Fundamentals of Structural Dynamics, 2nd Edition

Preface to Structural Dynamics—An Introduction to Computer Methods xi

Preface to Fundamentals of Structural Dynamics xiii

About the Authors xv

1 The Science and Art of Structural Dynamics 1

1.1 Introduction to Structural Dynamics 1

1.2 Modeling of Structural Components and Systems 2

1.3 Prototype Spring–Mass Model 7

1.4 Vibration Testing of Structures 12

1.5 Scope of the Book 12

1.6 Computer Simulations; Supplementary Material on the Website 15

References 16

Problems 16

Part I Single-Degree-of-Freedom Systems 19

2 Mathematical Models of SDOF Systems 21

2.1 Brief Review of the Dynamics of Particles and Rigid Bodies 21

2.2 Elements of Lumped-Parameter Models 24

2.3 Application of Newton’s Laws to Lumped-Parameter Models 27

2.4 Application of the Principle of Virtual Displacements to Lumped-Parameter Models 34

2.5 Application of the Principle of Virtual Displacements to Continuous Models: Assumed-Modes Method 41

References 50

Problems 51

3 Free Vibration of SDOF Systems 56

3.1 Free Vibration of Undamped SDOF Systems 58

3.2 Free Vibration of Viscous-Damped SDOF Systems 61

3.3 Stability of Motion 66

3.4 Free Vibration of an SDOF System with Coulomb Damping 70

3.5 Experimental Determination of the Natural Frequency and Damping Factor of an SDOF System 72

References 77

Problems 78

4 Response of SDOF Systems to Harmonic Excitation 81

4.1 Response of Undamped SDOF Systems to Harmonic Excitation 82

4.2 Response of Viscous-Damped SDOF Systems to Harmonic Excitation: Frequency-Response Functions 87

4.3 Complex Frequency Response 93

4.4 Vibration Isolation: Force Transmissibility and Base Motion 96

4.5 Vibration Measuring Instruments: Accelerometers and Vibrometers 101

4.6 Use of Frequency-Response Data to Determine the Natural Frequency and Damping Factor of a Lightly Damped SDOF System 104

