Vibration of Continuous Systems

Broad, up-to-date coverage of advanced vibration analysis by the market-leading author Successful vibration analysis of continuous structural elements and systems requires a knowledge of material mechanics, structural mechanics, ordinary and partial differential equations, matrix methods, variational calculus, and integral equations. Fortunately, leading author Singiresu Rao has created Vibration of Continuous Systems, a new book that provides engineers, researchers, and students with everything they need to know about analytical methods of vibration analysis of continuous structural systems. Featuring coverage of strings, bars, shafts, beams, circular rings and curved beams, membranes, plates, and shells-as well as an introduction to the propagation of elastic waves in structures and solid bodies-Vibration of Continuous Systems presents: * Methodical and comprehensive coverage of the vibration of different types of structural elements * The exact analytical and approximate analytical methods of analysis * Fundamental concepts in a straightforward manner, complete with illustrative examples With chapters that are independent and self-contained, Vibration of Continuous Systems is the perfect book that works as a one-semester course, self-study tool, and convenient reference.

Table of Contents:
Preface xv Symbols xix 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 8 1.5 Discrete and Continuous Systems 11 1.6 Vibration Problems 15 1.7 Vibration Analysis 16 1.8 Excitations 17 1.9 Harmonic Functions 18 1.10 Periodic Functions and Fourier Series 24 1.11 Nonperiodic Functions and Fourier Integrals 26 1.12 Literature on Vibration of Continuous Systems 29 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 Multidegree-of-Freedom Systems 43 2.3 Recent Contributions 60 References 61 Problems 62 3 Derivation of Equations: Equilibrium Approach 68 3.1 Introduction 68 3.2 Newton’s Second Law of Motion 68 3.3 D’Alembert’s Principle 69 3.4 Equation of Motion of a Bar in Axial Vibration 69 3.5 Equation of Motion of a Beam in Transverse Vibration 71 3.6 Equation of Motion of a Plate in Transverse Vibration 73 3.7 Additional Contributions 80 References 80 Problems 81 4 Derivation of Equations: Variational Approach 85 4.1 Introduction 85 4.2 Calculus of a Single Variable 85 4.3 Calculus of Variations 86 4.4 Variation Operator 89 4.5 Functional with Higher-Order Derivatives 91 4.6 Functional with Several Dependent Variables 93 4.7 Functional with Several Independent Variables 95 4.8 Extremization of a Functional with Constraints 96 4.9 Boundary Conditions 100 4.10 Variational Methods in Solid Mechanics 104 4.11 Applications of Hamilton’s Principle 115 4.12 Recent Contributions 119 References 120 Problems 120 5 Derivation of Equations: Integral Equation Approach 123 5.1 Introduction 123 5.2 Classification of Integral Equations 123 5.3 Derivation of Integral Equations 125 5.4 General Formulation of the Eigenvalue Problem 130 5.5 Solution of Integral Equations 133 5.6 Recent Contributions 147 References 148 Problems 149 6 Solution Procedure: Eigenvalue and Modal Analysis Approach 151 6.1 Introduction 151 6.2 General Problem 151 6.3 Solution of Homogeneous Equations: Separation-of-Variables Technique 153 6.4 Sturm–Liouville Problem 154 6.5 General Eigenvalue Problem 163 6.6 Solution of Nonhomogeneous Equations 167 6.7 Forced Response of Viscously Damped Systems 169 6.8 Recent Contributions 171 References 172 Problems 173 7 Solution Procedure: Integral Transform Methods 174 7.1 Introduction 174 7.2 Fourier Transforms 175 7.3 Free Vibration of a Finite String 181 7.4 Forced Vibration of a Finite String 183 7.5 Free Vibration of a Beam 185 7.6 Laplace Transforms 188 7.7 Free Vibration of a String of Finite Length 194 7.8 Free Vibration of a Beam of Finite Length 197 7.9 Forced Vibration of a Beam of Finite Length 198 7.10 Recent Contributions 201 References 202 Problems 203 8 Transverse Vibration of Strings 205 8.1 Introduction 205 8.2 Equation of Motion 205 8.3 Initial and Boundary Conditions 209 8.4 Free Vibration of an Infinite String 210 8.5 Free Vibration of a String of Finite Length 217 8.6 Forced Vibration 227 8.7 Recent Contributions 231 References 232 Problems 233 9 Longitudinal Vibration of Bars 234 9.1 Introduction 234 9.2 Equation of Motion Using Simple Theory 234 9.3 Free Vibration Solution and Natural Frequencies 236 9.4 Forced Vibration 254 9.5 Response of a Bar Subjected to Longitudinal Support Motion 257 9.6 Rayleigh Theory 258 9.7 Bishop’s Theory 260 9.8 Recent Contributions 267 References 268 Problems 268 10 Torsional Vibration of Shafts 271 10.1 Introduction 271 10.2 Elementary Theory: Equation of Motion 271 10.3 Free Vibration of Uniform Shafts 276 10.4 Free Vibration Response due to Initial Conditions: Modal Analysis 289 10.5 Forced