Study of Dynamic Stress and Life Estimation Approach

Study of Dynamic Stress and Life Estimation
for Viscoelastic Rotor-A Finite Element
Approach
A thesis submitted to National Institute of Technology, Rourkela in partial fulfilment
for the degree of
Master of Technology
in
Mechanical Engineering
by
Abhinav Khare
212ME1276
Department of Mechanical Engineering
National Institute of Technology, Rourkela
Rourkela - 769008, Odisha, India
June - 2014
Study of Dynamic Stress and Life Estimation
for Viscoelastic Rotor-A Finite Element
Approach
A thesis submitted to National Institute of Technology, Rourkela in partial fulfilment
for the degree of
Master of Technology
in
Mechanical Engineering
by
Abhinav Khare
212ME1276
Under the guidance of
Dr. H. Roy
Assistant Professor
Department of Mechanical Engineering
National Institute of Technology, Rourkela
Rourkela - 769008, Odisha, India
June - 2014
National Institute of Technology, Rourkela
CERTIFICATE
This is to certify that the thesis entitled, “Study of Dynamic Stress and Estimation
of Life for Viscoelastic Rotor-A Finite Element Approach”, which is submitted
by Mr. Abhinav Khare in partial fulfilment of the requirement for the award of
degree of M.Tech in Mechanical Engineering to National Institute of Technology,
Rourkela is a record of candidate‟s own work carried out by him under my
supervision. The matter embodied in this thesis is original and has not been used for
the award of any other degree.
Dr. H. Roy
Date:
Place: Rourkela
Assistant Professor
Mechanical Engineering Department
ACKNOWLEDGEMENTS
I am grateful to my supervisor Dr. Haraprasad Roy, whose valuable advice, interest
and patience made this work a truly rewarding experience on so many levels. I am
also thankful to my friends and colleagues for standing by me during the past
difficult times. Particularly, I am indebted to Mr. Saurabh Chandraker for his utterly
selfless help.
As for Yogesh Verma , Ranjan Kr. Behra, Rohit Kr. Singh, Abdul
Hussian,Naveen Tatapudi ,Mahesh Pol, Aniket Devekar: You were there for me when
really needed and I am yours forever.
Abhinav Khare
ABSTRACT
Inherent material damping causes energy dissipation alongside energy storage is
predominant phenomenon of viscoelastic material under dynamic loading. For this
reason viscoelastic materials are extensively used for vibration control and stability
problems. Firstly this work deals with the analytical study of stress developed in
viscoelastic beam. Instantaneous stress is obtained by operating instantaneous strain.
Modelling technique as ATF (Augmenting Thermodynamic Fields) displacements is
used to represent the constitutive relationship in time domain by using certain
viscoelastic parameters. This operator based constitutive relationship is used to bring
the equations of motion. Finite element technique is used to discretize the continuum
by using Euler Bernoulli Beam theory and study has been done for various cases of
beams by applying different boundary conditions, under different loading situation.
Time domain solution is calculated through state space representation of equation of
motion which is further utilized for getting the dynamic stress.
Further this study has been extended to develop a finite element mathematical
model of internally damped viscoelastic rotor which is subjected to combined
loading (i.e. due to bending and torsion) and also exposed to thermal environment.
The shaft damping is incorporated by considering Kelvin – Voigt model. The
bending stress & shear stress are calculated along various location as well as
different instant of the rotor. Maximum stress location decides the failure point have
also been calculated for our further design. Safe design of rotor parameters have
been predicted by comparing failure stresses of the system with the Goodman
diagram. Our thematic aim was to investigate fatigue behaviour and estimation of
rotor life with the help of the S-N curve, which necessitate the calculation of
endurance strength that can also be interpolated through Goodman diagram.
Numerical results show that the life of the rotor is affected by the temperature but
innocent with the axial torque in operating range. It has been concluded through the
resultant stresses the design is safe and hence rotor has infinite life of operation.
i
NOMENCLATURE
le
Length of an Element
L
Length of Beam or Shaft
DR
Diameter of Rotor Shaft
DD
Outer Diameter of a Disc
tD
Thickness of Disc
U
Unbalance of the Disc
R
Position Vector of Displaced Centre of Rotation
y
Lateral Distance from Neutral Axis of Shaft
y
Deflection in Lateral Direction
A
Area of Cross-Section
t
Time
q
Nodal Displacement Vector
I
Moment of Inertia of Rotor shaft
E
Young’s Modulus of Elasticity

Linear Strain

Strain Rate

Bending Stress
M y y , M zz
Bending Moment about Y and Z Axes

Spin Speed

Whirl Speed
ii
V
Shear Force
M
Bending Moment

Volume Density
v
Coefficient of Viscous Damping
F 
External Force Vector
MT 
Translatory Mass Matrix
M R 
Rotary Mass Matrix
 M    MT    M R 
Total Mass Matrix of an Element
G  ,  K 
Gyroscopic and Stiffness Matrices
 KB 
Bending Stiffness Matrix
 KC 
Skew Symmetric Circulatory Matrix
 KT 
Axial Torque Matrix
KA 
Axial Stiffness Matrix
 B
Strain Displacement Matrix
xy 
Hermite Shape Function in X-Y Plane
zx 
Hermite Shape Function in Z-X Plane
 
Shape Function
v, w
Displacement in Y and Z Axes
iii
 ,
Angle of Rotation about Y and Z Axes
Fs
Factor of Safety
u
Ultimate Strength
e
Endurance Strength
f
Failure Stress
Nf
Life of Component
 max , min
Maximum and Minimum Shear Stress
 max ,  min
Maximum and Minimum Bending Stress
 m , a
Mean and Amplitude Value of Shear Stress
 m , a
Mean and Amplitude Value of Bending Stress
  ,  
Mean and Amplitude Value of Equivalent Stress

Coefficient of Thermal Expansion
T
Temperature Difference
TR
External Torque
PA
Axial Load Due to Thermal Load
T
Average Temperature over Cross-Section
TO
Reference Temperature

Augmenting Thermodynamic Field
H
Affinity
eq m
eq a
iv
b
Inverse of Relaxation Time

Strength of Coupling between Mechanical Displacement Field
an Thermodynamic Field

