Vibration of Plates

by Chakraverty; S.

• ISBN13: 9781420053951
• ISBN10: 1420053957
• Format: Hardcover
• Copyright: 2008-12-16
• Publisher: CRC Press

Summary

Structural designs in aeronautical, marine, mechanical, and civil engineering often incorporate regular, irregular, and complex-shaped members to reduce costly material, lighten loads, provide ventilation, and alter structures' resonant frequencies. Vibration of Plates gives engineering and architecture professionals optimal methods of analyzing the vibration that can result from this practice to understand how it affects the behavior of complex structures.

Table of Contents

Preface	p. xi
Acknowledgments	p. xv
Background of Vibration	p. 1
Vibration Basics	p. 1
Causes of Vibration	p. 2
Requirements for Vibration	p. 2
Discrete and Continuous Systems	p. 3
Glossary of Some Terms	p. 3
Basic Vibration Model	p. 4
One Degree of Freedom Systems	p. 5
Simple Pendulum	p. 6
Metal Thin Strip with a Mass at One End	p. 7
Torsion of a Rod Having a Pulley at One End	p. 8
Electric Circuit Having Current, Capacitance, Inductance, and Voltage	p. 9
Two Degree of Freedom Systems	p. 10
Equation of Motion for Two Degree of Freedom System	p. 11
Example of Two Degree of Freedom System (with Damping and Force)	p. 11
Coordinate Coupling	p. 13
Multi-Degree of Freedom Systems	p. 13
Equation of Motion for Multi-Degree of Freedom System	p. 14
Equation of Motion for Multi-Degree of Freedom System with Damping and Force	p. 16
Continuous Systems	p. 17
Transverse Vibration of a String	p. 17
Longitudinal Vibration of a Rod	p. 19
Transverse Vibration of an Elastic Beam	p. 20
Vibration of Membrane	p. 21
Initial and Boundary Conditions	p. 24
Equation of Motion through Application of Energy Method	p. 26
Massless Spring Carrying a Mass m	p. 27
Simple Pendulum	p. 28
Spring of Mass m[subscript s] Carrying a Mass m	p. 29
Multi-Degree of Freedom System	p. 31
Vibration of String	p. 31
Vibration of Membrane	p. 31
Further Reading	p. 32
Methods of Analysis for Vibration Problems	p. 33
Single Degree of Freedom System	p. 33
Free Vibration without Damping	p. 34
Free Vibration with Damping without Force	p. 35
Forced Vibration in Single Degree of Freedom System	p. 39
Harmonic Distributing Force (with Damping)	p. 39
Undamped System with Sinusoidal Force	p. 41
Two Degree of Freedom System	p. 42
Multi-Degree of Freedom System	p. 44
Reduction to an Eigenvalue Problem for General System (Conservative)	p. 44
Orthogonality of the Eigenvectors	p. 46
Modal Matrix	p. 47
Relationship between [P], [S], and [lambda]	p. 48
Solution of the Dynamical Problem (Free Vibration)	p. 49
Classical Solution for Forced Vibration without Damping	p. 51
Modal Damping in Forced Vibration	p. 52
Normal Mode Summation	p. 53
Response Computation	p. 54
Continuous Systems	p. 56
Vibration of a Taut String	p. 56
Transverse Vibration of an Elastic Beam	p. 61
Vibration of Membrane	p. 64
Rectangular Membrane	p. 64
Circular Membrane	p. 66
Approximate Methods for Vibration Problems	p. 68
Rayleigh's Method	p. 68
Rayleigh-Ritz Method	p. 70
Further Reading	p. 73
Vibration Basics for Plates	p. 75
Stress-Strain Relations	p. 76
Engineering Constants	p. 80
Plane Stress	p. 81
Plate Theory	p. 83
Strain-Displacement Relations	p. 84
Compatibility Equations	p. 84
Kinematics of Deformation of Plates	p. 85
