Impact Mechanics

Impact Mechanics

Impact mechanics is concerned with the reaction forces that develop duringa collision and the dynamic response of structures to these reaction forces.The subject has a wide range of engineering applications, from designingsports equipment to improving the crash worthiness of automobiles.

This book develops a range of different methodologies that are used toanalyze collisions between various types of structures. These range fromrigid body theory for structures that are stiff and compact, to vibrationand wave analysis for flexible structures. The emphasis is on low-speedimpact where any damage is local to the small region of contact betweenthe colliding bodies. The analytical methods combine mechanics of contactbetween elastic-plastic or viscoplastic bodies with dynamics of structuralresponse. These methods include representations of the source of contactforces - the forces that cause sudden changes in velocity - consequentlythe analytical methods are firmly based on physical interactions and lessdependent on ad hoc assumptions than have been achieved hitherto.

Intended primarily as a text for advanced undergradute and graduatestudents, Impact Mechanics builds upon foundation courses in dynamicsand strength of materials. It includes numerous industrially relevant ex-amples and end-of-chapter homework problems drawn from industry andfrom sports such as golf, baseball, and billiards. Practicing engineers willalso find the methods presented in this book very useful in calculating theresponse of mechanical systems to impact.

Bill Stronge is a recognized expert on impact mechanics and his researchhas had a major influence on current understanding of collisions that involvefriction. He conducts research on impact response of plastically deformingsolids aimed at applications in the design of light, crashworthy structuresand energy-absorbing collision barriers. Dr. Stronge is the Reader of Ap-plied Mechanics in the Department of Engineering at the University ofCambridge and a Fellow of Jesus College.

Impact Mechanics

W. J. STRONGEUniversity of Cambridge

Of CAMBRIDGE\W UNIVERSITY PRESS

PUBLISHED BY THE PRESS SYNDICATE OF THE UNIVERSITY OF CAMBRIDGEThe Pitt Building, Trumpington Street, Cambridge, United Kingdom

CAMBRIDGE UNIVERSITY PRESSThe Edinburgh Building, Cambridge CB2 2RU, UK40 West 20th Street, New York NY 10011-4211, USA477 Williamstown Road, Port Melbourne, VIC 3207, AustraliaRuiz de Alarcon 13, 28014 Madrid, SpainDock House, The Waterfront, Cape Town 8001, South Africa

http ://www. Cambridge. org

© Cambridge University Press 2000

This book is in copyright. Subject to statutory exceptionand to the provisions of relevant collective licensing agreements,no reproduction of any part may take place withoutthe written permission of Cambridge University Press.

First published 2000

First paperback edition 2004

Typeface Times Roman 10.25/12.5 pt. System MTEX2£ [TB]

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

Library of Congress Cataloguing-in-Publication DataStronge, W J. (William James), 1937-

Impact mechanics / W.J. Stronge.p. cm.

ISBN 0 521 63286 2 hardback1. Impact. I. Title.

TA354 .S77 200062O.ri25-dc21

99-044947ISBN 0 521 63286 2 hardbackISBN 0 521 60289 0 paperback

To Katerina and Jaime

Contents

List of Symbols page xiiiPreface xvii

Chapter 1 Introduction to Analysis of Low Speed Impact 11.1 Terminology of Two Body Impact 2

1.1.1 Configuration of Colliding Bodies 21.1.2 Relative Velocity at Contact Point 31.1.3 Interaction Force 3

1.2 Classification of Methods for Analyzing Impact 31.2.1 Description of Rigid Body Impact 51.2.2 Description of Transverse Impact on Flexible Bodies 91.2.3 Description of Axial Impact on Flexible Bodies 91.2.4 Applicability of Theories for Low Speed Impact 9

1.3 Principles of Dynamics 111.3.1 Particle Kinetics 111.3.2 Kinetics for a Set of Particles 131.3.3 Kinetic Equations for a Rigid Body 141.3.4 Rate of Change for Moment of Momentum of a System

about a Point Moving Steadily Relative to an InertialReference Frame 17

1.4 Decomposition of a Vector 181.5 Vectorial and Indicial Notation 19

Chapter 2 Rigid Body Theory for Collinear Impact 212.1 Equation of Relative Motion for Direct Impact 212.2 Compression and Restitution Phases of Collision 232.3 Kinetic Energy of Normal Relative Motion 252.4 Work of Normal Contact Force 262.5 Coefficient of Restitution and Kinetic Energy Absorbed in Collision 262.6 Velocities of Contact Points at Separation 282.7 Partition of Loss of Kinetic Energy 30

Chapter 3 Rigid Body Theory for Planar or 2D Collisions 353.1 Equations of Relative Motion at Contact Point 353.2 Impact of Smooth Bodies 37

vii

viii Contents

3.3 Friction from Sliding of Rough Bodies 403.3.1 Amontons-Coulomb Law of Dry Friction 403.3.2 Equations of Planar Motion for Collision of Rough Bodies 413.3.3 Contact Processes and Evolution of Sliding during Impact 42

3.4 Work of Reaction Impulse 443.4.1 Total Work Equals Change in Kinetic Energy 443.4.2 Partial Work by Component of Impulse 453.4.3 Energetic Coefficient of Restitution 463.4.4 Terminal Impulse p/ for Different Slip Processes 47

3.5 Friction in Collinear Impact Configurations 553.6 Friction in Noncollinear Impact Configurations 59

3.6.1 Planar Impact of Rigid Bar on Rough Half Space 59

Chapter 4 3D Impact of Rough Rigid Bodies 634.1 Collision of Two Free Bodies 64

4.1.1 Law of Friction for Rough Bodies 674.1.2 Equation of Motion in Terms of the Normal Impulse 674.1.3 Sliding that Halts during Collision 684.1.4 Terminal Normal Impulse from Energetic Coefficient

of Restitution 694.2 Oblique Collision of a Rotating Sphere on a Rough Half Space 704.3 Slender Rod that Collides with a Rough Half Space 72

4.3.1 Slip Trajectories or Hodographs 744.4 Equilateral Triangle Colliding on a Rough Half Space 75

4.4.1 Slip Trajectories and Hodograph for Equilateral TriangleInclined at 6 = n/4 11

4.5 Spherical Pendulum Colliding on a Rough Half Space 794.5.1 Numerical Results for 0 = n/3 and TT/4 82

4.6 General 3D Impact 83

Chapter 5 Rigid Body Impact with Discrete Modeling of Compliancefor the Contact Region 86

5.1 Direct Impact of Viscoelastic Bodies 865.1.1 Linear Viscoelasticity - the Maxwell Model 875.1.2 Simplest Nonlinear Viscoelastic Deformable Element 895.1.3 Hybrid Nonlinear Viscoelastic Element for Spherical

Contact 915.1.4 Parameters of the Hybrid Nonlinear Element for Impact

on a Golf Ball 935.2 Tangential Compliance in Planar Impact of Rough Bodies 93

5.2.1 Dynamics of Planar Collision for Hard Bodies 945.2.2 Slip Processes 1005.2.3 Oblique Impact of an Elastic Sphere on a Rough Half Space 1045.2.4 Dissipation of Energy 1095.2.5 Effects of Tangential Compliance 1115.2.6 Bounce of a Superball 112

Contents ix

Chapter 6 Continuum Modeling of Local Deformation Near the ContactArea 116

6.1 Quasistatic Compression of Elastic-Perfectly Plastic Solids 1176.1.1 Elastic Stresses - Hertzian Contact 1176.1.2 Indentation at Yield of Elastic-Plastic Bodies 1196.1.3 Quasistatic Elastic-Plastic Indentation 1206.1.4 Fully Plastic Indentation 1236.1.5 Elastic Unloading from Maximum Indentation 124

6.2 Resolved Dynamics of Planar Impact 1266.2.1 Direct Impact of Elastic Bodies 1266.2.2 Eccentric Planar Impact of Rough Elastic-Plastic Bodies 129

6.3 Coefficient of Restitution for Elastic-Plastic Solids 1316.4 Partition of Internal Energy in Collision between Dissimilar

Bodies 1326.4.1 Composite Coefficient of Restitution for Colliding Bodies

with Dissimilar Hardness 1326.4.2 Loss of Internal Energy to Elastic Waves 134

6.5 Applicability of the Quasistatic Approximation 1376.6 Transverse Impact of Rough Elastic-Plastic Cylinders - Applicability

of Energetic Coefficient of Restitution 1376.6.1 Elastic Normal Compliance 1386.6.2 Yield for Plane Strain Deformation 1396.6.3 Elastic-Plastic Indentation 1396.6.4 Fully Plastic Indentation 1406.6.5 Analyses of Contact Forces for Oblique Impact of Rough

Cylinders 1406.6.6 Loss of Internal Energy to Elastic Waves for Planar (2D)

Collisions 1426.7 Synopsis for Spherical Elastic-Plastic Indentation 143

Chapter 7 Axial Impact on Slender Deformable Bodies 1467.1 Longitudinal Wave in Uniform Elastic Bar 146

7.1.1 Initial Conditions 1477.1.2 Reflection of Stress Wave from Free End 1507.1.3 Reflection from Fixed End 1527.1.4 Reflection and Transmission at Interface - Normal

Incidence 1527.1.5 Spall Fracture Due to Reflection of Stress Waves 153

7.2 Planar Impact of Rigid Mass against End of Elastic Bar 1567.2.1 Boundary Condition at Impact End 1577.2.2 Boundary Condition at Dashpot End 1577.2.3 Distribution of Stress and Particle Velocity 1587.2.4 Experiments 159

7.3 Impact, Local Indentation and Resultant Stress Wave 1607.4 Wave Propagation in Dispersive Systems 162

7.4.1 Group Velocity 163

x Contents

7.5 Transverse Wave in a Beam 1647.5.1 Euler-Bernoulli Beam Equation 1647.5.2 Rayleigh Beam Equation 1657.5.3 Timoshenko Beam Equation 1667.5.4 Comparison of Euler-Bernoulli, Rayleigh and

Timoshenko Beam Dynamics 168

Chapter 8 Impact on Assemblies of Rigid Elements 1738.1 Impact on a System of Rigid Bodies Connected by Noncompliant

Bilateral Constraints 1748.1.1 Generalized Impulse and Equations of Motion 1748.1.2 Equations of Motion Transformed to Normal and Tangential

Coordinates 1778.2 Impact on a System of Rigid Bodies Connected by Compliant

Constraints 1828.2.1 Comparison of Results from Alternative Analytical

Approximations for Multibody Systems with UnilateralConstraints 183

8.2.2 Numerical Simulation and Discussion of Multibody Impact 1888.2.3 Spatial Gradation of Normal Contact Stiffness K}, = x* 1908.2.4 Applicability of Simultaneous Impact Assumption 198

Chapter 9 Collision against Flexible Structures 2019.1 Free Vibration of Slender Elastic Bodies 201

9.1.1 Free Vibration of a Uniform Beam 2019.1.2 Eigenfunctions of a Uniform Beam with Clamped Ends 2029.1.3 Rayleigh-Ritz Mode Approximation 2039.1.4 Single Degree of Freedom Approximation 205

9.2 Transverse Impact on an Elastic Beam 2069.2.1 Forced Vibration of a Uniform Beam 2079.2.2 Impact of a Perfectly Plastic Missile 2079.2.3 Effect of Local Compliance in Structural Response to Impact 2099.2.4 Impact on Flexible Structures - Local or Global Response? 214

Chapter 10 Propagating Transformations of State in Self-OrganizingSystems 219

10.1 Systems with Single Attractor 22010.1.1 Ball Bouncing down a Flight of Regularly Spaced Steps 220

10.2 Systems with Two Attractors 22210.2.1 Prismatic Cylinder Rolling down a Rough Inclined Plane 22210.2.2 The Domino Effect - Independent Interaction Theory 22810.2.3 Domino Toppling - Successive Destabilization by Cooperative

Neighbors 23510.2.4 Wavefront Stability for Multidimensional Domino Effects 240

10.3 Approach to Chaos - an Unbounded Increase in Number of Attractors 24210.3.1 Periodic Vibro-impact of Single Degree of Freedom Systems 24210.3.2 Period One Orbits 244

Contents xi

10.3.3 Poincare Section and Return Map 24410.3.4 Stability of Orbit and Bifurcation 245

Appendix A Role of Impact in the Development of Mechanics Duringthe Seventeenth and Eighteenth Centuries 248

Historical References 264

Appendix B Glossary of Terms 266

Answers to Some Problems 269References 271

Index 277

List of Symbols

Man is not a circle with a single center; he is an ellipse with twofoci. Facts are one, ideas are the other.

Victor Hugo, Les Miserables

a radius of cylinder or sphere; radius of contact areaa = ac/aY, nondimensional maximum contact radiusb width, thicknessc dashpot force coefficientCo longitudinal wave speed, uniaxial stress (thin bars)cCT critical dashpot force coefficientcg = dco/dk, group velocity of propagating wavescp = co/k, phase velocity of propagating wavese, eo,e* kinematic, kinetic, energetic coefficient of restitution/ , g,h functionsho moment of momentum about point Og = 9.81 m s~2, gravitational constant/ = V— 1 imaginary unit; typical number in seriesk = 2n/X, wave numberkr area radius of gyration for cross-section of bar about centroidkr mass radius of gyration of body B for center of massm = (M~l + M'~l)~l, effective massniij inertia matrix for contact point Cm generalized inertia matrix (r x r, where r = number of generalized

coordinates)n number of particles in systemp = ps, normal component of reaction impulse at point of contactpc normal impulse at transition from period of compression to restitutionPf normal impulse at termination of restitution periodps normal impulse at termination of initial period of slidingq transverse force per unit lengthqr generalized coordinater radial coordinatert, r[ position vectors from centers of mass G and G to point of contact C

s = Jv\ + v\, sliding speed at any normal impulse p

xni

xiv List of Symbols

s = sgn(v\), direction of sliding (planar changes of velocity)t timet\ time of transition from initial stick to slidingt2 time of transition from sliding to stickUi components of displacementw/ particle velocity in wave incoming to interfaceuR particle velocity in wave reflected from interfaceuT particle velocity in wave transmitted through interfacev =Vi, normal component of relative velocity of coincident contact pointsVo normal component of relative velocity of contact points at incidenceV\, v2 tangential components of relative velocity of contact pointsVf normal component of relative velocity at termination of restitutionx axial coordinatey transverse coordinate, nondimensional indentationz depth coordinateA area of cross-sectionA i constantDi dissipation of energy from work of / th component of forceE, E' Young's moduli of material in bodies B, B '£* = EE'I(E + E'), effective Young's modulus at contactF = F 3 , normal force at contact pointFt components of contact forceG shear modulus of material/ moment of inertia for cross-section of beamIij, I[j moments, products of inertia for bodies B, B' about respective centers

of massL lengthM bending moment at section of beamM, M' masses of rigid bodies B, Br, respectivelyP , P' normal components of impulse acting on bodies B, B;, respectivelyR radius of cylinder, sphere/?* RR''/(R + R')9 effective radius of contact curvatureR* effective radius of contact curvature after plastic deformationS shear force at section of beamT kinetic energy of system of colliding bodies7o incident kinetic energy of systemTf final kinetic energy of system at termination of period of restitutionU potential energy (e.g. gravitational potential)Vt, V[ components of velocity at contact points C, CVt, V[ components of velocity at centers of mass of bodies B, Br

Wn, W3 work of normal component of reaction force at CW\, W2 work of tangential components of reaction force at CWc = Wi(pc), work of normal force during compressionWf = W3 (/?/), final work of normal forceWf — W\(pf) + W 2(pf) + Wi(pf)9 final total work of contact reactionX nondimensional displacementY yield stress

List of Symbols xv

Z = dX/d(cot), nondimensional velocitya = M/pAL, mass ratiofi\,fi2, fo inertia coefficients (planar changes in velocity)y = EI/pAy = p^/Pxfc, inertia parameterYR = ( r - l ) / ( r + 1), reflection coefficientYT = 2 (Ai r /A 2 ) / ( r + 1), transmission coefficientY\ = [p(tf) — p(t\)]/p c, ratio of impulse during final slip to p c

Yo shear warping at neutral axisy = Ey0, shear rotation of cross-section8 relative indentation at contact pointSijk permutation tensorSij components of strainjr = c/cCT, damping ratior] local coordinateK) = x — cot, Galilean coordinater] square root of ratio of tangential to normal compliance6 = dw/dx, rotation of section; inclination of body# ratio of kinetic energy of toppling group to that of leading element$Y ratio of mean fully plastic indentation pressure to uniaxial yield stressK stiffness coefficient of spring elementX wavelength of propagating disturbance/x0 dashpot force coefficient\i Amontons-Coulomb coefficient of limiting friction (dry friction)jl coefficient for stickv Poisson's ratio§ local coordinate£ = x + ct, Galilean coordinate| = 2d/a, characteristic depth for plane strain deformation fieldp mass densityGij components of stressr nondimensional time; characteristic time; +i(+), ratio angular speeds before and after impact0 = t a n ^ t ^ / ^ i X sliding direction in tangent plane0 isoclinic direction of slip0* separatrix direction of slip/ stiffness ratiox//o angle of incidence for relative velocity at contact point\JTf angle of rebound for relative velocity at contact pointco tangential resonant frequencyCOQ initial angular velocitycoo = K/m, characteristic frequency of oscillationcoc cutoff frequency for propagationodd damped resonant frequencycot, (D\ angular velocity vectors for bodies B , B 'coi angular speed for contact at point /F = A2P2C2/A1P1C1, impedance ratio

xvi List of Symbols

, dashpot force ratioS Timoshenko beam coefficientEo = Fo/A, negative pressure<$> matrix*I> geometry of polygonal solidQ normal resonant frequency

Vectors & Dyadics

e unit vector parallel to common tangent planeho moment of momentum about point Oh moment of momentum about center of massn = n3 unit vector normal to common tangent planer, position vector of /th particle relative to center of masss =\e/\\e\, direction of sliding (3D)v = V — V relative velocity across contact pointF/ force on /th particleI moment of inertia for center of massP,- impulse on /th particleV; velocity of /th particleV velocity of center of masspt position vector of /th particle relative to inertial reference frameu? angular velocity of rigid body

Preface

Caminante, no hay camino.Se hace camino al ander."Traveller, there is no pathPaths are made by walking."

A. Machado, popular song from Latin America

When bodies collide, they come together with some relative velocity at an initialpoint of contact. If it were not for the contact force that develops between them, thenormal component of relative velocity would result in overlap or interference near thecontact point and this interference would increase with time. This reaction force deformsthe bodies into a compatible configuration in a common contact surface that envelopesthe initial point of contact. Ordinarily it is quite difficult and laborious to calculate defor-mations that are geometrically compatible, that satisfy equations of motion and that giveequal but opposite reaction forces on the colliding bodies. To avoid this detail, severaldifferent approximations have been developed for analyzing impact: rigid body impacttheory, Hertz contact theory, elastic wave theory, etc. This book presents a spectrum ofdifferent theories for collision and describes where each is applicable. The question ofapplicability largely depends on the materials of which the bodies are composed (theirhardness in the contact region and whether or not they are rate-dependent), the geometricconfiguration of the bodies and the incident relative velocity of the collision. These factorsaffect the relative magnitude of deformations in the contact region in comparison withglobal deformations.

A collision between hard bodies occurs in a very brief period of time. The duration ofcontact between a ball and bat, a hammer and nail or an automobile and lamppost is nomore than a few milliseconds. This brief period has been used to justify rigid body impacttheory in which bodies instantaneously change velocity when they collide. As a conse-quence of the instantaneous period of contact, the bodies have negligible displacementduring the collision. For any analysis of changes in momentum occurring during impact,the approximation that displacements are negligibly small greatly simplifies the analysis.With this approximation, the changes in velocity can be calculated without integratingaccelerations over the contact period. Along with this simplification, however, there isa hazard associated with loss of information about the contact forces that cause thesechanges in velocity - without forces the changes in velocity cannot be directly associatedwith deformability of the bodies.

xvn

xviii Preface

In order to solve more complex problems, particularly those involving friction, wedevelop a method that spreads out the changes in velocity by considering that they are acontinuous function of impulse rather than time. With this approach, the approximationof negligible displacement during a very brief period of contact results in an equation ofmotion with constant coefficients; this equation is trivially integrable to obtain changesin momentum of each body as a function of the impulse of the contact force. This permitsthe analyst to follow the evolution of contact and variation in relative velocity across thecontact patch as a function of impulse.

For rigid body impact theory the equations of dynamics are not sufficient to solve forthe changes in velocity - an additional relation is required. Commonly this relation isprovided by the coefficient of restitution. Most books on mechanics treat the coefficient ofrestitution as an impact law; i.e., for the contact points of colliding bodies, they considerthe coefficient of restitution to be an empirical relationship for the normal componentof relative velocity at incidence and separation. This has been satisfactory for collisionsbetween smooth bodies, but for bodies with rough surfaces where friction opposes slidingduring impact, the usual kinematic (or Newton) coefficient of restitution has a seriousdeficiency. In the technical and scientific literature the topic of rigid body impact was re-opened in 1984 largely as a consequence of some problems where calculations employingthe kinematic coefficient gave solutions which were patently unrealistic - for collisions inwhich friction opposes small initial slip, such calculations predicted an increase in kineticenergy as a consequence of the collision. In order to rectify this problem and clearly sep-arate dissipation due to friction from that due to irreversible internal deformations nearthe contact point of colliding bodies, a different definition of the coefficient of restitution(termed the energetic coefficient of restitution) was proposed and is used throughout thisbook. In those problems where friction is negligible or where slip is unidirectional duringcontact, the energetic and kinematic coefficients are equal; if the direction of slip changesduring contact, however, these coefficients are distinct.

These methods will be illustrated by analyses of practical examples. Many of theseare taken from sport; e.g. the bounce of a hockey puck, the spin (and consequent hook orslice) resulting from mis-hitting a golf shot, and batting for maximum range.

While rigid body impact theory is effective for analyzing the response of hard bodies,more complex analytical descriptions are required if a colliding body is soft or deformable,i.e. if the collision generates significant structural deformations far from the contact region.This occurs if the impact occurs near a slender section of a colliding body or if the bodyis hollow, as in the case of an inflated ball. To calculate the response of deformablebodies a time-dependent analysis is required, since the contact force depends on localdeformation of the body. In this case the response depends on the compliance of the contactregions in addition to the inertia properties and initial relative velocities that determinethe outcome for rigid body impacts. The compensation for the additional complexity oftime-dependent impact analysis for deformable bodies is that an empirically determinedcoefficient of restitution is no longer required to relate the final and initial states of thesystem. This relationship can be calculated for any given material and structural properties.For colliding bodies that are compact in shape and composed of hard materials, the contactstresses rapidly diffuse, so that substantial deformations occur only near the point of initialcontact; in this case the change of state resulting from impact can be calculated on the basisof quasistatic continuum mechanics. On the other hand, for impacts that are transverse to

Preface xix

some slender member, the collision generates vibratory motion far from the site of impactso that the calculation must be based on structural dynamics of beams, plates or shells.Examples are provided for impact between elastic-plastic solids and for collisions againstslender elastic plates and beams.

This textbook evolved from lecture notes prepared for an upper division course pres-ented originally at the University of California-Davis. A later version of the coursewas tested on students at the National University of Singapore. Those notes were ex-panded with additional material developed subsequently by myself, my students and mycolleagues. I was aided in improving the presentation by helpful criticisms from MontHubbard, Chwee Teck Lim, Victor Shim and Jim Woodhouse. Our interest has been indeveloping more physically based analytical models in order to improve the accuracy ofcalculations of impact response and to increase the range of applicability for any measure-ment of collision properties of a system. In these respects this book is complementary tothe neoclassical treatise Impact, the Theory and Physical Behavior of Colliding Solids, byW. Goldsmith - a monograph which provides a wealth of experimental data on collisionbehavior of metals, glass and natural materials. The present text has stepped off fromthis base to incorporate the physically based knowledge of mechanics of collision thathas been developed in the last 40 years. In order to appreciate the analytical methods de-scribed here, the background required is an undergraduate engineering course includingdynamics, strength of materials and vibrations.

CHAPTER 1

Introduction to Analysis of Low Speed Impact

Philosophy is written in this grand book - I mean the universe -which stands continuously open to our gaze, but cannot be under-stood unless one first learns to comprehend the language in whichit is written. It is written in the language of mathematics and itscharacters are triangles, circles and other geometric figures, withoutwhich it is humanly impossible to understand a single word of it;without these one is wandering about in a dark labyrinth.

Galileo Galilei, Two New Sciences, 1632

When a bat strikes a ball or a hammer hits a nail, the surfaces of two bodies cometogether with some relative velocity at an initial instant termed incidence. After incidencethere would be interference or interpenetration of the bodies were it not for the interfacepressure that arises in a small area of contact between the two bodies. At each instantduring the contact period, the pressure in the contact area results in local deformationand consequent indentation; this indentation equals the interference that would exist ifthe bodies were not deformed.

At each instant during impact the interface or contact pressure has a resultant force ofaction or reaction that acts in opposite directions on the two colliding bodies and therebyresists interpenetration. Initially the force increases with increasing indentation and itreduces the speed at which the bodies are approaching each other. At some instant duringimpact the work done by the contact force is sufficient to bring the speed of approach ofthe two bodies to zero; subsequently, the energy stored during compression drives the twobodies apart until finally they separate with some relative velocity. For impact betweensolid bodies, the contact force that acts during collision is a result of the local deformationsthat are required for the surfaces of the two bodies to conform in the contact area.

The local deformations that arise during impact vary according to the incident relativevelocity at the point of initial contact as well as the hardness of the colliding bodies. Lowspeed collisions result in contact pressures that cause small deformations only; these aresignificant solely in a small region adjacent to the contact area. At higher speeds there arelarge deformations (i.e. strains) near the contact area which result from plastic flow; theselarge localized deformations are easily recognizable, since they have gross manifestationssuch as cratering or penetration. In each case the deformations are consistent with thecontact force that causes velocity changes in the colliding bodies. The normal impactspeed required to cause large plastic deformation is between 102 x VY and 103 x Vywhere VY is the minimum relative speed required to initiate plastic yield in the softer body

1 / Introduction to Analysis of Low Speed Impact

(for metals the normal incident speed at yield VY is of the order of 0.1 m s 1). This textexplains how the dynamics of low speed collisions are related to both local and globaldeformations in the colliding bodies.

1.1 Terminology of Two Body Impact

1.1.1 Configuration of Colliding Bodies

As two colliding bodies approach each other there is an instant of time, termedincidence, when a single contact point C on the surface of the first body B initially comesinto contact with point O on the surface of the second body B'. This time t — 0 is the initialinstant of impact. Ordinarily the surface of at least one of the bodies has a continuousgradient at either C or C (i.e., at least one body has a topologically smooth surface) sothat there is a unique common tangent plane that passes through the coincident contactpoints C and C The orientation of this plane is defined by the direction of the normalvector n, a unit vector which is perpendicular to the common tangent plane.

Central or Collinear Impact Configuration:If each colliding body has a center of mass G or G' that is on the common normal linepassing through C, the impact configuration is collinear, or central. This requires that theposition vector r c from G to C, and the vector r^ from G' to C, both be parallel to thecommon normal line as shown in Fig. 1.1a:

r c x n = r J : x n = 0.

Collinear impact configurations result in equations of motion for normal and tangentialdirections that can be decoupled. If the configuration is not collinear, the configuration iseccentric.

Eccentric Impact Configuration:The impact configuration is eccentric if at least one body has a center of mass that is off theline of the common normal passing through C as shown in Fig. 1.1b. This occurs if either

rcxn/0 or r^xn/0.

If the configuration is eccentric and the bodies are rough (i.e., there is a tangential force

tang, plane

(b)

Figure 1.1. Colliding bodies B and B' with (a) collinear and (b) noncollinear impact con-figurations. In both cases the angle of incidence is oblique; i.e. 0o 7 0-

1.2 / Classification of Methods for Analyzing Impact

of friction that opposes sliding), the equations of motion each involve both normal andtangential forces (and impulses). Thus eccentric impact between rough bodies involveseffects of friction and normal forces that are not separable.

1.1.2 Relative Velocity at Contact Point

At the instant when colliding bodies first interact, the coincident contact pointsC and C have an initial or incident relative velocity Vo = v(0) = Vc(0) — \'c(0). Theinitial relative velocity at C has a component Vo • n normal to the tangent plane and acomponent (n x vo) x n parallel to the tangent plane; the latter component is termedsliding. The angle of obliquity at incidence, x/fo, is the angle between the initial relativevelocity vector VQ and the unit vector n normal to the common tangent plane,

= tan"Y(n x vp) x n \\ v0 • n /

Direct impact occurs when in each body the velocity field is uniform and parallel to thenormal direction. Direct impact requires that the angle of obliquity at incidence equalszero ( "o = 0); on the other hand, oblique impact occurs when the angle of obliquity atincidence is nonzero (I/TQ ^ 0).

1.1.3 Interaction Force

An interaction force and the impulse that it generates can be resolved into com-ponents normal and parallel to the common tangent plane. For particle impact the impulseis considered to be normal to the contact surface and due to short range interatomic re-pulsion. For solid bodies, however, contact forces arise from local deformation of thecolliding bodies; these forces and their associated deformations ensure compatibility ofdisplacements in the contact area and thereby prevent interpenetration (overlap) of thebodies. In addition a tangential force, friction, can arise if the bodies are rough and thereis sliding in the contact area. Dry friction is negligible if the bodies are smooth.

Conservative forces are functions solely of the relative displacement of the interact-ing bodies. In an elastic collision the forces associated with attraction or repulsion areconservative (i.e. reversible); it is not necessary however for friction (a nonconservativeforce) to be negligible. In an inelastic collision the interaction forces (other than friction)are nonconservative, so that there is a loss of kinetic energy as a result of the cycle ofcompression (loading) and restitution (unloading) that occurs in the contact region. Theenergy loss can be due to irreversible elastic-plastic material behavior, rate-dependentmaterial behavior, elastic waves trapped in the separating bodies, etc.

1.2 Classification of Methods for Analyzing Impact

In order to classify collisions into specific types which require distinct methodsof analysis, we need to think about the deformations that develop during collision, thedistribution of these deformations in each of the colliding bodies, and how these de-formations affect the period of contact. In general there are four types of analysis forlow speed collisions, associated with particle impact, rigid body impact, transverse im-pact on flexible bodies (i.e. transverse wave propagation or vibrations) and axial impact


(a)

(c)

Figure 1.2. Impact problems requiring different analytical approaches: (a) particle impact(stereo-mechanical), (b) rigid body impact, (c) transverse deformations of flexible bodiesand (d) axial deformation of flexible bodies.

on flexible bodies (i.e. longitudinal wave propagation). A typical example where eachmethod applies is illustrated in Fig. 1.2.

(a) Particle impact is an analytical approximation that considers a normal compo-nent of interaction impulse only. By definition, particles are smooth and spherical.The source of the interaction force is unspecified, but presumably it is strong andthe force has a very short range, so that the period of interaction is a negligiblysmall instant of time.

(b) Rigid body impact occurs between compact bodies where the contact area re-mains small in comparison with all section dimensions. Stresses generated in thecontact area decrease rapidly with increasing radial distance from the contactregion, so the internal energy of deformation is concentrated in a small regionsurrounding the interface. This small deforming region has large stiffness andacts much like a short but very stiff spring separating the colliding bodies at the


contact point. The period of contact depends on the normal compliance of thecontact region and an effective mass of the colliding bodies.

(c) Transverse impact on flexible bodies occurs when at least one of the bodiessuffers bending as a result of the interface pressures in the contact area; bendingis significant at points far from the contact area if the depth of the body in thedirection normal to the common tangent plane is small in comparison with di-mensions parallel to this plane. This bending reduces the interface pressure andprolongs the period of contact. Bending is a source of energy dissipation duringcollision in addition to the energy loss due to local deformation that arises fromthe vicinity of contact.

(d) Axial impact on flexible bodies generates longitudinal waves which affect thedynamic analysis of the bodies only if there is a boundary at some distance fromthe impact point which reflects the radiating wave back to the source; it reflectsthe outgoing wave as a coherent stress pulse that travels back to its source essen-tially undiminished in amplitude. In this case the time of contact for an impactdepends on the transit time for a wave travelling between the impact surface andthe distal surface. This time can be less than that for rigid body impact betweenhard bodies with convex surfaces.

1.2.1 Description of Rigid Body Impact

For bodies that are hard (i.e. with small compliance), only very small defor-mations are required to generate very large contact pressures; if the surfaces are initiallynonconforming, the small deformations imply that the contact area remains small through-out the contact period. The interface pressure in this small contact area causes the initiallynonconforming contact surfaces to deform until they conform or touch at most if not allpoints in a small contact area. Although the contact area remains small in comparisonwith cross-sectional dimensions of either body, the contact pressure is large, and it givesa large stress resultant, or contact force. The contact force is large enough to rapidlychange the normal component of relative velocity across the small deforming region thatsurrounds the contact patch. The large contact force rapidly accelerates the bodies andthereby limits interference which would otherwise develop after incidence if the bodiesdid not deform.

Hence in a small region surrounding the contact area the colliding bodies are subjectedto large stresses and corresponding strains that can exceed the yield strain of the mate-rial. At quite modest impact velocities (of the order of 0.1 m s"1 for structural metals)irreversible plastic deformation begins to dissipate some energy during the collision; con-sequently there is some loss of kinetic energy of relative motion in all but the most benigncollisions. Although the stresses are large in the contact region, they decay rapidly withincreasing distance from the contact surface. In an elastic body with a spherical coordinatesystem centered at the initial contact point, the radial component of stress, a r, decreasesvery rapidly with increasing radial distance r from the contact region (in an elastic solid or

decreases as r~2 in a 3D deformation field). For a hard body the corresponding rapid de-crease in strain means that significant deformations occur only in a small region aroundthe point of initial contact; consequently the deflection or indentation of the contact arearemains very small.


Since the region of significant strain is not very deep or extensive, hard bodies havevery small compliance (i.e., a large force generates only a small deflection). The smallregion of significant deformation is like a short stiff spring which is compressed betweenthe two bodies during the period of contact. This spring has a large spring constant andgives a very brief period of contact. For example, a hard-thrown baseball or cricket ballstriking a bat is in contact for a period of roughly 2 ms, while a steel hammer striking a nailis in contact for a period of about 0.2 ms. The contact duration for the hammer and nail isless because these colliding bodies are composed of harder materials than the ball and bat.Both collisions generate a maximum force on the order of 10 kN (i.e. roughly one ton).

From an analytical point of view, the most important consequence of the small com-pliance of hard bodies is that very little movement occurs during the very brief periodof contact; i.e., despite large contact forces, there is insufficient time for the bodies todisplace significantly during impact. This observation forms a fundamental hypothesis ofrigid body impact theory, namely, that for hard bodies, analyses of impact can considerthe period of contact to be vanishingly small. Consequently any changes in velocity occurinstantaneously (i.e. in the initial or incident configuration). The system configuration atincidence is termed the impact configuration. This theory assumes there is no movementduring the contact period.

Underlying Premises of Rigid Body Impact Theory(a) In each of the colliding bodies the contact area remains small in comparison

with both the cross-sectional dimensions and the depth of the body in the normaldirection.

(b) The contact period is sufficiently brief that during contact the displacements arenegligible and hence there are no changes in the system configuration; i.e., thecontact period can be considered to be instantaneous.

If these conditions are approximately satisfied, rigid body impact theory can be applicable.In general this requires that the bodies are hard and that they suffer only small localdeformation in collision. For a solid composed of material that is rate-independent, a smallcontact area results in significant strains only in a small region around the initial contactpoint. If the body is hard, the very limited region of significant deformations causes thecompliance to be small and consequently the contact period to be very brief. This resultsin two major simplifications:

(a) Equations of planar motion are trivially integrable to obtain algebraic relationsbetween velocity changes and the reaction impulse.1

(b) Finite active forces (e.g. gravitational or magnetic attraction) which act duringthe period of contact can be considered to be negligible, since these forces dono work during the collision.

