FINITE ELEMENT BASED VIBRATION AND STABILITY ANALYSIS OF

FINITE ELEMENT BASED VIBRATION AND STABILITY ANALYSIS OF
FINITE ELEMENT BASED VIBRATION AND STABILITY ANALYSIS OF
FUNCTIONALLY GRADED ROTATING SHAFT SYSTEM UNDER
THERMAL ENVIRONMENT
A THESIS SUBMITTED IN THE PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF
Master of Technology
In
MECHANICAL ENGINEERING
[Specialization: Machine Design and Analysis]
By
Debabrata Gayen
211ME1156
Department Of Mechanical Engineering
National Institute of Technology Rourkela
Rourkela, Orissa, India – 769008
June, 2013
FINITE ELEMENT BASED VIBRATION AND STABILITY ANALYSIS OF
FUNCTIONALLY GRADED ROTATING SHAFT SYSTEM UNDER
THERMAL ENVIRONMENT
A THESIS SUBMITTED IN THE PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF
Master of Technology
In
MECHANICAL ENGINEERING
[Specialization: Machine Design and Analysis]
By
Debabrata Gayen
211ME1156
Under the Supervisions of
Prof. T. Roy and Prof. A. Mitra
Department Of Mechanical Engineering
National Institute of Technology Rourkela
Rourkela, Orissa, India – 769008
June, 2013
NATIONAL INSTITUTE OF TECHNOLOGY ROURKELA
CERTIFICATE
This is to certify that the thesis entitled, “Finite Element Based Vibration and Stability
Analysis of Functionally Graded Rotating Shaft System Under Thermal Environment”,
being submitted by Mr. DEBABRATA GAYEN in partial fulfillment of the requirements for
the
award
of
“MASTER
OF
TECHNOLOGY”
Degree
in
“MECHANICAL
ENGINEERING” with specialization in “MACHINE DESIGN AND ANALYSIS” at the
National Institute of Technology Rourkela (India) is an authentic Work carried out by him
under our supervisions.
To the best of our knowledge, the results embodied in the thesis has not been
submitted to any other University or Institute for the award of any Degree or Diploma.
Supervisor
Co ‐ Supervisor
(Prof. T. Roy)
(Prof. A. Mitra)
Department of Mechanical Engineering
Department of Mechanical Engineering
National Institute of Technology Rourkela
National Institute of Technology Rourkela
Orissa, India- 769008.
Orissa, India- 769008.
NIT ROURKELA
Page i
ACKNOWLEDGEMENTS
First and foremost, I wish to express my sense of gratitude and indebtedness to
my supervisors, Prof. Tarapada Roy and Prof. Anirban Mitra for their inspiring guidance,
encouragement, and untiring efforts throughout the course of this work. Their timely
help, constructive criticism and painstaking efforts made it possible to present the work
contained in this thesis.
Specially, I extend my deep sense of indebtedness and gratitude to Prof. Tarapada
Roy for his kindness in providing me an opportunity to work under his supervision and
guidance. He played a crucial role in the process of my research work. First of all, he allowed
me to join his research group; even two scholars were working under him. His advice to
harmonize theory and applications help me a lot in my research. He showed me different
ways to approach a research problem and the need to be persistent to accomplish my goal.
His keen interest, invaluable guidance and immense help have helped me for the successful
completion of the thesis.
After the completion of this Thesis, I experience a feeling of achievement and
satisfaction. Looking into the past I realize how impossible it was for me to succeed on my
own. I wish to express my deep gratitude to all those who extended their helping hands
towards me in various ways during my tenure at NIT Rourkela. I greatly appreciate and
convey my heartfelt thanks to my colleagues ‘flow of ideas, dear ones and all those who
helped me in the completion of this work. The beautiful weather of NIT Campus, kept me in
good health and high spirits throughout the research period.
I am also thankful to Prof. K. P. Maity, Head of the Department, Mechanical
Engineering, for his moral support and valuable suggestions regarding the research work.
I am especially indebted to my parents for their love, sacrifice and support.
They are my first teachers after I came to this world and have set great examples for
me about how to live, study and work.
DEBABRATA GAYEN
Roll No. 211MEE1156
Department of Mechanical Engineering
National Institute of Technology Rourkela
Orissa, India- 769008.
NIT ROURKELA
Page ii
Table of Contents
Certificate
i
Acknowledgements
ii
Contents
iii
List of Tables
v
List of Figures
vi
Abstract
vii
1.
2.
3.
Introduction
1
1.1
Background and Importance of Rotor Dynamic
1
1.2
Composite Materials
3
1.3
Drawback of Composite Materials
4
1.4
Conceptual Idea about FGMs
4
1.5
Applications of FGMs
5
1.6
Outline of the Present Work
5
Review of Relevant Literatures
7
2.1
Introduction
7
2.2
Composite Materials Structure
7
2.3
Functionally Graded Materials structure
8
2.4
Vibration and control
10
2.5
Motivation
11
2.6
Objectives of Present Work
12
Modeling for Effective Materials Properties of FG Shaft
13
3.1
Effective Materials Properties of FGM
13
3.2
Modeling for Material Properties of FG Rectangular Cross-Section
13
3.2.1
Power Law Gradation
14
3.2.2
Exponential Gradation Law
15
3.3
4.
Modeling of FGMs Properties for Circular Cross-Section
15
3.3.1
15
Power Law Gradation
Formulation for Governing Equations of Rotor Shaft System
18
4.1
Introduction
18
4.2
Mathematical Modeling of Functionally Graded Shaft
19
4.3
Strain Energy Expression for FG Shaft
20
4.4
Kinetic Energy Expression for FG Shaft
21
NIT ROURKELA
Page iii
5.
4.5
Kinetic Energy Expression for Disks
22
4.6
Work done Expression due to External Loads and Bearings
23
4.7
Governing Equations of Rotor-Shaft System
23
4.8
Solution Procedure
24
4.9
Contribution of Internal Damping
25
Results and Discussions
26
5.1
Problem Specifications and Summarized of Various Analyses
26
5.2
Code Validation
27
5.3
Temperature Distribution in a FG Shaft
28
5.4
Temperatures
28
5.5
Comparative Studies of FG Shaft over Steel Shaft
32
5.6
Comparative studies of FG shaft with and without Temperatures
35
5.7
5.8
5.9
6.
Variations of Mechanical Properties of FG Shaft with Positions and
The Effect of Different Gradient Indexes on Various Responses of FG
Shaft
The Effects of Different Temperatures and Gradient Indexes on Various
Responses of FG Shaft
Time Responses for FG Shaft System due to Unbalance Masses
38
41
45
Conclusions and Scope of Future Works
48
6.1
Conclusions
48
6.2
Scope of Future Works
49
Appendix
50
References
53
List of Publications
60
NIT ROURKELA
Page iv
List of Tables
Table 5.1 Mechanical properties and geometric dimension of steel rotor-shaft system [69]..26
Table 5.2 Material properties of FGM compositions [79]…………………………………...27
Table 5.3 Materials and temperature coefficients value for mechanical properties [34]…….27
Table 5.4 Temperature variation of FG shaft for different radial position and
power law gradient index  k  …………………………………………………….28
Table 5.5 Variation of Young’s modulus with different radial positions,
temperatures and power law gradient indexes of FG shaft………………………..30
Table 5.6 Variation of thermal conductivity with different radial positions,
temperatures and power law gradient indexes of FG shaft………………………..31
Table 5.7 First critical speed and maximum real part for different
power law gradient indexes………………………………………………………..40
Table 5.8 First critical speed and maximum real part for different
values of temperatures and power law gradient indexes  k  ……………………....45
Table 5.9 Maximum amplitudes for different temperatures and
power law gradient index (k ) ..…………………………………………………….46
NIT ROURKELA
Page v
List of Figures
Fig. 3.1 Volume fraction of ceramic throughout the FGM layer…………………………….14
Fig. 3.2 Volume fraction of ceramic and metal throughout the FGM layer………………….15
Fig. 4.1 Displacement and Rotational variables with coordinate systems…………………...18
Fig. 4.2 Schematic diagram of rotor-bearing system with coordinate systems………………18
Fig. 5.1 Campbell diagram for first two pairs of modes……………………………………..27
Fig. 5.2 Variation of temperature with radial position and power law gradient index……....28
Fig. 5.3 Variation of Young modulus with power law gradient index of FG shaft ….……...29
Fig. 5.4 Variation of Poisson ratio with power law gradient index of FG shaft……………..29
Fig. 5.5 Variation of density with power law gradient index of FG shaft ……………….…..30
Fig. 5.6 Variation of Poisson Ratio with different temperature and power law
gradient index through radial direction of FG shaft…………………………………..31
Fig. 5.7 Variation of Young modulus with different temperature and power law
gradient index through radial direction of FG shaft…………………………………32
Fig. 5.8 Comparison of Campbell diagrams of rotating shafts: (a) FG and (b) Steel………..33
Fig. 5.9 Variation of maximum real part against speed of rotation: (a) FG and (b) Steel.…..34
Fig. 5.10 Variation of damping ratio for first six modes of rotating shafts:
(a) FG and (b) Steel…………………………………………………………………..35
Fig. 5.11 Comparison of Campbell diagrams of rotating FG shafts:
(a) With Temperature and (b) Without Temperature ……………..…………………36
Fig. 5.12 Variation of maximum real part against speed of rotation of FG shaft:
(a) With Temperature and (b) Without Temperature ………………………………..37
Fig. 5.13 Variation of damping ratio for first six modes of rotating FG shaft:
(a) With Temperature and (b) Without Temperature………………………………...38
NIT ROURKELA
Page vi
Fig. 5.14 The Campbell diagram of FG shaft: (a) k  5 and (b) k  10 …………………….39
Fig. 5.15 The maximum real part against spin speed of FG shaft:
(a) k  5 and (b) k  10 ……………………………………………………………...40
Fig. 5.16 The damping ratio of first six modes of FG shaft: (a) k  5 and (b) k  10 .............41
Fig. 5.17 The Campbell diagram of FG shaft: (a) T  300K and (b) T  600K ……….…...42
Fig. 5.18 Maximum real part against spin speed of FG shaft:
(a) T  300K and (b) T  600K …………………………………………………….43
Fig. 5.19 Damping ratio for first six modes of FG shaft: (a) T  300K and (b) T  600K ...44
Fig. 5.20 Amplitudes vs. Time response along transverse direction of FG shaft:
(a) Stable response and (b) Unstable response……………………………………….47
NIT ROURKELA
Page vii
Abstract
The present work deals with the study of vibration and stability analyses of functionally
graded (FG) spinning shaft system under thermal environment using three nodded beam
element based on Timoshenko beam theory (TBT). Temperature field is assumed to be a
uniform distribution over the shaft surface and varied in radial direction only. Material
properties are assumed to be temperature dependent and graded in radial direction according
to power law gradation and exponential law gradation respectively. In the present analysis,
the mixture of Aluminum Oxide (Al2O3) and Stainless Steel (SUS304) is considered as FG
material where metal contain (SUS304) is decreasing towards the outer diameter of shaft. The
FG shafts are modeled as a Timoshenko beam by mounting discrete isotropic rigid disks on it
and supported by flexible bearings that are modeled with viscous dampers and springs. Based
on first order shear deformation (FOSD) beam theory with transverse shear deformation,
rotary inertia, gyroscopic effect, strain and kinetic energy of shafts are derived by adopting
three-dimensional constitutive relations of material. The derivation of governing equation of
motion is obtained using Hamilton’s principle and solutions are obtained by three-node finite
element (FE) with four degrees of freedom (DOF) per node. . In this work the effects of both
internal viscous and hysteretic damping have also been incorporated in the finite element
model. A complete code has been developed using MATLAB program and validated with the
existing results available in literatures. The analysis of numerical results reveals that
temperature field and power law gradient index have a significance role on the materials
properties (such as Young modulus, Poisson ratio, modulus of rigidity, coefficient of thermal
expansion etc.) of FG shaft. Various results have also been obtained such as Campbell
diagram, stability speed limit (SLS), damping ratio and time responses for FG shaft due
unbalance masses and also compared with conventional steel shaft. It has been found that the
responses of the FG spinning shaft are significantly influenced by radial thickness, power law
gradient index and internal (viscous and hysteretic) damping and temperature dependent
material properties. The obtained results also show that the advantages of FG shaft over
conventional steel shaft.
Keywords: Power law gradient index; Functionally graded shaft; Temperature dependent
material properties; Viscous and hysteretic damping; Rotor-Bearing-shaft system; Finite
element method; Campbell diagram; Damping ratio; stability speed limit (SLS)
NIT ROURKELA
Page viii
CHAPTER 1
INTRODUCTION
Composite materials and structures are more and more frequently used in
advanced engineering fields mainly because of their high stiffness-to-weight ratio that
is particularly favorable. However the main downside of composite materials is
represented by the weakness of interfaces between adjacent layers known as
delimitation phenomena that may lead to structural failure. To partially overcome these
problems, a new class of materials named Functionally Graded Materials (FGMs) has
recently been proposed whose various material properties vary through the radial and
thickness direction in a continuous manner and thus free from interface weakness. The
gradation of material properties reduces thermal stresses, residual stresses, and stress
concentrations. A functionally graded structure is defined as, those in which the volume
fractions of two or more materials are varied continuously as a function of position along
certain dimension (typically the radius and thickness) of the structure to achieve a require
function. FGMs can provide designers with tailored material response and exceptional
performance in thermal environments. For example, the Space Shuttle utilizes ceramic tiles as
thermal protection from heat generated during re-entry into the Earth’s atmosphere. An FGM
composed of ceramic on the outside surface and metal on the inside surface.
Due to high strength, high stiffness, and low density characteristics, FGMs rotor
shafts have been sought as new potential candidates for replacement of the conventional
metallic shafts in many application areas for the design of rotating mechanical components
such as, driveshaft for helicopters and cars and jet engine, commercial and military rotating
machines, aerospace and space vehicles etc. In Rotor-dynamic applications, composites have
been demonstrated both numerically and experimentally. Accompanied by the development
of many new advance composite materials and various mathematical models of rotor-shafts
were also developed by researchers.
1.1 Background and Importance of Rotor Dynamic
Rotor dynamics has a remarkable history, largely due to the interplay between its
theory and its practice. Rotor dynamics is a specialized branch of applied mechanics
concerned with the behavior and diagnosis of rotating structures. It is commonly used to
analyze the behavior of structures ranging from jet engines and steam turbines to auto
engines and computer disk storage. Basic level of Rotor Dynamic is concerned with rotor
and stator. Rotor is a rotating part of a mechanical device or structures supported by bearings
and influenced by internal phenomena that rotate freely about an axis fixed in space.
Engineering components concerned with the subject of rotor dynamics are rotors in
machines, especially of turbines, generators, motors, compressors, blowers, alternators,
pumps, brakes, distributors and the like.
NIT ROURKELA
Page 1
Rotor provides with materials bearings to constrain their spin axis in a more or rigid
way to a fixed position in space, are referred to as fixed rotor (which consider spin speed is
constant), where as those that are not considered in any way are referred free rotor (which
consider spin speed is governed by conservation of angular momentum). In operation Rotors
have a great deal of rotational energy and a small amount of vibrational (bending, axial and
torsional) energy.
In Rotor Dynamics field William John Macquorn Rankine (1869) performed the first
analysis of a spinning shaft. He chose a two-degrees-of-freedom model consisted of a rigid
mass whirling in an orbit, with elastic spring acting in the radial direction. He defined the
whirling speed of the shaft but he can be shown that beyond this whirling speed the radial
deflection of Rankine's model increases without limit and this speed is called threshold speed
for the divergent instability.
In 1883 Swedish engineer Carl Gustaf Patrik de Laval developed a single-stage steam
impulse turbine for marine applications and succeeded in its operation at 42,000 rpm. He first
used a rigid rotor, but latter used a flexible rotor and shown that it was possible to operate
above critical speed by operating at a rotational speed about seven times the critical speed.
In 1895, Stanley Dunkerley published a study of the vibration of shafts loaded by
pulleys. The first sentence of his paper reads, "It is well known that every shaft, however
nearly balanced, when driven at a particular speed, bends, and, unless the amount of
deflection is limited, might even break, although at higher speeds the shaft again runs true.
This particular speed or 'critical speed' depends on the manner in which the shaft is
supported, its size and modulus of elasticity, and the sizes, weights, and positions of any
pulleys it carries." In 1895 German civil engineer August Foppl who showed that an alternate
rotor model exhibited a stable solution above Rankine's whirling speed. In England W. Kerr
(1916) published experimental evidence that a second "critical speed" existed and it was
obvious to all that a second critical speed could only be attained by the safe traversal of the
first critical speed. In 1918 Ludwig Prandtl was the first to study a Jeffcott rotor with a noncircular cross-section.
In 1919, Henry Jeffcott modeled a simple rotor to study the flexural behavior of rotors
and dynamic behavior in “The lateral vibration of loaded shafts in the neighborhood of a
whirling speed-the effect of want of balance,” It is often referred to as Jeffcott rotor. But
Jeffcott's analytical model the disk did not wobble. As a result, the angular velocity vector
and the angular momentum vector were collinear and no gyroscopic moments were
generated.
However, this attribution is incorrect and Föppl (1895) published “Das problem der
laval’shen turbinewelle,” in which its behavior is correctly analyzed. After that Jeffcott
confirmed Foppl's prediction that a stable supercritical solution existed and he extended
Foppl's analysis by including external damping.
NIT ROURKELA
Page 2
In 1924 Aurel B. Stodola removed the Jeffcott’s restriction and developed dynamics
of elastic shaft with discs and continuous rotors without considering gyroscopic moment, the
secondary resonance phenomenon due to gravity effect, the balancing of rotors, and methods
of determining approximate values of critical speeds of rotors with variable cross sections,
supercritical solutions were stabilized by Coriolis accelerations.
Then Baker (1933) described self-excited vibrations due to contact between rotor and
stator. In 1933 David M. Smith obtained simple formulas that predicted how the threshold
spin speed for supercritical instability varied with bearing stiffness and with the ratio of
external to internal viscous damping.
Gradually, the Jeffcott rotor model with many variations came closer to the practical
needs of the rotor dynamic field of the day. But, not close enough. In 1945 Prohl's and
Myklestad's work led the Transfer Matrix Method (TMM) for analyzing instabilities and
modeling techniques of rotors.
After World War II, rotor dynamics had become an international endeavor, and
recognized by the Rotor Dynamics Committee of the International Federation of the Theory
of Machines and Mechanisms (IFToMM) and beginning in Rome (1982), Tokyo (1986),
Lyon (1990), Chicago (1994), Darmstadt (1998), Sydney (2002), and Vienna (2006).
For revolution in solution capability, In the 1960s numerical methods developed for
structural dynamics analysis and digital computer codes and rotor dynamics codes was based
on the TMM method but in the 1970s another underlying algorithm, the Finite Element
Method (FEM), became available for the solution of the prevailing beam-based models. Now,
in the beginning of the 21st century, rotor dynamics are combining the FEM and solids
modeling techniques to generate simulations that accommodate the coupled behavior of
flexible disks, flexible shafts and flexible support structures into a single, massive,
multidimensional-model.
1.2 Composite Materials
Composite materials are formed by combining two or more material on a micro scale
form and their constituents do not dissolve or merge into each other, to achieve superior
enhanced properties. These are widely used in a variety of structures, including army and
aerospace vehicles, nuclear reactor vessels, turbines parts, buildings and smart highways (i.e.
civil infrastructure applications) as well in sports equipment and medical prosthetics.
Laminated composite structures consist of several layers of different fiber-reinforced
laminate bonded together to obtain desired structural properties (e.g. stiffness, strength, wear
resistance, CTE, Thermal conductivity, damping, and so on). Varying the lamina thickness,
lamina material properties, and stacking sequence desired structural properties can be
achieved. The increased use of laminated composites in various types of structures led to
considerable interests in their analysis. Composite materials exhibit high strength-to weight
and stiffness-to-weight ratios, which make them ideally suited for use in weight sensitive
NIT ROURKELA
Page 3
structures. This weight reduction of structures leads to improvement of their structural
performance, especially in space applications.
1.3 Drawback of Composite Materials
Though laminated composites give numerous advantages over conventional materials,
their major downside is however represented by the repeated cyclic stress, impact load and so
on can causes to separate layers and weakness of interfaces between adjacent layers, known
as delamination phenomena (i.e. Mode of failure or failure mechanisms of composite
materials). It may lead to failure of the structure. Additional problems include the presence of
residual stresses due to the difference in coefficient of thermal expansion and coefficient of
moisture expansion of the fiber and matrix. For anisotropic constitution of laminated
composite structures often results in stress concentrations near material and geometric
interface that can lead to damage in the form of de-lamination, matrix cracking and adhesive
bond separation. These problems can be reduced if the sudden change of material properties
is somehow prevented.
1.4 Conceptual Idea about FGMs
First FGM concepts have come from Japan in 1984 during a space plane project.
There a combination of materials used would serve the purpose of a thermal barrier capable
of withstanding a surface temperature of 2000 K and a temperature gradient of 1000 k across
a 10 mm section. Recently FGMs concept has become more popular in Europe (Germany). A
collaborative research center Transregio (SFB Transregio) is funded since 2006 in order to
exploit the potential of grading mono-materials, such as steel, aluminum and polypropylene,
by using thermo mechanically coupled manufacturing processes
Functionally Graded Materials (FGMs) are those composite materials where the
composition or the microstructure is locally varied so that a certain variation of the local
material properties is achieved. FGM is also defined as, those in which the volume fraction of
two or more materials are achieved continuously as a function of position along certain
directions of the structure to achieve a required function (e.g. mixture of ceramic and metal).
It is materially heterogeneous which is defined for those objects with and/or multiple material
objects with clear material domain. By grading of material properties in a continuous manner,
the effect of inter-laminar stresses developed at the interfaces of the laminated composite due
to abrupt change of material properties between neighboring laminas is mitigated.
As many thin walled members, i.e., plates and shells used in reactor vessels, turbines
and other machine parts are susceptible to failure from buckling, large amplitude deflections,
or excessive stresses induced by thermal or combined thermo mechanical loading. Thus,
FGMs are primarily used in structures subjected to extreme temperature environment or
where high temperature gradients are encountered. Mainly they are manufactured from
isotropic components such as metals and ceramics, since role of metal portion is acts as
structure support while ceramics provides thermal protection in environments with severe
NIT ROURKELA
Page 4
thermal gradients (e.g. reactor vessels, semiconductor industry).In such conditions ceramic
provides heat and corrosion resistance, while the metal provides the strength and toughness.
