# Vibration of Continuous Systems

ISBN: 978-0-470-11786-6 February 2007 744 Pages

## Description

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.

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

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

Most up to date coverage by the market leading author in the area: S. S. Rao

Covers a broad range of structural elements including strings, bars, shafts, beams, circular rings, membranes, plates and shells in a self-contained format to allow for ease of instruction

Presents the exact analytical, approximate analytical, and numerical methods of analysis for all elements covered