4.7 Equivalent Viscous Damping 107

4.8 Structural Damping 111

References 112

Problems 113

5 Response of SDOF Systems to Nonperiodic Excitation 117

5.1 Response of a Viscous-Damped SDOF System to an Ideal Step Input 117

5.2 Response of Undamped SDOF Systems to Rectangular Pulse and Ramp Loadings 119

5.3 Response of Undamped SDOF Systems to a Short-Duration Impulse: Unit Impulse Response 123

5.4 Response of SDOF Systems to General Dynamic Excitation: Convolution Integral Method 125

5.5 Response Spectra 128

5.6 System Response by the Laplace Transform Method: System Transfer Function 136

References 142

Problems 143

6 Numerical Evaluation of the Dynamic Response of SDOF Systems 147

6.1 Integration of Second-Order Ordinary Differential Equations 148

6.2 Integration of First-Order Ordinary Differential Equations 159

6.3 Nonlinear SDOF Systems 171

References 181

Problems 182

7 Response of SDOF Systems to Periodic Excitation: Frequency-Domain Analysis 184

7.1 Response to Periodic Excitation: Real Fourier Series 184

7.2 Response to Periodic Excitation: Complex Fourier Series 189

7.3 Response to Nonperiodic Excitation: Fourier Integral 195

7.4 Relationship Between Complex Frequency Response and Unit Impulse Response 199

7.5 Discrete Fourier Transform and Fast Fourier Transform 200

References 205

Problems 205

Part II Multiple-Degree-of-Freedom Systems—Basic Topics 209

8 Mathematical Models of MDOF Systems 211

8.1 Application of Newton’s Laws to Lumped-Parameter Models 212

8.2 Introduction to Analytical Dynamics: Hamilton’s Principle and Lagrange’s Equations 218

8.3 Application of Lagrange’s Equations to Lumped-Parameter Models 223

8.4 Application of Lagrange’s Equations to Continuous Models: Assumed-Modes Method 228

8.5 Constrained Coordinates and Lagrange Multipliers 238

References 240

Problems 241

9 Vibration of Undamped 2-DOF Systems 248

9.1 Free Vibration of 2-DOF Systems: Natural Frequencies and Mode Shapes 249

9.2 Beat Phenomenon 254

9.3 Additional Examples of Modes and Frequencies of 2-DOF Systems: Assumed-Modes Models 258

9.4 Free Vibration of Systems with Rigid-Body Modes 266

9.5 Introduction to Mode Superposition: Frequency Response of an Undamped 2-DOF System 268

9.6 Undamped Vibration Absorber 272

Reference 275

Problems 275

10 Vibration Properties of MDOF Systems: Modes, Frequencies, and Damping 281

10.1 Some Properties of Natural Frequencies and Natural Modes of Undamped MDOF Systems 282

10.2 Model Reduction: Rayleigh, Rayleigh–Ritz, and Assumed-Modes Methods 298

10.3 Uncoupled Damping in MDOF Systems 302

10.4 Structures with Arbitrary Viscous Damping: Complex Modes 307

10.5 Natural Frequencies and Mode Shapes of Damped Structures with Rigid-Body

Modes 316

References 322

Problems 322

11 Dynamic Response of MDOF Systems: Mode-Superposition Method 325

11.1 Mode-Superposition Method: Principal Coordinates 325

11.2 Mode-Superposition Solutions for MDOF Systems with Modal Damping: Frequency-Response Analysis 330

11.3 Mode-Displacement Solution for the Response of MDOF Systems 342

11.4 Mode-Acceleration Solution for the Response of Undamped MDOF Systems 349

11.5 Dynamic Stresses by Mode Superposition 351

11.6 Mode Superposition for Undamped Systems with Rigid-Body Modes 353

References 359

Problems 360

Part III Continuous Systems 365

12 Mathematical Models of Continuous Systems 367

12.1 Applications of Newton’s Laws: Axial Deformation and Torsion 367

12.2 Application of Newton’s Laws: Transverse Vibration of Linearly Elastic Beams (Bernoulli–Euler Beam Theory) 374

12.3 Application of Hamilton’s Principle: Torsion of a Rod with Circular Cross Section 379

12.4 Application of the Extended Hamilton’s Principle: Beam Flexure Including Shear Deformation and Rotatory Inertia (Timoshenko Beam Theory) 382

References 385

Problems 385

13 Free Vibration of Continuous Systems 388

13.1 Free Axial and Torsional Vibration 388

13.2 Free Transverse Vibration of Bernoulli–Euler Beams 392

13.3 Rayleigh’s Method for Approximating the Fundamental Frequency of a Continuous System 398

13.4 Free Transverse Vibration of Beams Including Shear Deformation and Rotatory Inertia 400

13.5 Some Properties of Natural Modes of Continuous Systems 401

13.6 Free Vibration of Thin Flat Plates 405

References 409

Problems 409

Part IV Computational Methods in Structural Dynamics 415

14 Introduction to Finite Element Modeling of Structures 417

14.1 Introduction to the Finite Element Method 418

14.2 Element Stiffness and Mass Matrices and Element Force Vector 419

14.3 Transformation of Element Matrices 430

14.4 Assembly of System Matrices: Direct Stiffness Method 438

14.5 Boundary Conditions 445

14.6 Constraints: Reduction of Degrees of Freedom 447

14.7 Systems with Rigid-Body Modes 451

14.8 Finite Element Solutions for Natural Frequencies and Mode Shapes 453

References 462

Problems 463

15 Numerical Evaluation of Modes and Frequencies of MDOF Systems 469

15.1 Introduction to Methods for Solving Algebraic Eigenproblems 469

15.2 Vector Iteration Methods 471

15.3 Subspace Iteration 480

15.4 QR Method for Symmetric Eigenproblems 483

15.5 Lanczos Eigensolver 489

15.6 Numerical Case Study 496

References 498

Problems 498

16 Direct Integration Methods for Dynamic Response of MDOF Systems 500

16.1 Damping in MDOF Systems 501

16.2 Numerical Integration: Mathematical Framework 504

16.3 Integration of Second-Order MDOF Systems 510

16.4 Single-Step Methods and Spectral Stability 516

16.5 Numerical Case Study 525

References 527

Problems 528

17 Component-Mode Synthesis 531

17.1 Introduction to Component-Mode Synthesis 532

17.2 Component Modes: Normal, Constraint, and Rigid-Body Modes 534

17.3 Component Modes: Attachment and Inertia-Relief Attachment Modes 539

17.4 Flexibility Matrices and Residual Flexibility 544

17.5 Substructure Coupling Procedures 549

17.6 Component-Mode Synthesis Methods: Fixed-Interface Methods 557

17.7 Component-Mode Synthesis Methods: Free-Interface Methods 559

17.8 Brief Introduction to Multilevel Substructuring 564

References 571

Problems 572

Part V Advanced Topics in Structural Dynamics 577

18 Introduction to Experimental Modal Analysis 579

18.1 Introduction 580

18.2 Frequency-Response Function Representations 584

18.3 Vibration Test Hardware 590

18.4 Fourier Transforms, Digital Signal Processing, and Estimation of FRFs 594

18.5 Modal Parameter Estimation 604

18.6 Mode Shape Estimation and Model Verification 612

References 615

Problems 616

19 Introduction to Active Structures 617

19.1 Introduction to Piezoelectric Materials 617

19.2 Constitutive Laws of Linear Piezoelectricity 620

19.3 Application of Newton’s Laws to Piezostructural Systems 624

19.4 Application of Extended Hamilton’s Principle to Piezoelectricity 627

19.5 Active Truss Models 630

19.6 Active Beam Models 637

19.7 Active Composite Laminates 641

References 646

Problems 647

20 Introduction to Earthquake Response of Structures 650

20.1 Introduction 650

20.2 Response of a SDOF System to Earthquake Excitation: Response Spectra 652

20.3 Response of MDOF Systems to Earthquake Excitation 660

20.4 Further Considerations 664

References 665

Problems 666

A Units 667

B Complex Numbers 671

C Elements of Laplace Transforms 674

D Fundamentals of Linear Algebra 682

E Introduction to the Use of Matlab 697

Index 715