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 ( P1T 1 1 PT PT ) 1 PT 2 3 (4) Where P1 , 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 1u 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 ) 2I 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 2x 2 v0 2x 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 2x 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 5.676 105 m and for unstable response the maximum amplitude is 1.897 104 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 13 we 13 xe 13 ye T T T T 13 112 Mass Matrices: M v 33 033 M e 0 33 0 33 033 M w 33 033 033 033 M x 33 033 033 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 33 1212 xf N D xi xi 033 033 033 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 33 033 Ke T K v x 33 T K v y 33 xf 033 K v x 33 K ww 33 K w x 33 K x x 33 T K x y 33 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 33 Kvv K s A55 A66 ' xi T K w x 33 T NB ' K zzBi T x xBi dx i 1 xf xi K v y 33 K w y 33 K x y 33 K y y 33 1212 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 33 033 CeB 033 0 33 xf N B Cv xi i 1 Bi C yy 033 Cw 33 033 033 033 033 033 033 033 033 033 033 1212 xf N B x xBi dx Cw CzzBi T x xBi dx T xi i 1 Gyroscopic Matrix: 033 033 Ge 0 33 033 G I P x y 033 033 033 033 033 033 033 033 033 G x y 33 G x y 33 033 1212 T xf dx T xi I PD xf N D T x xDi dx i 1 xi xf Elemental circulation matrix KCir 1212 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" 412 0 0 EI 0 44 Page 52 References [1] Koizumi, M., 1993, “Concept of FGM,” Ceramic Trans., 34, pp. 3–10. 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[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

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