This book covers developments in the theory of oscillations from diverse viewpoints, reflecting the fields multidisciplinary nature. It introduces the state-of-the-art in the theory and various applications of nonlinear dynamics. It also offers the first treatment of the asymptotic and homogenization methods in the theory of oscillations in combination with Pad approximations. With its wealth of interesting examples, this book will prove useful as an introduction to the field for novices and as a reference for specialists.
This book covers developments in the theory of oscillations from diverse viewpoints, reflecting the fields multidisciplinary nature. It introduces the state-of-the-art in the theory and various applications of nonlinear dynamics. It also offers the first treatment of the asymptotic and homogenization methods in the theory of oscillations in combination with Pad approximations. With its wealth of interesting examples, this book will prove useful as an introduction to the field for novices and as a reference for specialists.
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.- 2.5 Nonstationary Nonlinear Systems.- 2.6 Parametric and Self-Excited Oscillation in a Three-Degree-of-Freedom Mechanical System.- 2.7 Modified Poincaré Method.- 2.8 Hopf Bifurcation.- 2.10 Normal Modes of Nonlinear Systems with n Degrees of Freedom.- 2.11 Nontraditional Asymptotic Approaches.- 2.12 Padé Approximants.- 3. Continuous Systems.- 3.1 Continuous Approximation for a Nonlinear Chain.- 3.2 Homogenization Procedure in the Nonlinear Dynamics of Thin-Walled Structures.- 3.3 Averaging Procedure in the Nonlinear Dynamics of Thin-Walled Structures.- 3.4 Bolotin-Like Approach for Nonlinear Dynamics.- 3.5 Regular and Singular Asymptotics in the Nonlinear Dynamics of Thin-Walled Structures.- 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.- 4. Discrete—Continuous Systems.- 4.1 Periodic Oscillations of Discrete-Continuous Systems with a Time Delay.- 4.2 Simple Perturbation Technique.- 4.3 Nonlinear Behaviour of Electromechanical Systems.- GeneralReferences.- Detailed References (d).
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