Biharmonic Equation	p. 87
Minimum Total Potential Energy Approach for Biharmonic Equation	p. 91
Equation of Motion for Vibration of Plates by Hamilton's Principle	p. 96
Differential Equation for Transverse Motion of Plates by Elastic Equilibrium	p. 98
Boundary Conditions	p. 99
Various Forms of Equation of Motion of a Plate in Cartesian Coordinates	p. 100
Formulations in Polar and Elliptical Coordinates	p. 101
Polar Coordinates	p. 101
Elliptical Coordinates	p. 103
Further Reading	p. 104
Exact, Series-Type, and Approximate Methods for Transverse Vibration of Plates	p. 105
Method of Solution in Polar Coordinates	p. 106
Circular Plate	p. 109
Circular Plate with Clamped Condition All Around	p. 110
Circular Plate with Simply Supported Condition All Around	p. 112
Circular Plate with Completely Free Condition All Around	p. 114
Annular Plates	p. 115
Circular Annular Plate with Outer and Inner Both Clamped (C-C)	p. 116
Circular Annular Plate with Outer Clamped and Inner Simply Supported (C-S)	p. 117
Circular Annular Plate with Outer Clamped and Inner Free (C-F)	p. 118
Circular Annular Plate with Outer Simply Supported and Inner Clamped (S-C)	p. 118
Circular Annular Plate with Outer and Inner Both Simply Supported (S-S)	p. 118
Circular Annular Plate with Outer Simply Supported and Inner Free (S-F)	p. 119
Circular Annular Plate with Outer Free and Inner Clamped (F-C)	p. 119
Circular Annular Plate with Outer Free and Inner Simply Supported (F-S)	p. 120
Circular Annular Plate with Outer and Inner Both Free (F-F)	p. 120
Method of Solution in Elliptical Coordinate System	p. 122
Method of Solution in Rectangular Coordinate System	p. 126
Approximate Solution Methods	p. 133
Rayleigh's Method for Plates	p. 134
Rayleigh-Ritz Method for Plates	p. 135
Bibliography	p. 141
Development of Characteristic Orthogonal Polynomials (COPs) in Vibration Problems	p. 143
Preliminary Definitions	p. 144
Linear Dependence and Linear Independence of Vectors	p. 145
Construction of Orthogonal Polynomials	p. 148
Three-Term Recurrence Relation	p. 148
Gram-Schmidt Orthogonalization Procedure	p. 149
Standard Orthogonal Polynomials	p. 151
Characteristic Orthogonal Polynomials	p. 151
Characteristic Orthogonal Polynomials in One Dimension	p. 152
Beam in [0,1] with Both Ends Clamped	p. 154
Condensation with COPs in the Vibration Problems	p. 155
Free Flexural Vibration of Rectangular Plate Using One-Dimensional Characteristic Orthogonal Polynomials	p. 158
Characteristic Orthogonal Polynomials in Two Dimensions	p. 160
Free Flexural Vibration of Triangular Plate Using Two-Dimensional Characteristic Orthogonal Polynomials	p. 163
Bibliography	p. 164
Boundary Characteristic Orthogonal Polynomials (BCOPs) in Vibration of Plates	p. 167
Boundary Characteristic Orthogonal Polynomials in n Dimensions	p. 168
Boundary Characteristic Orthogonal Polynomials in Two Dimensions	p. 169
Elliptic and Circular Domains	p. 172
Triangular Domains	p. 174
Parallelogram Domains	p. 176
Recurrence Scheme for the BCOPs	p. 178
Recurrence Relations for Multidimensional Orthogonal Polynomials	p. 179
Kowalski's Relations in Two Dimensions	p. 180
Matrix Form of Kowalski's Relations	p. 181
BCOPs in Terms of the Original Functions	p. 183
Generalization of the Recurrence Scheme for Two-Dimensional BCOPs	p. 185