Whatever problems arises in using composite materials those problems can be
reduced significantly by using FGMs instead of composite materials because FGMs changes
the material properties from surface to surface or layer to layer. FGMs are new advanced
multifunctional composites where volume fractions of the reinforcements phase(s) vary
smoothly. Additionally, FGM allows the certain superior and multiple properties without any
mechanically weak interface. This new concept of materials hinges on materials science and
mechanics due to integration of the material and structural considerations into the final design
of structural component. Moreover, gradual change of properties can be tailored to different
applications and service environments.
1.5 Applications of FGMs
Due to progressing of technology it is need for advanced capability of materials to
become a priority in engineering field for higher performance systems. FGMs are relatively
new materials and are being studied for the use in high temperature applications. FGM is an
extensive variety of applications in engineering practice which requires materials
performance to vary with locations within the component. The following applications are
noticeable such as,
1) Aerospace field (space planes, space structures, nuclear reactors, insulations for
cooling structures, Aerospace skins, Rocket engine components, Vibration control,
Adaptive structures etc.).
2) Engineering field (Turbine blade, shaft, cutting tool etc.)
3) Optical field (optical fiber, lens etc.)
4) Electronics field (sensor, graded band semiconductor, substrate etc.)
5) Chemical field (Heat Exchanger, Reactor Vessel, Heat Pipe etc.)
6) Biomaterial field (artificial skin, drug delivery system, prosthetics etc.)
7) Commodities (Building materials, Sports good, Car body etc.)
8) Energy conversion (Thermoelectric generator, Thermo ionic converter, Fuel cells,
Solar cells etc.)
9) Optoelectronics
10) Piezoelectricity
1.6 Outline of the Present Work
The outline of this thesis is divided into six chapters. Chapter 1 presents an
introduction of composite material, FG materials and also brief introduction to rotor
dynamics. The outline of the present work is also given in the Chapter 1.
Then Chapter 2 provides a state of the art of composite materials structure,
functionally graded materials structure and vibration and control of spinning shaft systems.
NIT ROURKELA
Page 5
In Chapter 3 represents the modeling of FG material to obtain the effective material
properties of FG shaft.
Chapter 4 presents a detailed formulation of the governing equations of spinning rotor
shaft system for analyzing vibration and stability of FGMs. Formulations of the equations of
motion are developed for a first order shear deformable beam by contributing internal
damping.
Chapter 5 discusses the results of the analyses performed in this project work
and detailed report of results and discussion has been given. Various type of analysis for FG
shaft have been studied and presented. Finally, Chapter 6 summarizes conclusions of this
project work and scopes for further work are suggested.
NIT ROURKELA
Page 6
CHAPTER 2
REVIEW OF RELEVANT LITERATURES
2.1 Introduction
Numerous research works have been done in the field of modeling and analysis of
functionally graded (FG) structures. Some important works based on the different analysis
have been presented in the following sections. Advanced composite materials offer numerous
superior properties over metallic materials, like high specific strength and high specific
stiffness. This has resulted in the extensive use of laminated composite materials in
aircraft, spacecraft and space structures. In an effort to develop the super heat resistant
materials, Koizumi [1] first proposed the concept of FGM. These materials are
microscopically heterogeneous and are typically made from isotropic components, such as
metals and ceramics. After the concept of FGMs was set by the Japanese school of material
science and confirmation of their potentials several branches of research originated and are
still being broadened by research groups all over the world.
2.2 Composite Materials Structure
Zinberg and Symonds [2] investigated the model of rotating anisotropic shafts and
compared the critical speeds of composite shaft over aluminum alloy shaft by using
equivalent modulus beam theory (EMBT). Nelson et al. [3] and Nelson [4] contributed a
substantial improvement in the computational analysis of rotating shafts incorporating the
effect of gyroscopic, rotary inertia, moment and shear deformation into the FE shafts model.
Rouch et al. [5] implemented dynamics behavior of a linearly tapered circular Timoshenko
beam formulation by numerical integration method to reduce the system bandwidth, system
degree of freedom without a significant loss of accuracy and verify the gyroscopic effects of
shafts rotation in rotor dynamic. Zorzi et al. [6] investigated the effect of constant axial
torque on the dynamics, reliability, safety and survivability of rotor-bearing systems using
FEM and determine the static bulking torque and critical speeds of long and short bearing. By
using Bernoulli-Euler beam theory, Bert [7] developed governing equation of composite
shaft, including effect of gyroscopic, bending and torsion coupling and determines critical
speeds of composite shaft. Kim and Bert [8] adopted Sanders best first approximation shell
theory to determine critical speeds of a rotating circular cylinder hollow shaft, containing
layers of arbitrary laminated composite material. Abramovich et al. [9] employed a special
orthogonality procedure which is applied to help understanding the dynamic behavior of the
non-symmetrical laminated beam. Further by using Bresse-Timoshenko beam theory, Bert
and Kim [10] employed Hamilton’s principle to derive equations of motion of composite
shafts and to find out critical speeds. Bert and Kim [11] analyzed the dynamic instability of a
composite drive shaft subjected to fluctuating torque and/ or rotational speed by using various
shell theories and they investigated effect of constant torque and rotational speed. Singh and
Gupta [12] implemented a Layer wise Beam Theory (LBT) with assuming a layer-wise
NIT ROURKELA
Page 7
displacement field and were extended to solve a composite rotor dynamic problem. Again
Singh and Gupta [13] analyzed and compared the conventional rotor dynamic parameter like
natural frequencies, critical speeds, damping factor, unbalance response (UBR) and threshold
stability by using EMBT and LBT. Forrai [14] implemented a finite element model for
stability analysis of self-excited bending vibrations of linear symmetrical rotor bearing
system with internal damping. Chatelet et al. [15] proposed a finite element modeling to
reduce the dynamic behavior of rotating structures and whole disk shaft assembly is supposed
to be cyclically symmetrically. By using FOSD beam theory (continuum based Timoshenko
beam theory), M. Y. Chang et al. [16] implemented the vibration behaviors of rotating
laminated composite shaft model where transverse shear deformation, rotary inertia,
gyroscopic effects and coupling effect are incorporated. Kapuria et al. [17] presents static and
dynamic electro-thermo-mechanical analysis of angle-ply hybrid piezoelectric beams using
an efficient coupled zigzag theory. Span-to-thickness ratio, type of loading and the
orientation of the angle-ply on the accuracy of the theories is investigated. Gubran et al. [18]
analyzed natural frequencies of composite tubular shafts by using modified EMBT with shear
deformation, rotary inertia and gyroscopic effects. Modifications take into account effects of
stacking sequence and different coupling mechanisms in composite materials. By using FEM,
Wang et al. [19] established a solution method for the one-dimensional transient temperature
and thermal stress fields in non-homogeneous materials. Syed et al. [20] investigated simple
analytical expressions for computing thermal stresses in fiber-reinforced composite beams
with rectangular and hat sections due to change of temperature. By using a homogenized
finite element beam model, Sino et al. [21] analyzed dynamic instability and natural
frequencies of an internally damped rotating composite shaft. The influence of laminate
parameters: stacking sequences, fiber orientation, transversal shear effect on natural
frequencies and instability thresholds of the shaft are considered.
2.3 Functionally Graded Materials Structure
Feldman and Aboudi [22] studied the elastic bifurcational buckling of FG plates under
in-plane compressive load. They concluded that with optimal non-uniform distribution of
reinforcing phases, the buckling load can be significantly improved for FG plate over the
plate with uniformly distributed reinforcing phase. Praveen and Reddy [23] investigated
static and dynamic response of the FG ceramic-metal plates using a simple power law
distribution and a finite element that accounted for the transverse shear strains, rotary inertia
and moderately large rotations in the von- Karman sense. Gasik [24] developed an efficient
micromechanical model for computing thermal stresses; evaluating dynamic stress/strain
distribution and inelastic behavior of FGMs with an arbitrary non-linear 3D-distribution
phases. Suresh and Mortensen [25] provide an excellent introduction to the fundamentals of
FGMs. Aboudi et al. [26] developed a more general higher-order theory for FG materials and
illustrated the utility of FG microstructures in tailoring the behavior of structural components
in various applications. Nakamura et al. [27] investigated Kalman filter technique which was
originally introduced for signal/digital filter processing, is used to estimate FGM throughthickness compositional variation and a rule-of-mixtures parameter that defines effective
properties of FGMs. Wang et al. [28] proposed a method to determine the transient and
NIT ROURKELA
Page 8
steady state thermal stress intensity factors of graded composite plate containing noncollinear cracks subjected to dynamic thermal loading. Woo and Meguid [29] presented an
analytical solution for the large deflections of plates and shallow shells made of FGMs under
the combined action of thermal and mechanical loads. Sankar [30] provided an elasticity
solution for FG beams in which the beam properties are graded in the thickness direction
according to an exponential law. Here FG Euler-Bernoulli beam theory considered under
mechanical loading. Sankar and Tzeng [31] expanded upon Sankar’s [30] earlier work by
investigating beams with through the thickness temperature gradients. Chakraborty et al. [32]
proposed a two-node beam element for FGMs based on FOSD theory and applied it to
static, thermal, free vibration and wave propagation problems. They assumed
displacement field of the element satisfies the general solution to the static part of the
governing equations. Nemta AIIa [33] introduced 2D-FGM, for withstand super high
temperatures and to give more reduction in thermal stresses. Reddy [34] worked on
characterizing the theoretical formulation of FGMs to include the derivations of
equations used to calculate material properties throughout the thickness of the material
based on the through-the-thickness distribution of materials. Na and Kim [35] studied the
thermo-mechanical buckling of FGMs using a finite element discretization method.
Przybyowicz [36] presented a problem of active stabilization of a rotating shaft made of a
three-phase FG Material with piezoelectric fraction and determination of such a distribution
of the volume fraction of the active phase within the shaft, which makes the system possibly
most resistant to self-excitation. Cooley [37] researched FGM shell panels under thermal
loading using the FEM. By using a multi-layered approach, Shao [38] analyzed the analytic
solutions of temperature, displacements, and thermal/mechanical stresses in a FG circular
hollow cylinder. Based on the FOSD theory and von-Karman nonlinear kinematics, WU et al.
[39] obtained a solution for the nonlinear static and dynamic responses of the FG materials
rectangular plate. Argeso and Eraslan [40] developed a computational model for the analysis
of elastic, partially plastic and residual stress states in long FG rotating solid shafts. Rahimi
and Davoodinik [41] developed the analysis of thermal behavior and distribution of material
properties of FG beam. For thermal loading the steady state of heat conduction with
exponentially and hyperbolic variations through the thickness of FGB, is considered. Piovan
and Sampaio [42] developed a rotating nonlinear FG beam model accounting for arbitrary
axial deformations. This model is also employed to analyze other simplified models based on
isotropic materials or composite materials. By using first-order shear deformation plate
theory Zhao et al. [43] presented the mechanical and thermal buckling analysis of FG
ceramic–metal plates. By using Ritz method and HOSD beam theories, Simsek [44]
investigated Static analysis of a FG Timoshenko simply-supported beam subjected to a
uniformly distributed load. By using Classical beam theories Giunta et al. [45] analyzed
linear static analysis of beams made of materials whose properties are graded along one or
two directions. Afsar and Go [46] developed the finite element analysis of thermo-elastic
field in a thin circular FGM disk subjected to a thermal load and an inertia force due to
rotation of the disk. By using different higher-order beam theories, Simsek [47] analyzed
fundamental frequency of FG beams having different boundary conditions. Alibeigloo [48]
developed analytical solution for FGM beams integrated with piezoelectric actuator and
sensor under an applied electric field and thermo-mechanical load. Kocaturk et al. [49]
NIT ROURKELA
Page 9
studied non-linear static analysis of a cantilever Timoshenko beam composed of FGM under
a non-follower transversal uniformly distributed load with large displacements and large
rotations. Mazzei and Scott [50] investigated the FGMs on resonance of bending shafts under
time-dependent axial loading. By differential quadrature method, Alashti and Khorsand [51]
carried out three-dimensional thermo- elastic analysis of a FG cylindrical shell with
piezoelectric layers under the effect of asymmetric thermo- electro-mechanical loads.
2.4 Vibration and Control
Dimentberg [52] studied both viscous and hysteretic internal damping and listed that
viscous damping is destabilizing at speeds beyond the first critical, hysteretic damping is
destabilizing at all speeds. Gunter and Trumpler [53] studied the influence of internal friction
on the stability of high speed rotors with anisotropic supports. Ruhl [54], Ruhl and Booker
[55] described a FEM for modeling a rotor-system only including translational inertia and
bending effects. Lund [56] analyzed stability of damped critical speeds of a flexible rotor
supported by identical fluid film bearings. Dimarogonas [57] devised a general method for
stability analysis of rotating shafts including rotary inertia, gyroscopic effect and internal
damping with a combination of a transfer matrix technique. Zorzi and Nelson [58] developed
a FE simulation of rotor-bearing systems with considering internal damping (viz. viscous and
hysteretic damping) effects. Dutt and Nakra [59] founded that viscoelastic supports
increase the stability limit compared to that with either viscously damped flexible
supports or elastic supports. Abduljabbar et al. [60] addressed an active vibration controller
for controlling the dynamics of a flexible rotor running in flexibly-mounted journal
bearings. Wettergren et al. [61] analyzed that instability can be avoided or minimized with
appropriate selection of external damping, asymmetry in shaft stiffness and non-isotropic
force coefficients. Qin and Mao [62] developed a new shaft element model with ten DOF for
coupled torsional-flexural vibration of rotor systems including the effects of translational
and rotational inertia, gyroscopic moments, bending, shear and torsional deformations,
internal damping, and mass eccentricity. Dutt and Nakra [63] used a popular minimization
technique to reduce rotor vibration by applying viscoelastic bearing supports. Ku [64]
formulated a C0 class Timoshenko beam FE model to analyze the dynamic characteristics of a
rotor bearing system with hysteretic internal damping. By using Mori-Tanaka mean field
theory, M. Y. Chang et al. [65] implemented the vibration analysis of rotating composite
shafts containing randomly oriented reinforcement. They investigated the natural frequency
of stationary shafts and whirling speeds as well as critical speeds of rotating shafts. Roy et al.
[66] proposed an augmenting thermodynamic field (ATF) technique for theoretical study of
the dynamics of a viscoelastic rotor-shaft system containing that internal material damping.
By using TBT, Xiang et al. [67] analyzed free and forced vibration of a laminated FG beam
of variable thickness under thermally induced initial stresses and both the axial and rotary
inertia of the beam are considered. The beam consists of a homogeneous substrate and two
inhomogeneous FG layers whose material composition follows a power law distribution in
the thickness direction in terms of the volume fractions of the material constituents. Based on
the three-dimensional linear elasticity theory, Li et al. [68] analyzed free vibration of FG
material sandwich rectangular plates with simply supported and clamped edges. Das et al.
NIT ROURKELA
Page 10
[69] proposed an active vibration control scheme for controlling transverse vibration of a
rotor shaft due to unbalance forces and theoretical study. Using Bernoulli-Euler beam theory
(BEBT) and the rotational spring model, free vibration and buckling response of FG beams
with edge cracks were considered by Yang and Chen [70]. Based on TBT, Ke et al. [71]
studied free vibration and elastic buckling of beams made of FGMs containing open edge
cracks. It is assumed that the material properties follow exponential distributions along beam
thickness direction. By using multiple scales method, Hosseini et al. [72] studied free
vibrations of an in-extensional simply supported rotating shaft with nonlinear curvature and
inertia. Rotary inertia and gyroscopic effects are included, but shear deformation is neglected.
Boyaci et al. [73] investigated critical bifurcations emanating destructive limit-cycle
oscillations of higher amplitudes and the influence of the shaft elasticity on the critical limitcycle oscillations. By using HOSD, Mahi et al. [74] studied exact solutions of the free
vibration of a beam made of symmetric FG materials. Here material properties are taken to be
temperature-dependent and vary continuously through the thickness according to a power law
distribution (P-FGM), or an exponential law distribution (E-FGM) or a sigmoid law
distribution (S-FGM). By using boundary element method (BEM), Sapountzakis et al. [75]
developed a general flexural-torsional vibration problem of Timoshenko beams of arbitrarily
shaped composite cross-section taking into account the effects of warping stiffness, warping
and rotary inertia and shear deformation. Boukhalfa and Hadjoui [76] employed hp -version,
hierarchical finite element model and concerned with the dynamic behavior of the rotating
composite shaft on rigid bearings. Thermal buckling behaviour of FG beams associated with
different boundary conditions were investigated by Kiani and Eslami [77] by using the
EBBT. Shahba et al. [78] developed free vibration and stability analysis of axially FG
homogeneous tapered Timoshenko beams through a finite element approach. By using
EBBT, Alshorbagy et al. [79] presented the dynamic characteristics of FG beam by FEM
with material graduation in axially or transversally through the thickness based on the power
law. By EBBT and von Karman geometric nonlinearity, Rafiee et al. [80] investigated effects
of material property distribution on the nonlinear dynamic behavior of FG beams and effects
of different parameters on the frequency-response. Kumar et al. [81] developed analytical
solution for the free vibration analysis of FGM plate without enforcing zero transverse shear
stress conditions on the top and bottom surfaces of the plate using higher order displacement
model.
2.5 Motivation
Although literature review reveals that a lot of research work has been done on
vibration analysis of composite and functionally graded structures. The studies on the
analysis of vibration and stability of rotor shaft system with FGMs based on the Timoshenko
beam under thermo-mechanical loading yet have not been discussed. This present work will
explore vibration and stability analyses of functionally graded rotating shaft system based on
finite element method under thermal environment. FGM performance is first characterized
under thermal environments and mechanical loading in order to understand the unique
characteristics of FGMs and to compare FGM structural response to traditional metal
structure. The main advantage of using FGMs instead of traditional materials is that the
NIT ROURKELA
Page 11
internal composition of their component materials can be tailored to satisfy the requirements
of a particular structure. Although this technology has not fully implemented so internal
structure of the material could be prepared to manufacture pressure vessels and other thermal
structure. This work is an important step in being able to properly design mechanical
structure using a FGMs system. The core portion of structure is made FGMs could resist high
temperature and mass, structure size requirements can be reduced by tailoring the ingredient
of each component based upon load and stress interaction in different areas of mechanical
structure. FGMs are of increasing importance as designers seek a way to address structure
under combined mechanical and thermal loads. The finite element method is commonly
employed to analyze structure like beam, plate, shell and solid elements. But for FGMs it is
important step to achieve this goal, a first order shear deformation (FOSD) FG shaft model is
formulated and applied to shaft subjected to temperatures and mechanical loading.
2.6 Objectives of Present Work
The specific objectives of the present thesis have been laid down as
 Development of material modeling for FG shaft based on the different laws of
gradation
 Modeling of FG shaft with temperature dependent material properties
 Effects of different temperatures and power law gradient indexes on variation of
mechanical properties through the radial direction of the FG shaft.
 Finite modeling of FG spinning shaft system (i.e. rotor-bearing-shaft system) in order
to study the vibration behavior of this shaft system
 To study the vibration and stability analysis of FG shaft system by incorporating
internal viscous and hysteretic damping
 To comparative study of the various responses of FG shaft over steel shaft
 To study the effects of different temperatures and power law gradient indexes on the
various responses of the FG shaft.
 To study the dynamic behaviors (i.e. critical speed, fundamental frequencies,
Campbell diagram, Damping ratio, Time response and Stability Limit Speed) of
rotating FG shaft system under thermal and mechanical loadings by incorporating
internal viscous and hysteretic damping
NIT ROURKELA
Page 12
CHAPTER 3
MODELING FOR EFFECTIVE MATERIALS PROPERTIES OF FG
SHAFT
This chapter modeling of FG material to obtain the effective material properties of FG
shaft by considering power law gradation and exponential law gradation.
3.1 Effective Materials Properties of FGM
As FGMs are heterogeneous materials so there is need for the determination of
effective material properties. To achieve best performance, accurate material property
estimation is essential for analysis and design of FG structures/system. There are various
models developed to determine the material properties of FGM such as
1) Rules of mixtures: Linear rule of mixtures and Harmonic rule of mixtures
2) Variational approach
3) Micromechanical approaches
Rules of mixtures employ bulk constituent properties assuming no interaction
between phases. This approach derived from continuum mechanics and is free from empirical
considerations. In variational approach, variational principles of thermo-mechanics used to
derive the bounds for effective thermo-physical properties. Micromechanical approaches
include information about spatial distribution of the constituent materials. Standard
micromechanical approach is based on concept of unit cell or representative volume element
(RVE) to represent the microstructure of composite.
3.2 Modeling for Material Properties of FG Rectangular Cross-section
A FGM beam is considered having with finite length L and Thickness t and also made
of a mixture of ceramics (aluminum oxide) and metal (stainless steel). Here material in top
t
t
surface  z   and in bottom surface  z    of the shaft is ceramic and metal respectively.
2
2




