Nonlinear Dynamics: Between Linear and Impact Limits

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Nonlinear Dynamics: Between Linear and Impact Limits

Author: Valery N. Pilipchuk
Professor (Research) Mechanical Engineering - Wayne State University
ISBN: 978-3-642-12798-4
e-ISBN: 978-3-642-12799-1
Pdf file format - 366 pages - Quality: 10

Copy of Preface:
The main objective of this book is to introduce a unified physical basis for analyses of vibrations with essentially unharmonic, non-smooth or may be discontinuous time shapes. It is known that possible transitions to nonsmooth limits can make investigations especially difficult. This is due to the fact that the dynamic methods were originally developed within the paradigm of smooth motions based on the classical theory of differential equations.
From the physical standpoint, these represent low-energy approaches to modeling dynamical systems. Although the impact dynamics has also quite a long pre-history, any kind of non-smooth behavior is often viewed as an exemption rather than a rule. Similarly, the classical theory of differential equations usually avoids non-differentiable and discontinuous functions. To-date, however, many theoretical and applied areas cover high-energy phenomena accompanied by strongly non-linear spatio-temporal behaviors making the classical smooth methods inefficient in many cases. For instance, such phenomena occur when dealing with dynamical systems under constraint conditions, friction-induced vibrations, structural damages due to cracks, liquid sloshing impacts, and numerous problems of nonlinear physics. Similarly to the wellknown analogy between mechanical and electrical harmonic oscillators, some electronic instruments include so-called Schmitt trigger circuits generating nonsmooth signals whose temporal shapes resemble mechanical vibro-impact processes. In many such cases, it is still possible to adapt different smooth methods of the dynamic analyses through strongly non-linear algebraic manipulations with state vectors or by splitting the phase space into multiple domains based on the system specifics. As a result, the related formulations are often reduced to discrete mappings in a wide range of the dynamics from periodic to atochastic. Possible alternatives to such approaches can be built on generating models developing essentially nonlinear/unharmonic behaviors as their inherent properties. Such models must be general and simple enough in order to play the role of physical basis. As shown in this book, new generating systems can be found by intentionally imposing the ‘worst case scenario’ on conventional methods in anticipation that failure of one asymptotic may point to its complementary counterpart. However, the related mathematical formalizations are seldom straightforward and require new principles. For instance, the tool developed here employs nonsmooth (impact) systems as a basis to describing not only impact but also smooth or even linear dynamics. This is built on the idea of non-smooth time substitutions/transformations (NSTT) proposed originally for strongly nonlinear but still smooth models.
On the author’s view, the methodological role of NSTT is to reveal explicit links between impact dynamics and hyperbolic algebras analogously to the link between harmonic vibrations and conventional complex analyses. In particular, this book gives the first systematic description for NSTT and related analytical and numerical algorithms. The text focuses on methodologies and discussions of their physical and mathematical basics. Detailed applications are mostly excluded from this book, however, necessary references on journal publications are provided.

Short 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 of non-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.

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Nonlinear Dynamics: Between Linear and Impact Limits - Valery N. Pilipchuk

Author: Valery N. Pilipchuk | Size: 17.1 MB | Format: PDF | Publisher: Springer | Year: 2010 | pages: 360 | ISBN: 3642127983

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 of non-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.

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