The Behaviour of Nonlinear Vibrating Systems

The Behaviour of Nonlinear Vibrating Systems : Volume II: Advanced Concepts and Applications to Multi-Degree-of-Freedom Systems

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The purpose of this book is to provide students, practicing engineers and scientists with a treatment of nonlinear phenomena occurring in physical systems. Although only mechanical models are used, the theory applies to all physical systems governed by the same equations, so that the book can be used to study nonlinear phenomena in other branches of engineering, such as electrical engineering and aerospace engineering, as well as in physics. The book consists of two volumes. Volume I is concerned with single- degree-of-freedom systems and it presents the fundamental concepts of nonlinear analysis. Both analytical methods and computer simulations are included. The material is presented in such a manner that the book can be used as a graduate as well as an undergraduate textbook. Volume II deals with multi-degree-of-freedom systems. Following an introduc- tion to linear systems, the volume presents fundamental concepts of geometric theory and stability of motion of general nonlinear systems, as well as a concise discussion of basic approximate methods for the response of such systems. The material represents a generalization of a series of papers on the vibration of nonlinear multi-degree-of-freedom systems, some of which were published by me and my associates during the period 1965 - 1983 and some are not yet published.
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Product details

  • Hardback | 336 pages
  • 159 x 243.3 x 25.4mm | 712.15g
  • Dordrecht, Netherlands
  • English
  • 1990 ed.
  • XX, 336 p.
  • 0792303695
  • 9780792303695

Table of contents

1/Normal Oscillations in Autonomous Conservative Systems.- 1.1. Harmonic solution by the Ritz method.- 1.2. Harmonic solution by the averaging and asymptotic methods.- 1.3. Comparison of methods: nonlinear normal modes and nonlinear normal coordinates.- 1.4. Examples - theoretical and computer simulation analysis.- 1.5. Harmonic plus constant term solutions - systems with quadratic nonlinearities.- 2/Normal Oscillations of Elastic Nonlinear Continuous Systems.- 2.1. Harmonic solution - `Linear Normal Mode' approach.- 2.2. Harmonic solution - `Nonlinear Normal Mode' approach.- 2.3. The generalized Ritz method for a beam with nonlinear boundary conditions.- 3/Free Oscillations with Arbitrary Initial Conditions.- 3.1. The multi-frequency almost-periodic solution by the harmonic balance method.- 3.2. The almost-periodic multi-frequency solutions by the averaging method and the asymptotic method.- 3.3. The almost-periodic oscillations in a two-degree-freedom system.- 4/Harmonic Solution in Nonautonomous Systems and Its Local Stability.- 4.1. The Ritz method and variational coupled Hill's equations.- 4.2. The first order unstable regions by the perturbation procedure based on the Floquet theory.- 4.3. The first order unstable regions by the asymptotic and averaging method.- 4.4. First order unstable regions of harmonic solution in a two- degree-of-freedom system - theoretical and computer simulation analysis.- 4.5. Harmonic plus constant term solution - systems with quadratic nonlinearity.- 5/Principal Resonances.- 5.1. The mode shape of resonant vibrations in undamped systems.- 5.2. The single nonlinear mode method in weakly damped systems.- 5.3. The combined Ritz-averaging method.- 5.4. Some remarks on nonlinear normal coordinates.- 6/Principal Resonances - Examples of Theoretical and Computer Simulation Analysis.- 6.1. Two-degree-of-freedom systems - problem of coupling of normal coordinates.- 6.2. Three-degree-of-freedom systems - the single nonlinear mode method.- 6.3. Homogeneous system with two-degrees-of-freedom.- 7/Secondary Resonances (Periodic and Almost-Periodic).- 7.1. Harmonic balance method: steady-state solution and its local stability.- 7.2. The averaging method: steady-state solution and its local stability.- 7.3. The combined harmonic balance - averaging procedure.- 7.4. Determination of types of secondary resonances associated with given nonlinear characteristics.- 7.5. Basins of attraction of the secondary resonances.- 7.6. Two-degree-of-freedom system: steady-state secondary resonances - theoretical and simulation analysis.- 7.7. Two-degree-of-freedom system: basins of attraction.- 8/Internal Resonances.- 8.1. Interaction of principal and internal resonances by the averaging method.- 8.2. The harmonic balance method and types of internal resonances associated with given nonlinear characteristics.- 8.3. Interaction of external and internal resonances in a two- degree-of-freedom system: theoretical and simulation results.- 9/Parametric Resonances.- 9.1. Parametric resonances in linear systems - survey of methods.- 9.2. Determination of the combination parametric resonance by the harmonic balance method.- 9.3. First order parametric resonances in nonlinear systems.- 9.4. Parametric resonances in a two-degree-of-freedom system - theoretical and computer simulation analysis.- References.
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