Thermo-mechanical Stress Analysis of Functionally Graded Tapered Shaft System

Thermo-mechanical Stress Analysis of Functionally Graded Tapered Shaft System
Thermo-mechanical Stress Analysis of Functionally
Graded Tapered Shaft System
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
Dinesh Patil
(213ME1385)
DEPARTMENT OF MECHANICAL ENGINEERING
NATIONAL INSTITUTE OF TECHNOLOGY
ROURKELA – 769008
JUNE, 2015
Thermo-mechanical Stress Analysis of Functionally
Graded Tapered Shaft System
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
Dinesh Patil
(213ME1385)
Under the Supervision of
Prof. Tarapada Roy
DEPARTMENT OF MECHANICAL ENGINEERING
NATIONAL INSTITUTE OF TECHNOLOGY
ROURKELA – 769008
JUNE, 2015
National Institute of Technology
Rourkela
CERTIFICATE
This is to certify that the thesis entitled, “Thermo-mechanical Stress
Analysis of Functionally Graded Tapered Shaft System” submitted by
Mr. Dinesh Patil, Roll No. 213ME1385 in partial fulfilment of the
requirements for the award of Master of Technology Degree in
Mechanical Engineering with specialization in “Machine Design and
Analysis” at National Institute of Technology, Rourkela is an authentic
work carried out by him under my supervision and guidance. To the best
of my knowledge, the matter embodied in this thesis has not been
submitted to any other university/ institute for award of any Degree or
Diploma.
Date: 01 June 2015
Prof. Tarapada Roy
Dept. of Mechanical Engineering
National Institute of Technology,
Rourkela -769008
DEPARTMENT OF MECHANICAL ENGINEERING
NATIONAL INSTITUTE OF TECHNOLOGY
ROURKELA 769008
ACKNOWLEDGEMENT
It gives me immense pleasure to express my deep sense of gratitude to my supervisor Prof.
Tarapada Roy for his invaluable guidance, motivation, constant inspiration and above all for his
ever co-operating attitude that enabled me to bring up this thesis to the present form.
I express my sincere thanks to the Director, Prof. S.K.Sarangi, National Institute of Technology,
Rourkela for motivating me in this endeavour and providing me the necessary facilities for this
study.
I would like to extend my thanks to Ph.D. scholars, D Koteswara Rao and Ashirbad Swain,
Department of Mechanical Engineering, NIT Rourkela for their support and guidance during my
project work.
Last but not the least; I would like to express my love, respect and gratitude to my parents and
my brother-in-law, who have always supported me in every decision I have made, guided me in
every turn of my life, believed in me and my potential and without whom I would have never been
able to achieve whatsoever I could have till date.
Place: Rourkela
Date: 01 June 2015
Dinesh Patil
M. Tech., Roll No: 213ME1385
Machine Design and Analysis
Department of Mechanical Engineering
National Institute of Technology, Rourkela
Page i
CONTENTS
CONTENTS
ii
List of Figures
iv
List of Tables
v
ABSTRACT
vi
CHAPTER 1
1
INTRODUCTION
1.1 Background of Rotor Dynamics
1.2 Composite Materials
1.3 Drawbacks of Composite Materials
1.4 Theoretical Understanding about FGMs
1.5 Practical Applications of FGMs
CHAPTER 2
LITERATURE REVIEW
2.1 Introduction
2.2 Functionally Graded Materials
2.3 Stresses in FGMs
2.4 Rotor Dynamics
2.5 Motivation
2.6 Aim of Present Work
CHAPTER 3
MATERIAL MODELLING FOR TAPERED FG SHAFT
3.1 Actual Material Properties of FGM
3.2 Material Modelling of FGMs
3.2.1 Laws of Gradation
3.2.1.1 Power Law Gradation
3.2.1.2 Exponential Law of Gradation
3.3 Modelling of Material Properties Applicable To tapered FG Shaft
3.3.1 Power Law
CHAPTER 4
FORMULATION FOR TAPERED FG SHAFT
4.1 Introduction
1
1
3
4
4
5
6
6
6
6
7
8
9
10
11
11
11
11
12
12
13
13
14
16
16
16
Page ii
4.2 Finite Element Modelling of Shaft
4.2.1 Kinetic Energy Expression of Shaft
4.2.2 Strain Energy Equation for FG Shaft
4.2.3 Kinetic energy expression for disks on shaft
4.3 Expression for work done to external load and bearings
4.4 Governing equation of rotor shaft
4.5 Contribution of internal damping
17
18
19
19
20
20
22
CHAPTER 5
23
RESULTS AND DISCUSSION
5.1 Problem Description and Summarization of Discussion
5.2 Validation of Code
5.3 Temperature Distribution in Tapered FG Shaft
5.4 Material properties of tapered FG shaft depends on temperature and power law index
5.5 Stress analysis in tapered FG shaft
5.5.1 Comparative study of tapered FG shaft over steel tapered shaft
5.5.2 Variation of stresses for different values of ‘k’ in radial direction
5.5.3 Transient uncoupled stress analysis for different value of power law index
5.5.4 Transient coupled stress analysis for different value of power law index
CHAPTER 6
CONCLUSION AND SCOPE OF FUTURE WORK
6.1 Conclusions
6.2 Scope of future work
23
23
24
26
27
29
29
31
34
37
40
40
40
40
Appendix
41
References
45
Page iii
List of Figures
Figure 3. 1 Volume fraction of metal in FGM rectangular cross-section.
12
Figure 3. 2 Volume fraction of metal in tapered FG shaft
14
Figure 4. 1 Displacement variables
16
Figure 4. 2 Diagram showing tapered shaft and bearing system
16
Figure 5. 1 Campbell diagram for laminated graphite-epoxy composite material.
25
Figure 5. 2 Temperature variation in mid-section of tapered FG shaft.
26
Figure 5. 3 Volume fraction of ceramic material along radius for power law index.
27
Figure 5. 4 Variation of young’s modulus along radius for power law index.
28
Figure 5. 5 Variation of Poisson’s ratio along radius for power law index.
28
Figure 5. 6 Variation of CTE along radius for power law index.
29
Figure 5. 7 Stress developed in tapered Steel shaft along radius.
30
Figure 5. 8 Normal stress in tapered FG shaft along radius.
30
Figure 5. 9 Shear stress in tapered FG shaft along radius.
31
Figure 5. 10 Normal stress in tapered FG shaft along radius:
(a) At 6000 RPM, (b) At 12000 RPM
32
Figure 5. 11 Shear stress in tapered FG shaft along radius.
(a) At 6000 RPM, (b) at 12000 RPM
33
Figure 5. 12 Transient uncoupled normal stress in tapered FG shaft:
(a) At 6000 RPM, (b) at 12000 RPM
35
Figure 5. 13 Transient uncoupled shear stress in tapered FG shaft in theta direction:
(a) At 6000 RPM, (b) at 12000 RPM
36
Figure 5. 14 Transient uncoupled shear stress in tapered FG shaft in radial direction:
(a) At 6000 RPM, (b) at 12000 RPM
37
Figure 5. 15 Transient coupled normal stress in tapered FG shaft:
(a) At 6000 RPM, (b) At 12000 RPM
38
Figure 5. 16 Transient coupled shear stress in tapered FG shaft in theta direction:
(a) At 6000 RPM, (b) At 12000 RPM
39
Page iv
List of Tables
Table 5. 1 Geometric dimensions of steel and FG tapered shaft
23
Table 5. 2 Material properties of FGM [43]
24
Table 5. 3 Materials and temperature coefficient of mechanical properties [28]
24
Table 5. 4 Dimensions and properties of laminated graphite-epoxy shaft [35]
25
Table 5. 5 Temperature variation in mid-section of tapered FG shaft.
26
Page v
ABSTRACT
Present work deals with the study of stresses developed in tapered functionally graded (FG)
shaft system under both thermal and mechanical environment for three nodded beam element by
using Timoshenko beam theory. The temperature distribution in radial direction is assumed based
on one dimensional steady state temperature field by Fourier heat conduction equation without
considering heat generation. Temperature dependent material properties are varied along the radial
direction using power law gradation. Tapered FG shaft consists of rigid disk attached at its centre
and shaft is mounted on two flexible bearings acts as spring and damper, inner radius of the tapered
shaft is varying in x direction keeping thickness of hollow tapered shaft is constant. For the present
analysis the Mixture of Stainless steel (SUS304) and Aluminum oxide (Al2O3) are considered as
inner and outer surface material of the FG shaft. Three dimensional constitutive relations are
derived based on first order shear deformation theory (FSDT) for Timoshenko beam element
considering rotary inertia, strain and kinetic energy of shaft and gyroscopic effect. In present study,
structural and hysteretic damping are incorporated. Hamilton’s principle is used to derive
governing equation of motion for three nodded beam element for six degree of freedom per node.
Complete MATLB code is generated and shows that temperature field and power law gradient
index have important part on material properties. Comparative study is carried out for Stainless
steel and FG tapered shaft, shows that stress developed in FG shaft is comparatively lower than
Steel shaft. Various results are obtained for coupled and uncoupled environment. Transient stress
are obtained for varying power law index value and speed as a parameter. Stress amplitude
increases for increase in speed and power law index. Results achieved for FG shaft shows
advantages over steel shaft.
Keywords: Functionally graded materials (FGMs), Power law index, Tapered shaft, Timoshenko
beam theory (TBT), Three nodded beam element, Finite element method, Thermo-mechanical,
Stress analysis.
Page vi
CHAPTER 1
INTRODUCTION
Composite materials are materials, composed of two or more fundamental materials with
different properties, when combined to get a material with different properties than that of
individual constituents. Composite material structures are more frequently used in engineering
fields as their high strength to weight ratio and high stiffness to weight ratio is basically favourable
for material selection. Main disadvantage with composite material is, weakness in interface
between neighbouring layers, which is popularly known as delamination phenomenon that may
cause structural failure. To overcome this problem, a new class of material presented, named as
Functionally Graded Materials (FGMs). FGMs are recognised as, whose material properties are
varying in certain direction and thus overcome interface weakness. FGMs are defined as, the
materials whose volume fractions of two or more materials are varied continuously along certain
direction to attain required purpose. FGMs provide better material response and excellent
performance in thermal environments like thermal barrier and space application, where it is used
to protect space shuttle from heat generated during re-entry to Earth’s atmosphere by modelling
ceramic material at outer surface metal at inside surface.
Because of high strength, stiffness and low density material characteristics, brings an idea
for replacing conventional metallic shafts with FGMs rotor shaft in many application areas like
design of spinning components such as driveshaft in automobiles, jet engines and helicopters,
turbine shafts and other rotating machineries. Composite materials has been validated both
numerically and experimentally in rotor dynamics applications. Along with this various new
advanced composite materials and material models for rotor shaft has been developed by