Material Property Relating the Changes in H to 
v
CONTENTS
Abstract
i
Nomenclature
ii
List of Figures
vii
1.
Introduction
1.1 Background and Importance
1.2 Viscosity
1.3 Dynamic Stress and Fatigue
1.4 Overview Of Available Literature
1.4.1 History
1.4.2 Recent Research in the Field of Fatigue Analysis
1.5 Analysis Objective
1.6 Thesis Outline
Dynamics of Beams
2.1 Introduction
2.2 Equation of Motion Undamped of Euler Bernoulli beam
2.3 Elastic Beam Finite Element Formulation
2.4 Incorporation of Damping Matrix
2.5 Numerical Problem for Beam
1
2
3
6
6
8
10
11
2.
Fatigue Analysis of Rotors
3.1 Introduction
3.2 Mathematical Formulation
3.1.1 Finite Element Model
3.1.2 Fatigue Analysis
3.2 Numerical Problem
3.2.1 Validation of Finite Element Code
3.2.2
Single Disc Rotor
4
Conclusion and Future Scope
4.1 Conclusion
4.2 Future Scope
4.3 References
4.4 Appendix
12
13
15
16
18
3
vi
24
25
25
27
31
31
34
41
42
43
45
LIST OF FIGURES
S. No.
Caption
Page No.
1.1
Stress-Strain Graph for Elastic & Viscoelastic Material
2
2.1
Schematic Diagram for Euler-Bernoulli Beam Section
13
2.2
Schematic Diagram of Cantilever Beam
18
2.3
Mode Shapes for Cantilever Beam
19
2.4
Frequency Response Plot for Free End of Beam
19
2.5
Response-Time Plot for Free End of Beam
20
2.6
Stress-Time Plot for Free End of Beam
20
2.7
Stress-Strain Plot Considering Undamped Beam
21
2.8
Response-Time Plot for Free End of Damped Beam
22
2.9
Stress-Time Plot for Free End of Damped Beam
22
2.10
Stress-Strain Plot for Considering Damped Beam
23
3.1
Displaced Position of the Shaft Cross-Section
25
3.2
Goodman Diagram for Interpolating Endurance Strength
30
3.3
Stress-Cycle (S-N) Diagram
30
3.4
Schematic Diagram of Lalanne Rotor
31
3.5
Campbell Diagram
33
3.6
Mass Unbalance Response
33
3.7
Schematic Diagram for Single Disc Rotor
35
3.8
Decay Rate plot
36
3.9
SLS for Different Axial Torque and Temperature
Difference respectively
37
3.10
Variation of Bending Stress, Shear Stress, Equivalent
Stress along the Length of the Rotor Shaft
39
vii
Chapter 1
INTRODUCTION
1.1 Background and Importance
A rotor is considered as long slender one dimensional structure, is mounted
by one or more number of bearings through which it can rotate. Earlier it was
considered as single mass in the form of a point mass, a rigid disc or a long rigid
shaft. Rotating machines ranging from very large systems like power plant rotors, for
example, a turbo generator, ship propeller to very small systems like a tiny dentist‟s
drill, with a variety of rotors such as pumps, compressors, steam turbines, motors,
turbo pumps etc. are used as example in process industry. The principal components
of a rotor-dynamic system are the shaft or rotor with disk and the bearing and shaft
or rotor is the rotating component of the system.
Rotors are the major sources of vibration in most of the machines. As the
rotation speed increases, the amplitude of vibration often passes a maximum that is
called a critical speed. If the amplitude of vibration at a critical speed is excessive,
then catastrophic failure can occur. Another phenomenon occurs quite often in
rotating machinery is instability. By and large every material is classed as
viscoelastic, because damping exhibit in all kind of materials. The internal damping
in the shaft or rotating damping generates a tangential force which proposional to the
spin speed. Rotors may develop unstable behaviour after certain spin speed. Due to
unstable nature and resonance at critical speed, performance or output of rotor is
restricted.
Unlike non rotating structure, the rotor is subjected to repetitive loading.
Thus there is a chance of damaging the system before yielding due to fatigue stress
develop and rotor life reduces. To overcome those problems or to achieve safe
operation of the rotor, it is essential to study of rotor and its dynamics.
1
1.2 Viscoelasticity
Most of the time engineering material show different behaviour as the pure
elastic material and deviates from the Hook‟s law. This is due to some inheritant
property of the material which is known as viscoelasticity. Viscoelasticity, as the
name implies, is a property that combines elasticity and viscosity. A material, which
is viscoelastic in nature stores and also dissipates energies and therefore the stresses
in such materials are not in phase with the strain. For this reason, it is extensively
used in various engineering applications for controlling the amplitude of resonant
vibrations and modifying wave attenuation and sound transmission properties which
helps in increasing structural life through reduction in structural fatigue.
The classical theory of elasticity states that for sufficiently small strains, the
stress in an elastic solid is proportional to the instantaneous strain and is independent
of the strain rate. In a viscous fluid, according to the theory of hydrodynamics, the
stress is proportional to the instantaneous strain rate and is independent of the strain.
Viscoelastic materials exhibit solid and fluid behaviour. Such materials include
plastics, amorphous, polymers, ceramics, glasses and biomaterials such as muscle.
Viscoelastic materials are characterized by constant-stress creep and constant-strain
relaxation. The Stress-Strain Curves for a purely elastic and a viscoelastic material
are shown in fig.1.



(b)
(a)

