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Nonlinear Dynamics

Between Linear and Impact Limits

  • Book
  • © 2010

Overview

Authors:
  1. Valery N. Pilipchuk

    1. Professor (Research) Mechanical Engineering, Wayne State University, Detroit, USA

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  • First systematic description for the non smooth time transformations and related analytical and numerical algorithms
  • Bridges the gap between linear and strongly nonlinear dynamics by showing how to switch the physical basis when approaching the area of severe (impact) nonlinearities
  • Presents a unified physical basis for analyses of vibrations with essentially non-harmonic or discontinuous temporal shapes

Part of the book series: Lecture Notes in Applied and Computational Mechanics (LNACM, volume 52)

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Table of contents (14 chapters)

  1. Front Matter

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  2. Introduction

    • Valery N. Pilipchuk
    Pages 1-36

  3. Smooth Oscillating Processes

    • Valery N. Pilipchuk
    Pages 37-49

  4. Nonsmooth Processes as Asymptotic Limits

    • Valery N. Pilipchuk
    Pages 51-91

  5. Nonsmooth Temporal Transformations (NSTT)

    • Valery N. Pilipchuk
    Pages 93-129

  6. Sawtooth Power Series

    • Valery N. Pilipchuk
    Pages 131-144

  7. NSTT for Linear and Piecewise-Linear Systems

    • Valery N. Pilipchuk
    Pages 145-178

  8. Periodic and Transient Nonlinear Dynamics under Discontinuous Loading

    • Valery N. Pilipchuk
    Pages 179-193

  9. Strongly Nonlinear Vibrations

    • Valery N. Pilipchuk
    Pages 195-239

  10. Strongly Nonlinear Waves

    • Valery N. Pilipchuk
    Pages 241-244

  11. Impact Modes and Parameter Variations

    • Valery N. Pilipchuk
    Pages 245-264

  12. Principal Trajectories of Forced Vibrations

    • Valery N. Pilipchuk
    Pages 265-273

  13. NSTT and Shooting Method for Periodic Motions

    • Valery N. Pilipchuk
    Pages 275-294

  14. Essentially Non-periodic Processes

    • Valery N. Pilipchuk
    Pages 295-303

  15. Spatially-Oscillating Structures

    • Valery N. Pilipchuk
    Pages 305-337

  16. Back Matter

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Keywords

About this book

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

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)

Authors and Affiliations

  • Professor (Research) Mechanical Engineering, Wayne State University, Detroit, USA

    Valery N. Pilipchuk

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