Lecture Notes in Applied Mechanics Volume 1 Series Editor Prof. Dr.-Ing. Friedrich Pfeiffer Springer Berlin Heidelberg New York Barcelona Hong Kong London Milan Paris Singapore Tokyo Set-Valued Force Laws Dynamics of Non-Smooth Systems Christoph Glocker Springer Professor Dr.-Ing. CHRISTOPH GLOCKER Institute of Mechanical Systems Dept. of Mechanical and Process Engineering ETR Zentrum - CLA J2l.l CR -8092 Zurich SWITZERLAND email: [email protected] With 65 Figures ISBN 978-3-642-53595-6 Cataloging-in-Publication Data applied for Glocker, Christoph: Set valued force laws: dynamics of non-smooth systems I Christoph Glocker. - Berlin; Heidelberg; New York; Barcelona; Hong Kong; London; Milan; Paris; Singapore; Tokyo: Springer, 2001 (Lecture notes in applied mechanics; Vol. 1) ISBN 978-3-642-53595-6 ISBN 978-3-540-44479-4 (eBook) DOI 10.1007/978-3-540-44479-4 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 http://www.springer.de © Springer-Verlag Berlin Heidelberg 2001 Softcover reprint of the hardcover 1st edition 2001 The use of general descriptive names, registered names, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protec­ tive laws and regulations and free for general use. The publisher cannot assume any legal responsibility for given data, especially as far as direc­ tions for the use and the handling of chemicals and biological material are concerned. This information can be obtained from the instructions on safe laboratory practice and from the manufacturers of chemicals and laboratory equipment. Cover design: Design & Production GmbH, Heidelberg Typesetting: Digital data supplied by author SPIN: 10792269 62/3020xv-5 4 3 2 1 0 - printed on acid-free paper Preface As one of the oldest natural sciences, mechanics occupies a certain pioneering role in determining the development of exact sciences through its interaction with mathematics. As a matter of fact, there is hardly an area in mathematics that hasn't found an application of some form in mechanics. It is thus almost inevitable that theoretical methods in mechanics are highly developed and laid out on different levels of abstraction. With the spread of digital processors this goes as far as the implementation in commercial computer codes, where the user is merely con­ fronted on the surface with the processes that run in the background, i.e. mechan­ ics as such: in teaching and research, as well as in the context of industry, me­ chanics is much more, and must remain much more than the mere production of data with the help of a processor. Mechanics, as it is talked about here, tradition­ ally includes a wide spectrum, ranging from applied mechanics, analytical and technical mechanics to modeling.and experimental mechanics, as well as technical realization. It also includes the subdisciplines of rigid body mechanics, continuum mechanics, or fluid mechanics, to mention only a few. One of the fundamental and most important concepts used by nearly all natural sciences is the concept of linearization, which assumes the differentiability of mappings. As a matter of fact, all of classical mechanics is based on the avail­ ability of this quality. The most simple examples, however, such as the absolute value at the origin show that linearization may fail. The next possible structural level is convexification, where linear spaces are replaced by convex cones. Ine­ quality problems, as they occur in mechanics with dry friction or problems of contact, can be dealt with in this way, while they have to be excluded in classical mechanics. In good textbooks on the dynamics of rigid bodies one can thus find a general treatment of multibody systems which enables the student to deal with complex mechanical systems. Yet questions of dry friction and unilateral contact, along with the problem of impacts are merely addressed in specialized, simple ex­ amples. The reason for this lies in the fact that the mathematical methods for the description and treatment of inequalities have only been developed during the last decades and are still in a state of flux. Classical mechanics, on the other hand, had already reached one of its primes towards the end of the 18th century. It is there­ fore not surprising that commercially available computer software only allows the treatment of inequalities in a rudimentary fashion, quasi as an appendage. A gen­ eral implementation would require deep structural