Figure1.1: Stress-Strain Graph for Elastic and Viscoelastic Material
2
Due to loss of energy during loading and unloading time, the stress strain
curve for viscoelastic material is elliptic in nature. The area enclosed by the ellipse is
a hysteresis loop and shows the amount of energy lost as heat in a loading and
unloading cycle.
The study for rotary machines, requires a careful and detailed analysis, as the
rotation movement of the rotor appreciably influences the dynamic comportment of
the system, making the modal parameters dependent on the rotation of the machine.
The gyroscopic effect couples the rotation movement and, it is dependent on the
rotation speed of the rotor. Therefore it is expected that the natural frequencies and
vibration modes of a rotating machine also depend on the system speed.
1.3 Dynamic Stress and Fatigue
When load is applied on the body a resistive force set up in the body and this
resistive force per unit area of cross section of body is known as stress. The nature of
stress depends upon the nature of applied load. In a part subjected to some forces,
stresses are generally distributed as a continuously varying function within the
continuum of material. Every infinitesimal elements of the material can conceivably
experience different stresses at the same time. Normal stresses acts perpendicular
(i.e. normal) to the surface of the body and tend to pull it out (i.e. tensile normal
stress) or push it in (i.e. compressive normal stress). Shear stresses acts parallel to the
surface of the body, in pairs on opposite faces, which tends to distort the cube into a
rhomboidal shape. When body is subjected to loads transverse to its length cause
body to bend and thus resultant stresses due this type of loading is called bending
stress.
Most of the time machine parts are rarely loaded in as simple manner as the
specimens used to get the tensile strength. Henceforth if the tensile strengths is taken
as design criteria, then they should correlate in some way the actual load which a
part is capable of carrying. Component parts such as crankshaft, gears, springs,
3
shafts, rotor, connecting rod, turbine, etc., are subjected to combined bending and
torsion stress as well as to stress repeated in cycles Slaymaker (1959) [1].
There are different types of stresses which a machine components encounters
during operation under dynamic loading. (i) The stresses which vary from zero to
maximum value are known as repeated stresses. (ii)The stresses which vary from a
certain minimum value to a certain maximum value of same nature are known as
fluctuating stresses. (iii)The stresses which vary from a minimum value to a
maximum of the opposite nature (i.e. from a certain minimum compressive to a
certain maximum tensile or from a minimum tensile to a maximum compressive) are
known as alternating stresses.
Ductile material include majority of the metals and polymers. Ductile
materials have same tensile strength as compressive strength and are not as
susceptible to stress raisers as are brittle materials. Two popular theories of yield
criteria are presented: the maximum shear stress theory and the distortion energy
theory.
Maximum-shear-stress theory was first proposed by Coulomb (1773) but was
independently discovered by Tresca (1868) is therefore often called the Tresca yield
criterion. This theory states that a part subjected to any combination of loads will fail
by yielding or fracture whenever the maximum shear stress exceeds a critical value.
The critical value can be determined from standard uniaxial tension tests.
Maximum distortion energy criterion or Von Mises criterion, states that for a
given structural material is safe as long as the maximum value of distortion energy
per unit volume in that material remains smaller than distortion energy per unit
volume required to cause yield in a tensile-test specified of the same material. It is
experimentally deduced that distortion energy theory gives best results for ductile
material which is subjected to combined loading.
Factor of safety may be defined as a figure used in structural applications that
provides a design margin over the theoretical design capacity. It is also known as the
4
safety factor, it allows for uncertainty in the design process, such as calculations,
strength of materials, quality and duty. To ensure good service from a part the
condition to which it is subjected should be less severe than the condition which
would cause it to fail. Failure occurs when factor of safety becomes less than unity.
High stress is generally associated with failure; hence the common conception of
factor of safety is ratio of maximum stress to working or design stress.
In fatigue failure most of the failures occurs due to time-varying load rather
to static loads. These failures typically occur at stress levels which are significantly
lower than the yield strength of the materials. Fluctuating load induce fluctuating or
cyclic stress that often result in failure by means of cumulative damage. Fatigue
failure is the single largest cause in metals estimated to be 90% of all metallic
failures as stated by Bernard (2004) [2]. Fatigue failure are catastrophic and
insidious, occurring suddenly and often without warning. Fatigue failure is
applicable on both microscopic and macroscopic scales. Fatigue is a complex
phenomenon. It is crack propagation, initially on a micro scale and then extremely
rapid as the fatigue crack reaches a critical length. Fatigue is concerned whenever
cyclic stress are present. Through experiments it has been concluded that fatigue
crack generally begins at the surface and propagate through the bulk, unless large
subsurface flaws or stress raiser exits in the substrate.
The fatigue or endurance limit of a material is defined as the maximum
amplitude of completely reversed bending stress that the standard specimen can
withstand for an unlimited number of cycles (106 -107) without fatigue failure or
appearance of crack. For loading other than reversed bending load it known as
endurance strength.
Fatigue life is defined as the number of stress cycles that the standard
specimen can complete during the test before the appearance of the first fatigue
crack. Based on the number of stress cycle that the part is expected to undergo in its
lifetime, fatigue is classified as either low cycle fatigue which is generally below 103
5
stress cycles and high cycle fatigue which is generally above 103 but less than 106
cycles Khurmi and Gupta (2009) [3],Bhandari (2010) [4]. For determination of life
of a component, stress-life approach has been employed. This method is the oldest
one and used for high cycle fatigue where assembly is expected to last for more than
about 103 cycles of stress. In this method stress in the system are determined and
these stress value are plotted on S-N diagram also called Wohler diagram in order to
get life of a machine component corresponding to failure stress in the system. It is
best when the load is predictable and consistent over the life of the part as in Norton
(2003) [5]. It is a stress-based model, which seeks to determine a fatigue strength or
endurance limit for the material so that the cyclic stress can be kept below that level
and failure can be avoided for the required number of cycles. The part is then
designed based on the fatigue strength or endurance limit of the material and factor
of safety.
1.4 Overview of Available Literature
1.4.1 History
The present day requirement for ever-increasing reliability in the field of
rotor dynamics is now more important than before and continues to grow constantly.
Advances are continually being made in this area, due to the consistent demand from
the power-generation and transportation industries. Because of progress made in
engineering and materials science, rotatory machinery is becoming faster and lighter,
as well as being required to run for longer period of time. All of these factors mean
that the direction, location and analysis of faults play a vital role in the field of rotor
dynamics.
The rotor-bearing system of modern rotating machines constitutes a complex
dynamic system. The modelling of rotors and their associated support structures has
been developed to a high degree of sophistication over the past twenty years
especially by the use of finite element analysis. The challenging nature of rotor
dynamics problems have attracted many scientists and engineers whose
6
investigations have contributed to the impressive progress in the study of rotating
systems. With the advancement in high-speed machinery and increases in their
power/weight ratio, the determinations of the rotor dynamic characteristics through
reliable mathematical models gains prime importance. The advancement in modern
instrumentation and computational capabilities has helped in implementing
simulation techniques of these complex models. Modern machinery is bound to fulfil
increasing demands concerning durability as well as safety requirements.
Jeffcott (1999) [6] provided a very basic model of a rotor. Initially, he made
three assumptions: (i) No damping is associated with the rotor, (ii) Axially
Symmetric rotor, and (iii) The rotor carries a point mass. Later, the model was
expanded to take care of damping. Although the Jeffcott rotor model is an
oversimplification of real-world rotors, it retains some basic characteristics and
allows us to gain a qualitative insight into important phenomena typical of rotor
dynamics, while being much simpler than more realistic models.
Nelson and McVaugh (1976) [7] written extensively history of rotor
dynamics and most of the work is based on Finite Element Methods. There are many
software packages that are capable of solving the rotor dynamic system of equations.
Rotor dynamic specific codes are more versatile for design purposes. These codes
make it easy to add bearing coefficients, side loads, and many other items.
To reduce the vibration problems in the high strength heavy rotors, researcher
focused to use the light weight and damped material rotors in spite of using the heavy
rotors. This make another revolution in the field of rotor dynamics, through this not
only the size of the heavy machine reduces as well as the new concept of using the
damped material like viscoelastic are predominant in rotor dynamics. Voigt model (2element model) was used to represent the shaft material constitutive relationship by
Zorzi and Nelson (1976) [8] who discretized shaft continuum using finite beam
elements to derive equations of motion and study dynamic behavior of rotor-shaft
systems. Researchers like Dutt and Nakra [14], Gunter [24], Genta [23], and Lalanne
7
and Ferraris [18], studied the stability of the rotor system with internal damping.
They obtained the results in form of Campbell diagrams and decay rate plots.
Unbalance response and the threshold spin speed known as Stability Limit of Spin
Speed were taken as indices of stability.
1.4.2 Recent Research in the Field of Fatigue Analysis
Gujar (2013) [9] calculated the stress induced in stepped shaft of an Inertia
dynamometer. System, forces, torque acting on a shaft are also taken into account.
Finite element method is employed for stress analysis. Stress concentration is
occurred at the stepped, keyways, shoulders, sharp corners etc. caused fatigue failure
of shaft. Due to stress concentration which arises due to various factor results in
lowering the endurance limit which has been calculated using Modified Goodman
Method. Factor of safety and theoretical number cycles for the shaft before failure is
estimated.
Elevator drive shaft has been analyzed for failure by Goksenli (2009) [10].
Failure occurred at the keyway of the shaft due to initiation of crack at the edge of
keyway. Forces and torque on the shaft are determined and stresses occurring at the
failure surface are calculated. Stress analysis is also carried out by using finite
element method (FEM) and the results are compared with the calculated values.
Endurance limit and fatigue safety factor is calculated using Goodman diagram,
fatigue cycle of the shaft is estimated through Wohler diagram. Fracture failure
results due to low radius of curvature.
Mahesh (2013) [11] studied the crankshaft for estimation of life through FEM
and for this purpose it was necessary to calculate dynamic load, stresses in the
system. Finite element analysis is performed on forged steel crankshaft of four stroke
engines to obtain the variation of stress magnitude at critical locations. The dynamic
force analysis is carried out analytically using MATLAB program, FE model in
ANSYS, and boundary conditions are applied according to the engine mounting
conditions. The analysis is carried out for different engine speeds and at different
8
ratio of fillet radius to diameter of crank pin which gives the critical location on the
crankshaft. Analysis has been done to calculate fatigue life using stress-cycle(S-N)
approach of crankshaft under complex loading conditions. Due to the repeated
bending and twisting, results in fatigue as the cracks form in the fillet area.
During operation of rotor the effect of axial torque effect on lateral dynamics
has to be taken into account. Therefore Zorzi and Nelson (1977) [25] simulated a
finite element model to study the effect of constant axial torque on the dynamics of
rotor bearing system. Formulating finite element equation of motion which includes
axial torque gives rise to an incremental torsional stiffness matrix. This matrix is
circulatory due to non-conservative nature of system.
A stress analysis method for web-core sandwich beams was carried out by
Jani et al. [12]. The beam is a transverse cut from a sandwich plate. The structural
analysis of the homogenized beam follows thick face plate kinematics giving the
deflection, bending moment and shear force distributions. Then the normal stress
components can be calculated accurately by reconsidering the periodic structure of
the beam. An analytical elastic–plastic stress analysis was done on the metal-matrix
composite by Onur and Unrau [13].
A time domain solution is achieved by Roy et al. (2009) [15] using two
different
displacement
modelling
techniques
namely
ATF
(Augmenting
Thermodynamics Fields) and ADF (Anelastic Displacement Fields) which uses
certain viscoelastic parameters. Constitutive relationship are used to obtain equation
of motion of the continuum after discretizing it with finite beam elements. Study of
dynamic behaviour of composite beams and rotors can be done utilizing this method.
Unbalance such as shaft bow or mass unbalance etc. in the rotor results in the
whirling of rotor shaft and due to the whirling orbits are forms which can be
elliptical or circular in nature. Analysis for stresses in the rotor has been done by
Muszynska [16] when simultaneous backward and forward whirl occurs at different
section of the rotor under influence of mass imbalance and shaft bow imbalance
9
located in two axial plane. Zhenxing et al. (2013) [20] developed a model that can be
applied to study the influence of thermal effect on flexible multi-body dynamics. A
concept of absolute nodal coordinate formulation (ANCF) was used to analyze the
coupled thermal-structure using Euler-Bernoulli beam model. Space structures such
as satellite, etc. are subjected to mechanical, thermal and dynamic loads. Therefore
Narashimha et al. (2010) [21] simultaneously solve heat transfer and structural
problem by combining structural displacement and distribution of temperature and
with this temperature distribution thermal moment was evaluated. The stiffness of
the system changes when it is thermally loaded due to temperature difference and
hence this effect is included by Robert D Cook et al. (2002) [22] using finite element
method.
1.5 Analysis Objective
In the era of advancement where every care has been taken off to reduce the
defects for smooth operation of machinery with low maintenance and for longer life
without failure, but there always exists some difference between the actual
manufactured machine component and its prototype i.e. manufacturing defect always
occurs in each and every product be it due to machining process or may be because
of human error etc. In rotor system these inaccuracies may include shaft bow,
bearing defects, defects in disk or blades, etc. During assembly or installation of the
components of rotor there may occur misalignment between shaft and bearing axis or
disc is not line with shaft axis i.e. eccentricity, such type of errors results in
unbalance of rotor and the vibrations in the system increases. Eccentricity might
result in the overheating at the bearing ends. During operation at high speed these
errors cannot be neglected as stress are setup at different section of the rotor which
causes instability in the rotor. These stress are dynamic and cyclic in nature which
may lead to early failure, so are matter of concern.
10
It becomes necessary to find stresses arises from various factors as
eccentricity, irregularity in material surface, forces, torque, heat generation or
eccentric loading etc. Design point must be the highly stressed location in the rotor
and analysis must be based on this design point. The nature of stresses plays an
important role as in dynamic systems cyclic stresses occurs which are different from
static stress and has more detrimental effect. These cyclic stresses causes fatigue in
the system which results in catastrophic failure. Thus there is a need to analyze the
system for fatigue failure too as it helps to find life of component in terms of
revolutions. Catastrophic failure can be prevented by analyzing system for fatigue
failure and thus loss which could have been done by this failure can also be saved.
Fatigue analysis helps to decide the time for maintenance and therefore ensuring
longer life with smoother operation.
1.6 Thesis Outline
Chapter 1 gives a brief introduction of rotor dynamics which is followed by a
brief overview of the development of dynamics of rotor shaft systems. It discusses
the various rotor models and stability study of systems under various internal and
external effects.
Chapter 2 develops the equation of motion for an Euler Bernoulli beam. It
discusses the finite element modelling of the system which is later used in the
modelling of rotor. It discuss about the bending stresses in beam under dynamic
static condition for both undamped and damped beam.
Chapter 3 first develops finite element formulation of the internally damped
rotor shaft system and then validates the finite element code for the rotor shaft
system with the published results which available in the literature. Further this study
is for modelling single disc rotor to find cyclic stresses in the system and to finally
fatigue analysis of the rotor shaft.
Finally, in chapter 4, the conclusions, future scope of the work and references
along with appendix are included.
11
Chapter 2
DYNAMICS OF BEAMS
This chapter gives the detailed study of beam. Equation of motion is derived
for elastic beam using Euler Bernoulli Beam theory and afterward damping is
incorporated through ATF (Augmenting Thermodynamic Field) technique. Initially
an undamped beam model is taken and mode shape and frequency response are
plotted. Further, equations of motion are used to obtain state space form to find time
domain solution. Normal strain and bending stress are determined with the help of
these time domain response. Similar study is done for damped system as for
undamped system.
2.1 Introduction
A beam can be defined as a structure having one of its dimensions much
larger than the other two. The axis of the beams is defined along that longer
dimension and cross section normal to the axis is assumed to smoothly vary along
the span or length of the beam. Engineering structures often consists of an assembly
or grid of beams with cross sections having shapes such as I‟s or T‟s. A large number
of machine parts also are beam like structures: lever arms, shaft, turbines, ship
propeller and quite a lot of aeronautical structures such as wings.
The solid mechanics theory of beams, commonly referred to as „beam theory‟
which plays an important role in structural analysis because it provides the designer
with simple tool to analyze innumerable structures. Although more sophisticated
tools, such as the finite element method etc., are now widely available for the stress
analysis of complex structures or beams models, are often used as pre design stage
because they provide valuable insight into the behaviour of structures. Such
calculations are also quite useful for validating computational solution.
12
Several beam theories have been developed based on various assumptions,
and lead to different levels of accuracy. One of the simplest and most useful of these
theories was first given by Euler and Bernoulli and is commonly called as EulerBernoulli beam theory. A fundamental assumption of this theory is that the cross
section of the beam is infinitely rigid in its own plane, i.e. no deformation occurs in
the plane of the cross section. Consequently, in the plane displacement field can be
represented by two rigid body translation and one body rotation as in Bauchau and
Craig [17]. Two additional assumption deals only with in-plane displacement of the
cross-section: during deformation, the cross section is assumed to remain plane and
normal to the deformation axis of the beam.
Another beam theory developed is Timoshenko beam theory and it was
developed Stephen Timoshenko in the early 20th century. The modal takes into
account shear deformation and rotational inertia effects, making it suitable for
describing the behavior of short beams, sandwich beams or beams subjected to highfrequency excitation when the wavelength approaches the thickness of the beam.
2.2 Equation of Motion of Undamped Euler Bernoulli Beam
A beam element is shown in fig. 2.1 which subjected to moment and shear
force along with the external force. The equation of motion has been derived using
Euler-Bernoulli beam theory and by applying force balance and moment balance
along the axis of the beam.
F ( x, t )
M
M
V
dx
V
M
dx
x
V
dx
x
Figure 2.1: Schematic diagram of Euler Bernoulli Beam Section
13
Now applying force equilibrium on beam section
F
x
=0
V 
2 y