researchers.
1.1 Background of Rotor Dynamics
Rotor dynamics has a significant history, mainly due to its relationship with theory and
practice. Rotor dynamics is a particular branch of applied mechanics deals with the performance
and analysis of rotor assemblies. Rotor dynamics mostly used to analyse the performance of a
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turbine shafts, jet engine to auto engines and computer storage disks. Basically Rotor dynamics
deals with rotor and stator. Rotating part in mechanical devices are called rotors, which are
supported on bearings, thus shaft rotate freely about its axis. Many engineering components are
deals with the subject of Rotor dynamics and which gives better solution for components like
turbines, compressors, alternators, blowers, motors, pumps, brakes etc.
Rotor delivers with behaviour of materials to limit their spin axis in a more or rigid way to
a fixed position in space, those are mentioned to as fixed rotor (considering spin speed is constant),
while rotors which are not considering in any way are mentioned as free rotors (considering spin
speed is governed by conservation of angular momentum). In process, Rotors have excessive deal
with rotational energy and small amount of vibrational energy.
In the field of Rotor Dynamics William John Macquorn Rankine (1869) implemented the
first analysis of rotating shaft. Considering two degree of freedom shaft model attached with rigid
mass whirling in an orbit, having elastic spring acting in radial direction. Whirling speed of the
shaft has defined, shows that radial deflection of Rankin’s model increases beyond this whirling
speed, this speed is termed as threshold speed for the divergent instability.
Swedish engineer Carl Gustaf Patrik de Laval (1833), for marine application he established
a single-stage steam impulse turbine and achieved 42000 RPM. He used initially a rigid rotor and
then used flexible rotor to operate above critical speed by running at a speed around seven times
the critical speed.
Stanley Dunkerley (1895) studied the pulley loaded vibration of shafts. It is known that
shafts are well balanced when rotating at particular speed, bends except the amount of deflection
is restricted, even shafts may break, though shafts runs at high speed. This critical speed is depends
on the way in which shaft is supported, modulus of elasticity, size, weight of the shaft and position
of mass (pulleys). German civil engineer August Foppl (1895) presented another rotor model
showing stable response above whirling speed. W. kerr (1916) showed experimentally that, second
critical aped will occurred when rotor crosses first critical speed safely. Ludwig Prandtl (1918)
studied non-circular cross section Jeffcott rotor.
Henry Jeffcott (1919) modelled and studied behaviour of simple spinning rotor under
flexural and dynamic behaviour. Actually in Jeffcott model disk do not wobble. As a result, the
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angular velocity vector and angular momentum vector are collinear thus no gyroscopic moments
are generates.
Aurel B. Stodola (1924) developed dynamics of elastic continuous rotor having discs
without considering gyroscopic moment, balancing of shaft, secondary resonance phenomenon
due to gravity effect and methods to determine critical speeds of shafts for variable cross sections,
also by using Coriolis accelerations supercritical solutions can be stabilized.
Baker (1933) found and defined that because of contact between rotor and stator system
exhibits self-exited vibrations. David M. Smith (1933) found formulas for predicting threshold
spin speed for supercritical instability varied through bearing stiffness and also with ratio between
external to internal viscous damping. Many variations came closer to practical needs of the rotor
dynamic field for Jeffcott rotor model. Prohl’s and Myklestad’s (1945) analysed instabilities and
modelling methods in rotors dynamics by Transfer Matrix Method (TMM).
In 1960s, for exact solution capabilities, numerical methods are established for structural
dynamic analysis, rotor dynamics codes and digital computer codes were constructed on TMM
method. In 1970s alternative fundamental procedure developed that is Finite Element Method
(FEM), developed for solution of beam type of models. In 21st century, rotor dynamics are
combined FEM and solids modelling methods to create simulations that adapt the coupled
behaviour of disks, elastic shafts and elastic support assemblies into a single, multidimensionalmodel.
1.2 Composite Materials
Composite materials are materials made by combining two or more materials, in a micro
scale form and their elements do not dissolve or fuse into each other, to achieve greater improved
properties. These materials are broadly used in many applications like aerospace vehicles, nuclear
reactors, buildings, automobile vehicles, turbine parts, medical instruments, sports components
and in many civil applications. Laminated composite materials contains of several layers of
different fibre reinforced materials, bonded together to get the required properties like strength,
stiffness, coefficient of thermal expansion, damping and wear resistance. By changing lamina
thickness, material properties and stacking sequence preferred properties of the material can be
achieved. As composite materials gives high stiffness to weight ration and high strength to weight
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ratio, which motives to use in weight sensitive structures. These kind of structures carry
improvement of their structural functions especially in aircraft and space applications.
1.3 Drawbacks of Composite Materials
Even though composite materials gives many advantages over other conventional
materials, their main disadvantage are impact load, repeated stress cycles and many more, these
causes the separation of layers and weakness at interfaces between neighbouring layers, this type
of failure mechanism is known as delamination phenomenon, which may seriously cause failure
of structures. Other problems of composite materials are, variation in coefficient of thermal
expansion and coefficient of moisture of expansion within the materials will leads to residual
stress. These residual stress may cause failure of structures. Stress concentrations near material,
geometric interface, which will cause the damage in the form of delamination, matrix cracking and
parting of adhesive bond, for anisotropic constitution of laminated composite materials. These
difficulties can overcome by avoiding rapid changes in material properties.
1.4 Theoretical Understanding about FGMs
Concept of FGM came for the First time while space plane project was going on in year
1984 in Japan. Materials of kind, would serve the purpose of withstanding a surface temperature
of around 2000 K with temperature gradient of 1000 K through a 10mm thickness. Nowadays
FGMs are becoming a more popular in Germany. FGMs are those of composite materials where
the microstructures or composition of materials are locally different so that a definite variation of
indigenous material properties are attained, variation properties are along certain directions. By
using gradation factor in FGMs, it can be eliminate the sudden change of material properties as in
conventional materials, this gradation will helps to eliminate delamination phenomenon, inter
laminar stresses and gives better bonding within material as a whole.
Many components like thin shells, plates, turbine blades and many more machine parts are
subjected thermal or combined effect of thermo mechanical loading are more likely to fail by large
deflections, buckling and excessive stresses. Thus, in such a condition, FGMs can be used where
high temperature environment or high temperature gradient. FGMs are largely manufactured from
isotropic constituents such as ceramics and metals. Here ceramic portion acts as thermal barrier
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and metallic portion serves as structural support. In this condition ceramic portion delivers heat
resistance and corrosion resistance, metallic portion gives strength and toughness.
Many problems arises for traditional materials and composite materials can be significantly
solved by using FGMs, as gradual variation of material properties in FGMs gives better stability.
This new class of materials and their gradual changes in properties are used to design many
components and applicable to many areas.
1.5 Practical Applications of FGMs
As technology and innovations are growing rapidly, it is desired to meet materials and their
applications as required. FGMs are new class of materials, which are partially fulfil the present
need and requirement in engineering field. FGMs are mainly used in high temperature
environment, as properties are required to vary inside the materials. Following are some of
noticeable applications for FGMs,
i.
Aerospace application (space components, planes, insulations for body, structures, Rocket
engine. Aerospace skins, nuclear reactors, vibration control etc.).
ii.
Engineering applications (rotor shaft system, cutting tools, turbine blades, valves etc.).
iii.
Electronic applications (semiconductors, sensors, substrates etc.).
iv.
Chemical industries (Rector Vessels, Heat Exchangers, Heat Pipe etc.).
v.
Goods materials (Sports, building materials etc.).
vi.
Energy exchangers (Thermo ionic converters, solar cells, thermoelectric generators, Fuel
cells etc.).
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CHAPTER 2
LITERATURE REVIEW
2.1 Introduction
Great number of research has been done in the field of modelling and analysis of FGM’s.
Some of important workout has been done and presented in following section. Composite materials
possess high strength and stiffness. This is an outcome for the use of composite materials in field
of aircraft and space shuttles. In an attempt to develop heat resistant materials FGMs are
developed. Composition of two or materials and structures changes over volume by using some
gradation laws such as, exponential law power law, stepwise variation and continuous variation
[1].
2.2 Functionally Graded Materials
Schmauder et al. [2] investigated mechanical behaviour of ZrO2/NiCr 80 20 compositions
FGMs are analysed and compared with experimental results. And also found that new parameter
matricity controls the stress level of composite, globally and also locally. Sladek et al. [3] analysed
time dependent heat conduction in nonhomogeneous FGMs. Laplace transforms technique is used
to solve initial boundary value problem. Results obtained for finite strip and hollow cylinder
having exponential variation of material properties. Shao et al. [4] presented stress analysis of FG