The effective material properties P can be written as,
P  PcV c  P mV m
(1)
Where, P c and P m are the material properties of the ceramic and metal respectively.
Now V c and V m are the volume fractions of ceramic and metal respectively and they are
related by,
V c V m  1
NIT ROURKELA
(2)
Page 13
3.2.1 Power Law Gradation
In Fig. 3.1 V c describes the volume fraction of ceramic at any point z throughout the
thickness t according to volume fraction index n which controls the shape of the function and
variation is assumed to be in terms of a simple power law. The power law is expressed by,
 2z  t 
V c( z )  

 2t 
k
(3)
Where k is the volume fraction index ( k  0 ).
Fig. 3.1 Volume fraction of ceramic throughout the FGM layer
The material properties P that are temperature dependent can be written as,
2
3
P  P0 ( P1T 1  1  PT
 PT
)
1  PT
2
3
(4)
Where P1 , P1 , P2 and P3 are the coefficient of temperature T 1 , T , T 2 and T 3
respectively. From equation (1) to (4), the modulus of elasticity  E  , Poisson’s ratio   ,
coefficient of thermal expansion   , thermal conductivity
 K  and
the density    are
written as,
 2z  t 
E  z , T    Ec T   Em T  
  Em  T 
 2t 
k
 2z  t 
 ( z , T )   c T   m T  
   m T 
 2t 
k
 2z  t 
  z , T    c T    m T  
   m T 
 2t 
k
(5)
 2z  t 
K  z , T    K c T   K m T  
  K m T 
 2t 
k
 2z  t 
  z    c  m  
  m
 2t 
k
Here only density    is assumed to be a constant and only it is vary along radial
position and power law gradient index and independent of temperature.
NIT ROURKELA
Page 14
3.2.2 Exponential Gradation Law
In exponential gradation the materials properties are assume to vary according to
P  z   P0e
t

k z 
2

(6)
Where P0 refers the material properties of bottom surface of the FGM shaft and ‘ k ’ is
the parameter describing the gradation across the thickness direction. Now modulus of
elasticity, coefficient of thermal expansion, thermal conductivity and density of the FGM
shaft varies as,
E ( z )  E0 e
t 

k z 
2

K ( z )  K 0e
,  ( z )   0e
t 

k z 
2

t 

k z 
2

,  ( z )  0e
t 

k z 
2

(7)
Here Poisson ratio assumed as constant. This law reflects the simple rule of mixtures
used to obtain properties of FGM shaft and computational effort is to be reduced.
3.3 Modeling for Material Properties of FG Circular Cross-section
A FGM shaft is considered with finite length L, inner radius (ri ) and outer radius (ro ) .
Material of the shaft is considered in top surface ( z  ri / 2) as ceramic and in bottom surface
( z  ro / 2) as metal.
Fig. 3.2 Volume fraction of ceramic and metal throughout the FGM layer
3.3.1 Power Law Gradation
In Fig. 3.2, V c and V m describes the volume fraction of ceramic and metal
respectively at any point z throughout the radius r according to power law and power law
gradient index  k  controls the shape of the function. The power law is expressed by,
NIT ROURKELA
Page 15
 r  rm 
V m z  

 rc  rm 
k
(8)
Where power law gradient index is considered greater than and equal to zero i.e.
k 0
Now according to power law distribution variations of temperature dependent
material properties (Young’s modulus  E  , Poisson’s ratio   , Coefficient of thermal
expansion   and Thermal conductivity  K  ) along radial direction of FG circular shaft
becomes,
k
 r  rm 
E ( z , T )   Ec T   Em T  
  Em  T 
 rc  rm 
k
 r  rm 
 ( z, T )   c T   m T  
   m T 
 rc  rm 
k
 r  rm 
 ( z, T )   c T    m T  
   m T 
 rc  rm 
(9)
k
 r  rm 
K ( z , T )  K c T   K m T  
  K m T 
 rc  rm 
k
 r  rm 
 ( z )   c  m  
  m
 rc  rm 
Here only density    is assumed to be a constant and only it is vary along radial
position and power law gradient index.
Now, nonlinear temperature distribution (NLTD) is assumed to occur in the radial
direction of FG shaft where the temperature Tc and Tm are in ceramic-rich and metal-rich
surfaces respectively. In the absence of heat generation, the temperature field for the steadystate one dimensional Fourier equation of heat conduction law becomes as,
d 
dT 
K ( z)
0

dz 
dz 
(10)
Where, at z  (ri / 2) , T  Ttop and at z  (ro / 2) , T  Tbottom Now the solution of
equation obtained by polynomial series and written by,
T ( z )  Tm  (Tc  Tm ) ( z)
NIT ROURKELA
(11)
Page 16
Where,
k 1
2 k 1
3 k 1
2
3
 r  r 
 r  rm 
 r  rm  
K cm  r  rm 
K cm
K cm
m




 



 
(2k  1) K m2  rc  rm 
(3k  1) K m3  rc  rm  
1  rc  rm  (k  1) K m  rc  rm 
 ( z)  

4 k 1
5 k 1
4
5
C
 r  rm 
 r  rm 
K cm
K cm



5 
  (4k  1) K 4  r  r 

(5k  1) K m  rc  rm 
m  c
m 


C  1
2
3
4
5
Kcm
Kcm
K cm
K cm
K cm




(k  1) K m (2k  1) K m2 (3k  1) K m3 (4k  1) K m4 (5k  1) K m5
Kcm  Kc  Km .
NIT ROURKELA
Page 17
CHAPTER 4
FORMULATION FOR GOVERNING EQUATIONS OF ROTOR SHAFT
SYSTEM
4.1 Introduction
Based on the FOSD theory, shaft is modeled as a Timoshenko beam with considering
rotary inertia and gyroscopic effect. The shaft is considered uniform circular cross-section
and it is rotate at constant speed about its longitudinal axis. The displacements variables and
schematic diagram of rotor-bearing system are shown with coordinate systems in Fig. 4.1 and
Fig. 4.2 respectively.
Fig. 4.1 Displacement and rotational variables with coordinate systems
Fig. 4.2 Schematic diagram of rotor-bearing system with coordinate systems
NIT ROURKELA
Page 18
4.2 Mathematical Modeling of Functionally Graded Shaft
The displacement fields are assumed as follow
u x ( x, y , z , t )  z  x ( x , t )  y  y ( x , t )
v y ( x, y, z, t )  v0 ( x, t )
(12)
wz ( x, y, z, t )  w0 ( x, t )
where u x , vx and wx are the flexural displacements of any point on the cross-section
of the shaft in the x , y and z directions , the variables v0 and w0 are the flexural
displacements of the shaft’s axis, while  x and  y are the rotation angles of the cross-section ,
about the y and z axis, respectively. Now strain components in the cylindrical system can be
written in terms of the displacement variables as follow,
u
x
1  1 u
v
w 
 x  
 cos  sin 

2  r x
x
x 
1u
v
w 
 xr    cos  sin 

2 r
x
x 
 xx 
(13)
After simplifying,

 x
 r cos  y
x
x
  r  0
 xx  r sin 
 rr  
1
w
v 
1
v
w 
 xr   rx    x sin    y cos   sin  0  cos  0 
2
x
x 
(14)
 x   x    y sin    x cos   sin  0  cos  0 
2
x
x 
Now the strain terms can be represents in matrix form as,

0

  xx  
     sin  
 x  
x
  xr  
 cos  
x

NIT ROURKELA
0

x

sin 
x
cos 
r sin 

x
cos 
sin 

x   v0 
 
w0
sin    
 x 
 
 cos     y 

r cos 
(15)
Page 19
The stress–strain relations of the rth layer expressed in the cylindrical coordinate
system can be expressed as
 xx  C11r xx  ks C16r x
 x    x  ks C16r xx  ks C 66r x
(16)
 xr   rx  ks C 55r xr
In matrix form stress-strain relations of rth layer expressed in the cylindrical
coordinate system can be expressed as
 xx   C11r
    k C
 x   s 16r
  xr   0

ks C16r
ks C 66r
0
0    xx 

0   x 

ks C 55r    xr 
(17)
Stress-strain relation matrix,
 C11r

D   ks C16r

 0
ks C16r
ks C 66r
0
0 

0 

ks C 55r 
Where k s the shear correction factor and C ijr is the constitutive matrix which is
related to elastic constants of principle axes.
4.3 Strain Energy Expression for FG Shaft
Now Strain energy expression of FG shaft can be obtains as follows,
1
1
T
U s       dV   ( xx xx   rr  rr      2 xr  xr  2 x  x  2 r  r )dV
2V
2V
 rr     r  0
So strain energy expression can be rewritten as
Us 
1
 ( xx xx  2 xr  xr  2 x  x )dV
2V
(18)
Where total volume of the element V  (rd dxdr )
NIT ROURKELA
Page 20
Now,
 (
 )dV   (C11n xx  ks C16n x ) xx dV   (C11n xx 2  ks C16n x  xx )dV
xx xx
V
V
V
2
L
 L   2
 L  x   L  y   L v0  x   L w0  y  
  y   1
x
 B11   
 dx    x  dx   2 ks A16    y x dx      x x dx     x x dx     x x dx  
 0  x 
 
0
 0
 0
 0
 
 0
 (2
V
xr
 xr )dV   (2ks C 55 n xr ) xr dV   ks C 55 n ( xr ) 2 dV
V
V
2
L
 2
 L
  L

v0
w0
 v0 
 w0 
2
 ks A55    x dx    y dx   
dx   
dx   2  y
dx    2  x
dx  


x 
x 
x
x
 0
0
0
0
 0
  0
 
L
L
 (2    )dV   2(k C
x
V
2
L
x
s
  ks C 66n x ) x dV
16n xx
V
L
1

  L  y   L v0  x   L w0  y  


dx    
dx    
dx  
 ks A16    y x dx      x

x   0
x   0 x x   0 x x  
2

 0


2
2
L
L
L
 L 2
 L w0   L v0  

 v0 
 w0 
2
  ks A66    x dx    y dx   
 dx    x  dx   2  x x dx    2  y x dx  

x



0
0
0
 0
  0
 
 0

Now taking first variation of above strain energy expression and obtains,
L
  L   x  x 
 L   x  x


  y  y


dx



dx


 x dx
  
  y

 x

y


x
x
x
x

1
  0  x x 

0
0 


 ks A16  L
 B11  L  

L
2
 y  y  w0   
y 
    y
   v0  x   x  v0 dx   w0

dx


0  x x  x x dx  
  0  x x  
 0  x x
x x 

U S  
L
L
L




 v0  v0 

 (19)
   x x dx    y y dx   

dx

x

x



0
0
0

  ks  A55  A66   L

L
L



   w0  w0 dx    y  v0  v0  y dx    x  w0  w0  x dx 








x

x

x

x

x

x






0
0
 0



The terms A16, A55, A66 and B11 are given in Appendix.
4.4 Kinetic Energy Expression for FG Shaft
The kinetic energy of the rotating composite shaft including the effects of translatory
and rotary can be written as follow,





1 L

Ts   Im (v0 2  w0 2 )  I d (  x 2   y 2 )  2I p  x  y 2  2 I d (  x 2   y 2 )  2 I p dx
20

NIT ROURKELA
(20)
Page 21
Where Ω is the rotating speed of the shaft which is assumed constant, L is the total

length of the shaft, the gyroscopic effect denotes by 2 P  x  y and rotary inertia effect is

 

representing by I d   x 2   y 2  . The mass moment of inertia denotes by I m and the


diametrical mass moments of inertia represent by I d and polar mass moment of inertia I p of
rotating shaft per unit length is defined in the appendix. As the term 2 I d   x 2   y 2  is far
smaller than 2 I p so it will neglect in further analysis. The terms I m , I d and I p are given in
appendix.
Now taking first variation of the kinetic energy of shaft obtains as,
 
    v0   w0 
   x
y
 w0
 Id  x
 y
Im  v0

L
t
t 
t
t


 Ts   
0
 y 
 
 I p   y  x   x

t 





dx



(21)
4.5 Kinetic Energy Expression for Disks
Here assumption is made that disks are fixed to the shafts and material of the shafts
are considered as isotropic material and kinetic energy expression of the disks can be obtains
as ,

1 L ND  D  2  2  D   2  2 
D
2
2 D
Td     Imi
v

w

I




2

I


0
0 
di  x
y 
pi x y   I pi   x  xDi  dx

2 0 i 1  




(22)
D
D
Where disks position is denotes by i(=1,2,3….) and I mi , I di and I piD are denotes the
mass moment of inertia , the diametrical mass moment of inertia and the polar mass moment
of inertia respectively. The symbol   x  xDi  denotes the one dimensional spatial Dirac
delta function and N D is the number of discrete disks which is attached with shaft and xDi is
the location of the disk.
Now taking the first variation of the kinetic energy of the disk, obtains
 D    v0   w0  D    x   y
 w0
 I di   x
 y
 Imi  v0
L ND
t
t 
t
t



 Td    
0 i 1
 y 
 
 I piD   y  x   x


t 

NIT ROURKELA



  x  xDi  dx


(23)
Page 22
4.6 Work done Expression due to External Loads and Bearings
Here Ry , Rz is assumed the external force intensities (force per unit length) which is
subjected on the shaft and M x , M y is the external torque intensities (moment per unit length),
which is distributed along the shaft. Now virtual work-done by the external loads can be
represents as follow,
 WE    Ry v0  Rz w0  M y y  M x x dx
L
(24)
0
Now bearings are modeled by springs and viscous dampers. Virtual work done by
springs and dampers can be represents as,
Bi
  K yy
v0 v0  K zyBi v0 w0  K yzBi w0 v0  K zzBi w0 w0 
( x  x )dx
 WB    








