Asymptotic Approaches in Nonlinear Dynamics: New Trends and Applications

ISBN-13: 9783642720819 / Angielski / Miękka / 2012 / 310 str.

Jan Awrejcewicz;Igor V. Andrianov;Leonid I. Manevitch

Opis

How well is Nature simulated by the varied asymptotic models that imaginative scientists have invented? B. Birkhoff 52J This book deals with asymptotic methods in nonlinear dynamics. For the first time a detailed and systematic treatment of new asymptotic methods in combination with the Pade approximant method is presented. Most of the basic results included in this manuscript have not been treated but just mentioned in the literature. Providing a state-of-the-art review of asymptotic applications, this book will prove useful as an introduction to the field for novices as well a reference for specialists. Asymptotic methods of solving mechanical and physical problems have been developed by many authors. For example, we can refer to the excel- lent courses by A. Nayfeh 119-122]' M. Van Dyke 154], E.J. Hinch 94J and many others 59, 66, 95, 109, 126, 155, 163, 50d, 59dJ. The main features of the monograph presented are: 1) it is devoted to the basic principles of asymp- totics and its applications, and 2) it deals with both traditional approaches (such as regular and singular perturbations, averaging and homogenization, perturbations of the domain and boundary shape) and less widely used, new approaches such as one- and two-point Pade approximants, the distributional approach, and the method of boundary perturbations.

Spis treści

1. Introduction: Some General Principles of Asymptotology..- 1.1 An Illustrative Example.- 1.2 Reducing the Dimensionality of a System.- 1.3 Continualization.- 1.4 Averaging.- 1.5 Renormalization.- 1.6 Localization.- 1.7 Linearization.- 1.8 Padé Approximants.- 1.9 Modern Computers and Asymptotic Methods.- 1.10 Asymptotic Methods and Teaching Physics.- 1.11 Problems and Perspectives.- 2. Discrete Systems.- 2.1 The Classical Perturbation Technique: an Introduction.- 2.2 Krylov-Bogolubov-Mitropolskij Method.- 2.3 Equivalent Linearization.- 2.4 Analysis of Nonconservative Nonautonomous Systems.- 2.4.1 Introduction.- 2.4.2 Nonresonance Oscillations.- 2.4.3 Oscillations in the Neighbourhood of Resonance.- 2.5 Nonstationary Nonlinear Systems.- 2.6 Parametric and Self-Excited Oscillation in a Three-Degree-of-Freedom Mechanical System.- 2.6.1 Analysed System and Equation of Motion.- 2.6.2 Transformation of the Equations of Motion to the Main Coordinates.- 2.6.3 Zones of Instability of the First Order.- 2.6.4 Calculation Examples.- 2.7 Modified Poincaré Method.- 2.7.1 One-Degree-of-Freedom System.- 2.7.2 General Nonlinear Systems.- 2.8 Hopf Bifurcation.- 2.9 Stability Control of Vibro-Impact Periodic Orbit.- 2.9.1 Introduction.- 2.9.2 Control of Vibro-Impact Periodic Orbits.- 2.9.3 Stability Control.- 2.9.4 Simulation Results.- 2.10 Normal Modes of Nonlinear Systems with n Degrees of Freedom.- 2.10.1 Definition.- 2.10.2 Free Oscillations and Close Natural Frequencies.- 2.11 Nontraditional Asymptotic Approaches.- 2.11.1 Choice of Asymptotic Expansion Parameters.- 2.11.2 ?-Expansions in Nonlinear Mechanics.- 2.11.3 Asymptotic Solutions for Nonlinear Systems with High Degrees of Nonlinearity.- 2.11.4 Square-Well Problem of Quantum Theory.- 2.12 Padé Approximants.- 2.12.1 One-Point Padé Approximants: General Definitions and Properties.- 2.12.2 Using One-Point Padé Approximants in Dynamics.- 2.12.3 Matching Limit Expansions.- 2.12.4 Matching Local Expansions in Nonlinear Dynamics.- 2.12.5 Generalizations and Problems.- 3. Continuous Systems.- 3.1 Continuous Approximation for a Nonlinear Chain.- 3.2 Homogenization Procedure in the Nonlinear Dynamics of Thin-Walled Structures.- 3.2.1 Nonhomogeneous Rod.- 3.2.2 Stringer Plate.- 3.2.3 Perforated Membrane.- 3.2.4 Perforated Plate.- 3.3 Averaging Procedure in the Nonlinear Dynamics of Thin-Walled Structures.- 3.3.1 Berger and Berger-Like Equations for Plates and Shells.- 3.3.2 “Method of Freezing” in the Nonlinear Theory of Viscoelasticity.- 3.4 Bolotin-Like Approach for Nonlinear Dynamics.- 3.4.1 Straightforward Bolotin Approach.- 3.4.2 Modified Bolotin Approach.- 3.5 Regular and Singular Asymptotics in the Nonlinear Dynamics of Thin-Walled Structures.- 3.5.1 Circular Rings and Axisymmetric Cylindrical Shells.- 3.5.2 Reinforced and Isotropic Cylindrical Shells.- 3.5.3Nonlinear Oscillations of a Cylindrical Panel.- 3.5.4 Stability of Thin Spherical Shells Under Dynamic Loading.- 3.5.5 Asymptotic Investigation of the Nonlinear Dynamic Boundary Value Problem for a Rod.- 3.6 One-Point Padé Approximants Using the Method of Boundary Condition Perturbation.- 3.7 Two-Point Padé Approximants: A Plate on Nonlinear Support.- 3.8 Solitons and Soliton-Like Approaches in the Case of Strong Nonlinearity.- 3.9 Nonlinear Analysis of Spatial Structures.- 3.9.1 Introduction.- 3.9.2 Modified Envelope Equation.- 4. Discrete—Continuous Systems.- 4.1 Periodic Oscillations of Discrete-Continuous Systems with a Time Delay.- 4.1.1 The KBM Method.- 4.2 Simple Perturbation Technique.- 4.3 Nonlinear Behaviour of Electromechanical Systems.- 4.3.1 Introduction.- 4.3.2 Dynamics Equations.- 4.3.3 Averaging.- 4.3.4 Numerical Results.- General References.- Detailed References (d).