hollow circular cylinder in combined mechanical and thermal environment by considering linearly
increasing temperature. Temperature dependent material properties are considered and solution
for ordinary differential equations are solved by Laplace transforms technique. Farhatnia et al. [5]
presented stress distribution for composite beam having FGM in middle layer. Temperature
dependent material properties are considered for uniform temperature gradient. Jyothula et al. [6]
presented nonlinear analysis of FGMs in thermal environment by changing material variation
parameter, aspect ratio, and boundary condition re analysed with higher order displacement model.
Nonlinear simultaneous equation are obtained by Navier’s method and equations are solved by
Newton Raphson iterative method. Callioglu [7] presented thermoelasticity solution for FG disc.
By using infinitesimal deformation theory and power law distribution used to get solution. Stress
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and displacement variation are presented along radial position due to centrifugal action, steady
state temperature, internal and external pressure. Abotula et al. [8] studied stress field for curving
cracks in FGMs for thermo-mechanical loading. Using strain energy density criterion effect of
curvature parameters, temperature gradients on crack growth directions, non-homogeneity values
are found and discussed. Bhandari et al. [9] studied parametric study of FGM plate by varying
volume fraction distribution and boundary conditions. Static analysis of FGM plate has studied by
sigmoid law and compared with literature. Kursun et al. [10] presented stress distribution in a long
hollow FG cylinder under thermomechanical environment. By using infinitesimal deformation
theory, solution for displacement model are found.
2.3 Stresses in FGMs
Woo et al. [11] reveals effect of thermomechanical coupling in FGMs plays an important
role. Using von Karman theory, fundamental equation for shallow shells are obtained. Material
properties and thermomechanical stress field are determined. Reddy and Cheng [12] studied
thermomechanical deformation of FG simply supported plates, properties of material are valued
by Mori-Tanaka scheme. Temperature, displacement and stress distribution are computed for
different volume fraction. Jin and Paulino [13] studied edge crack in a FGM strip under thermal
environment. Thermal properties of FGM vary over thickness direction, Young’s modulus and
Poisson’s ratio are assumed to be constant. Temperature solutions are obtained for short time by
using Laplace transform and asymptotic analysis. Chakraborty et al. [14] examined stress variation
in FGMs by use of both power law and exponential variation of material properties. New beam
element is developed for behavioural study of FGMs. Senthil et al [15] presented
thermomechanical deformation of a simply supported FG plate subjected to thermal loads on its
top and bottom surfaces. Transient displacement and thermal stresses are obtained for several
critical location of plate subjected to time dependent temperature and heat flux. Wang et al. [16]
developed meshless algorithm to simulate thermal stress distribution in two-dimensional FGMs.
Displacement components are determined by governing equations and boundary condition. Tahani
et al. [17] presented dynamic characteristics of FG thick hollow cylinder under loading.
Temperature dependent material properties are considered and vary long radial direction. Dynamic
behaviour of thermo elastic stresses are discussed for various grading index. Gupta et al. [18]
studied dynamic crack growth behaviour of FGMs under transient thermo-mechanical loading.
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Principal stress and circumferential stress are discussed and are associated with propagating crack
tip.
2.4 Rotor Dynamics
Zorzi and Nelson [19] studied damped rotor stability including hysteric and internal
viscous damping using linear finite element concept. Rouch and Kao [21] presented cubic function
for mass, stiffness and gyroscopic matrices for a beam element used for transverse displacement.
Kim and Bert [22] presented critical speed of hollow cylindrical shaft for laminated composite
materials using thin and thick shell theories. Obtained results are compared with classical beam
theory, results are well accurate. Bert and Kim [23] presents buckling torque for cylindrical hollow
laminated composite shaft material. Obtained results are compared with experiments, results are
well accurate. Dimarogonas [24] reviewed vibration response for cracked structural member.
Based on vibration amplitude and speed of rotation, crack will open and close. Singh and Gupta
[25] presented dynamic analysis of composite rotor applying layerwise beam theory and
conventional equivalent modulus beam theory. Wettergren and Olsson [26] studied instabilities of
horizontal rotor supported on flexible bearings. Found that critical speed can be reduced
significantly by internal damping. Abduljabbar et al. [27] presented dynamic vibration control of
flexible rotor mounted on journal bearing by using feedback controller device and feed forward
controller device. Reddy and Chin [28] studied thermoelastic response of FG cylinders and plates
in dynamic in condition. First order shear deformation plate theory is used for transverse shear
strains, coupled with heat conduction equation. Liew et al. [29] analysed thermomechanical
behaviour of FG cylinders. Solutions are achieved by novel limiting process. Lin et al. [30]
presents sensitivity analysis, dynamic behaviour of high speed spindle in thermo-mechanical
environment. Spindle stiffness is determined for different speed effect, appropriate cooling effect
and bearing preload. Chang et al. [31] studied laminated composite spinning shaft using first order
shear deformation theory. Governing equation for rotor derived by employing Hamilton’s
principle. Shokrieh et al. [32] analysed torsional stability for rotating composite shaft. Effect of
stacking sequence and boundary conditions on strength and buckling torque of composite drive
shaft has been calculated using finite element analysis. Shao [33] presented solution for
displacement, temperature, thermal and mechanical stresses for FG circular hollow cylinder using
multi-layered method based on laminated composites model. Temperature dependent material
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properties are assumed along radial direction and equal in each layer. Shao and Ma [34] studied
stress analysis in FG hollow cylinder subjected to coupled thermal and mechanical environment
for linearly varying temperature field. Applying Laplace transform technique, solution for time
dependent temperature field and thermo mechanical stress variation has been calculated. Das et al.
[35] studied vibration control of transverse vibration of rotor shaft system due to unbalance.
Vibration control is done by electromagnets. Xiang and Yang [36] studied free and forced vibration
of laminated FG beam of variable thickness for thermally induced stresses using TBT. Roy et al.
[37] studied dynamic behaviour of viscoelastic rotor shaft system introducing internal damping of
material. Critical speed of rotor can be increased by introducing composite material, aluminium
matrix with carbon fibre. Bayat et al. [38] presented thermo elastic analysis for FG rotating disks.
Temperature dependent material properties are considered along radial direction with variable
thickness of disk. Badie et al. [39] examines natural frequency, buckling strength, failure modes,
torsional stiffness and fatigue life of composite drive shaft by changing fibre stacking angle and
orientation angle using finite element analysis (FEA). Poursaeidi and Yazdi [40] presented causes
of extreme bends in rotor shaft and straightening methods by choosing hot spotting process.
Sheihlou et al. [41] studied torsional vibration of FG micro-shaft using Hamilton’s principle.
Vibrations equations are solved by Galerkin’s weighted residual technique. Also studied effect of
volume fraction and boundary condition on natural frequency and frequency response of micro FG
shaft. Rao et al. [42] analysed dynamic behaviour of FG shaft using TBT. Material properties are
assumed to be vary according to exponential law.
2.5 Motivation
Though literature review discloses a lot of research work has been done on thermo
mechanical stress analysis of composites and FGMs. Research on stress analysis of FG tapered
shaft system based on TBT has not been yet discussed. Considering shaft rotating in high
temperature environment like turbine shaft, rocket engine components FGMs gives better solution
over traditional composite materials. Stress distribution in tapered FG shaft under thermal and
mechanical environment compared with traditional materials. Present work discloses stress
analysis of rotor shaft system with FGMs under both thermal and mechanical environment.
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2.6 Aim of Present Work
Main objectives of present work has been laid down here,
i.
Material modelling for tapered FG shaft based on power law gradation.
ii.
Modelling of temperature dependent material properties for FG shaft.
iii.
Variation of mechanical properties with respect to temperature and power law indexes
along radial direction.
iv.
To study stresses developed in tapered shafts made of FGMs.
v.
Comparative study between tapered FG shaft and tapered Stainless Steel shaft.
vi.
To study the stresses developed both in thermal and mechanical environment for different
speed, varying power law index value.
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CHAPTER 3
MATERIAL MODELLING FOR TAPERED FG SHAFT
Material modelling of FG tapered shaft is explained in detail in this chapter by taking power
law gradation and exponential gradation.
3.1 Actual Material Properties of FGM
Properties of FGMs are changing along certain directions, so that it is required to find
effective material properties of FGMs applying to shaft. For exact analysis of FGMs it is required
to find accurate material properties. Many simulations are established for determining properties
of FG shaft. Bulk constituent properties assumes no interaction between phases by employing rule
of mixture. Thermo physical properties are derived by variational approach. Spatial distribution of
constituent materials are having information about micromechanical approach.
3.2 Material Modelling of FGMs
Considering FG beam of finite length and thickness, made of Aluminum Oxide (Al2O3) as
a ceramic material and Stainless Steel (SUS304) as a metal. Here material are varying along y