dx   V  Fdx   A 2 dx
V
x
t


(2.1)
V
2 y
dx  Fdx   A 2 dx
x
t
Simplifying and rearranging the above equation
V
2 y
 A
 F
x
t2
(2.2)
Now applying moment equilibrium on beam
M
x
=0
M 
dx

d x   M  Vd x  F d x  0
M 
x 
2

(2.3)
M
dx
d x  F d x  Vd x  0
x
2
(2.4)
In above equation the term
V=
dx
is negligible. Simplifying eq. (2.4) we get
2
M
x
(2.5)
According to beam theory, the bending moment can be written as
2 y
M =-E I 2
x
(2.6)
Then
3 y
V = -E I 3
x
(2.7)
Differentiating eq. (2.7) w.r.t x
14
V
4 y
=-E I 4
x
x
(2.8)
Using eq.(2.8) in eq.(2.2) we get
 4y
 2y
-E I 4 -  A 2 = - F
x
t
(2.9)
Rearranging eq.(9) we get
EI
 4y
 2y
+

A
=F
x4
t2
(2.10)
Thus, governing equation for free vibration of Euler Bernoulli beam
 2y E I  4y

=0
 t 2 A  x 4
2.3 Elastic Beam Finite Element Formulation
The Galerkin residual method is employed in the preceding equation to
obtain the finite element form.
le
T
0 [xy ]  EIy   Ay  dx  0
(2.11)
With EI constant, and (xy )i are the usual cubic shape functions, two integration by
parts of the fourth-derivative term in eq. (2.11) yields
le
le
le
T
T
T
xy  ydx  xy  ydx   xy  y  xy  y
 
 
 
  
 0
0
0

T

(2.12)
Substituting eq. (2.6), (2.7) and eq. (2.12) in eq. (2.11) gives
le
  T

 xy

0

EIy   A xy  y dx  0   xy  V  xy  M 
T
T


T
le
0
(2.13)
The term V and M becomes a part of load vector  F 
y  [xy ]{q} and y  [B]{q} where [B]  [xy ],
15
(2.14)
{q}  [v1 1 v2  2 ]T
and
y  [xy ]{q}
Substituting eq. (2.14) in eq. (2.13) and assembling elements, we obtain
[k ]{q} j  [m]{q} j  {F}
(2.15)
With the element matrix expanded to „structural size‟, eq. (2.15) is recognized as the
standard equation of motion as
[K]{q} j  [M]{q} j  {F}
(2.16)
where [K] and [M] are respectively the global stiffness and global mass matrix.
2.4 Incorporation of Damping Matrix
The damping is incorporated through a modelling technique namely (ATF)
Augmenting Thermodynamic Fields as given in Roy et al. [15]. The mechanical and
thermal stress, called as affinity are written as
  E  
(2.17)
    
 is the value of  at equilibrium and is obtained as
    0


(2.18)

  b(   )
(2.19)
Using eq. (2.18) in eq. (2.19) and rearranging we get
  b 
b

 (b  D) 

b

(2.20)