Bi
Bi
Bi
Bi
Bi


0 i 1

C
v

v

C
v

w

C
w

v

C
w

w
yy
0
0
zy
0
0
yz
0
0
zz
0
0


L NB
(25)
Where, N B denotes number of bearings, xBi denotes the positions of bearings, K Bi
denotes the equivalent stiffness of i th bearings, C Bi denotes the equivalent damping
coefficient of i th bearings.
4.7 Governing Equations of Rotor-Shaft System
The governing equations of rotor-shaft system can be derived using equation number
(19), (21), (23), (24) and (25) and applying Hamilton’s principle which is,
t2
  T   U s   WE   WB dt  0
t1
Since total kinetic energy of the shaft and disks is T  Ts  Td
t2
  (Ts  Td )   U s   WE   WB dt  0
t1
Now simplifying arranging the above equation the motion equations obtains as,
  y  2v0  1
 2v0
 2  x N D D  2v0
 v0 : Im 2  ks  A55  A66  
 2   ks A16 2   Imi 2   x  xDi   Pvb0  Ry
i 1
t
x
t
 x x  2
 2  y N D D  2 w0
  2 w0  x  1
 2 w0
b
 w0 : Im 2  ks  A55  A66   2 
  ks A16 2   Imi 2   x  xDi   Pw0  Rz
i 1
t
x  2
x
t
 x
NIT ROURKELA
Page 23
 x : I d
 y 1
2x
 2 v0
2x
 w


I


k
A

B
 k s  A55  A66   0   x 
p
s 16
11
2
2
2
t
t
2
x
x
 x

 y 
 D 2x
D
   I di


  x  xDi   M x
pi
i 1 
t 2
t 
2 y
2 y
 x 1
 2 w0
v 

 y : I d
 I p
 k s A16
 B11
 k s  A55  A66    x  0 
2
2
2
t
t
2
x
x
x 

ND 

2 y
D
D  x
   I di


  x  xDi   M y
pi
i 1 
t 2
t 

ND
4.8 Solution Procedure
Finite element analysis aim is to find out the field variable (displacement) at nodal
points by approximate analysis, assuming at any point inside the element basic variable is a
function of values at nodal points of the element, this function is called shape function or
interpolation function. Now here employing the approximate solution method the governing
equations are approximated by a system of ordinary differential equations. In the present
finite element model, the three-nodded one-dimensional line elements are consider, each
node having four degrees of freedom.
Lagrangian interpolation functions are used to approximate the displacement fields of
shaft. The element’s nodal degree of freedom at each node is v0 , w0 ,  x and  y . Now
displacement field variable can be represents as,
v0   v0k  t  k   ,  x    xk  t  k ( )
r
r
k 1
k 1
(26)
w0   w0k  t  k   ,  y    yk  t  k ( )
r
r
k 1
k 1
Now one dimensional Lagrange polynomial defined as,
  m
m 1   
k
m
r
Lk ( )  
(27)
m k
Obviously equation (equation 27) takes the value equal to zero at all points except at
point k (=1). Here Lk (k=1, 2, 3....r), is Lagrange polynomial and  k (k=1, 2, 3….) is the
interpolation function or approximation function and  is the natural Coordinate whose
value varies between -1 and 1. Now General Lagrange polynomial is written as follow,
Lk ( ) 
(  1 )(  2 )......(  k 1 )(  k 1 )......(  r )
(k  1 )(k  2 )......(k  k 1 )(k  k 1 )......(k  r )
NIT ROURKELA
(28)
Page 24
Now for three nodded element (r = 3), shape function or interpolating function can be
written as follow,
1 
 1   
 1   
,  2  1  2 ,  3 
2
2
Now putting the above expression of displacement variables into the governing
equations, obtains the following equation of motion of the spinning shaft system as,
M 




q  C    G  q   K q  F 
(29)
Where,  M  = Mass matrix,
G  = Gyroscopic matrix, C  = Total damping
matrix,  K  = Structural Stiffness matrix, q = Nodal Displacement vector,  F  = External
force vector. All the elemental matrices and vectors are given in appendix.
4.9 Contribution of Internal Damping
Based on Zorzi and Nelson [58], Qin and Mao [62] the derivation of rotor dynamic
Lagrangian equation of motion including the contribution of both internal viscous and
hysteretic damping of shaft disk element can be extended as,
M 


 1  

H
q  C    G   V  K  q  

 1   H2


Where KCir is the


H
  K   V  


1   H2




  K Cir  q  F  (30)