h
2


h
2
direction, treating top  y    surface as ceramic and bottom  y    surface as metal.
Considering P as an actual material properties,
P  PmVm  PV
c c
Where Pm ,
Vm
and Pc ,
(1)
Vc
are material properties, volume fraction of metal and ceramic
respectively. Also sum of volume fraction of metal Vm  and volume fraction of ceramic Vc  are
always unity at any graded direction. It is related as,
Vm  Vc  1
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3.2.1 Laws of Gradation
There are many laws for varying volume fraction of materials namely, power law
gradation, exponential law of gradation, step wise gradation and continuous gradation etc. Mainly
researchers are using power law gradation and exponential law.
3.2.1.1 Power Law Gradation
Here volume fraction of materials changes along certain direction, by using index called as
power law index k. This factor controls volume fraction of any materials and controls shape,
strength of material. Figure 3. 1 shows volume fraction of metal in FGMS. This is expressed for
rectangular block as,
 2y  h 
Vc  y   

 2h 
k
(3)
Where k  0
Figure 3. 1 Volume fraction of metal in FGM rectangular cross-section.
If p is temperature dependent material properties, it can be written as,
P  P0  P1 1  1  P1  P2 2  P3 3 
(4)
Where P-1, P1, P2 and P3 are temperature coefficients θ-1, θ1, θ2 and θ3 respectively and P0
material properties at ambient temperature. Here material properties are function of temperature
and certain direction and it is given by,
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 2y  h 
E  y,   Ec    Em   
  Em  
 2h 
k
 2y  h 
  y,    c    m   
  m  
 2h 
k
 2y  h 
  y,    c     m   
   m  
 2h 
k
(5)
 2y  h 
  y    c   m  
  m
 2h 
k
Density is assumed to be not dependent on temperature and it is vary along certain
directions only.
3.2.1.2 Exponential Law of Gradation
In this gradation material properties are vary along certain directions as,
P  y   P0e
 h
k y 
 2
(6)
Where P0 denotes, bottom surface material properties of FGM, ‘k’ is the factor which
controls gradation across thickness ‘h’. Young’s modulus thermal conductivity, coefficient of
thermal expansion and density of the FG material are given as,
E  y   E0e
h