(2.21)
Rearranging eq. (2.21) we get
16

b

(b  D)

(2.22)
Using eq. (2.22) in eq. (2.17), neglecting the term
D
, as b,  0 , the constitutive
b
relationship is rewritten as
  (ao  a1D)  E()
where ao  E 
(2.23)
2
E
and a1 

b
Operator based equation is achieved to obtain stress. This operator based approach is
further employed as
Mq+K()q=0
(2.24)
l
E () I e
where K () 
[xy ]T [xy ]dx
3

le 0
Solving eq. (2.24) we get equation of motion for damp system, where damping
matrix is derived from modifying stiffness matrix, is given as
C  
EI le
Ble3
 
 xy
0 
T
xy  dx
(2.25a)
I   2  le  T  
  xy
  dx
 K   l 3  E     xy
0

e 
(2.25b)
Mechanical strain and stress in finite element form is given by eq. (2.26)
 
x  y xy
x
T
q ,
(2.26a)
T
  {q}
 E y xy
(2.26b)
17
2.5 Numerical Problem for Beam
A cantilever beam of rectangular cross section is taken and is discretized into
finite elements. Each element has two nodes and four degree of freedom. This beam
is symmetric about x-axis and loaded harmonically at its free end. A finite element
analysis based on Euler-Bernoulli beam theory has been done to determine the mode
shape and frequency response. For this purpose a matlab code is developed to find
the eigenvalues, eigenvectors and response pattern. Then bending stress for
undamped and damped system is determined using time domain response. The
schematic diagram in Fig. 2.2 shows a cantilever beam.
F
L
Figure 2.2: Schematic Diagram of Cantilever Beam
Data for the mild steel cantilever beam is density (  )  7800kg/m3 , modulus
of Elasticity ( E )  2  10e11 Pa, amplitude of applied force ( F )  1N, beam
length ( L)  0.6m, Width= 0.04m, Depth=0.07m
Fig. 2.3 shows first three mode shapes plotted from Eigen vector of
undamped beam system. A normal mode of a vibrating or oscillating system is a
pattern of motion in which all parts of the system move sinusoidally with the same
frequency and with a fixed phase relation.
18
Mode Shape
2
Ist modeshape
IInd modeshape
1.5
IIIrd modeshape
Displacement
1
0.5
0
-0.5
-1
-1.5
-2
2
4
6
8
10
No.of nodes
12
14
16
18
Figure 2.3: Mode shapes of Cantilever Beam
The frequencies of the normal modes of a system are known as its natural
frequencies or resonant frequencies and fig. 2.4 shows frequency response of the free
end of the beam, when an external cosine function is applied. It is obtained by
calculating displacement (response) from equation of motion for different frequency.
-4
10
-6
Response Amplitue(m)
10
-8
10
-10
10
-12
10
-14
10
-16
10
0
0.5
1
1.5
Frequency (Hz)
2
2.5
3
Figure 2.4: Frequency Response at Free End of Beam
19
4
x 10
Now time dependent response for beam tip is obtained for undamped system
by formation of the state space form. Fig.2.5 and Fig.2.6 gives the displacement and
stress variation with respect to time for undamped beam and the nature of both the
curves are same indicating the steady state fluctuation.
6
1
x 10
0.8
0.6
Displacement(m)
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
0
0.2
0.4
0.6
0.8
1
Time(s)
1.2
1.4
1.6
1.8
2
Figure 2.5: Response-Time Plot for Free End of Beam
17
4
x 10
3
2
 (Pa)
1
0
-1
-2
-3
-4
0
0.2
0.4
0.6
0.8
1
Time(s)
1.2
1.4
1.6
1.8
Figure 2.6: Stress-Time Plot for Free End of Beam
20
2
Fig. 2.7 shows the stress vs. strain. The plot is a straight line between stress and
strain for the cantilever beam and is similar to elastic stress-strain plot.
17
4
x 10
3
2
 (Pa)
1
0
-1
-2
-3
-4
-2
-1.5
-1
-0.5
0

0.5
1
1.5
2
6
x 10
Figure 2.7: Stress-Strain Plot Considering Undamped Beam
Study of undamped system is further extended to damped system and
damping is achieved in the system as given section 2.4, where b,  , and δ are ATF
parameters and their values for steel are given in Roy et al. [15] are b = 1.807e5;
 =1.807e5; δ=3.186e7. The response and stress plot for damped system shows the
constant amplitude after some time, this shows that the system is under steady state
excitation.
21
-6
1
x 10
0.8
0.6
Displacement (m)
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
0
0.2
0.4
0.6
0.8
1
Time(s)
1.2
1.4
1.6
1.8
2
Figure 2.8: Response-Time Plot at Free End of Damped Beam
4
1.5
x 10
1
 (Pa)
0.5
0
-0.5
-1
-1.5
0
0.2
0.4
0.6
0.8
1
1.2
Times(s)
1.4
1.6
1.8
2
Figure 2.9: Stress-Time Plot at Free End of Damped Beam
22
4
1.5
x 10
1
 (Pa)
0.5
0
-0.5
-1
-1.5
-2.5
-2
-1.5
-1
-0.5
0

0.5
1
1.5
2
2.5
-8
x 10
Figure 2.10: Stress-Strain Plot Considering Damped Beam
In undamped system the stress-strain graph in fig 2.7 shows a straight line
thus follows hook‟s law for elastic material. Whereas in damped system, due to
energy dissipation stress-strain plot is elliptical nature as shown in fig. 2.10.
Therefore results for damped beam or viscoelastic beam are as expected and are in
accordance with the theory.
23
Chapter 3
FATIGUE ANALYSIS OF ROTORS
In this section mathematical formulation of finite element model for rotor is
done and the gyroscopic effect due to disc and inertia due to rotation is also taken
into account. Formulation for fatigue analysis is also given in this section. The
motion is considered in two dimensional plane. The work starts with the validation
of finite element code with the published results and further this work is extended for
fatigue analysis of single disc rotor system.
3.1
Introduction
A rotor is a body suspended through a set of cylindrical hinges or bearings
that allows it to rotate freely about an axis fixed in space. Jeffcott rotor can be stated
as the simplest example of rotor which consists of a point mass attached to the
massless shaft or the rotor is considered as single mass in the form of a point mass, a
rigid disc or a long rigid shaft. Rotors fall into two groups. First one is where the
rotor is rigid and does not deflect up to and including the operating speed. The other
group comprises flexible rotors that “bow” up to the operating speed.
Rotating shaft are employed in industrial machine such as steam and gas
turbines, turbo generators, internal combustion engines, reciprocating and centrifugal
compressor or automobile machinery for power transmission. To overcome the ever
increasing demand for power and high speed transportation, the rotors of these
machines are made extremely flexible, which makes the study of vibratory motion an
essential part of design.
24
3.2 Mathematical Formulation
In this section the mathematical modelling of viscoelastic rotor shaft is
represented. The finite element model of the viscoelastic rotor shaft system is based
on the Euler-Bernoulli beam theory. The equation of motion is obtained from the
constitutive relation where the damped shaft element is assumed to behave as Voigt
model i.e., combination of a spring and dashpot in parallel
3.2.1 Finite Element Model
The displaced position of the shaft cross section is shown by fig. 3.1 as in
[26]. The displacement of the shaft center along Y and Z direction is indicated by (v,
w) and an element of differential radial thickness dr at a distance r (where r varies
from 0 to ro ) subtending an angle d (t ) where  is the spin speed in rad/sec and
 t varies from 0 to 2 at any instant of time
' t ' . Due to transverse vibration the
shaft is under two types of rotation simultaneously, i.e., spin and whirl
whirl speed.
Figure 3.1: Displaced Position of the Shaft Cross - Section
25
 is the
The dynamic longitudinal stress and strain induced in the infinitesimal area are  x
and
x
respectively. The expression of  x and
x
at an instant of time are given
as Zorzi and Nelson [8].
 x  E   v  ,  x   r cos      t 
2
 R ( x, t )
x
(3.1)
2
Following [8] the bending moments at any instant of time about the y and z-axes are
expressed as
2  ro

