Nonlinear dynamics. Between linear and impact limits
type: Item
instance of: scholarly article
title: Nonlinear dynamics. Between linear and impact limits
zbMATH Open document ID: 1202.37004
DOI: 10.1007/978-3-642-12940-7
author: V. N. Pilipchuk
published in: Lecture Notes in Applied and Computational Mechanics
publication date: 2010-06-25T00:00:00Z
review text: 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 a subgroup of rigid-body rotations. As follows from the first two chapters of the book, these most elementary macro-level processes invoke conventional tools of elliptic complex numbers, trigonometric expansions, and the related analytical algorithms of averaging and asymptotic integration. Further, it is shown that another subgroup of rigid-body motions, namely translations with reflections, generates hyperbolic complex numbers whenever the inner non-smooth time of the reflection is introduced. This remarkable observation opens an alternative way to building constructive methods of dynamic analyses for strongly nonlinear and essentially anharmonic processes. Mathematical properties of non-smooth temporal substitutions are discussed in Chapters 4 and 5 with necessary proofs; however Section 3 of the Introduction provides necessary information for a quick start of practical applications. Then, Chapters 6, 7 and 8 illustrate applications of the developed methods to linear, weakly nonlinear and strongly nonlinear vibrating systems, respectively. In particular, applying the non-smooth temporal substitutions to linear and weakly nonlinear systems may be very effective in those cases when non-smooth or discontinuous loadings are present and thus harmonic or quasi harmonic approaches require long expansions for capturing essential features of the dynamics. Chapters 9 through 11 deal with the concept of nonlinear normal modes. In particular, formulations of the normal mode problem for impact systems are introduced. Chapter 12 presents a semi-analytical approach combining NSTT with the shooting method that essentially extends the area of applications. Chapter 13 describes a physical basis for non-smooth time substitutions for the case of essentially non-periodic processes. Fina…
reviewed by: A. A. Martynyuk
Mathematics Subject Classification ID: 37-02
Mathematics Subject Classification ID: 70-02
Mathematics Subject Classification ID: 70Kxx
Mathematics Subject Classification ID: 37N20
Mathematics Subject Classification ID: 70-08
Mathematics Subject Classification ID: 37N05
zbMATH DE Number: 5726191
zbMATH Keywords: elliptic complex numbers
zbMATH Keywords: trigonometric expansions
zbMATH Keywords: algorithms of averaging
zbMATH Keywords: asymptotic integration
zbMATH Keywords: anharmonic processes
zbMATH Keywords: strongly nonlinear vibrating systems
zbMATH Keywords: nonlinear normal modes
zbMATH Keywords: non-periodic processes
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/978-3-642-12940-7
OpenAlex ID: W2486438414
description: scientific article; zbMATH DE number 5726191