MATHEMATICAL MODELLING OF COMPLEXMECHANICAL SYSTEMSVolume 1: Discrete Models

MATHEMATICS AND ITS APPLICATIONSSeries Editor: G. M. BELLEmeritus Professor of Mathematics, King's College London, University of London

STATISTICS, OPERATIONAL RESEARCHAND COMPUTATIONAL MATHEMATICS SectionEditor: B. W. CONOLLY,Emeritus Professor of Mathematics (Operational Research), Queen Mary College, University ofLondon

Mathematics and its applications are now awe-inspiring in their scope, variety and depth. Not only is there rapid growth in puremathematics and its applications to the traditional fields of the physical sciences, engineering and statistics, but new fields ofapplication are emerging in biology, ecology and social organization. The user of mathematics must assimilate subtle newtechniques and also learn to handle the great power of the computer efficiently and economically.

The need for clear, concise and authoritative texts is thus greater than ever and our series endeavours to supply this need. It aimsto be comprehensive and yet flexible. Works surveying recent research will introduce new areas and up-to-date mathematicalmethods. Undergraduate texts on established topics will stimulate student interest by including applications relevant at thepresent day. The series will also include selected volumes of lecture notes which will enable certain important topics to bepresented earlier than would otherwise be possible.

In all these ways it is hoped to render a valuable service to those who learn, teach, develop and use mathematics.

Mathematics and its ApplicationsSeries Editor: G. M. BELLProfessor of Mathematics, King's College London, University of London

Anderson, I. COMBINATORIAL DESIGNS: Construction MethodsArtmann, B. CONCEPT OF NUMBER: From Quaternions to Monads and Topologies] FieldsArczewski, K. & Pietrucha, J. MODELLING OF COMPLEX MECHANICAL SYSTEMS:

Volume 1: Discrete ModelsArczewski, K. & Pietrucha, J. MATHEMATICAL MODELLING OF COMPLEX MECHANICAL SYSTEMS:

Volume 2: Continuous ModelsBainov, D.D. & K. Covachev THE AVERAGING METHOD AND ITS APPLICATIONSBainov, D.D. & Simeonov, P.S. SYSTEMS WITH IMPULSE EFFECT: Stability, Theory and ApplicationsBaker, A.C. & Porteous, H.L. LINEAR ALGEBRA AND DIFFERENTIAL EQUATIONSBalcerzyk, S. & Josefiak, T. COMMUTATIVE RINGSBalcerzyk, S. & Josefiak, T. COMMUTATIVE NOETHERIAN AND KRULL RINGSBaldock, G.R. & Bridgeman, T. MATHEMATICAL THEORY OF WAVE MOTIONBall, M.A. MATHEMATICS IN THE SOCIAL AND LIFE SCIENCES: Theories, Models and MethodsBarnett, S. SOME MODERN APPLICATIONS OF MATHEMATICSBartak, J., Herrmann, L., Lovicar, V. & Vejvoda, D. PARTIAL DIFFERENTIAL EQUATIONS OF EVOLUTIONBejancu, A. FINSLER GEOMETRY AND APPLICATIONSBell, G.M. & Lavis, D.A. STATISTICAL MECHANICS OF LATTICE MODELS, Vols. 1 & 2Berry, J.S., Burghes, D.N., Huntley, I.D., James, D.J.G. & Moscardini, A.O. MATHEMATICAL MODELLING COURSESBerry, J.S., Burghes, D.N., Huntley, I.D., James, D.J.G. & Moscardini, A.O. MATHEMATICAL MODELLING

METHODOLOGY, MODELS AND MICROSBerry, J.S., Burghes, D.N., Huntley, I.D., James, DJ.G. & Moscardini, A.O. TEACHING AND APPLYING

MATHEMATICAL MODELLINGBrown, R. TOPOLOGY: A Geometric Account of General Topology, Homotopy Types and the Fundamental GroupoidBurghes, D.N. &Borrie,M. MODELLING WITH DIFFERENTIAL EQUATIONSBurghes, D.N. & Downs, A.M. MODERN INTRODUCTION TO CLASSICAL MECHANICS AND CONTROLBurghes, D.N. & Graham, A. INTRODUCTION TO CONTROL THEORY, INCLUDING OPTIMAL CONTROLBurghes, D.N. & Wood, A.D. MATHEMATICAL MODELS IN THE SOCIAL, MANAGEMENT AND