During the contact period the only significant active forces are reactions at points ofcontact with other bodies; these reactions are induced by displacement constraints.

Figure 1.3 shows a collision where application of rigid body impact theory is appropri-ate. This series of high speed photographs shows development of a small area of contactwhen an initially stationary field hockey ball is struck by a hockey stick at an incident

1 Because velocity changes can be obtained from algebraic relations, rigid body impact was one of themost important topics in dynamics before the development of calculus in the late seventeenth century.

Figure 1.3. High speed photographs of hockey stick striking at 18 m s ' (40 mph) againsta stationary field hockey ball (dia. D' = 14 mm, mass M'' = 130 g). Interframe periodx = 0.0002 s, contact duration tf ^ 0.0015 s, and maximum normal force Fc % 3900 N.

speed of 18 m s"1. During collision the contact area increases to a maximum radiusac that remains small in comparison with the ball radius R'\ in Fig. 1.3, ac/Rf < 0.3.The relatively small contact area is a consequence of the small normal compliance (orlarge elastic modulus) of both colliding bodies and the initial lack of conformation of thesurfaces near the point of first contact.

A useful means of postulating rigid body impact theory is to suppose that two collidingbodies are separated by an infinitesimal deformable particle.2 The deformable particle is

2 The physical construct of a deformable particle separating contact points on colliding rigid bodies ismathematically equivalent to Keller's (1986) asymptotic method of integrating with respect to time theequations for relative acceleration of deformable bodies and then taking the limit as compliance (orcontact period) becomes vanishingly small.

8 1 / Introduction to Analysis of Low Speed Impact

located between the point of initial contact on one body and that on the other, althoughthese points are coincident. The physical construct of an infinitesimal compliant elementseparating two bodies at a point of contact allows variations in velocity during impact tobe resolved as a function of the normal component of impulse. This normal componentof impulse is equivalent to the integral of the normal contact force over the period oftime after incidence. Since collisions between bodies with nonadhesive contact surfacesinvolve only compression of the deformable particle - never extension - the normalcomponent of impulse is a monotonously increasing function of time after incidence.Thus variations in velocity during an instantaneous collision are resolved by choosing asan independent variable the normal component of impulse rather than time. This givesvelocity changes which are a continuous (smooth) function of impulse.

There are three notable classes of impact problems where rigid body impact theory isnot applicable if the impact parameters representing energy dissipation are to have anyrange of applicability.

(a) The first involves impulsive couples applied at the contact point. Since the contactarea between rigid bodies is negligibly small, impulsive couples are inconsistentwith rigid body impact theory. To relate a couple acting during impulse to phys-ical processes, one must consider the distribution of deformation in the contactregion. Then the couple due to a distribution of tangential force can be obtainedfrom the law of friction and the first moment of tractions in a finite contact areaabout the common normal through the contact point.

(b) A second class of problems where rigid body impact theory does not apply is axialimpact of collinear rods with plane ends. These are problems of one dimensionalwave propagation where the contact area and cross-sectional area are equalbecause the contacting surfaces are conforming; in this case the contact areamay not be small. For problems of wave propagation deformations and particlevelocities far from the contact region are not insignificant. As a consequence, forone dimensional waves in long bars, the contact period is dependent on materialproperties and depth of the bars in a direction normal to the contact plane ratherthan on the compliance of local deformation near a point of initial contact.

(c) The third class of problems where rigid body theory is insufficient are transverseimpacts on beams or plates where vibration energy is significant.

Collisions with Compliant Contact Region Between Otherwise Rigid BodiesWhile most of our attention will be directed towards rigid body impact, there are caseswhere distribution of stress is significant in the region surrounding the contact area.These problems require consideration of details of local deformation of the collidingbodies near the point of initial contact; they are analyzed in Chapters 6 and 8. The mostimportant example may be collisions against multibody systems where the contact pointsbetween bodies transmit the action from one body to the next; in general, this case requiresconsideration of the compliance at each contact. Considerations of local compliance maybe represented by discrete elements such as springs and dashpots or they can be obtainedfrom continuum theory.

For collisions between systems of hard bodies, it is necessary to consider local dis-placement in each contact region although global displacements are negligibly small;i.e. different scales of displacement are significant for different analytical purposes. The


relatively small displacements that generate large contact forces are required to analyzeinteractions between spatially discrete points of contact. If the bodies are hard however,these same displacements may be sufficiently small so that they have negligible effecton the inertia properties; i.e. during collision any changes in the inertia properties areinsignificant despite the small local deformations.

1.2.2 Description of Transverse Impact on Flexible Bodies

Transverse impact on plates, shells or slender bars results in significant flexuraldeformations of the colliding members both during and following the contact period. Inthese cases the stiffness of the contact region depends on flexural rigidity of the bodiesin addition to continuum properties of the region immediately adjacent to the contactarea; i.e., it is no longer sufficient to suppose that a small deforming region is surroundedby a rigid body. Rather, flexural rigidity is usually the more important factor for contactstiffness when impact occurs on a surface of a plate or shell structural component.

1.2.3 Description of Axial Impact on Flexible Bodies

Elastic or elastic-plastic waves radiating from the impact site are present in everyimpact between deformable bodies - in a deformable body it is these radiating waves thattransmit variations in velocity and stress from the contact region to the remainder of thebody. Waves are an important consideration for obtaining a description of the dynamicresponse of the bodies, however, only if the period of collision is determined by waveeffects. This is the case for axial impact acting uniformly over one end of a slender barif the far, or distal, end of the bar imposes a reflective boundary condition. Similarly,for radial impact at the tip of a cone, elastic waves are important if the cone is truncatedby a spherical surface with a center of curvature at the apex. In these cases where theimpact point is also a focal point for some reflective distal surface, the wave radiatingfrom the impact point is reflected from the distal surface and then travels back to thesource, where it affects the contact pressure. On the other hand, if different parts of theoutgoing stress wave encounter boundaries at various times and the surfaces are not normalto the direction of propagation, the wave will be reflected in directions that are not towardsthe impact point; while the outgoing wave changes the momentum of the body, this waveis diffused rather than returning to the source as a coherent wave that can change thecontact pressure and thereby affect the contact duration.

1.2.4 Applicability of Theories for Low Speed Impact

This text presents several different methods for analyzing changes in veloc-ity (and contact forces) resulting from low speed impact, i.e. impact slow enough thatthe bodies are deformed imperceptibly only. These theories are listed in Table 1.1with descriptions of the differences and an indication of the range of applicability for each.

The stereomechanical theory is a relationship between incident and final conditions; itresults in discontinuous changes in velocity at impact. In this book a more sophisticatedrigid body theory is developed - a theory in which the changes in velocity are a continuousfunction of the normal component of the impulse p at the contact point. This theory

Table 1.1. Applicability of Theories for Oblique, Low Speed Impact

Independent Coeff. of

Angle of Spatial GradientIncidence of Contactat Impact Point,*7 Compliance,0

Impact Theory

StereomechanicaF

Rigid body^

Compliant contact^

Continuum^

Variable

None

ImpulseP

Timet

Timet

Restitution*

e,e0

None

(Impact PointCompliance)/(Structural ComputationalCompliance)^ Effort Illustration

All

All

» 1

<3Cl (simultaneous)

All » 1

All All

Low

Low

Moderate

High

a e, eo,e* = kinematic, kinetic, energetic coefficients of restitution.b \x — Amontons-Coulomb coefficient of limiting friction; /3\, 3 = inertia coefficients.c Distributed points of contact.d Flexible bodies.6 Nonsmooth dynamics.'f Smooth dynamics.8 Or negligible tangential compliance.

1.3 / Principles of Dynamics 11

results from considering that the coincident points of contact on two colliding bodiesare separated by an infinitesimal deformable particle - a particle that represents localdeformation around the small area of contact. With this artifice, the analysis can followthe process of slip and/or slip-stick between coincident contact points if the contact regionhas negligible tangential compliance. Rigid body theories are useful for analyzing twobody impact between compact bodies composed of stiff materials; however, they havelimited applicability for multibody impact problems.

When applied to multibody problems, rigid body theories can give accurate resultsonly if the normal compliance of the point of external impact is very small or large incomparison with the compliance of any connections with adjacent bodies. If compliance ofthe point of external impact is much smaller than that of all connections to adjacent bodies,at the connections the maximum reaction force occurs well after the termination of contactat the external impact point, so that the reactions essentially occur sequentially. Smallimpact compliance results in a wave of reaction that travels away from the point of externalimpact at a speed that depends on the inertia of the system and the local compliance ateach connecting joint or contact point. On the other hand, if the normal complianceof the point of external impact is very large in comparison with compliance of anyconnections to adjacent bodies, the reactions at the connections occur simultaneously withthe external impact force. Only in these limiting cases can the dynamic interaction betweenconnected bodies be accurately represented with an assumption of either sequential orsimultaneous reactions. Generally the reaction forces at points of contact arise frominfinitesimal relative displacements that develop during impact; these reaction forces arecoupled, since sometimes they overlap.

If however other points of contact or cross-sections of the body have compliance of thesame order of magnitude as that at any point of external impact, then the effect of theseflexibilities must be incorporated into the dynamic model of the system. If the compliantelements are local to joints or other small regions of the system, an analytical modelwith local compliance may be satisfactory; e.g. see Chapter 8. On the other hand, if thebody is slender, so that significant structural deformations develop during impact, eithera wave propagation or a structural vibration analysis may be required; see Chapter 7or 9. Whether the distributed compliance is local to joints or continuously distributedthroughout a flexible structure, these theories require a time-dependent analysis to obtainreaction forces that develop during contact and the changes in velocity caused by theseforces.

Hence the selection of an appropriate theory depends on structural details and thedegree of refinement required to obtain the desired information.

1.3 Principles of Dynamics

1.3.1 Particle Kinetics

The fundamental form of most principles of dynamics is in terms of the dynamicsof a particle. A particle is a body of negligible or infinitesimal size, i.e. a point mass.The particle is the building block that will be used to develop the dynamics of impactfor either rigid or deformable solids. A particle of mass M moving with velocity Vhas momentum MV. If a resultant force F acts on the particle, this causes a change inmomentum in accord with Newton's second law of motion.

Vit)

Pit

Figure 1.4. Change in velocity of particle with mass M resulting from impulse P(0-

Law II: The momentum MS of a particle has a rate of change with respect to timethat is proportional to and in the direction of any resultant force F(t) acting on theparticle3:

d(M\)/dt = F (1.1)

Usually the particle mass is constant, so that Eq. (1.1) can be integrated to obtain thechanges in velocity as a continuous function of the impulse P(t):

Jo¥{t')dtf = M~lP(t) (1.2)

This vector expression is illustrated in Fig. 1.4.The interaction of two particles B and B' that collide at time t = 0 generates active

forces F(t) and F'(0 that act on each particle respectively, during the period of interaction,0 < t < tf - these forces of interaction act to prevent interpenetration. The particularnature of interaction forces depends on their source: whether they are due to contactforces between solid bodies that cannot interpenetrate, or are interatomic forces actingbetween atomic particles. In any case the force on each particle acts solely in the radialdirection. These interaction forces are related by Newton's third law of motion.

Law III: Two interacting bodiesmagnitude, opposite in direction

F = - F

haveand

forces of actioncollinear:

and reaction that are equal

(1

in

.3)

Laws II and III are the basis for impulse-momentum methods of analyzing impact. Letparticle B have mass M, and particle B' have mass M'. Integration of (1.3) gives equalbut opposite impulses —P'(0 = P(f), s o that equations of motion for the relative velocityv = V — V can be obtained as

v(0 = v(0) + m~lP(t), m~l = AT1 + M' (1.4)

3 Newton's second law is valid only in an inertial reference frame or a frame translating at constantspeed relative to an inertial reference frame. In practice a reference frame is usually considered to befixed relative to a body, such as the earth, which may be moving. Whether or not such a referenceframe can be considered to be inertial depends on the magnitude of the acceleration being calculatedin comparison with the acceleration of the reference body, i.e. whether or not the acceleration of thereference frame is negligible.


Figure 1.5. (a) Equal but opposite normal impulses P on a pair of colliding bodies withmasses M and M' result in velocity changes M - 1 P and — M'~ XV respectively, (b) Thelight lines are the initial and the final velocity for each body, while the heavy lines are theinitial relative velocity v(0), the final relative velocity v(P) and the change m - 1 P in relativevelocity.

where m is the effective mass. The change of variables from velocity \(t) in an inertialreference frame to relative velocity \(t) is illustrated in Fig. 1.5. Equation (1.4) is an equa-tion of relative motion that is applicable in the limit as the period of contact approacheszero (tf -> 0); this equation is the basis of smooth dynamics of collision for particles andrigid bodies.

Example 1.1 A golf ball has mass M — 61 g. When hit by a heavy club the ball acquiresa speed of 44.6 m s"1 (100 mph) during a contact duration tf = 0.4 ms. Assume that theforce-deflection relation is linear, and calculate an estimate of the maximum force Fmax

acting on the ball.

Solution

Effective mass m = 0.061 kg.Initial relative velocity v(0) = VQ = —44.6 ms" 1 .

(a) Linear spring =>> simple harmonic motion for relative displacement 8 at fre-quency co where cotf =n.

(b) Change in momentum of relative motion = impulse, Eq. (1.4):

mv0 = F(t)dt= Fmaxsin(cot)dtJo Jo

Fmax = 21.4kN(^2tons)

1.3.2 Kinetics for a Set of Particles

For a set of n particles where the /th particle has mass M,- and velocity V,- theequations of translational motion can be expressed as

/=i k=\


where F/ is an external force acting on particle / and F- is an internal interaction forceof particle k on particle /. Since the internal forces are equal but opposite (F^ = — F' ki\the sum of these forces over all particles vanishes; hence

1=1 1=1

The moment of momentum ho of particle / about point O is defined as ho = pt x Mt V;,where pt is the position vector of the particle from O and Mt is the mass of the particle.Thus the set of particles has a moment of momentum about O,

ho = ^2 Pi x M/V/.i=i

For a set of n particles the rate of change of moment of momentum about O is related tothe moment about O of the external forces acting on the system:

xF, . (1.6)

If the configuration of the system does not change during the period of time t, integrationof (1.6) with respect to time gives

ho(0 - MO) = Ypt x / F / W = Y] Pi x P/(f). (1.7)7=1 Jo 7=1

1.3.3 Kinetic Equations for a Rigid Body

A rigid body can be represented as a set of particles separated by fixed distances.When the body is moving, the only relative velocity between different points on the bodyis due to angular velocity UJ of the body and the distance between the points.

Trans lational Momentum and Moment of MomentumSuppose a body of mass M is composed of n particles with individual masses M;, / =1, . . . , n where the position vector of each particle pt can be expressed as a set of coordi-nates in an inertial reference frame with origin at O. The location p of the center of massof this set of particles is given by

n n

p = M~l Y^ MtPi, where M = Y^ Mti=\ i=\

as illustrated in Fig. 1.6. The center of mass has velocity V = dp/dt = M~l YH=i M* V/.Hence for the set of particles that constitute this body, the translational equation of motion(1.5) can be expressed as

i n

- (MV) = ^ F , - . (1.8)i=\

For a rigid body with mass M, this integrated form of Newton's law states that the temporalrate of change of translational momentum M\ is equal to the resultant of external forcesacting on the body, Y^i=\ ^ •

A/

Figure 1.6. Elementary rigid body consisting of particles B\ and B2 that are linked by alight rigid bar. The particles have masses M\ and M2 respectively; the particles are subjectto forces F\ and F2.

The moment of momentum about O for the body can be expressed in terms of themoment of the translational momentum of the body MY plus the couple from the rotationalmomentum of the body. First note that the position pt of a particle can be decomposedinto the position vector p of the center of mass plus the position vector r; of the /th particlerelative to the center of mass; i.e., pt = p + r,-. Thus the moment of momentum about Obecomes

where h is the moment of momentum for the system about the center of mass G,

h = x MiV/ = Ti x (u; x r,-) = I1=1

x F,, (1.9)

and I is the inertia dyadic for the center of mass G.4

Thus the equation of motion (1.6) can be expressed as

dho _ d , d_, - V -~dT~d~t(pxM ) + Tt( •a ; ) - Z > + r<)

This decomposition of the particle position vector pt = p + r,- has separated the equationfor the rate of change of moment of momentum about O into a term for the moment oftranslational momentum of the body acting at the center of mass G and a second term forthe moment of momentum relative to G.

Noting that the differential of an applied impulse dP[ = F; dt we obtain from (1.8)and (1.9) that for a rigid body there are three independent equations of motion in termsof the applied impulse dP = YH=\ P*>

d(M\) = dPd(p x MY) =pxdP

n

d(I • LJ) = ] T r, x dPt

(1.10a)

(1.10b)

(1.10c)

4 For body-fixed Cartesian coordinates aligned with mutually perpendicular unit vectors n ; , 7 = 1, 2, 3,and origin at G, the inertia dyadic is expressed as I = /^nyn*, where subscripts denote Cartesiancoordinate directions. This rotational inertia of the body has coefficients obtained by integrating overthe mass of the body, Ijj = / ( r • r — rjrf)dM and Ijk = —J rjrk dM, j ^ k, where r, is the y'thcomponent of the position vector r in a Cartesian coordinate system and no summation is implied byrepeated subscripts.


Equation (1.10b) is the differential of the moment about O of translational momentum ofthe body.

Kinetic EnergyFor some problems it is preferable to use a scalar measure of activity of a body; e.g.the kinetic energy T rather than the vectorial representations (1.10). Consider a bodycomposed of n particles so that the kinetic energy T is expressed as

The kinetic energy of a rigid body T can be resolved into translational kinetic energy Tv

and rotational kinetic energy T^

\ \ V X + V2-V2 + V3-V3) (1.11a)

= ^((Ojlu + Cojl22 + C0JI33 + 2o)\CO2I\2 + 2&>2&>3 23 + 2o)3COiI3^ (1.11b)

where V is the velocity of the center of mass, u is the angular velocity of the body and Iis the inertia dyadic for the center of mass.

Taking the scalar product of (1.10a) with V and (1.10c) with UP we obtain the equationof motion,

r , x dPt. (1.12)

These equations of motion have a right-hand side that is the differential of the rate-of-work of applied impulses and the differential of the rate-of-work of applied torques aboutthe center of mass, respectively. Expressions (1.12) are scalar equations for the changeof state of a rigid body subject to a number n of active impulses P/.

Example 1.2 A cube resting on a flat level plane has mass M and sides of length 2a.The cube has slightly convex sides so only the edges touch the plane. One edge of thecube is raised slightly and then released so that when the opposite edge C strikes thesurface, the cube is rotating with angular velocity &>_. The impact is perfectly plastic sothere is no bounce of edge C. Find the angular velocity co+ immediately after impact at C

and calculate the part of the initial kineticenergy that is lost at impact, (71 — T+)/T-.

2aSolution

Moment of momentum about C:

hc(r) = /33w(O + rG/c x

Polar moment of inertia for G:

133 = 2Ma2/3

MV(r).

Velocity of center of mass:

V_ = VB-V+ = Vc+

_ x rG/B = aco-(—n\ - n 2)+ x rG / c = aco+(-ni + n2).


Moment of momentum about impact point C:

2Ma2 / x 2Ma2co-

hC- = &>-n3 + a(ii\ + n2) x Maco-(—n\ — n2) = n3

2Ma2 / x 8 M a V

hc+ = — — co+n3 + a(ni +n 2 ) x Maco+(-n{ + n2) = n3.

Impulsive moment about impact point C equals zero during impact [Eq. (1.6)]:

h c + = h c _ => CD+/CD- = 1/4.

Kinetic energy (planar motion):

T = |

71 = Ma2co2_ + \Ma2(o2_, T+ = Ma2co2+ + \Ma2a>2

+.

Part of initial kinetic energy absorbed in impact:

r_ - r+ _ 1571 ~ 16'

Equation (1.6) relates the moment of forces about the origin to the rate of change ofmoment of momentum about the origin. Is there a similar relation for moment of forcesabout a point that is moving?

1.3.4 Rate of Change for Moment of Momentum of a System about a PointMoving Steadily Relative to an Inertial Reference Frame

Let point A be coincident with point O but moving steadily at velocity VA relativeto an inertial reference frame, and denote the position vector of the /th particle relativeto A by Pi. If hA denotes the moment of momentum of the system of n particles withrespect to A, then Eq. (1.6) gives the following theorem.

For a system of particles, the rate of change of moment of momentum with respectto point A is equal to the moment of external forces about A if and only if

(i) VA = 0, so that point A is fixed in an inertial reference frame; or(ii) VA and V are parallel.

In either case,

dhA n

During a period t in which the configuration of the system does not vary, Eq. (1.13) canbe integrated with respect to time to give

hA(r) - hA(0) = £ p , - x P,(0- (1.14)

Example 1.3 A system consists of two particles A and B that are connected by a lightinextensible string. Particle A with mass 2M is located at pA = (1, 1, 0) and has velocityVA = (1, 0, 0) in a Cartesian reference frame, while particle B with mass M is located atpB = (3, 2, 0) and has velocity VB = (\, y, 0). Find (i) the component of velocity y andthe out-of-plane component co of the angular velocity of the string; and (ii) the momentof momentum of the system about the origin O.

SolutionPosition of B relative to A,

PB/A = PB ~ PA = (2, 1. 0).

Velocity of B relative to A,

Inextensible string requires

o = V B / A - P B / A =

Thus (i) velocity of B is

Angular velocity u; causesVB/A = u x p B / A = (0, 0, a>) x (2, 1, 0)

= <w(-1,2,0)

, » = i.Thus (ii) the moment of momentum about O is

2h o = J^p , . x Af/V,-

= 2Af(0, 0, - 1 ) + Af(0, 0, 2) = 0.

1.4 Decomposition of a Vector

Any vector v can be decomposed into a component vnn in direction n and acomponent vte in a direction perpendicular to n:

v = v • nn + (n x v) x n = vnn + vtewhere

vn = n • v, vt = |n x v|, e = ^ ( n x v) x n.

This relation is particularly helpful for analyzing oblique impact of rigid bodies; it is usedto resolve the relative velocity and impulse at the contact point into components normaland tangential to the surfaces of the colliding bodies.

It is worth noting that the vector v has a component perpendicular to n that can beexpressed alternatively as

vte = ( n x v ) x n = v — v- nn. (1-15)

Problems 19

1.5 Vectorial and Indicial Notation

Frequently there is a need to express a vector variable such as the velocity V ofa point in terms of the components of velocity in some reference frame. Let n, be a setof mutually perpendicular unit vectors fixed in reference frame *R. The components ofvelocity in this reference frame have magnitudes that can be expressed as Vl; = V • nt,/ = 1, 2, 3. In vectorial notation the vector is expressed in terms of its componentsV = 5Zf=1V • liiiii = ViUi, where the last equality follows from repeated subscriptsimplying summation over the range of spatial coordinates. On the other hand, in indicialor shorthand notation V = V/, i.e., unit vectors are implied rather than written explicitly.Similarly, if r denotes the position of C relative to a point O fixed in 9*, then in vectorialnotation r = r,-n/ and in indicial notation r = r/.

As a consequence of these definitions, the inner or dot product can be expressed inalternative forms,

3

r.y = riVi^J2nVi (1.16)1=1

where a repeated subscript (e.g. /) implies summation over the range of spatial coordinates.Likewise the vector or cross product can be expressed as

3 3

r x V - sijkrjVk = ^£eijhrjVk (1.17)j=\ k=\

where stjk is the permutation tensor which has values e^ = +1 if indices are in cyclicorder, stjk = — 1 if indices are in anticyclic order and £;y* = 0 if an index is repeated.

PROBLEMS

1.1 A wood block of mass M is struck at the center of one side by a bullet of mass M'that strikes at a normal incident speed Vb. The bullet is captured by the block. Findthe final speed of the block and the fraction of the bullet's initial kinetic energy 7bthat is transformed to heat.

1.2 Direct impact of a body with mass M (g) against a rigid wall at an incident velocityVb (m s"1) results in a contact duration tf (ms). For collision conditions (M, Vb, (/•)>calculate an estimate of the maximum force Fmax (N) during a collision of each ofthe following bodies: tennis ball (45, 44, 4.0); golf ball (61,44, 0.6); Ping Pong ball(2.5, 44, 0.5), steel hammer (103, 1, 0.2).

1.3 For the two particle system in Ex. 1.3, find the center of mass p of the system, thevelocity VG of the center of mass, and the inertia dyadic I and moment of momentumh relative to the center of mass. Verify that these relations satisfy h = I • UJ.

1.4 A rigid uniform bar of mass M and length 3L lies across two parallel rails B and B'that are separated by width L. The bar is transverse to the rails and centered. The endclosest to B' is raised a small distance and then released so that the bar is rotatingwith angular speed co- when it strikes B'.(a) Find the ratio &>+/&>_, where 00+ is the angular speed of the bar at the instant

immediately after impact, if the impact with rail B' is perfectly plastic.(b) Find co+/a>- if the impact with rail B' is elastic.


(c) Obtain the fraction of the incident kinetic energy 7!_ that finally is lost as a resultof collision (a).

1.5 A prismatic cylinder with polygonal cross-section has n equal sides, where n > 4.Between adjacent sides a regular polygon has an included angle 2a, where a = n/n.Let the prismatic cylinder of mass M have a radius a from the center O to each vertexQ. Find that this prismatic cylinder has a polar moment of inertia / = Ma2(2 +cos 2a)/6 about the center of mass O.

If the cylinder is rolling on a flat and level surface and before vertex Q impactswith the surface it has angular speed &>_, obtain an expression for the angular speedco+ the instant after Q strikes the surface. Assume that after impact, Q remains incontact with the surface.

1.6 A wheel of radius a and radius of gyration kr = (I/M)l/2 about the center of massrolls upon a rough horizontal surface which may be idealized as a series of uniformserrations of pitch 2b. The wheel rolls without slip or elastic rebound on the tips ofthe serrations under the action of a constant horizontal force F applied at its center.(a) Show that at each impact the ratio of the angular speed at separation to that at

incidence satisfiesco+ _k2 + a2-2b2

co~ k2 + a2

(b) Find that when steady conditions are reached the wheel moves with a fluctuatingforward velocity and

Mbv2 k2 + a2-b2

F~a2-b2 a2 + k2

where V\ is the maximum forward component of velocity in each cycle.1.7 For vectors r = {rx, r2, r 3 } r and V = {V\, V2, V?,}T verify that r x V = eijkrj Vk.

CHAPTER 2

Rigid Body Theory for Collinear Impact

The value of a formalism lies not only in the range of problems towhich it can be successfully applied but equally in the degree towhich it encourages physical intuition in guessing the solution ofintractable problems.

Sir Alfred Pippard, Physics Bulletin 20, 455, 1969.

Two bodies, labeled B and B', collide when they come together with an initialdifference in velocity. Ordinarily they first touch at a point that will be termed the contactpoint C. During a very brief period of contact, the point C on the surface of body B iscoincident with point C on the surface of body B'. If at least one of the bodies, B orB', has a surface that is topologically smooth at the contact point (i.e., the surface hascontinuous curvature), there is a plane tangent to this surface at C; the coincident contactpoints C and C lie in this tangent plane. If both bodies are convex and the surfaces havecontinuous curvature near the contact point, then this tangent plane is tangential to bothsurfaces that touch at C; i.e., the surfaces of the colliding bodies have a common tangentplane. The direction of the normal to the tangent plane is specified by a unit vector n;this direction is termed the common normal direction. The contact force and changesin relative velocity at the contact point C will be resolved into components normal andtangential to the common tangent plane.

2.1 Equation of Relative Motion for Direct Impact

Consider two colliding bodies B and B' that have masses M and M' and time-dependent velocities V(t) and V'(t) in the direction parallel to n. In a direct collisionthese bodies are not rotating when they collide, so that the velocity is uniform in eachbody (i.e. the same at every point). During contact there are equal but opposite com-pressive reaction forces which develop at contact points C and C ; these forces opposeinterference or overlap of the contact surfaces. In the case of direct impact betweencollinear bodies the relative velocity between the contact points C and C remains par-allel to the common normal direction throughout the contact period. A reaction forcedevelops at the contact point as a consequence of compression of the local contactregion; this force opposes relative motion during contact. In a direct collision the re-action force acts in the normal direction; i.e. parallel to the velocities, as illustrated inFig. 2.1. If the colliding bodies are hard, the contact force is very large in comparison withany body force; consequently, in rigid body impact theory any body or applied contact

21

22 2 / Rigid Body Theory for Collinear Impact

V1

Figure 2.1. Collinear impact of two rigid bodies with contact points separated by aninfinitesimal deformable particle. The particle represents small local deformation of thecontact region.

forces of finite magnitude are negligibly small in comparison with the reaction at thecontact point C. The finite body forces are ignorable because they do no work during thevanishingly small displacements that develop during an almost instantaneous collision.This is why a body force such as gravity does not affect the changes in velocity occurringin a collision. During impact between hard bodies the only active forces are reactionsat points of contact. These reaction forces are indefinitely large; nevertheless, they pro-duce a finite impulse that continuously changes the relative velocity during the instant ofcontact.

The assumption that the deformation of colliding bodies can be lumped in an infinites-imal deformable particle between the contact points is a key to obtaining changes invelocity as a function of impulse during the infinitesimal contact period. At the contactpoint, bodies B and B' are subjected to contact forces ¥(t) and F'(0 which have normalcomponents of force F(t) = F • n and F\t) = F' • n respectively. These reactions generatenormal components of impulse P(t) and P\t) where

dP = Fdt and ' = F'dt.

The translational equation of motion for each body in direction n can be expressed as

MdV=dP and Mr dV = dP'.

Let the normal component of relative velocity across the deformable particle at the contactpoint be v = V - V .

This modeling recognizes that because the deformable particle has negligible mass,the reaction impulses acting on either side of this particle are equal in magnitude butopposite in direction. Thus the same reaction impulse acts on each colliding body but thedirections of these impulses are opposed,

dP = -dP\

Defining the impulse on body B as positive, dp = dP, and substituting the translational

2.2 / Compression and Restitution Phases of Collision 23

equations of motion into the definition of relative velocity between the contact pointsgives the differential equation for changes in the normal component of relative velocity,

= m~ldp (2.1)

where the effective mass m is defined as

m = (M-1 + M'- ')-1 = - ^ - (2.2)M + M

After integration of Eq. (2.1) and application of the initial condition i?0 = v(0) = V(0) —Vf(0), the normal relative velocity v(p) is obtained as a function of normal impulse p,

v = vo + m~lp, where v0 < 0 (2.3)

Thus during collision the normal component of relative velocity is a linear function ofthe normal impulse.

The key to calculating changes in velocity during impact is to find a means of evaluatingthe terminal impulse /?/ at separation. The theory of rigid body impact will be more usefulif the terminal impulse can be based on physical considerations - more useful in the sensethat if experimentally obtained values of physical parameters are representative of theunderlying sources of dissipation, these impact parameters will be applicable within arange of incident velocities. In Sect. 2.5 the terminal impulse is related to the energeticcoefficient of restitution; this coefficient represents dissipation of (kinetic) energy due toinelastic deformation in the region surrounding the contact point.

2.2 Compression and Restitution Phases of Collision

After the colliding bodies first touch, the contact force F(t) rises as the de-formable particle is compressed. Let 8 be the indentation or compression of the deformableparticle. (The particle represents the compliance of the small part of the total mass whichhas significant deformation, i.e. the region of the bodies that surrounds the contact pointC.) Without detailed information about the compliance of the colliding bodies there is noway of obtaining 8 directly. Nevertheless, if compliance is rate-independent, the maxi-mum indentation and maximum force occur simultaneously when the normal componentof relative velocity vanishes.1 Figure 2.2a illustrates the normal contact force as a func-tion of indentation 8, while Fig. 2.2b shows this force as a function of time. The lattergraph shows the separation of the contact period into an initial phase of approach orcompression and a subsequent phase of restitution. During compression, kinetic energyof relative motion is transformed into internal energy of deformation by the contact force- the contact force does work that reduces the initial normal relative velocity of the col-liding bodies while simultaneously an equal but opposite contact force does work thatincreases the internal deformation energy of the deformable particle. The compression

1 Chapter 5 considers impact of bodies composed of viscoelastic or rate-dependent materials. For impactswhere the contact region is viscoelastic the maximum force does not necessarily occur simultaneouslywith vanishing of the normal relative velocity.


Fc

F(6)

Fit)

compression restitu-tion

(b)

p(t)

(c)

(d)

I restitu-compression^ | tion

Vf

I 1Pc

(e)Pf

Figure 2.2. Normal contact force F as a function of (a) relative displacement 8 and (b)time t; (c) normal impulse /?(>) as function of time £; and changes of normal velocities V, Vof colliding bodies as functions of (d) time / and (e) normal impulse p.

phase terminates and restitution begins when the normal relative velocity of the contactpoints vanishes. During the subsequent phase of restitution, the elastic part of the internalenergy is released. Elastic strain energy stored during compression generates the forcethat drives the bodies apart during restitution - the work done by this force restores partof the initial kinetic energy of relative motion. The compliance of the deforming regionduring restitution is smaller than that during compression, so when contact terminatesthere is finally some residual compression 8f of the deformable particle.

At any time t after incidence, the normal component of contact force F has an im-pulse p which equals the area under the curve of force shown in Fig. 2.2b. Since normalforce is always compressive, the normal component of impulse increases monotoni-cally as illustrated in Fig. 2.2c. Thus the normal impulse p can replace time t as an

2.3 / Kinetic Energy of Normal Relative Motion 25

independent variable. During compression the increasing impulse slows body B' and in-creases the speed of B as illustrated in Fig. 2.2d. Let the instant when indentation changesfrom compression to restitution be tc. The colliding bodies have a relative velocity be-tween contact points that vanishes at the end of compression: v(tc) = 0; i.e., compressionterminates when the contact points have the same speed Vc in the normal direction.Figure 2.2e illustrates that the contact point of each body experiences a change in ve-locity that is directly proportional to the normal reaction impulse at the contact point Cas expressed by Eq. (2.3). The reaction impulse pc = fQ

c F(t)dt which brings the twobodies to a common speed is termed the normal impulse for compression; this impulse isa characteristic which is useful for analyzing collision processes. The normal impulse forcompression is obtained from Eq. (2.3) and the condition that at the end of compressionthe normal component of relative velocity vanishes [v(pc) = 0]. Hence the normal im-pulse for compression is the product of the effective mass and the initial relative velocityatC,

pc = —mvo, where v 0 < 0. (2.4)

2.3 Kinetic Energy of Normal Relative Motion

During collision each body undergoes changes in velocity that can be expressedas a function of the normal component of impulse, p:

V = V0 + M-lp(2.5)

V' = Vi - M'-Xp.

These equations give changes in the normal component of relative velocity v(p):

v = vo + m~lp, where v = V — Vr

Compression terminates when both bodies have a common normal component of velocityVc = V(pc) = V'{pc)', this occurs when the normal impulse for compression pc = —mvo.At this normal reaction impulse the bodies have normal velocities that can be expressedas

V(pc) = Vo-%-vo and V \pc) = Vo' + - % 0 . (2.6)M M'

If the system has an initial kinetic energy 7b, where

| 0 | 0

then at the transition from compression to restitution the system kinetic energy Tc =T(pc) = (M + MOV^/2 will have been reduced to

TC = TO- \mv\. (2.7)

The kinetic energy lost during compression is referred to as the kinetic energy ofnormal relative motion at C. For collinear impact between bodies B and Br, this equalsthat part of the initial kinetic energy of relative motion which is due to the normalcomponent of relative velocity. During compression contact forces do work on collidingbodies that transforms this kinetic energy of normal relative motion into internal energyof deformation. For elastic deformations, this internal energy that is stored in the bodies

is known as strain energy. The elastic strain energy is the source of the normal componentof force that drives the contact points apart during the restitution phase of collision.