k y 
2

K  y   K0e
 h
k y 
 2
  y    0e
  y   0 e
h

k y 
2

h

k y 
2

(7)
(8)
This simple rule of mixture is assumes poison’s ratio is constant.
3.3 Modelling of Material Properties Applicable To tapered FG Shaft
Tapered shaft with finite length L, inner radius at beginning and end of shaft are R0 and R1
respectively having constant thickness of t throughout the tapered shaft. Top surface of shaft is of
ceramic rich and inner surface of shaft is metal rich.
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Figure 3. 2 Volume fraction of metal in tapered FG shaft
3.3.1 Power Law
Figure 3. 2 Volume fraction of metal in tapered FG shaft across the radius of tapered shaft,
it is considered as Two Dimensional variation. FG rotating shaft is a composed of Stainless steel
(SUS3O4) and Aluminum oxide (Al2O3) as metal and ceramic materials. Volume fraction of these
materials are varied along Y direction using power law and gradation factor k. Here radius of shaft
is a function in x as radius is going to change along x direction. Volume fraction for ceramic is
given by,
 r  rm 
Vc  y   

 rc  rm 
k
(9)
Where, k  0
Here, sum of volume fraction of metal and ceramic is unity at any particular radius of shaft i.e.
Vm  Vc  1
(10)
Let ‘P’ be temperature dependent material properties and it can be written as,
P  P0  P1 1  1  P1  P2 2  P3 3 
(12)
Where P-1, P1, P2 and P3 are temperature coefficients θ-1, θ1, θ2 and θ3 respectively and P0
material properties at ambient temperature. Here material properties are function of temperature
and certain direction and it is given by,
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k
 r  rm 
E  y,   Ec    Em   
  Em  
 rc  rm 
k
 r  rm 
  y,    c    m   
  m  
 rc  rm 
(13)
k
 r  rm 
  y,    c     m   
   m  
 rc  rm 
k
 r  rm 
  y    c   m  
  m
 rc  rm 
Density is assumed to be not dependent on temperature and it is vary along certain
directions only. One Dimensional steady state temperature field is assumed by Fourier heat
conduction equation without considering heat generation is given by,
d 
d 
K  y    0;    c

dy 
dy 
at
y  rc ;    m
at
y  rm
Where θm and θc are the temperatures in metal-rich and ceramic-rich surfaces respectively.
The temperature variation is assumed to occur in the radial direction only, and the temperature
field is assumed by considering the following polynomial series [43]
  y    m   c   m   ( y )
(14)
Where

K cm
K 2 cm
K 3cm
k 1
2 k 1
3 k 1 
r

r

r

r








2
3  
(k  1) K m
(2k  1) K m
(3k  1) K m
1

 ( y)  
4
5

C
K cm
K cm
4 k 1
5 k 1
r

r






4
(5k  1) K 5 m
 (4k  1) K m

Where Kcm
 Kc  K m ,
 r  rm 
r 
 and
 rc  rm 


K cm
K 2 cm
K 3cm
K 4 cm
K 5cm
C  1 




2
3
4
5 
 (k  1) K m (2k  1) K m (3k  1) K m (4k  1) K m (5k  1) K m 
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CHAPTER 4
FORMULATION FOR TAPERED FG SHAFT
4.1 Introduction
FG tapered shaft consists of three nodded Timoshenko beam, based on the First order shear
deformation theory considering both Gyroscopic and rotary inertia effect. Hollow circular cross
section shaft is considered for analysis and it is rotating about its longitudinal axis. Figure 4. 1
shows displacement variables and Figure 4. 2 shows diagram representing the shaft.
Figure 4. 1 Displacement variables
Figure 4. 2 Diagram showing tapered shaft and bearing system
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4.2 Finite Element Modelling of Shaft
FG rotating shaft has modelled using Finite element method for three nodded Timoshenko
beam element having six degrees of freedom at each node. By applying linear elastic and small
deflection theory is assumed in present work.
Assumed displacement field as given below [31],
ux ( x, y, z, t )  u ( x, t )  z  x ( x, t )  y  y ( x, t ) 

u y ( x, y, z, t )  v( x, t )  z ( x, t )


uz ( x, y, z, t )  w( x, t )  y ( x, t )

(15)
Strain-displacement relations can be written in Cartesian coordinate system as,


u
1
v
  
 z x  y y ;  xy     y   z 
x
x
x
2
x
x  

1
w
 

 xz    x 
 y  ;  yy   zz   yz  0

2
x
x 
 xx 
(16)
These strain relations can be transformed in to cylindrical coordinate system by using
transformation matrix. Strain-displacement relations can now be written in cylindrical coordinate
system by taking y = r cosθ, z = r sinθ, m = cosθ and n = sinθ.
 xx  1 0 0   xx 
   0 n m   
 x  
  xy 
 xr  0 m n   xz 
(17)


u
 r sin  x  r cos  y ;  rr      r  0
x
x
x
1
v
w
 
 x    y sin    x cos   sin   cos 
r

2
x
x
x 
1
w
v 
 xr    x sin    y cos   sin 
 cos  
2
x
x 
(18)
Where
 xx 
And the above strain displacement relations in cylindrical coordinate can be written in
matrix form as,
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 
 2 x

 xx 

   1  0
 x  2 
 xr 

 0

0
0

x

cos 
x
 sin 

x

sin 
x
cos 
2r sin 

x
2r cos 
cos 
sin 
sin 
 cos 

x
 u 
0  v 
 
  w 
r
 
x    x 

 
0   y 
 
 
(19)
Stress strain relations in any layer of the FG shaft can be written as [31],
 xx   C11r
    k C
 x   s 16 r
 xr   0
ks C16 r
ks C66 r
0
0   xx 
 
0   x 
ks C55r   xr 
(20)
Where ks is the shear correction factor and Cijr represents constitutive element, related to
elastic constants for transversely isotropic material.
Coupled (thermo-mechanical) stress strain relations in any layer of FG shaft can be written as,
 xx   C11r
    k C
 x   s 16 r
 xr   0
ks C16 r
ks C66 r
0
0    xx  T  


0    x    0  

ks C55r  
  xr   0  
(21)
4.2.1 Kinetic Energy Expression of Shaft
The effect of both rotary and translation of FG shaft are considered for deriving kinetic
energy expression, it is written as,
2
2
2
2
2
L
1  I m (u  v  w )  I d (  x   y )  2I p  x  y 
Ts   
 dx
2 0   I p 2  2I p  2 I p  2 Id (  x 2   y 2 ) 


(22)
Where,
Ω is rotating speed of the shaft, L is length of the shaft, Ip, Im and Id are polar mass moment
of inertia, mass moment of inertia and diametrical mass moment of inertia respectively. In above
equation neglecting some small terms, first variation of kinetic energy is written as,
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   u
 y
  x
 v
 w 
I
u

v

w

I





m
d
x
y


L
t
t 
t
t
 t

 Ts   
  y

0
  I p  I p   I d   x
  y x 
t






dx



(23)
4.2.2 Strain Energy Equation for FG Shaft
Strain energy of FG shaft is given by,
Us 
As
1
1
T
   dV    xx xx   rr rr      2 xr xr  2 x  x  2 r  r dV

2V
2V
(24)
 rr     r  0
Strain energy can be rewritten as,
Us 
1
( xx xx  2 xr xr  2 x  x )dV
2 V
(25)
By taking variation in strain energy expression we get,


 x 
 u
  x sin    y cos    

 x  r sin  x  

  xx 
   xr 
w
v   




 cos 


 sin 
 
 U s     r cos  y

x
x  dV

x


V

v
w
  
 
 x   y sin    x cos   sin  x  cos  x  r x  
 
 
(26)
Where, dV  rd drdx
4.2.3 Kinetic energy expression for disks on shaft
Disks fixed on shaft are treated as isotropic material. Expression for kinetic energy of disks
is written as,


D
2
2
2
D
2
2
L

1 ND  I mi  u  v  w   I di  x   y
 ( x  xDi )dx
Td    
2 0 i 1  2I diD  x  y  I piD 2  2I piD  2 I piD 


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(27)
Page 19
D
D
D
Where I mi , I di and I pi mass moment of inertia, diametrical and polar mass moment of
inertia of ith disk respectively. The term   x  xDi  represents one dimensional spatial Dirac delta
function. xDi gives location ith disk and ND is the number of disks which are attached to shaft.
Variation of the kinetic energy of disk given by,
 D   u
 y
 v
 w  D   x
v
w
 y
 I mi  u
  I di   x
t
t 
t
t
 t

