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Chapter 1 Introduction 1

1.1 General Introduction 1

1.2 One-dimensional Vibration 1

1.3 Two-dimensional Vibration 2

1.4 Three-dimensional Vibration 3

Chapter 2 Chebyshev Polynomials 6

2.1 Introduction 6

2.2 Definition 6

2.3 Basic Properties 7

2.4 Some Advantages 8

Chapter 3 Ritz Method 13

3.1 Introduction 13

3.2 Basic Formulation 14

3.3 Construction of Solutions 17

3.4 Construction of Boundary Characteristic Functions 20

Chapter 4 3-D Vibration of Rectangular Plates 29

4.1 Introduction 29

4.2 Mathematical Formulation 30

4.3 Convergence Study and Comparison 36

4.4 Numerical Results 40

4.5 Conclusions 47

Chapter 5 3-D Vibration of Rectangular Plates On Pasternak Foundation 48

5.1 Introduction 48

5.2 Formulation 49

5.3 Convergency and Comparison 51

5.4 Numerical Results 58

5.5 Conclusions 63

Chapter 6 3-D Vibration of Circular and Annular Plates 65

6.1 Introduction 65

6.2 Theoretical Formulation 66

6.3 Convergence and Comparison Study 71

6.4 Numerical Results 78

6.5 Conclusions 84

Chapter 7 3-D Vibration of Circular Plates on Pasternak Foundation 86

7.1 Introduction 86

7.2 Formulation 88

7.3 Convergence and Comparison Study 91

7.4 Parametric Study 94

7.5 Conclusions 102

Chapter 8 3-D Vibration of Solid and Hollow Circular Cylinders 104

8.1 Introduction 104

8.2 Formulation 105

8.3 Convergence and Comparison Studies 107

8.4 Numerical Results 113

8.5 Concluding Remarks 119

Chapter 9 3-D Vibration of A Torus with Circular Cross-Section 120

9.1 Introduction 120

9.2 Theoretical Formulation 122

9.3 Convergence Study 129

9.4 Eigenfrequencies and Mode Shapes 132

9.5 Conclusions 138

Chapter 10 3-D Vibration of Isosceles Triangular Plates 140

10.1 Introduction 140

10.2 Theoretical Formulation 142

10.3 Convergence and Comparison Studies 146

10.4 Numerical Results 150

10.5 Concluding Remarks 157

Chapter 11 3-D Vibration of Prisms with Isosceles Triangular Cross-sections 158

11.1 Introduction 158

11.2 Formulation 160

11.3 Convergence and Comparison Study 162

11.4 Numerical Results 166

11.5 Conclusion 171

Chapter 12 3-D Vibration of Skew Thick Plates 172

12.1 Introduction 172

12.2 Theoretical Formulation 173

12.3 Convergence and Comparison Studies 177

12.4 Numerical Results 188

12.5 Conclusions 192

Chapter 13 3-D Vibration of Prisms with Skew Cross-section 193

13.1 Introduction 193

13.2 Mathematical Formulation 195

13.3 Convergence and Results 196

13.4 Conclusions 202

Chapter 14 3-D Vibration of Generalized Super Elliptical Plates 203

14.1 Introduction 203

14.2 Formulation 204

14.3 Convergence and Comparison Study 206

14.4 Parametric Study 212

14.5 Concluding Remark 217

Chapter 15 3-D Vibration of Thick Circular Plates with Built-in Edges 218

15.1 Introduction 218

15.2 Theoretical Formulation 220

15.3 Convergence and Comparison Studies 226

15.4 Parametric Studies 230

15.5 Conclusions 239

Chapter 16 3-D Vibration of Rectangular Plates with Mixed Boundary Conditions 240

16.1 Introduction 240

16.2 Model 242

16.3 Basic Formulae 244

16.4 Boundary Characteristic Functions 244

16.5 Convergence Study 246

16.6 Comparison Study 249

16.7 Numerical Results 252

16.8 Conclusions 256

References 257