2.4 Work of Normal Contact Force

Work done by the normal contact force during separate phases of compressionand restitution gives a relationship between impulse applied during compression, pc, andthe final or terminal impulse at separation, pf. During compression the normal contactforce does work on the deformable particle (in fact, on the small deforming region ineach body surrounding the initial point of contact); this work deforms the particle andraises its internal energy. Of course the counterpart to the force that compresses theparticle is the equal but opposite force that reduces the kinetic energy of normal rela-tive motion during compression. Part of the energy absorbed during compression of theparticle is recoverable during restitution; the recoverable part is known as elastic strainenergy.

The work Wn done on the compressible particle by the normal component of force Fcan be calculated by recognizing that the force is related to the differential of impulse,dp = Fdt, so that

Wn= f Fvdt' = f vdp'.Jo Jo

(2.8)

2.5 Coefficient of Restitution and Kinetic Energy Absorbed in Collision

Unless the impact speed is extremely small, there is energy dissipated in a col-lision. This can be due to plastic deformation, elastic vibrations and also rate-dependentprocesses such as viscoplasticity. Whatever the cause, it results in an inelastic or irre-versible relation between the normal component F of the contact force and the com-pression 8. This dissipation results in smaller compliance during unloading (restitution)than was present during loading (compression); i.e., the force-deflection curve given in

F(6) v(p}

vf

(a) (b)

Figure 2.3. (a) Work Wc done by normal contact force F against bodies during periodof compression, and work Wf — W c recovered during restitution, as functions of normalrelative displacement 8 at contact point, (b) Work of normal contact force related to changesin normal relative velocity during periods of compression (p < pc) and restitution (pc <P < Pf)-

2.5 / Coefficient of Restitution and Kinetic Energy Absorbed in Collision 27

Fig. 2.3a exhibits hysteresis. The kinetic energy of relative motion that is transformed tointernal energy of deformation during loading equals the area under the loading curvein Fig. 2.3a; this area is denoted by Wc = Wn(pc). On the other hand, the area underthe unloading curve equals the elastic strain energy released from the deforming regionduring restitution; in Fig. 2.3a this is denoted by Wf — Wc = Wn(pf) — Wn(pc). In therestitution phase the contact force generated by elastic unloading increases the kineticenergy of relative motion. These transformations of energy are due to work done by thecontact force. This work done by the reaction force can easily be calculated for the sepa-rate phases of compression and restitution if changes in relative velocity are obtained as afunction of normal impulse as illustrated in Fig. 2.3b; after initiation of contact the workdone during these separate phases is proportional to the area between the horizontal axisand the line describing the normal relative velocity at any impulse.

In a direct collision the reaction force is normal to the common tangent plane; duringcompression the impulse of the normal contact force does work Wn(pc) on the rigidbodies that surround the small deforming region - this work equals the internal energyof deformation absorbed in compressing the deformable region. An expression for thiswork is obtained by integrating (2.3) and recalling that pc = —

Wn(pc) = I" v(p)dp = vopc + \m-lp2c = -\mvl. (2.9)

Jo z zThis is just the kinetic energy of normal relative motion that is lost during compression.During the succeeding phase of restitution the rigid bodies regain some of this kineticenergy of normal relative motion due to the work Wn(pf) — Wn(pc) done by contact forces,

Wn(pf)-Wn(Pc)= f(vo+m-lp)dp=r^(l-^), vo<O. (2.10)

This work comes from and is equal to the elastic strain energy released during restitution.These expressions for work done by the contact force during separate parts of the

collision (period) are used to express the part of the initial kinetic energy of normalrelative motion that is lost due to hysteresis of contact force. Expressions (2.9) and (2.10)give the part of this transformed energy that is irreversible, and this can be used to definean energetic coefficient of restitution, e*.

Definition. The square of the coefficient of restitution, e\, is the negative of theratio of the elastic strain energy released during restitution to the internal energy ofdeformation absorbed during compression,

2 _ Wn(pf) - Wn(pc)Wn(Pc) • (

This coefficient has values in the range 0 < e* < 1, where 0 implies a perfectly plasticcollision (i.e., no final separation, so that none of the initial kinetic energy of normalrelative motion is recovered), while 1 implies a perfectly elastic collision (i.e., no lossof kinetic energy of normal relative motion). For direct impact the final impulse pf andthe total loss of kinetic energy are both directly related to the coefficient of restitutione* by expressions (2.9) and (2.10). Using the negative of the square root of Eq. (2.11),

the impulse at separation is obtained as

pf= - mvo(l + e*) = pc(l + <?•). (2.12)

For direct collinear impact the normal component of relative velocity at separation isgiven by

pf. (2.13)

Hence the ratio of final to initial relative velocity and the ratio of normal impulse duringrestitution to that during compression are also directly related to e*\

Vf A Pf ~ PCe* = — — and e* = pc

The first of the expressions above is the same as the definition of the kinematic co-efficient of restitution, e = —Vf/vQ. Newton first defined this impact parameter on thebasis of his measurements of energy loss in collisions between identical balls. Newtonpresumed (incorrectly) that the coefficient of restitution is a material property. In hisPrincipia (1686) he gave values for spheres made of steel (e = 5/9), glass (e = 15/16),cork (e = 5/9) and compressed wool (e = 5/9). The relationship between the normalimpulse for restitution and that for compression is termed the kinetic coefficient of restitu-tion, e0 = (pf — pc)/Pc This coefficient was defined by Poisson (1811), who recognizedthat it is equivalent to the kinematic coefficient of restitution for direct impact betweenrough bodies if the direction of slip is constant.2 Subsequently, in Chapter 3, it will beshown that the kinematic, kinetic and energetic definitions of the coefficient of restitutionare equivalent unless the bodies are rough, the configuration is eccentric and the directionof slip varies during collision.

2.6 Velocities of Contact Points at Separation

With terminal impulse p/, at each contact point C or C the normal componentof velocity can be obtained from Eq. (2.6) and (2.12). At separation each contact pointhas a velocity Vf = V(pf) or V'f = V\pf) that is given by

Vf = Vo + M-lPf = V0-

(2.14)

At separation Eq. (2.13) gives a normal relative velocity Vf = v(pf) as

vf =vo + m~lpf = -e*v0. (2.15)

This is the same as that given by the kinematic definition of the coefficient of restitution.Alternatively the velocity changes during restitution can be obtained from the changes

in kinetic energy during this phase of impact. The final velocities are expressed in a formsomewhat like that of (2.6),

Vf = Vc + (m/M)vf and V'f = Vc - (m/M')vf. (2.16)

2 Rough bodies have a tangential force - friction - that opposes relative tangential motion, or slip, at thecontact point C.

2.6 / Velocities of Contact Points at Separation 29

This gives a terminal kinetic energy

Tf = Tc+mv2f/2

where Tc = (M + Mf) V?/2. Hence for a frictionless collinear impact the terminal relativevelocity Vf is related to changes in kinetic energy by

u . (2,7)If irreversibility of the contact force is the only source of dissipation during collision,

then at separation the system suffers a final loss of kinetic energy,

This loss of energy depends directly on the kinetic energy of normal relative motion atthe contact point C.

Example 2.1 A stationary sphere of mass M is struck at a normal angle of obliquityby an identical ball moving at 10 m s"1. (i) Calculate the work done by the contact forceduring compression Wc and the part of the initial kinetic energy To that is transformedduring the period of compression, (ii) Calculate the work done during restitution, Wf — Wc.(iii) Obtain an expression for the terminal impulse pj from the energetic coefficient ofrestitution and the ratio of work done during restitution to the negative of that done duringcompression, (iv) Calculate the relative velocity between the spheres at separation andthe final velocity of each sphere as functions of the coefficient of restitution e*.

Solution

Effective mass:

m = M/2.

Initial relative velocity:

v0 = 0 - V'(0).

Initial momentum:

MV(0) + M'V'(O) = 0 - 2mv0.

Initial kinetic energy:

Equation of relative motion:

v(p) = vo + m~lp.

Impulse terminating compression:

pc = -mv0, v(pc) = 0.

(i) Work during compression:

Wc= f Cv( 2

Jo

Kinetic energy at end of compression:

Tc = To + Wc = mvlll.

Part of initial kinetic energy TQ transformed during compression:To - Tc = 1

To 2'

(ii) Work during restitution:

wf-wc == (Pf - Pcf/Im.

(iii) Definition of coefficient of restitution,

2._ Wf-Wc (Pf- Pc)2

wc PI •gives impulse at separation

Pf — —mvo(l + e*), Vo < 0.

(iv) Relative velocity at separation:

Velocity of center of inertia:

vc = v(Pc) =Velocity at separation:

Vf = V(pf) =VC + ^vf - V'(0).

2.7 Partition of Loss of Kinetic Energy

For a collision between dissimilar bodies, the loss of kinetic energy duringthe period of compression can be divided into a kinetic energy loss for each body byconsidering separately the work done on each body by contact forces. In order to achievethis partition the change in velocity of each body relative to the center of inertia must beconsidered separately. The center of inertia is a reference frame moving in the commonnormal direction with constant speed Vc throughout the period of contact. This speed isobtained from the principle of conservation of translational momentum and recognitionthat at the termination of the compression phase, the two bodies have a common velocity

MV(0) + M'V'(0)Vc = — M T W ' — • (2>18)

During a collision between bodies B and Br, let the normal component of velocitiesrelative to the center of inertia be defined as V\ = V — V c and v2 = Vc — V. After

2.7 / Partition of Loss of Kinetic Energy 31

substituting into (2.14), a relation between the momenta of relative motion is obtained,Mv\ = M'v2. Hence the kinetic energy T(p) can be expressed as

T = 0.5M(Vc + vxf + 0.5M'(Vc - v2f

= 0.5 [(M + M')V2 + Mv\ + M'v2]

i.e., the kinetic energy separates into a part associated with the velocity of the center ofinertia (which does not change during impact) and parts due to the relative motion of eachbody separately. After using the momentum relation, the final loss of kinetic energy forthe impact pair will be

To - Tf = 0.5 ( l + ^j M[v2(0) - v2(pf)]

= 0.5 (l + ^ \ M'[v2(0) - v22(pf)]. (2.19)

Each body dissipates some kinetic energy of relative motion (relative to the velocity ofthe steadily moving center of inertia). At the termination of compression there is a normalimpulse pc and the loss of kinetic energy for each body can be expressed as

To - Tc = 0.5 f 1 + — \ Mv 2(0) = 0.5 ( 1 + — j M'v 2(0).

Hence the final loss of energy for the system in comparison with the energy absorbedduring compression is obtained as

= 1 - V-^LL (2.20)To - Tc vf (0) yj(0)

implying that Vi(pf)/v\(0) = v2(pf)/v2(0). Consequently the final change in relativevelocity for each body is in proportion to the incident velocity relative to the velocity ofthe center of inertia. This partition of change of relative velocities is independent of anydifference in compliance of the bodies.

Example 2.2 Smooth billiard balls B and B' with equal masses M are rolling on a leveltable before they collide; at incidence the centers of mass have initial velocities Vo andVo, respectively, while the point where each ball touches the table is stationary. Beforeimpact the centers of mass are moving in directions at angles 6 and 0', respectively,relative to the common normal direction, as shown in Fig. 2.4. Find at the termination ofimpact the velocity of the point on each ball which touches the table. These velocities arethe initial post-impact sliding velocity of each ball on the table.

Solution During impact the normal component of relative velocity betweencontact points C and C is v = V • n — V' • n. The balls have a center of mass O or O\respectively, and the initial velocities of the contact points are given by

/ . pVo = Vo sin 0 ei + [Vo cos 0 + -^

Vo = t/or sin 0 ' e i + (?<( cos 9' - ^


Figure 2.4. Plan view of initial velocities for centers of two colliding spheres.

Since the balls are spheres, the centers of mass and the contact points C and C have thesame normal component of relative velocity v. During impact this is a function of theimpulse,

v = Vo + p/m

v0 = % cos 6 — VQ COS 0''.

At the end of compression the normal impulse is pc = —mvo. Thus at separation thecenters of mass have velocities

/ . mvo\V/ = Vb sin 0 ei + I Vo cos 0 — (1 + £*) )n

V M )\f = VQ sin 0f ei + I % cos 0f + (1 + e

and at the contact point there is a normal relative velocity Vf = (V/ — Vp • n = — e*Vo.Meanwhile the points of contact between the spheres and the table that initially werestationary, now slide at velocities

v,--(. + . ) £V, = (!+,.)!=%.

J M

PROBLEMS

Normal Incidence

2.1 A railway boxcar with mass 50,000 kg is rolling at 2 m s"1 when it is struck by anovertaking boxcar of mass 100,000 kg rolling in the same direction at 3 m s"1. Fora coefficient of restitution e* = 0.5, find the velocity of each boxcar at separationand calculate the loss of kinetic energy in the collision.

Problems 33

2.2 Regulations require that a baseball has mass M = 150 g and a coefficient of restitu-tion e* = 0.546 it 0.03 at an impact speed of 26 m s~l. Suppose a baseball is thrownat 40 ms" 1 and is hit by a bat with mass M' = 800 g; the ball is hit at the center ofmass of the bat which is traveling at a speed of 25 m s"1 in a direction opposite tothat of the ball. Find the speed of bat and ball at the instant of separation. Also showthat the part of the incident kinetic energy that is dissipated in the collision equalsI —el. (Assume the angle of incidence is normal, the rotational kinetic energy of thebat is negligible and the coefficient of restitution is independent of the impact speed.)

2.3 Three elastic spherical balls Bi, B2 and B3 have their centers aligned. Balls B2and B3 are stationary and slightly separated before ball Bi, traveling at speed Vb,collides against B2. Prove that the final speed V?>{pf) of ball B3 is a maximum ifball B2 has mass M2 equal to the geometric mean of the masses of Bi and B3; i.e.M2 — (MiM 3)1/2. Find the maximum value of the ratio V3 (/?/)/ Vb by letting themass ratio M3/M1 = a2 (proof originally by Huyghens).

2.4 In Problem (2.3) let the number of balls be an arbitrary number n rather than 3. Toobtain a maximum ratio of the final to the initial speed Vn(pf)/Vo, prove that themass of each ball is related to that of its neighbors by M,- = (Mj-iAfz+i)1/2, where1 < i < n. Find the maximum ratio Vn(p/)/ Vb as a function of n.

Oblique Incidence

2.5 An incident particle B' scatters through angle 0f when it strikes a stationary particleB. After this elastic impact the velocity of particle B makes an angle 0 with the initialdirection of the incident particle. Find the mass ratio of the two particles W/M asa function of angles 0 and 0'. (Hint: first obtain that for each body the change inmomentum in the common normal direction equals 2m VQ COS 0.)

2.6 A deuteron is the nucleus of an atom of heavy hydrogen in a molecule of heavy water;it is composed of a proton and a neutron and has about double the mass of a neutron.If a neutron traveling at 100 m s"1 in a nuclear reactor suffers an elastic collision witha deuteron that is initially at rest and the neutron is scattered through an angle of 30°,find the scattering angle of the deuteron. Also find the final speed of each particle.

2.7 A smooth sphere traveling at an initial speed Vb collides against an initially stationaryidentical sphere with angle 6 between the common normal direction and the pathof the traveling sphere. Show that the course of the initially moving sphere will bedeflected through an angle n/2 - 0 - tan-1[(l - e*)(2 tan 0)"1]. For the second(initially stationary) sphere, find the final direction of travel as an angle measuredfrom the initial course of the first sphere.

2.8 A compound pendulum with mass M' and radius of gyration kr rotates freely aboutits pivot with angular speed COQ. A stationary sphere of mass M has a direct collisionwith the pendulum at a distance X from the pivot. If the collision is elastic and atseparation the pendulum is stationary while the sphere has speed XCOQ, show that theimpact occurs at distance X = kr from the pivot. If instead the collision is inelasticwith coefficient of restitution e*, find the angular speed of the pendulum and thespeed of the ball at separation if the point of impact is located at X = kr.

2.9 A uniform rod of mass M' and length 2L represents a baseball bat. The bat has aradius of gyration kr about the center of mass, kr = L/>/3. The rod pivots about africtionless pin at one end. Assume that the initially stationary bat is driven about the

pin by a couple C that is constant. After rotation through an angle 0 = n/2 the batstrikes a stationary ball of mass M. (The rotational speed of the bat when it strikesthe ball is given by <9_ = ^JnC/M\L2 + £r

2).)(a) Find the impact point on the bat where a direct collision produces no reaction

at the pin. Show that this point, the center of percussion, is located a distance§ + L away from the pivot where £ L = kf.

(b) For a bat that strikes the ball at the center of percussion, find that a maximumimpulse is imparted to the ball if the mass ratio of bat to ball, Mf/M = 4. Forthis optimal bat striking a baseball, find an expression for the maximum speedof the batted ball if the collision is elastic.

2.10 Your local Little League team has asked you to provide advice on the optimal lengthof baseball bats in order to hit the ball as far as possible. Let the bat have a uniformdensity per unit length p and length L, and suppose that the kinetic energy of thebat at impact is a constant 7b. The ball with mass M has speed wo when it is struckby the bat. Assume that the batter's hands can be considered to be a pinned jointthat cannot restrain the changes in velocity during impact. Furthermore, assume thatthere is direct impact between ball and bat and that the coefficient of restitution is e*.(a) Explain why optimal range is achieved if the ball is struck near the end of the bat.(b) Find the length L of the bat for maximum range.

CHAPTER 3

Rigid Body Theory for Planar or 2D Collisions

Nature uses only the longest threads to weave her patterns, so eachsmall piece of her fabric reveals the organization of the entire tapestry.

R.R Feynman, The Character of Physical Law, 1965

Eccentric impact configurations have the center of mass of at least one ofthe colliding bodies off the common normal line which passes through the contactpoint. A consequence of impact in an eccentric configuration is that the impulse act-ing at the contact point gives an impulsive moment about the center of mass. Thiscauses changes of the angular as well as the translational velocities during the period ofcontact.

3.1 Equations of Relative Motion at Contact Point

Let bodies B and B' with masses M and M' collide at point C. Let the rotationalinertia for the center of mass of each body about the axis normal to the plane be specified byradii of gyration1 kr and k'r and let the impact configuration be defined by position vectorsrt = (n , r3) and r[ — (rj, r'3) from the center of mass of each body to the contact pointC. The bodies have angular velocities coi and co'; and velocities Vt and V{ at the centersof mass, respectively. These position and velocity vectors are defined in a Cartesiancoordinate system n\,n^, where n^ = n is normal to the common tangent plane throughthe contact point C and n \ is tangential to this plane, as depicted in Fig. 3.1. At the contactpoint of each body, a reaction force Ft or FJ develops that opposes interpenetration ofthe bodies during impact. These forces give differentials of impulse dPt and dP[ at C andC, respectively:

dPt = Ft dt and dP[ = F[dt, i = 1, 3.

With these terms defined, Newton's second law gives equations of motion for translationof the centers of mass,

dVi =M~xdPi(3.1)

dv; Mfxdp;

1 Here the diacritical mark A over a symbol expresses that this variable is for the center of mass or foraxes passing through the center of mass. Usually, for kr, the radius of gyration about the center of mass,this diacritical mark will be suppressed.

35

36 3 / Rigid Body Theory for Planar or 2D Collisions

deformable.particle

Figure 3.1. Colliding rigid bodies with contact points separated by deformable particle.

and for planar rotation of the bodies,

(3.2)

where a repeated index indicates summation and the permutation tensor is given bySjjk = +1 if the indices are in cyclic order, eijk = — 1 if the indices are in anticyclic order andSijk = 0 if there are repeated indices. (The transformation from vector to index notation isr = rt and P == Pt. In index notation the vector product is expressed as r x dV = Sijk^j dPk.)The normal (or polar) radius of gyration about the center of mass of some common shapesof bodies can be found in Table 3.1.

With the construct of an infinitesimal deformable particle between the points of contact,changes in relative velocity between the bodies at the contact point C are obtained as afunction of impulse Pt during the contact. Since these are rigid bodies, the velocity ofthe contact point on each body is related to the velocity of the center of mass of the samebody by

Vt = V( + sijkcojrk, Vj = V- + Sijkd)'/^ (3.3)

The relative velocity at C is defined as the velocity difference

v. = y _ y!

After substituting Eqs (3.1)—(3.3) into an expression for the differential of relative velocityat the contact point dv( = dVt — dV{, the differential equations for velocity changesare obtained in terms of the components of impulse. In addition, since the deformableparticle has negligible mass, the differential impulses on either side of the particle areequal but opposite: dpt =dP[ = —dP(. Consequently, for planar velocity changes weobtain dvt = mj"1 dp;, or

dv3\ = mPi -h~\\dp (3.4)

3.2 / Impact of Smooth Bodies 37

Table 3.1. Radius of Gyration k for Axes through Center of Mass

Plane Sections

Shape

Rectangle - height a, base bThin circular disk - radius aThin circular ring - radius a,

thickness bTriangle - height a, base bEllipse - axes a, b

3D Bodies

Body

Circular cylinder - radius a, length LThin-walled circular cylinder - radius a,

thickness b, length LSphere - radius aThin-walled spherical shell - radius a,

thickness bEllipsoid - axes 2a, 2b, 2c

AreaA

abna2

2nab

ab/2nab

VolumeV

na2L2nabL

4na3/34na2b

4nabc/3

In-plane Radiusof Gyration,k

a/V\2a/2a/2

a/y/lSa/2 orb/2

Cross-section Radiusof Gyration,k

y/(a2+b2)/l2yj{6a2 + L2)/12

a^/2/5a^/2/3

y/(b2 + C2)/5

Polar Radiusof Gyration,k2=kr

y/(a2+b2)/l2a/V2a/V2

y/(a2+b2)/lSy/a2 + b2/2

Axial Radiusof Gyration,k2

a/V2a

aJT/5

where the elements fa of the configuration matrix can be expressed as

Pi = 1 + mrl/Mi* + mr^/M'k?

(3.5)

= l+ mr\lMk2r

To proceed, the tangential and normal components of reaction force need to be relatedin order to express the differential equations (3.4) in terms of a single, monotonouslyincreasing independent variable - the normal impulse p = p3.

3.2 Impact of Smooth Bodies

Smooth surfaces have negligible friction or adhesion; in this case the only com-ponent of force is that which is normal to the tangent plane. In collisions between bodieswith smooth contact surfaces, the tangential differential impulse vanishes:

dpi = dp2 — 0 for smooth contact surfaces. (3.6)

Example 3.1 A smooth solid sphere Br with mass M and radius R that is rolling on arough level surface collides with an identical ball B that is initially stationary. Before thecollision the center of mass G has speed V'(0). Suppose the collision is elastic (e* = 1).

The contact between the balls is smooth, but for sliding between the ball and the tablethere is a coefficient of friction //. Find the final velocity of each center of mass after thecollision at a time when sliding ceases.

Solution To analyze the oblique impact shown in Fig. 3.2, establish unit vectorsn\, n3 in the plane of the table with «3 = n in the direction of the normal to the commontangent plane. Let 0 be the angle of obliquity between the normal to the tangent planeand the initial velocity vector V^O) for the moving center of mass, so that V/(0) =Vr(0)(— sin 6n\+ cos 0 n3). Ball B' is initially rolling (no slip), so that the point on theball which touches the table A' is stationary; hence, at incidence the ball has componentsof angular velocity

o/,(0) = ^ - ^ cos 9,R

(o'3(0) = ^ - ^ sin 0.RR

When impact commences the contact point C on ball B' has tangential and normalcomponents of velocity

where C is located at r[ = GC relative to the center of mass G. At incidence the contactpoint C has components of relative velocity vt = ViC — V{ c, expressed as

vi (0) = 0 + V'(0) sin 0 = Vo' tan 0

v2(0) = 0+V/(0)cos6 = VQ

p3(0) = 0 - V'(0)cos<9 = -VQ

where VQ is the normal component of incident relative velocity at Cr. Notice that thecoordinate system has been set up so that the initial value of the normal component ofrelative velocity 1^(0) < 0 while the in-plane component ^i(O) > 0.

For impact between two identical bodies with smooth (frictionless) contact surfacesthe normal component of the equation of motion can be integrated over an initial part ofthe contact period to give

v3(p) = v3(0) + m m-x =

Figure 3.2. Oblique impact of a rolling sphere against a stationary sphere, where the normalcomponent of velocity of G' at impact is VQ. Arrows indicate (a) incident velocity of centerof mass, and (b) components of incident velocity (dashed) and change in velocity at G andG' during impact (solid).

3.2 / Impact of Smooth Bodies 39

Attheendof compression the normal component of relative velocity vanishes [v3(pc) = 0],giving a normal impulse for compression PC = MVQ/2 and, for elastic impact wheree* — 1, a normal impulse at separation pj = (1 + e*)/?c = MVQ. Hence the colliding bod-ies suffer equal but opposite changes in velocity,

AVt = MVQH3, AV] = - M V ^ .

In each ball these changes in velocity are uniform. Thus the initially stationary ball movesoff in direction 6 with a uniform velocity MV^ri^; i.e., at separation the struck ball is slidingon the table in direction 0 and it has not begun to rotate.

At the instant of separation, the ball B' that was initially rolling has a stationarycenter of mass and a velocity —VQH^ at point A' that is touching the table; i.e., thisball now slides in direction — n3 with the same angular velocity it had before impact,o)i(O) = (VQ/R, 0, (VQ tan 0)/R). After separation from impact, changes in velocity ofthe center of mass of ball B' occur in the n\,n$ plane due to friction at point A wherethe ball is sliding on the table. If friction satisfies the Amontons-Coulomb law with alimiting coefficient of friction /x and the collision occurs on a level surface where thegravitational constant is g, then for the period of sliding r that immediately followsseparation, integration of components of equations of translational and rotational motionof the ball give the following normal component of velocity of the center of mass and thecomponent of angular velocity in direction n \,

MVf3 =

2MR2 , 2MR2

t < r.

Sliding continues until the angular speed has slowed to the angular speed for rolling,co[ = Vf

3/R. Together with the preceding equations, this gives a period of sliding rB' =2 1 ^ / 7 ^ , and at the end of this period, a velocity VJ(rB0 = ( - VQ tan 6, 0, 21^/7) forthe center of mass. Figure 3.3a shows the velocity of the center of mass of each sphere atseparation and the change of velocity that occurs as the sphere slides on the table. At timerB' after impact the ball once again begins to roll. While the ball is sliding the frictionforce is constant, consequently, ball Br travels a distance X3 while sliding and

x3 = 2V20/49fig.

(a) (b)

Figure 3.3. Oblique impact of a rolling sphere against a stationary sphere: (a) velocity ofG and Gr at separation (dashed arrows) and change in velocity during postimpact sliding(solid arrows); (b) terminal velocities of G and G' at termination of sliding.


This distance is not very far. For VQ = 1 m s"1, a coefficient of limiting friction /x = 0.1and g = 9.81 m s~2 we obtain a distance x^ = 40 mm, which is less than the diameterof a billiard ball. As a consequence of the collision the path of ball B' has been divertedthrough an angle 6>B' = tan"1 [(sin 20)/(1.8 - cos 20)].

3.3 Friction from Sliding of Rough Bodies

3.3.1 Amontons-Coulomb Law for Dry Friction

Dry friction due to roughness of contacting surfaces can be represented by theAmontons-Coulomb law of sliding friction (Johnson, 1985). This law relates tangentialand normal components of reaction force at the contact point by introducing a coefficient oflimiting friction \± which acts if there were sliding. For negligible tangential compliancethe sliding speed s is given by s = Jv\ + v\. Denoting the normal component of adifferential increment of impulse by dp = dP =n • dpt, the law of friction takes theform

if v2x+v2

2=0 (3.7)

dp, dP2 = — ^Vl dp if v\ + v\>0. (3.8)

jEquation (3.7) expresses an upper bound on the ratio of tangential to normal force forrolling contact; i.e., if a force ratio less than /x can satisfy the constraint of zero sliding,then if prior to separation the sliding speed vanishes (s = Jv\ + v\ = 0), subsequentlythe contact sticks.

If sliding is present (s > 0), Eq. (3.8) represents a tangential increment of impulse orfriction force which, at any impulse, acts in a direction directly opposed to sliding. Thesliding direction can be defined by angle 0 = tan" l (v2/v\) which is the angle in the tangentplane measured from n {. Thus in three dimensional problems the components of slip are

V\ = s cos 0, v2 = s sin0

and (3.8) can be expressed as

dp\ = —[JL cos (pdp, dp2 — —/JL sin (j) dp, s > 0.

For planar impact v2 = 0 and (p = 0 or n.For sliding contact the Amontons-Coulomb law expresses that the ratio of tangential

to normal force is a constant; i.e., friction is independent of both sliding speed and normalpressure. During impacts of hard bodies with convex contact surfaces the normal pressurein the contact area is very large - certainly large enough to increase the true contact areadue to plastic deformation of surface asperities (see Greenwood, 1996). It was Morin(1845) who, at the suggestion of Poisson, performed experiments which showed that thedynamic coefficient of friction for impact was equal to that measured in quasistatic testsat steady, relatively low speeds of sliding.

During sliding the tangential force is related to the normal force by the law of friction(3.7)-(3.8), hence the equations of motion (3.4) can be expressed in terms of a single

3.3 / Friction from Sliding of Rough Bodies 41

independent variable - the normal reaction impulse. Since the normal contact force isalways compressive, this impulse is a monotonously increasing scalar function during thecontact period; i.e., the collision process can be resolved as a function of the independentvariable p.

3.3.2 Equations of Planar Motion for Collision of Rough Bodies

The law of friction is the key that relates tangential to normal impulse if thecontact is sliding. On the other hand, if friction is sufficient to prevent the development ofsliding, this law provides the coefficient of limiting friction for stick - i.e., it representsa constraint on friction force. At any impulse during contact, a means of discriminatingwhether the next increment of impulse is sliding or sticking is obtained by assuming thatthere is no tangential component of relative velocity (or slip) at C and then comparingthe ratio of the differentials of tangential constraint impulse to normal impulse for stickwith the coefficient of friction /x.

Equations of Planar Motion for StickFor an arbitrary normal impulse p = /?3, if there is no tangential acceleration, then Eq. (3.4)gives

dvi = m-\px dpx - fodp) = 0. (3.9)

Suppose any initial sliding vanishes V\ (p) = 0 at impulse ps, where 0 < ps < pf. In orderto maintain stick during p> ps,a specific ratio of tangential to normal reaction force isrequired; this is termed the coefficient for stick, \x. Thus

dp\ = ft dp =>> dv\ = 0 during p > ps

where for planar impact, jl is defined as

fi = fo/Pi- (3.10)The coefficient for stick depends solely on the distribution of mass; it can be either positiveor negative.

Stick occurs if the tangential relative velocity V\(p) = 0; this implies that the coef-ficient of limiting friction JJL > |/2| since otherwise the contact begins to slide.2 If thecontact sticks (v3 — 0 and /x > |/2|) during the period p > p s the ratio of reaction forcesat C is /2 and the direction of the tangential force depends on sgn(/32). While the contactpoint sticks, any changes in the relative velocity at C satisfy the following equations ofmotion:

dvJdp = 0(3.11)

dv3/dp = m l(p3 &)

Equations of Planar Motion for SlidingOn the other hand, if the contact is already sliding (/2/x), then the law of friction re-lates the tangential and normal components of reaction force: dp\ = —s/xdp, where

2 Notice that fi\ > 1 but for collinear impact, #2 = 0. Hence for a collinear collision, if initial sliding isbrought to a halt before separation, subsequently the contact sticks.

v3(0)i^(0)

0

• r

=N^3to)

X,

"3(0)

iSlO)_ ^ — • •

v3l0)

Figure 3.4. Changes in normal and tangential components of relative velocity at C asfunctions of normal impulse p for Coulomb friction and initial slip that is first brought toa halt at impulse ps and then reverses in direction.

s = sgn(v\) = V\l\v\|. For Eq. (3.4) this gives

dv3/dp =

(3.12a)

(3.12b)

These equations can be integrated to give the relative velocity at any impulse during aninitial period of unidirectional slip. For a normal component of relative velocity that isnegative at incidence, 03 (0) < 0, this gives

vi(p) = 01 (0) -

v3(p) =Henceforth in this chapter (and without loss of generality), the coordinate system willbe set up so that at incidence•, the tangential component of relative velocity is positive:i?i(0) > 0. This convention will eliminate the need to carry along in calculations the signof the current direction of slip, s.

With expressions (3.12)—(3.13), the equations of motion can be examined to identifythe range of inertia and friction parameters for different contact processes. Figure 3.4illustrates the changes in both normal and tangential components of relative velocity asa function of normal impulse p for an eccentric collision in which initial sliding ^i(0) isbrought to a halt at impulse ps and then reverses. The rates of change of velocity withimpulse are constant for ranges of impulse 0 < p < ps and ps 0, ^3 > 0 and^1A > P2 - Moreover, in order for normal force to oppose indentation, Eq. (3.12b) requires3

> - 1 to give dv3/dp > 0. (3.14)

3 For eccentric impact configurations (/% ^ 0) inequality (3.14) provides an upper bound on the coeffi-cient of friction in order to avoid jam (Stronge, 1990).

3.3 / Friction from Sliding of Rough Bodies 43

0 . 5 -

Figure 3.5. Regions of incident angle of obliquity and friction for different types of slidingprocesses at the contact point C.

The process of slip is described by (3.12a). For s > 0 slip is retarded if dv\/dp < 0,whereas for s < 0 retardation requires dv\/dp > 0. If initial retardation causes slip tovanish at impulse ps before separation, reversal in direction of slip requires dv\/dp > 0during p> ps. Conditions of this type give the range of parameters wherein it is possibleto have a particular slip process; e.g., the direction of tangential acceleration is con-stant so if sliding vanishes before separation, thereafter either the contact sticks or slipreverses:

can stick if 1 <

can reverse if 0 < < 1.

(3.15a)(3.15b)

Reversal can occur only if JJL > 0 [assuming s(0) > 0]. Figure 3.5 illustrates the range ofthese different types of contact processes as a function of a parameter /x//Z that combinesfriction and inertia properties. In this figure the vertical bands are given by inequalities(3.14) and (3.15).

Stick or slip reversal can occur only if the speed of slip vanishes before separation; oth-erwise sliding continues in the initial direction throughout the contact period. Integrationof (3.12) indicates that sliding vanishes during compression if

Y = (3.16a)

(3.16b)

where pc is the normal impulse at transition from compression to restitution and pf is theterminal normal impulse at separation. When sliding vanishes [V\(ps) = 0], the normal

v3(0) /33(1 + KM/A)while sliding vanishes during restitution if

r p\{\ + n/n) l o,(o) r ftd +11/a) 1

impulse ps can be obtained from (3.13a),l ) (3.17)

and the normal component of relative velocity is V3(ps) = v^iO) I m " 1 ^ +If sliding vanishes during compression, an additional impulse pc — ps occurs beforecompression terminates. For parameters that give slip reversal during compression, thecompression phase of collision has normal impulse

mv3(0) [1 , ( 2/zfe \vl(0)]11 + U ) } 0<Ps<Pc (3'18a)whereas for unidirectional slip

Pc = -(f t + Mft)~W3(0), ps > pc. (3.18b)

In Fig. 3.5 the diagonal line represents the ratio of incident velocities V\ (0)/y3(0) = tan ^0 ,where the initial slip is brought to a halt just as the contact points separate [Eq. (3.16b)].Angles of incidence that give initial sliding speeds outside this line, \\J/Q\ > tan"1 [(1 + e*)x (1 + /x//Z)(l + y/z//!)"1], result in slip that will slow but not halt before separation.