 Td    
 y  D

0 i 1
D
I piD   y x   x
  I pi  I pi 
t 


L ND
 
 
 
( x  xDi )dx



(28)
4.3 Expression for work done to external load and bearings
Rx, Ry and Rz are external force intensities, Mx, My and Mxθ are external intensities of
moments distributed along shaft length. Virtual work done by external loads is given by,
L
 We    Rx u  Ry v  Rz w  M y y  M x x  M x  dx
(29)
0
Spring and viscous damper bearing are modelled here and virtual work done by springs
and dampers are given by,
  K yyBi v v  K zyBi v w  K yzBi w v  K zzBi w w
 WB     Bi
( x  xBi )dx
Bi
Bi
Bi

C
v

v

C
v

w

C
w

v

C
w

w

0 i 1 
zy
yz
zz
 yy
L NB
(30)
Where KBi and CBi represents equivalent stiffness and equivalent damping coefficient of ith
bearing.
4.4 Governing equation of rotor shaft
The governing equation of rotating shaft can be derived by using above strain energy
expression, kinetic energy of shaft and disks, work done expressions by external loads and bearings
by applying Hamilton’s principle. Which is given by,
t2
  T  U
s
  We   WB dt  0
(31)
t1
Total kinetic energy of shaft and disks is given by T  Ts  Td
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t2
  T  T   U
s
d
s
  We   WB dt  0
(32)
t1
Finite element analysis is to find field variables at each nodal points by approximate
analysis, taking nodes within the elements. These variables is a function of values at nodal points
of the element, this function is called as interpolation function or shape function. In present model
three node one dimensional Timoshenko beam element is considered having six degrees of
freedom at each node.
Lagrangian interpolation function is used to approximate the displacement fields rotating
shaft. Elemental nodal degree of freedom at each node are u, v, w, βx, βy, ϕ. Displacement field
variables are given by,
r
u   u k  t  k   ;
k 1
r
x   
k 1
r
v   v k  t  k   ;
k 1
r
k
x
r
w   wk  t  k  
k 1
(33)
r
 t  k   ;  y     t  k   ;     t  k  
k 1
k
y
k
k 1
One dimensional Lagrangian polynomial is given by,
  m
m 1   
m
m k k
r
Lk    
Above equation gives value zero at all points except k = 1, Lk is Lagrangian polynomial
and ψk (k = 1, 2, 3…) interpolation function and η is the natural coordinate whose value varies
from -1 to +1. For three node element r is equal to three, shape function can be written as,
1 
 (1   )
;
2
 2  1  2 ;  3   (1   )
2
(34)
Now by putting above displacement variables into the governing equations we get
following equation of motion for FG spinning shaft.
 M q  C    G  q   K q  F 
(35)
Where [M] is mass matrix, [G] is gyroscopic matrix, [C] is total damping matrix, [K] is
structural stiffness matrix, {q} is nodal displacement vector and {F} is the external force vector.
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4.5 Contribution of internal damping
By including both internal viscous and hysteresis damping [19] of shaft and disk elements
extended final equation of motion can written as,
 1  
H


