Spectral Element Method in Str - Usik Lee

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Résumé :

Preface xi

Part One Introduction to the Spectral Element Method and Spectral Analysis of Signals 1

1 Introduction 3

1.1 Theoretical Background 3

1.1.1 Finite Element Method 3

1.1.2 Dynamic Stiffness Method 4

1.1.3 Spectral Analysis Method 4

1.1.4 Spectral Element Method 5

1.1.5 Advantages and Disadvantages of SEM 6

1.2 Historical Background 8

2 Spectral Analysis of Signals 11

2.1 Fourier Series 11

2.2 Discrete Fourier Transform and the FFT 12

2.2.1 Discrete Fourier Transform (DFT) 12

2.2.2 Fast Fourier Transform (FFT) 16

2.3 Aliasing 17

2.3.1 Aliasing Error 17

2.3.2 Remedy for Aliasing 20

2.4 Leakage 20

2.4.1 Leakage Error 20

2.4.2 Artificial Damping 23

2.5 Picket-Fence Effect 25

2.6 Zero Padding 25

2.6.1 Improving Interpolation in the Transformed Domain 26

2.6.2 Remedy for Wraparound Error 27

2.7 Gibbs Phenomenon 29

2.8 General Procedure of DFT Processing 30

2.9 DFTs of Typical Functions 34

2.9.1 Product of Two Functions 34

2.9.2 Derivative of a Function 36

2.9.3 Other Typical Functions 36

Part Two Theory of Spectral Element Method 39

3 Methods of Spectral Element Formulation 41

3.1 Force-Displacement Relation Method 41

3.2 Variational Method 58

3.3 State-Vector Equation Method 68

3.4 Reduction from the Finite Models 75

4 Spectral Element Analysis Method 77

4.1 Formulation of Spectral Element Equation 77

4.1.1 Computation of Wavenumbers and Wavemodes 79

4.1.2 Computation of Spectral Nodal Forces 81

4.2 Assembly and the Imposition of Boundary Conditions 82

4.3 Eigenvalue Problem and Eigensolutions 83

4.4 Dynamic Responses with Null Initial Conditions 86

4.4.1 Frequency-Domain and Time-Domain Responses 86

4.4.2 Equivalence between Spectral Element Equation and Convolution Integral 87

4.5 Dynamic Responses with Arbitrary Initial Conditions 89

4.5.1 Discrete Systems with Arbitrary Initial Conditions 90

4.5.2 Continuous Systems with Arbitrary Initial Conditions 99

4.6 Dynamic Responses of Nonlinear Systems 104

4.6.1 Discrete Systems with Arbitrary Initial Conditions 105

4.6.2 Continuous Systems with Arbitrary Initial Conditions 107

Part Three Applications of Spectral Element Method 111

5 Dynamics of Beams and Plates 113

5.1 Beams 113

5.1.1 Spectral Element Equation 113

5.1.2 Two-Element Method 114

5.2 Levy-Type Plates 119

5.2.1 Equation of Motion 119

5.2.2 Spectral Element Modeling 120

5.2.3 Equivalent 1-D Structure Representation 125

5.2.4 Computation of Dynamic Responses 126

Appendix 5A: Finite Element Model of Bernoulli-Euler Beam 130

6 Flow-Induced Vibrations of Pipelines 133

6.1 Theory of Pipe Dynamics 133

6.1.1 Equations of Motion of the Pipeline 134

6.1.2 Fluid-Dynamics Equations 136

6.1.3 Governing Equations for Pipe Dynamics 137

6.2 Pipelines Conveying Internal Steady Fluid 138

6.2.1 Governing Equations 138

6.2.2 Spectral Element Modeling 139

6.2.3 Finite Element Model 144

6.3 Pipelines Conveying Internal Unsteady Fluid 146

6.3.1 Governing Equations 146

6.3.2 Spectral Element Modeling 147

6.3.3 Finite Element Model 153

Appendix 6.A: Finite Element Matrices: Steady Fluid 157

Appendix 6.B: Finite Element Matrices: Unsteady Fluid 159

7 Dynamics of Axially Moving Structures 163

7.1 Axially Moving String 163

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