 
M z z     v  r cos  t  x  r d r  d (  t )
0 0
2  ro

 
(3.2)
M y y    w  r sin  t  x  r d r  d (  t )
0 0
After following Zorzi and Nelson [8] and utilizing equation (3.2), the governing
differential equation for one shaft element is given as:
 M    M q    K    G q   K      K    K    K q   F 
T
R
V
B
B
V
C
T
A
(3.3)
In the preceding equation  M T  ,  M R  , G  ,  K B  ,  KC  ,  KT  ,  K A  are
translational mass matrix, rotary inertia matrix, gyroscopic matrix, bending stiffness
matrix, skew symmetric circulatory matrix, stiffness matrix due to externally applied
 81 is the nodal
torque , axial stiffness matrix due to thermal load respectively. q
displacement vector and V is coefficient of viscous damping. After referring to [25]
and [22],  KT  ,  K A  are incorporated. It is due to externally applied axial torque
and axial elongation for temperature difference between shaft and surrounding. The
26
matrix expression used in eq. (3.3) are given below following [27]. The elements in
the matrices are given in detail in the appendix.
le
le
 M T     A   x    x  dx,
 M R     I    x     x  dx
le


T
G    2 I    x   0 1   x  dx,
1 0
0
le
T
 K B    EI    x      x   dx,
0
le
T  0
1
  
 KT    TA    x  

 1 0   x  dx,
0


le
T
 0 1
 KC    EI  ( x) 
  ( x) dx
0
1 0
T
0
0
le
 K A   PA     x     x  dx ,
T
0
which,
the
xy  x


 x   



 


 
0
Hermite


,
zx x 

0

shape
T
q  v1

T
1 w1 1 v2  2 w2 2 , in
function
matrix  ( x) , is
given
by
with subscripts in the elements showing the respective
 
planes.
The axial load due to thermal expansion is given as PA  E T To  where 
is coefficient of thermal expansion, T is the average temperature over the crosssection and To is the reference temperature. TA is the magnitude of applied external
axial torque.  is the mass density. A is area of cross-section and I is moment of
inertia of shaft. E is the Young‟s Modulus
3.2.2 Fatigue Analysis
Axial strain and strain rate of the viscoelastic rotor shaft system due to transverse
loading in terms of shape function and nodal coordinate can be expressed as
x  y  ( x)
 q ,
T
x  y  ( x)
27
 q
T
(3.4)
where, y is the lateral distance from the neutral axis of shaft.
Bending stress develop in the axial direction is given as

    ( x) {q}
 x  E y   ( x) {q}
T
T
v
(3.5)
According to beam theory, bending moment and shear force are written as
M x
M

, and V 
I
y
x
(3.6)
After utilizing the preceding equation, the shear force for viscoelastic material is
given as



 ( x) {q}    ( x) {q}
V  EI 
T
T
v
(3.7)
The shaft having circular cross-section, maximum shear stress is given by
 max  43  avg
(3.8)
where average shear stress,  avg 
V
A
Endurance strength of the rotor shaft is determined with help of Goodman equation,
so for this it is necessary to find mean and amplitude values of bending and shear
stress as
m 
 max  min
2
Subscript
,
a 
 max  min
2
(3.9)
m stands for mean or average value and a is for amplitude or
variable value. Similarly mean shear stress  m and amplitude shear stress  a can be
calculated from the shear stress fluctuation. After calculating all stresses from the
above relations, equivalent mean stress and equivalent amplitude stress are
calculated from distortion energy theory as in reference [5].
28
( eq )m 
 m 
2
 3  m  , ( eq )a 
2
 a 
2
 3  a 
2
(3.10)
And now endurance strength  e  can be determined from Goodman equation as
given in eq. (3.11) with some assumed value of factor of safety referring to [10].
 eq a  eq m
e



u
1
(3.11)
Fs
For fatigue analysis the primary condition is to know the value of stress
which results in failure and for safe operation of the system the operating stresses
must be kept below this failure stress ( f ) . Fig.3.2 shows the Goodman diagram
whose ordinate and abscissa indicate the amplitude and mean stress. Goodman line is
drawn for interpolation purpose, which is between the ultimate and endurance stress
point. The operating stress point X is obtained from the coordinate values of ( eq )m
and ( eq )a . For safe design the point X should be below the Goodman line. The
failure stress point is achieved by line joining between ultimate stress point to point
operating stress point X and extending it to ordinate. Thus from this diagram failure
stress can be interpolated using similar triangles as given in [11]. Interpolated
relation for failure stress is given as follows.
f 
  u    eq a
  u     eq m
(3.12)
29
a
e
f
( eq ) a
X
u
( eq ) m
m
Figure 3.2: Goodman Diagram for Interpolating Endurance Strength
S-N curve also known as Wohler diagram for high cycle fatigue, is shown in fig.3.3,
which is the stress vs. operating cycle diagram in logarithmic scale. Life can be
determined corresponding to the failure stress  f  through Wohler diagram again
by using similarity of triangles and the interpolated relation is
Life N f  10
p
(3.13)
where P  3   6  3
 