 1   H2
M
q

C


G


K
q

        V      

V  




K  



 q  F 


H
  K cir 

1   H2 
(36)
Where Kcir is the skew-symmetric circulation matrix [16].
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CHAPTER 5
RESULTS AND DISCUSSION
A complete MATLAB code has been developed and validated for above formulation.
Results are presented for different stresses, speed and power law index value based on problem
specified below.
5.1 Problem Description and Summarization of Discussion
Tapered shaft consists of rigid disk attached at its centre, supported on two similar bearings.
Shaft is modelled for present analysis using three nodded beam element. Dimensions of shaft and
rigid disk are presented in Table 5. 1. Tapered shaft is divided into 10 elements, and 8 equal thick
layers. Stiffness and damping coefficients for two identical bearings are taken as Kyy=7 × 107
N/m, Kzz=5 × 107 N/m, Cyy=700 Ns/m, Czz=500 Ns/m.
Table 5. 1 Geometric dimensions of steel and FG tapered shaft
Parameter
Shaft length (m)
FG Shaft
Disk
1.0
Beginning radius, Ro (m)
0.035
End radius, Rs (m)
0.015
Thickness of hollow shaft, t (m)
0.03
Coefficient of viscous damping (Ns/m)
0.0002
Coefficient of hysteric damping (Ns/m)
0.0002
Density (Kg/m3)
7800
Outer diameter (m)
0.24
Thickness (mm)
5.0
Developed MATLAB code has been validated for present available literatures. Several results for
FG tapered shaft has presented in this chapter. Primarily, temperature distribution in FG shaft
along radial direction is shown for different power law index value. Table 5. 2 and Table 5. 3 are
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values for temperature dependent material properties such as young’s modulus, Poisson’s ratio,
coefficient of thermal expansion (CTE) and density of material are plotted along radial direction
for different power law index. Next, comparative study between stainless steel shaft and FG shaft.
Then, transient stress analysis is carried out for different speed and power law index value.
Table 5. 2 Material properties of FGM [43]
Properties
Stainless Steel
Aluminum oxide
Young’s Modulus (GPa)
210
390
Poisson’s ratio
0.3
0.26
7800
3960
Density
Table 5. 3 Materials and temperature coefficient of mechanical properties [28]
Property
E (Pa)
K (W/mK)
CTE (1/K)
Poisson Ratio
Material
P0
P-1
P1
P2
P3
SUS304 201.035×109
0
3.079×10-4
-6.533×10-7
0
Al2O3
349.548×109
0
-3.853×10-4
4.027×10-7
-1.673×10-10
SUS304
15.3789
0
-0.00126
0.209×10-5
-7.22×10-10
Al2O3
-14.087
-1123.6
0.00044
0
0
SUS304
12.33×10-6
0
0.0008
0
0
Al2O3
6.827×10-5
0
0.00018
0
0
SUS304
0.3262
0
-2.001×10-4
3.797×10-7
0
Al2O3
0.26
0
0
0
0
5.2 Validation of Code
To verify the developed code, uniform shaft made of graphite epoxy composite material
with disk at centre of shaft [31] (dimensions are in Table 5. 5). Obtained results are well agreement
with literature. Figure 5. 1 shows Campbell diagram for first four pairs of modes attained an
excellent match with published result [31]. Subsequently temperature dependent material property
are discussed, these Figures are also match with already published results, thus developed codes
are validated for correctness.
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Table 5. 4 Dimensions and properties of laminated graphite-epoxy shaft [35]
Parameters
Composite shaft
Length (m)
0.72
Inner diameter (m)
0.028
Outer diameter (m)
0.048
Shear correction factor
0.56
Model damping ratio
0.01
I mD (Kg)
e (10-5 m)
I pD
Disk
2.4364
5.0
(Kg m2)
0.3778
m2)
0.1901
I dD (Kg
Bearing
Kyy=Kzz (107 N/m)
1.75
Cyy=Czz (102 Ns/m)
5.0
Figure 5. 1 Campbell diagram for laminated graphite-epoxy composite material.
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5.3 Temperature Distribution in Tapered FG Shaft
Temperature variation in tapered FG shaft is shown in Figure 5. 2. As material properties
are functions of temperature and radial direction, presented here temperature distribution. This
variation is due to thermal conductivity, CTE, young’s modulus of material. Temperature variation
is obtained at mid-section of tapered shaft, it is observed that, for k zero to one temperature
decreases gradually and k greater than one temperature increases.
Table 5. 5 Temperature variation in mid-section of tapered FG shaft.
Radius (m)
0.0279
0.0316
0.0354
0.0391
0.0429
0.0466
0.0504
0.0541
k
T1(K)
T2(K)
T3(K)
T4(K)
T5(K)
T6(K)
T7(K)
T8(K)
0
471.88
515.63
559.38
603.13
646.88
690.63
734.38
778.13
0.2
469.67
510.19
552.50
595.90
640.15
685.10
730.68
776.81
0.5
468.78
507.56
548.46
591.04
635.11
680.60
727.45
775.60
1
468.82
507.20
547.11
588.66
632.01
677.30
724.73
774.50
5
470.66
511.98
553.32
594.76
636.55
679.24
724.03
773.58
10
471.20
513.60
556.00
598.40
640.82
683.41
726.87
774.48
Figure 5. 2 Temperature variation in mid-section of tapered FG shaft.
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5.4 Material properties of tapered FG shaft depends on temperature and power
law index
Tapered FG shaft is modelled by taking Aluminum oxide as a ceramic and Stainless steel
as a metal, these are rich at top and bottom surfaces respectively. Figure 5. 3 shows volume fraction
of ceramic material of FGM. Properties of material are changes along radius of shaft, here power
law index is significant factor. According to above consideration and formulation, as ‘k’ value
approaches to zero, material becomes fully ceramic and as ‘k’ value approaches to infinity material
becomes fully metal. Linear variation of material is obtained by taking k=1. Since shaft is in
thermal environment, it is necessary to find properties depends on temperature. Figure 5. 4 Figure
5. 5 and Figure 5. 6 are modulus elasticity, Poisson’s ratio and coefficient of thermal expansion
(CTE) respectively. Here properties are changing for each element as analysis is progressed. Also
assumed that properties are average of inner and outer surface of a particular layer is at the middle
of the layer.
Figure 5. 3 Volume fraction of ceramic material along radius for power law index.
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Figure 5. 4 Variation of young’s modulus along radius for power law index.
Figure 5. 5 Variation of Poisson’s ratio along radius for power law index.
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Figure 5. 6 Variation of CTE along radius for power law index.
5.5 Stress analysis in tapered FG shaft
Objective of present study is to analyse the coupled thermo-mechanical stresses in tapered
FG shaft. Initially comparative study of FG shaft has carried out over steel shaft. Then, stress
results are plotted for different values of power law index and speed and also time dependent stress
are also presented.
5.5.1 Comparative study of tapered FG shaft over steel tapered shaft
It is essential to compare results of stainless steel and FG shaft to show effects of FG shaft
over steel shaft. In this comparative study section temperature assumed is linear. Material
properties are functions of temperature and radial direction only. Keeping all parameters (as in
Table 5. 1) are same for both FG and steel analysis is done. Figure 5. 7 shows normal and shear
stress in x and theta direction respectively for Stainless steel material. Normal stress is increasing
along radius negatively, shear stress also increasing along radius positively. Figure 5. 8 and Figure
5. 9 shows normal and shear stress in x and theta direction respectively for FGM.
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Figure 5. 7 Stress developed in tapered Steel shaft along radius.
Results are presented for shaft running at 6000 RPM, different ‘k’ values. It is easily seen from
these Figures that, stress developed in FG shaft is lesser than Stainless steel shaft near outer surface
of shaft. This conclusion influence to study FGMs.
Figure 5. 8 Normal stress in tapered FG shaft along radius.
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Figure 5. 9 Shear stress in tapered FG shaft along radius.
5.5.2 Variation of stresses for different values of ‘k’ in radial direction
In coupled thermal and mechanical environment, thermal strain is in coupled with only
normal stress and shear stress in theta direction as in equation (21). Temperature dependent
material properties are considered and temperature variation is as shown in Figure 5. 2. Figure 5.
10 (a) and (b) shows normal stress on plane perpendicular to x axis in x direction at 6000 RPM
and 12000 RPM respectively. Maximum stress value is obtained at time t=0.008 sec and t=0.044
sec for 6000 RPM and 12000 Rpm respectively. Stress increases along radius of shaft negatively
as thermal stress dominates mechanical stress in coupled environment. Also considering at a
particular radius, normal stress is increases as ‘k’ value increases, because volume fraction of steel
material is increases as power law index (k) increases. Fig (a) and Fig (b) are almost same but
difference is, as speed increases amplitude in stress is more, which is shown and explained in
succeeding section.
Figure 5. 10 (a) and (b) shows shear stress on plane perpendicular to x axis in theta direction
at 6000 RPM and 12000 RPM respectively. Maximum stress values are obtained at time t=0.008
sec and t=0.044 sec for 6000 RPM and 12000 Rpm respectively. Shear stress increases along radius
of shaft positively as thermal stress dominates mechanical stress in coupled environment. Also
considering a particular radius, shear stress is increases as ‘k’ value increases, because volume
fraction of steel material is increases as power law index (k) increases. Fig (a) and Fig (b) are
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almost same but difference is, as speed increases fluctuation in stress is more, which is shown and
explained in succeeding section.
Figure 5. 10 Normal stress in tapered FG shaft along radius:
(a) At 6000 RPM, (b) At 12000 RPM
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Figure 5. 11 Shear stress in tapered FG shaft along radius.
(a) At 6000 RPM, (b) at 12000 RPM
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5.5.3 Transient uncoupled stress analysis for different value of power law index
Uncoupled (without considering thermal strain) transient stress obtained by
equation (20) at the top of the surface. Considering temperature dependent material properties and
temperature variation as in Figure 5. 2. Figure 5. 12 Shows normal stress on plane perpendicular
to x axis in x direction, Figure 5. 13 and Figure 5. 14 shows shear stress on plane perpendicular to
x axis in theta and radial direction respectively also (a) and (b) represents shaft running at 6000
RPM and 12000 RPM respectively. Maximum stress amplitude developed in tapered shaft for
initial small time interval, then stress amplitude is decreases as time increases and maintain
constant amplitude at speed 6000 RPM. For initial small time interval stress amplitude is smaller,
as time increases stress amplitude increases then attains almost constant amplitude for shaft
running at 12000 RPM. Also it can be seen that, as power law index increases stress amplitude
increases, this is based on displacement. Also by looking at y axis values, stress amplitude
increases as speed increases.
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Figure 5. 12 Transient uncoupled normal stress in tapered FG shaft:
(a) At 6000 RPM, (b) at 12000 RPM
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Figure 5. 13 Transient uncoupled shear stress in tapered FG shaft in theta direction:
(a) At 6000 RPM, (b) at 12000 RPM
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Figure 5. 14 Transient uncoupled shear stress in tapered FG shaft in radial direction:
(a) At 6000 RPM, (b) at 12000 RPM
5.5.4 Transient coupled stress analysis for different value of power law index
Coupled (with considering thermal strain) transient stress obtained by equation (21) at the
top of the surface. Considering temperature dependent material properties and temperature
variation as in Figure 5. 2. Figure 5. 15 Shows normal stress on plane perpendicular to x axis in x
direction, Figure 5. 16 shows shear stress on plane perpendicular to x axis in theta direction also
(a) and (b) represents shaft running at 6000 RPM and 12000 RPM respectively. Maximum stress
amplitude developed in tapered shaft for initial small time interval, then stress amplitude is
decreases as time increases and maintain constant amplitude at speed 6000 RPM. For initial small
time interval stress amplitude is smaller, as time increases stress amplitude increases then attains
almost constant amplitude for shaft running at 12000 RPM. Also it can be seen that, as power law
index increases stress amplitude increases, this is based on displacement. Also by looking at y axis
values, stress amplitude increases as speed increases.
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Figure 5. 15 Transient coupled normal stress in tapered FG shaft:
(a) At 6000 RPM, (b) At 12000 RPM
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Figure 5. 16 Transient coupled shear stress in tapered FG shaft in theta direction:
(a) At 6000 RPM, (b) At 12000 RPM
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CHAPTER 6
CONCLUSION AND SCOPE OF FUTURE WORK
Important conclusions are drawn in this chapter based on above discussed results.
Opportunity of future work is also been presented in this chapter.
6.1 Conclusions
Present study supports to draw following important conclusions.
i.
Three nodded Timoshenko beam element has been implemented for modelling and analysis
of FG tapered shaft by taking into account of structural damping and hysteretic damping
in temperature environment.
ii.
The temperature distribution is assumed based on one Dimensional steady state
temperature field by using Fourier heat conduction equation without considering heat
generation.
iii.
Temperature dependent material properties are established by taking different power law
index value.
iv.
Stress values are compared between steel and FG shaft by taking temperature dependent
material properties for linear variation of temperature, it is found that stresses developed
in FG shaft is lower than Steel shaft.
v.
It is also noted that stress increases as power law index increases.
vi.
Stress amplitude increases as speed of the shaft increases.
6.2 Scope of future work
i.
Nonlinear modelling of FG shaft.
ii.
Active vibration control of FG shaft.
iii.
Analysis and control of breathing crack in FG shaft.
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Appendix
Now simplifying and arranging the above equation gives,
n
I m  x      (r 2i 1  r 2i ); I d  x  
i 1
i
 n
 n
4
4

