Tractable Models of Solid Mechanics

Formulation, Analysis and Interpretation
1. Auflage 2011
ISBN: 978-3-642-15372-3
Verlag: Springer
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 302 Seiten

This book describes significant tractable models used in solid mechanics - classical models used in modern mechanics as well as new ones. The models are selected to illustrate the main ideas which allow scientists to describe complicated effects in a simple manner and to clarify basic notations of solid mechanics. A model is considered to be tractable if it is based on clear physical assumptions which allow the selection of significant effects and relatively simple mathematical formulations. The first part of the book briefly reviews classical tractable models for a simple description of complex effects developed from the 18th to the 20th century and widely used in modern mechanics. The second part describes systematically the new tractable models used today for the treatment of increasingly complex mechanical objects – from systems with two degrees of freedom to three-dimensional continuous objects.

Autoren/Hrsg.

Inhaltsverzeichnis

1;Preface;4
2;Abbreviations;7
3;Contents;8
4;1 Introduction: Historical Development of Tractable Models;11
4.1;References;21
5;2 Discrete Finite Systems;22
5.1;2.1 Linear Oscillators;22
5.1.1;2.1.1 Linear Conservative Oscillator;23
5.1.2;2.1.2 Linear Oscillator with Viscous Damping;24
5.1.2.1;2.1.2.1 Strong Energy Dissipation (e >> 1);25
5.1.2.2;2.1.2.2 Weak Energy Dissipation (e<<1);28
5.1.2.3;2.1.2.3 Exercise;30
5.1.3;2.1.3 Linear Oscillator with Viscous Damping and Periodic (Harmonic) Forcing;30
5.1.3.1;2.1.3.1 Exercise;32
5.1.4;2.1.4 Two Coupled Oscillators;33
5.1.4.1;2.1.4.1 Weakly Coupled Oscillators with Strongly Different Frequencies;33
5.1.4.2;2.1.4.2 Exercise;36
5.1.4.3;2.1.4.3 Weakly Coupled Oscillators with Close Frequencies;36
5.1.4.4;2.1.4.4 Exercise;40
5.1.4.5;2.1.4.5 Strongly Coupled Oscillators with Essentially Different Frequencies;40
5.1.4.6;2.1.4.6 Exercise;42
5.1.4.7;2.1.4.7 Strongly Coupled Oscillators with Close Frequencies;43
5.1.4.8;2.1.4.8 Exercise;43
5.2;2.2 Single-DOF Nonlinear Oscillator;43
5.2.1;2.2.1 Quasilinear Oscillator;45
5.2.2;2.2.2 Strongly Nonlinear Oscillator and Vibro-Impact Approximation;48
5.2.2.1;2.2.2.1 Special Solutions for Vibro-Impact Motions;48
5.2.2.2;2.2.2.2 Vibro-Impact Systems Treatable by Analytic Functions;51
5.2.3;2.2.3 Oscillator with Multiple Equilibriums;55
5.3;2.3 Forced Nonlinear Oscillator;56
5.3.1;2.3.1 General Remarks;56
5.3.2;2.3.2 Governing Equations;57
5.3.3;2.3.3 The Dynamics of the Oscillator without Dissipation and the LPT;59
5.3.4;2.3.4 The Transient Dynamics of a Weakly Damped Oscillator;62
5.3.4.1;2.3.4.1 Non-smooth Temporal Transformations;63
5.3.4.2;2.3.4.2 The Construction of a Generating Solution;64
5.3.4.3;2.3.4.3 Calculation of A0(t0) and .0(t0);67
5.3.5;2.3.5 Quasi-Linear Oscillations;70
5.4;2.4 Entrainment, Synchronization and Resonance Capture;73