LIFE SCIENCESButkovskiy, A.G. GREEN'S FUNCTIONS AND TRANSFER FUNCTIONS HANDBOOKCartwright, M. FOURIER METHODS: for Mathematicians, Scientists and EngineersCerny.I. COMPLEX DOMAIN ANALYSISChorlton, F. VECTOR AND TENSOR METHODSCohen, D.E. COMPUTABIUTY AND LOGICCordier, J.-M. & Porter, T. SHAPE THEORY: Categorical Methods of ApproximationCrapper, G.D. INTRODUCTION TO WATER WAVESCross, M. & Moscardini, A.O. LEARNING THE ART OF MATHEMATICAL MODELLINGCullen, M.R. LINEAR MODELS IN BIOLOGYDunning-Davies, J. MATHEMATICAL METHODS FOR MATHEMATICIANS, PHYSICAL SCIENTISTS AND

ENGINEERSEason, G., Coles, C.W. & Gettinby, G. MATHEMATICS AND STATISTICS FOR THE BIOSCIENCESHJai, A. & Pritchard, A.J. SENSORS AND CONTROLS IN THE ANALYSIS OF DISTRIBUTED SYSTEMSExton, H. MULTIPLE HYPERGEOMETRIC FUNCTIONS AND APPLICATIONSExton, H. HANDBOOK OF HYPERGEOMETRIC INTEGRALS

series continued at back of book

MATHEMATICALMODELLING OF COMPLEXMECHANICAL SYSTEMSVolume 1: Discrete Models

Dr K. ARCZEWSKIInstitute of Aeronautics and Applied Mechanics,Warsaw University of TechnologyandDr J. PIETRUCHAInstitute of Aeronautics and Applied Mechanics,Warsaw University of TechnologyTranslation Editor:Dr C. M. LEECHDepartment of Mechanical Engineering,Applied Mechanics Division, UMIST

ELLIS HORWOODNEW YORK LONDON TORONTO SYDNEY TOKYO SINGAPORE

First published in 1993 byELLIS HORWOOD LIMITEDMarket Cross House, Cooper Street,Chichester, West Sussex, PO19 1EB, England

A division ofSimon & Schuster International GroupA Paramount Communications Company

© Ellis Horwood Limited, 1993

All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, ortransmitted, in any form, or by any means, electronic, mechanical, photocopying, recording orotherwise, without the prior permission, in writing, of the publisher

Printed and bound in Great Britainby Bookcraft, Midsomer Norton

British Library Cataloguing in Publication Data

A catalogue record for this book is available from the British Library

ISBN 0-13-563750-3

Library of Congress Cataloging-in-Publication Data

Available from the publisher

To our teachers, from whose wisdom we drew, and to our students, to whom wewant to transmit our experience

Wisdom begins when you sincerely want to learn. To desire Wisdom is to love herWisdom of Solomon 6:17

Table of contentsPreface 10

1. Basic notions of modelling 151.1 Model, object, phenomenon 151.2 Investigation of phenomena by means of models 18

1.2.1 Physical modelling 191.2.2 Mathematical modelling 211.2.3 Identification as a method of model formulation 221.2.4 Investigation of the mathematical model 231.2.5 Verification of solutions 231.2.6 Investigation of phenomena—summary 24

1.3 An empirical and a causal model 251.4 The influence of purpose of modelling on the final form of the model 281.5 Discrete and continuous models 341.6 Stochastic versus deterministic models 401.7 Models related to the differential model 45

2 The framework for modelling 522.1 Relationships between mechanics and technology 522.2 The fundamental notions of classical mechanics 53

2.2.1 The mechanical system 532.2.2 Constraints and their classification 542.2.3 Generalized coordinates 582.2.4 The number of degrees of freedom 652.2.5 Representations of the motion in space 672.2.6 Quasi-coordinates 70