3.4 Work of Reaction Impulse

3.4.1 Total Work Equals Change in Kinetic Energy

The total work done on the colliding bodies by contact forces Wf = E?=1

can be calculated from the sum over all components of the partial work done by eachseparate component of impulse,

[*tf C Pi \Pf )

Jo JoWf= / FiVidt= / Vidpt. (3.19)

This work equals the change AT in the kinetic energy of the system. For a collisionbetween bodies B and Br that occurs at contact point C, the change in kinetic energy andhence the work can be expressed as

M[Vi{tf) - Vi(0)][Vi(tf)

h ( ' ) a>(0)][o>(t) + coj(0)]

M'

where matrices /,-_,• and 7y contain the moments (and products) of inertia of each bodyfor axes through each respective center of mass. During the brief period tf the reactionforce at contact point C imparts an impulse to each body, P,(^/) = M[Vi(tf) — Vi-(O)]and P((tf) — M[V/(tf) — Vj'(O)], and an impulsive torque about each center of mass,

k = I u[coj(tf)-coj(0)] and djkr'jP^ = t'u[a>'j(ff)-a>'j<!Oj], respectively. Thus the

3.4 / Work of Reaction Impulse 45

change in kinetic energy can be written as

[V f VJ(O)]

+ y [V/(tf) + t> ^

Together with the expression for equal but opposed contact forces pt = Pi = — PI andthe relation for relative velocity across the intermediate deformable particle, [V((t) =[Vj(t) — etjkrjCOkit)] — [V/(t) — eijkrj(o'k(t)], the preceding equation gives the total workdone by reaction forces,

Wf = ^[vi(tf) + vi(0)]. (3.20)

This relation was derived first by Thomson and Tait (1879) and subsequently appeared inRouth (1905, Art 346). The total work equals the sum of the work done by individual com-ponents of impulse Wf = E?=1 Wl(pf)\ in general, however, no expression such as (3.20)applies to the partial work Wt(pf) done by the /th component of the reaction impulse.

3.4.2 Partial Work by Component of Impulse

The energetic coefficient of restitution depends on the work done by the normalcomponent of contact force during the period of contact. A useful method of calculatingthis work is to use the following theorem for each separate period of slip (wherein thecomponents of contact force are proportional) and then sum the results for the period ofcollision.

The partial work Wn done on colliding bodies by the component of reaction impulsein direction ht during any period of unidirectional slip At = t2 —1\ equals the scalarproduct of this component of reaction impulse Apa and half the sum of the com-ponents in direction hi of the initial and final relative velocities across the contactpoint:

Wn = Apn[Vn(t2) + Vn{t\)]/2 (no summation on h) (3.21)

where Ap^ = «/[/?/fe) — Piih)]* Vn(t) = hiVi(t) and hi is a unit vector «//z, = 1.

PROOF Let Ap( be the impulse acting on a body during a period t2 — h • The aim is tocalculate the partial work Wn done by the component of impulse that acts in a directionparallel to a unit vector ht .4 At the contact point, changes in relative velocity are obtainedfrom the second law of motion,

Vi(t) = Vi{tx) + mjj1 APj, t > h (3.22)

where m^1 is the inverse of the inertia matrix for C that was given in Eq. (3.4). Herewe assume that the contact period is brief so that during contact the bodies do notchange in orientation. The impulse Api has a component in direction ht that equals

4 Components of vectors are defined in relation to a reference frame composed of a triad of unit vectors «/.


(Apn)hi = (hj Apj)ht = hiTi3j=lhj Apj. If the components of Apj increase proportion-ally during t >tu then an expression similar to (3.19) can be integrated to give the partialwork done by the component of impulse acting in direction h(,

Wn = (hj Apj^fiilViih) + m;kl Apk/2]}.

With (3.22) and after rearranging, this can be expressed as

Wn = hi APi {hj[Vj(t2) + Vj(ti)]}/2 = Aph [vn(t2) + vn(ti)]/2. q.e.d.

Theorem (3.21) is especially useful for impulsive forces, i.e. in the limit as tf —> 0.For a single point of collision between two smooth (frictionless) bodies there is only anormal component of impulse, so (3.20) and (3.21) are equivalent. With friction, however,tangential sliding during contact results in tangential impulses in addition to the normalimpulse. For friction that is in accord with the Amontons-Coulomb law, the tangentialimpulse increases in proportion to the normal impulse during any period of unidirectionalsliding. Hence for collisions between rough bodies, theorem (3.21) is applicable onlyduring a period of unidirectional sliding. Generally in order to calculate the partial workof a component of impulse using (3.21), the contact period must be separated into a seriesof discrete periods of unidirectional sliding.

In some cases it can be helpful to recognize that according to (3.21) the work Wt doneby the /th component of impulse is

so that the total work of three mutually perpendicular components of impulse can beexpressed as

W = ^[vjih) - Vj(tx)][vk(t2) + vk(h)]

^ i (3.23)

According to (3.20), however, this total work is independent of whether or not the impulseis unidirectional; i.e., the total work depends solely on the differences between the initialand final states of impulse and relative velocity at the contact point. On the other hand,for eccentric collision configurations (m,j ^ 0, / ^ j) Eq. (3.21) for the partial work ofany component of impulse is applicable only within periods where the contact force isconstant in direction (Stronge, 1992).

As a consequence of (3.23), Thomson and Tait (1879, Art. 309) correctly say "that ifany number of impacts be applied to a body, their whole effect will be the same whetherthey be applied together or successively (provided that the whole time be infinitely short),although the work done by each particular impact is in general different according to theorder in which the several impacts are applied" (italics added).

3.4.3 Energetic Coefficient of Restitution

A coefficient of restitution relates the normal impulse applied during the resti-tution phase to that during the compression phase; the sum of these impulses gives theterminal normal impulse pf when the contact points finally separate. For situations where


the direction of slip varies during collision, the only energetically consistent definition ofthis coefficient is the so-called energetic coefficient of restitution. This directly relates thecoefficient of restitution to irreversible deformation in the contact region. For analysesemploying impulse as an independent variable, this definition of the coefficient of resti-tution explicitly separates the dissipation due to hysteresis of contact forces from that dueto friction between the colliding bodies.

The work W3 done on the bodies by the normal component of reaction impulse p = p3

is given by

/

tip) rp

F3.vtdt= / v3dp.Jo

During the compression phase of collision, the work done by the normal componentof reaction impulse decreases the sum of the kinetic energies of the colliding bodiesand brings the normal component of relative velocity to rest; if tangential compliance isnegligible, this work equals the internal energy gained by the bodies during this period. Thepart of this internal energy that is recoverable is known as elastic strain energy. This strainenergy drives the contact points apart during the subsequent restitution phase of collisionand thereby restores some of the kinetic energy of relative motion that was absorbed duringcompression. The difference between the work done to compress the bodies and the workdone by the release of strain energy during restitution is the collision energy loss due tointernal irreversible deformation. This work is used to define an energetic coefficient ofrestitution. For an impulse-dependent analysis, this coefficient is independent of frictionand the process of slip.

The square of the coefficient of restitution, e\, is the negative of the ratio of theelastic strain energy released during restitution to the internal energy of deformationabsorbed during compression,

2 _ W3(pf) - W3(Pc)W3(Pc) f0

Pcv3(p)dP

where a characteristic normal impulse for the compression phase pc is defined as theimpulse that brings the normal component of relative velocity to rest, i.e. v3(pc) = 0.This characteristic impulse for the compression phase can be calculated from the lawsof motion. If the normal component of relative velocity is plotted as a function of im-pulse as shown in Fig. 3.4, the energetic coefficient of restitution is just the square rootof the ratio of areas under the curve before and after pc. Consequently, if the inter-nal energy loss parameter e* is known, the terminal impulse at separation py can becalculated.

3.4.4 Terminal Impulse pf for Different Slip Processes

In order to obtain the terminal impulse pf corresponding to any value of theenergetic coefficient of restitution e*, the work done by the normal reaction impulse iscalculated separately for each phase of collision in accord with Eq. (3.21). For planarimpact the work done on the colliding bodies by any component of reaction impulse can

be calculated most easily by reference to the diagram in Fig. 3.4, which plots changesin relative velocity across the contact point as a function of impulse. Each line segmenton this diagram is the solution of the equations of motion (3.4) during a separate periodof slip. In the diagram, there is a crosshatched area beneath a relative velocity curve;this area equals the work done by the normal component of reaction impulse. Duringcompression the normal component of impulse does work (on the deformable particle)that increases with increasing impulse until at impulse pc the work equals W3(pc). Sub-sequently, during restitution, the rate of work is negative, so that finally at separation thepart of the kinetic energy of normal relative motion that has been restored by the normalimpulse during restitution equals W3(p/) — W3(pc). The difference in sign for work doneduring compression and restitution is signified by different crosshatching.

Unidirectional Slip d u r i n g C o n t a c t : - a f f i > ( l + e § ^ |In the case of dry friction the tangential component of reaction force acts in a directionopposed to sliding in the contact region. For two-dimensional problems, the equations ofmotion give a constant direction of sliding if either the initial sliding speed is sufficientlylarge so that slip does not halt during the contact period (impulse) or the initial slidingspeed is zero. For this case the characteristic normal impulse for compression, pc, isobtained from the condition that the normal component of relative velocity vanishes atthe termination of compression,

pc = ^f*] • (3.18b)ft + sfip

Here, if 0 < 0 (i.e. ft < 0) or if the initial direction of sliding is such that s < 0, then thesystem jams if the coefficient of friction is sufficiently large, /x > ft / f t . Otherwise thecontact slides continuously in the initial direction and the normal component of relativevelocity at any impulse p is obtained as

v3(p) = 03(0) + m-*(ft + siLh)P-

Hence the partial work done by the normal impulse W3 during the period of compressionpc can be calculated from (3.21),

•=/'

Jo ft +Similarly, for the subsequent period of restitution the partial work done by the normalimpulse is

W3(Pf) - W3(pc) = / v3(p)dp = — ( f t + snh)(p f - Pc)2. (3.26)Jpc l

Here we note that the normal reaction does negative partial work during compression,and this reduces the kinetic energy of relative motion. During restitution the partial workof normal impulse is positive, and this restores some of the kinetic energy of relativemotion that was transformed to strain energy during compression.

The partial work of the normal impulse during compression and that during restitutionare used to evaluate the energetic coefficient of restitution e*, or rather, to determine the

terminal impulse pf in terms of the coefficient of restitution:2- W3(Pc)] ^(P£__x

?c) \PcW3(pc

pf = Pcd+ej = V T ' 7 X " 7 - <3-27)

When contact terminates at impulse pj, the contact points separate with the followingratios for changes in relative speeds during impact:

= -e*. (3.28)

During contact there is also work done by the tangential component of impulse andthis always decreases the kinetic energy of relative motion. For monotonous sliding thepartial work done by friction is obtained as

Wi(pf)= I"1Vi(p)dp =Jo

Slip Reversal during Compression: 0For initial sliding in direction z;i(0)>0 that is brought to a halt during compressionand then reverses, use first the equation (3.13) for changes in sliding speed in order todetermine the normal impulse ps that brings sliding to a halt:

vx(ps) = O = vl<P)-m-\fo + iLfa)ps => Ps = ™ VmQ . (3.29)

At impulse ps the normal component of relative velocity has changed to

v3(Ps) = 03(O) + f ^ 3 ^ 2 ) ?i(0). (3.30)

Noting that slip reverses at normal impulse ps, the characteristic impulse for compressionis again found from the condition that v3(pc) = 0, so that

or

Al + \ (3-18a)Thus slip reversal occurs at a part of the normal compression impulse that is proportionalto

Pc h + flfil + 2fJLP2V{(0)/V3(0) 'To calculate the partial work done by the normal component of impulse using (3.21),

the impulse must be separated into the part occurring during each period of unidirectionalslip. Thus at impulse ps where sliding halts,

W3(Ps) = [2i>3(0) +

Subsequently, for impulse p> ps there is a period of reverse sliding if n < jl. Duringthis period there is additional work done by the normal reaction impulse. This furtherreduces the kinetic energy of relative motion:

W3(Pc) - W3(Ps) = ~ Ps)2.

The sum of these two expressions gives the partial work done by the normal componentof impulse during compression, W3(pc):

(3-31)

The partial work done by the normal reaction impulse during restitution can be obtainedas

W3(Pf) - W3(Pc) = - Pc) = ^ - ( f t - - pcf (3.32)

The energetic coefficient of restitution e* provides a relationship between the partial workW$(pf) — W3(pc) done by normal impulse during restitution and the partial work W3(pc)done during compression. This relationship gives the terminal impulse pf in terms ofthe coefficient of restitution. In comparison with the characteristic normal impulse forcompression, the terminal impulse is obtained as

This impulse is used to calculate the velocity components at separation,

Vl(Pf) = -™~\Pl ~ H>P\)(Pf ~ Ps)

V^iPf) =m

1 + -2T^k{j)1/2

so that

»3(0)_ f 1 ,

€*L 1 1 I + M£l 03(0) J 1 ++

1/2

(3.34)

This expression for the change in the normal component of relative velocity depends oninertia properties, the angle of obliquity at incidence and the coefficient of friction inaddition to the energetic coefficient of restitution (Stronge, 1993).

To calculate dissipation from friction, the partial work of the tangential impulse isseparated into that before and that after the impulse ps where the direction of slidingreverses:

- ufaXPf -

The initial kinetic energy To = mv^(0)[ 1 +v^(0)/Vj(0)] can be used to nondimensionalize

the expressions for work done by the contact force during impact:

r

To

To

Slip-Stick Transition during Compression: M > M = |f > — ^ < | ^ |The equations of motions indicate that the speed of initial sliding decreases if /x > jl =/?2/A. If the initial speed of sliding is small, so that slip halts during the contact period,then during the remainder of the contact period the contact points stick if the coefficientof friction is sufficiently large (/x > jl)\ otherwise, the direction of slip reverses. Againtaking the case of S(0) > 0, the normal impulse required to bring slip to a halt and thenormal component of velocity v3(ps) at this impulse are obtained as

•*>> "«» + ( ^ f W o , (335)For the subsequent period of stick the tangential force is only that which is requiredto provide dv\(p) = 0, p > ps; i.e., the equations of motion for stick are obtained byreplacing the coefficient of friction by jl\

-mv3(ps)P p

This gives a normal impulse for compression,

-mv3(0)( 3 - 3 6 )

During the parts of the compression period before and after slip halts, the normalimpulse does work on the colliding bodies given by

W3(Ps) = EL [203(0) + nC\h + fi/h)ps]

W3(pc)-W3(Ps)= 2(p3 -11 fh)

whereas during restitution, if the contact points stick, the normal work will be

W3(Pf) - WM - m ^ ^ \ P f - Pcf. (3.38)

Together with the energetic coefficient of restitution, the normal work during these sep-arate parts of the contact period give a terminal impulse p/ which can be expressed inrelation to the normal impulse for compression pc,

. (3.39)

With this terminal impulse, at the contact point the components of terminal velocity givethe following velocity ratios:

Vl(Pf) = o

= Qi + P)h(p,X\ [ Qi + iDhviiO)] ( 3 4 0 )* ( ft - Aft \Pc)) 1 (ft + ufa) vm j '

/am: (32 < 0, /x > - /3 3 / /3 2

Jam (or self-locking) is a process where during an initial period of sliding the normalcomponent of relative velocity increases due to a positive normal acceleration at thecontact point - during jam the rate of indentation increases. This acceleration is mostlydue to rotational acceleration that is generated by a large friction force. Jam occursonly if there is an eccentric impact configuration with the center of mass at a smallnegative inclination relative to the common normal ft < 0, a large coefficient of friction/x > —ft/ft, and initial sliding in the positive direction. This process persists until initialsliding is brought to a halt; thereafter, the contact points stick and are driven apart by thenormal contact force (Batlle, 1998).

During jam Eq. (3.12) gives

dvi/dp = -m-\fa + iifa) < 0, dvs/dp = m~\fa + /xft) < 0. (3.41)

Since ft ft > /*f, this gives /x > — ft/ft > — ft/ft and /x > /x; consequently, after initialslip is brought to a halt at impulse ps, the contact points stick. Initial sliding ceases whenthe normal impulse equals

ft + MPi

At impulse ps when initial sliding has been brought to a halt, the normal relative velocityhas increased in magnitude to

- i;3(0) -

Subsequently the contact sticks, since /x > /x, and further impulse accelerates the contactpoints apart:

v3(p) = v3(ps) + m - 1 ( f t - /xft)(P - ps)

= i?3(0) + m~\ix + p,)faps + m~\p3 - £LP2)P, P > Ps-

The normal impulse for compression, pc, is obtained from the condition that v3(pc) = 0,


so that

Pc =-mv3(0)

ft - Aft ft - AftThe terminal normal velocity v3(p/) is given by

= in (ft — AftX/?/ — Pc)'

(3.43)

(3.44)

To obtain the terminal impulse, the energetic coefficient of restitution e* can beused. This requires separate evaluation of the partial work done by the normal impulseduring compression, W3(ps) + [W3(pc) - W3(ps)], and that done during restitution,W3(pf) - W3(Pc):

Pcv3{ps)y3\pc —

. PcV3(Ps)) = 1

-1 /2 f (/x + /2)ft(ft + iLfa)mv\(0)ft - Aft (ft +

(0)^(0)

W3(pf) - W3(pc) =

ft + iiPi(Pf ~ Pc)V3(Pf)

m - l

= — ( f t - filhXPf - Pc).Hence during restitution the change in normal impulse is obtained from (3.24),

Pf PcA)ft(ft , 1/2

ft-Aft T " (ft + Mft)^3(0) ' (ft + /zft)2i732(O) J

(3.45)

Figure 3.6 shows the changes in relative velocity and the work done during different

MO)

Figure 3.6. Changes in relative velocity for eccentric collision with initial period of jam.The shaded areas indicate the normal work done during compression and that recoveredduring restitution. Note that if the configuration and coefficient of friction are sufficient tocause jam, the sliding process is jam-stick.

Table 3.2. Terminal Normal Impulse Ratio P//pcfar Planar Impact, z?i(O) > 0, z;3(0) < 0

Relative velocityat incidence,

Process

Friction Impulse that Normal impulsecoeff. halts init. slip, of compression,

ps/mvi(0) -pc/mv3(0)

Ratio of terminal tocomp. normal impulse,pf/pc

Continous stick 0Continous slip (1+e*)— <

PbS l ip_s t i c k incompression

Slip reversal incompression

Slip-stickin ^ < _restitution Pb

> 1 1 + e*

1 + e*

v3(0)4)^2 J

< ,

) ^ > 1(PS - Pc)2

r e s t l t u t i o n

_ . , . , . ., ... , . f/i, ^ < \p,\, sliTangential impulse at separation pi(pf) = ~nps + fi(pf - ps) if ps < pf and (i = j > sli

slip reversal,slip_stick

3.5 / Friction in Collinear Impact Configurations 55

periods of impulse for an eccentric collision with initial jam. The dynamics of jam havebeen extended to 3D collisions by Batlle (1998).

Grazing Incidence: f32 < 0, // > — /3 3 / /3 2 , *>i(0)/i>3(0) —> ooJam can result in an impact process in the limiting case of vanishing normal componentof velocity ^(0) -> 0; this is termed grazing incidence. In this case the normal impulsethat brings initial sliding to a halt ps is given by (3.42). Subsequently the contact stickswhile the normal contact force drives the bodies apart. Hence the normal impulse pc atthe end of compression is obtained from (3.43),

_ -(M + iL)fapsPc~ h- fifa

while the impulse during restitution pf — pc,

= e+mvm f (M + ji)lh(ft3 + jxft) 11 / 2

Pf Pc A - A A l (A + ^ i ) 2 J 'Grazing incidence finally results in a positive normal component of relative velocityalthough the incident relative velocity has no normal component.

Terminal Impulse p/ for Arbitrary Initial ConditionsFor various ratios of initial tangential to normal incident speeds, Table 3.2 lists the impulseps at which initial slip is halted, the compression impulse pc and the terminal impulsePf. The range of angles of incidence wherein each relation applies depends on inertiaproperties of the bodies and the coefficient of friction as well as the initial ratio of tangentialto normal incident speed.

3.5 Friction in Collinear Impact Configurations

Collinear impact configurations always give ft = 0 and ft = 1, so that the equa-tions of relative motion (3.13) simplify to

vi(p) = Pi(0) - Sm-lfiPip (3.46a)

m-lp. (3.46b)

The sphere shown in Fig. 3.7 has mass M and radius R ; it is rotating about a transverseaxis when it collides with a stationary half space at a contact point C. At any reactionimpulse p during contact the sphere has angular velocity u;(p) = (0, oo, 0) and the center

\PFigure 3.7. Oblique impact of rotating sphere on rough half space.

of mass G has translational velocity \(p) = (z>i, 0, v3). At the contact point the frictionforce satisfies Coulomb's law of friction. For this system and slip at C in the positivedirection (i.e. V\ > 0), the differential equations of motion for changes in components ofvelocity are given in terms of the normal component of impulse p:

Mdvx = -fi dp, dv3 = dp, Mk2 dco = -fiR dp (v{ >0).

The components of relative velocity \{p) = (v\, 0, v3) at C are obtained from the follow-ing expressions that apply in this case of collinear collision:

vi(p) = vx(p) + Rco(p), v3(p) = v3(p).

These translations and the differential equations give differential equations for changesin relative velocity at C that can readily be integrated. On applying the initial condi-tions t?!(0) = fli(0) + Rco(0), t73(O) = v3(0) and noting that v3(0) < 0, these integrationsgive

Vl(p) = R2/k2r)p, v3(p) = v3(0) + M~lp 0).

A transition from compression to restitution occurs at impulse pc when the normalvelocity vanishes [v3(pc) = 0], so that pc = — Mv3(0). Hence the previous expressionscan be divided by ^ (0) to give nondimensional expressions,

vi(p)pc Pc

These changes in relative velocity at C are illustrated in Fig. 3.8. For the planar changesin velocity that result from this collinear configuration, the changes in velocity are linearfunctions of impulse ratio p/pc.

The second equation above can be equated to zero to give the impulse ratio pslpc whensliding terminates:

Rco{0)]/V3(0)Ps = -Pi(0)/P3(0) _ -

A collinear collision terminates at impulse p/ — (1 + e*)p c when the contact points sepa-rate. If sliding halts before separation (ps < pf), then when sliding halts, the components

V3(Q)<0

• ^ 1

P/Pc

(1-eJ

Figure 3.8. Changes in components of velocity and impulse as a function of impulse actingon a rotating sphere. The speed of slip V\ (p) vanishes at impulse ps and then sticks, whereasthe characteristic impulse for compression pc occurs at the transition from compression torestitution.

3.5 / Friction in Collinear Impact Configurations 57

of velocity at the center of mass are

{ Ps_ VsiPs) = liPs Rco(Ps) = RcojO) | R^fr^ P' V(0) A ' V(0) V(0) ^ £? P "V3(0) V3(0) ^ Pc' V3(0) A:' V3(0) V3(0)

After sliding halts (ps < p < pf), there are no additional changes in translational velocityVi or angular speed to if the collision configuration is collinear. For a collinear configura-tion there is no friction force or additional tangential impulse required to maintain stick;i.e., v\(p) = 0, Fi(p) = 0 and px = -jips during ps < p < pf.

Example 3.2 A rigid sphere of radius R is rotating at an initial angular speed co(0)about a horizontal axis when it strikes the level surface of a rough elastic half space atan incident speed 1^(0). Before impact the horizontal translational velocity of the centerof mass is zero: Vi(0) = 0. Find (i) the angle of rebound ^ / and (ii) the coefficient offriction \x which causes slip to cease at the instant of separation.

Solution Impulse ratio when slip stops:

ps -Rco(0)/V3(0)

(i) If ps < Pf, any initial slip is brought to a halt during contact, so that the final velocityat G is given by

Vi(Ps) = A siPs) = (Ps) = () ( ^ AV3(0) ~ M ^ c ' V3(0) " **' V3(0) ~ V3(0) ^PrPc'

Hence the angle of rebound &f as defined in Fig. 3.9 is given by

V3(Pf) e*(l

(ii) If ps > Pf, slip continues throughout contact, giving a final velocity at G

ROJ(PS) RCO(0) R2

/x(+6*), £*, 7V3(0) ^ V3(0) V3(0) V3(0)

and an angle of rebound ^ / = tan-1[—/x(l + ^~1)]. For an initially rotating sphere with

Figure 3.9. Rotating sphere dropped vertically onto a half space, illustrating angle ofrebound \j/f at separation.

1.0

0.5

- sphere

j i i L

Figure 3.10. Rebound angle for rotating sphere (or cylinder) as function of the angle ofincidence at the contact point.

no tangential speed, sliding at the contact point does not stop during impact if

Rco(0)/V3(0)

Figure 3.10 shows the rebound angle *Pf of a dropped ball as a function of the rate ofrotation. The angle is limited by gross slip vanishing before separation.

Example 3.3 A sphere (or cylinder) of radius R rolls on a level surface without slipbefore it collides against a ramp inclined at angle n/2 — &Q a s depicted in Fig. 3.11a.Before impact the center of mass has a horizontal speed RO)Q. Find the maximum coeffi-cient of friction \x for gross slip and the angle of rebound *Pf if /x < /L

Solution At incidence the normal and tangential components of velocity areas follows:

y3(0) = -Ro)0 COS tf'o, Vi(0) = -Rco(0) sin VO.

Initial slip persists until the normal impulse equals ps, i.e.

Ei = —Pc

Initial slip is stopped during collision if ps/pc <l+e*; otherwise there is gross slip. Therange of values for coefficients of friction and restitution that result in slip-stick is shownin Fig. 3.12. If the coefficient of friction is large enough to cause gross slip, at separationthe center of mass of the sphere is traveling at an angle tan"1 [e~l cot 0 + /x( 1 + e~l)] fromnormal. This is an upper bound on the change in angle for given values of the coefficientsof friction and restitution.

3.6 / Friction in Noncollinear Impact Configurations 59

(b)

Figure 3.11. (a) Rolling sphere with center of mass traveling at speed s0 when it collidesagainst inclined bumper, and (b) components of incident and terminal velocities at centerof mass G.

0.5-

0.25-

0 Ti/o ^ /3

Figure 3.12. Angles of inclination of bumper and coefficient of friction giving either slip-stick or gross slip behavior during impact.

M (1+eJ

slip-stick

gross

1

* / /

Slip

\Q

3.6 Friction in Noncollinear Impact Configurations

3.6.1 Planar Impact of Rigid Bar on Rough Half Space

The following is a straightforward example illustrating some effects of frictionfor oblique impact of eccentric rigid bodies. The example of a slender rigid bar has beenchosen for simplicity and clarity of presentation. The reader should recognize, however,that if the eccentricity of the collision configuration is large, rigid body theory is not veryaccurate for slender bodies, because that theory neglects transverse vibrations, which areimportant for slender bodies.5 Here we consider impact of a rigid bar of mass M andlength 2L with an end that strikes against the surface of a rough half space. At impactthe bar is inclined at angle 0 from the vertical as shown in Fig. 3.13. The bar has a radiusof gyration kr = L/V3 about a transverse axis through the center of mass. The center ofmass of the bar has an initial velocity V(0) = (V\, V3), and the bar is not rotating, so thatthe initial angular velocity o;(0) = 0. For this impact configuration the effective massm — M and the elements of the inverse of inertia matrix are

Px = l+3cos 2 (9 = ( 5 + 3 cos 2<9)/2

P2 = 3 sin 0 cos 0 = (3 sin 260/2

p3 = l+3sin2<9 = (5-3cos26>)/2

5 Experiments by Stoianovici and Hurmuzlu (1996) have shown that the effect of transverse vibrationscan be important for impact of a slender bar in an eccentric configuration.

- -•1.5

To

(b)

l

N^lissipation [30°

-1 v,[0)lv3i0)

Figure 3.13. Oblique elastic impact of inclined rod (0 = n/6) on rough half space{jX = 0.4): (a) components for terminal velocity of contact point; (b) terminal kineticenergy Tf = T(pf) and energy dissipated by friction To — Tf.

where fi\ > 0 and fi3 > 0. For dry friction with a coefficient of friction \x the equations ofmotion for frictional impact (3.12) give the following differential equations for changesin relative velocity at the contact point C as a function of normal impulse p:

dv\ = -m~x(pdv3 =m-l(p3

Integration of these equations requires consideration of the different slip processes whichcan occur for different parts of the range of angle of incidence #o = tan"1 [Ui(0)/i?3(0)].

Terminal Velocity at Contact Point and Energy Dissipation during ImpactFigure 3.13 illustrates calculated values for changes in relative velocity at the contactpoint and energy loss due to friction of a rigid rod inclined at an angle of 0 = +30° thatcollides against a stationary half space. The values were calculated from expressions inSect. 3.3.2. The calculations are for a coefficient of friction equal to unity so that all energyloss is due to friction. Two values of the coefficient of friction have been examined; thegraphs on the left are for light friction \i < jl, where small initial slip can reverse duringcollision, while those on the right are for heavy friction fz > /x, where small initial slipcan be halted and then stick.

This example clearly shows that if the coefficient of friction is based on work doneby the normal component of contact force (i.e. the energetic coefficient of friction), thenthe normal component of relative velocity at separation, v3(pf), is not directly related tothe normal component of relative velocity at incidence, ^3(0). The ratio of these speedsequals the coefficient of restitution only if the direction of slip is constant or if thereis no slip during the collision. For light friction the speed of sliding either decreases orincreases from its initial value depending on the direction of sliding in relation to the angleof inclination of the center of mass; initial sliding in the direction away from the center

Problems 61

of mass decelerates only if friction is heavy (/x > /x). The large coefficient of frictionix = 0.6 brought initial sliding to a halt during collision for —1 < V\(0)/v3(0) < 1. Withheavy friction, small initial sliding results in slip-stick during contact.

On the plots of energy loss in the collision, the dashed line represents the coefficient ofrestitution. The dependence of frictional dissipation on the direction of slip relative to theangle of inclination of the center of mass is particularly noticeable. There is much morefrictional dissipation when the bar strikes with the end that leads the center of mass, muchas a javelin does at the end of its flight. If these calculations were done with a coefficientof restitution of less than unity (e* < 1), then the height of the dashed horizontal linewould be reduced.

PROBLEMS

3.1 If bodies B and B' with masses M and Mf collide at C, show that the two centers ofmass have a relative velocity fy = V/ — V( i = 1, 3 which satisfies the differentialequation dVj = m~x dpi. Separate this differential equation for changes in relativevelocity into a part for each mass. Do the same for the relative velocity V[ at contactpoint C. For an eccentric impact between smooth bodies, use the difference betweenchanges in the normal components of relative velocity at C and the center of massto obtain the change in angular velocity of body B as a function of the coefficient ofrestitution e*, the radius of gyration kr and the eccentricity r\.

3.2 For two colliding bodies, the center of mass of the system has a normal component oftranslational velocity that is constant. Express this velocity V3(pc) = V^iPc) in termsof the component of initial velocity for bodies with masses M and M' respectively.

3.3 A billiard ball of radius R is initially at rest on a level table. An impulse is impartedto the sphere by striking it with a cue stick. Find that the sphere begins to roll on thetable without slip if the line of action of the impulse passes through a point Ac locateda distance £ above the center of mass (G), where %/R = k^/R2 and kr is the radiusof gyration of the sphere about G. Point Ac is termed the center of percussion for thecontact point C. An impulse applied through the center of percussion Ac generatesno reaction at C in a direction perpendicular to line AcC.

3.4 For a smooth billiard ball that rolls into an identical ball and collides at an angleof obliquity #o a s specified in Example 3.1, find the velocity of the center of theinitially stationary ball B immediately following a post-impact period of sliding ifthe limiting coefficient of friction between the ball and the table is /x. Also find anexpression for the total sliding distance X^B-

3.5 To obtain the largest angle of deflection for the rolling ball, pool or billiard playersfrequently aim the centerline of the path of this ball at the lateral surface of thestationary ball which is to be hit. For smooth solid spherical balls, compare theangular deflection of the rolling ball using this tactic with the maximum achievableangular deflection.

3.6 For impact between spherical balls as described in Problem 3.4, calculate the trans-lational momentum parallel and perpendicular to the initial path (a) at the instantof separation from impact and (b) at the instant when sliding terminates and rollingresumes. Explain the difference.

3.7 Prove that after initial slip is brought to a halt at impulse ps, slip cannot reinitiate inthe original direction during ps < p < pf.


3.8 A golf ball of mass M and radius R is struck by a club of much larger mass. The clubhas a face inclined at angle & = 7t/6 from vertical, and it moves horizontally withspeed Vb when it strikes the ball. Assume the tangential compliance is negligible,(a) Derive equations of relative motion for the tangential and normal components

V\(p) and Vi(p) of the relative velocity, and show that these can be expressed as

vi = Pi (0) + \m-x px

V3 =

(b) Obtain the coefficient of Coulomb friction /x that is just sufficient to bring initialsliding to a halt at the transition from compression to restitution.

(c) Write an expression for the work done by the contact force during compressionif the coefficient of friction /x = 2\/3/21.

3.9 In the case of slip that halts during the restitution phase of contact, obtain an analyt-ical expression for the terminal normal impulse p/ at separation. Show that this isin agreement with the expression given in Table 3.2.

3.10 For a slender rod with one end striking a heavy half space, find the angle of inclina-tion 0 where jam requires the smallest coefficient of friction. Evaluate the smallestcoefficient of friction that gives jam. Explain why jam necessarily terminates duringcompression.

3.11 A solid rectangular block has edges of length b parallel to the surface of a level rigidhalf plane and height a. A bottom edge A of the block initially is in contact with theplane, and the block has an angular velocity coo about A at the instant edge B strikesthe plane. Assume that contact with the plane occurs only along the parallel edgesA and B.(a) Assuming edge B does not rebound and there is no slipping during impact, show

that after impact the body starts to rotate about B with angular velocityco0(2a2 -b2) r-——— -^— for aV2 > b.

2(a2 + b2)(b) Assume instead that the impact at B is perfectly elastic, so that the velocity of

the impact point B reverses during the impact. Write down the velocity of B justbefore and after impact. Calculate the angular velocity of the block just afterthe impact, using momentum considerations. Confirm that energy is conservedduring impact.

(c) From the changes in translational momentum obtain the ratio of tangential to ver-tical impulse to show that in order to prevent slip during impact, the coefficientof dry friction \x must satisfy /x > 0.6.

3.12 A smooth (frictionless) sphere of mass M and radius R rolls on a level plane atangular speed COQ before it collides with a small vertical step of height h. The impactwith the step has a coefficient of restitution e*.(a) Find an expression for AT, the energy dissipated in the collision.(b) Obtain a lower bound on the initial speed of the center of mass, Rcoo, if the sphere

is to pass over the step.

CHAPTER 4

3D Impact of Rough Rigid Bodies

Like a ski resort full of girls hunting for husbands, and husbandshunting for girls, the situation is not as symmetrical as it might seem.

