Vibration of Continuous Systems

A revised and up-to-date guide to advanced vibration analysis written by a noted expert

The revised and updated second edition of Vibration of Continuous Systems offers a guide to all aspects of vibration of continuous systems including: derivation of equations of motion, exact and approximate solutions and computational aspects. The author—a noted expert in the field—reviews all possible types of continuous structural members and systems including strings, shafts, beams, membranes, plates, shells, three-dimensional bodies, and composite structural members.

Designed to be a useful aid in the understanding of the vibration of continuous systems, the book contains exact analytical solutions, approximate analytical solutions, and numerical solutions. All the methods are presented in clear and simple terms and the second edition offers a more detailed explanation of the fundamentals and basic concepts. Vibration of Continuous Systems revised second edition:

• Contains new chapters on Vibration of three-dimensional solid bodies; Vibration of composite structures; and Numerical solution using the finite element method
• Reviews the fundamental concepts in clear and concise language
• Includes newly formatted content that is streamlined for effectiveness
• Offers many new illustrative examples and problems
• Presents answers to selected problems

Written for professors, students of mechanics of vibration courses, and researchers, the revised second edition of Vibration of Continuous Systems offers an authoritative guide filled with illustrative examples of the theory, computational details, and applications of vibration of continuous systems.

Singiresu S. Rao is a Professor in the Mechanical and Aerospace Engineering Department at the University of Miami. His main areas of research include structural dynamics, multi objective optimization and development of uncertainty models in engineering modeling, analysis, design and optimization. He is a Fellow of ASME and an Associate Fellow of the AIAA.

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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