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Nonlinear Oscillations Exact Solutions and their Approximations

Nonlinear Oscillations

Ivana Kovacic

Nonlinear Oscillations Exact Solutions and their Approximations

123

Ivana Kovacic Faculty of Technical Sciences Centre for Vibro-Acoustic Systems and Signal Processing University of Novi Sad Novi Sad, Serbia

ISBN 978-3-030-53171-3 ISBN 978-3-030-53172-0 https://doi.org/10.1007/978-3-030-53172-0

(eBook)

© Springer Nature Switzerland AG 2020 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface

This book has several representative nonlinear oscillators in its core and the rest of the material is ‘knitted’ around them. The mechanical models of these oscillators are collected in Chap. 1, starting with a linear oscillator, which is given as a reference point. Their corresponding mathematical models (equations of motion) are given, for which the reader needs to be familiar with the laws of mechanics and analytical dynamics. Note that for all of them, the nonlinearity stems from the restoring force. Their solutions for motion are shown in the following chapter, and what is common for all of them, is that they represent exact solutions for motion, and these exact solutions are the main focus and characteristic of this book—they are used through the rest of the book as a basis for other solutions and the design of systems that can exhibit them, free damped oscillators with one degree of freedom, forced oscillators with one degree of freedom, chains of oscillators, continuous systems as well as for nonlinear isochronous oscillators. It should be pointed out that although the exact solutions are given in forms of certain special functions, the prior knowledge of these functions is not needed,