Alan Lindsay Mackay, Lecture, Birkbeck College,University of London, 1984

Three-dimensional (3D, or nonplanar) changes in velocity occur in collisionsbetween rough bodies if the configuration is not collinear and the initial direction ofsliding is not in-plane with two of the three principal axes of inertia for each body. Incollisions between rough bodies, dry friction can be represented by Coulomb's law. If thereis a tangential component of relative velocity at the contact point (sliding contact) this lawrelates the normal and tangential components of contact force by a coefficient of limitingfriction. The friction force acts in a direction opposed to sliding. For a collision with planarchanges in velocity, sliding is in either one direction or the other on the common tangentplane. In general however, friction results in nonplanar changes in velocity. Nonplanarvelocity changes give a direction of sliding that continuously changes, or swerves, duringan initial phase of contact in an eccentric impact configuration. This chapter obtainschanges in relative velocity during rigid body collisions as a function of the impulse Pdue to the normal component of the reaction force. The changes in velocity depend on twoindependent material parameters - the coefficient of friction and an energetic coefficientof restitution.

During moderate speed collisions between two hard bodies there are continuouschanges in relative velocity which can easily be calculated if we recognize that the bodiesare not entirely rigid - there is a small region of deformation that surrounds the initialcontact point. At the contact point there is a point of contact on each colliding body; thesepoints are coincident, but they have different velocities. By considering an infinitesimallysmall deformable region located between the bodies at the contact point, changes in rela-tive velocity vt across the contact point can be expressed as a function of the impulse dueto the reaction force. The reaction impulse Pt that develops during a collision depends onthe initial relative velocity vt (0) and both material and inertia properties of each body at theimpact point. The relative velocity between the colliding bodies and the reaction impulsecan be resolved into components normal and tangential to the common tangent planefor the contact surfaces at the point of contact. Let the unit vector n be oriented normalto the tangent plane. The normal component of the impulse, P = Pi nn, monotonicallyincreases in magnitude during contact because the normal force F is compressive; thus,changes in relative velocity are obtained as a function of the scalar independent variable P.

63

64 4 / 3D Impact of Rough Rigid Bodies

For bodies composed of rate-independent materials, the maximum normal force andmaximum compression occur at the impulse Pc where the normal component of relativevelocity vanishes. During an initial period of contact P < Pc, the normal component ofcontact force F = dP/dt does negative work on the bodies1; this work compresses thebodies in a small region around the contact point while decreasing the sum of their kineticenergies. Following maximum compression P > Pc, the normal component of contactforce decreases as it drives the bodies apart while the compressed region undergoes arelease of elastic strain energy. The work done on the bodies by the normal force Fis positive during the restitution period P > PC9 and this restores that part of the initialkinetic energy which was transformed into elastic strain energy during compression.

In addition to the normal component of relative velocity at the contact point thereis also a tangential component of relative velocity called slip. If the bodies are rough, atangential contact force, termed friction, opposes any slip. This friction force complicatesthe analysis of impact problems, especially if friction and normal force components areinterdependent due to coupling in the equations of motion. This coupling occurs if theimpact configuration is eccentric or noncollinear, i.e. if the center of mass for each bodyis not on the common normal line through the contact point. The analytical complicationsdevelop if the direction of slip varies during the contact period.

For eccentric collisions with changes in relative velocity that are planar, complica-tions due to friction have been overcome by dividing the total impulse into parts whichcorrespond to successive phases of unidirectional slip. In general, however, collisionsbetween rough bodies result in changes V((P) — vt(0) in relative velocity at the contactpoint that are nonplanar unless the initial relative velocity i?/(0) lies in the same plane astwo principal axes of inertia for each body. We will show that in most cases the directionof slip changes continuously during contact if the collision configuration is eccentric. Ageneral formulation for impact of two unconstrained rough rigid bodies will be devel-oped. Then the effects of friction in 3D (nonplanar) collisions will be illustrated by threeexamples: the oblique impact of a sphere that has a component of initial rotation aboutthe direction of initial translation, the impact of an inclined rod on a half space and acollision of a spherical pendulum on a half space. These examples represent collinearand two eccentric impact configurations, respectively. The colliding sphere is an exampleof a collinear impact configuration, while the collision of an inclined rod and that of aspherical pendulum are examples of eccentric impact configurations.

4.1 Collision of Two Free Bodies

Consider two bodies that collide at contact point C; the bodies have no displace-ment constraints except that they are mutually impenetrable at C. If the surface of at leastone of the bodies has continuous curvature at C , there is a common tangent plane thatcontains point C. First define a common normal direction n that is perpendicular to thecommon tangent plane. Let nt, i = 1,2,3, be a set of mutually perpendicular unit vectorswith n\ and n2 in the tangent plane while n3 = n is normal to this plane, as shown inFig. 4.1. The bodies have centers of mass located at G and G' respectively. There is a

1 Relative to either body, during compression the normal contact force is opposed to the normal compo-nent of relative velocity across the deformable particle at the contact point.

4.1 / Collision of Two Free Bodies 65

Figure 4.1. Collision between two rough bodies. The rigid bodies have contact points Cand C that are separated by a deformable particle.

position vector r, from G to the contact goint C, while r[ locates C from G'. The bodieshave masses M and M\ inertia tensors //,- and //y for second moments of their massesat G and G' respectively . Let V( and V( be the velocities of the centers of mass, while &>;and &>• are the corresponding angular velocities for the bodies in reference frame nt . Atthe contact points C and C the bodies are subjected to contact forces Ft and F[\ thesecontact forces are reactions that apply an impulse to each body. Denote these impulsesas Pt(t) and P((t), where

= Ftdt and dP[ = F[dt. (4.1)

The first aim is to obtain equations of motion in terms of relative velocities at the contactpoint, since this is where relative displacement (interpenetration) is resisted by reactionforces.

At the two centers of mass the equations of translational and rotational motion for eachbody can be expressed as

MdVt = dPt (4.2a)

and

Iij dcoj = 8ijkrj dPk (4.3a)

M'dVi = dP[ (4.2b)

I-j dco'j — £ijhrj dPk (4.3b)where a repeated index (e.g. j or k) indicates summation and the permutation tensor etjktakes the values Sijk = +1 if the indices are in cyclic order, Sijk = — 1 if the indices arein anticyclic order and sijk = 0 for repeated indices. Thus in index notation the vectorproduct rj x dPk — £ij krj dPk. For a body of volume V with density p, the elements ofthe inertia matrix 1^ for axes through the center of mass are defined as the moments andproducts of inertia. Typically,

n = fJv

= -[ rxr2pdV.J

During collision the reaction forces that act at the contact point are large if the bodiesare hard; in particular, these forces are very large in comparison with any body force.Consequently it can be assumed that the only forces acting during a collision are thereactions at C and O ; the impulse of these reactions depends on changes in relativevelocity across a small deforming region at C . The effect of a small deforming regioncan be represented by assuming that during a collision the contact points C and C on thecolliding bodies are separated by a negligibly small deformable element.2 The velocityof each contact point, V/ or V/, can be obtained from the velocity of the respective centerof mass and the relationship between velocities of two points on a rigid body,

Vi = Vt: + £ijkCOjrk and V{ = V? + sijkcoyk.

Let the relative velocity vt between the contact points C and C be defined as

vt = Vt- Vf. . (4.4)

Any incremental changes in reaction impulse acting on the rigid bodies are equal inmagnitude but opposite in direction if the infinitesimally small deforming element hasnegligible mass; i.e.

dPi = dPi = -dP[. (4.5)

Changes in relative velocity at C can be related to changes in impulse of the reaction bysubstituting (4.2) and (4.3) into (4.4) and thence (4.5). This gives an equation of motionfor changes in relative velocity vi9

dvi = mjj1 dpj (4.6)

where the elements of the inverse inertia matrix for C are given by

and 8ij is the Kronecker delta defined as Sij = 1 if / = j and <5/7 = 0 if / ^ j . This inverseinertia matrix is symmetric, mjj1 = m^.1, The following are representative elements:

U = (rlr3/231 - rtI2\ - rr2l

= {nril r\l

In these expressions notice that the matrix //,- of moments and products of inertia has aninverse which is denoted by I^1 ; e.g. I^1 = (/13/23 — ^12^33)/ det(//y).

2 The physical construct of a deformable particle separating contact points on colliding rigid bodies ismathematically equivalent to Keller's (1986) asymptotic method of integrating with respect to time theequations for relative acceleration of deformable bodies and then taking the limit as the compliance(or contact period) becomes vanishingly small.


4.1.1 Law of Friction for Rough Bodies

Dry friction between colliding bodies can be represented by the Amontons-Coulomb law of sliding friction (Johnson, 1985). This law relates the tangential com-ponent to the normal component of reaction force at the contact point by introducing acoefficient of limiting friction /x which acts if there is sliding, i.e. v\-\-v\ > 0. Denot-ing the magnitude of the normal component of a differential increment of impulse bydp = dp3 = dPn, this law takes the form

if v2x+vl = 0 (4.8a)

dpi = /~"A dP} dp2 = " ' " tip if v\ + v\> 0. (4.8b)

Equation (4.8a) expresses an upper bound on the ratio of tangential to normal forcefor rolling contact; for ratios of tangential to normal contact force that are less than /nthe sliding speed s = Jv\ + v\ vanishes. If sliding is present (s > 0), the tangentialincrement of impulse or friction force at any impulse acts in a direction directly opposedto sliding and has a magnitude that is directly proportional to the normal force.3 Thesliding direction can be defined by the angle (p measured in the tangent plane from n\\thus 0 = t a n " 1 ^ / ^ ) and

Z71=scos0, V2 =s sin (p. (4.9)

Since the normal contact force must be compressive, the impulse of the normal componentof reaction is a monotonously increasing scalar function during the collision period. Hencerates of change for relative velocity at the contact point C can be expressed as a functionof the rate of change of impulse for the normal component of reaction; i.e., the collisionprocess can be resolved as a function of the independent variable p.

4.1.2 Equation of Motion in Terms of the Normal Impulse

For sliding in direction 0(/?) the equations of motion can be obtained in termsof the impulse of the normal component of reaction:

m\l (4.10a)

m^ (4.10b)

dv^/dp = -/xm3"11 cos0 — /zra^1 sin0 + m^1. (4.10c)

These equations of motion are not separable into independent expressions for each com-ponent of velocity unless /x = 0 or mjj1 = 0 for / ^ j ; i.e., either the contact surfacesare perfectly smooth or the impact configuration is collinear and the sliding velocity isin-plane with two principal axes of inertia for the center of mass of each body. Since theinertia terms in (4.10a) and (4.10b) are not proportional, the rates of change are generallydifferent for each tangential component of slip; thus for nonplanar changes in relativevelocity the direction of slip (f)(p) continually varies while s > 0.

3 To simplify the presentation and reduce the number of parameters, any distinction between static anddynamic coefficients of friction has been neglected.

Alternatively, the equations of motion for slip (4.10a, b) can be expressed in terms ofvariables^, 0) rather than (i?j, v2). In this manner, Keller (1986) obtained

— = m^} cos 0 -f m 2} sin0 — /xm[/ cos2 0 — 2ixm\2x sin0 cos 0

- /xm^1 sin2 </> = g(fjL, 0) (4.11a)

^/0 _j _. . _..5 — = — m^ 3 sin0 + m23 cos0 + fiym^ — m22) sin0 cos 0

+ /xra~1(sin20 - cos20) = /*(/x, 0). (4.11b)

With these definitions, the sliding speed s can be expressed as a function of the currentdirection of slip 0:

4.1.3 Sliding that Halts during Collision

If the collision is eccentric and the initial speed of sliding is small enough, slipcan halt before separation. After slip halts, the contact patch either sticks or resumessliding in a new direction. For a sufficiently large coefficient of friction, the contact patchsticks; i.e., after slip halts, dv\/dp = dv2/dp = 0. This velocity constraint is imposedby a tangential force which corresponds to a differential impulse (dp\ + dp\)l/1 = (I dpwhere jl < /x. The constraint force has a direction 0 — n and a ratio of tangential tonormal force /x that can be obtained by equating (4.1 la) and (4.1 lb) to zero:

min m,o —

[(mnm

22m13(4.12)

The ratio of tangential to normal contact force, /2, is termed the coefficient for stick. If thecoefficient of friction is larger than the coefficient for stick (/x > //), the contact motionis termed slip-stick. In this case, if friction brings initial slip to a halt before separation,subsequently the contact points stick (roll without sliding). On the other hand, if /x < jl,there is a second phase of slip that begins when the initial slip vanishes. In planar collisionsthis second phase is termed slip reversal.

If slip resumes immediately after halting, it does so in a direction 0(/x) which is a rootof /z(/x, 0) = 0 as expressed in Eq. (4.1 lb). In this second phase of slip the direction ofslip depends on the coefficient of friction if /x < /x.4 For a coefficient of friction that isslightly less than the coefficient for stick (/x = /x — s), however, the direction of secondphase sliding is opposite to the constraint force for stick; i.e. lim£^0 0 = 0. In general,during any second phase of slip, the direction 0(/x) is constant, since in Eq. (4.11b) his independent of s. The direction of second phase slip is one of a set of characteristic

4 During any second phase of slip the direction is given also by <j> = tan~l(dv2/dv\), since h = — sin <f>dv\ I dp + cos 0 dV2/dp.


directions, termed isoclinics, where the direction of slip is constant; in the slip plane V\, i72

the isoclinic lines 4>(/JL) depend on the impact configuration and the coefficient of friction.Batlle (1996) has shown that if the isoclinic directions are distinct, there is only one alongwhich the relative acceleration is positive (ds/dp > 0). Consequently the direction ofsecond phase slip is unique.

4.1.4 Terminal Normal Impulse from Energetic Coefficient of Restitution

The collision process terminates at a normal impulse p/ that can be obtainedfrom the energetic coefficient of restitution that was defined in Eq. (3.24). The terminalimpulse /?/ is obtained from the energetic coefficient of restitution by first separating theterminal work done by the normal component of force, W3 (/?/), into the work done duringcompression, Wc = Ws(pc), and the additional work done during restitution, W3(pf) —Wi(pc). The energetic coefficient of restitution e* is defined as the ratio

W3(pf) - W3(Pc)W3(pc)

(4.13)

For 3D collisions where the direction of slip is continually varying as a function ofimpulse, the integration required to calculate these terms is nontrivial because the normalrelative velocity v$(p) is not simply a linear function of normal impulse. In Fig. 4.2 thechanges in this normal component of velocity as a function of normal impulse are shownas a curve connecting the initial and final states. According to Eq. (4.10c), at every impulsethe slope of this curve depends on the current direction of slip, (j)(p).

Further understanding of the process of slip during collision can be obtained fromthe following three examples. The first is a central (collinear) collision of a rotatingsphere on a rigid half space; because of the initial rotation, the tangential component oftranslational relative velocity for the mass center is not parallel to the initial directionof slip. The second and third examples are noncollinear collisions where the resultant offorces acting during contact is not in-plane with two of the principal axes of inertia forC, so the changes in velocity are nonplanar.

Figure 4.2. Changes in the normal component of relative velocity during collision. Theslope of the curve changes when the direction of slip changes. The cross-hatched areas underthe curve are equal to the work done by the normal component of force during compressionand restitution, respectively.


4.2 Oblique Collision of a Rotating Sphere on a Rough Half Space

Consider a rigid sphere of radius R, mass M and moments of inertia In = I22 =Mk2 about the center of mass G. The center of mass is moving with translational relativevelocity V/(0) when it collides with a rough rigid half space at a contact point C . Beforethe collision the sphere is rotating with angular velocity &>;(0). Let the center of mass Ghave components of velocity Vi and V2 in the tangent plane and % in the normal directionn as shown in Fig. 4.3. At the contact point the reaction force gives a normal componentof impulse p3 = p and tangential components p\ and p2. The equations of motion forthis sphere can be written as

dVi=M~ldpu

dcox = M-lRk;2

= M~ldp2, d% = M~x dp

dco2 = -M-lRk;2dpu dco3 = 0(4.14)

where the reaction is assumed to give no couple. Although some experiments on spinningspheres by Horak (1948) have measured a mean reaction couple during collision, thiscouple is a consequence of development of a finite radius for the contact patch. A time-dependent analysis somewhat like the simulation of Brach (1993) is required to obtainthe frictional reaction couple as a function of deformation of the colliding bodies. Hencethese experimental results are outside the realm of rigid body impact theory (see Lim andStronge, 1994).

At the contact point C the relative velocity i?/ between the sphere and the surface ofthe half space has a normal component v3 = v3 = V3 and components of slip

= V\ — Rco 2, Rco\

The differential equations for change in relative velocity at the contact point are

dv\dv2

1M

0R2I 0

1

dpidp2

dp3

(4.15)

4%

Figure 4.3. Rotating sphere colliding with a rough half space at oblique angle of incidence.

4.2 / Oblique Collision of a Rotating Sphere on a Rough Half Space 71

The slip speed s{p) and the angle of slip (j){p) are defined as follows:

s2 = v\ + vj, 0 = t^n~l(v2/v\)

so thatVi = scoscp, v2 = ssincj).

If tangential components are related to the normal component of impulse by theAmontons-Coulomb law, then while the contact point is slipping the components ofchanges in tangential impulse are related to the differential of normal impulse by

dp\ = —/ncoscpdp, dp2 = —/jisirupdp. (4.16)

Instead of expressing slip in terms of velocity components V\ and v2 in the tangentplane, the equations of motion for a sphere can be written directly in terms of the slipspeed s and angle of slip 0. Noting that the moments of inertia about diametral axes of asphere are I\\ = I22 = Mk2, either Eqs. (4.11) or (4.15) result in differential equationsfor slip of a sphere,

ds/dp = -/zAT1 (1 + R2/k2), d(f)/dp = 0. (4.17)

Thus, for collinear or central collision of bodies that are axisymmetric about the commonnormal direction, the direction of slip 0 does not vary.5

From Eq. (4.17) the slip speed can be obtained as a function of impulse:

s(p) = s(0) - /xM"1 (1 + R2/k2)p.

The transition impulse pc when compression terminates and separation begins is given bypc = — Mv3(0), whereas in a collinear collision the final impulse at separation is givenby pf = (1 + e*)pc. This terminal impulse provides an upper limit on the change in slipvelocity.

If slip continues throughout the contact period, a solid sphere (with R2/k2 = 5/2) hasa final speed of slip

s(pf) = s(0) + 3.5/x(l + e*)v3(0) (4.18)

where the coordinate system in Fig. 4.3 gives a normal component of relative velocitythat is negative at incidence: 1 3(0) < 0.

Changes in tangential relative velocity of the contact point C and the center of mass Gthat develop during collision are illustrated in Fig. 4.4. The slip velocity is directed towardsthe origin. Slip continues throughout the collision if s(0)/v3(0) > —3.5/x(l + e*), buthalts during collision and subsequently sticks if the initial slip speed is smaller than thislimiting value. Let the normal impulse where slip halts be denoted by ps. If the collisionconfiguration is collinear, there is no tangential force after slip halts (p > ps), so thereare no further changes in tangential relative velocity. At the center of mass G, the changein tangential velocity is parallel to that for the contact point C, but the change in speed isonly 2/7 as large as at C. After slip halts, the sphere rolls in the direction of the tangentialcomponent of velocity at the center of mass G; the terminal velocity of the center of

5 For central impact of ellipsoidal bodies with I\ \ i^ hi, the angle of slip is constant only if the directionof slip is parallel to a principal axis of inertia; i.e.,0(0) = 0, n/2, n, where /12 = h?> = 31 = 0-

Figure 4.4. Slip trajectory st (p) and tangential velocity of center of mass, V, (p) — Vt (p)-nn,during impact of a solid sphere on a rough half space. The cross-hatched region denotes thedomain where initial speed of slip s(0) is halted by friction before separation.

mass has a tangential component V\{ps)n\ + V2(Ps)^2- For this collinear configuration,the direction of slip is constant during collision, and all changes in velocity are in thissame direction (i.e. planar). Nevertheless, most points in a rotating sphere do not have aninitial velocity in the plane that contains both the normal to the common tangent planeand the initial direction of slip; for points where the initial velocity is out of this plane, thevelocity continuously changes in direction while the slip speed is changing. The directionof the tangential component of the translational velocity for G changes smoothly betweenthe initial direction 0(0) and terminal direction 0(pf) as shown in Fig. 4.4.

The center of mass has components of final velocity at separation Vtf = Vi(pf) thatare given by

yi{pf)

V3(0)

%ipf)

V,(0)V3(0)

<^(0)

t>3(0)

—e*

2 o,-(0)7 t>3(0)

l-M(l+e

V3(0)where

if

if

*(0)t>3(0)

5(0)V3(0)

<3.5/i(l

0,.= JT/2-I

(«• = !, 2)

(4.19)

/ = 1i = 2 .

4.3 Slender Rod That Collides with a Rough Half Space

As a first example of 3D impact, consider a slender rod that is inclined at an angle0 from the normal direction n when it strikes a massive half space as shown in Fig. 4.5.Let the center of mass of the rod be located at the origin of a Cartesian coordinate system

4.3 / Slender Rod That Collides with a Rough Half Space 73

Figure 4.5. Inclined rod colliding with a rough half space. At the contact point C there isinitial slip in the tangent plane.

which has unit vectors n3 = n normal and nx, n2 tangential to the surface of the halfplane; the rod lies in the nx, n3 plane. The rod is assumed to have length 2L, mass perunit length M/2L and, for the n2 axis which is transverse to the rod, a radius of gyrationk = L/y/3 about the center of mass. Relative to the center of mass G, the contact pointC is located at r,-, where r\ = —LsinO,r 2 = 0, r3 — —LcosO. At the contact pointthere are normal and tangential forces; in an increment of time dt these forces produce adifferential of impulse dpt. Let the center of mass of the rod have translational relativevelocity Vt and the angular velocity of the rod be denoted by coi; then the equations oftranslational and rotational motion can be expressed as

V , | idPl

MdVt = M {dV2\ = I dp2 \ (4.20a)dV3 I I dpi

dcoj = Mk2

Here the inertia matrix is singular, so it is not possible to obtain the inverse matrix formoments and products of inertia /V71 as required by Eq. 4.7. This is because the rodhas been assumed to have a negligible moment of inertia about its longitudinal axis;consequently, in the expression above, the first and last equations are linearly dependent.To eliminate this dependence, we assume that the rod has a constant rate of rotation aboutits longitudinal axis; i.e.,

cos200

sin 0 cos 0

010

— sin 0 cos 00

sin26>

\d<oA\dco2\\dco3

\r2dp3= {r3dPl

\rxdp2

- r3 dp2

- n dp3

- r2 dpx

giving dco3 = — tan 0 dcox — dcox. (4.20b)0 = sinO dcox + cos 0 dco3,

After substitution in the previous expession, the first two equations result in

(4.20c)

In a coordinate system with origin at the center of mass, the contact point C is located

Mk2 dco\dco2

\r2 dp3 - r3 dp2

{ r3 dpi - rx dp3

at r/. Thus, since only the rod is moving, the relative velocity at C is given by

vi = Vi = Vi+eijka>jrk. (4.20d)

Hence substituting (4.20a,b,c) into (4.20d) results in a set of equations identical to (4.6):

wheredvt = mi/dpj

y 1~ M

1~ 2M

0 1. -nr3/k2

"5 + 3 cos 200

- 3 sin 20

+ (r

080

0

0rf)/**

1

- 3 sin 200

5 - 3 cos 20

0+ r?/*2_

(4.21a)

(4.21b)

(4.21c)

Because of symmetry about the n \, n3 plane, the off diagonal terms m 12 = m23 = 0 andthe equations of motion can be expressed as

dv\/dp = —\jim\l cos0 + m^1

dv2/dp = -

dv3/dp = —

sin0

cos0

(4.22a)

(4.22b)

(4.22c)

4.3.1 Slip Trajectories or Hodographs

On the relative velocity plane for slip (v\, v^) there are lines termed isoclinicswhere the direction of slip is a constant. In general the orientation of the isoclinic linesdepends on the inertia properties of bodies and the coefficient of friction; i.e. the orienta-tion of isoclinic lines are characteristic values for the system. These lines are asymptotesfor the direction of slip, so that the direction of slip flows towards an isoclinic as thenormal impulse increases.

Along an isoclinic, changes in velocity are parallel to the current velocity, so that (4.22)gives

dv2/dpdvi/dp

sin0 (4.23)

that is either,

sin 0 = 0 or cos 0 = m/x(mn

1-m221)'

For the present impact configuration, which is symmetrical with respect to n2, Eq. (4.23)has roots 0 = 0, it for all values of the coefficient of friction \i ; the symmetry of theseroots is due to the inertia being symmetrical, which gives m\l = m^ = 0. It is worth not-ing that for the rod, along 0 = n (i.e. V\ < 0, v2 = 0) Eq. (4.22a) gives dv\ /dp > 0. This isthe case where slip speed decreases irrespective of the value of the coefficient of friction /x.On the other hand, slip on the isoclinic 0 = 0 results in increasing speed (dv\ /dp > 0) iffji < Im^l/m^j1 and decreasing speed {dv\/dp <0) if \± > \m\l\/m\l. Thus if the

4.4 / Equilateral Triangle Colliding on a Rough Half Space 75

Table 4.1. Critical Friction

Initial Directionof Slip,

0(0)(deg)

030456090

Coefficient fi*

0 = 30°

1.732.002.453.46oo

e for Inclined Rod

Critical Friction Coefficient /A*

0=45°

1.01.151.412.0oo

6> = 60°

0.580.670.821.15oo

coefficient of friction is sufficiently large, any initial slip will decrease in magnitudeduring the contact period; in this case, if the slip vanishes before separation it subse-quently sticks rather than beginning a second phase of slip. If however the coefficientof friction is small, the direction of slip always evolves towards n and the speed in-creases. For the present case of an inclined rod, Eq. (4.12) gives a coefficient for stick of/x = m\llm\l = (3 sin 29)/(5 + 3 cos 20).

If the coefficient of friction satisfies the condition /x > cot 0, then Eq. (4.23) has a thirdroot,6 0* = cos~l [[JL~1 m~[2 /(m~[i — m^)] = cos"1 (/x"1 cot#) This again is an isoclinicwhere the direction of slip remains constant - it will be termed a separatrix. A separatrixseparates two regions of initial velocity in which the direction of slip asymptoticallyapproaches different isoclinics as the normal impulse increases (or as the sliding speeddecreases). For any initial velocity the separatrix passing through this point in phase spaceis given by the critical coefficient of friction /x* = cot 9/ cos 0(0). This is the value ofthe coefficient of friction which causes the sliding speed to continually decrease until slipvanishes simultaneously with termination of the compression period. Typical values forthe critical coefficient of friction of an inclined slender rod are listed in Table 4.1.

Figure 4.6 illustrates some flow-lines evolving from various initial velocities of slipfor a rod inclined at 45° and three different coefficients of friction, /x = 0.5, 1.0 and 1.5.For small friction, /x < 0.6, slip does not stop but tends to an isoclinic direction $ = n,whereas for larger friction, slip will vanish if the contact period is sufficiently long. InFig. 4.6 the shaded region indicates initial velocities where slip is brought to a halt duringcompression. By extrapolation, this gives an indication of the range of initial conditionswhere slip will vanish before separation. These regions do not appear if /x < /x, becausethis friction is insufficient to slow initial slip. For a large coefficient of friction, /x = 1.5,there is a separatrix shown as a dashed line; this separates regions of sliding velocitywhich have different directions of approach as slip vanishes. The flow (or evolution ofslip) in more complex examples is described by Mac Sithigh (1996) and Batlle (1996).

4.4 Equilateral Triangle Colliding on a Rough Half Space

A thin plate provides an example of 3D impact which does not suffer from asingular inertia matrix of the type illustrated in Sect. 4.3. Consider a thin uniform plateof equilateral triangular shape which has one corner that collides against the surface of a

6 This lower bound for the coefficient of friction /x is required in order that — 1 < cos 0 < 1.


v2/v2[Q)\

-1 0

(b)

-1

Figure 4.6. Hodograph, phase plane, of slip trajectories for a slender rod inclined at 45°and (a) /x = 0.5, (b) /x = 1.0 and (c) /x = 1.5. Isoclines 0 = 0,7r are the same for all /x, whilea separatrix 0* occurs only if /x > cot#. For this symmetrical configuration, the hodographis symmetrical about the V\/\v3(0)\ axis, but only one half is shown. For initial slip in theshaded region, slip stops during compression.

massive half space. Let the plate have mass M and sides of length 2L. At incidence theplate is perpendicular to the surface and the center of mass is inclined at angle 0 from thenormal to the surface, n^=n. Axes are chosen such that the plate lies in the n\, n?> planeas shown in Fig. 4.7.

In order to obtain the inertia matrix for the plate, consider first the coordinate system§, x) with an origin at the contact point C and the direction of § on an axis of symmetryof the plate. These axes are in the n\, n3 plane, and the § axis is inclined at angle 0 fromthe normal n3=n. For these axes the moments and products of inertia are given by

= \ML\ = \ML\ hi = = I2n = /*„ = 0.

These principal moments of inertia for C can be transformed to rotated coordinates n \, n3

using Mohr's circle or otherwise, to obtain the inertia matrix Itj (/, j = 1, 2, 3) for the

4.4 / Equilateral Triangle Colliding on a Rough Half Space 11

Figure 4.7. Equilateral triangle plate with one corner striking a rough half space.

contact point C of an equilateral triangular plate,

ML2 "100

020

0"01

The inverse of this matrix for moments of inertia, IT.1, as required by (4.7), is

ML2

200

010

002

4.4.1 Slip Trajectories and Hodograph for Equilateral Triangle Inclinedat 0 = TT/4

In this section slip trajectories are calculated for the example of one corner ofan equilateral triangle colliding against a massive body. Calculations are performed foran impact configuration which has a specific angle of eccentricity for the center of mass,0 = 7r/4. For the equilateral triangular plate with side lengths 2L the vector rt from thecenter of mass to the contact point C has components

-2L sin(7r/4) -2Lcos(;r/4)

This vector and Itj combine to give the inverse of inertia matrix m^ for point C accordingto Eq. (4.7),

3 0 - 2 "0 9 0

- 2 0 3

Together with the equations of motion (4.21), this completes the preparation of the differ-ential equations which describe motion during contact between a corner of an equilateraltriangular plate and a rough half plane.

As in the case of the inclined rod, slip at the contact point of the triangular plate hasisoclinics 0 = 0, n that are in-plane with the plate. In this case Eq. (4.12) gives a coeffi-cient for stick of \x — 2/3 ^ 0.66. An additional isoclinic, the separatrix, is obtained fromEq. (4.1 lb) only if /z > m\ll(m\l — nc£)\ the separatrix direction in phase space is atan angle 0* = cos~l [/JL~1 m^/(m^ — m^1)] = cos~l(l/3fi). These features of the sliptrajectories are illustrated in Fig. 4.8. In both Figs. 4.6 and 4.8 note that for small friction/x < /x, sliding never entirely vanishes unless it approaches the origin along the isoclinic0 = 0; even in that case, if sliding vanishes, it reverses immediately and accelerates alongan isoclinic pathline. On the other hand, if friction is moderately large (IJL > /x), all slip tra-jectories converge towards zero slip speed. Whether slip vanishes or not in a particular casedepends on the initial slip velocity and the distance from the origin along the pathline(i.e. the change in normal impulse during contact). For the present impact configura-tion the range of initial conditions where slip vanishes during compression is shaded inFig. 4.8b and c.

-1

i = 0.66= p.

dVi/dp =

(b)

v2/|v3(0)l

: 1 . 0 > j i / . : \ . . . . .

-1 0 v1/|v3(0)l(c)

Figure 4.8. Hodograph, or phase plane of slip trajectories for an eccentric triangular plate(0 = 7r/4). For initial slip in the shaded region, friction is sufficient to stop slip duringcompression.

4.5 / Spherical Pendulum Colliding on a Rough Half Space 79

For slip along isoclinics where the direction of slip remains constant, the normalimpulse at the termination of compression, pc, is obtained by setting v^(pc) = 0 inEq. (4.22c):

~Pc 1 M- 1 cos0

Thus along any isoclinic, for a terminal impulse pf there is a change in the componentof tangential velocity V\ that is in-plane with the plate,

-V\ -I m13 —- 1 COS0

v3(0) - 1 COS0 Pc

Whether or not this is sufficient to bring slip to a halt before separation depends on theratio z?i(0)/z;3(0) between components of initial velocity and on the impulse ratio p//pc.

4.5 Spherical Pendulum Colliding on a Rough Half Space

As an example of an eccentric impact configuration with a velocity constraint atone point (rather than a free body), consider a spherical simple pendulum that collideson a rough half space. The pendulum is a rigid body of length L with one end pivoted ata stationary point O. At the instant of impact the pendulum is inclined at an angle 0 fromthe normal to a massive half space, so the support is located a perpendicular distanceL cos 6 from the plane surface of the half space as illustrated in Fig. 4.9.

During collision the pendulum is subjected to reactions at both the contact point C andthe fixed support O, so the free body impact equations (4.6)-(4.7) are not directly appli-cable. The impulse of the reaction at O can be eliminated from consideration, however,by obtaining equations of motion in terms of moments about this fixed point. Since themotion of the pendulum is a pure rotation about O, it is convenient to use a cylindricalcoordinate system with axis n directed normal to the surface of the half space and aradial unit vector n\ directed from the projection of O on the surface towards the contact

Figure 4.9. Spherical simple pendulum colliding with a rough half space. Initially thependulum is rotating about both n2 and n3.

point C . The differential equations for rotations about these unit vectors can be obtainedas follows:

Iij dcoj = £ijkrj dpk

where dpt is the differential of impulse at C, rt is the position vector of C from the fixedpoint O, and 7,-y is the second moment of mass for point O in reference frame nt. Thencethe relative velocity vt at the contact point can be calculated from

Vi=£ijkQ)jrk

and the differential equations for changes in relative velocity can be expressed in termsof the differential impulse of the reaction at C,

dvt = Sikm^nnhirmrn dpj. (4.24)

In this simple pendulum, however, the bob is assumed to be a particle of mass M atthe contact point C. Like the previous example, this idealization has a negligibly smallmoment of inertia about line OC, so an additional constraint is required to obtain a uniquesolution (this constrains rotations about OC). Let this constraint on the angular velocitybe co\ = 0.

This additional constraint reduces the number of equations of motion to the number ofdegrees of freedom, namely two. Nevertheless, separate equations for the rates of changeof components of relative velocity at the contact point as a function of the reaction impulsecan be obtained:

Mdv\ = cos2 0 dp i + sin 0 cos 0 dp

Mdv2 = dp2

Mdv3 = sin 0 cos 0 dp\ + sin2 0 dp

where in reference frame ni9 the spherical pendulum has moments of inertia about thepivot point O given by

In = ML2 cos2 0, I22 = ML2 and I33 = ML2 sin2 0.

If there is slip at the contact point, v\ + v\> 0; in this case the Amontons-Coulomblaw of friction gives

Mdvi/dp = sin# cos# — /x cos2 6 cos0 (4.25a)Mdv2/dp — -pi sin 0 (4.25b)Mdv3/dp = sin2 0 — \x sin 0 cos 6 cos 0 (4.25c)

where (j)(p) = tzn~l(v2/vi) is the angle of slip. These equations can be integrated to givethe components of velocity as a function of normal impulse p,

p /xcos20 fp

V\ (p) = V\ (0) + —- sin 0 cos 0 —— / cos </>(/?) dpM M Jo

v2(p) = 02(0) - ^ / sin (f)(p) dp (4.26)M JP V fp

v3(p) = v3(0) H sin 0 sin 0 cos 0 I cos (p(p) dpMM Jo

4.5 / Spherical Pendulum Colliding on a Rough Half Space 81

where the initial conditions for a spherical pendulum (0, co2(0), co3(0)) give the normaland tangential components of initial relative velocity at C,

i?!(0) = -Lco2(0) cos0 , i?2(0) = Lco3(0) s in9, i?3(0) = -Lco2(0) sinO.

