Spectral Element Method in Structural Dynamics is a concise and timely introduction to the spectral element method (SEM) as a means of solving problems in structural dynamics, wave propagations, and other related fields. The book consists of three key sections. In the first part, background knowledge is set up for the readers by reviewing previous work in the area and by providing the fundamentals for the spectral analysis of signals. In the second part, the theory of spectral element method is provided, focusing on how to formulate spectral element models and how to conduct spectral element analysis to obtain the dynamic responses in both frequency- and time-domains. In the last part, the applications of SEM to various structural dynamics problems are introduced, including beams, plates, pipelines, axially moving structures, rotor systems, multi-layered structures, smart structures, composite laminated structures, periodic lattice structures, blood flow, structural boundaries, joints, structural damage, and impact forces identifications, as well as the SEM-FEM hybrid method. - Presents all aspects of SEM in one volume, both theory and applications
- Helps students and professionals master associated theories, modeling processes, and analysis methods
- Demonstrates where and how to apply SEM in practice
- Introduces real-world examples across a variety of structures
- Shows how models can be used to evaluate the accuracy of other solution methods
- Cross-checks against solutions obtained by conventional FEM and other solution methods
- Comes with downloadable code examples for independent practice
Spectral Element Method in Structural Dynamics can be used by graduate students of aeronautical, civil, naval architectures, mechanical, structural and biomechanical engineering. Researchers in universities, technical institutes, and industries will also find the book to be a helpful reference highlighting SEM applications to various engineering problems in areas of structural dynamics, wave propagations, and other related subjects. The book can also be used by students, professors, and researchers who want to learn more efficient and more accurate computational methods useful for their research topics from all areas of engineering, science and mathematics, including the areas of computational mechanics and numerical methods.
Table of Contents:
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
7.1.1 Equation of Motion 163
7.1.2 Spectral Element Modeling 165
7.1.3 Finite Element Model 170
7.2 Axially Moving Bernoulli–Euler Beam 172
7.2.1 Equation of Motion 172
7.2.2 Spectral Element Modeling 174
7.2.3 Finite Element Model 178
7.2.4 Stability Analysis 178
7.3 Axially Moving Timoshenko Beam 181
7.3.1 Equations of Motion 181
7.3.2 Spectral Element Modeling 183
7.3.3 Finite Element Model 188
7.3.4 Stability Analysis 189
7.4 Axially Moving Thin Plates 192
7.4.1 Equation of Motion 192
7.4.2 Spectral Element Modeling 195
7.4.3 Finite Element Model 204
Appendix 7.A: Finite Element Matrices for Axially Moving String 209
Appendix 7.B: Finite Element Matrices for Axially Moving Bernoulli–Euler Beam 210
Appendix 7.C: Finite Element Matrices for Axially Moving Timoshenko Beam 210
Appendix 7.D: Finite Element Matrices for Axially Moving Plate 212
8 Dynamics of Rotor Systems 219
8.1 Governing Equations 219
8.1.1 Equations of Motion of the Spinning Shaft 220
8.1.2 Equations of Motion of Disks with Mass Unbalance 223
8.2 Spectral Element Modeling 228
8.2.1 Spectral Element for the Spinning Shaft 228
8.2.2 Spectral Element for the Disk 237
8.2.3 Assembly of Spectral Elements 239
8.3 Finite Element Model 242
8.3.1 Finite Element for the Spinning Shaft 243
8.3.2 Finite Element for the Disk 246
8.3.3 Assembly of Finite Elements 247
8.4 Numerical Examples 249
Appendix 8.A: Finite Element Matrices for the Transverse Bending Vibration 253
9 Dynamics of Multi-Layered Structures 255
9.1 Elastic–Elastic Two-Layer Beams 255
9.1.1 Equations of Motion 255
9.1.2 Spectral Element Modeling 258
9.1.3 Spectral Modal Analysis 263
9.1.4 Finite Element Model 266
