Nonlinear Dynamics
Between Linear and Impact Limits
1st Edition
Published on 9. May 2010
XI, 364 pages
978-3-642-12799-1 (ISBN)
Description
Nonlinear Dynamics represents a wide interdisciplinary area of research dealing with a variety of "unusual" physical phenomena by means of nonlinear differential equations, discrete mappings, and related mathematical algorithms. However, with no real substitute for the linear superposition principle, the methods of Nonlinear Dynamics appeared to be very diverse, individual and technically complicated. This book makes an attempt to find a common ground for nonlinear dynamic analyses based on the existence of strongly nonlinear but quite simple counterparts to the linear models and tools. It is shown that, since the subgroup of rotations, harmonic oscillators, and the conventional complex analysis generate linear and weakly nonlinear approaches, then translations and reflections, impact oscillators, and hyperbolic (Clifford's) algebras must give rise to some "quasi impact" methodology. Such strongly nonlinear methods are developed in several chapters of this book based on the idea ofnon-smooth time substitutions. Although most of the illustrations are based on mechanical oscillators, the area of applications may include also electric, electro-mechanical, electrochemical and other physical models generating strongly anharmonic temporal signals or spatial distributions. Possible applications to periodic elastic structures with non-smooth or discontinuous characteristics are outlined in the final chapter of the book.
Reviews / Votes
From the reviews:
"This book is based on a series of papers which the author published earlier. The main subject of this book is the concept of non-smooth time transformations (NSTT) to describe periodic solutions of essentially nonlinear differential equations. . This book is suitable for scientists interested in applied mathematics, in particular those interested in constructing (approximations of) periodic solutions for differential equations." (Wim T. van Horssen, Mathematical Reviews, Issue 2012 j)
"The content of this book is built on a new physical idea that the effectiveness of linear and weakly nonlinear dynamic theories is due to the spatio-temporal nature of harmonic motions associated with subgroup of rigid-body rotations. . this original book will be of interest to the experts, professors and post-graduate students in various areas of nonlinear physics, fundamental and engineering mechanics, applied mathematics, and other fields of research dealing with nonlinear dynamic models, non-smooth or discontinuous processes." (Anatoly Martynyuk, Zentralblatt MATH, Vol. 1202, 2011)
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05/2010
1st Edition
Springer
€160.49
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Content
Smooth Oscillating Processes.- Nonsmooth Processes as Asymptotic Limits.- Nonsmooth Temporal Transformations (NSTT).- Sawtooth Power Series.- NSTT for Linear and Piecewise-Linear Systems.- Periodic and Transient Nonlinear Dynamics under Discontinuous Loading.- Strongly Nonlinear Vibrations.- Strongly Nonlinear Waves.- Impact Modes and Parameter Variations.- Principal Trajectories of Forced Vibrations.- NSTT and Shooting Method for Periodic Motions.- Essentially Non-periodic Processes.- Spatially-Oscillating Structures.
"Chapter 1 Introduction (p. 1-2)
Abstract. This chapter contains physical and mathematical preliminaries with di?erent introductory remarks. Although some of the statements are informal and rather intuitive, they nevertheless provide hints on selecting the generating models and corresponding analytical techniques. The idea is that simplicity of a mathematical formalism is caused by hidden links between the corresponding generating models and subgroups of rigid-body motions.
Such motions may be quali?ed indeed as elementary macro-dynamic phenomena developed in the physical space. For instance, since rigid-body rotations are associated with sine waves and therefore (smooth) harmonic analyses then translations and mirror-wise re?ections must reveal adequate tools for strongly unharmonic and nonsmooth approaches. This viewpoint is illustrated by physical examples, problem formulations and solutions.
1.1 Brief Literature Overview
Analytical methods of conventional nonlinear dynamics are based on the classical theory of di?erential equations dealing with smooth coordinate transformations, asymptotic integrations and averaging. The corresponding solutions often include quasi harmonic expansions as a generic feature that explicitly points to the physical basis of these methods namely - the harmonic oscillator. Generally speaking, some of the techniques are also applicable to dynamical systems close to integrable but not necessarily linear.
However, nonlinear generating solutions are seldom available in closed form [9]. As a result, strongly nonlinear methods usually target speci?c situations and are rather di?cult to use in other cases. Generating models for strongly nonlinear analytical tools with a wide range of applicability must obviously 1) capture the most common features of oscillating processes regardless their nonlinear speci?cs, 2) possess simple enough solutions in order to provide e?ciency of perturbation schemes, and 3) describe essentially nonlinear phenomena out of the scope of the weakly nonlinear methods.
So the key notion of the present work suggests possible recipes for selecting such models among so-called non-smooth systems while keeping the class of smooth motions still within the range of applicability. Note that di?erent non-smooth cases have been also considered for several decades by practical and theoretical reasons. On the physical point of view, this kind of modeling essentially employs the idea of perfect spatio-temporal localization of strong nonlinearities or impulsive loadings. For instance, sudden jumps of restoring force characteristics are represented by absolutely sti? constraints under the assumption that the dynamics in between the constraints is smooth and simple enough to describe. As a result, the system dynamics is discretized in terms of mappings and matchings di?erent pieces of solutions."
Abstract. This chapter contains physical and mathematical preliminaries with di?erent introductory remarks. Although some of the statements are informal and rather intuitive, they nevertheless provide hints on selecting the generating models and corresponding analytical techniques. The idea is that simplicity of a mathematical formalism is caused by hidden links between the corresponding generating models and subgroups of rigid-body motions.
Such motions may be quali?ed indeed as elementary macro-dynamic phenomena developed in the physical space. For instance, since rigid-body rotations are associated with sine waves and therefore (smooth) harmonic analyses then translations and mirror-wise re?ections must reveal adequate tools for strongly unharmonic and nonsmooth approaches. This viewpoint is illustrated by physical examples, problem formulations and solutions.
1.1 Brief Literature Overview
Analytical methods of conventional nonlinear dynamics are based on the classical theory of di?erential equations dealing with smooth coordinate transformations, asymptotic integrations and averaging. The corresponding solutions often include quasi harmonic expansions as a generic feature that explicitly points to the physical basis of these methods namely - the harmonic oscillator. Generally speaking, some of the techniques are also applicable to dynamical systems close to integrable but not necessarily linear.
However, nonlinear generating solutions are seldom available in closed form [9]. As a result, strongly nonlinear methods usually target speci?c situations and are rather di?cult to use in other cases. Generating models for strongly nonlinear analytical tools with a wide range of applicability must obviously 1) capture the most common features of oscillating processes regardless their nonlinear speci?cs, 2) possess simple enough solutions in order to provide e?ciency of perturbation schemes, and 3) describe essentially nonlinear phenomena out of the scope of the weakly nonlinear methods.
So the key notion of the present work suggests possible recipes for selecting such models among so-called non-smooth systems while keeping the class of smooth motions still within the range of applicability. Note that di?erent non-smooth cases have been also considered for several decades by practical and theoretical reasons. On the physical point of view, this kind of modeling essentially employs the idea of perfect spatio-temporal localization of strong nonlinearities or impulsive loadings. For instance, sudden jumps of restoring force characteristics are represented by absolutely sti? constraints under the assumption that the dynamics in between the constraints is smooth and simple enough to describe. As a result, the system dynamics is discretized in terms of mappings and matchings di?erent pieces of solutions."
