Nonlinear Dynamics of Active and Passive Systems of Vibration Protection PDF
Preview Nonlinear Dynamics of Active and Passive Systems of Vibration Protection
Foundations of Engineering Mechanics M. Z. Kolovsky, Nonlinear Dynamics of Active and Passive Systems of Vibration Protection Springer-Verlag Berlin Heidelberg GmbH M. Z. Kolovsky Nonlinear Dynamics of Active and Passive Systems of Vibration Protection Translated by A.K. Belyaev With 156 Figures ' Springer Series Editors: V. I. Babitsky J. Wittenburg Loughborough University Universitiit Karlsruhe (TH) Department of Mechanical Engineering lnstitut fiir Mechanik LE11 3TU Loughborough, Leicestershire Kaiserstrasse l2 Great Britain D-76128 Karlsruhe I Germany Author: Mikhail Z. Kolovsky Kondratievsky 56 -24 195197 St. Petersburg I Russia Translator: A.K. Belyaev Johannes-Kepler-Universitat Linz Institut fiir Mechanik und Maschinenlehre A-4040 Linz I Austria Cataloging-in-Publication Data applied for Die Deutsche Bibliothek-CIP-Einheitsaufnahme Kolovskij; Mikhail Z.: Nonlinear dynamics of active and passive systems of vibration protection I M.Z. Kolovsky. Transl. by A.K. Belyeav. (Foundations of engineering mechanics) ISBN 978-3-662-22236-2 ISBN 978-3-540-49143-9 (eBook) DOI 10.1007/978-3-540-49143-9 This work is subject to copyright. All rights are reserved, whether the whole or part oft he material is concerned, specificallythe rights oftranslation, reprinting, reuseofillustrations, recitation, broadcasting, reproduction on microfilm or in other ways, and storage in data banks. Duplication ofthis publication or parts thereofis 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. Berlin Heidelberg GmbH. Violations are liable for prosecution act under German Copyright Law. © Springer-Verlag Berlin Heidelberg 1999 Originally published by Springer-Verlag Berlin Heidelberg New York in 1999 Softcover reprint of the hardcover 1st edition 1999 The use of general descriptive names, registered names, trademarks, 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. Typesetting: Camera-ready copy from translator Cover-Design: de'blik, Berlin SPIN 10705979 6213020 54 3 2 1 0 Printed on acid-free paper Preface With progress in technology, the problem of protecting human-beings, ma chines and technological processes from !>Ources of vibration and impact has become of utmost importance. Traditional "classical" methods of pro tection, based upon utilising elastic passive and dissipative elements, turn out to be inefficient in many situations and can not completely satisfy the complex and often contradictory claims imposed on modern vibration protection systems which must provide high performance at minimum di mensions. For these reasons, active vibration protection systems, which are actually systems of automatic control with independent power sources, are widely used nowadays. Appearing and developing active systems require that traditional ap proaches to the analysis and synthesis of vibration protection systems must be revised. Firstly, there exists the necessity to re-state the problem of vi bration protection from mechanical actions as an equivalent problem in closed-loop control systems design, which is to be solved by the methods of control theory. Furthermore, it turns out that certain inherent proper ties of active systems must be taken into account for a proper design. In the majority of cases, the dynamic models of the objects to be protected and the bases to which these objects are to be attached must be revised. They are no longer considered as rigid bodies but elastic bodies with weak dissipation. The dynamic modelling of vibration protection systems is also compli cated by the necessity to include nonlinearity. The classical linear theory of elastic suspension, implying that each isolator possesses linear elastic and dissipative properties, is limited due to dimensional constraints. Any 6 Preface actual isolator can have a linear characteristic only for a certain range of deflections, the latter being referred to as the region of linearity. Any actual vibration protection system has restrainers limiting the maximum admis sible geometric size of the linearity region. It is clear that the linear theory can be applied only for those excitations for which the isolators'defl ections remain within the linearity region. However, it is not always feasible to meet this condition. Dimensional constraints lead to a series of phenomena of apparent non linear character which do not manifest themselves in the framework of a linear dynamic model. In particular, the assumption that the steady-state forced vibration of a system under harmonic or polyharmonic excitation is independent of initial conditions is no longer valid. In turn this means that the uniqueness of the steady-state forced vibration is no longer ensured. Any vibration protection system does not function properly when its vi bration exceeds the linearity region. The very possibility of the occurrence of such vibrations must be excluded for a properly designed system. Analy sis shows that this can be achieved by increasing the dissipative forces. How ever, the performance of vibration protection systems deteriorates with the growth of damping. This gives rise to the problem of seeking the optimal damping, and one of the most important tasks of the theory of vibration protection is to find the solution. The above problems came into being in the early sixties because of the need to design facilities aimed at protecting items of equipment mounted aboard aircraft, spacecraft, vessels and other moving objects from vibration and impact excitations. The results of these theoretical and experimen tal investigations were covered in two monographs by the present author, namely M.Z. Kolovsky "Nonlinear theory of vibration protection systems", Nauka, Moscow, 1966 and M.Z. Kolovsky "Automatic control of vibration protection systems", Nauka, Moscow, 1976. The contents of these books is the basis for the present book which is offered to the reader in Eng lish. The author anticipates that the statements contained in the present book this will be useful for a wide circle of readers whose scientific interests are related to the