Asymptotic Methods in the Theory of Plates with Mixed Boundary Conditions

Preface ix

List of Abbreviations xiii

1 Asymptotic Approaches 1

1.1 Asymptotic Series and Approximations 1

1.1.1 Asymptotic Series 1

1.1.2 Asymptotic Symbols and Nomenclatures 5

1.2 Some Nonstandard Perturbation Procedures 8

1.2.1 Choice of Small Parameters 8

1.2.2 Homotopy Perturbation Method 10

1.2.3 Method of Small Delta 13

1.2.4 Method of Large Delta 17

1.2.5 Application of Distributions 19

1.3 Summation of Asymptotic Series 21

1.3.1 Analysis of Power Series 21

1.3.2 Padé Approximants and Continued Fractions 24

1.4 Some Applications of PA 29

1.4.1 Accelerating Convergence of Iterative Processes 29

1.4.2 Removing Singularities and Reducing the Gibbs-Wilbraham Effect 31

1.4.3 Localized Solutions 32

1.4.4 Hermite-Padé Approximations and Bifurcation Problem 34

1.4.5 Estimates of Effective Characteristics of Composite Materials 34

1.4.6 Continualization 35

1.4.7 Rational Interpolation 36

1.4.8 Some Other Applications 37

1.5 Matching of Limiting Asymptotic Expansions 38

1.5.1 Method of Asymptotically Equivalent Functions for Inversion of Laplace Transform 38

1.5.2 Two-Point PA 41

1.5.3 Other Methods of AEFs Construction 43

1.5.4 Example: Schrödinger Equation 45

1.5.5 Example: AEFs in the Theory of Composites 46

1.6 Dynamical Edge Effect Method 49

1.6.1 Linear Vibrations of a Rod 49

1.6.2 Nonlinear Vibrations of a Rod 51

1.6.3 Nonlinear Vibrations of a Rectangular Plate 54

1.6.4 Matching of Asymptotic and Variational Approaches 58

1.6.5 On the Normal Forms of Nonlinear Vibrations of Continuous Systems 60

1.7 Continualization 61

1.7.1 Discrete and Continuum Models in Mechanics 61

1.7.2 Chain of Elastically Coupled Masses 62

1.7.3 Classical Continuum Approximation 64

1.7.4 "Splashes" 65

1.7.5 Envelope Continualization 66

1.7.6 Improvement Continuum Approximations 68

1.7.7 Forced Oscillations 69

1.8 Averaging and Homogenization 71

1.8.1 Averaging via Multiscale Method 71

1.8.2 Frozing in Viscoelastic Problems 74

1.8.3 The WKB Method 75

1.8.4 Method of Kuzmak-Whitham (Nonlinear WKB Method) 77

1.8.5 Differential Equations with Quickly Changing Coefficients 79

1.8.6 Differential Equation with Periodically Discontinuous Coefficients 84

1.8.7 Periodically Perforated Domain 88

1.8.8 Waves in Periodically Nonhomogenous Media 92

References 95

2 Computational Methods for Plates and Beams with Mixed Boundary Conditions 105

2.1 Introduction 105

2.1.1 Computational Methods of Plates with Mixed Boundary Conditions 105

2.1.2 Method of Boundary Conditions Perturbation 107

2.2 Natural Vibrations of Beams and Plates 109

2.2.1 Natural Vibrations of a Clamped Beam 109

2.2.2 Natural Vibration of a Beam with Free Ends 114

2.2.3 Natural Vibrations of a Clamped Rectangular Plate 118

2.2.4 Natural Vibrations of the Orthotropic Plate with Free Edges Lying on an Elastic Foundation 123

2.2.5 Natural Vibrations of the Plate with Mixed Boundary Conditions "Clamping-Simple Support" 128

2.2.6 Comparison of Theoretical and Experimental Results 133

2.2.7 Natural Vibrations of a Partially Clamped Plate 135

2.2.8 Natural Vibrations of a Plate with Mixed Boundary Conditions "Simple Support-Moving Clamping" 140

2.3 Nonlinear Vibrations of Rods, Beams and Plates 144

2.3.1 Vibrations of the Rod Embedded in a Nonlinear Elastic Medium 144

2.3.2 Vibrations of the Beam Lying on a Nonlinear Elastic Foundation 153

2.3.3 Vibrations of the Membrane on a Nonlinear Elastic Foundation 155

2.3.4 Vibrations of the Plate on a Nonlinear Elastic Foundation 158

2.4 SSS of Beams and Plates 160

2.4.1 SSS of Beams with Clamped Ends 160

2.4.2 SSS of the Beam with Free Edges 163

2.4.3 SSS of Clamped Plate 166

2.4.4 SSS of a Plate with Free Edges 170

2.4.5 SSS of the Plate with Mixed Boundary Conditions "Clamping-Simple Support" 172

2.4.6 SSS of a Plate with Mixed Boundary Conditions "Free Edge-Moving Clamping" 180

2.5 Forced Vibrations of Beams and Plates 184

2.5.1 Forced Vibrations of a Clamped Beam 184

2.5.2 Forced Vibrations of Beam with Free Edges 189

2.5.3 Forced Vibrations of a Clamped Plate 190

2.5.4 Forced Vibrations of Plates with Free Edges 194

2.5.5 Forced Vibrations of Plate with Mixed Boundary Conditions "Clamping-Simple Support" 197

2.5.6 Forced Vibrations of Plate with Mixed Boundary Conditions "Free Edge - Moving Clamping" 202

2.6 Stability of Beams and Plates 207

2.6.1 Stability of a Clamped Beam 207

2.6.2 Stability of a Clamped Rectangular Plate 209

2.6.3 Stability of Rectangular Plate with Mixed Boundary Conditions "Clamping-Simple Support" 211

2.6.4 Comparison of Theoretical and Experimental Results 219

2.7 Some Related Problems 221

2.7.1 Dynamics of Nonhomogeneous Structures 221

2.7.2 Method of Ishlinskii-Leibenzon 224

2.7.3 Vibrations of a String Attached to a Spring-Mass-Dashpot System 230

2.7.4 Vibrations of a String with Nonlinear BCs 233

2.7.5 Boundary Conditions and First Order Approximation Theory 238

2.8 Links between the Adomian and Homotopy Perturbation Approaches 240

2.9 Conclusions 263

References 264

Index 269