Additional information about behavior during impact can be gained by consideringthe slip trajectory dv2/dv\ at any impulse p,

/xsec2 (9 sin 0=

dv\ tan 9 — /z COS 0Suppose that the angle of slip </>(/?) is a constant throughout the compression phase ofcollision and that slip vanishes simultaneously with the normal component of relativevelocity at impulse pc, i.e. that the incident angle of slip is coincident with a separatrix0(0) = 0*. For 0 < 0 < 7r this condition can be expressed as

dv2(p)/dptan</>(p) = ) y " f = tan0(0), 0 < p < pc.dvi(p)/dp

After substitution from (4.25), this constant direction of slip is associated with a criticalvalue /z* for the coefficient of friction,

/z* = - cot 0/ cos 0(0). (4.28)

Note that the spherical pendulum has initial conditions V\ (0) < 0 and v2{0) > 0, which give—1 < cos 0(0) < 0 and consequently a critical coefficient of friction which is positive(M*>0).

At this point we can anticipate a few results. If the coefficient of friction is large(/z > /z*), the circumferential component of slip vanishes at impulse pc where compres-sion terminates, while if fi < JJL* and /z < tan0, the circumferential component of slipasymptotically approaches zero but never completely vanishes during the collision. Ineither case the direction of slip is continuously changing unless the coefficient of frictionequals the critical value /z = /z*.

For the spherical pendulum the kinematic constraints give

03(O)/i>i(O) = tan (9, v3(0)/v2(0) = tan 0 cot 0(0).

With these constraints, the components of velocity (4.26) can be expressed as a functionof the normal component of reaction impulse p:

= L psin0 /z cos2

1L psin20 /z cos2fl fp cos0(p) J1 Mv3(0) ii*Mv3(O)Jo cos 0(0) P\C°

dp\ cot^tan0(O) (4.29)sin 0(0)

v3(p) psin29 /z COS20 fp ^Vjr /= 1 -| 1 / dp.

Mv3(0) /z* Mv3(0) J0 cos 0(0)Here it is worth noting that an alternative derivation of the equations of motion in

terms of generalized coordinates can result in an expression directly in terms of angularspeeds rather than the normal and slip velocity components at the contact point. For the


spherical simple pendulum this gives

[l[o

j [l Olfrf^l |-Lcos0 0 L s i n f l l H H ,. ,„,722 [o s i n ^ J j ^ H 0 Lsine 0 J j ^ j " (430)

These equations relating changes in generalized momentum with differential incrementsof impulse can also be obtained from Smith's (1991) general formulation of equa-tions of impulsive motion for colliding bodies with velocity constraints. If the rates ofchange of tangential components of impulse are related to the normal component by theAmontons-Coulomb law, then dp\ = —//, COS0 dp and dp2 = —/x sirup dp. Hence,

r + A c o s 0 ( P ) c o t 2 idpI IL* COS 0(0) J p c

dco2 = —j i H — cor v I ^7-[ /z* cos 0(0) J pc

(4.31)dco3 fi sin0(/?) 2 dp

esc 6 ^—esc 6 ^co3(0) /z* sin 0(0) pc

where a characteristic normal impulse for compression pc that brings the normal compo-nent of relative velocity to a halt in the absence of friction is obtained as

_ Mk22co2(0) Mv3(0)

To integrate these differential equations, the angle of slip (p(p) must be expressed in termsof the rotation rates,

co3 sin 6 —co 2 cos 0cos0 =

Jco\ cos2 e+coj sin2 0 Jcoj cos2 0 + a>\ sin2 ©

The components of slip are directly related to the angular velocities, so a comparison of(4.31) with the combination of (4.25), (4.29) and (4.32) gives

dv3 dco2 dv2 —dco 3

^(0) = ^ ( 0 ) ' ^(0) = co2(0)'

The terminal impulse p/ that corresponds to any specific value of the energetic coef-ficient of restitution e* can be calculated either by integrating the last equation in (4.29)or by double integration of the first equation in (4.31). This terminal impulse is the upperlimit of integration for each component of relative velocity; it depends on any variationsin the angle of slip.

4.5.1 Numerical Results for 6 = TT/3 and TT/4

The values of the critical friction coefficient /z* that give a steady direction ofslip during compression are listed in Table 4.2 for a variety of initial conditions andimpact configurations. For most of these conditions the critical value is rather large incomparison with typical friction measurements.

During collision the normal and tangential components of velocity are continuouslychanging as a function of impulse. The changes in components of slip are illustrated inFig. 4.10; unless /x//z* = 1.0 or the slip direction is an isoclinic, the direction of slip iscontinuously varying. In Fig. 4.10 the radial component of slip V\ is negative during the

4.6 / General 3D Impact 83

Table 4.2.

Inclinationat Impact,0 (deg)

45

60

Critical Friction Coefficient /x* for Spherical Pendulum

Initial Directionof Slip,0(0)(deg)

165154150135120117

165150139135120106

Ratio of InitialComponentsof Slip,02(O)M(O)

-0.27-0.50-0.58-1.00-1.75-2.00

-0.28-0.59-0.86-1.00-1.74-3.47

Ratio of InitialAngular Speedsco3(0)/co2(0)

0.270.500.581.001.752.00

0.160.340.500.581.012.0

Critical Coeff.of Friction,At*

1.031.121.161.422.012.24

0.590.670.760.821.162.10

compression and positive during the restitution phase of collision. The curvature of theslip trajectories indicates that the pendulum bob in this system has a radial accelerationthat is larger than the circumferential acceleration during the compression phase, butthe circumferential acceleration is the largest component during the restitution phase.If /JL < fi* and jji < tan 6, the circumferential component of velocity monotonically de-creases throughout the collision, whereas if tan# < /x < /z*, the circumferential slip van-ishes during restitution. After circumferential slip vanishes, the slip velocity is radial anddecreasing. Furthermore, if /x > /z*, the circumferential and radial components of slipvanish simultaneously at the end of the compressive phase of collision. In the latter case,Eqs. (4.25a) and (4.27) show that friction is sufficient to prevent the radial component ofslip from reversing. Consequently, if/x > /z*, there is no final separation, as the pendulumis stuck in the compressed state.

The present analysis gives ratios of angular speeds at separation to those at incidence(final/initial angular speed) that can be obtained from Fig. 4.10 for several initial condi-tions. The separation velocity is diminished if either the coefficient of friction increasesor the coefficient of restitution decreases. If friction dissipates some of the initial kineticenergy of relative motion, then at separation both the radial and the circumferential slipspeeds are smaller than the initial values even if the coefficient of restitution e* = 1.

4.6 General 3D Impact

Some of the results presented in this chapter have been known for a long time. Itwas Coriolis in his Jeu de Billiard (1835) who first proved that during a collision betweentwo rough spheres the contact points on the two bodies have the same direction of slipthroughout any collision. Later Ed. Phillips generalized this result in Liouville 's Journal,Vol. 14 (1849). He showed that in a collinear collision between two rough bodies, thedirection of slip is constant if each body has a principal axis of inertia in the direction of


v2/W3(0)\

v>3(0)l-0.8 -0.4 0 0.4 0.8

(b)

Figure 4.10. Slip trajectories for spherical pendulum: (a) 9 =45° , .0(0) =135°; (b) 0 ••45°, 0(0) =154°; (c) 9 = 60°, 0(0) = 135°.

initial slip. In all other cases, however, the contact points have a rate of change of relativetangential velocity with impulse, d(vt — vt • nn)/dp, that is not parallel to the direction ofslip and therefore the direction of slip changes continuously. While the direction of slipswerves only if the impact configuration is eccentric, the tangential velocity of the centerof mass swerves whenever there is friction and the body has a component of initial angularvelocity parallel to the initial tangential velocity of the center of mass. These changesin direction occur smoothly as a function of impulse. Slip can halt before separationonly if the coefficient of friction is larger than a characteristic value. Slip that halts canimmediately resume in a different direction only if the impact configuration is eccentric.

Problems 85

In this chapter, changes in relative velocity that occur during an instant of impact havebeen obtained from ordinary differential equations of motion using the normal impulseof the reaction at the contact point as an independent variable. These equations representcollisions between hard bodies where deformation is limited to an infinitesimally smallregion around the contact point. By considering that contact points on two colliding bodiesare separated by a deformable particle, we have been able to follow the process of slip. Thisgives a method of dividing the energy loss into separate parts due to friction and irreversibledeformations, e.g., by calculating separately the work done by normal and tangentialcomponents of the reaction impulse. In fact, the advance of the present analytical method,compared with a semigraphical method presented by Routh (1905) and an analyticalmethod given by Keller (1986), is in the use of an energetic coefficient of restitution e* tocalculate the terminal impulse at final separation. This coefficient of restitution representshysteresis of contact force due to internal irreversible deformation. It is calculable fromthe kinetic equations for rigid body motions. This method of obtaining changes in velocityas a function of impulse represents collisions between slightly deformable solids wherethe contact patch remains small.

PROBLEMS

4.1 A rigid rod of mass M and length 2L is inclined at angle 0 from the vertical when itstrikes a massive half space. For a small coefficient of friction /x, find the isoclinics(lines along which slip has a constant direction). For these isoclinic lines find thelimiting coefficient of friction which, if slip vanishes during collision, prevents asecond phase of slip. On the isoclinic for initial slip that approaches the origin, findthe largest ratio of slip to normal velocity at incidence which results in slip vanishingsimultaneously with the termination of compression.

4.2 For the spherical pendulum in Sect. 4.5, find the isoclinic lines. Also find the limitingcoefficient of friction which prevents second phase slip. For /x = 0.5 sketch flow linesfor 0(0) = 45, 90, and 135 deg.

4.3 A homogeneous sphere of radius R rolls on a level table at an initial angular speedCOQ before striking a rough vertical wall. At incidence i^0 is the angle between thevelocity of the center of mass and the normal to the wall. Let /x be the coefficient offriction and e* the coefficient of restitution between the sphere and the wall.(a) Find that at the impact point, RCDQ is the initial slip speed and the angle of

incidence is inclined at \j/0 from vertical.(b) Show that in order for slip to stop during compression, the coefficient of friction

must satisfy/x > 2/7 if $ 0 < 7r/4

/x > 2 tan($0)/7 if ^ 0 < nl^-(c) Show that the center of mass has a rebound angle \jr f that satisfies

if 0<ps/pc<l+e*

. tan(iA0) /x(l+e*)tan(i/f f) = if ps/pc > 1 + e*.3 e* e*

(d) Sketch regions of slip-stick and continuous slip for e* = 0,1 onaplotof^o vs/x.

CHAPTER 5

Rigid Body Impact with Discrete Modeling ofCompliance for the Contact Region

In ancient days two aviators procured to themselves wings. Daedalusflew safely through the middle air and was duly honoured on hislanding. Icarus soared upwards to the sun till the wax melted whichbound his wings and his flight ended in fiasco . . . The classicalauthorities tell us, of course, that he was only 'doing a stunt'; butI prefer to think of him as the man who brought to light a seriousconstructional defect in the flying-machines of his day.

So, too, in science. Cautious Daedalus will apply his theorieswhere he feels confident they will safely go; but by his excess ofcaution their hidden weaknesses remain undiscovered. Icarus willstrain his theories to the breaking-point till the weak joints gape. Forthe mere adventure? Perhaps partly, that is human nature. But if heis destined not yet to reach the sun . . . we may at least hope to learnfrom his journey some hints to build a better machine.

Sir Arthur Eddington, Stars and Atoms, 1927.

In this chapter lumped parameter models for compliance of the deforming regionare used to examine the influence of factors which previously in this book were assumedto be negligibly small - namely the effects of (a) a viscoelastic or rate-dependent normalcompliance relation and (b) tangential compliance. Because these factors depend on theinteraction force and not simply the impulse, the analysis of their effects necessarilyuses time rather than normal impulse as an independent variable. Thus these analyses areclosely akin to problems of vibration of one and two degree of freedom systems where thedependent variables (i.e. displacements and velocities) depend on the initial conditionsas well as the system parameters.

5.1 Direct Impact of Viscoelastic Bodies

Although most of this book describes the dynamics of collision between elas-tic or elastic-plastic bodies, nevertheless there are many examples of impact betweennonmetallic bodies where the contact force is rate-dependent or viscoelastic. This is cer-tainly true of golf balls, and it may be the best representation for other hard balls thatare hit by a bat, e.g. baseballs and cricket balls. Impact mechanics of bodies representedby nonlinear viscoelastic relations has been investigated by Hunter (1956) and by Huntand Crossley (1975). Here a simpler development is based on a linear viscoelastic com-pliance relation for the deformable element in order to illustrate some consequences ofmaterial rate dependence.

86

5.1 / Direct Impact of Viscoelastic Bodies 87

x+x0

Figure 5.1. Collinear collision of bodies separated by a Maxwell linear viscoelastic ele-ment.

5.1.1 Linear Viscoelasticity - the Maxwell Model

The Maxwell model is the simplest viscoelastic element which represents thecontact force arising from mutual compression of colliding bodies1; this element has alinear spring and dashpot in series as shown in Fig. 5.1. For this model the complianceof the deforming region gives a normal force which increases smoothly with normalcompression and some kinetic energy of normal relative motion that is restored duringrestitution; i.e., the model gives coefficients of restitution in a range from 0 to 1. Letbodies B and B' be separated by a Maxwell element; the spring has a spring constantK and uncompressed length x0, while the dashpot has a damping force constant c anduncompressed length y0. The relative displacement of the bodies x and a part of thisdisplacement that is due to the compression of the dashpot y give the normal force Facting on body B'; the same force acts in both the spring and dashpot,

F = -K(X -y) = -cy. (5.1)

If the colliding bodies B and B' have masses M and M' respectively, then the effectivemass m can be obtained from the definition m~l = M~l + M'~x. Thus there is an equationof relative motion,

m'x = —K(X — y).

By differentiating Eq. (5.1) and adding this to the equation of motion, another equationof motion is obtained in terms of the spring extension z = x — y,

z + + = 0, = mcoo/2c. (5.2)

To represent a collision between bodies with an initial normal relative speed i?0, the initialconditions are

*(0) = y(0) = z(0) = 0-v0, y = 0.

Equation (5.2) gives simple harmonic motion for spring extension during the contactperiod, i.e. the period in which the spring is compressed:

z = -c sin(codt), COdt < TV (5.3)

1 The Kelvin-Voight solid is an alternative elementary viscoelastic model that has a linear spring anddashpot in parallel. With an initial discontinuity in relative velocity at the contact point, however, thismodel gives a normal force which jumps to a finite value at the instant of incidence.

oc

5 / Rigid Body Impact with Discrete Modeling of Contact Compliance

1.0C =co

0 1.0 2.0time

3.0

Figure 5.2. For Maxwell model the maximum force occurs during compression, beforethe normal relative velocity vanishes at time tc. (On each curve the nondimensional timeo)tc is indicated by a cross.)

where the damped natural frequency a>d = cooy/l — f2. Consistent with the usual as-sumption of no tensile normal force, separation is assumed to occur at time tf = n/(Od.At separation the normal relative velocity Jif =x(tf) is obtained as

xf = z{tf) + -z(tf) = voe~ (5.4)

The normal force between the two bodies during collision is given by either the forcein the spring or that in the dashpot - these forces are the same:

F = -KZ = (1 - f 2)~l/2mcoov0e~^ot sin(codt), codt < n. (5.5)

Figure 5.2 illustrates this force for both an elastic collision c = oo (or f = 0 ) and acollision with a large damping ratio c = mco (or £ = 0.5). The Maxwell model gives anasymmetrical force with a contact period tf that increases with the damping ratio f. Themaximum force is reduced as a result of the compliance of the dashpot. This force has anormal impulse

p(t) = mvo{l - - f 2

The transition from the compression to restitution phases of impact occurs at time tc

when the normal impulse has eliminated the initial normal relative momentum; i.e.,pc = p(tc) = mvo at time tc = a>^bodies separate, the final impulse

n — tan-1(f ~ly/\ — £2)]. Thus at time tf when theimparted to body B' is given by the ratio

= l+^ / v w . (5.6)

For a direct impact this gives a coefficient of restitution e* that agrees with that obtainedfrom the ratio of normal components of relative velocity at incidence and separation,

e* = e-*nWl-t . (5.7)

The Maxwell viscoelastic model results in a coefficient of restitution e* that is independentof the incident relative velocity.

In a collision between viscoelastic bodies, the compliance relation is rate-dependent;consequently the transition from the compression to the restitution phase of contact doesnot occur at the instant of maximum force when the spring compression is maximum.With the Maxwell model the dashpot continues to compress throughout the entire contactperiod. The analysis above indicates that the transition from compression to restitutionoccurs when the normal impulse is equal in magnitude to the initial momentum of rela-tive motion mvo and that this impulse causes the normal relative velocity to vanish;i.e. xc=x(tc) = 0.

5.1.2 Simplest Nonlinear Viscoelastic Deformable Element

The linear viscoelastic model gives a coefficient of restitution that is independentof the normal component of relative velocity at incidence; i.e. it is independent of thenormal impact speed. For many collision pairs a coefficient of restitution that decreaseswith increasing relative velocity at incidence is required in order to represent experimentalmeasurements. A nonlinear viscoelastic model is a means of achieving this velocitydependence.

One possibility is a linear spring in parallel with a nonlinear dashpot; in this materialmodel the dashpot provides a force that depends on both the relative velocity and thedisplacement. The proposed material model gives a normal force F that depends onthe normal relative displacement x and the relative velocity x across the deformableelement,

F = -C\X\X-KX. (5.8)

This is a simple example of a more general nonlinear model that was proposed byWalton (1992). Similar linear-spring-nonlinear-dashpot models have been employed byStoianovici and Hurmuzlu (1996) and Chatterjee (1997). For a collinear collision betweenbodies B and B' with points of contact C and C that are separated by this viscoelasticelement, the equation of relative motion is

m'x - cxx + KX = 0, i(0) = -v0 < 0 (5.9)

where m is the effective mass, m~l = M~l + M'~x. In (5.9) the absolute value signhas been omitted after taking into account the initial conditions i(0) = — VQ < 0 andjt(O) = 0. To analyze the dynamic response of this system it is useful to change theindependent variable from time t to displacement x and rewrite the equation of motionas

m dx ( cx\-x—+x 1 =0.K dx \ K J

A nondimensional displacement X and velocity Z are defined as follows:

ex ccox CX 9 KX = —= = , Z = —, co2 = - . (5.10)

Vra/c K K m

In terms of these nondimensional variables, the equation of relative motion becomes

dZZ —

dX

90 5 / Rigid Body Impact with Discrete Modeling of Contact Compliance

0.5 0.75indentation

Figure 5.3. Nonlinear viscoelastic element gives a hysteresis loop for contact force as afunction of deflection. The size of the loop increases with incident relative velocity.

This equation is separable and can be integrated to give

--— (5.11)

where initially, the displacement X(0) = 0 and the relative velocity Z(0) = Zo =—CK~ lV0 < 0.

For this model the nondimensional force across the deformable element is given by

c(mK3yl/2F = -X(l - Z). (5.12)Figure 5.3 plots this force for two different initial velocities. Because of the velocity-dependent element the maximum force occurs substantially before the maximum relativedeflection. This maximum force increases roughly in proportion to the incident relativevelocity. For loading followed by unloading the force-deflection curve forms a loop.With increasing impact speed there is an increase in the area within this hysteresis looprelative to the area under the curve during loading, i.e. in the part of the initial kineticenergy of relative motion that is dissipated increases with impact speed. Consequentlythis nonlinear viscoelastic element gives a coefficient of restitution that decreases as thenormal incident relative velocity increases.

For a collinear collision the coefficient of restitution can be calculated from the ratio ofnormal components of relative velocity at separation and at incidence, i.e. the kinematiccoefficient of restitution. For any initial velocity Zo separation occurs when the forcevanishes, i.e. X = 0. Hence the separation velocity Zf is a root of (5.11) with the rightside of the equation set equal to zero; i.e., Zf is obtained from

Zf - Zo = - In - f , 0 < Zf < -Zo (5.13)\ l — z 0 /

and the coefficient of restitution e* is given by

e* = -Zf/Z0.

The coefficient of restitution for this nonlinear viscoelastic element is shown in Fig. 5.4.

-cvo

0 1.0 2.0 3.0normal incident relative speed

Figure 5.4. Nonlinear viscoelastic element gives a coefficient of restitution that decreaseswith increasing normal component of incident relative velocity.

Chatterjee has identified a close approximation for this curve,

Thus for large impact speeds Zo <C — 1 the coefficient of restitution is inversely propor-tional to the normal relative speed at incidence; i.e. e* ~ — Z^1.

5.1.3 Hybrid Nonlinear Viscoelastic Element for Spherical Contact

For elastic spheres the Hertz contact relation gives a normal force proportionalto the 3/2 power of indentation or relative displacement. In cases where during thecycle of loading and unloading the elastic restoring force is dominant but there also is asmall velocity-dependent dissipation of energy, Simon (1967) suggested a hybrid relationbetween contact force F and relative displacement x,

F = -K\X\1/2(X + C|JC|JC), JC(O) = - i?0 < 0 (5.14)

where the elastic stiffness K can be obtained from the Hertz relation K = 4E*Rj /3 [seeEq. (6.8)] and c is a damping coefficient.

For collinear collision between two bodies with spherical contact surfaces and x < 0the Simon relation gives an equation of relative motion,

x =xdx/dx = -m~lK\x\l/2(x - cxx). (5.15)

The analytical results can be expressed most compactly in terms of nondimensionalvariables, viz. a nondimensional relative displacement X, velocity Z, and time r:

X=x/R^ Z = dX/dx = cx, z = (cR*ylt.

With these variables the response of the system depends on only a single nondimensionalcomposite parameter f which represents the energy loss.2 The ordinary differential equa-tion for relative displacement can then be expressed in nondimensional form as

m2 The principal advantage of nondimensionalization is that the parameter £ is itself nondimensional and

thus independent of the set of units used for measurement.

where indentation results in a force of positive sign. After integrating and applying initialconditions X(0) = 0 and Zo = Z(0) < 0, the nondimensional velocity is obtained as

This can be rearranged to obtain the displacement at any nondimensional velocity,

| 2/5

(5.17)

which has a negative sign for displacement to indicate indentation. The correspondingnondimensional expression for the contact force is

F/KR3J2 = \X\3/2(l - Z). (5.18)

The latter can be compared with Eq. (5.12) from the previous nonlinear viscoelasticmodel.

The contact force as a function of displacement X, shown in Fig. 5.5, displays ahysteresis loop similar to that in Fig. 5.3. The size of this loop in comparison with thearea under the loading curve is equal to the part of the normal partial work that is dissipatedby the damper, i.e. to the coefficient of restitution.

The terminal velocity at separation Z / or the coefficient of restitution e* = —Zf/Z 0

can be obtained from Eq. (5.17) by recognizing that when X = 0 the contact force vanishes.This condition, however, gives an expression identical to (5.11). Thus the relation betweencoefficient of restitution and nondimensional incident speed for this viscoelastic model isidentical to that shown in Fig. 5.4; only the nondimensionalization of velocity is different.Hence for simulation of experiments, the damping coefficient c is obtained from the curvein Fig. 5.4 with measurements of the impact speed t?0 and the coefficient of restitution e*.

o£oc

0.05-

0.2 0.3indentation

Figure 5.5. Load-deflection curve for hybrid nonlinear viscoelastic model and two differ-ent initial velocities: (a) Zo = -0.38, f = 2.5 (solid curve) and (b) Zo = -0.50, f = 2.5(dashed curve).

5.2 / Tangential Compliance in Planar Impact of Rough Bodies 93

Table 5.1. Viscoelastic Parameters for Two-Piece Golf Ball from DynamicMeasurements

(ms-1)

36.642.7

^max tf(N) (M

e* cs) (sm"1)

12,260 450 0.81 0.010514,980 — 0.81 0.0089

K ZQ r(106Nm-3/2)

20.820.9

0.38 2.50.38 1.8

0.1890.230

Note: Size of golf ball: radius R = 0.0213 m, mass M = 61 g.Source: Experimental data from Johnson and Lieberman (1996).

15

o

aEoc

10

0 100 200time

300 400 t [p.s)

Figure 5.6. Contact force F(0 for a two-piece golf ball struck by a heavy body with normalincident speed i(0) = v0 = —36.6 ms"1 (lower curve) or —42.7 ms" 1 (upper curve).

5.1.4 Parameters of the Hybrid Nonlinear Element for Impact on a Golf Ball

There are two types of golf balls in common use: (a) two-piece balls with a solidcore of cross-linked rubber and an ionomer cover, and (b) wound balata balls with eithera durable ionomer or a balata cover. The results in Table 5.1 are for two-piece balls.3

The dynamic behavior of these balls is accurately represented by the hybrid nonlinearviscoelastic relation for spherical contacts as shown in Fig. 5.6. This figure comparescalculated contact forces for two collisions with experimental measurements.

While this analysis has focused on impact between a golf ball and a club head, theimpact behavior of the ball bouncing off the turf also has been related to viscoelastic pa-rameters. Haake (1991) measured the incident and separation velocities of rotating golfballs striking a turf green at different speeds and used these measurements to derive con-stants for a linear three parameter viscoelastic model. Data for similar lumped parametermodels have been obtained by Ujihashi (1994) and Johnson and Liebermann (1996).

5.2 Tangential Compliance in Planar Impact of Rough Bodies

For oblique impact between rough bodies there are both normal and tangentialcomponents of contact force; the tangential force (friction) opposes the tangential relativevelocity, which is termed sliding or slip. Effects of dry friction on changes in velocityduring impact can be obtained from Coulomb's law only by summation of changes during

3 Wound balata balls have a more nonlinear compliance that is not so well represented by the Simon twoparameter hybrid viscoelastic relation.


successive stages of unidirectional slip (Goldsmith, 1960; Stronge, 1990; Brach, 1993).In previous chapters of this book the analyses have assumed that tangential complianceof the bodies is negligible in the contact region. While this assumption is an asymptoticlimit that represents large initial slip, it may not be accurate for small initial slip wherethe direction of slip can change during contact. Maw, Barber and Fawcett (1976, 1981)performed a dynamic analysis of oblique impact between rough elastic spheres usingHertz contact theory to obtain the normal tractions in the contact area. For deformablebodies they found that at small sliding speeds the contact area had an outer annuluswhich was sliding. The annulus surrounded a central area where there was no tangentialcomponent of relative velocity, i.e., the central area was sticking. This combined state ofslip and stick was termed microslip. Both the analysis and experiments of Maw, Barberand Fawcett showed that if initial slip was small, the direction of slip could be reversed dur-ing collision by tangential compliance. With negligible tangential compliance such rever-sals are not possible if the impact configuration is collinear (i.e. if the centers of mass of thecolliding bodies are on the line of the common normal passing through the contact point).

There have been several attempts to develop approximations which produce the effectof slip reversal for small angles of incidence in collinear as well as noncollinear collisions.Bilbao, Campos and Bastero (1989) defined a kinematic tangential coefficient of resti-tution which varies exponentially with the coefficient of friction and the ratio of normalto tangential components of incident velocity. Smith (1991) defined a kinetic coefficientof restitution that relates the tangential impulse to the coefficient of friction, the normalimpulse and an average velocity of sliding during collision. Brach (1989) used two linearrelations for changes in tangential velocity; these employed a kinematic coefficient ofrestitution (negative) at very small angles of incidence and a kinetic coefficient of resti-tution at larger angles. All of these approaches were designed to produce at large anglesof incidence a ratio of tangential to normal impulse equal to the coefficient of friction;they do not represent Coulomb's law of friction in general.

In contrast to the elastic continuum approach of Maw, Barber and Fawcett (1976) andthe approximations above, the present section uses a lumped parameter representationfor compliance of the contact region. Equations of motion are developed from a fewphysical laws, and system characteristics are expressed in terms of coefficients that areindependent of the angle of incidence. This model yields either slip or stick at the contactpoint, depending on whether the ratio of tangential to normal contact force is as large as thecoefficient of friction. During stick the tangential force depends on relative displacement,so a time-dependent analysis is required to resolve the changes in velocity that occur ina collision; nevertheless, we assume that the total period of contact is so brief that thereis no change in configuration during collision. Comparison will reveal that the presentlumped parameter modeling gives velocity changes that are almost the same as those inexperiments by Maw, Barber and Fawcett (1981) and contact forces that are similar tomeasurements by Lewis and Rogers (1988, 1990). In addition the present analysis yieldsresults for a model of inelastic collisions with tangential compliance.

5.2.1 Dynamics of Planar Collision for Hard Bodies

To focus on the effects of tangential compliance during collision, consider twobodies with masses M and M' that collide at contact point C as shown in Fig. 5.7.

Figure 5.7. Discrete parameter model for impact of a rough, compliant body on a halfspace.

At C the contact areas of the colliding bodies have a common tangent plane. Let unitvectors n\ and n3 be oriented in directions tangent and normal to this plane respec-tively. At incidence (time t = 0) a point on each body comes into contact; at incidencethese points have a relative velocity z?/(0) with tangential and normal components V\(0)and i>3(0) respectively. The orientation of the coordinate system is defined such thatat incidence both normal and tangential components of relative velocity are negative:V\ (0) < 0 and ^(0) < 0. The collision period is separated into an initial period of normalcompression and a subsequent period of separation. The compression period terminatesat time tc when the normal component of relative velocity vanishes, v3(tc) = 0. Thecontact points separate at time tf when the final relative velocity has a normal com-ponent v&f) > 0. Hence during compression, kinetic energy is absorbed by deforma-tion of the bodies, while during restitution, elastic strain energy generates the force thatdrives the bodies apart and restores some of the kinetic energy that was absorbed duringcompression.

To simplify the dynamic analysis we assume that both bodies are rigid except for aninfinitesimally small deformable region that separates the bodies at the contact point. Letthe bodies have masses M and M' and radii of gyration kr and k!r about their respectivecenters of mass. From the center of mass of each colliding body, the contact point islocated by a position vector 77 or r[ with components in directions «/, / = 1,3. Thus thecolliding bodies have an inverse of inertia matrix for C

t]where

Pi = 1 + mr2/Mk2 + mrr2/M'k'2

p2 = mrxri/Mk2 + mr^/M'k'2

fc = 1 + mr2/Mk2 + rnr'2IM'k!2.

Notice that if the collision configuration is collinear (fi2 = 0), then in the inverse ofthe inertia matrix the terms off the principal diagonal all vanish. Hence for a collinear


collision configuration, the effects of normal and tangential impulse on changes in thecomponents of relative velocity at C are decoupled.

The infinitesimal deforming region around C is modeled by assuming that one of thecolliding bodies is connected to a massless particle located at C as illustrated in Fig. 5.7.The connection is via two independent compliant elements - one tangential and onenormal to the tangent plane. This particle has a tangential component of displacementu\(t) relative to the body that depends on the tangential force at C and the tangentialcompliance. The contact force F; acts at the contact point; it has components in directionsnt. This force applies a differential impulse dpt = Ft dt in an increment of time dt. Thusthe equations of planar motion for the rigid body can be expressed as

To progress further it is necessary to be specific about the compliance in the contactregion so that contact force can be calculated. The components of this force are neededin order to distinguish between periods of slip and stick at C. In order to calculate thechanges in relative velocity that occur while the contact sticks, it is necessary first toobtain the components of contact force.

Linear Compliance ModelHere we assume that both normal and tangential compliant elements are piecewise linear.During compression let the normal element have an arbitrary stiffness K while the tangen-tial element has stiffness K/TJ2 as shown in Fig. 5.8. During compression the parameter r\2

is a ratio of normal to tangential stiffness at the contact point; this ratio depends on thestructure of the colliding body in the deforming region. These spring constants yieldcomponents of force at the contact point which, together with the inverse mj^ of theinertia matrix, give an equation of motion in terms of relative displacements u\ and M3 atthe contact point. This equation of motion is expressed as follows:

i « 3 i \-P2r1- & j i « 3 r (5-19)

This system has two natural frequencies, a> and Q, which are the eigenvalues of the

Figure 5.8. Stiffness of normal and tangential compliant elements during loading andunloading. The energy dissipated by irreversible internal deformations is proportional tothe ratio of the area under the curve during unloading to that during loading.


frequency equations,

Ico2/a>20]

\ 2 2 \ i ±

\

l --Pin K

m

For a collinear impact configuration and linear stiffness, the colliding bodies undergoindependent simple harmonic motions (SHMs) in the normal and tangential directionswhile the contact point sticks. These motions have frequencies that depend on the stiffnessand the components /J,- of the inverse of inertia matrix that were given in (5.14). For acollinear impact configuration the natural frequency of normal motion (£2) and that oftangential motion (co) are given by

co = (5.20)

where time tc is the instant when the compression period terminates. At this instant thenormal compliant element has a maximum compression u^{tc) = 0.

For normal indentation by a rigid circular punch on an elastic half space, Johnson(1985) gives expressions for the normal stiffness K = Ea/(\ — v 2) and the tangentialstiffness /c/rj2 = 2Ea/(2 — v)(l + v), where a is the radius of the punch. These relationsgive rj2 = (2 - v)/2(l - v).

Coefficient of RestitutionThe coefficient of restitution e* can be defined as the square root of the ratio of the elasticstrain energy released at the contact point during restitution to the kinetic energy absorbedby internal deformation during compression. For negligible tangential compliance, theloss of kinetic energy due to irreversible internal deformations in the contact region canbe obtained from the work done on the bodies by the normal component of contactforce. Equation (3.24) defined the energetic coefficient of restitution e* according to thiswork. For negligible tangential compliance, this is the only part of the work done on thebodies that goes into deformation. Consequently, it is the only part that can be storedas elastic strain energy if tangential compliance is negligible - the work done by thetangential component of force is all dissipated by friction. If tangential compliance isnot negligible, however, there is also energy absorbed by tangential deformations u\.Here it is assumed that this deformation is entirely elastic. This assumption is based onconsidering the coefficient of restitution as representing nonfrictional energy losses thatare due primarily to contained plastic deformation. In an initial range of elastic-plasticdeformation the region of plasticity is contained beneath the surface of a deforming body;this contained, or subsurface, plasticity has very little effect on tangential compliance.Hence for elastic-plastic bodies which collide at low speeds, the energetic coefficient ofrestitution still applies; i.e.

With linear compliance the normal component of force does work W^ifc) on the bodiesduring compression; this is simply the area under the normal compression line in Fig. 5.8,

W3(tc) = F3(tc)u3(tc)/2. In the present model the effect of the coefficient of restitution e*is obtained by changing the stiffness of the normal compliant element at the transitiontime tc when compression terminates. Hence at the instant of maximum compression, tc,the stiffness of the normal element increases from K to ic/el. For changes in the normalcomponent of relative velocity, this change in stiffness makes the frequency of SHM largerduring restitution than it was during compression. The collision terminates and separationoccurs at a final time tf\ for collinear collisions tf = (1 + e*)tc. At separation the normalcomponent of relative displacement has a terminal value u3(tf) = (1 — el)u3(tc).

Normal Components of Velocity and Force in Collinear CollisionHenceforth in this section the analysis is limited to collinear collisions (i.e. fc = 0), sothat the normal and tangential equations of motion (5.19) are decoupled. For a collinearcollision the normal force (and hence changes in the normal component of relative ve-locity) is independent of the process of slip or stick at the contact point. For a linearcompliant element, the colliding bodies undergo separate stages of SHM during com-pression and restitution periods of the collision. Thus at any time during the collision, thenormal component of relative velocity is as follows:

v3(t) = i?3(0) cos Qt, 0 < t < tc (5.21a)

v3(t) = e*v3(0)cos(— + J(1 - e~ x)) , tc < t <tf. (5.21b)

This normal component of velocity is continuous at the time of maximum compressiontc when the frequency of SHM increases from £2 to Q/e*. The normal component ofimpulse that causes these changes in velocity can be obtained directly, and this impulsecan be differentiated to obtain the normal component of contact force. The expressionsfor these variables are listed in Table 5.2.