Vibration of a Uniform Shaft: Modal Analysis 292 10.6 Torsional Vibration of Noncircular Shafts: Saint-Venant’s Theory 295 10.7 Torsional Vibration of Noncircular Shafts, Including Axial Inertia 299 10.8 Torsional Vibration of Noncircular Shafts: Timoshenko–Gere Theory 300 10.9 Torsional Rigidity of Noncircular Shafts 303 10.10 Prandtl’s Membrane Analogy 308 10.11 Recent Contributions 313 References 314 Problems 315 11 Transverse Vibration of Beams 317 11.1 Introduction 317 11.2 Equation of Motion: Euler–Bernoulli Theory 317 11.3 Free Vibration Equations 322 11.4 Free Vibration Solution 325 11.5 Frequencies and Mode Shapes of Uniform Beams 326 11.6 Orthogonality of Normal Modes 339 11.7 Free Vibration Response due to Initial Conditions 341 11.8 Forced Vibration 344 11.9 Response of Beams under Moving Loads 350 11.10 Transverse Vibration of Beams Subjected to Axial Force 352 11.11 Vibration of a Rotating Beam 357 11.12 Natural Frequencies of Continuous Beams on Many Supports 359 11.13 Beam on an Elastic Foundation 364 11.14 Rayleigh’s Theory 369 11.15 Timoshenko’s Theory 371 11.16 Coupled Bending–Torsional Vibration of Beams 380 11.17 Transform Methods: Free Vibration of an Infinite Beam 385 11.18 Recent Contributions 387 References 389 Problems 390 12 Vibration of Circular Rings and Curved Beams 393 12.1 Introduction 393 12.2 Equations of Motion of a Circular Ring 393 12.3 In-Plane Flexural Vibrations of Rings 398 12.4 Flexural Vibrations at Right Angles to the Plane of a Ring 402 12.5 Torsional Vibrations 406 12.6 Extensional Vibrations 407 12.7 Vibration of a Curved Beam with Variable Curvature 408 12.8 Recent Contributions 416 References 418 Problems 419 13 Vibration of Membranes 420 13.1 Introduction 420 13.2 Equation of Motion 420 13.3 Wave Solution 425 13.4 Free Vibration of Rectangular Membranes 426 13.5 Forced Vibration of Rectangular Membranes 438 13.6 Free Vibration of Circular Membranes 444 13.7 Forced Vibration of Circular Membranes 448 13.8 Membranes with Irregular Shapes 452 13.9 Partial Circular Membranes 453 13.10 Recent Contributions 453 References 454 Problems 455 14 Transverse Vibration of Plates 457 14.1 Introduction 457 14.2 Equation of Motion: Classical Plate Theory 457 14.3 Boundary Conditions 465 14.4 Free Vibration of Rectangular Plates 471 14.5 Forced Vibration of Rectangular Plates 479 14.6 Circular Plates 485 14.7 Free Vibration of Circular Plates 490 14.8 Forced Vibration of Circular Plates 495 14.9 Effects of Rotary Inertia and Shear Deformation 499 14.10 Plate on an Elastic Foundation 521 14.11 Transverse Vibration of Plates Subjected to In-Plane Loads 523 14.12 Vibration of Plates with Variable Thickness 529 14.13 Recent Contributions 535 References 537 Problems 539 15 Vibration of Shells 541 15.1 Introduction and Shell Coordinates 541 15.2 Strain–Displacement Relations 552 15.3 Love’s Approximations 556 15.4 Stress–Strain Relations 562 15.5 Force and Moment Resultants 563 15.6 Strain Energy, Kinetic Energy, and Work Done by External Forces 571 15.7 Equations of Motion from Hamilton’s Principle 575 15.8 Circular Cylindrical Shells 582 15.9 Equations of Motion of Conical and Spherical Shells 591 15.10 Effect of Rotary Inertia and Shear Deformation 592 15.11 Recent Contributions 603 References 604 Problems 605 16 Elastic Wave Propagation 607 16.1 Introduction 607 16.2 One-Dimensional Wave Equation 607 16.3 Traveling-Wave Solution 608 16.4 Wave Motion in Strings 611 16.4.1 Free Vibration and Harmonic Waves 611 16.5 Reflection of Waves in One-Dimensional Problems 617 16.6 Reflection and Transmission of Waves at the Interface of Two Elastic Materials 619 16.7 Compressional and Shear Waves 623 16.8 Flexural Waves in Beams 628 16.9 Wave Propagation in an Infinite Elastic Medium 631 16.10 Rayleigh or Surface Waves 635 16.11 Recent Contributions 643 References 644 Problems 645 17 Approximate Analytical Methods 647 17.1 Introduction 647 17.2 Rayleigh’s Quotient 648 17.3 Rayleigh’s Method 650 17.4 Rayleigh–Ritz Method 661 17.5 Assumed Modes Method 670 17.6 Weighted Residual Methods 673 17.7 Galerkin’s Method 673 17.8 Collocation Method 680 17.9 Subdomain Method 684 17.10 Least Squares Method 686 17.11 Recent Contributions 693 References 695 Problems 696 A Basic Equations of Elasticity 700 A.1 Stress 700 A.2 Strain–Displacement Relations 700 A.3 Rotations 702 A.4 Stress–Strain Relations 703 A.5 Equations of Motion in Terms of Stresses 704 A.6 Equations of Motion in Terms of Displacements 705 B Laplace and Fourier Transforms 707 Index 713

About the Author :
Singiresu S. Rao, PhD, is Professor and Chairman of the Department of Mechanical Engineering at the University of Miami in Coral Gables, Florida. He has authored a number of textbooks, including the market-leading introductory-level text on vibrations, Mechanical Vibrations, Fourth Edition.