problems of vibration theory and control theory and his practical recommendations will be used by engineers developing modern vibration protection systems. The present book consists of six Chapters. The first two Chapters are concerned with the theory of linear active and passive systems. Some ap proximate methods of analysis for nonlinear systems are outlined in Section 1 of Chapter 3. Here, along with the widely used methods of harmonic bal ance, Galerkin's method, the method of linearisation with respect to the distribution function is developed. Application of this method is especially efficient for analysis of polyharmonic processes. The method of statistical linearisation is also explained. The rest of Chapter 3, as well as Chapters 4 and 5, deal with the analysis of nonlinear active and passive systems of Preface 7 vibration protection. Chapter 6 is devoted to the optimisation of vibration protection systems. At the time when the monographs in Russian were first prepared and published, the author was supported by his teachers A.l. Lurie and LB. Barger, the bright images of whom have been held in remembrance through out the author's life. The author is also deeply grateful to A.A. Pervozvan sky for his valuable advice and to V.I. Babitsky who edited both of the original monographs and contributed much to their improvement. Compo sition of the present book involved the tasks of combining the above two monographs and translating them from Russian into English. Both of these turned out to be challenging problems. A great deal of help in these tasks was given by A.K. Belyaev, to whom the author is very thankful. Translator's Preface Firstly, I would like to thank the author for giving me the opportunity to translate into English the two monographs of his which result in the present book. From a professional perspective, this was a very interesting and cognitive experience. Secondly, I am thankful to Professor Hans Irschik, from the Johannes Kepler University of Linz, Austria. The translation was carried out during my stay at his Institute, throughout which he provided invaluable advice and contributed greatly to the improvement of the manuscript. Thirdly, I appreciate the assistance of my son, Nikita Belyaev, from the Technical University of St. Petersburg, Russia for the considerable technical support he gave during the translation. And finally, I would like to express my sincere gratitude to Dr. Stewart McWilliam, from the University of Nottingham, UK who took the trouble of editing the manuscript which I translated into English. I am greatly obliged to him not only for his thorough correction of the galley-proofs but also for his many useful and profound suggestions on the manuscript. Contents Preface 5 Translator's preface 8 1 Dynamic characteristics and efficiency of vibration protec- tion systems 13 1.1 Statement of the problem of protection from dynamic exci- tations . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.1.1 One-dimensional dynamic vibration absorber 15 1.1.2 Multidimensional dynamic vibration absorber 17 1.1.3 Uniaxial vibration isolator . . . . . . . . . . . 17 1.2 Dynamic characteristics of the vibration-isolated system and the base . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 1.2.1 Example 1: The dynamic compliance matrix for a free rigid body . . . . . . . . . . . . . . . . . . . . . 35 1.2.2 Example 2. The dynamic compliance of a two degree of-freedom system . . . . . . . . . . . . . . . . . . . 36 1.3 Efficiency conditions in the case of harmonic excitation . . . 39 1.4 Efficiency in case of polyh armonic, random and non-stationary excitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 2 Linear active systems 61 2.1 Structure and elements of active systems . 61 2 .1.1 Transfer functions of active systems 61 10 Contents 2 ol. 2 Sensors 0 0 0 0 0 0 0 0 0 0 0 0 64 201.3 Compensators and amplifiers 67 201.4 Actuators 0 0 0 0 0 0 0 0 0 0 0 70 202 One-dimensional linear active systems 81 203 Conditions for stability of active systems 0 108 2.4 Systems with several measurement points 122 205 Transient processes in active systems and protection from impacts 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 129 206 Work and power in active systems 147 207 Multidimensional systems 0 0 0 0 0 156 3 Nonlinear passive single-degree-of-freedom systems 167 301 Methods of analysis for nonlinear system 0 0 0 0 0 0 0 0 167 302 Forced vibration under harmonic excitation and linear damp- ing 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 183 303 Forced vibrations under harmonic excitation and Coulomb friction 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 199 304 Forced vibration in a system with internal friction 0 208 305 Comparative study of the various types of damping 0 212 306 Forced vibration under polyharmonic excitation 0 0 0 218 30601 System with nonlinear elastic force and linear damping219 30602 System with linear elastic force and dry friction 0 0 0 224 30603 General solution for the problem of polyharmonic driving 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 227 30 7 Subharmonic resonances in vibration protection systems 230 308 Subharmonic resonance in systems with rigid stops 242 309 Forced random vibrations 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 249 4 Nonlinear passive multi-degree-of-freedom systems 257 401 Static analysis of nonlinear elastic suspensions 0 0 0 0 257 402 Small vibration of elastically suspended rigid body 0 263 403 Vibration of an object mounted on nonlinear elastic isolators 272 4o4 Free vibration of a nonlinear vibration protection system 0 282 405 Resonant vibration 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 293 406 Forced vibration in systems with Coulomb friction 300 40 7 Forced vibration of elastic bodies 0 0 0 0 0 0 0 0 0 0 306 5 Nonlinear active systems 319 501 Resonant vibrations in nonlinear systems under harmonic excitation 0 0 0 0 0 0 0 0 0 0 0 0 0 319 502 Subharmonic vibrations 0 0 0 0 0 0 0 0 0 0 332 503 Influence of nonlinearities in feedback 342 5.4 Stability of vibration in nonlinear systems 354 6 Optimal systems of vibration protection 365
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