Notice that to maintain contact without interpenetration (or separation), we must havev3(t) + u3(t) = 0, where normal compression gives u3 > 0, and where U3(t) = du3/dt.

Tangential Velocity and Force During StickDuring the period in which the contact point sticks, SHM applies also to tangential changesin velocity if the tangential compliant element is linear. The tangential oscillations duringstick occur with frequency co. It is convenient to express the velocity and force duringstick in terms of an initial displacement u\(t2) and an initial velocity V\(t2) at time t = t2

Table 5.2.

Quantity

Normal Displacement,

Compression, 0 <

Velocity,

t <tc

Force and Impulse during

Formula

Restitution, tc <t <

Collision

Displacement u3(t) = -Q~lv3(0)sinQt u3(t) = -elQ~lv3(0)sin\ h —(1 - e~l))\ e* 2 /

Velocity v3(t) = z;3(0)cos^r v3(t) = e*i?3(0)cosf — + ^-(1 - e~l) JmQv3(0) mQv3(0) * / Qt n , \

Force F3(t) = ^ sinftf > 0 F3(t) = ^ sin( — + - ( 1 - e~ l) > 0P3 p3 \e* 2 /

Impulse p(t) = 1 ^ (1 - cos flr) p(t) = f ^ 1 - e*cos — + - ( 1 - e~ x)P3 P3 L \^* 2 /J

when stick begins. While the contact point sticks, there is no sliding [v\(t) + u\{t) = 0,where u\ = du\/dt]; thus the tangential displacement, the velocity and the contact forceon C can be written as

ux{f) = wife)cosco(t - t2) - o)-xvx(t2) sinco(t - t2) (5.22a)

Vi(t) = cou\{t2) sinco(t - t2) + V\(t2) cos co(t - t2) (5.22b)

F\(t) = ra/Jf1 co2wife)cosco(t — t2) — mfi^ lcDV\(t2)sinco{t — t2), t > t2.

(5.22c)

The state of stick persists while the ratio between tangential and normal components ofcontact force satisfies |Fi | /F 3 < /x.

Velocity Changes during Sliding with FrictionFor dry friction that can be represented by Coulomb's law, sliding occurs if the ratiobetween the tangential and normal components of force is equal to the coefficient offriction /A. Thus sliding depends on the tangential compliance and relative displacementwi(r) between the particle at C and the body; if there is sliding contact, the particle issliding at velocity V\(t) + wi(0- On the other hand, if \F\ | /F 3 < /x then the contact issticking, so V\ (t) = — u \ (t). For simplicity, coefficients of static and dynamic friction areassumed to be equal. While the contact slides in direction s = sgn(v\ + wi), the contactforce applies a differential impulse

dpj = {—/JLS 1}T dp, j = 1, 3

where dp = dp3. Consequently, in a collinear collision the differential equation for slidingcan be expressed in terms of a different independent variable - the normal component ofimpulse p rather than time:

f dvi/dp 1 _ _! \P\r]~2

~m oInitial Stick or Slip?With the present model, the contact point C either sticks or slips, depending on the ratiobetween the components of the contact force, viz. whether or not the contact force is insidethe cone of friction.4 First suppose that stick begins at the initial instant of contact andthen test whether this satisfies the limiting force ratio. An initial period of stick terminatesat a time t\ when the ratio of tangential to normal force first becomes as large as /z; i.e.,time t\ is obtained from

i co sin Qt\

\Fi\ 1 i?i(0) ft sincoti= 1, tc<ti<tf (5.24b)

4 This model gives a period of initial stick if the incident tangential velocity is small - initial stick occursbecause the particle at C has negligible mass. Mindlin and Deresiewicz (1953) have shown that duringan early stage of collision for elastic bodies, any finite tangential velocity results in central stick plusa peripheral annulus of slip around the small contact patch.

The process of initial stick takes place if t\ > 0, i.e. if in the limit as t\ -> 0 the forceratio is inside the cone of friction. This requires an angle of incidence such that

?i (0) 2

Thus initial stick occurs if the angle of incidence is small, i.e. if at incidence the ratio ofthe tangential to normal component of relative velocity at C (which equals the tangent ofthe angle of incidence ^o) is within a range 0 < i?i(0)/z?3(0) < IAT)2 that is bounded bythe product of the coefficient of friction and the ratio of normal to tangential stiffness. Forcollisions with an angle of incidence at C which is larger in magnitude than tan"1 (the contact point begins sliding when contact initiates.

5.2.2 Slip Processes

Small Angle of Incidence, z>i(0)/z?3(0) < /j,rj2

For small angles of incidence, stick initiates at initial contact and continues until time t\,when slip begins. Thereafter slip continues until separation at time tf. Figure 5.9 sketches

1.0

0.5

- 0 . 5 -

-1.0-

+1.0-

stick slip (e*= 1.0)

c

* \ ^ ^ j h-slip ( =05)

compression -~- - restitution

(b)

1.5 2.0 f/t

-1.0-

Figure 5.9. (a) Normal and tangential forces and (b) tangential velocity during collisionfor small angle of incidence i>i(0)/z?3(0) < /JLTJ2 and frequency ratio co/Q = 1.7. Forces areillustrated for both e* = 0.5 and 1.0. For each coefficient of restitution, the final velocity atseparation is indicated by a cross x.

the components of force for this case of initial stick and terminal slip. At time t\ whenslip begins, the relative velocity at C has a tangential component

V$t$ = Vi(0)cOSCOti.

Thus from (5.24) the terminal tangential relative velocity can be obtained:

vi(tf) = vx(h) - ^ [p(tf) - p(h)] (5.25)

where the change in normal impulse during the final period of slip is given by

p(tf) - p(h) = -

The impulse ratio during slip, y\, is the ratio of the normal impulse applied during periodtf — t\ to the normal impulse during compression,

This ratio depends on initial conditions. In the subsequent analysis we consider the ratioof compliances co/Q > 1, since this applies to most elastic bodies.5 For this case slipbegins during restitution at a time t\ > tc when u \ (t\) > 0, so that s = +1 and the terminalvelocity can be expressed as

V\(tf) = Vi(ti) A

Intermediate Angle of Incidence /IT/2 < v\(0)/v3(0) < //(I + e*)(3i/f33

If the angle of incidence is larger than tan~!(^^2), initially there is sliding at the contactpoint as illustrated in Fig. 5.10. During the initial period of sliding the body has a tangentialcomponent of relative velocity at C that is given by

(5.27)m

This initial sliding terminates and stick begins at time t2 when subsequent sliding andstick give the same rate of change for the tangential force; i.e., the transition to stickoccurs when

lim — 1 - ± — - = [i—*-j—- .e^o \_ ds ds J

Differentiation of Eq. (5.22c) gives for the period t > t2,

mco3ui(t2) .. mohh(t2)sin co(t — t2) cos co(t — t2)dt Pi Pi

where the dynamics for sliding during t < t2 give transition values at the beginning of

5 For elastic spheres with Poisson's ratio v = 0.3, the ratio of stiffnesses r]2 = 1.21 and elements ofthe inverse of inertia matrix give y#i//?3 = 3.5, so that co/Q = 1.7 (e.g. see Johnson, 1985), whereasfor incompressible elastic spheres, ^2 = 1.5 and co/Q = 1.53. Furthermore, in the asymptotic limit ofnegligible tangential compliance, r\2 -> 0, so that co ->• oo.

stick slip (g = 1.0)

Figure 5.10. (a) Normal and tangential forces and (b) tangential velocity during collisionfor intermediate angle of incidence \xr\2 < £>i(0)/p3(0) < /x(l + e*)/*i/#3 and frequencyratio co/Q = 1.7. Forces are illustrated for both e* = 0.5 and 1.0. For each coefficient ofrestitution, the final velocity at separation is indicated by a cross x.

stick,i £203(O) . o- sin £2t2u\(t2) =

vi(t2) = 0i(0) - ^v3(0) [1 - cos Qt2], h < tc

P3A £203(0) s i n ( - + - ( l - , ;

+ | d - > tc.

The normal force during restitution is given in Table 5.2; hence by differentiation weobtain

dF3(t2) „_,,dt

Q.t2, t2 < tc

dF3(t2) £22mv3(O) (Qt2 nt2 > tc.


Thus after equating the rates of change for components of force, we obtain the expressionto be solved for Qt2 by taking the limit as t —• t2,

Qt2 = cos"1!

0i (0)= - — U - ^ ' J - h C O S | : —; | ,

The limits of applicability for these equations have been expressed in terms of the ratioof velocity components by noting that the transition time t2 = tc occurs if Z7i(O)/z73(O) =

For intermediate angles of incidence the contact point sticks at time t2 and then thetangential compliant element begins a period of SHM. During the period of sticking thetangential components of velocity and force are given by Eq. (5.22):

V\ (t) = cou i (t2) sin co(t — t2) + V\ (t2) cos co(t — t2)

F{(t) = m^la)2ui(t2)cosco(t - t2) - m^lcov{(t2) sinco{t - t2).

This period of stick terminates and slip begins again at time t3 when the ratio of com-ponents of force next becomes as large as the coefficient of friction, \F\ \/F3 = /x. Thisgives a final phase of slip which begins at time t3 — a time that can be obtained from

cos co(t3 — t2) sinoo{t3 — t2)co/jiV3(0)

Sticking ceases at time t3 when a second phase of slip begins. For this second phase of slipthe initial tangential components of velocity and force, V\{t3) and F\{t3), are given above.

The second phase of slip terminates at separation. During this final phase of slip theparticle at the contact point is sliding as the elastic strain energy in the compliant elementsis decreasing to zero. Thus changes in velocity during this phase are given by (5.23). Thefinal tangential velocity of the contact point at time tf = (1 + e*)tc is given by

V\(tf) = V\(t3) — [ifi\m [p(tf) — p(t3)]

where s(tf) = + 1 , since the final direction of slip is opposite to the initial direction asillustrated in Fig. 5.10. The nondimensional tangential velocity at separation is

vi(tf) _ vx{t3) t rv M _ , V W | ( 5 2 8 )

Large Angle of Incidence, Pi(O)/z73(O) > /x(l + e^)/31/f33

If initial slip does not cease before separation, the transition time t2 > (1 + e*)tc\ i.e.,slip continues in the initial direction throughout the entire contact period - a contactprocess that is sometimes termed gross slip. For this case, at separation Eq. (5.23) givesthe tangential velocity V\(tf) as follows:

= vx(0) + iiPim-\l + e*)p(tc)


where S(*y) = S(0) = —1. This can be rewritten as

(5.29)

For gross slip, the direction of sliding is constant - friction merely reduces the speed.

5.2.3 Oblique Impact of an Elastic Sphere on a Rough Half Space

Combined effects of friction and tangential compliance have been evaluated foroblique impact of a sphere on a half space. For a solid sphere composed of material withPoisson's ratio v = 0.3, the ratio of stiffnesses is rj2 = 1.21 while the ratio of elementsof inertia is fi\ /fo = 3.5, so that the ratio of frequencies is co/ £2 = 1.7. These values areused in the following examples of oblique impact of an elastic sphere against a massivehalf space.

Tangential Velocity at SeparationFor a sphere striking a half space at angle of incidence i/ro = tan^fi^OVi^O)],Eqs. (5.26), (5.28) and (5.29) were used to calculate the tangential relative velocity forcontact point C at separation. In Fig. 5.11, the results from the present discrete parametermodel are compared with the elasticity solution given by Maw, Barber and Fawcett (1976)and experimental measurements by K.L. Johnson (1983) for a rubber sphere (Poisson'sratio v = 0.5) striking a heavy steel plate at a small speed. The elastic solution and thediscrete parameter model each have similar processes that develop at the contact pointin three parts of the range of angle of incidence. The predictions of these two modelsare most different for small and intermediate angles of incidence where the discrete pa-rameter model has a final period of slip that is prolonged by elastic strain energy storedin the tangential compliant element. Throughout most of the range of small to interme-diate angles of incidence, both the elastic continuum and the discrete parameter modelsof sphere impact have a tangential relative velocity at separation that is in the oppositedirection to the incident tangential velocity. For a collinear collision this velocity reversal

initial stickterminal slip slip- stick- reverse slip continuous

slip(lumped parameter model)

- 3 - (elastic sphere V= 0.3)

Figure 5.11. Tangential velocity of contact point on elastic sphere at instant of separationas a function of the angle of incidence. Solid curve, lumped parameter model; dashed curve,elastic continuum analysis; dot-dash curve, analysis for negligible tangential compliance;circles, experiments with a rubber ball.

- 3 -

Figure 5.12. Tangential velocity of contact point on sphere at separation as function of theangle of incidence for coefficients of restitution e* = 0, 0.5 and 1.0.

at C is entirely due to tangential compliance. In almost all respects the results of thesemodels are practically identical. Figure 5.11 also shows the final velocities calculatedfor negligible tangential compliance; in this case the tangential component of relativevelocity at C is zero unless the angle of incidence is large enough to cause gross slip, i.e.Vi(0)/v3(0)>ii(l+e*)pl/p3.

The effect of coefficient of restitution e* on the change in the tangential componentof relative velocity at C is shown in Fig. 5.12. The angle of incidence for gross slipdecreases with increasing internal dissipation. The coefficient of restitution e* affectsonly the impulse imparted during restitution; the restitution impulse (and changes invelocity during restitution) decreases with decreasing e*. This causes the shift in thecurve for separation velocity that is apparent in Fig. 5.12.

Angle of Incidence for Maximum FrictionExperiments using repeated impacts on steel tubes at oblique angles of incidence wereperformed by Ko (1985). He showed that for relatively small normal impact speeds,i?3 < 1 m s"1, the wear rate of steel tubes is closely correlated with the maximum tangen-tial force Fimax and that for any colliding missile this force varies with angle of obliquity.For the present model, the tangential force F\{t) can be calculated as a function of theangle of incidence, t2Ln~l[v\(0)/v3(0)]. Irrespective of the angle of incidence, the largestvalue of friction, Fi max, occurs during the compression period if co/ £2 > 1; consequently,the maximum tangential force is independent of the coefficient of restitution. For any im-pact speed v(0) = [vj(O) + vj(0)]l/2 the tangential component of force can be comparedwith the largest normal force F3max = F3(fc), where

^3 max sr11/2- (5.30)

Expressions for the maximum tangential force are given below for different ranges of theangle of incidence.

Small Angle of Incidence, z>i(0)/z?3(0) < /J,TJ2

For small angles of obliquity the contact point initially sticks and only begins to slideduring the restitution period. The maximum tangential force occurs during compressionwhen the tangential velocity reverses in direction at time r = Qtc/(Q. At this time the

contact is sticking, so the maximum tangential force can be obtained from Eq. (5.22c):

^lmax = -OOmVi(O)/0i.

This maximum tangential component of force can be expressed as a nondimensional ratiol imaxl /&mv(Q). Thus for small angles of obliquity

_ [hQmv(0) £2t7(O) rj 1/2'

Intermediate Angle of Incidence, fir]2 < V\(0)/v3(0) < /i(l + e*)j3\/ f33

At intermediate angles of obliquity there is initial sliding, but then stick begins at time t2

during the compression period. When stick begins, the contact point is still moving in theinitial direction; i.e., V\(t2) > 0. Maximum friction develops shortly after the period ofstick begins and before the instant of maximum compression. The friction force duringsticking can be expressed as

Fx(t) ii0 \Qux(t2) [fo vY(t2) . 1

At the transition from sliding to stick the displacement u\(t2) and velocity V\(t2) dependon the coefficient of friction. The transition velocity V\(t2) is obtained from (5.23), whilethe displacement u\(t2) is calculated from the friction law F\ = —fisF 2 and the force F2

given in Table 5.2. Thus

2 • o v&*> 0i(0) 0in o ,= rjz sin Qt2, — = — - —(1 - cos Qt2).

nv(0) fiv(O) 0

rj sin Qt2,liv3(0) nv3(0) fiv3(O) 03

Large Angle of Incidence, ^i(0)/z;3(0) > /x(l +If the direction of slip is constant throughout the collision period, then the maximumfriction force is directly proportional to the normal force and the coefficient of friction:

"lmax _ H<P3110-- 1

Qmv(0) 1/2-

For gross sliding this maximum tangential force occurs simultaneously with the largestnormal force; i.e., the tangential force is a maximum at time tc when the compressionperiod terminates.

These expressions for the maximum values of components of contact force have beenused to calculate the largest normal and tangential forces which occur during obliquecollisions. The largest values vary with the angle of incidence and the coefficient offriction as shown in Fig. 5.13. The largest values for the peak force occur for collisions atintermediate angles of incidence. The angle of incidence where the peak force is largestincreases from about 20° for a coefficient of friction /x = 0.1 to almost 60° for ji = 1.0.

Maximum Friction for Negligible Tangential ComplianceIf tangential compliance is negligible, oblique collision always results in an initial periodof sliding; this sliding is halted before separation, and there is a subsequent final periodof stick unless the angle of incidence is large enough to cause gross slip. Since we areconsidering central or collinear collisions, slip reversal does not occur. Consequently, if

0 0.2 (U 0.6 1.0 2.0 4.0

30° 60°angle of incidence,

90°

Figure 5.13. Maximum normal component of force (bold solid curve) and tangential com-ponent of force (dashed curves) during oblique impact of a sphere. The light solid curvesshow the maximum force if tangential compliance is negligible.

initial slip comes to a halt during compression, the peak tangential force occurs at theinstant t2 when sliding terminates:

3 - 1 sin(0)]1/2' h <tc

where

£2t2 = cos- l

On the other hand, if the bodies are still sliding when compression terminates at time tc,then the largest friction force occurs at this instant simultaneously with the largest normalforce:

lmax

Qmv(0) 1/2' tc.

In Fig. 5.13 the dashed lines show the maximum friction force for a compliant solidsphere, whereas the light extensions to the left of these curves are results for similarcollisions between spheres with negligible tangential compliance. For solid spheres, thelargest tangential force for a compliant body is substantially less than the largest forcecalculated with the assumption of negligible tangential compliance. In either case thelargest tangential force occurs in the range of small to intermediate angles of incidence. Atlarge angles of incidence there is gross slip; in this case the maximum tangential force isindependent of tangential compliance.

Comparison with Measurement of Peak Force during Oblique ImpactLewis and Rogers (1988) performed impact experiments in which a 25.4 mm diametersteel sphere collided against a heavy steel plate at angles of incidence that varied between0° and 85° from normal.The impact speeds were small, being in the range 0.01-0.05 ms" 1 .The sphere was attached at the free end of a 1.8 m long pendulum by a steel "ball holder".

30° 60angle of incidence,

90°

Figure 5.14. Comparison of the maximum normal and tangential components of contactforce with experimental measurements by Lewis and Rogers (1988). At each angle ofincidence, the normal and friction forces on a 101 g sphere colliding against a half spacewere calculated for an incident speed 0.048 m s"1 and coefficient of friction \JL = 0.18.

Piezoelectric force transducers were used to make separate measurements of normaland tangential components of contact force during impact. For gross slip or continuoussliding of this ball on the plate, Lewis and Rogers reported a coefficient of dynamicfriction \± = 0.179.

In Fig. 5.14 experimental data taken from collisions at an impact speed of 0.048 m s"1

are compared with normal and tangential components of force calculated by the presenttheory (using a coefficient of friction /x = 0.18). The calculations depend on an estimateof the mass of the "ball holder". The agreement between experiment and theory shown inFig. 5.14 was achieved by increasing the mass of the ball by 50% in order to allow forinertia of the support system. In addition the calculations used a relative compliance ratiox]1 = 1.21 and ratio of elements of inertia /?i//?3 = 3.5 that are representative of solidspheres. This resulted in both qualitative and quantitative agreement between the calcu-lations and the experiments for the full range of possible angles of incidence. Four seriesof tests using different impact speeds each gave a largest measurement of tangential forceat an angle of incidence of about 40°. For /z = 0.18 the present lumped parameter modelgives a largest value of peak tangential force at about 35° irrespective of impact speed.[For any angle of incidence, the ratio between peak tangential and normal components offorce can only be as large as the coefficient of friction if these peak values occur simul-taneously, i.e. if i?i(0)/z;3(0) > /x/3i//?3 or the angle of incidence \/fo > 32° for a roughsolid sphere with \x — 0.18.]

Comparison with Measurements of Impulse Ratio at SeparationUsing thin pucks on an air table, Chatterjee (1997) and colleagues have measured anglesof incidence and reflection for uniform circular disks colliding against a heavy steel bar.The disks were made from a polymer, Delrin. They struck the fixed bar at angles ofincidence from normal in the range 0 < X/TQ < 85°. At impact the disks were translatingat speeds in the range 0.25 < v(0) < 0.75 ms" 1 , and they had negligible angular velocity.For incident angles in the range 0 < Vo < 50° two similar pucks gave measurements ofthe coefficient of restitution e* = 0.95 ± 0.3. For larger angles of incidence XJTQ > 50°,the coefficient of restitution was slightly larger.

Measurements of the angle of incidence and the angle of rebound for the center ofmass were used to obtain the ratio of the tangential to the normal component of impulse

0 2 r

0.5 1 1.5

incidence angle, tan"1 lv^[0)/v3[0)] (rad)

Figure 5.15. Ratio of tangential to normal impulse for a circular Delrin puck collidingagainst a steel half space at angle of incidence tan"1 [vi(0)/v3(0)]. Vertical bars describe therange of experimental values. The cross and circle symbols designate two pucks which areidentical. The dashed line indicates the calculation using the bilinear spring model.

at separation. If the coefficient of friction is independent of sliding speed and the angleof incidence is large enough to induce gross sliding, this ratio should be a constant thatis equal to the coefficient of sliding friction.

Figure 5.15 compares the measurements of Chatterjee (1997) with calculations basedon the bilinear spring model in this chapter. The calculations presented here assumea coefficient of sliding friction6 î — 0.13 and a coefficient of restitution e* = 0.95,which are consistent with Chatterjee's measurements. For the thin uniform disk, a ratioof normal to tangential stiffness r\2 = 1 was used on the basis of an estimate from the2D finite element calculations of Lim (1996); together with the radius of gyration for acircular disk, this gives a frequency ratio co/ £2 ^ \ / 3 .

The bilinear spring model accurately represents the experimental results other thannear the transition to gross sliding; this transition occurs at an angle of incidence ô =tan"1 [î(0)/i^3(0)] = 0.66 rad. Near this transition the tangential impulse is smaller thanthe calculated value, indicating that there is stick for more of the contact period than iscalculated on the basis of the bilinear spring model. Chatterjee's experiments showed nodependence of the coefficient of friction on either the normal force or the sliding speed.

5.2.4 Dissipation of Energy

Internal Dissipation from Hysteresis of Normal forceDuring partly elastic collisions (e* < 1) there is always irreversible internal deformationthat dissipates a part of the initial kinetic energy To. In the present model this internal

Crude measurements indicated that the coefficient of sliding friction /x < 0.2.

dissipation D3 is entirely due to hysteresis of the normal component of force. It can beobtained as the negative of work done by this component of force,

D3(tf)* 2ft

since in collinear collisions (ft = 1) the part of the initial kinetic energy that is dissipatedinternally by irreversible deformations can be expressed as

(5.31)

Frictional DissipationFriction dissipates energy only during periods of slip. During these periods the tangentialforce does some work that changes the tangential strain energy and also some that isdissipated by friction D\. Whereas the total work done on the body by the tangentialforce depends on the tangential relative velocity v\ the frictional energy loss dependsonly on the sliding speed v\+u\. The remainder of the work done by the tangential forceis stored as elastic strain energy in the tangential compliant element and later recoveredas the normal force decreases before separation. Thus for small angles of incidence wherethe contact point slides only after an initial period of sticking, the tangential dissipationduring sliding is calculated from7

P fJpih)

Using Eq. (5.20), this gives

2Di(tf) 2M2

K

On the other hand, if the angle of incidence is intermediate, the contact point slides priorto time t2 and again slides after time f3. Thus for an intermediate angle if < tC9

2Di(tf) = /^f t [ P?(0) _ vj(h) vj(t3) _ v\(U) 1

A2(0) ^1(0) ^ K 0 ) 2^2(0)JFinally, if the angle of incidence is large so that there is gross sliding, the part of theenergy dissipation due to friction can be expressed as

^ . ( 1 + , . ^ f ^ . A ( 1 + ^ l .v3

Equations (5.32a-c) were used to evaluate the part of the initial kinetic energy of relativemotion that is dissipated by friction 2D\ (tf)/mv2(0). In Fig. 5.16 this frictional dissipationis plotted as a function of angle of incidence for coefficients of friction \x = 0.1 and 0.5.If the contact region has nonnegligible tangential compliance there is almost no frictionaldissipation if the angle of incidence is small; i.e., not much of the kinetic energy To is

7 During any period tb — ta with a constant direction of sliding the part of the total energy dissipationD(tb) — D(ta) that is due to a component in direction w,- of the contact force Fj can be calculatedfrom theorem (3.20), D/(te) - A0a ) = ni[pj(tb)- pj(ta)][Vj(tb) + Vj(ta)]/2, where at any time t theforce has provided an impulse Pj{t). For a collinear impact configuration this gives Di(tb) — Di(ta) =mJjX[v2j{tb) — Vj(ta)]/2 if the direction of sliding is constant.


0.2 O.I* 0.6 1.0 2.0

30° 60°angle of incidence,

90°

Figure 5.16. The part of the initial kinetic energy that is dissipated by friction duringimpact of a sphere for various angles of incidence and coefficients of restitution e* = 1 (solidcurves) and e* = 0.5 (dashed curves). For gross slip the frictional dissipation is independentof tangential compliance.

required to bring small initial slip to a halt before separation. On the other hand, if thereis gross slip due to a large angle of incidence, tangential compliance has no effect onfrictional dissipation. The fraction of the initial kinetic energy which is dissipated byfriction is maximum at an angle of incidence slightly larger than the smallest angle givinggross slip.

These results are based on the supposition that the coefficient of friction is a parameterthat is constant during impact. For impact speeds v3(0) > 50 m s"1 where indentation ofmetals results from uncontained plastic deformation, Sundararajan (1990) has pointed outthat the tangential force can be increased by finite indentation and decreased by frictionalheating.

5.2.5 Effects of Tangential Compliance

For almost all angles of incidence, the response of this simple lumped parame-ter model is identical with that of the quasistatic (Hertz) elastic analysis. The microslippresent in the continuum analysis has no significant effect on changes in velocity of thecolliding bodies. Both the elastic continuum analysis and the lumped parameter modelshow that if slip is brought to a halt during collision, tangential compliance can subse-quently reverse the direction of slip. Slip can be brought to a halt during collision onlyif the tangential component of incident velocity is not too large, i.e. if

For oblique impacts, the largest tangential force generated by friction during collisionoccurs during compression. This maximum frictional force is independent of the coeffi-cient of restitution. In a compliant body the largest force is somewhat smaller than thatwhich occurs if the contact region has negligible tangential compliance. Although thepresent calculation of the largest force is based on a model with normal compliance equalto a constant rather than the nonlinear compliance suggested by Hertz type analysis, thedetails of normal compliance have only a very small effect on the largest tangential force


in a collision. In this respect a more significant factor is the ratio of the tangential to thenormal compliance, rj2.

Although the present analysis has considered collinear impact configurations (andconsequently planar changes in velocity), the same framework can be used to analyzenoncollinear collisions. In a noncollinear collision, the normal and tangential motionsare coupled so components of relative displacement w, and relative velocity V[ do notundergo SHM; nevertheless, the equations of motion can be integrated numerically toobtain changes in contact force and velocity during separate phases of stick or slip.

In this model only the normal compliant element is irreversible; consequently, energylosses due to internal hysteresis and those due to friction remain decoupled. While thisis representative of dissipation due to contained elastic-plastic deformation where in-dentation is barely perceptible, it is unlikely to be accurate at higher impact speeds. Inelastic-plastic bodies, if the impact energy is large enough to develop significant per-manent indentation (uncontained plastic deformation), the inelastic internal deformationdepends on both normal and tangential components of contact force. Consequently, forimpact energies that produce significant indentation, sources of dissipation are no longerassignable to separate components of force, nor representable by coefficients which areindependent of angle of incidence.

5.2.6 Bounce of a Superball

A Superball is a solid rubber ball that is highly elastic. When such a ball is gentlylaunched in a horizontal direction and simultaneously given backspin about a transverseaxis, when the falling ball strikes a level floor the horizontal component of velocity atthe center of mass can reverse direction simultaneously with a reversal in the directionof rotation. This behavior is illustrated in Fig. 5.17. For a spherical ball these reversalsare solely due to tangential compliance of the contact region; in Fig. 5.11 they occurin the region where the tangential velocity is negative. K.L. Johnson (1983) analyzedthe bounce of a Superball as an example of an observable effect entirely attributable totangential compliance and friction; this effect cannot be obtained from rigid body impacttheory.

Figure 5.17. Spherical rubber ball with backspin. During collision, friction and tangentialcompliance combine to reverse directions of both translation and spin.


For each bounce the changing velocity of the ball can be plotted on a graph of theincident horizontal (tangential) velocity of the center of mass, £>i(0), and the incidentangular speed about the transverse axis, &>(0); in Fig. 5.18 each of these variables hasbeen nondimensionalized by the product of the coefficient of Coulomb friction /x and thenormal component 1)3(0) of the incident velocity of the center of mass. The moment ofmomentum hc about the contact point C on the lower surface of the ball during impactcan be expressed as

2hc(t) = mRvi(t) + mk2

rco(t) = mRvx(t) + -mR2oo(t).

If the contact area remains small in comparison with the radius R, the reaction force atC has no resultant moment; hence the initial and terminal moments of momentum aboutthe contact point are equal:

he _ *>i(0) 2Rco(0) = vx(tf) 2Rco(tf)/nmRv3(0) /zfr>(0) 5 /xf>3(0) fif>3(0) 5

During impact the velocity changes from an initial state (z>i(0), &>(0)) to a final state(v\(tf), co{tf))\ these changes occur along lines of constant moment of momentum hc.Constant moment of momentum he is represented by lines of slope —5/2 that are labeledacross the bottom of Fig. 5.18.

Any initial state (i>i(0), &>(0)) gives a specific tangential velocity V\{0) at contactpoint C:

vx(0) = f>i(0) - Rco(0)

or

PI(Q) = 0i(Q) _ **>(0) ( 5 3 4 )

flf>3 /XZ>3 IAV3

Lines on the graph with slope +1 have constant values of i7i(0)//xz>3(0); these have beenlabeled along the left side of the figure. The initial and terminal tangential velocities at Care directly related by Fig. 5.12 for three different values of the coefficient of restitution.In Fig. 5.18, the terminal tangential velocities at C for a coefficient of restitution e* = 1are labeled along the right hand side of the figure. Thus for any incident tangential velocityat C the terminal tangential velocity at C is given at the right hand end of this diagonalline; this terminal velocity equals the incident tangential velocity for the succeedingimpact.

In Fig. 18 a numbered series of impacts (1,2,3,...) is shown; these begin with initialconditions [v\ (0)//zz>3(0) = —1, Rco(0)/'ixv3{0) = 4] which result in the tangential velocityat C changing from vx (0)/fiv3(0) = - 5 to V\ (tf)/fiv3(O) = 1.7. In this case, when the ballundergoes the first impact, the velocity jumps from the fourth to the second quadrant; thusboth tangential and angular velocities change sign. In the second impact the direction ofrotation changes once again, but the direction of the tangential velocity does not change.This second bounce gives, for the third impact, an incident tangential velocity at Cof VI(0)//JLV3(0) = —0.3. The shaded field in this figure represents the range of incidentspeeds which give simultaneous reversal of direction for both the tangential and rotationalvelocities. For most initial conditions there are only a few simultaneous reversals beforethe ball bounces off in one direction.

Ru){0)

Figure 5.18. Graph for changes in tangential and rotational velocities of sphere during impact wheree* = 1. The incident tangential velocity of the center of mass is on the horizontal axis while that of thecontact point C is along the left border. The dashed line represents a series of three collisions whichoccur on the line he = 0.6. Incident velocities in the shaded teardrop-shaped regions give reversalfor both translational and rotational velocities.

114

Effects of Tangential Compliance 115

PROBLEMS

5.1 For the Maxwell solid, find the relative velocity xc = x(tc) at the instant when thenormal impulse is equal to the initial difference in translational momenta, mVQ. (Thisproves that the transition from compression to restitution occurs simultaneously withmaximum relative indentation.)

5.2 For the Maxwell solid, find the following expression for the nondimensional timecotc when the normal component of relative velocity vanishes [x(tc) = 0]:

cotc = n tan -

Hence show that for c — mco (f = 0.5), the relative velocity vanishes at a timecotc = 2.42.

CHAPTER 6

Continuum Modeling of Local DeformationNear the Contact Area

Only those bodies which are absolutely hard are exactly reflectedaccording to these rules. Now the bodies here amongst us (being anaggregate of smaller bodies) have a relenting softnesse and springy-nesse, which makes their contact be for some time and in more pointsthan one. And the touching surfaces during the time of contact doeslide one upon another more or lesse or not at all according to theirroughnesse. And few or none of these bodyes have a springynessesoe strong as to force them one from another with the same vigorthat they came together.

Isaac Newton, Laws of Motion Paper, MS. Add 3958,Cambridge University.

In practice the bodies that are colliding are composed of elastic, elastic-plasticor viscoplastic materials, so that the large contact forces acting during a collision induceboth local deformations near the contact point and global deformations (vibrations) ofthe entire body. This chapter focuses on the local deformations in a contact region thatcan be represented as an elastic-perfectly plastic solid; the additional effect of globaldeformations will be introduced in Chapter 7.

For collisions between hard bodies, the analysis of changes in velocity during col-lision is simplified by assuming that the initial point of contact is surrounded by aninfinitesimally small deforming region. For other purposes, however, it is necessaryto consider deformations in the small region surrounding a finite area of contact. Onesuch purpose is to relate the coefficient of restitution e* to energy dissipated by plas-tic deformation in the contact region. Here we analyze details of deformation in thecontact region and relate these to interface pressure between the bodies. The aim isto express hysteresis of contact forces as a function of impact parameters and proper-ties of the colliding bodies, i.e. to obtain a theory for estimating nonfrictional energylosses in collisions between elastoplastic solids. In the first instance this theory is basedon the assumption that nonfrictional energy loss is entirely due to plastic deformation.Such a theory is useful for identifying the range of impact speeds where a particu-lar form of material behavior is representative of the physics of deformation. Subse-quently, effects of friction and additional energy losses associated with elastic waves areconsidered.

116

6.1 / Quasistatic Compression of Elastic-Perfectly Plastic Solids 111

6.1

Figure 6.1. Compression and indentation of spherical contact surfaces.

Quasistatic Compression of Elastic-Perfectly Plastic Solids

6.1.1 Elastic Stresses - Hertzian Contact

A continuum analysis of contact forces and the deformations that arise fromquasistatic compression of elastic, elastic-plastic or perfectly plastic bodies can be usedto develop a theory of impact for hard bodies composed of rate-independent materials. Inthis theory deformations are negligible outside a small contact region, and the deformingregion acts as a nonlinear inelastic spring between two rigid bodies; the mass of thedeforming region is assumed to be negligible. Hertz1 (1882) first developed this quasistatictheory for elastic deformation localized near the contact patch and applied it to thecollision of solid bodies with spherical contact surfaces. Hertz's theory provides a verygood approximation for collisions between hard compact bodies where the contact regionremains small in comparison with the size of either body.