2.3 Selected notions of integrated mechanics 742.3.1 A physical system 742.3.2 Generalized constraints 74233 Physical variables .' 752.3.4 The number of describing functions 77

8 Table of contents

2.3.5 Eulerian and Langrian description of the continuum motion 792.3.6 The state space 83

3. Modelling by means of balance laws 873.1 Conservation laws versus balance laws 873.2 Two introductory examples 923.3 Methodology of modelling by means of balance laws 95

3.3.1 The tetrahedron—a mnemonic aid in the modelling process 953.3.2 Basic laws of mechanics 963.3.3 Body 1003.3.4 Force 1063.3.5 Motion 1083.3.6 Specific laws of mechanics 1153.3.7 Characteristics of bodies in motion 1163.3.8 Final hints 117

3.4 Applications 1183.4.1 Parachute with a payload 1183.4.2 Elevator hydraulic amplifier 1213.4.3 The ablation at the nose of the re-entry vehicle 128

4. Modelling using variational principles 1354.1 From Newtonian to variational mechanics 135

4.1.1 Why variational principles? 1354.1.2 Postulating or deducing the principles? 1364.1.3 Verbal formulation of Hamilton's principle 140

4.2 Basic variational principles 1454.2.1 Types of principles 1454.2.2 Fundamental concepts 1464.2.3 Differential variational principles 1614.2.4 Integral variational principles 170

4.3 Modelling of holonomic systems 1754.3.1 Lagrange equations of the second kind 1754.3.2 The Boltzmann-Hamel equations 1774.3.3 The Lagrange-Maxwell equations 1804.3.4 Case studies 183

4.4 Modelling of nonholonomic systems 1954.4.1 Introductory remarks 1954.4.2 Lagrange equations of the first kind with multipliers 1974.4.3 Maggi equations 1994.4.4 The Gibbs-Appell equations ] 2014.4.5 Case studies 204

5. Modelling by means of graphs 2215.1 Basic notions and concepts of graph theory 221

5.1.1 What is a graph? 221

Table of contents 9

5.1.2 Different kinds of graph 2225.1.3 Matrix representation of a directed graph 228

5.2 A brief history of graph theory 2305.3 The linear graph modelling method 235

5.3.1 System, components and terminals 2355.3.2 Terminal representation 2375.3.3 A system graph 2425.3.4 Formulating techniques 2445.3.5 The limits of a method 249

5.4 Modelling of rigid-body systems 2515.4.1 Introductory remarks 2515.4.2 The key idea 2585.4.3 Basic notation conventions 2605.4.4 The mathematical description of the interconnection structure 2625.4.5 The kinetic energy 2675.4.6 The potential energy of gravity forces 2725.4.7 The equations of motion 2745.4.8 Example 2785.4.9 Concluding remarks 284

Postscript 285

References 288

Index 290

Preface

This book has arisen from efforts to formulate a rational methodology for modellingcomplex mechanical systems. A stimulus for these efforts was provided by two observa-tions made during more than twenty years of our didactic and scientific work in thePower and Aeronautical Faculty of the Warsaw University of Technology. The first ofthese two observations concerns the results of the traditional teaching process that istypical of many technical universities. In this process a student of a mechanical facultygains knowledge within the confines of numerous separate courses, such as mechanicalengineering, thermodynamics, theory of elasticity, fluid mechanics and theory of vibra-tions, and then of even more specific courses such as flight mechanics, heat transmission,magnetohydrodynamics or aeroelasticity. Knowledge gained in this manner, althoughquite broad, often lacked adequate integrity. This means that laws that have commonroots, or are simply different forms of the same law (e.g. rate of change of linear momen-tum, or conservation of energy) were perceived by students as rather different, andhaving not too much in common. A student educated in this manner was usually helplesswhen confronted with the problem of creating a model of a complex phenomenon, inwhich one had to apply knowledge exceeding the domain of just one narrow discipline.Similarly, much difficulty was caused by questions of the type:

— Why does the continuity equation, so commonly used in the mechanics of fluids, seemto be completely ignored in the theory of elasticity?