skew-symmetric circulation matrix. After assembly of the all
elemental matrices, the equation of motion of the FG shaft can be obtained.
NIT ROURKELA
Page 25
CHAPTER 5
RESULTS AND DISCUSSIONS
Based on the above formulation a complete MATLAB program has been developed
and validated. Different types of analysis have been carried out and presented based on
following problem specification.
5.1 Problem Specifications and Summarized of Various Analyses
Three disks rigidly mounted on shaft supported by two identical bearings has been
modeled and analyzed using beam elements. All dimensions of this system are given in the
Table 5.1. The shaft has been discretized by 13 elements (it is decided after convergence
study). The stiffness and damping coefficients of each bearing are taken as Kyy = 7×107
N/m, Kzz =5×107 N/m, Cyy =700Ns/m, and Czz =500Ns/m at both ends. All the necessary data
used in the present different analyses are given in Table 5.2 and Table 5.3.
The recent developed code has been validated with the results available in literatures.
Various type of analysis for FG shaft have been studied and presented in the following
sections. Firstly, it is shown that how temperature is distributed in radial direction of the FG
shaft for different temperatures and power law gradient indexes. Secondly, the variation of
mechanical properties (such as Young modulus, Poisson ratio, Thermal conductivity,
coefficient of thermal expansion and mass density etc.) is presented for different radial
position, different temperatures and power law gradient indexes. Thirdly, comparative study
has been presented and discussed for FG shaft over steel shaft. Fourthly, the effects of power
law gradient indexes have been analyzed for various responses of FG shaft. Fifthly, the
effects of both power law gradient indexes and temperatures variations have been analyzed
discussed for various responses of FG shaft. Finally, effect of both internal viscous and
hysteretic damping have been carried out on the Campbell diagram.
Table 5.1 Mechanical properties and geometric dimension of steel rotor-shaft system [69]
Parameter
Shaft
Rotor shaft length (m)
1.3
Rotor shaft diameter (mm)
100
Young’s Modulus (Gpa)
200
Coefficient of viscous damping(s) 0.0002
Coefficient of hysteretic damping 0.0002
Eccentricity (m)
Density (Kg/m3)
7800
Outer diameter (m)
Thickness (mm)
Position from left end of rotor (m)
NIT ROURKELA
Disk 1
Disk 2
7800
0.24
5.0
0.2
0.0002
7800
0.40
5.0
0.5
Disk 3
7800
0.40
6.0
1.0
Page 26
Table 5.2 Material properties of FGM compositions [79]
Properties
Stainless Steel (SUS304)
Young’s Modulus (GPa)
Density (Kg/m3)
Poisson Ratio
Aluminum oxide (Al2O3)
210
7800
0.3
390
3960
0.26
Table 5.3 Materials and temperature coefficients value for mechanical properties [34].
Property Material
E (Pa)
SUS304
Al2O3
K
SUS304
(W/m K) Al2O3
CTE
SUS304
(1/K)
Al2O3
Poisson SUS304
ratio
Al2O3
P0
P-1
P1
201.0354e9
0
3.079296e-4
349.5486e9
0
-3.853206e-4
15.37895
0
-0.001264
-14.087
-1123.6 0.00044
12.33e-6
0
0.0008
6.827e-5
0
0.00018
0.32622351
0
-2.001822e-4
0.26
0
0
P2
P3
-6.533971e-7
0
4.026993e-7 -1.6734e-10
0.20923e-5
-0.0722e-8
0
0
0
0
0
0
3.797358e-7
0
0
0
5.2 Code Validation
In order to verify the FE developed code the following dimensions and mechanical
properties were considered for steel shaft in Das [69] (details of which are given in Table
5.1). In order to convergence study of the result, it has been observed that result from the
present code has been achieved an excellent agreement with the already published results of
Das [69] and thus validates the correctness of the developed code. It is shown in Fig. 5.1.
Whirling Speed (rpm)
3
x 10
4
2.5
2
1.5
1
0.5
0
0
0.5
1
1.5
2
Rotating Speed (rpm)
2.5
3
x 10
4
Fig. 5.1 Campbell diagram for first two pairs of modes
NIT ROURKELA
Page 27
5.3 Temperature Distribution in a FG Shaft
The temperature distribution in FG layer is nonlinear which is clearly shown in Fig. 5.
2. This is due to that of thermal conductivity, coefficient of thermal expansion, modulus of
elasticity and density which are the function of the radius of shaft only. For k  0 and k   ,
the temperature distribution is a straight line and does not depend upon the material properties
of the shaft. For other values of k , the temperature distribution depends upon radial positions
and material properties and also the law of gradation and it is presented in Table 5.4. From
the Table 5.4, it is clear that from zero to one and one to ten values of k , temperature
gradually decreases and increases respectively.
Table 5.4 Temperature variation of FG shaft for different radial position and power law
gradient index  k 
r (m)
0.05
0.06
0.07
0.08
0.09
0.1
k
0.0
0.5
1.0
3.0
5.0
10
T1 (K)
300
300
300
300
300
300
T2 (K)
360
352.5686
352.2281
355.1254
356.6653
358.1465
T3 (K)
420
409.6354
407.8254
410.7097
413.3858
416.2933
T4 (K)
480
470.0942
467.2561
468.0016
470.6490
474.4676
T5 (K)
T6 (K)
540
600
533.6039
600
531.0844
600
529.4601
600
530.5007
600
533.2644
600
Temperature Variation (T)
600
550
500
k = 0.0
k = 0.5
k = 1.0
k = 3.0
k = 5.0
k = 10.0
450
400
350
300
0.05
0.06
0.07
0.08
Position (r)
0.09
0.1
Fig. 5.2 Variation of temperature with radial position and power law gradient index
5.4 Variations of Mechanical Properties of FG Shaft with Positions and Temperatures
In the FG shaft model, shaft is composed of ceramic (aluminum oxide (Al2O3)) and
metal (stainless steel (SUS304)) and its material property vary according to power law
NIT ROURKELA
Page 28
gradation. Also aluminum oxide and stainless steel (SUS304) are considered at the top and
the bottom surfaces of the shaft respectively. Now based on the above formulation it has been
found that, for k  0 the material approaches to a homogeneous ceramic, and for k   the
material becomes entirely metal, for 0  k   , the material will contain both metal and
ceramic and for k  1 ceramic and metal contribution is equal in the material. For a variation
of mechanical properties, the required material properties and temperature coefficients of
aluminum oxide and stainless steel are given in Table 5.1, Table 5.2 and Table 5.3
respectively. Now the Fig. 5.3, Fig. 5.4 and Fig. 5.5 show the mechanical properties of the
FG shaft with various positions and power law gradient indexes (k ) .
Radial Position (m)
0.05
0.04
0.03
k=0.3
k=0.6
k=1.0
k=2.0
k=5.0
k=10
0.02
0.01
0
2
2.5
3
3.5
Modulus of Elasticity (Pa)
4
11
x 10
Fig. 5.3 Variation of Young modulus with power law gradient index of FG shaft
0.05
Radial Position (m)
k = 10
0.04
k=5
0.03
k=2
0.02
k=1
0.01
k = 0.6
k = 0.3
0
0.26
0.27
0.28
Poission Ratio
0.29
0.3
Fig. 5.4 Variation of Poisson ratio with power law gradient index of FG shaft
NIT ROURKELA
Page 29
Radial Position (m)
0.05
0.04
0.03
0.02
0.01
0
3000
k=0.3
k=0.6
k=1.0
k=2.0
k=5.0
k=10
4000
5000
6000
Density (Kg/m3)
7000
8000
Fig. 5.5 Variation of density with power law gradient index of FG shaft
The mechanical properties of FG shaft with changing position, different temperatures
and law of gradation index values are presented in the Table 5.5 and Table 5.6.
Table 5.5 Variation of Young’s modulus with different radial positions, temperatures and
power law gradient indexes of FG shaft
k
T(K)
0
0.01
0.02
0.03
0.04
0.0
300
420
600
3.2023
3.1348
3.0678
3.2023
3.1348
3.0678
3.2023
3.1348
3.0678
3.2023
3.1348
3.0678
3.2023
3.1348
3.0678
3.2023
3.1348
3.0678
0.5
300
420
600
2.0778
2.0386
1.9089
2.5807
2.5288
2.4272
2.7890
2.7319
2.6418
2.9489
2.8877
2.8065
3.0836
3.0190
2.9454
3.2023
3.1348
3.0678
1.0
300
420
600
2.0778
2.0386
1.9089
2.3027
2.2579
2.1407
2.5276
2.4771
2.3724
2.7525
2.6963
2.6042
2.9774
2.9155
2.8360
3.2023
3.1348
3.0678
5.0
300
420
600
2.0778
2.0386
1.9089
2.0782
2.0390
1.9093
2.0894
2.0499
1.9208
2.1653
2.1239
1.9990
2.4463
2.3978
2.2886
3.2023
3.1348
3.0678
10
300
420
600
2.0778
2.0386
1.9089
2.0778
2.0386
1.9089
2.0780
2.0388
1.9090
2.0846
2.0453
1.9159
2.1986
2.1563
2.0333
3.2023
3.1348
3.0678
NIT ROURKELA
r (m)
E(1011Pa)
0.05
Page 30
Table 5.6 Variation of thermal conductivity with different radial positions, temperatures and
power law gradient indexes of FG shaft
k
T(K)
r (m)
K(1/K)
0
0.01
0.02
0.03
0.04
0.05
0.0
300
420
600
36.814
20.9958
8.5743
36.814
20.9958
8.5743
36.814
20.9958
8.5743
36.814
36.814
20.9958 20.9958
8.5743
8.5743
36.814
20.9958
8.5743
0.5
300
420
600
12.1434
12.0680
12.9010
23.1765 27.7465
16.0606 17.7144
10.9661 10.1646
31.2532 34.2095
18.9835 20.0533
9.5496 9.0311
36.8140
20.9958
8.5743
1.0
300
420
600
12.1434
12.0680
12.9010
17.0775
13.8536
12.0357
22.0117
15.6391
11.1703
26.9458
17.4247
10.3050
31.8799
19.2102
9.4396
36.8140
20.9958
8.5743
5.0
300
420
600
12.1434
12.0680
12.9010
12.1513
12.0709
12.8997
12.3961
12.1594
12.8567
14.0618
12.7622
12.5646
20.2275
14.9935
11.4832
36.8140
20.9958
8.5743
10
300
420
600
12.1434
12.0680
12.9010
12.1434
12.0680
12.9010
12.1460
12.0690
12.9006
12.2926
12.1220
12.8749
14.7924
13.0266
12.4365
36.8140
20.9958
8.5743
0.05
Radial Position (m)
0.04
0.03
0.02
0.01
0
0.26
Power law gradation
without temperature
Power law gradation
with T = 300 K
Power law gradation
with T = 420 K
Power law gradation
with T = 600 K
0.27
0.28
0.29
0.3
0.31
0.32
0.33
0.34
Poission Ratio
Fig. 5.6 Variation of Poisson ratio with different temperature and power law gradient index
through radial direction of FG shaft
NIT ROURKELA
Page 31
The Fig. 5.6 and Fig. 5.7 show the variation of Poisson’s ratio and Young’s Modulus
with and without the temperature effect considering k  5.0 . From Fig. 5.6 It is clearly
noticed that without the temperature, the values of Poisson’s ratio less than that of
considering temperature and it is increases with increasing temperature. From Fig. 5.7 It is
clearly noticed that without the temperature, the values of Young’s Modulus more than that
of considering temperature and it is decreases with increasing temperature.
Radial Position (m)
0.05
0.04
0.03
Power law gradation without temperature
Power law gradation with T = 300 K
Power law gradation with T = 420 K
Power law gradation with T = 600 K
0.02
0.01
0
2
2.5
3
Modulus of Elasticity (Pa)
3.5
11
x 10
Fig. 5.7 Variation of Young modulus with different temperature and power law gradient
index through radial direction of FG shaft
5.5 Comparative Studies of FG Shaft over Steel Shaft
For a comparative study of FG shaft over steel shaft all necessary data are given in
Table 5.1, Table 5.2 and Table 5.3. Fig. 5.8 (a) and (b) show the comparison of Campbell
diagram of FG shaft over steel shaft and it has been observed that the first critical speed
occurs at 3891 rpm for steel shaft where as for FG shaft the first critical speed occurs at
4534 rpm. Fig. 5.9 (a) and (b) show the variation of maximum real part against speed of
rotation for the FG and steel shafts respectively. It has been noticed that the maximum real
part for steel shaft gives 17.45 rpm where as for the FG shaft it becomes 23.99 rpm. Fig.
5.10 (a) and (b) show the variation of damping ratio for first six modes of FG and steel shaft
respectively. It has been observed that for the first forward mode of whirling damping ratio
becomes negative at around 4756 rpm for steel shaft and beyond this speed it is unstable. And
also for the first forward mode of whirling damping ratio becomes negative at around 10220
rpm for the FG shaft and beyond this speed it is unstable. So from the Fig. 5.8 (a) and (b),
Fig. 5.9 (a) and (b) and Fig. 5.10 (a) and (b) it is evidently clear that the FG shaft is more
stable than the steel shaft in same operating conditions.
NIT ROURKELA
Page 32
Whirling Speed (rpm)
3.5
x 10
4
FGM
3
III F Mode
2.5
III B Mode
2
II F Mode
1.5
1
II B Mode
I F Mode
0.5
0
0
3
0.5
x 10
I B Mode
1
1.5
2
2.5
3
4
Rotor Spin Speed (rpm)
x 10
4
Whirling Speed (rpm)
STEEL
2.5
2
1.5
1
III F Mode
III B Mode
II F Mode
II B Mode
I F Mode
0.5
I B Mode
0
0
0.5
1
1.5
2
Rotor Spin Speed (rpm)
2.5
3
x 10
4
Fig. 5.8. Comparison of Campbell diagrams of rotating shafts: (a) FG and (b) Steel
NIT ROURKELA
Page 33
Maximum real part of eigenvalus
Maximum real part of eigenvalus
40
FGM
20
0
-20
-40
0
0.5
1
1.5
2
Rotating Speed (rpm)
2.5
3
x 10
4
80
STEEL
60
40
20
0
-20
0
0.5
1
1.5
2
Rotating Speed (rpm)
2.5
3
x 10
4
Fig. 5.9 Variation of maximum real part against speed of rotation: (a) FG and (b) Steel
NIT ROURKELA
Page 34
0.6
Mode 5
FGM
Damping Ratio
0.4
Mode 6
Mode 3
0.2
Mode 4
Mode 1
0
Mode 2
-0.2
0
2000
4000
6000
8000
Rotating Speed (rpm)
10000
12000
0.6
Mode 5
STEEL
Damping Ratio
0.4
Mode 6
Mode 3
0.2
Mode 1
Mode 4
0
Mode 2
-0.2
0
5000
10000
Rotating Speed (rpm)
15000
Fig. 5.10 Variation of damping ratio for the first six modes of rotating shafts:
(a) FG and (b) Steel
5.6 Comparative Studies of FG Shaft with and without Temperatures