log  0.9 u   log  f
log  0.9 u   log  e 
log10 
log100.9 u 
log10  f 
log10 e 
3
6
log10 N f
Figure 3.3: Stress-Cycle(S-N) Diagram
30
log10 N
3.3 Numerical Problem
3.3.1 Validation of Finite Element Code
The schematic diagram of three disc rotor shaft system is shown in fig. 3.4 as
given in Lalanne and Ferraris [18]. A rotating shaft system with simply supported
ends is considered without axial load and axial torque i.e. PA  0, TA  0 . Few
numerical results are obtained for validation purpose.
Figure 3.4: Schematic Diagram of Lalanne Rotor
The material properties of the steel rotor are shown in table 3.1 and table 3.2
shows the dimensions and mass unbalance of these three discs. Following Lalanne
and Ferraris [18], initially the discs are placed at nodes 3, 6 and 11. The unbalance is
considered at disc-2 and it is 200 gm. mm
Material
Density
(kg/m3)
Young’s
Modulus
(GPa)
Length
(m)
Diameter
(m)
Coefficient
of Viscous
Damping
(s)
Mild steel
7800
200
1.3
0.2
0
Table 3.1: Rotor Material and its Properties
31
Disc
Diameter
(m)
Thickness
(m)
Mass Unbalance
1
0.24
0.05
0
2
0.40
0.05
2  10-4
3
0.40
0.06
0
(kg-m)
Table 3.2: Disc Dimensions and Unbalance for Three Disc
The Campbell diagram is plotted in fig. 3.5 which give the critical
frequencies of the system. It shows the forward and backward whirl lines and gives
the critical frequency when whirl lines are cut by line inclined at 45°, known as
synchronous whirl line (SWL). Frequency response plot is shown in fig. 3.6. Peaks
in the graph shows the resonance in the system at corresponding frequency i.e.
system is unstable at these frequency because frequency of the forcing function
coincides with the natural frequency of the system and this condition is called
resonance. When the system is under resonance condition large amplitude vibrations
takes which may lead to failure and thus system must be avoided to run at critical
frequencies to avoid failure.
32
1200
Natural frequency(Hz)
1000
800
600
400
200
0
0
0.5
1
1.5
2
2.5
Excitation Frequency(rpm)
3
3.5
4
x 10
Figure 3.5: Campbell Diagram
-2
10
L&F(1998)
Present
-4
Amplitude (m)
10
-6
10
-8
10
-10
10
0
0.5
1
1.5
Rotational Speed (rpm)
2
2.5
4
x 10
Figure 3.6: Mass Unbalance Response
The eigenvalues and eigenvectors obtained to find out the stability of the
system. Close match between the computed and reported results in Lalanne [18]
validate the correctness of the code as shown by table below
33
S. No.
Frequency
Present Result
L&F (1998)
1.
F1
54
55.41
2.
F2
68
67.20
3.
F3
154
157.9
4.
F4
196
193.6
5.
F5
232
249.9
6.
F6
415
407.5
7.
F7
444
446.7
8.
F8
599
714.9
9.
F9
751
622.7
10.
F10
1089
1076
Table 3.3: Frequency Validation
3.3.2 Single Disc Rotor
A rotor shaft system having AISI/SAE 1050 cold drawn steel shaft with
simply supported bearing ends with disc at the center is considered which is
subjected to thermal load due to temperature difference with the environment and an
external axial torque is applied. The bearing is modelled as a flexible damped
support having stiffness and damping coefficients (k yy , cyy ) and (k zz , czz ) in the x-y and
x-z planes respectively. The schematic diagram for rotor shaft system is shown in fig.
3.7.
34
tD
y
l
L
C
A
DR
kyy
dyy
B
kyy
DD
x
dyy
z
Figure 3.7: Schematic Diagram of Single Disc Rotor
Data for the system are k yy  kzz  7e7 N/m, cyy  czz  7e2 N/m, Length of
rotor is L  1.0 m, diameter of rotor is DR  0.07 m, diameter and thickness of disc
are DD  0.4 m, t D  0.05 m respectively, and the unbalance of the disc, U = 200
gm-mm. Material property are as follows E  2e11Pa,   7850kg/m3 , V  2e-4 sec,
Ultimate strength of
steel ( u )  690e6 Pa, Coefficient of thermal expansion
( )  13e-6 K .
°
-1
An internally damped rotor shaft with disc at the centre is taken for fatigue
analysis. The shaft is discretized into ten elements with disc placed at centre node
and the unbalance is considered at disc and it is 2 104 kg-m.
A MATLAB code is written based on the FE formulation for numerical
simulation. The eigen values of the system are determined and the maximum real
part of all eigen values are plotted against spin speed, this graph is called decay rate
plot as shown in fig. 3.8. It is seen from the plot that the maximum real part increases
from negative to positive with spin speed. After certain spin speed it touches zero
line and system become unstable. The corresponding spin speed is called stability
limit of spin speed.
35
40
30
Real Part
20
Unstable
Region
10
0
-10
-20
0
1000
2000
3000
4000
5000
6000
7000
8000
9000
10000
 (rpm)
Figure 3.8: Decay Rate Plot
Stability limit of spin speed (SLS) of the rotor shaft system is drawn for
various axial torque and various values of temperature difference. The SLS is
obtained from the eigenvalues, when all the eigenvalues have negative real part, the
system is stable. When the real part touches the zero line, the corresponding spin
speed is called SLS. Fig. (3.9 a) shows the SLS vs axial torque, when the
temperature is remain constant ( T  30°K). The SLS for various values of
temperature difference is plotted in fig. (3.9 b) for a constant value of axial torque
( TA  1000N-m). It can be seen that SLS is almost innocent with the increase of
torque but decrease with the increase in temperature. With increase in temperature,
reduction in system effective stiffness, results to decrease the SLS. Therefore system
stresses are found out at spin speed of 3100 rpm with axial torque equal to 1000N-m
and at temperature difference of 30° K.
36
At  T=30K
3104.375
3104.3745
SLS (rpm)
3104.374
3104.3735
3104.373
3104.3725
3104.372
3104.3715
1000
1100
1200
1300
1400
1500 1600
TA (N-m)
1700
1800
1900
2000
44
46
48
50
(a)
At T A=1000N-m
3200
3100
3000
SLS (rpm)
2900
2800
2700
2600
2500
2400
30
32
34
36
38
40
42
T K
(b)
Figure 3.9: SLS for different Axial Torque and Temperature Difference Respectively
Fig. 3.10.(a, b, c) shows the time varying bending stress, shear stress and
their equivalent stress for various non-dimensional location denoted as (L*) of the
37
rotor shaft respectively where 0 and 1 on abscissa denote left hand side and right
hand side of shaft respectively.
5
0.5
x 10
0
 (Pa)
-0.5
-1
-1.5
-2
-2.5
t=0.0040s
t=0.0020s
-3
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
L*
(a)
4
1
x 10
0.8
0.6
0.4
 (Pa)
0.2
0
-0.2
-0.4
-0.6
t=0.0040s
t=0.0020s
-0.8
-1
0
0.1
0.2
0.3
0.4
0.5
L*
(b)
38
0.6
0.7
0.8
0.9
1
5
3
x 10
t = 0.0040s
t = 0.0020s
2.5
eq
2
1.5
1
0.5
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
L*
(c)
Figure 3.10: Variation of Bending Stress, Shear Stress and Equivalent Stress along the length of rotor
From the equivalent stress plot shown below it can be seen that rotor shaft is
highly stressed at its mid-point, hence center of the shaft length is taken as design
point for the further analysis.
Rotor shaft is designed on the basis of fatigue analysis. For this purpose
endurance strength, is calculated with Goodman relation for assumed value of factor
of safety Fs
 1.5, failure stress is also interpolated through Goodman diagram on the
basis of maximum stress develop at mid-point. Thereafter life corresponding to this
failure stress is evaluated through Wohler diagram. Life is defined as the number of
cycles that a mechanical component can complete before the appearance of the first
fatigue crack over its surface.
It can be seen from the fatigue results in Table 3.4 that failure stress  f
comes out to be less than the endurance strength  e hence design parameters of rotor
operation are safe as stress are within safe limit of operation.
39
Endurance Strength  e
Failure Stress  f
Life N f
(MPa)
(MPa)
(Cycles)
110
74
5.0848  106
Table 3.4: Fatigue Analysis Results
For steel if life for the system lies in the range 106 to 107 which can be taken as
infinite, as from the Wohler diagram in fig. 3.3 it can be deduced that when life
comes out to be greater than 106 cycles then it lies in the region of high cycle fatigue
and component has an infinite life. Therefore our results are in accordance with
given theory.
40
Chapter 4
CONCLUSION AND FUTURE SCOPE
4.1 Conclusion
A literature survey has been carried out to investigate the developments in
modelling for beam. It is observed that a huge amount of work has been done in
modelling different kind beams such as sandwich beam, etc. and most of them
involved the Finite Element Modelling. But it was felt that not much has been done
involving determination of stresses that are developed during operation of beams and
rotor.
Through this work a finite element modelling of beam is achieved and results
are validated for mode shapes of undamped Euler Bernoulli beam by formulating
equation of motion for beam. Stresses and strain due to transverse loading which
causes bending are determined from time domain solution. This has been done for
both undamped and damped model. Damped system is achieved by employing either
by augmenting thermodynamic fields (ATF) or anelastic displacement fields (ADF)
technique. The stress-strain curve for undamped and damped system are in
accordance with the supported theory and thus validates the finite element
formulation which is done in matlab.
Modelling of viscoelastic three disc rotor has also been done for validation
purpose and further this finite element formulation is utilized for fatigue analysis of
single disc rotor which subjected to thermal loading and external torque. In addition
to time varying bending stress, shear stress as well as equivalent stress are also
determined for various location of the shaft. The equivalent stress shows the centre
of the rotor shaft is subjected to maximum stress and indicates failure point or design
point. Fatigue analysis has been carried out considering design point and the results
shows that the design parameters are safe and rotor shaft has infinite life.
41
4.2) Future Scope
In this work stresses are function of time response i.e. time domain response
has been taken to determine stresses in the system. In future this work can be carried
out in modal plane which means modal response can be used instead of time
response to determine modal distribution of stresses. We have found out most
stressed location as a function of beam length similarly this location can be deduced
as a function of mode when analysis is based on modal plane i.e. we will find the
highly stressed mode and aiming to control the value of stresses in that particular
mode through control theory.
This work has lot of scope in future as using this model more realistic
conditions of operation can be achieved by incorporating the effect of stress raiser
such as notch, fillet, and keyways, etc. in both beams and rotors. Disc mounted on
the shaft is considered as lumped mass and but in future, modelling can be done by
taking into account the effect turbine blades which were earlier considered as disc on
the shaft. If rotor is considered as turbine then turbulence of steam or water and its
effect on fatigue life could be taken into account and there are numerous such
conditions and effects which could be considered for future work.
The modelling of present work is based on Euler Bernoulli beam theory
which neglects shear deformation of the cross-section thus this model is applicable
where slenderness ratio (L/DR) is near to 15 or long rotor shaft. To make this model
suitable for short beams Timoshenko Beam theory can be applied which incorporates
shear deformation and thus short beams as well as long beams can be analysed
through this model.
As there is a great demand of high strength to weight ratio, therefore stress
analysis of fibre reinforced composite shaft made of either unidirectional fibre or
bidirectional fibres. Manufacturing of rotor using composites becomes economical
and is resistant to atmospheric conditions.
42
4.3) References
1] Slaymaker.R.R., “Mechanical Design and Analysis‟‟, John Willey & Sons, 1959.
2] Bernard J. Hamrock. and Jacobson Bo. and Steven R.Schmid., “Fundamentals of
Machine
Element‟‟, McGraw-Hill Higher Education, 2004.
3] Khurmi R.S. and Gupta J.K., “Machine Design‟‟, S. Chand Publishers, 2009.
4] Bhandari V.B., “Design of Machine Elements‟‟, Tata McGraw-Hill Education,
2010.
5] Norton R.L., “Machine Design-An Integrated Approach‟‟, Pearson Education,
2003.
6] Jeffcott, H., “The lateral vibration of loaded shafts in the neighbourhood of a
whirling speed-the effect of want of balance”, Phil. Mag., 37 (6), pp. 304-314, 1919.
7] Nelson H.D. and McVaugh J.N., “The Dynamics of Rotor-Bearing System using
Finite Elements”, Journal of Engineering for Industry, vol. 98, pp. 593-600, 1976.
8] Zorzi E.S. and Nelson H.D., “Finite Element Simulation of Rotor-Bearing
Systems with Internal Damping”, Journal of Engineering for Power, Transactions of
the ASME, vol. 99, pp. 71-76, 1977.
9] Gujar R.A. and Bhaskar S.V., “Shaft Design under Fatigue Loading by Using
Modified Goodman Method”, International Journal of Engineering Research and
Applications, vol. 3(4), pp.1061-1066, 2013.
10] Goksenli A. and Eryurek I.B., “Failure analysis of an elevator drive shaft”,
Journal of Engineering Failure Analysis, vol. 16, pp. 1011–1019, 2009.
11] Raotole L. Mahesh, Sadaphale D. B., Chaudhari J. R., “Prediction of Fatigue
Life of Crank Shaft using S-N Approach”, International Journal of Emerging
Technology and Advanced Engineering, Volume 3, Issue 2, February 2013.
12] Jani Romanoff. and Petri Varsta and Alan Klanac., “Stress Analysis of
Homogenized Web-Core Sand Sandwich Beams‟, Journal of composite structure 79
pp 411-4222, 2007.
43
13] Onur Sayman, Umran Esendemir., “An elastic–plastic stress analysis of simply
supported metal-matrix composite beams under a transverse uniformly distributed
load”, Journal of composite science and technology 62 pp 265-273, 2002.
14] Dutt J.K. and Nakra B. C., “Stability of rotor systems with viscoelastic
supports”, Journal of Sound and Vibration, 153 (1), pp. 89-96.
15] Roy H., Dutt J. K. and Datta P. K., “The Dynamics of Multi-layered Viscoelastic
Beams”, Journal of Structural Engineering and Mechanics, vol. 33, pp 391-406,
2009.
16] Muszynska A. “Forward and Backward Precession of a Vertical Anistropically
Supported Rotor”, Journal of Sound and Vibration, vol 192(1), pp 207-222, 1996.
17] Bauchau O.A.,Craig J.I., “Euler-Bernoulli beam theory”, Journal of Solid
Mechanics and its application, Volume 163, pp 173-221, 2009.
18] Lalanne M. and Ferraris G., “Rotor Dynamics Prediction in Engineering”, John
Wiley and Sons, 1998.
19] Ogata K., “Modern Control Engineering”, Pearson Education International,
2002.
20] Zhenxing Shen. and Qiang Tian. and Xiaoning Liu. and Gengkai Hu.,
“Thermally induced vibrations of flexible beams using Absolute Nodal Coordinate
Formulation”, Journal of Aerospace Science and Technology 29 pp 386-393, 2013.
21] Narasimha Marakala. and Appu K.K. and Ravikiran Kadoli., “Thermally induced
vibration Of a simply supported beam using finite element method”, International
Journal of Engineering Science and Technology vol 2 (12), pp 7874-7879, 2009.
22] Robert D. Cook, Davis S.M., Michael E.P. and Robert J.W., “ Concept And
Application of Finite Element Analysis”, John Wiley & Sons,2002 4th edition.
23] Genta Giancarlo., Dynamics of rotating systems, Springer Verlag, 2009.
24] Gunter Edgar J. Jr., “Rotor bearing stability”, Proceedings of the First TurboMachinery Symposium.
44
25] Zorzi E.S. and Nelson H.D. “The dynamics of rotor-bearing systems with axial
torque - A Finite Element Approach”. Journal of Mechanical Design, vol. 102,
pp158-61, 1988.
26] Roy H. and Dutt J. K. and Datta P. K., “Dynamics of Viscoelastic Rotor Shaft
Using Augmenting Thermodynamics Fields- A Finite Element Approach”,
International Journal of Mechanical Science, vol. 50, pp 845-853,2008.
4.5 Appendix
Translatory Mass Matrix  M T 
22le
0
0
54 13le
0
0 
156

