular meshes on the master plane. The element has plasticity,
hyper elasticity, stress stiffening, large deflection and large strain capabilities. It has also
mixed-formulation capability for simulating deformations. The disc is modelled with
MASS21 element. This element is defined by a single node, concentrated mass
components (f. " # ) in the element coordinate directions and rotary inertias (f. " # ) about
the element coordinate axes.
COMBIN14 is taken for modeling the bearing elements. The element represents a
2-D element and lies in a constant plane. This gives the longitudinal spring-damper option
in a uniaxial tension-compression element. The mass for spring-damper element is
negligible. Masses are added by using the appropriate mass element.
The system is analysed for transient response for the spin speed range of 0 - 30000
rpm. Bearing reactions for the three time stage period of 0.01 sec with the force of 1 KN is
found and plotted in Figure 4.33.
76
Figure 4.33 Bearing reaction forces – Transient analysis
Figure 4.33 shows the bearing reaction forces performed by transient analysis for
the time period of 0.01 sec, which acts in the left and right bearings in both X and Z
directions.
77
4.6. Observations
The flexible multi disk rotor-bearing system is analyzed by finite element method
for a typical configuration as shown in Fig. 4.1, which includes a transverse crack and
internal damping. The formulation for the crack and internal dampings are made by using
finite element method. The analysis carried for three different cases. The effect of crack
depths, crack location and rotor speeds are considered as vital parameters in the analysis.
As shown in Figs. 4.3 & 4.12, the critical speeds of the rotor-bearing system for the
cracked and uncracked system on orthotropic bearings reveals that the critical speed is
reduced for the cracked system with increased value of crack depth. The system shows the
behavior due to the effect of shaft bending and the time periodic stiffness change. Figs. 4.4
and 4.13 shows the natural whirl frequencies of cracked and uncracked rotor-bearing
systems. The natural whirl frequency is less in case of orthotropic bearing due to the fact
that they have different stiffness along three mutually perpendicular directions which are
independent of each other.
The starting points of the instability regions for the periodically time-varying
system are found out and they are shown in Figs. 4.5 and 4.16. It is seen that the starting
points of the instability regions are close for the cracked system as compared to the
uncracked system. This is due to the whirling speed of the shaft. The frequency response
and phase-plane diagrams were carried out by using Houbolt’s implicit time integration
scheme to study the effect of spin speed with 1000 rpm and 5000 rpm. When the spin
speed is increased from 1000 rpm to 5000 rpm the data series in the phase – plane
diagrams are disturbed and the frequency domain has a single influenced frequency. The
increments of spin speed of the system results in the chaotic motion. The results observed
from the finite element approach are compared with the results obtained by Nelson and
Vaugh [11] as shown in Table. 4.5. They are found to be in good agreement.
78
Chapter-5
Conclusions and Future scope
5.1
Conclusions
The present study simulates the dynamics of a multi disk, variable cross section rotor
system supported on two bearings at stations four and six, respectively. The theoretical
analysis is carried out using FEM approach which offers significant benefits in
understanding the dynamic behaviour of rotor-bearing systems. Generally, analysis of
higher order sets of equations formulated with finite element approach clearly
demonstrates the power of the method and understanding. For the complex rotor system,
formulation of the equation of motion, natural whirl frequencies, unbalance response, and
the effect of crack depths, crack location and rotor speed are carried for the analysis. The
conclusions drawn from the results and discussions are depicted below.
(a) Unbalance response for the uncracked rotor bearing system with variable cross
sections supported on isotropic and orthotropic bearings for the spin speed range of
0-30000 rpm has been found, and the results are validated with the single disc
system [11] for the same speed range.
(b) The natural whirl frequencies for the first three modes of the uncracked rotor
bearing system with variable cross sections were found and the convergence study
made with three sets of elements. It is observed that the frequency of the first three
modes converges with 30 numbers of elements.
79
(c) Natural whirl speeds are calculated with the help of 6 station finite element model
which includes the 19 sub elements. The forward and the backward whirl modes are
obvious due to the gyroscopic effect at all the natural frequencies.
(d) The critical speeds of the system were found with isotropic bearings at 1.621 x 1TY
rpm and orthotropic bearings at 1.5885 x 1TY rpm respectively. It concludes that,
the rotor bearing system should surpass at these critical speeds to avoid the
catastrophic failure.
(e) The open crack on the rotor-disk-bearing system seems to have greater impact on
the system instability. The instability region frequently raises when the crack depth
grows, and the parametric instability swing to fall in minor rotating speed domain.
(f)
The starting points of the system instability regions which is related to the first two
forward whirl modes for the system without transverse crack were found and given
as Ω p = 3799 rpm and Ω c = 9690 rpm, and for the system with transverse crack
were found and given as Ω p = 3742 rpm and Ω c = 9774 rpm.
(g)
The analysis of open crack with the asymmetric angle (s=TU, the non dimensional
crack depth µ and the crack locations on the rotor systems are investigated to show
their effects on the system instability regions. The interest rotating speed ranges
were found for the PIR SU U with speed [3470, 3742] and CIR (U ) with speed
[9660, 9774] rpm.
(h) The frequency response and phase-plane diagrams were carried out by using
Houbolt’s implicit time integration scheme to study the effect of spin speed with
1000 rpm and 5000 rpm. When the spin speed is increased from 1000 rpm to 5000
rpm the data series in the phase – plane diagrams were disturbed and the frequency
domain has a single influenced frequency and the increments of spin speed of the
system will result in the chaotic motion.
80
(i)
Transient response analysis was performed for the spin speed range of 0 - 30000
rpm with the help of ANSYS® – v13. Bearing reactions for the three time stage
period of 0.01 sec with a force of 1 kN was found and plotted in two directions for
the left and right bearings.
The frequency response was derived from Houbolt’s implicit time integration
scheme using an interactive script written in MATLAB® numerical computing software.
The frequency response and phase diagrams were obtained at a two specified operating
speeds with disc eccentricity of 1x10-3 m. The presence of transverse crack in the system
has a greater impact in the starting point of the system instability regions and when the
non-dimensional crack depth ‘µ’ increases the system natural whirl frequencies falls in the
minor rotating speed domain. The results obtained for the rotor system indicates that it can
be analysed further with various forms of internal dampings and shear deformation.
81
5.2
Future scope
The dynamic simulations of a linear system can be analyzed further by incorporating active
magnetic bearing (AMB)/fluid film bearings and external dampings into the system
configuration. This work can be extrapolated to deal with a non-linear system with the
above mentioned systems for the bearing nonlinearities and other factors influenced by the
viscous medium. The crack is taken as open in nature here. A breathing crack with time
varying function can be used to study the various dynamic behaviour of the system. Multidisc rotor with disks of asymmetrical inertia and shapes can also be studied. The effects of
various disc eccentricities and the operating speeds on the system frequencies and global
system vibration response can be properly speculated.
There is a wide scope of analysis on open and breathing transverse cracks for the
rotor system. Various effects such as disc eccentricities and operating speeds on the
frequencies and overall vibration response can be studied. For the detection of cracks, this
can be done inversely with the help of cracked excitation frequencies, crack location and
the crack depths by adopting suitable methods.
82
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List of Publications
S. Bala Murugan and R.K. Behera, “Analysis of flexible rotor-bearing systems with finite
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91

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