Numerical Procedure for Generalization of the Recurrence Scheme for Two-Dimensional BCOPs	p. 186
Generation of BCOPs as Per Grades of the Monomials	p. 188
References	p. 191
Transverse Vibration of Elliptic and Circular Plates	p. 193
Introduction	p. 193
Generation of BCOPs for Elliptic and Circular Plates with Constant Thickness	p. 194
Rayleigh-Ritz Method for Elliptic and Circular Constant Thickness Plates	p. 196
Some Numerical Results of Elliptic and Circular Plates	p. 202
Clamped Boundary	p. 202
Simply Supported Boundary	p. 208
Completely Free Boundary	p. 212
Conclusion	p. 220
References	p. 221
Triangular Plates	p. 223
Introduction	p. 223
Mapping of General Triangle onto a Standard Triangle	p. 224
Generation of the BCOPs over the Triangular Domain	p. 225
Rayleigh-Ritz Method in Triangular Plates	p. 226
Some Numerical Results and Discussions for Triangular Plates	p. 227
Bibliography	p. 237
Rectangular and Skew Plates	p. 239
Introduction	p. 239
Mapping of General Skew Domain into a Standard Unit Square	p. 240
Generation of the BCOPs in the Standard Square Domain	p. 241
Rayleigh-Ritz Method for Skew Plates	p. 242
Some Numerical Results and Discussions for Rectangular and Skew Plates	p. 244
References	p. 259
Circular Annular and Elliptic Annular Plates	p. 261
Introduction	p. 261
Domain Definition	p. 262
Governing Equations and Method of Solution	p. 263
Generation of the BCOPs in Annular Domains	p. 265
Some Numerical Results and Discussions for Annular Plates	p. 266
Convergence Study	p. 266
Comparison between Exact and BCOPs Results	p. 266
Annular Circular Plate	p. 267
Annular Elliptic Plate	p. 273
References	p. 283
Plates with Nonhomogeneous Material Properties	p. 285
Introduction	p. 285
Basic Equations and Method of Solution	p. 286
Type 1 Nonhomogeneity	p. 286
Type 2 Nonhomogeneity	p. 287
Orthogonal Polynomials Generation	p. 289
Some Numerical Results and Discussions	p. 290
Results for Type 1 Nonhomogeneity	p. 290
Special Case (when [alpha] = [beta] = 0)	p. 309
Results for Type 2 Nonhomogeneity	p. 309
Bibliography	p. 317
Plates with Variable Thickness	p. 319
Introduction	p. 319
Generation of BCOPs for Variable Thickness Plates	p. 320
Rayleigh-Ritz Method in the Variable Thickness Plates	p. 324
Numerical Results for Variable Thickness Plates	p. 332
Variable Thickness (Case 1)	p. 332
Variable Thickness (Case 2)	p. 344
Bibliography	p. 353
Plates with Orthotropic Material Properties	p. 355
Introduction	p. 355
Domain Definitions	p. 357
Elliptic Orthotropic Plates	p. 357
Annular Elliptic Orthotropic Plates	p. 357
Basic Equations and Method of Solutions	p. 357
Generation of BCOPs	p. 360
Orthogonal Polynomials for Elliptic Orthotropic Plates	p. 360
Orthogonal Polynomials for Annular Elliptic Orthotropic Plates	p. 361
Numerical Results and Discussions	p. 361
Results for Elliptic and Circular Plates with Rectangular Orthotropy	p. 362
Clamped Boundary	p. 362
Simply Supported Boundary	p. 366
Free Boundary	p. 366
Results for Annular Elliptic Plates with Rectangular Orthotropy	p. 374
Bibliography	p. 381
Plates with Hybrid Complicating Effects	p. 385
Introduction	p. 385
Basic Equations for the Hybrid Complicating Effects	p. 386
Generation of BCOPs	p. 388
Some Numerical Results and Discussions	p. 389
Conclusion	p. 402
Bibliography	p. 402
Index	p. 405
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