9.2 Elastic–Viscoelastic–elastic–Three-Layer (PCLD) Beams 269
9.2.1 Equations of Motion 269
9.2.2 Spectral Element Modeling 272
9.2.3 Spectral Modal Analysis 279
9.2.4 Finite Element Model 283
Appendix 9.A: Finite Element Matrices for the Elastic–Elastic Two-Layer Beam 288
Appendix 9.B: Finite Element Matrices for the Elastic–VEM–Elastic Three-Layer Beam 289
10 Dynamics of Smart Structures 293
10.1 Elastic–Piezoelectric Two-Layer Beams 293
10.1.1 Equations of Motion 293
10.1.2 Spectral Element Modeling 297
10.1.3 Spectral Element with Active Control 300
10.1.4 Spectral Modal Analysis 301
10.1.5 Finite Element Model 303
10.2 Elastic–Viscoelastic–Piezoelctric Three-Layer (ACLD) Beams 305
10.2.1 Equations of Motion 305
10.2.2 Spectral Element Modeling 308
10.2.3 Spectral Element with Active Control 312
10.2.4 Spectral Modal Analysis 313
10.2.5 Finite Element Model 315
11 Dynamics of Composite Laminated Structures 319
11.1 Theory of Composite Mechanics 319
11.1.1 Three-Dimensional Stress–Strain Relationships 319
11.1.2 Stress–Strain Relationships for an Orthotropic Lamina 320
11.1.3 Strain–Displacement Relationships 322
11.1.4 Resultant Forces and Moments 323
11.2 Equations of Motion for Composite Laminated Beams 324
11.2.1 Axial–Bending–Shear Coupled Vibration 325
11.2.2 Bending–Torsion–Shear Coupled Vibration 327
11.3 Dynamics of Axial–Bending–Shear Coupled Composite Beams 330
11.3.1 Equations of Motion 330
11.3.2 Spectral Element Modeling 330
11.3.3 Finite Element Model 336
11.4 Dynamics of Bending–Torsion–Shear Coupled Composite Beams 339
11.4.1 Equations of Motion 339
11.4.2 Spectral Element Modeling 339
11.4.3 Finite Element Model 346
Appendix 11.A: Finite Element Matrices for Axial–Bending–Shear Coupled Composite Beams 349
Appendix 11.B: Finite Element Matrices for Bending–Torsion–Shear Coupled Composite Beams 351
12 Dynamics of Periodic Lattice Structures 355
12.1 Continuum Modeling Method 355
12.1.1 Transfer Matrix for the Representative Lattice Cell (RLC) 356
12.1.2 Transfer Matrix for an ET-Beam Element 361
12.1.3 Determination of Equivalent Continuum Structural Properties 362
12.2 Spectral Transfer Matrix Method 365
12.2.1 Transfer Matrix for a Lattice Cell 366
12.2.2 Transfer Matrix for a 1-D Lattice Substructure 367
12.2.3 Spectral Element Model for a 1-D Lattice Substructure 368
12.2.4 Spectral Element Model for the Whole Lattice Structure 369
13 Biomechanics: Blood Flow Analysis 373
13.1 Governing Equations 373
13.1.1 One-Dimensional Blood Flow Theory 373
13.1.2 Simplified Governing Equations 375
13.2 Spectral Element Modeling: I. Finite Element 376
13.2.1 Governing Equations in the Frequency Domain 377
13.2.2 Weak Form of Governing Equations 378
13.2.3 Spectral Nodal DOFs 379
13.2.4 Dynamic Shape Functions 380
13.2.5 Spectral Element Equation 381
13.3 Spectral Element Modeling: II. Semi-Infinite Element 384
13.4 Assembly of Spectral Elements 385
13.5 Finite Element Model 386
13.6 Numerical Examples 388
Appendix 13.A: Finite Element Model for the 1-D Blood Flow 391
14 Identification of Structural Boundaries and Joints 393
14.1 Identification of Non-Ideal Boundary Conditions 393
14.1.1 One-End Supported Beam 394
14.1.2 Two-Ends Supported Beam 397
14.2 Identification of Joints 404
14.2.1 Spectral T-Beam Element Model for Uniform Beam Parts 404
14.2.2 Equivalent Spectral Element Model of the Joint Part 405
14.2.3 Determination of Joint Parameters 407
15 Identification of Structural Damage 413
15.1 Spectral Element Modeling of a Damaged Structure 413
15.1.1 Assembly of Spectral Elements 413
15.1.2 Imposition of Boundary Conditions 414
15.1.3 Reordering of Spectral Nodal DOFs 415
15.2 Theory of Damage Identification 416
15.2.1 Uniform Damage Representation 416
15.2.2 Damage Identification Algorithms 417
15.3 Domain-Reduction Method 425
15.3.1 Domain-Reduction Method 425
15.3.2 Three-Step Process 427
16 Other Applications 429
16.1 SEM–FEM Hybrid Method 429
16.2 Identification of Impact Forces 434
16.2.1 Force-History Identification 435
16.2.2 Force-Location Identification 436
16.3 Other Applications 439
References 441
Index 449