— What do Kepler's second law, which states that the areal velocity of each planet isconstant, and the postulate of symmetry for a stress tensor in continuous media have incommon?

— In the first course of mechanics two basic laws of particle motion, i.e. the oneconcerning the time rate of change of angular momentum and that of work and kineticenergy, are usually derived by means of a simple transformation of the law of timerate of change of linear momentum (Newton's second law), and this strongly suggestsmutual dependence between these three laws. At the same time these three laws aretreated as apparently independent when the mathematical models of heat conductingfluids are formed. The question arises: are the laws of energy conservation, linear andangular momentum change interdependent or not?

Preface 11

Should one, though, be surprised that such questions cause difficulties for students,who are just acquiring knowledge, when many textbooks refer to the symmetry of thestress tensor either as an axiom, a law, or a property of the medium, and do not show thatthis symmetry is a consequence of the angular momentum balance law and of the as-sumption that a given medium cannot possess an internal angular momentum (spin)? If,on the other hand, a given medium does not satisfy this assumption, i.e. it can store spin,the stress tensor can be arbitrary and the corresponding medium is called a micropolarmedium, or alternatively a Cosserat continuum. Moreover, in the literature—and this isthe second observation—models are often presented in their final, neat and elegant form.In reality there are many steps, choices, doubts and even errors before the modellerreaches a satisfactory model. These features are usually hidden, and hence the beginnerfails to see them. Due to this fact the usefulness of results presented in some scientificpapers is either very limited, or not evident to the reader. These two observations formedthe motive for creating a facultative course, usually taken at the beginning of the secondhalf of studies, entitled Physical System Modelling—Theory and Practice, whose twomain but equivalent objectives are: (1) Integration and ordering of knowledge gained todate through, for example, noting common roots of methods developed in particulardomains of mechanics; (2) provision of a rational methodology for creating mathematicalmodels for various levels of complex physical phenomena, e.g. of the thermo-aero-elasticvibrations of a turbine blade. The objectives of both volumes of the present book areexactly the same. These objectives have significantly influenced the shape and contentsof the book, and will in turn greatly influence the potential readership.

The book is addressed to students in the senior years of technical universities, toengineers and applied scientists. It is also addressed to teachers all over the world whoare involved in teaching mathematical modelling. It is assumed that the reader has studiedmathematics through a first course in ordinary differential equations and has passedexaminations in engineering mechanics and strength of materials. Although it is notabsolutely necessary, a reader would find it helpful to have studied fluid mechanics,classical elasticity and thermodynamics. As this implies, although we use multiple no-tions, information and results coming from the domains of the disciplines mentionedabove, we do not explain or derive them. Nevertheless, some of them are accompanied bycomments explaining their utility in the process of modelling.

The book is structured into two volumes. The first volume is concerned with discretemodels and the second with continuous ones. The present volume is composed of fivechapters, the first two of which are of preparatory character.

In the first chapter we explain what modelling is. There we define the notions funda-mental to modelling, present modelling against the background of other activities inengineering, describe subsequent stages of modelling and indicate to what extent thepurpose of modelling influences the shape of a model. We have also tried to show thateach step in the process of modelling requires a deep understanding of a variety ofconcepts and specific approaches blended with a combination of experience, intuition andforesight. This makes model building both a science and an art.

The art aspect of modelling is difficult to teach in any formal sense. Thus, we havestriven to transmit our experience to the students through numerous comments accompa-nying the notions, concepts and laws presented, as well as through detailed presentation

12 Preface

of a wide scope of case studies in which subsequent stages of model creation have beenhighlighted.

A special difficulty in the teaching of the science aspects is the breadth and depth ofmaterial needed for model building. The approach we have taken in Chapter 2 and, tosome extent, in subsequent chapters is as follows. We indicate common roots, similaritiesand a common basis, in the form of balance laws, for a wide range of phenomena actingas subjects for modelling, then we identify five sets of topics whose elements appear ineach causal model of a complex phenomenon.

