Bifurcations in Discontinuous Mechanical Systems of Filippov-Type pptx

Filippov systems form a subclass of discon-tinuous dynamical systems which can be described by a set of rst-order ordinary di erential equations with a discontinuous right-hand side.. Th

Bifurcations in Discontinuous Mechanical Systems of Filippov-Type

From: Codex Madrid I (8937), original title: `Tratado de Estatica y Mechanica enItaliano', (1493).

CIP-DATA LIBRARY TECHNISCHE UNIVERSITEIT EINDHOVEN

Leine, Remco I

Bifurcations in discontinuous mechanical systems of Filippov-type/

by Remco Ingmar Leine

-Eindhoven: Technische Universiteit Eindhoven, 2000

discontin-Printed by the Universiteitsdrukkerij TU Eindhoven, The Netherlands

Cover design by Ben Mobach

Typeset by the author with LATEX2"

Copyright c 2000 by R I Leine

All rights reserved No parts of this publication may be reproduced or utilized inany form or by any means, electronic or mechanical, including photocopying, record-ing or by any information storage and retrieval system, without permission of thecopyright holder

This research was nanced by the Dutch Technology Foundation (STW)

Bifurcations in Discontinuous

Mechanical Systems of Filippov-Type

PROEFSCHRIFT

ter verkrijging van de graad van doctor

aan de Technische Universiteit Eindhoven,

op gezag van de Rector Magni cus, prof.dr M Rem, voor een commissie aangewezen door het College voor Promoties

in het openbaar te verdedigen op donderdag 15 juni 2000 om 16.00 uur

door Remco Ingmar Leine geboren te Bleiswijk

prof.dr.ir D.H van Campen

en

prof.dr H Nijmeijer

Copromotor:

dr.ir A de Kraker

`L'essentiel est invisible pour les

yeux'Antoine de Saint-Exupery,

Le Petit Prince

Voor Ilse

1.1 Motivation 1

1.2 Nonlinear Dynamics and Bifurcations 2

1.3 Discontinuous Systems 3

1.4 Literature Survey 5

1.5 Objective and Scope of the Thesis 7

1.6 Outline of the Thesis 7

2 Filippov Theory 9 2.1 The Construction of a Solution 9

2.2 Numerical Approximation 18

2.3 Dry Friction Models and Dierential Inclusions 19

2.4 Example: The Stick-slip System 22

3 Fundamental Solution Matrix 27 3.1 Fundamental Solution Matrix for Smooth Systems 27

3.2 Jumping Conditions: A Single Discontinuity 28

3.3 Construction of Saltation Matrices 31

3.4 Example I: The Stick-slip System 34

3.5 Example II: The Discontinuous Support 37

4 Non-smooth Analysis of Filippov Systems 41 4.1 Linear Approximations at the Discontinuity 41

4.2 Generalized Dierentials 47

5 Bifurcations of Fixed Points 49 5.1 Smooth Systems 50

5.2 Discontinuous Bifurcation: The Basic Idea 54

5.3 Saddle-node Bifurcation 56

5.4 Transcritical Bifurcation 58

5.5 Pitchfork Bifurcation 61

5.6 Hopf Bifurcation 64

5.7 Hopf{Pitchfork Bifurcation 66

5.8 Discussion and Conclusions 69

6 Bifurcations of Periodic Solutions 73 6.1 Discontinuous Systems 73

6.2 Bifurcations in Smooth Systems 74

6.3 Discontinuous Bifurcation: The Basic Idea 75

6.4 The Poincare Map 77

6.5 Intersection of Hyper-surfaces of Discontinuity 80

6.6 Fold Bifurcation; Trilinear Spring System 84

6.7 In nitely Unstable Periodic Solutions 89

6.8 Symmetry-Breaking Bifurcation; Forced Vibration with Dry Friction 92 6.9 Flip Bifurcation; Forced Stick-slip System 95

6.10 Discussion and Conclusions 100

7 Concluding Remarks 103 7.1 Overview and Summary of Contributions 103

7.2 Recommendations 105

A Application to a Simple Model of Drillstring Dynamics 109 A.1 Motivation 109

A.2 Principles of Oilwell Drilling 110

A.3 Downhole Measurements 112

A.4 Modeling of Stick-slip Whirl Interaction 116

A.5 Fluid Forces 116

A.6 Contact Forces 118

A.7 Whirl Model 119

A.8 Stick-slip Model 125

A.9 Stick-slip Whirl Model 126

A.10 Discussion and Conclusions 131

A.11 Coordinate Systems 132

B Some Theoretical Aspects of Periodic Solutions 133 B.1 Transformation to an Autonomous System 133

B.2 The Variational Equation 134

Physical phenomena such as dry friction, impact and backlash in mechanical systems

or diode elements in electrical circuits are often studied by means of mathematicalmodels with some kind of discontinuity Filippov systems form a subclass of discon-tinuous dynamical systems which can be described by a set of rst-order ordinary

dierential equations with a discontinuous right-hand side

Mechanical systems with dry friction constitute an important example of pov systems The presence of dry friction-induced self-sustained vibrations can behighly detrimental to the performance of mechanical systems Many other practicalous characteristics of physical phenomena It is therefore desirable to know whetherperiodic solutions of a system with dry friction (or Filippov systems in general) existfor a certain parameter set and how these periodic solutions change for a varyingparameter of the system Such parameter studies are usually conducted by means ofpath-following techniques where a branch of xed points or periodic solutions is fol-lowed while varying a parameter A branch of xed points or periodic solutions canfold or can split into other branches at critical parameter values This qualitativechange in the structural behaviour of the system is called `bifurcation' Bifurcations

Filip-in smooth systems are well understood but little is known about bifurcations Filip-indiscontinuous systems

The objective of the thesis is to investigate dierent aspects of bifurcations of xed points in non-smooth continuous systems and of periodic solutions in Filippovsystems Filippov systems expose non-conventional bifurcations called `discontin-uous bifurcations', being dierent from the conventional bifurcations occurring insmooth systems In this thesis, Filippov theory, generalized derivatives and Floquettheory are combined, which leads to new insight into bifurcations in discontinuoussystems

First, Filippov's solution concept for dierential equations with discontinuousright-hand side is reviewed Systems with dry friction require special attention Dif-ferential equations with discontinuous right-hand side are extended to dierentialinclusions with Filippov's convex method Existence of solutions to dierential in-clusions is guaranteed under additional conditions but no uniqueness of solutions isguaranteed Non-uniqueness plays an important role in the bifurcation behaviour ofFilippov systems

The local stability of a periodic solution is governed (for the hyperbolic case) bythe fundamental solution matrix The fundamental solution matrix is also essential

for the understanding of bifurcations of periodic solutions Discontinuities of thevector eld in Filippov systems cause jumps in the fundamental solution matrix.The discontinuous behaviour of the fundamental solution matrix is explained andelucidated in mechanical examples.

The jumps in the fundamental solution matrix can be analyzed by the linearapproximation method which approximates a discontinuous system by a non-smoothcontinuous system The linear approximation method replaces a discontinuity in thevector eld by a boundary layer with a vector ... objective of the thesis is to investigate dierent aspects of bifurcations of

1 xed points in non-smooth continuous systems

2 periodic solutions in discontinuous systems of Filippov-type. .. of periodicsolutions in Filippov systems

bifur -Bifurcations of xed points of non-smooth continuous systems are treated and it

is shown that discontinuous bifurcations of xed points... related to bifurcations of xed points in non-smooth continu-ous systems (discontinuous Jacobian) We will therefore also address bifurcations of xed points in non-smooth continuous systems

The