5.4.1;2.4.1 Pendulum with Constant External Torque;74
5.4.2;2.4.2 Entrainment of the Van der Pol Oscillator by External Harmonic Force;77
5.4.3;2.4.3 Synchronization of Oscillators and Related Models;84
5.4.4;2.4.4 Resonance Capture;86
5.4.4.1;2.4.4.1 Exercises;89
5.4.5;2.4.5 Forced Oscillator with Multiple States of Equilibrium;90
5.5;2.5 Symmetric Systems of Coupled Nonlinear Oscillators Beating Phenomena;93
5.5.1;2.5.1 Exercise;102
5.6;2.6 2DOF Systems of Nonlinear Oscillators with Essential Asymmetry Targeted Energy Transfer (TET);103
5.6.1;2.6.1 Targeted Energy Transfer in an Unforced 2DOF System;103
5.6.2;2.6.2 Targeted Energy Transfer in Forced 2DOF System;114
5.7;2.7 Coupled Nonlinear Oscillators with Time Delays;142
5.7.1;2.7.1 Analytic Model;142
5.7.2;2.7.2 Numeric Verification -- Straight Modes;150
5.7.3;2.7.3 Numeric Verification -- "Oval"-- Modes and Phase -- Locked Solutions;153
5.8;2.8 Low-DOF Discrete Nonlinear Systems;156
5.9;2.9 Concluding Remarks;169
5.10;References;170
6;3 Infinite Discrete Systems;175
6.1;3.1 Dynamics of Infinite Nonlinear Chains;175
6.1.1;3.1.1 Long-Wavelength Approximation. Equation of Supersonic Extension Solitons in an Infinite FPU Chain;175
6.1.2;3.1.2 Zigzag Chain and Long-Wave Solitons;177
6.1.3;3.1.3 Envelope Solitons;179
6.1.4;3.1.4 Optical Breathers in a Zigzag Chain;184
6.1.5;3.1.5 Torsional Solitons;194
6.1.6;3.1.6 Approximation of Immobile Neighbour Chains;196
6.2;3.2 Dynamics of Essentially Nonlinear and Vibro-Impact Chains;200
6.2.1;3.2.1 Oscillatory Chain with Rigid Barriers;201
6.2.2;3.2.2 Discrete Breathers in a Vibro-Impact Chain;206
6.2.2.1;3.2.2.1 System of Klein-Gordon (KG) Type;207
6.2.2.2;3.2.2.2 System of Fermi-Pasta-Ulam (FPU) Type;211
6.3;3.3 The Problem of Heat Conduction in Dielectrics;215
6.4;3.4 Solitons in Energetically Nondegenerate Quasi-One-Dimensional Models;225
6.4.1;3.4.1 Quasi-One-Dimensional Model of a Molecular Crystal: Soliton Modes of Motion in a Bistable Nonlinear System;227
6.5;3.5 Dynamics of Ensembles of Interacting Nonlinear Chains;233
6.6;3.6 Concluding Remarks;241
6.7;References;241
7;4 Continuous Systems;245
7.1;4.1 One-Dimensional Models;245
7.1.1;4.1.1 Bolotin Model;245
7.1.1.1;4.1.1.1 Exercises;249
7.1.2;4.1.2 Simplification of the Timoshenko Beam;249
7.2;4.2 The Planar Dynamical Problem and Tractable One-Dimensional Models of an Elastic Solid;256
7.2.1;4.2.1 Exercises;261
7.3;4.3 The Two-Dimensional Orthotropic Model and Its Application to a Complex Contact Problem;261
7.3.1;4.3.1 Basic Asymptotic Decomposition of the Orthotropic Plate Problem;261
7.3.2;4.3.2 The Contact Problem for a Planar Orthotropic Strip;267
7.4;4.4 Models of Elastic Foundation;272
7.4.1;4.4.1 General Equations and Asymptotic Analysis;272
7.4.2;4.4.2 Example -- Dynamical Problem;276
7.4.3;4.4.3 Example -- An Axisymmetric St278
7.5;4.5 On the Concept of Solids;289
7.6;4.6 Models of Non-Fourier Heat Conduction;292
7.7;4.7 Concluding Remarks;299
7.8;References;299
8;Afterword;302
9;Index;304