For a comparative study of FG shaft over steel shaft, all necessary data are given in
Table 5.1, Table 5.2 and Table 5.3. Fig. 5.11 (a) and (b) show the Campbell diagram of the
FG shaft system considering with and without temperature effects and it has been observed
that with temperature the first critical speed occurs at 4625.62 rpm where as for without
temperature it becomes 4963.31 rpm. Fig. 5.12 (a) and (b) also show the variation of
maximum real part against speed of rotation of FG shaft considering with and without
temperature effect respectively. It has been noticed that with temperature the maximum real
part comes 25.187 rpm where as for without temperature it becomes 29.215 rpm.
NIT ROURKELA
Page 35
Fig. 5.13 (a) and (b) show the variation of damping ratio for the first six modes of FG
shaft considering with and without the temperature effect respectively. It has been observed
that for the first forward mode of whirling damping ratio becomes negative at around 9853
rpm for with temperature where as at around 12000 rpm for without temperature and beyond
this speed it is unstable. So from the Fig. 5.11 (a) and (b), Fig. 5.12 (a) and (b) and Fig. 5.13
(a) and (b) , it is evidently clear that without temperature the FG shaft is more stable.
Whirling Speed (rpm)
3.5
x 10
4
3
2.5
2
With Temperature effect
1.5
1
0.5
0
0
Whirling Speed (rpm)
3.5
x 10
0.5
1
1.5
2
Rotating Speed (rpm)
2.5
3
x 10
4
4
3
2.5
2
1.5
1
With out Temperature effect
0.5
0
0
0.5
1
1.5
2
Rotating Speed (rpm)
2.5
3
x 10
4
Fig. 5.11 Comparison of Campbell diagrams of rotating FG shafts: (a) with temperature and
(b) without temperature
NIT ROURKELA
Page 36
Maximum real part of eigenvalus
Maximum real part of eigenvalus
60
40
With Temperature effect
20
0
-20
-40
0
0.5
1
1.5
2
Rotating Speed (rpm)
2.5
3
x 10
4
40
With out Temperature effect
20
0
-20
-40
0
0.5
1
1.5
2
Rotating Speed (rpm)
2.5
3
x 10
4
Fig. 5.12 Variation of maximum real part against speed of rotation of FG shaft:
(a) with temperature and (b) without temperature
NIT ROURKELA
Page 37
0.6
Damping Ratio
0.4
With Temperature effect
0.2
0
-0.2
0
2000
4000
6000
8000
Rotating Speed (rpm)
10000
12000
10000
12000
0.6
Damping Ratio
0.4
With out Temperature effect
0.2
0
-0.2
0
2000
4000
6000
8000
Rotating Speed (rpm)
Fig. 5.13 Variation of damping ratio for the first six modes of rotating FG shaft:
(a) with temperature and (b) without temperature
5.7 The Effect of Different Gradient Indexes on Various Responses of FG Shaft
In order to study the responses of FG shaft all others necessary data are given in Table
5.1, Table 5.2 and Table 5.3. From the Fig. 5.14 (a) and (b), it has been found that the first
critical speed occurs at 4581.56 rpm for k  10 where as for k  5 the first critical speed
occurs at 4687.68 rpm. Fig. 5.15 (a) and (b) show that the maximum real part is 25.67 rpm
for k  5 and for k  10 it comes 24.67 rpm. From the Fig. 5.16 (a) and (b), it is clear that
the first forward mode of whirling damping ratio becomes negative at around 11180 rpm for
k  5 and for k  10 , it comes 10580 rpm and beyond these speeds it is unstable.
NIT ROURKELA
Page 38
The Table 5.7 indicates the significance of the power law gradient index in vibration
and stability analysis of FG shaft. It is observed from this analysis, the less value of the
power law gradient index gives more stable system than that the higher value of the power
law gradient index.
Whirling Speed (rpm)
3.5
x 10
4
III F Mode
3
III B Mode
2.5
2
K=5
1
II B Mode
I F Mode
0.5
0
0
3.5
Whirling Speed (rpm)
II F Mode
1.5
x 10
3
0.5
I B Mode
1
1.5
2
Rotor Spin Speed (rpm)
2.5
3
x 10
4
4
III F Mode
III B Mode
2.5
K=10
2
1.5
II F Mode
1
II B Mode
I F Mode
0.5
0
0
0.5
I B Mode
1
1.5
2
2.5
3
4
Rotor Spin Speed (rpm)
x 10
Fig. 5.14 The Campbell diagram of FG shaft: (a) k  5 and (b) k  10
NIT ROURKELA
Page 39
Maximum real part of eigenvalus
Maximum real part of eigenvalus
40
K=5
20
0
-20
-40
0
0.5
1
1.5
2
Rotating Speed (rpm)
2.5
3
x 10
4
40
K = 10
20
0
-20
-40
0
0.5
1
1.5
2
Rotating Speed (rpm)
2.5
3
x 10
4
Fig. 5.15 The maximum real part against spin speed of FG shaft: (a) k  5 and (b) k  10
Table 5.7 First critical speed and maximum real part for different power law gradient indexes
k
0.0
0.5
1.0
3.0
First critical
speed (rpm)
5025.45
4963.31
4911.76
4770.93
NIT ROURKELA
Max. Real
part
30.216
29.215
28.382
26.465
k
5.0
7.0
10.0
15.0
First critical
speed (rpm)
4687.68
4633.50
4581.56
4534.16
Max. Real
part
25.672
25.176
24.674
23.990
Page 40
0.6
Damping Ratio
k=5
0.4
Mode 5
Mode 6
Mode 4
0.2
Mode 3
Mode 1
0
Mode 2
-0.2
0
2000
4000
6000
8000
Rotating Speed (rpm)
10000
12000
0.6
Damping Ratio
k = 10
0.4
Mode 5
Mode 6
Mode 4
0.2
Mode 3
Mode 1
0
Mode 2
-0.2
0
2000
4000
6000
8000
Rotating Speed (rpm)
10000
12000
Fig. 5.16 The damping ratio of first six modes of FG shaft: (a) k  5 and (b) k  10
5.8 The Effects of Different Temperatures and Gradient Indexes on Various Responses
of FG Shaft
In order to obtain the FG shaft responses the power law gradient index is considered
k  0.5 and all others necessary data are given in Table 5.1, Table 5.2 and Table 5.3. From
the Fig. 5.17 (a) and (b) , it has been found that first critical speed occurs at 4625.62 rpm for
T  300K where as for T  600K the first critical speed occurs at 4554.67 rpm. Fig. 5.18
(a) and (b) show that the maximum real part is 25.19 rpm for T  300K where as for
T  600K it becomes 24.26 rpm. Now from Fig. 5.19 (a) and (b), it is clear that the first
forward mode of whirling damping ratio becomes negative at around 9877 rpm where as for
T  300K and for T  600K it becomes 9623 rpm and beyond these speeds it is unstable.
NIT ROURKELA
Page 41
The Table 5.8 indicates the significance of both the values of temperature and the
power law gradient index on the vibration and stability analysis. The temperature and power
law gradient index have a significant role in the responses of FG shaft. It is also clear from
this analysis that less value of temperature and the power law gradient index gives more
stable system than that of higher values of temperature and power law gradient index.
Whirling Speed (rpm)
3.5
x 10
4
3
2.5
2
T = 300K
1.5
1
0.5
0
0
Whirling Speed (rpm)
3.5
x 10
0.5
1
1.5
2
Rotor Spin Speed (rpm)
2.5
3
x 10
4
4
3
2.5
2
T = 600K
1.5
1
0.5
0
0
0.5
1
1.5
2
Rotor Spin Speed (rpm)
2.5
3
x 10
4
Fig. 5.17 The Campbell diagram of FG shaft: (a) T  300K and (b) T  600K
NIT ROURKELA
Page 42
Maximum real part of eigenvalus
Maximum real part of eigenvalus
60
40
20
T = 300K
0
-20
-40
0
0.5
1
1.5
2
Rotating Speed (rpm)
2.5
3
x 10
4
60
40
20
T = 600K
0
-20
-40
0
0.5
1
1.5
2
Rotating Speed (rpm)
2.5
3
x 10
4
Fig. 5.18 Maximum real part against spin speed of FG shaft:
(a) T  300K and (b) T  600K
NIT ROURKELA
Page 43
Damping Ratio
0.6
0.4
T = 300K
0.2
0
-0.2
0
2000
4000
6000
8000
Rotating Speed (rpm)
10000
12000
10000
12000
Damping Ratio
0.6
0.4
T = 600K
0.2
0
-0.2
0
2000
4000
6000
8000
Rotating Speed (rpm)
Fig. 5.19 Damping ratio for first six modes of FG shaft: (a) T  300K and (b) T  600K
NIT ROURKELA
Page 44
Table 5.8 First critical speed and maximum real part for different values of temperatures and
power law gradient indexes  k 
k
0.0
0.5
1.0
T(K)
300
360
420
480
540
600
First critical
speed (rpm)
4733.41
4716.22
4701.49
4688.93
4678.25
4669.09
Max. Real k
part
26.269
26.113
25.981 3.0
25.869
25.774
25.693
T (K)
300
360
420
480
540
600
First critical
speed (rpm)
4286.47
4272.23
4256.29
4238.41
4218.31
4195.69
300
360
420
480
540
600
4131.17
4118.06
4101.88
4082.37
4059.25
4032.19
19.406
19.290
19.147
18.976
18.773
18.538
300
360
10.0 420
480
540
600
3918.55
3907.02
3890.65
3869.21
3842.39
3809.84
17.534
17.439
17.303
17.127
16.907
16.644
300
360
420
480
540
600
4625.62
4609.06
4593.98
4580.08
4567.09
4554.67
25.187
25.042
24.823
24.621
24.434
24.256
300
360
420
480
540
600
4536.28
4520.29
4504.95
4489.97
4475.07
4459.94
24.112
23.881
23.661
23.449
23.239
23.029
5.0
Max. Real
part
20.948
20.752
20.563
20.398
20.213
20.007
5.9 Time Responses for FG Shaft System due to Unbalance Masses
The Fig. 5.20 (a) and (b) show the displacement histories (stable and unstable
responses) due to unbalance masses in transverse directions for the index of k  0.5 and
T  300K . Here, the time responses in the transverse directions of this shaft have also been
obtained considering the unbalance mass of the disk 2 and the responses of this system have
been calculated with a time step of τ/10 s (where τ is the time period corresponding to the
first natural frequency of the system). Finally from the Fig. 5.20 (a) and (b) , it is clear that
for stable response the maximum amplitude is 5.676 105 m and for unstable response the
maximum amplitude is 1.897 104 m and also from Table 5.9, it is found that for both the
less values of the power law gradient index as well as the temperature variations, the
maximum amplitude is less, thus it promotes more stable system than that of higher values of
power law gradient and temperature variations.
NIT ROURKELA
Page 45
Table 5.9 Maximum amplitudes for different temperatures and power law gradient index (k )
k
T(K)
Stable response
Max. Amplitude
(10-5 m)
V- direction
W- direction
Unstable response
Max. Amplitude
(10-4 m)
V- direction
W- direction
0.0
300
420
600
8.762
8.330
7.980
6.935
6.593
6.351
2.406
2.896
3.801
2.004
2.359
3.205
0.5
300
420
600
7.164
6.883
6.371
5.676
5.556
5.291
2.207
2.651
3.297
1.897
2.418
2.806
1.0
300
420
600
6.188
5.959
5.391
5.005
4.868
4.729
2.530
3.139
3.832
2.229
2.716
3.168
3.0
300
420
600
4.230
4.084
3.950
3.844
3.795
3.629
5.327
6.520
7.137
5.023
5.975
6.727
5.0
300
420
600
3.579
3.539
3.376
3.268
3.092
3.023
10.06
11.75
13.68
9.428
9.800
12.67
10.0
300
420
600
2.806
2.678
2.602
2.729
2.413
2.202
23.55
26.97
31.43
21.92
24.91
29.41
NIT ROURKELA
Page 46
Transverse direction amplitude (m)
Transverse direction amplitude (m)
6
x 10
-5
Stable Response
4
2
0
-2
-4
-6
0
2
x 10
0.1
0.2
0.3
Time (s)
0.4
0.5
0.4
0.5
-4
Unstable Response
1
0
-1
-2
0
0.1
0.2
0.3
Time (s)
Fig. 5.20. Displacement histories due to unbalance masses along the transverse direction of
FG shaft: (a) stable response and (b) unstable response
NIT ROURKELA
Page 47
CHAPTER 6
Conclusions and Scope of Further Works
This chapter presents few important observations based on the vibration and stability
analysis of the FG spinning shaft system using developed MATLAB code. Scope of further
work in this direction has also been presented at the end of this chapter.
6.1 Conclusions
The present work enables to arrive at the following important conclusions:
 A three nodded beam finite element has been implemented for modeling and vibration
analysis of the FG shaft system by incorporating both the internal viscous and
hysteretic damping in the thermal environment.
 The temperature distribution is nonlinear along the radial direction of the crosssection of FG shaft.
 The material property distribution of the FG shaft has been performed very smoothly
along the radial direction by accounting different temperatures and power law
gradient indexes.
 From the comparison of various responses between steel and FG shaft, it has been
found that FG shaft is more stable than the steel shaft.
 From the comparison of various responses of the FG shaft without and with
temperatures consideration, it has been noticed that the FG shaft is more stable in case
of without temperature consideration than that of with temperature consideration.
 The power law gradient index plays an important role in the responses (viz. Campbell
diagram, damping ratio, critical speed, stability limit speed and time responses) of the
FG shaft system.
 It is observed that the less value of temperature and the power law gradient index
promotes more stable system than that of higher values of temperature and power law
gradient index
 Finally, it can be concluded that the present work can be used for modeling and
vibration analysis of the FG shaft system considering with or without temperature
dependent material properties according to power law gradation by incorporating both
internal viscous and hysteretic damping.
NIT ROURKELA
Page 48
6.2 Scope of Future Works
 Study of vibration and stability analysis for FG rotor shaft system under electrothermo-mechanical environment.
 Study of vibration and stability analysis by using a fluid film journal bearing for this
present model
 Active vibration control of FG rotor shaft system
 Nonlinear modeling of FG shaft and
 Multiobjective optimization of FG shaft system
NIT ROURKELA
Page 49
Appendix
The terms A11, A55, A66, B11
of the equation (19) and I m , I d , I p of the equation (20)
given as follows:
 k
 k
2
2
A55   C 55r  r i 1  r i  , A66   C 66 r  r 2i 1  r 2i 
2 i 1
2 i 1
2 k
 k
3
3
A16 
C
r

