Non-Smooth Dynamical Systems
The book provides a self-contained introduction to the mathematical theory of non-smooth dynamical problems, as they frequently arise from mechanical systems with friction and/or impacts. It is aimed at applied mathematicians, engineers, and applied scien
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1744
Springer Berlin Heidelberg New York Barcelona Hong Kong London Milan Paris Singapore Tokyo
Markus Kunze
Non-Smooth Dynamical Systems
Springer
Author Markus Kunze Mathematisches Institut Universitat Kaln Weyertal86 50931 Kaln, Germany E-mail: [email protected]
Cataloging-in-Publication Data applied for Die Deutsche Bibliothek - CIP-Eioheitsaufnahrne Kunze, Markus: Non-smooth dynamical systems I Markus Kunze. - Berlin; Heidelberg; New York; Barcelona; Hong Kong; London; Milan; Paris; Singapore ; Tokyo: Springer, 2000 (Lecture notes in mathematics; 1744) ISBN 3-540-67993-6
Mathematics Subject Classification (2000): 34Cxx, 34Dxx, 37-xx, 70Exx, 70Kxx ISSN 0075-8434 ISBN 3-540-67993-6 Springer-Verlag Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned. specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. Springer-Verlag Berlin Heidelberg New York a member of BertelsmannSpringer Science-Business Media GmbH © Springer-Verlag Berlin Heidelberg 2000 Printed in Germany
Typesetting: Camera-ready TEX output by the author SPIN: 10724224 41/3142-543210 - Printed on acid-free paper
The course of true love never did run smooth.
Shakespeare, A midsummer night's dream
Preface
There are many concrete problems in life, and particularly in mechanics and in the engineering sciences, where non-smooth phenomena play an important role: one might think of the noise of a squeaking chalk on a black-board or, sometimes more pleasantly, the sounds of stringed instruments like a violin. More relevant applications include noise generation in railway wheels, the chattering of machine tools, grating brakes, impact print hammers, percussion drilling machines, etc. Physically speaking, these effects often are due to the fact that there are rigid bodies which are in contact (they "stick"), whereas these contact phases are interrupted by "slip" phases during which one of the bodies moves relative to another. In addition to such behaviour mainly induced by friction, there may also be impacts between different parts of the system. From a mathematical viewpoint, problems of this kind are not easy to handle, since the resulting models are dynamical systems whose right-hand sides are not continuous or not differentiable. In many cases the solutions have to observe additional restrictions that frequently appear in the form of inequality constraints. Since many concepts from classical dynamical systems theory do rely on the smoothness of the underlying system or (semi-) flow, it was necessary to generalize those concepts to cov
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