Asymptotic Approaches in Nonlinear Dynamics

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).