Deterministic Chaos In One Dimensional Continuous Systems

by Jan Awrejcewicz, Vadim A Krysko, Irina V Papkova, Anton V Krysko

Overview

This book focuses on the computational analysis of nonlinear vibrations of structural members (beams, plates, panels, shells), where the studied dynamical problems can be reduced to the consideration of one spatial variable and time. The reduction is carried out based on a formal mathematical approach aimed at reducing the problems with infinite dimension to finite ones. The process also includes a transition from governing nonlinear partial differential equations to a set of finite number of ordinary differential equations.Beginning with an overview of the recent results devoted to the analysis and control of nonlinear dynamics of structural members, placing emphasis on stability, buckling, bifurcation and deterministic chaos, simple chaotic systems are briefly discussed. Next, bifurcation and chaotic dynamics of the Euler-Bernoulli and Timoshenko beams including the geometric and physical nonlinearity as well as the elastic-plastic deformations are illustrated. Despite the employed classical numerical analysis of nonlinear phenomena, the various wavelet transforms and the four Lyapunov exponents are used to detect, monitor and possibly control chaos, hyper-chaos, hyper-hyper-chaos and deep chaos exhibited by rectangular plate-strips and cylindrical panels.The book is intended for post-graduate and doctoral students, applied mathematicians, physicists, teachers and lecturers of universities and companies dealing with a nonlinear dynamical system, as well as theoretically inclined engineers of mechanical and civil engineering.

Product Details

ISBN-13:	9789814719698
Publisher:	World Scientific Publishing Company, Incorporated
Publication date:	05/20/2016
Series:	World Scientific Series On Nonlinear Science Series A , #90
Pages:	576
Product dimensions:	6.20(w) x 9.10(h) x 1.10(d)

Table of Contents

Prefac v

1 Bifurcational and Chaotic Dynamics of Simple Structural Members 1

1.1 Beams 2

1.2 Plates 5

1.3 Panels 9

1.4 Shells 12

2 Introduction to Fractal Dynamics 14

2.1 Cantor Set and Cantor Dust 14

2.2 Koch Snowflake 17

2.3 ID Maps 20

2.4 Sharkovsky's Theorem 23

2.5 Julia Set 25

2.6 Mandelbrot's Set 27

3 Introduction to Chaos and Wavelets 31

3.1 Routes to Chaos 31

3.2 Quantifying Chaotic Dynamics 46

4 Simple Chaotic Models 86

4.1 Introduction 86

4.2 Autonomous Systems 87

4.3 Non-Autonomous Systems 102

5 Discrete and Continuous Dissipative Systems 105

5.1 Introduction 105

5.2 Linear Friction 106

5.3 Nonlinear Friction 109

5.4 Hysteretic Friction 116

5.5 Impact. Damping 117

5.6 Damping in Continuous 1D Systems 118

6 Euler Bernoulli Beams 129

6.1 Introduction 130

6.2 Planar Beams 140

6.3 Linear Planar Beams and Stationary Temperature Fields 169

6.4 Curvilinear Planar Beams and Stationary Temperature Fields 207

6.5 Flexible Curvilinear Beam in Stationary Temperature and Electrical Fields 233

6.6 Beams with Elasto-Plastic Deformations 238

6.7 Multi-Layer Beams 269

7 Tinioshenko and Sheremetev-Pelekh Beams 307

7.1 The Timoshenko Beams 307

7.2 The Shernmetev-Pelekh Beams 324

7.3 Concluding Remarks 339

8 Panels 340

8.1 Infinite Length Panels 341

8.2 Cylindrical Panels of Infinite Length 438

9 Plates and Shells 459

9.1 Plates with Initial Imperfections 460

9.2 Flexible Axially-Symmetric Shells 491

Bibliography 530

Index 551