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More About This Title Vibration of Continuous Systems, Second Edition
English
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.
English
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.
English
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
