Vibration of Continuous SystemsISBN: 9780471771715
744 pages
February 2007

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 shellsas well as an introduction to the propagation of elastic waves in structures and solid bodiesVibration 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 selfcontained, Vibration of Continuous Systems is the perfect book that works as a onesemester course, selfstudy 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 SingleDegreeofFreedom System 33
2.2 Vibration of MultidegreeofFreedom 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 HigherOrder 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: SeparationofVariables 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: SaintVenant’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 InPlane 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 InPlane 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 OneDimensional Wave Equation 607
16.3 TravelingWave Solution 608
16.4 Wave Motion in Strings 611
16.4.1 Free Vibration and Harmonic Waves 611
16.5 Reflection of Waves in OneDimensional 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
Author Information
The Wiley Advantage
Covers a broad range of structural elements including strings, bars, shafts, beams, circular rings, membranes, plates and shells in a selfcontained format to allow for ease of instruction
Presents the exact analytical, approximate analytical, and numerical methods of analysis for all elements covered
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