4le 2
0
0
13le 3le 2
0
0 


156 22le 2 0
0
54 13le 


2
4le
0
0
13le 3le 2 
  A  le 

MT  

SYMMETRIC
156 22le
0
0 
420


2
4le
0
0 


156 22le 


4le 2 

Rotary Mass Matrix  M R 
3le
0
0
36

2
4le
0
0


36 3le 2

4le 2
 I


M R  
SYMMETRIC
30  le 





45
36
3le
0
3le
le 2
0
0
36
0
3le
4le 2
0
0 
0
0 
36 3le 

3le le 2 
0
0 

0
0 
36 3le 

4le 2 
Bending Stiffness Matrix  K B 
12
4le 2
0
0

2
4le
0
0


12 6le

4le 2
EI 
 KB   3  
le
SYMMETRIC





12
6le
0
0
6le
2le 2
0
0
12
6le
4le 2
0
0 

0
0 
12 6le 

6le 2le 2 
0
0 

0
0 
12
6le 

4le 2 
Skew Symmetric Circulatory Matrix  KC 
0
0
12
6le
0
0
6le
2le 2
12
6le
0
0
0
0
0
12
6le
0
0
36 3le
0

0
3le 4le 2


0
0

SKEW
0
2     I 

G  
 SYMMETRIC
30  le





0
0
36
3le
0
0
3le
le 2
36
3le
0
0
0
0
0
36
3le
0
0
12
0

0
6le


0

SKEW
EI
 KC   3  
SYMMETRIC
le





6le
4le 2
0
0
6le 
2le 2 
0 

0 
6le 

4le 2 
0 

0 
Gyroscopic and Stiffness Matrices G 
46
3le
le 2
0
0






3le 

4le 2 
0 

0 
Axial Stiffness Matrix  K A 
KA  
PA
30  le
3le
0
0
36

2
4le
0
0


36 3le 2

4le 2



SYMMETRIC





36
3le
0
3le
le 2
0
0
36
0
3le
4le 2
0
0 
0
0 
36 3le 

3le le 2 
0
0 

0
0 
36 3le 

4le 2 
Axial Torque Matrix  KT 
 0



 0


 0

 1
 l
 KT   TR   e
 0


 0


 0

 1

 le
0
0
1
0
0
0

le
0
1
1
1
le
2
0

0
0
0
le

1

0
2
0
0

1

0

1
1
1
le
2
0
0

le


0 


0 

1 

le 
1
 
2

0 


0 

0
0
0
0
0
le
0

1

1
1
le
2
0
0
le

1
1
le
0
1
0
le
0
0
2
47
1
1
le
2
1
1
le
0
1
le 
0
2

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