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Preface xv Acknowledgments xix About the Author xxi 1 Introduction: Basic Concepts and Terminology 1 1.1 Concept of Vibration 1 1.2 Importance of Vibration 4 1.3 Origins and Developments in Mechanics and Vibration 5 1.4 History of Vibration of Continuous Systems 7 1.5 Discrete and Continuous Systems 12 1.6 Vibration Problems 15 1.7 Vibration Analysis 16 1.8 Excitations 17 1.9 Harmonic Functions 17 1.10 Periodic Functions and Fourier Series 24 1.11 Non periodic Functions and Fourier Integrals 25 1.12 Literature on Vibration of Continuous Systems 28 References 29 Problems 31 2 Vibration of Discrete Systems: Brief Review 33 2.1 Vibration of a Single-Degree-of-Freedom System 33 2.2 Vibration of Multi degree-of-Freedom Systems 43 2.3 Recent Contributions 60 References 61 Problems 62 3 Derivation of Equations: Equilibrium Approach 69 3.1 Introduction 69 3.2 Newton’s Second Law of Motion 69 3.3 D’Alembert’s Principle 70 3.4 Equation of Motion of a Bar in Axial Vibration 70 3.5 Equation of Motion of a Beam in Transverse Vibration 72 3.6 Equation of Motion of a Plate in Transverse Vibration 74 3.7 Additional Contributions 81 References 81 Problems 82 4 Derivation of Equations: Variational Approach 87 4.1 Introduction 87 4.2 Calculus of a Single Variable 87 4.3 Calculus of Variations 88 4.4 Variation Operator 91 4.5 Functional with Higher-Order Derivatives 93 4.6 Functional with Several Dependent Variables 95 4.7 Functional with Several Independent Variables 96 4.8 Extremization of a Functional with Constraints 98 4.9 Boundary Conditions 102 4.10 Variational Methods in Solid Mechanics 106 4.11 Applications of Hamilton’s Principle 116 4.12 Recent Contributions 121 Notes 121 References 122 Problems 122 5 Derivation of Equations: Integral Equation Approach 125 5.1 Introduction 125 5.2 Classification of Integral Equations 125 5.3 Derivation of Integral Equations 127 5.4 General Formulation of the Eigenvalue Problem 132 5.6 Recent Contributions 149 References 150 Problems 151 6 Solution Procedure: Eigenvalue and Modal Analysis Approach 153 6.1 Introduction 153 6.2 General Problem 153 6.3 Solution of Homogeneous Equations: Separation-of-Variables Technique 155 6.4 Sturm-Liouville Problem 156 6.5 General Eigenvalue Problem 165 6.6 Solution of Nonhomogeneous Equations 169 6.7 Forced Response of Viscously Damped Systems 171 6.8 Recent Contributions 173 References 174 Problems 175 7 Solution Procedure: Integral Transform Methods 177 7.1 Introduction 177 7.2 Fourier Transforms 178 7.3 Free Vibration of a Finite String 184 7.4 Forced Vibration of a Finite String 186 7.5 Free Vibration of a Beam 188 7.6 Laplace Transforms 191 7.7 Free Vibration of a String of Finite Length 197 7.8 Free Vibration of a Beam of Finite Length 200 7.9 Forced Vibration of a Beam of Finite Length 201 7.10 Recent Contributions 204 References 205 Problems 206 8 Transverse Vibration of Strings 209 8.1 Introduction 209 8.2 Equation of Motion 209 8.3 Initial and Boundary Conditions 213 8.4 Free Vibration of an Infinite String 215 8.5 Free Vibration of a String of Finite Length 221 8.6 Forced Vibration 231 8.7 Recent Contributions 235 Note 236 References 236 Problems 237 9 Longitudinal Vibration of Bars 239 9.1 Introduction 239 9.2 Equation of Motion Using Simple Theory 239 9.3 Free Vibration Solution and Natural Frequencies 241 9.4 Forced Vibration 259 9.5 Response of a Bar Subjected to Longitudinal Support Motion 262 9.6 Rayleigh Theory 263 9.7 Bishop’s Theory 265 9.8 Recent Contributions 272 References 273 Problems 273 10 Torsional Vibration of Shafts 277 10.1 Introduction 277 10.2 Elementary Theory: Equation of Motion 277 10.3 Free Vibration of Uniform Shafts 282 10.4 Free Vibration Response due to Initial Conditions: Modal Analysis 295 10.5 Forced Vibration of a Uniform Shaft: Modal Analysis 298 10.6 Torsional Vibration of Noncircular Shafts: Saint-Venant’s Theory 301 10.7 Torsional Vibration of Noncircular Shafts, Including Axial Inertia 305 10.8 Torsional Vibration of Noncircular Shafts: The Timoshenko-Gere Theory 306 10.9 Torsional Rigidity of Noncircular Shafts 309 10.10 Prandtl’s Membrane Analogy 314 10.11 Recent Contributions 319 References 320 Problems 321 11 Transverse Vibration of Beams 323 11.1 Introduction 323 11.2 Equation of Motion: The Euler-Bernoulli Theory 323 11.3 Free Vibration Equations 331 11.4 