Fundamentals of Structural Dynamics

FUNDAMENTALS OF STRUCTURAL DYNAMICS From theory and fundamentals to the latest advances in computational and experimental modal analysis, this is the definitive, updated reference on structural dynamics. This edition updates Professor Craig’s classic introduction to structural dynamics, which has been an invaluable resource for practicing engineers and a textbook for undergraduate and graduate courses in vibrations and/or structural dynamics. Along with comprehensive coverage of structural dynamics fundamentals, finite-element–based computational methods, and dynamic testing methods, this Second Edition includes new and expanded coverage of computational methods, as well as introductions to more advanced topics, including experimental modal analysis and “active structures.” With a systematic approach, it presents solution techniques that apply to various engineering disciplines. It discusses single degree-of-freedom (SDOF) systems, multiple degrees-of-freedom (MDOF) systems, and continuous systems in depth; and includes numeric evaluation of modes and frequency of MDOF systems; direct integration methods for dynamic response of SDOF systems and MDOF systems; and component mode synthesis. Numerous illustrative examples help engineers apply the techniques and methods to challenges they face in the real world. MATLAB® is extensively used throughout the book, and many of the .m-files are made available on the book’s Web site. Fundamentals of Structural Dynamics, Second Edition is an indispensable reference and “refresher course” for engineering professionals; and a textbook for seniors or graduate students in mechanical engineering, civil engineering, engineering mechanics, or aerospace engineering.

Table of Contents:
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