r
,
B

C11r  r 4i 1  r 4i 
16
r



i 1
i
11
3 i 1
4 i 1
k
 k
 k
2
2
4
4
I m    i  r i 1  r i  , I d   i  r i 1  r i  , I p   i  r 4i 1  r 4i 
4 i 1
2 i 1
i 1
E
E
E
C11r 
,
C

,
C

C

16
r
55
r
66
r
1  2
1  2
2 1  
Where k is the number of layer in the laminate and ri 1 and ri are represents outer
and inner radii of the i th layer of the laminate respectively.  i is the density of the composite
shaft of i th layer. Now various matrices of the equation (34) can be written as follows:
Nodal displacements vector:
qe 
T

 ve 13 we 13  xe 13  ye 
T
T
T
T

13 112
Mass Matrices:
 M v 33

  033
 M e    0
33

  0
33

033
 M w 33
033
033
033
M  
 x  33
033
033
xf
 M v   I m        dx 
T
I mD
  T      x  xDi dx

i 1
xi
xf
 M w   I m        dx 
T
NIT ROURKELA






M   
 y  33 1212
xf N
D
xi
xi
033
033
033
I mD
xf N
D
  T      x  xDi dx

i 1
xi
Page 50
M    Id
 x
xf
      dx 
T
I dD
xf N
D
xi
M    Id
 y
  T     x  xDi dx

i 1
xi
xf
       dx 
T
I dD
xi
xf N
D
  T      x  xDi dx

i 1
xi
Stiffness Matrix:
 K 
 vv 33

  033
 Ke   
T
  K v 
x  33


T


K
  v y 
33

xf
033
 K v 
x  33

 K ww 33
 K w 
x  33

K  
 x x  33
T
K  
 x y  33

NB

T

Bi
  '    K yy



x

x






Bi  dx
 

i 1

T

xf

 K ww     K s  A55  A66   ' 

xi
 K   
 x x

xf

xi
 K   
 y y
T
 K w 
y 

33
 Kvv     K s  A55  A66   ' 

xi
T
 K w 
x  33

T
NB

  '    K zzBi   T      x  xBi   dx
 
i 1


xf

xi
 K v  
y 

33 
 K w  

y 

33

K   
 x y  33 

K   
 y y  33 1212

T
K s  A55  A66         D11   '    '  dx
T
K  A
s
xf
T
1
 Kv     K s B16     '    ' dx
x

 2
 xi    
 K w   K s  A55  A66 
x

xf
     dx
xi
xf

T
 A66         D11  '   '  dx
T
55

' T
 K v    K s  A55  A66 
y

 K w     1 K s B16 
y

 2

xf
xf

 ' 
 
T
 dx
xi
' T
  
  ' dx
 
xi

T
 K     1 K s B16   '        T   '  dx
  
 
 x y 2
x
i
NIT ROURKELA
Page 51
Damping Matrix of Bearing:
Cv 33

  033
CeB   


  033
  0
33

xf N
B
Cv    
xi i 1
Bi
C yy
 033
Cw 33
 033
 033
033
 033
 033
 033
033 

033 
033 
033 1212
xf N
B
       x  xBi dx Cw     CzzBi  T     x  xBi dx
T
xi i 1
Gyroscopic Matrix:
 033

 033
Ge   0
33


 033
G    I P
 x y
033
033
033
033
033
033
033
033
033
 G  
x y 

33




G   
 x y  33

033 
1212
T
xf
       dx 
T
xi
I PD
xf N
D
  T      x  xDi dx

i 1
xi
xf
Elemental circulation matrix
 KCir 1212   M T  Mdx
xi
Where,
 1'  2'  3' 0 0 0

0 0 0  1'  2'  3'