The intention behind Chapter 2 was to create a basis of notions for the purpose offormulating both discrete and continuous models for a broad class of mechanical phe-nomena. In Chapter 2 we recall several notions of classical mechanics, adding to themcomprehensive comments concerning their use in model development. A present-dayengineer has to cope more and more often with situations in which notions and methodsof classical mechanics are insufficient. Thus, having in mind the needs of modelling ofphenomena exceeding the scope of narrowly understood mechanics, we have provided asection in which we define the notions of so-called 'integrated mechanics'.

It appears that in all the domains of classical physics there exist two main differentapproaches to the formulation of equations describing the dynamic behaviour of a givensystem. The first approach is based on the use of balance laws, while the second refers toso-called variational principles. And so it is in mechanics. The first approach was estab-lished by Sir Isaac Newton in his famous Philosophiae Naturalis Principia Mathematica(1687), and then developed by Leonard Euler. This approach is alternatively called geo-metrical or Newtonian.

The variational approach is intimately linked with the names of Johann Bernoulli, Jeanle Rond d'Alembert, Joseph Louis Lagrange and Sir William Rowan Hamilton. It wasLagrange who summarized the subsequent epoch of important discoveries, forming anentirely different (to that founded by Newton) variational approach, in his monumentalMecanique Analitique (1788), and its subject matter is called today either Lagrangian oranalytical mechanics.

The division sketched out here into two methods of generating equations of motions iscompletely reflected in the book. Thus, Chapter 3 is devoted to modelling by means ofbalance laws, and Chapter 4 presents the variational approach to the modellingproblem.

In cases when a system becomes complex, due to a large number of elements and adifferentiated structure of connections between them (in other words the system has anintricate topology) a graph reflecting the structure becomes an extremely valuable tool ofmodelling. Methods which make use of graphs (irrespective of the fact that they can alsorefer to balance or variational methods for the development of mathematical model ofsystem components) are called graph-theoretical methods and two of them are presentedin Chapter 5.

Numerous examples are provided in the book, mostly from the field of aircraft engi-neering, but many examples may be seen to be directly relevant to civil and mechanicalengineering. The examples provided, however, are of distinctly different levels of diffi-culty. Simple examples are used only for the purpose of illustrating the correspondingtheoretical considerations. Problems are also considered, though, taken from a wide

Preface 13

variety of contemporary engineering situations. Some of them were the subjects of scien-tific publications, and the others might be. The range of topics appearing as examples inthe book is drawn from many university-level disciplines and it is also an objective ofthis book to demonstrate that the principles of mechanics and the methods of modellingare remarkably universal for such a range of topics, in spite of the impression often madewhen the topics are studied in separate courses.

In spite of its strong connections with mechanics the present book cannot be regardedto be a manual of either engineering mechanics or of any discipline of mechanics. Thebook, though, undoubtedly recapitulates and integrates knowledge from several disci-plines of mechanics, organizing it around the fundamental subject of modelling. Attentivetracking of the subjects treated in this book would certainly contribute to a deeper under-standing of many specific disciplines of mechanics and to a broadening of the field ofapplication of knowledge already acquired.

It is a pleasure to acknowledge the help we have received from different individualsand institutions. We are particularly indebted to Professor R. Gutowski who first intro-duced us to the methods of analytical mechanics, and to J. T. Szuster who first drew ourattention to the important problem of modelling. This work has benefited greatly fromdiscussions with our colleagues Dr F. Dul, Dr Z. Goraj and W. Michalski, to each ofwhom we express our thanks.

The authors wish to give special recognition to Dr C. M. Leech of UMIST for hisdetailed review of the manuscript and for his numerous helpful suggestions.

The final stage of manuscript preparation work was supported by grant No.3 0955 91 01. The authors wish to acknowledge the State Committee for Scientific Re-search which offered this grant.

We would like also to express our gratitude to the editorial staff of Ellis Horwood fortheir invaluable assistance in the finalization of this project.

Both authors also wish to acknowledge the encouragement, patience, and assistance oftheir wives during the preparation of this book.

Warsaw, 11 November 1990 Krzysztof P. ArczewskiJ6zef A. Pietrucha