changes, i.e. changes in the core of extensive computer programs. It is however foreseeable that this situation will change within the next decade. Today, the attempt has been made to use smooth functions obtained from regularization. This approximate description is unsatis- fying, both from a theoretical as well as from a physical point of view, because it does not produce the desired but merely a similar behavior, not to mention the numerical difficulties which then express themselves in the form of stiff differen­ tial equations. One thing is true in any case: We cannot think of the inclusion of ine­ qualities in mechanics as a small supplemental step. We must rather speak ofa ba­ sic extension of the classical theories, even of a completely new direction in me­ chanics, a new area of extremely fertile basic research that is developing and spreading at high speed. Used correctly, inequalities are an incredibly powerful tool. They further complete the range of what we are able to model and justice can be done to minimal models, i.e. models that portray, in a condensed form, the es­ sential dependencies on parameters in relation to a given set of questions in accor­ dance with Hertz's third requirement. In this context, we will have to rethink and reinterpret the physical content of non-uniqueness and non-existence of solutions, as well as in dynamics the lost property of continuous dependence of the trajecto­ ries on the initial conditions. Two names especially should be mentioned in connection with inequality problems in mechanics: Professor Jean Jacques Moreau from Montpellier, who likely made the most important contribution to finite freedom dynamics - and much more - in his already 1988 published CISM-course "Unilateral Contact and Dry Friction in Finite Freedom Dynamics", and Professor Panagiotis D. Panagiotopoulos, who passed away much too early. Professor Panagiotopoulos' work on nonconvex variational inequalities in the context of his 1982 theory on "hemivariational inequalities" was widely circulated in mechanics and mathemat­ ics. Almost all the ideas developed in the present treatise on "Multibody Dynamics with Unilateral Contacts", started in the extremely creative environment at his in­ stitute in Thessaloniki, where I was a guest for a year as fellow of the Feodor­ Lynen-Program supported by the Alexander von Humboldt-Stiftung. Last but not least, I would like to thank my doctoral supervisor, Proffessor Friederich Pfeiffer, who provided me with the opportunity to complete this work at the Institute B of Mechanics at the Technical University of Munich, and who over the years, with his extensive experience in the area of machine dynamics, was able to convey the necessary practical background. Munich, January 2001 Christoph Glocker Contents 1. Introduction............................... 1 1.1 Friction Laws . 1 1.2 Literature Survey. 5 1.3 Subjects and Contents. 9 2. Fundamental Concepts ................................... 17 2.1 Internal and External Forces . .. 17 2.2 The Law of Interaction. .. 19 2.3 The Dynamic Equilibrium. .. 20 2.4 The Virtual Work of a Dynamic System. .. 24 2.5 Resultant Force and Inertia Terms. .. 29 3. Rigid Body Systems . .. 33 3.1 Preliminaries on the Vector Product. .. 33 3.2 Rigid Body Kinematics. .. 34 3.3 Rigid Body Kinetics .................................... 37 3.4 The Dynamic Equilibrium of a Rigid Body ................ 39 3.5 The Virtual Work of a Rigid Body System. .. 43 3.6 Classical Bilateral Constraints ........................... 45 3.7 Generalized Coordinates. .. 51 4. Motion and Discontinuity Events . .. 55 4.1 Preliminaries on Integration of Functions. .. 55 4.2 Displacements, Velocities, and Accelerations ............... 61 4.3 Restriction to Finite Numbers of Discontinuities. .. 65 5. Displacement and Velocity Potentials.. .. .. .... .. .. .. 69 5.1 Directional Newton-Euler Equations. .. 69 5.2 Set-Valued Force Laws. .. 71 5.3 Scalar Potential Functions. .. 73 5.4 On the Modeling of Force Laws. .. 75 6. Representation of Scalar Force Laws. .. 79 6.1 Decomposition into Unilateral Primitives. .. 79 6.2 Variational Formulations and Upper Subderivatives . .. 82 6.3 The Convex Case: Conjugate Potentials and Duality . .. 84 6.4 Force Elements in Engineering Dynamics . .. 88 7. Force Laws on Different Kinematic Levels. .. .. .. .... .. 91 7.1 Continuity Properties of the Trajectories. .. 91 7.2 Displacement