M
 0 0 0  1"  2''  3"
 ''
"
"
 1  2  3 0 0 0
GA
0
 0
 GA 0
0
 
 0
0
0

0  EI
 0
NIT ROURKELA
0
 1'
 1"
0
0
 2'
 2''
0
0  1'  2'  3' 

 3' 0 0 0 
 3'' 0 0 0 

0  1"  2"  3"  412
0
0 
EI 

0  44
Page 52
References
[1] Koizumi, M., 1993, “Concept of FGM,” Ceramic Trans., 34, pp. 3–10.
[2] Zinberg, H., Symonds, M.F., 1970, “The Development of an Advanced Composite Tail
Rotor Driveshaft,” Presented at the 26th Annual Forum of the American helicopter Society,
Washington, D.C, June.
[3] Nelson, H.D., Mcvaugh, J.M., 1976, “The dynamics of rotor-bearing systems using finite
element method,” Journal of Engineering for Industry, 98, pp. 593–600.
[4] Nelson, H.D., 1980, “A finite rotating shaft element using Timoshenko beam element,
102, pp. 793-803.
[5] Rouch, K.E., Kao, J.S., 1979, “A tapered beam finite element for rotor dynamics
analysis,” Journal of Sound and Vibration, 66(1), pp.119-140.
[6] Zorzi, E.S., Nelson, H.D., 1980, “The dynamic of Rotor-bearing systems with axial torque
a finite element approach,” Journal of Mechanical Design, 102, pp. 158-161.
[7] Bert, C.W., 1992, “The effect of bending–twisting coupling on the critical speed of a
driveshafts,” Proceedings of the 6th Japan-US Conference on Composite Materials, Orlando,
FL. Technomic, Lancaster, PA, pp. 29-36.
[8] Kim, C.D., Bert, C.W., 1993, “Critical speed analysis of laminated composite hollow
drive shaft,” Composite engineering, 3(7-8), pp. 633-643.
[9] Abramovich, H., Livshits, A., 1993, “Dynamic behavior of cross-ply laminated beams
with piezoelectric layers,” Composite Structures, 25, pp. 371-379.
[10] Bert, C.W., Kim, C.D., 1995, “Whirling of composite material driveshaft including
bending, twisting coupling and transverse shear deformation,” Journal of Vibration and
Acoustics, 117, pp. 17-21.
[11] Bert, C.W., Kim, C.D., 1995, “Dynamic instability of composite-material drive shaft
subjected to fluctuating torque and/or rotational speed,” Dynamics and Stability of Systems,
2, pp. 125-147.
[12] Singh, S.P., Gupta, K, 1996, “Composite shaft rotor dynamic analysis using layer wise
theory,” Journal of Sound and Vibration, 191(5), pp. 739-756.
[13] Singh, S.P., Gupta, K., 1996, “Dynamic Analysis of composite rotors,” International
Journal of Rotating Machinery, 2(3), pp. 179-186.
[14] Forrai, L., 2000, “A finite element model for stability analysis of symmetrical rotor
system with internal damping,” Journal of Computational and Applied Mechanics, 1 (1), pp.
37-47.
NIT ROURKELA
Page 53
[15] Chatelet, E., Lornage, D., Jacquet-richardet, G., 2002, “A three dimensional modeling of
the dynamic behavior of composite rotors,” International Journal of Rotating Machinery,
8(3), pp. 185-192.
[16] Chang, M.Y., Chen, J.K., Chang, C.Y., 2004, “A simple spinning laminated composite
shaft model,” International Journal of Solids and Structures, 41, pp. 637–662.
[17] Kapuria, S., Ahmed, A., Dumir, P.C., 2004, “Static and dynamic thermo
electromechanical analysis of angle ply hybrid piezoelectric beams using an efficient coupled
zigzag theory,” Composites Science and Technology, 64, pp. 2463–2475.
[18] Gubran, H.B.H., Gupta, K., 2005, “The effect of stacking sequence and coupling
mechanisms on the natural frequencies of composite shafts,” Journal of Sound and Vibration,
282, pp. 231-248.
[19] Wang, B.L., Mai, Y.W., 2005, “Transient one dimensional heat conduction problems
solved by finite element,” International Journal of Mechanical Sciences, 47, pp. 303-317.
[20] Syed, K.A., Su, C.W., Chan, W.S., 2007, “Analysis of Fiber Reinforced Composite
Beams under Temperature Environment,” Proceedings of the Seventh International Congress
on Thermal Stresses, Taipei, Taiwan.
[21] Sino, R., Baranger, T.N., Chatelet, E., Jacquet, G., 2008, “Dynamic analysis of a rotating
composite shaft,” Journal of Composites Science and Technology, 68, pp. 337–345.
[22] Feldman, E., Aboudi, J. 1997, “Buckling analysis of functionally graded plates subjected
to uniaxial loading,” Composite Structures, 38, pp. 29–36.
[23] Praveen, G.N.; Reddy, J. N., 1998, “Nonlinear transient thermo elastic analysis of
functionally graded ceramic metal plates,” International Journal of Solids and Structures,
35(33), pp. 4457–4476.
[24] Gasik, M.M., 1998, “Micromechanical modeling of functionally graded materials,”
Computational Materials Science, 13 (1), pp. 42–55.
[25] Suresh, S., Mortensen, A., 1998, “Fundamentals of functionally graded materials”,
London, UK: IOM Communications Limited.
[26] Aboudi, J., Pindera, M.J., Arnold, S.M., 1999, “Higher-order theory for functionally
graded materials,” Composites, Part B: Engineering, 30 (8), pp.777–832.
[27] Nakamura, T., Wang, T., Sampath, S., 2000, “Determination of properties of graded
materials by inverse analysis and instrumented indentation,” Acta mater, 48, pp. 4293–4306.
[28] Wang, B.L., Han, J.C., Du, S.Y., 2000, “Crack problems for Functionally Graded
Materials under transient thermal loading,” Journal of Thermal Stresses, 23 (2), pp. 143–
168.
NIT ROURKELA
Page 54
[29] Woo, J., Meguid, S. A. 2001, “Nonlinear analysis of functionally graded plates and
shallow shells,” International Journal of Solids and Structure, 38, pp. 7409–74021.
[30] Sankar, B. V., 2001, “An elasticity solution for functionally graded beams,” Composites
Science and Technology, 61, pp. 689–696.
[31] Sankar, B. V., and Tzeng, J. T. 2002, “Thermal Stresses in Functionally Graded Beams,”
AIAA Journal, 40(6), pp. 1228-1232.
[32] Chakraborty, A., Gopalakrishnan, S., Reddy, J. N., 2003, “A New Beam Finite Element
for the Analysis of Functionally Graded Materials,” International Journal of Mechanical
Sciences, 45(3), pp. 519-539.
[33] Nemta-AIIa, M., 2003, “Reduction of thermal stresses by developing two dimensional
functionally graded materials,” International journal of solids and structures, 40, pp. 73397356.
[34] Reddy J.N., 1998, “Thermo-mechanical behavior of functionally graded materials,”
AFOSR Grant F49620-95-1-0342, Washington, D.C, August, pp. 1-78.
[35] Na, K. S., Kim, J. H. 2005, “Three-Dimensional Thermo mechanical Buckling of
Functionally Graded Materials,” AIAA Journal, 43(7), pp. 1605-1612.
[36] Przybyowicz, Piotr M., 2005, “Stability of activity controlled rotating shaft made of
functionally graded materials,” Journal of theoretical and applied mechanics, 43(3), pp. 609630.
[37] Cooley, William G., 2005. Application of functionally graded materials in aircrafts
structures, M.S Thesis, Air Force Institute of Technology, Wright Patterson AFB OH.
[38] Shao, Z.S., 2005, “Mechanical and thermal stresses of a functionally graded circular
hollow cylinder with finite length,” International Journal of Pressure Vessels and Piping, 82,
pp. 155-163.
[39] WU, Tsung-Lin., Shukla, K.K., Huang, J.H., 2006, “Nonlinear static and dynamic
analysis of functionally graded plates,” International Journal of applied mechanics and
engineering 11(3), pp. 679-698.
[40] Argeso, H., Eraslan, Ahmet N., 2007, “A Computational Study on Functionally Graded
Rotating Solid Shafts,” International Journal for Computational Methods in Engineering
Science and Mechanics, 8, pp. 391–399.
[41] Rahimi, G.H., Davoodinik, AR., 2008, “Thermal behavior analysis of the functionally
graded Timoshenko’s beam,” IUST International Journal of Engineering Science, 19(5-1),
pp. 105-113.
NIT ROURKELA
Page 55
[42] Piovan, M.T., Sampaio, R., 2009, “A study on the dynamics of rotating beams with
functionally Graded properties,” Journal of Sound and Vibration, 327, pp. 134-143.
[43] Zhao, X., Lee, Y.Y., Liew, K.M., 2009, “Mechanical and thermal buckling analysis of
functionally graded plates,” Journal of Composite Structures, 90, pp. 161-171.
[44] Simsek, M., 2009, “Static analysis of a functionally graded beam under a uniformly
distributed load by Ritz method,” International Journal of Engineering and Applied Sciences,
1(3), pp. 1-11.
[45] Giunta, G., Belouettar, S., Carrera, E., 2010, “Analysis of FGM beams by means of
classical and advanced theories,” Journal of Mechanics of Advanced Materials and
Structures, 17, pp. 622-635.
[46] Afsar, A.M., Go, J., 2010, “Finite element analysis of thermo elastic field in a rotating
FGM circular disk, Journal of Applied Mathematical Modelling,” 34, pp. 3309-3320.
[47] Simsek, M., 2010, “Fundamental frequency analysis of functionally graded beams by
using different higher order beam theories, Journal of Nuclear Engineering and Design,” 240,
pp. 697-705.
[48] Alibeigloo, A., 2010, “Thermo elasticity analysis of functionally graded beam with
integrated surface piezoelectric layers,” Journal of Composite Structures, 92, pp. 1535-1543.
[49] Kocaturk,T., Simsek, M., Akbas, S.D., 2011, “Large displacement static analysis of a
cantilever Timoshenko beam composed of functionally graded material,” Journal of Science
and Engineering of Composite Materials, 18, pp. 21-34.
[50] Mazzei, Arnaldo J., Scott, Richard A., 2011, “Effect of Functionally Graded Materials
on Resonances of Bending Shafts under Time-Dependent Axial Loading,” Journal of
Vibration and Acoustics, 133, pp. 061005 (1-10).
[51] Alashti, R. A., Khorsand, M., 2011, “Three-dimensional thermo-elastic analysis of a
functionally graded cylindrical shell with piezoelectric layers by differential quadrature
method,” International Journal of Pressure Vessels and Piping, 88, pp. 167-180.
[52] Dimentberg, F. M., 1961, “Flexural vibrations of rotating shafts”, Butterworths, London.
[53] Gunter, E. J. Jr., Trumpler, P. R., 1969, “The influence of internal friction on the
stability of high speed rotors with anisotropic supports,” Journal of Engineering for Industry,
pp. 1105-1113.
[54] Ruhl, R. L., 1970, “Dynamics of distributed parameter rotor systems: transfer matrix and
finite element techniques,” PhD dissertation, Cornell University.
NIT ROURKELA
Page 56
[55] Ruhl, R.L., Booker, J.F., 1972, “A finite element model for distributed parameter turbo
rotor system,” Journal of Engineering for Industry, 94(1), pp. 128-132.
[56] Lund, J. W., 1974, “Stability of damped critical speeds of a flexible rotor in fluid film
bearings,” Journal of Engineering for Industry, 96, pp. 509-517.
[57] Dimarogonas, A. D., 1975, “A general method for stability of rotating shafts”, Ingenieur
Archiv, 44, pp. 9-20.
[58] Zorzi, E. S., Nelson, H. D.,1977, “Finite element simulation of rotor-bearing systems
with internal damping”, Journal of Engineering for Power, pp.71-76.
[59] Dutt, J. K., Nakra, B. C.,1992, “Stability of rotor systems with viscoelastic supports”,
Journal of Sound and Vibration, 153(1), pp. 89-96.
[60] Abduljabbar, Z., EIMadany, M.M., AI-Bahkali, E, 1995, “On the vibration and control
of a flexible rotor mounted on fluid film bearing,” the fourth Saudi engineering conference
Nov., 4, pp.101-111.
[61] Wettergren, H. L., Olsson, K. O., 1996, “Dynamic instability of a rotating asymmetric
shaft with internal viscous damping supported on anisotropic bearings”, Journal of Sound and
Vibration, 195 (1), pp. 75-84.
[62] Qin, Q. H., Mao, C. X., 1996, “Coupled torsional-flexural vibration of shaft systems in
mechanical engineering-1 finite element model”, Journal of Computer & Structures, 58 (4),
pp. 835-843.
[63] Dutt, J. K., Nakra, B. C., 1996, “Stability characteristics of rotating systems with journal
bearings on viscoelastic support”, Journal of Mechanism and Machine Theory, 31(6), pp.
771-779.
[64] Ku, D. M., 1998, “Finite element analysis of whirl speeds for rotor-bearing systems with
internal damping”, Journal of Mechanical Systems and Signal Processing, 12 (5), pp. 599610.
[65] Chang, C.Y., Chang, M.Y., Huang, J.H., 2004, “Vibration analysis of rotating composite
shafts containing randomly oriented reinforcements,” Composite Structures, 63, pp. 21-32.
[66] Roy, H., Dutt, J.K., Datta, P.K., 2008, “Dynamics of a viscoelastic rotor shaft using
augmenting thermodynamic fields-A finite element approach”, International Journal of
Mechanical Sciences, 50, pp. 845-853.
[67] Xiang, H.J., Yang, J., 2008, “Free and forced vibration of a laminated FGM Timoshenko
beam of variable thickness under heat conduction,” Composites: Part B, 39, pp.292–303.
[68] Li, Q., Iu, V.P., Kou, K.P., 2008, “Three-dimensional vibration analysis of functionally
graded material sandwich plates,” Journal of Sound and Vibration, 311, pp. 498 – 515.
NIT ROURKELA
Page 57
[69] Das, A.S., Nighil, M.C., Dutt, J.K.; Irretier, H., 2008, “Vibration control and stability
analysis of rotor-shaft system with electromagnetic exciters,” Journal of Mechanism and
Machine Theory, 43, pp. 1295–131.
[70] Yang, J., Chen, Y., 2008, “Free vibration and buckling analyses of functionally graded
beams with edge cracks”. Composite Structures, 83, pp. 48-60.
[71] Ke, L.L., Yang, J., Kitipornchai, S., Xiang, Y., 2009, “Flexural Vibration and Elastic
Buckling of a Cracked Timoshenko Beam Made of Functionally Graded Materials,” Journal
of Mechanics of Advanced Materials and Structures, 16, pp. 488–502.
[72] Hosseini, S.A. A., Khadem, S.E., 2009, “Free vibrations analysis of a rotating shaft with
nonlinearities in curvature and inertia”, Journal of Mechanism and Machine Theory, 44, pp.
272-288.
[73] Boyaci, A., Seemann, W., Proppe, C., 2009, “Nonlinear stability analysis of rotorbearing systems,” Journal of Proceedings in Applied Mathematics and Mechanics, 9, pp. 279
-280.
[74] Mahi, A., Adda Bedia, E.A., Tounsi, A., Mechab, I., 2010, “An analytic method for
temperature dependent free vibration analysis of functionally graded beams with general
boundary conditions,” Journal of Composite Structures, 92, pp. 1877-1887.
[75] Sapountzakis, E. J., Dourakopoulos, J. A., 2010, ‘Shear Deformation Effect in Flexural
torsional Vibrations of Composite Beams by Boundary Element Method (BEM),” Journal of
Vibration and Control, 16, pp. 1763-1789.
[76] Boukhalfa, A., Hadjoui, A., 2010, “Free vibration analysis of an embarked rotating
composite shaft using the hp - version of the FEM,” Latin American Journal of Solids and
Structures, 7(2), pp. 105-141.
[77] Kiani, Y., Eslami, M.R., 2010, “Thermal buckling analysis of functionally graded
material beams’, International Journal of Mechanics and Materials in Design, 6, pp. 229-238
[78] Shahba, A., Attarnejad, R., Marvi, M. T., Hajilar, S., 2011, “Free vibration and stability
analysis of axially functionally graded tapered Timoshenko beams with classical and nonclassical boundary conditions,” Composites: Part B 42, pp. 801-808.
[79] Alshorbagy, Amal E., Eltaher, M.A., Mahmoud, F.F., 2011, “Free vibration
characteristics of a functionally graded beam by finite element method,’ Journal of Applied
Mathematical Modelling, 35, pp. 412-425.
NIT ROURKELA
Page 58
[80] Rafiee, M., Kalhori, H., Mareishi, S., 2011, “Nonlinear resonance analysis of clamped
functionally graded beams,” 16th International Conference on Composite Structures.
[81] Kumar, J.S., Reddy, B.S., Reddy, C.E., Reddy, K.V.K., 2011, “Higher order theory for
free vibration analysis of functionally graded material plates,” Journal of Engineering and
Applied Sciences, 6 (10), pp. 105-111.
NIT ROURKELA
Page 59
LIST OF PUBLICATIONS
 International Journals
 Debabrata Gayen and Tarapada Roy, “Hygro-Thermal Effects on Stress
Analysis of Tapered Laminated Composite Beam,” Int. Journal of Composite
Materials, Vol. 3, No. 3, pp. 46-55, 2013, DOI: 10.5923/j.cmaterials.2013030
3.0 2.
 Debabrata Gayen and Tarapada Roy, “Hygro-Thermal Stress Analysis of
Tapered Laminated Composite Beam,” Int. Journal of Mechanical Engineering
and Research (IJMER), Vol. 3, No. 3, pp. 236-240, 2013.
 Debabrata Gayen and Tarapada Roy, “Finite Element based Vibration
Analysis of Functionally Graded Spinning Shaft System”, Int. Journal of
Mechanical Sciences. (Submitted)
 Debabrata Gayen and Tarapada Roy, “Vibration and Stability Analysis of
Temperature Dependent Functionally Graded Rotating Shaft System Based on
Finite Element Approach”, Mechanism and Machine Theory. (To be
Submitted)
 International conferences
 Debabrata Gayen, D. Koteswara Rao, Tarapada Roy, “Thermo Mechanical
Vibration Analysis of Functionally Graded Rotating Shaft Using Timoshenko
Beam Element,” 1st Int. Conf. on ICMMME, Goa, 31st March, 2013.
 D. Koteswara Rao, Tarapada Roy, Debabrata Gayen and Prasad K. Inamdar,
“Finite Element Analysis of Functionally Graded Rotor Shaft Using
Timoshenko Beam Theory,” 2nd Int. Conf. of ICMPE, Hotel Lindsay, Kolkata,
15th February, 2013.
NIT ROURKELA
Page 60
Was this manual useful for you? yes no
Thank you for your participation!

* Your assessment is very important for improving the work of artificial intelligence, which forms the content of this project

Related manuals

Download PDF

advertisement