(
r

r
);
I
x



 i1 i p
  (r 4i1  r 4i )
4 i 1 i
2 i 1 i
 u : Im
 2u
 2u
 2 ND D  2v

A

k
B
  I mi 2   x  xDi   Rx
11
s 16
t 2
x 2
x 2 i 1
t
 v : Im
 
 2  x ND D  2v
 2v
 2v  1
 ks  A55  A66   y  2   k s B162
  I mi 2   x  xDi   Pvb  Ry
2
2
t
x
t
i 1
 x x  2
 w : Im
  2 w  x
2w

k
A

A



s
55
66 
2
t 2
x
 x
 2  y ND D  2 w
 1

k
B
  I mi 2   x  xDi   Pwb  Rz

s 16
2
2

x
t
i 1

 y 1
 2x
2x
 2v
 w

I

I


k
B

D
 ks  A55  A66  
  x 
p
s 16
11
d 2
2
2
t
t 2
x
x
 x

 x : 
 Mx
2
N
D


 y
 D  x 
   I di
 ks B16

   x  xDi 
x i 1 
t 2 


2
 2 y
2 y
 x 1
v  

2  w

I


k
B

D

k
A

A




Id
p
s
16
11
s
55
66
y


2
t 2
x 2
x 2
x  

 t
 y : 
  My
ND 
2 y 


D
x
  ks B16

   I di
   x  xDi 
x i 1 
t 2 


 : I p
 2
 2u
 2 N D 
 2 
 ks B16 2  ks D66 2    I diD 2    x  xDi   M x
2
t
x
x
t 
i 1 
Where
A55  x  
 n
 n
2
2
C
r

r
A
x

C66i  ri 21  ri 2 


55i  i 1
i ;
66  
2 i 1
2 i 1
 C16i  ri31  ri3  ; D11  x  
 n
 n
4
4
C
r

r
D
x

C66i  ri 41  ri 4 


11i  i 1
i ;
66  
4 i 1
2 i 1
A11  x     C11i  ri 21  ri 2  ;
n
i 1
B16  x  
2
3
n
i 1
Mass, stiffness, circulation and gyroscopic matrices of FG Timoshenko beam element.
NIT ROURKELA
Page 41
 
The general displacement matrix q e is given by
q 
e T

 U e  V e  W e   xe   ye   e 
T
T
T
T
T
T

118
Elemental mass matrix
  M 11 
 33

  0
33

  0
33
 M e   
  0
33

  0
33

 0
  33
033
033
033
 M 22 
33
033
033
033
033
033
033
033
 M 33 
33
033
033
033
033
033
033
033
 M 44 
33
033
033
 M 55 
33
033
033
033
033
033
033










66
 M  
33 1818
Where
xb
M   I m i j dx  M
11
ij
xb
22
ij
 M ;M
33
ij
  I d i j dx  M ; M
44
ij
55
ij
xa
xb
66
ij

xa
 I 
p
i
j
dx
xa
Elemental stiffness matrix
  K 11 
   33
  0
33

  0
33
e
 K   
  0
33

  0
33

  K 61 
 
 33
033
033
033
 K 22 
33
033
033
 K 24 
33
 K 25 
33
033
 K 33 
33
 K 34 
33
 K 35 
33
 K 42 
33
 K 43 
33
 K 44 
33
 K 45 
33
 K 52 
33
 K 53 
33
 K 54 
33
 K 55 
33
033
033
033
033
 K 16  
33

0
 33 

033 
033 
033 
 K 66  
33 1818
Where
 i  j
K   A11
dx;
x x
xa
xb
11
ij
xb
K   ks B16
16
ij
xa
  j
  ks  A55  A66  i
dx;
x x
xa
 i  j
dx
x x
xb
K
22
ij
NIT ROURKELA
 i  j
1
   ks B16
dx
2
x x
xa
xb
K
24
ij
Page 42
xb
K
25
ij
  ks  A55  A66 
xa
 i
 j dx
x
xb
 i
  j
1
35
K   ks  A55  A66 
 j dx Kij    ks B16 i
dx
x
2
x x
xa
xa
  j
K   ks  A55  A66  i
dx;
x x
xa
xb
xb
33
ij
xb
 i  j
1
dx;
; Kij42    ks B16
2
x x
xa
34
ij

  j 
xa ks  A55  A66  i j  D11 xi x dx;
xb
 ks  A55  A66  i
Kij52 
xa
 j
x
xb
K ij53 
dx;
xb
K ij45 
1
2k B
s
16
xa
 j 
 1
 i
1
   ks B16
 j  ks B16 i
dx;
2
x
2
x 
xa 
K
xb
Kij61   ks B16
xa
 i  j
dx;
x x
xb
Kij66   ks D66
xa
dx
1
  2 k B
s
16
xa

 k  A
xb
K
 j 
 i
1
 j  k s B16 i
dx
x
2
x 
 i  j
dx
x x
xb
54
ij
x
xa
xb
Kij44 
 j
xb
Kij43   ks  A55  A66  i
55
ij

s
xa
55
 A66  i j  D11
 i  j 
dx
x x 
 i  j
dx
x x
Circulatory stiffness matrix
xj
 Kcir 1818   M T  Mdx
xi
Where
0
0
0
0 0 0 0 0 0
 1  2  3 0 0 0 0 0 0
 0 0 0 ' ' ' 0 0 0
0
0
0  1'  2'  3' 0 0 0 
1
2
3

 0 0 0 0 0 0  '  '  '  '  '  ' 0 0 0 0 0 0 
 M    0 0 0 0 0 0  1''  2''  3''  1''  2''  3'' 0 0 0 0 0 0 
1
2
3
1
2
3


 0 0 0  1''  2''  3'' 0 0 0
0
0
0  1''  2''  3'' 0 0 0 


0
0
0
0 0 0  1  2  3  618
 0 0 0 0 0 0 0 0 0
0
0
0
0
0
0
GA 0

0 GA 0
0
   
0
0
0
0
0
0
0  EI

0
0
0
0
NIT ROURKELA
0
0
0
EI
0
0
0
0 
0

0
0

0  66
Page 43
Gyroscopic matrix
 033

 033
 0
33
e
G    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
xb
Gij45   I p i j dx ;
xa
NIT ROURKELA
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
033
033
033
G 45 
33
G 54 
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 1818
xb
Gij54    I p i j dx
xa
Page 44
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