Free Vibration Solution 331 11.5 Frequencies and Mode Shapes of Uniform Beams 332 11.6 Orthogonality of Normal Modes 345 11.7 Free Vibration Response due to Initial Conditions 347 11.8 Forced Vibration 350 11.9 Response of Beams under Moving Loads 356 11.10 Transverse Vibration of Beams Subjected to Axial Force 358 11.11 Vibration of a Rotating Beam 363 11.12 Natural Frequencies of Continuous Beams on Many Supports 365 11.13 Beam on an Elastic Foundation 370 11.14 Rayleigh’s Theory 375 11.15 Timoshenko’s Theory 377 11.16 Coupled Bending-Torsional Vibration of Beams 386 11.17 Transform Methods: Free Vibration of an Infinite Beam 391 11.18 Recent Contributions 393 References 395 Problems 396 12 Vibration of Circular Rings and Curved Beams 399 12.1 Introduction 399 12.2 Equations of Motion of a Circular Ring 399 12.3 In-Plane Flexural Vibrations of Rings 404 12.4 Flexural Vibrations at Right Angles to the Plane of a Ring 408 12.5 Torsional Vibrations 413 12.6 Extensional Vibrations 413 12.7 Vibration of a Curved Beam with Variable Curvature 414 12.8 Recent Contributions 423 References 424 Problems 425 13 Vibration of Membranes 427 13.1 Introduction 427 13.2 Equation of Motion 427 13.3 Wave Solution 432 13.4 Free Vibration of Rectangular Membranes 433 13.5 Forced Vibration of Rectangular Membranes 444 13.6 Free Vibration of Circular Membranes 450 13.7 Forced Vibration of Circular Membranes 454 13.8 Membranes with Irregular Shapes 459 13.9 Partial Circular Membranes 459 13.10 Recent Contributions 460 Notes 461 References 462 Problems 463 14 Transverse Vibration of Plates 465 14.1 Introduction 465 14.2 Equation of Motion: Classical Plate Theory 465 14.3 Boundary Conditions 473 14.4 Free Vibration of Rectangular Plates 479 14.5 Forced Vibration of Rectangular Plates 489 14.6 Circular Plates 493 14.7 Free Vibration of Circular Plates 498 14.8 Forced Vibration of Circular Plates 503 14.9 Effects of Rotary Inertia and Shear Deformation 507 14.10 Plate on an Elastic Foundation 529 14.11 Transverse Vibration of Plates Subjected to In-Plane Loads 531 14.12 Vibration of Plates with Variable Thickness 537 14.13 Recent Contributions 543 References 545 Problems 547 15 Vibration of Shells 549 15.1 Introduction and Shell Coordinates 549 15.2 Strain-Displacement Relations 560 15.3 Love’s Approximations 564 15.4 Stress-Strain Relations 570 15.5 Force and Moment Resultants 571 15.6 Strain Energy, Kinetic Energy, and Work Done by External Forces 579 15.7 Equations of Motion from Hamilton’s Principle 582 15.8 Circular Cylindrical Shells 590 15.9 Equations of Motion of Conical and Spherical Shells 599 15.10 Effect of Rotary Inertia and Shear Deformation 600 15.11 Recent Contributions 611 Notes 612 References 612 Problems 614 16 Vibration of Composite Structures 617 16.1 Introduction 617 16.2 Characterization of a Unidirectional Lamina with Loading Parallel to the Fibers 617 16.3 Different Types of Material Behavior 619 16.4 Constitutive Equations or Stress-Strain Relations 620 16.5 Coordinate Transformations for Stresses and Strains 626 16.6 Lamina with Fibers Oriented at an Angle 632 16.7 Composite Lamina in Plane Stress 634 16.8 Laminated Composite Structures 641 16.9 Vibration Analysis of Laminated Composite Plates 659 16.10 Vibration Analysis of Laminated Composte Beams 663 16.11 Recent Contributions 666 References 667 Problems 668 17 Approximate Analytical Methods 671 17.1 Introduction 671 17.2 Rayleigh’s Quotient 672 17.3 Rayleigh’s Method 674 17.4 Rayleigh-Ritz Method 685 17.5 Assumed Modes Method 695 17.6 Weighted Residual Methods 697 17.7 Galerkin’s Method 698 17.8 Collocation Method 704 17.9 Subdomain Method 709 17.10 Least Squares Method 711 17.11 Recent Contributions 718 References 719 Problems 721 18 Numerical Methods: Finite Element Method 725 18.1 Introduction 725 18.2 Finite Element Procedure 725 18.3 Element Matrices of Different Structural Problems 739 18.4 Dynamic Response Using the Finite Element Method 753 18.5 Additional and Recent Contributions 760 Note 763 References 763 Problems 765 A Basic Equations of Elasticity 769 A.1 Stress 769 A.2 Strain-Displacement Relations 769 A.3 Rotations 771 A.4 Stress-Strain Relations 772 A.5 Equations of Motion in Terms of Stresses 774 A.6 Equations of Motion in Terms of Displacements 774 B Laplace and Fourier Transforms 777 Index 783

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