Asymptotic approaches in nonlinear dynamics : new trends and applications

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 Pade 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 Poincare 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 Pade Approximants. - 2.12.1 One-Point Pade Approximants: General Definitions and Properties. - 2.12.2 Using One-Point Pade 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 Pade Approximants Using the Method of Boundary Condition Perturbation. - 3.7 Two-Point Pade 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).