Let nonconforming elastic bodies B and B' come into contact at a point C; in a neigh-borhood of C the surfaces of the bodies have radii of curvature RB and /?Bs as described inFig. 6.1. If these bodies are compressed by force F = F3 in the normal direction, Hertzshowed that the contact region spreads to radius a and within the contact area there is anelliptical distribution of contact pressure

= po(l-r2/a2)1/2r < a (6.1)

where r is a radial coordinate originating at the center and po = p(0) is the pressure at thecenter of the contact area. This contact pressure generates local elastic deformations andsurface displacements that cause initially nonconforming surfaces to touch or conformwithin a contact area. This pressure distribution results in a compressive reaction forceF on each body,

-FJo

p(r)2nrdr = -—poa 2 (6.2)

1 Hertz developed this theory during Christmas vacation 1880; he was 23 at the time and studying withKirchoff. Although this theory was initially dismissed by Kirchoff, subsequently it has proven to beextremely useful.

118 6 / Continuum Modeling of Local Deformation Near the Contact Area

The mean pressure p is two-thirds of the pressure at the center of the contact circle, p =2po/3. For the pressure distribution given in Eq. (6.1), Hertz obtained the normal displace-ment wt(r) at the surface of body / (/ = B, B') from the Boussinesq solution for a forceapplied normal to the surface of an elastic half space (Timoshenko and Goodier, 1970):

wt(r) = 0.25(1 - vf)7tap0E7l(2 - r2/a\ r < a (6.3)

where compressive displacements are positive. In this expression the elastic moduli ofbody / are given as Young's modulus Et and Poisson's ratio v,-.

The compression of each body 8( is equivalent to the relative displacement between theinitial contact point C and the center of mass, <5; = w,(0). Thus for axisymmetric bodieswith convex contact surfaces of curvature RJ~l, if the contact area is small in comparisonwith the cross-section, the radial distribution of the normal displacement can be expressedas

The total indentation from compression 8 = 8B + $Bf can be related to the pressure mag-nitude po at the center of the contact area by summing the individual effects expressedby Eq. (6.3):

8 = napo/2E* (6.4)

where an effective radius R* and modulus E* have been defined as

R* = (R-1 + R-})-1

The size of the contact area can be determined from Eqs. (6.3) and (6.4); the contactradius a is then related to contact force F using Eq. (6.2):

S a2 3FR* Rl 4aE*R*'

Rearranging, we obtain

a_ _ / 3FT*~ \4E~R~l

x l / 3

J (6.5)

3F \ 2 / 3

(6.6)

po _ 3F _ / 6F \ 1 / 3

The mean pressure in the contact region p and a compliance relation for interaction forceF(S) are obtained from Eqs. (6.6) and (6.7):

F 4 /P 3 / 2

where R* is the effective radius of curvature in the contact area before compression.This force can be integrated to obtain the work W done by the normal contact force incompressing the small deforming region to any indentation 8,

—— = / * d(8'/R*) = — I — J (6.9)E*R* Jo E*Rl 15 \RJ

6.1.2 Indentation at Yield of Elastic-Plastic Bodies

Elastic indentation continues until some point in the contact region has a state ofstress satisfying the yield criterion of a constituent material. If plasticity (i.e. irreversibledeformation) initiates at a uniaxial yield stress F, the elliptical (Hertzian) contact pressuredistribution for a spherical contact surface gives solely elastic deformation if the meanpressure/? < 1.1Y (Johnson, 1985). The transition pressure pY = 1.17 = §YY results inyield at a point beneath the contact surface for either von Mises or Tresca yield criteria.2

This transition pressure occurs at a limiting indentation for elastic deformation 8Y thatcan be obtained from

Thus the nondimensional indentation SY/R, normal force FY/YR2 and work WY/YR3

required to initiate yield are material properties,

( 6 1 O b )

(6.10c)

With this definition of the indentation at yield 8Y, the contact radius, normal force andwork done by normal force during elastic deformation 8 < 8Y can be expressed as

Contours of maximum shear stress (Tresca yield criterion) are illustrated in Fig. 6.2for an elastic solid compressed by a spherical indenter. The location of the maximumshear stress where yield initiates is substantially beneath the surface of the body.

2 The Hertz pressure distribution causes yield to initiate at a point below the contact surface at a nondi-mensional depth X?,/a = 0.45. The plastically deforming region expands from this point in a lenticularshape as the mean pressure increases above the yield pressure pY. Nevertheless, the plastically de-forming region remains contained below the surface until the mean pressure is as large as p = 2.8pY',consequently, after loading into this elastoplastic range and subsequent unloading, there is very smallfinal deflection at the surface.

p[r)

Figure 6.2. Contours of maximum shear stress beneath spherical indenter.

6.1.3 Quasistatic Elastic-Plastic Indentation

For colliding bodies with spherical contact surfaces which are composed of ma-terial that can be represented as an elastic-perfectly plastic solid with uniaxial yield stressF, plastic deformation initiates beneath the contact surface when the mean contact pres-sure equals p y = 1.1F; at this pressure plastic flow begins at a nondimensional depthx3/a = 0.45 beneath the contact surface. This depth is less than the contact radius a. Al-though the plastically deforming region enlarges as contact pressure increases, it remainsconfined below the surface for pressures throughout most of the range 1.1 < p/ Y < 2.8;this state is termed contained plastic deformation. In this elastoplastic range the observablepermanent indentation of the surface is small because plastic deformation is incompress-ible and the plastically deforming region is encased within an otherwise elastic body. Forcontact pressures in the elastoplastic range, Fig. 6.3 shows the development of the plasticregion and the evolution of the distribution of contact pressure. The analytical solutionsshown in this figure are contours for the second stress invariant J2 (equivalent to thevon Mises yield criterion) at the perimeter of the plastically deforming region for eachload F > FY.

While the shape of the evolving plastic zone can be calculated using the finite elementmethod, it is useful to have an analytical approximation that can estimate this behavior.Following observations by Mulhearn (1959) that any blunt indenter (pyramid, cone orsphere) produced roughly spherical displacements below the surface, K.L. Johnson (1985)suggested a simplified spherical expansion model for elastoplastic indentation. This modelconsists of an incompressible hemispherical core of radius a beneath the indenter; withinthis core the state of stress is hydrostatic pressure p. Surrounding the core is a plasticallydeforming thick hemispherical shell wherein the radial stress decreases. The outer surfaceof the shell is at radius c, where stresses satisfy the yield condition.

The model gives the radial and tangential components of stress in the plasticallydeforming zone as

or 2 \a - r - ' c

and an effective stress a = (crr + 2ao)/3 which can be expressed as

a < r < c.

(6.11a)

(6.11b)

elastic fn

elastic-plastic i mi

fullyplastic

• z1:0x0x0:0:

. r

• • ' . • . /

Figure 6.3. Contact pressure distribution and region of plastic deformation for indentation,giving elastic, contained plastic or uncontained fully plastic deformations.

Outside the plastic region r > c there is an elastic zone where

c \ 3 oe 1 / c \ 3

(6.11c)

Beneath the indenter the core pressure p is assumed to be uniform, so it is given by theradial component of stress at the hemispherical surface of the core,

y y -yar'r <612)

The rigid core and the plastically deforming region both increase in size as the indentationincreases. From elastic compressibility of the plastic regions < r < c, Hill (1950, p. 101)obtained a ratio between incremental changes in the radius of contact a and the radius ofthe elastic-plastic interface c,

da Y [3(1 -v)c2 2(1 - 2v)a

az cWithin the plastic region a similar ratio relates differential increments of radial displace-ment u(r) to incremental changes in the elastic-plastic interface,

du Y

If the material is rigid-plastic, the core and plastically deforming region are incompress-ible; for a rigid-plastic material the previous equation gives

a,

1/3(6.13)


where a equals the contact radius. Hence the core pressure is obtained as

(6.14)

In fact the stresses in the core are also at yield and therefore not hydrostatic. FromEq. (6.11a) a better estimate for the mean contact pressure is given by the tangentialcomponent of stress in the elastoplastic region, p & —a r(a) = p+2Y/3', this is discussedby Johnson (1985, p. 175). Hence for an elastic-plastic boundary located at r = c, theequilibrium of forces on the core requires a mean pressure p in the contact area given by

) (6.15)pY 3$Y \aY/

Yield begins at a mean pressure p/Y = $Y where an initially spheroidal contact surfacehas fry = 1.1 at a contact radius a(8Y) = aY. Notice that the change in indentation modelat yield results in a slight discontinuity in indentation force F at the transition from elasticto elastic-plastic behavior.

During elastic-plastic indentation the contact force F increases with indentation:

F/FY = (a/aYf[l + 0.61i}Yl ln(a/aY)]. (6.16)

The normal relative displacement of the colliding bodies (indentation) for this phase isobtained by assuming that there is negligible elastic deformation in the material surround-ing the contact patch; i.e., at the edge of the contact area the surface neither sinks in norpiles up. Then the total indentation is given by

8/8Y =0.5(a2/aY + l). (6.17)

(Recall that in the elastic range 8/8Y = a2/aY.) Although approximation (6.17) results inthe contact area being a discontinuous function of indentation at yield, both indentation 8and contact force F are continuous functions of the nondimensional contact radius a/aY.Thus for the range of indentation where deformations are elastic-plastic, Eq. (6.17) givesa ratio of the contact radius a to the contact radius at yield aY as a function of theindentation 8,

a/aY = (28/8Y - 1)1/2

Hence (6.15) can be expressed also as

p/pY = [l + O ^ ) " 1 ln(28/8Y - 1)]. (6.18)

This gives the following expression for the normal contact force F:

F/FY = (28/8Y - l){l + (3$Y)~l ln(28/8Y - 1)}.

The total work done by the normal contact force during indentation into the elastoplas-tic range is obtained by integrating the product of this force and the differential incrementof normal relative displacement:

W f8/8Y

After integration, substitution from (6.16) and recognizing that WY = ^FY8Y, the work

during indentation into the elastic-plastic range is obtained as

The elastic-plastic phase continues until the mean contact pressure satisfies p/Y =2.8. This contact pressure results in a nondimensional contact radius a=ap where thecontact pressure is fully plastic. Indentation experiments show that thereafter (a > ap)any additional indentation occurs without further increase in mean pressure p.

At a/aY ^ 12.9 = ap/aY or 8P/8Y & 84 Eq. (6.15) or (6.18) gives a mean pressureequal to that for fully plastic indentation, p/Y = 2.8. Further indentation a > ap is inthe regime of uncontained plastic deformation, where there is a different load-deflectionrelation.

6.1.4 Fully Plastic Indentation

In the previously discussed range of contained plastic deformation the mean con-tact pressure p increases with increasing indentation. This behavior has an upper limitwhere the plastic deformation is no longer contained beneath the contact surface; forspheroidal contact surfaces this occurs at a contact pressure of about p = 2.SY. Through-out the range of fully plastic indentation the mean contact pressure p is constant; thispressure is the same as that measured in a Brinell hardness test.

Uncontained plastic deformation or fully plastic indentation begins at a contact radiusa/aY % 12.9 where the contact pressure p = 2.8F. The force Fp at the transition fromelastoplastic to fully plastic indentation is given by

Fp/FY « 424 (6.20)

so the elastoplastic range of force spans more than two orders of magnitude, 1 < F/FY <424.3

The transition from contained to uncontained plastic deformation that occurs at 8P/8Y &84 requires a large amount of plastic work in comparison with the work to initiate yield:

WP/WY =41 .5 x 103. (6.21)

In the fully plastic range the contact pressure is uniform and remains constant p = 2.8while the contact force increases as the contact area continues to increase:

fL = ^ H _ , ) . „ » , , . (6.22,FY &Y \8Y )

In the fully plastic range the indentation 8/8Y = 0.5(a2/aY + 1) is given by the samecondition as that used in the range of elastoplastic indentation, Eq. (6.17). This forceresults in the work done during compression given by

W

3 Finite element analysis of elastic-perfectly plastic bodies with initially spherical contact surfaces givesfully plastic indentation initiating at Fp/Fy % 650 and 8p/8y ^140 .


P\r)/Y

Figure 6.4. Finite element calculation of contact pressures and region of plastic deforma-tion for a sphere indenting an elastic-perfectly plastic solid (Hardy, Baronet and Tordion,1971).

While this model gives reasonable estimates of final indentation and energy dissipatedby indentation, the transition from elastic-plastic to fully plastic indentation occurs ata much larger force and indentation than are indicated by some numerical simulations(Hardy, Baronet and Tordion, 1971; Follansbee and Sinclair, 1984). These numericalanalyses show that FP/FY ^ 20 rather than 420.

Figure 6.3 illustrates the distribution of pressure on the surface and the extent of theplastically deforming region for the elastic, elastic-plastic and fully plastic ranges ofindentation by a spherical indenter. The ranges of applicability for contained elastic-plastic and uncontained fully plastic deformations have been obtained from the analyticalapproximations. These conceptual images for the stress distribution can be compared withresults from a finite element analysis of indentation in an elastic-perfectly plastic solidby a spherical indenter that are shown in Fig. 6.4 (Hardy, Baronet and Tordion, 1971).

6.1.5 Elastic Unloading from Maximum Indentation

The work done on the deforming region by contact force during compressiongoes into deformation; part of this work is absorbed by elastic strain energy, and part isdissipated by plastic deformation. Immediately following the period of compression, theelastic strain energy sustains the normal contact force that drives the bodies apart duringthe period of restitution.

During unloading from maximum compression the compliance relation for the contactregion is elastic. Complete unloading from a maximum compressive force Fc or indenta-tion 8C that is in the plastically deforming range results in a change in indentation 8r, sothat when the compressed bodies separate there is a final indentation 8f = 8C — 8r.

To obtain an expression for the change in indentation we recognize that as a conse-quence of plastic deformation during loading, the contact area has an effective curvaturethat has changed from the initial value R ~l to a new unloaded curvature R ~ . The transitionis assumed to occur at maximum indentation. This curvature of the deformed surface

depends on whether the deformed bodies are both convex or whether one has becomeconcave:

\RR'/(R + R\ R>0, R'>0*• = { - - / - - / T . , " (6.24)

\RR/(R-R), -R > R >0.During elastic unloading the changes in the contact region are geometrically similar tothe changes that occur during loading; thus

8Y/R* = 8r/R*. (6.25)

Unloading results in a change in indentation 8r from the maximum indentation 8C. Forcontact forces in the plastically deforming range F > Fy these indentations are relatedto the respective contact radii by

8r/R* = a?/Rl, 28C/R* = a\lR\ + 8Y/R*.The assumption used to derive the ratio between indentation 8r recovered during unloadingand maximum indentation 8C is that during unloading the change in contact radius ar

equals the contact radius at maximum indentation ac\ hence

R,/R, = 8r/8Y = (28C/8Y - 1)1/2. (6.26)

This expression applies to unloading from either elastoplastic or fully plastic indentation.During unloading (i.e. the period of restitution), strain energy of elastic deformation

provides the power that is transformed into relative kinetic energy of the colliding bodies.This energy transformation is achieved by means of work done on the bodies by the contactforce. The work done on the colliding bodies by the contact force during unloading canbe obtained by integrating the unloading normal force F = \E*Rj (8 — 8f) over therange 8f < 8 < 8C. This work is negative, since the deforming region is expandingin the normal direction during restitution and this expansion is opposed by the normalcomponent of contact force. Since 8r = 8C — 8/, this integration results in

j ^ _ _[>'<*• (4 EA/R.}3 (±\V2,(±\ _ _8_E1 3 5/2

YRl ~ h V3 Y )\Rj \RJ a\Rj 15 YBy recalling the geometric relation for unloading (6.25) and expressions for indentationand loading work at yield, we obtain

SY Sr (3n\2 (VYY\2 WY 8 £ , /«5y\5/2

{ ) U r J and W = T 5 {These expressions can be substituted into the relation for work done by the contact forceduring unloading Wr to obtain the recovered energy as4

£•-£)'•-(£-)4 An alternative expression that at maximum indentation satisfies continuity of force (but not contact

area) is the following:

The outcome of the difference from (6.22) is only a small decrease in the normal impact speed toinitiate yield Vy, so the former relation is retained in order to provide continuity at the elastic limit.


10 10° 10indentation

Figure 6.5. Indentation force as a function of indentation for contact of elastoplastic spher-ical bodies. (Dashed line is for an elastic-perfectly plastic approximation that neglects theintermediate range of contained plastic deformation.)

The relationship between the indentation and the normal force can be represented asbeing within one of three successive ranges on indentation - elastic, elastic-plastic orfully plastic, where the latter two represent contained and uncontained plastic deformationrespectively. Figure 6.5 illustrates the normal contact force as a function of indentation.In this figure elastic unloading is illustrated also. Notice that because the contact surfacecurvature for unloading, R*, does not equal the initial contact radius curvature R*, theunloading line is not parallel to the elastic loading line.

6.2 Resolved Dynamics of Planar Impact

6.2.1 Direct Impact of Elastic Bodies

At each instant during collision the rate of change of the normal component ofrelative velocity depends on the interaction force F and hence on the current relativedisplacement 8. Accelerations during a collision depend on the relative displacement orinterference between the colliding objects. Here only the normal component of transla-tional relative velocity is considered. Assuming that the deforming region surroundingthe initial contact point C is sufficiently small so that it has negligible mass in compari-son with the remainder of the body, the mass of body / moves uniformly in the normaldirection n3. As the contact region is compressed during a collision, the approach of onecenter of mass relative to the other, 8 = 8& + fa, results in a reaction force F at C thatopposes the approach of the two centers of mass,

F = - B' = KS83/2 (6.28)

where KS is a stiffness parameter for nonconforming spherical contact,

KS = \ ?!/2

6.2 / Resolved Dynamics of Planar Impact 127

that depends on material properties in the deforming (contact) region. Here another as-sumption has been tacitly introduced - namely, that the compliance relation is the sameas that obtained from quasistatic compression. This assumption is valid if the small de-forming region is composed of material with a rate-independent constitutive relation.5

For a central or collinear collision, an effective mass m is defined in terms of theindividual masses w/ by

m~l = m#l + rri^} or m =

This gives an equation of relative motion

m8 = m8d8/d8 = -KS83/2.

Integration of this equation and subsequent application of the initial conditions 8(0) =—v(0) = —Vo and 8(0) = 0 result in a relative velocity given by

S2 = vl - P-85'2. (6.29)5m

The contact period is separated into a period of compression and a period of restitution.The compression phase of collision terminates at time tc when 8(tc) = 0. At this timethe compressive relative displacement between the centers of mass has its largest value,8C = 8(tc); likewise, for these rate-independent materials, the interaction force Fc = F(tc)is a maximum at the time when the normal relative velocity vanishes:

8C/R = R-X(5mvl/AKS)2/5 = (l5mi^/16£*/^)2/5 (6.30a)

where at incidence the kinetic energy of normal relative motion is To = mv^/2. Bynumerical integration of the relative velocity (6.29) Deresiewicz (1968) obtained theperiod of elastic compression as

- p-&5llY'2d& = 1.47^ = 1.43 (1^—) l/5 (6.31)

Alternatively this integral can be expressed in terms of gamma functions.The variation of the relative displacement 8(t) with respect to time is obtained from

the nonlinear relation (6.28); this is not very different from that given by the compres-sion of a linear spring between two rigid bodies that collide with an initial differencein momentum — mv 0 (see Fig. 5.1). The linear approximation involves a spring forceF = Fc sin(0.5nt/tc). Using the compression period obtained in Eq. (6.31), this linearapproximation gives a largest reaction force Fc % 1.48/c5

2/5ro3/5; i.e. roughly 14% less

than the force Fc calculated with Eq. (6.30b). Since the Hertz theory and the approximationobtained with linear compliance have the same normal impulse during compression, thiscomparison of maximum force Fc implies that the linear approximation gives a contactperiod that is slightly longer than the contact period for collision between elastic bodies.

5 Wagstaff (1924) and Andrews (1931) conducted experiments that demonstrated the validity of theHertz theory for contact force during impact. Additional validation is shown in Table 6.2 below.


An expression for the impact speed which is just sufficient to initiate yield vY can beobtained from (6.10a) with 8C = 8Y and (6.30a). This gives

i.e., for direct collisions the normal impact speed vY where yielding initiates is a materialproperty that is directly related to the indentation 8Y that initiates yield.

Example 6.1 Suppose a steel sphere with Young's modulus EB = 210 x 109 N m~2

and Poisson's ratio vB = 0.3 is dropped onto a flat steel anvil from a height h = 50 mm.If the sphere has radius RB = 10 mm, it will have mass mB = 32.4 g, an impact speedv0 = 1.0 m s " 1 and kinetic energy at impact 7o = mBvl/2 = 0.016 J. Assuming elasticdeformations, what are the largest contact force Fc, the maximum size of the contactregion ac and the elastic contact period 2tcl

Solution:

m = (m~l + m"1)"1 = 0.0324 kg E* = 0.5EB (l - v2)"1

= 115 x 109Nm"2.

R* = (R~l + R-,l)~l = 0.01 m KS = \E*RXJ2 = 1.51 x 1010 N m"3/2

Hence the largest force, contact radius, indentation and pressure are given by

Fc = 1.73/c52/5r0

3/5 = 1.71 kN ac/R+ = {FC/KS)X/3R~1/1 = 0.048

po(tc) = \.5Fc/na2c = 3540 N mm"2 8C/R* = (ac/R*)2 = 2.34 x 10~3

The duration of the contact period is less than 100 /xs:

2tc = 2(lA78c/v0) =69/xs.

For mild steel the quasistatic yield stress is Y = 1000 N mm"2; i.e., this very modest dropheight develops a maximum pressure po that is substantially in excess of the uniaxialyield stress.

In the above example it is interesting to note that Eq. (6.31) gives a contact period thatincreases in proportion to the radius of the sphere. Hertz estimated that for a low impactspeed z>o = l m s"1, an elastic sphere with radius equal to that of the earth would have acontact period of slightly more than one day.

It is worth noting that if a body is dropped onto an anvil from height h , the changein potential energy during the drop equals the work done during compression; for ahomogeneous solid sphere this work Wc can be expressed as Wc = (4jr/3)pghRl. Hencethe drop height for initial yield hY is given by pghY/Y = (9/5n2)(3Y/E*)4. The dropheight required for initial yield of metals is astonishingly small; for example a steel spheredropped onto a steel anvil has hY^3 mm. Measurements of indentation reported by Tabor(1951) have verified these predictions.6

6 Local impact damage to contact surfaces has been considered by Evans, Gulden and Rosenblatt (1978)and Engel (1976).

6.2 / Resolved Dynamics of Planar Impact 129

6.2.2 Eccentric Planar Impact of Rough Elastic-Plastic Bodies

Continuous changes in relative velocity across the small deforming region thatsurrounds the contact area can be obtained by supposing that the deforming region is aninfinitesimal deformable particle located between the colliding rigid bodies at contactpoint C. This particle is assumed to have negligible tangential compliance. In contrastto the method developed in the previous section, this gives changes in velocity that areindependent of any compliance relation for the contact region. Instead, the coefficientof restitution is used to relate work done by the normal component of impulse duringthe separate periods of restitution and compression. This relationship requires separa-tion of the normal impulse into that acting during compression and that acting duringrestitution - a separation that can be made a priori only for rate-independent compliancerelations.

Let Vc and Vc be the velocities of the two bodies at C, and let the relative velocityv = (v\,v3) across the deformable particle be defined as v = Vc — Vc. This relativevelocity is resolved into a component V\ in the common tangent plane and a componentv3 normal to this plane; the relative velocity V\ is termed slip. The coordinate system isoriented so that at incidence V\ > 0 and v3 < 0. An equation of motion for this systemcan be expressed in terms of components of impulse p = (p\, p3) of the reaction forceF = (Fi, F3) at the contact point C:

[-fa fa]\dp3\where the inertia parameters fa, fa, fa are a rearrangement of parameters defined byWang and Mason (1992). The inertia parameters depend on the masses M, M' of thebodies, their radii of gyration kr, k'r about their centers of mass, and the locations r, r' ofthe centers of mass relative to the contact point C, where r = (n, r3):

fa = \ + mr2/Mk2 + mrf2/Mfkf2

fa = mrxr3/Mk2 + mr[r^/Mfk'r2

fa = 1 + mr2/Mk2 + mr'2IM'kJ.

The coordinate system for describing the configuration is shown in Fig. 3.1.

Equations of Relative MotionWith Coulomb's law of friction the equations of motion (6.33) can be expressed in terms ofa monotonically increasing independent variable p = p3. The motion depends on whetherthe contact point is sliding or sticking. It is sliding if V\ ^ 0 or \x < \fa\/fa:

dvi/dp = -(Sfifa + fa)m~l

dv3/dp = (fa + s/j,fa)m~l.

It is sticking if V\ = 0 and /x > \fa\/fa:

where s = sgn (z?j) is the direction of sliding and /z is the coefficient of friction. (Forsimplicity the static and kinetic coefficients of friction are assumed to be equal.) These


equations can be integrated to give the relative velocity at any impulse p during a periodof unidirectional slip in direction s:

vx (p) = v{ (0) - (SiiPi + h)p/m (6.34a)

vs(p) = VsiO) + (ft + Siifop/m. (6.34b)

Hence the normal compression impulse/^ can be obtained from the condition i?3 (pc) — 0:

Pc = - ( f t + S/zft)-W3(0). (6.35)

Work and Indentation at Transition from Compression to RestitutionThe contact force does work on the rigid bodies that is the negative of the work Wc doneon the deforming region during the same period. For each separate period of unidirec-tional sliding this work can be calculated by a theorem (Stronge, 1992) that goes back toKelvin. Here for example we consider the limiting case of unidirectional sliding. Anal-yses for sliding that halts before separation, however, have been described in Chapter 3.For continuous slip the work done by the normal force during compression is as follows:

WY

1

ft +(6.36)

where again we employ the definition (6.32) for the normal incident speed at yield. Forcollinear collisions, ft = 0 and ft = 1, so that effects of normal and tangential forceare decoupled. Thus irrespective of the initial slip velocity, at the end of compressionpc = — my3(0) for a collinear collision,

vj(O)WY

(6.37)

Expression (6.36) or (6.37) can be equated with (6.19) or (6.24) to obtain the maximumindentation 8c/8y for any particular geometric configuration and incident velocity. Thismaximum indentation is required to calculate the part of the normal energy of relativemotion that is recovered during restitution, Wr.

Table 6.1 lists the smallest normal impact speed that initiates yield, vY, for several dif-ferent metals. This was calculated from measurements of the energy loss in collisions of

Table 6.1. Material Properties from Indentation and Impact Tests

Material

Mild steel:As receivedWork-hardened

Brass (drawn)Aluminum:

1180-H142014-T6

Densityp(g cm"3)

7.87.88.5

2.72.8

Young'sModulusE( (GPa)

210210100

6969

StaticYield

600650200

110410

DynamicYield

) Yd(MPa)

583-7801160250

130410

Impact Speedto Initiate Yield^ ( m s - 1 )

.049-. 101

.055

.007

.004

.076

Source

Author (KY = 2.8)Tabor(1951)Tabor(1948)

6.3 / Coefficient of Restitution for Elastic-Plastic Solids 131

small spheres at incident speeds of 0.1 < |z73(O)| < 3 m s *. The dynamic yield stress Yd

that corresponds to each vY has been calculated according to Eq. (6.32). For rate-dependentmaterials, Yd is larger than the yield stress Ys obtained from quasistatic indentation tests.

6.3 Coefficient of Restitution for Elastic-Plastic Solids

The energetic coefficient of restitution is the ratio of work done on the smalldeforming region during compression to work done by this region on the surroundingrigid bodies during restitution; i.e., it is a measure of that part of the kinetic energy ofnormal relative motion (the energy transformed to internal energy of deformation duringcompression) which is recoverable during restitution. Section 3.4.3 gave the followingdefinition: The square of the coefficient of restitution, e\, is minus the ratio of the elasticstrain energy released at the contact point during restitution to the energy absorbed byinternal deformation during compression. This energy ratio can be calculated from thework done by the normal component of contact force if tangential compliance is negligible(Stronge, 1995):

(6.38)

This final expression combines Eq. (6.27) with the energy transformed in compression,WC/WY, obtained from Eq. (6.19) or (6.24). Together with (6.36), it relates e* to thedamage number pv\l Y defined by W. Johnson (1972). The result is similar to that obtainedby Adams and Tran (1993), but here expression (6.38) explicitly incorporates the effectof friction in various possible slip processes. For elastic-perfectly plastic solids with aconvex spherical contact surface, the relation between the coefficient of restitution andthe normal impact speed is shown in Fig. 6.6.

In the limit of WC/WY » 1 Eq. (6.38) indicates that e* ^ [v2(0)/vY]l/4, where vY

depends on effective mass, contact curvature and material properties but is independent

102 k(0)IAy

Figure 6.6. The coefficient of restitution for impact of elastic-perfectly plastic solids de-pends on the eccentricity of the impact configuration as well as the normal impact speed. Fora rigid rod inclined at 9 = TT/4 or 0 the lines compare the analytical expressions with twosets of experimental data (0 = 0). For the eccentric configuration the narrow band at largespeeds indicates the range of values for opposing directions of gross slip if the coefficientof friction jji = 0.5.


3

2

1z

0

t>O

§32

1

0

-

-

-

1 1\? 0.2

f

-J

1 I

/ ~ \/

I 10.4

steel

1

lead sphere

i i0.6

sphere

I |

0 0.2 0.4 0.6time, ms

Figure 6.7. Force from coaxial impact of steel and lead spheres on end of steel rod at2.3 m s"1. Steel and lead spheres have masses of 64.2 and 73.3 g respectively.

of impact configuration. This functional relation between coefficient of restitution andnormal impact speed agrees with measurements on a wide range of metals reportedby Goldsmith (1960). The present expression neglects strain hardening and effects ofhigh strain rates; for some materials these effects are important if the impact speed ismoderately large (Mok and Duffy, 1965; Davies, 1949).

Figure 6.7 shows the stress pulses resulting from coaxial impact of a sphere againstthe end of a strain-gauged steel rod. The contact force as a function of time is shownfor both steel and lead spheres striking the bar at 2.3 ms" 1 . At this impact speed bothspheres suffer plastic deformation, but since the lead sphere has a smaller yield stress,this reduces the maximum force and prolongs the contact period in comparison with thesteel sphere. Table 6.2 provides a comparison between the measured values and Hertzelastic analysis for these collisions.

6.4 Partition of Internal Energy in Collision between Dissimilar Bodies

6.4.1 Composite Coefficient of Restitution for Colliding Bodieswith Dissimilar Hardness

Where colliding bodies are dissimilar, the loss of kinetic energy due to irre-versible internal deformation can be divided into the losses in the two bodies by consider-ing separately the work done on each body by contact forces. This separation associatesthese energy losses with properties in the contact region - it is independent of the par-titioned loss of kinetic energy obtained in Sect. 2.7. If each body has a coefficient ofrestitution e* or e^ obtained for a collision at the same impact speed against a body

Table 6.2. Measurements from Coaxial Collision of Spheres on a 24.5 mm Dia. Steel Bar

Sphere

Steel:D =M =E =

Lead:D =M =E =

25 mm64.2 g210kNmm-2

23 mm73.3 g14.7kNmm"2

ImpactSpeed

(ms"1)

0.911.151.411.692.0

0.911.151.411.692.0

ReboundSpeedvf(ms"1)

0.680.820.911.051.20

0.110.130.160.180.20

CompressionImpulse pc (N s)

Calc.

0.0580.0740.0910.1080.128

0.0670.0850.1030.1230.146

Expt.

0.0510.0730.0750.0920.126

0.0650.0920.1020.1350.130

RestitutionImpulse pr (N s)

Calc.

0.0430.0520.0580.0680.077

0.0080.0100.0110.0130.015

Expt.

0.0460.0730.0750.0820.100

0.0240.0280.0390.0610.062

MaximumForce Fc (kN)

Hertz

2.403.184.055.026.16

1.291.712.182.703.31

Expt.

1.943.183.704.034.58

0.800.850.920.901.16

CompressionPeriod tc ([As)

Hertz

4543414038

9591878481

Expt.

5246414641

120179181235193

Expt.ContactPeriod

(IIS)

10091818679

168232259301273

C.O.R.vf

e* —

0.740.710.650.620.60

0.120.120.110.110.10

Expt.ImpulseRatio

Prt U Pe

0.91.01.00.890.79

0.370.300.380.450.48

composed of the same material and geometrically similar to itself, then these coefficientscan be combined to calculate an effective coefficient of restitution e* for collision betweendissimilar bodies. In order to achieve this amalgamation, we consider for each body theratio of the work Wr done by the normal contact force during recovery to the work Wc doneduring compression, and note that this ratio is equivalent to the coefficient of restitution:

el = -Wr/Wc, e'^=-WfJWfc, while el = -Wr/Wc. (6.39)

During collision the normal component of contact force does work on each separatebody; this work is in proportion to their respective indentations <$; , /= B, B', becauseequal but opposed forces act on the colliding bodies. Hence

el = -$r/$c, K2 = ~K/~K and el = -8r/8c

so if, for example, the bodies remain elastic during compression, Eq. (6.4) gives theindividual indentations as 8t = napo(l — vf)/E(. Since the part of the total indentationfor each body is approximately equal to the part of the deformation (or strain) energy,the internal deformation energy is distributed between two colliding bodies in inverseproportion to the ratio of their elastic moduli Et/{\ — vf). This energy distribution isindependent of the relative curvature of the contact surfaces or the size of the bodies.For either frictionless or collinear collisions this implies that the composite coefficient ofrestitution e* is obtained from

- ^ = - ^ = + - ^ r (6.40)E* E E'

Hence colliding bodies composed of the same material absorb equal parts of the kineticenergy of normal relative motion irrespective of any difference in size. This is a conse-quence of the deforming region being a negligibly small part of the mass of either body.In Fig. 6.8 this theory is compared with experimental measurements of the coefficient ofrestitution obtained for collisions between pairs of spheres that are identical and othercollisions between pairs of spheres composed of dissimilar materials.

6.4.2 Loss of Internal Energy to Elastic Waves

During any collision between nonconforming bodies, the bodies come togetherwith some relative speed, and it is the local deformation of the bodies that generatesthe contact pressures that act to prevent mutual interference (or overlapping) betweenthe bodies. The stresses generated by local deformation cause the stress waves that radiateaway from the contact region. These stress waves transmit changes in velocity fromthe contact area and cause the momentum of the bulk of each colliding body to change.Unless the surfaces of the colliding bodies have a very particular shape, elastic waves arereflected from different parts of the surface at different times and in different directions,so that there is not a coherent reflected wave that returns to the contact region to relieve thecontact pressure. Rather, the duration of contact is controlled by the time required for thecontact pressure to accelerate the two bodies until they separate. Because the region withsignificant deformation is ordinarily very small in comparison with both the cross-sectionand the depth normal to the contact surface, the contact duration depends on the effectivemass m = MBM&/(MB + M B ) . That is, unless the bodies are bounded by a surface witha focal point in the contact region, the contact duration for a collision is determined by

6.4 / Partition of Internal Energy in Collision between Dissimilar Bodies 135

0.8

0.6

0.4

0.2

ivory / ivory

bronze/bronze

bronze/brass

brass/brassPbl ivory

Pb/Pb

, ms-

Figure 6.8. Coefficient of restitution e* for direct impact of spheres of equal size. The solidlines are a best fit to experimental data collected by Goldsmith (1960), while the dashedlines for dissimilar materials are from Eq. (6.40).

quasistatic compliance and the effective mass m of the colliding bodies rather than wavepropagation considerations.

Elastic stress waves play a part in energy dissipation during collision only if the relativesize of the bodies, RB/R&, is quite different from unity. In this case, there may be timefor elastic waves to redistribute stresses and velocities in the smaller body in accord withrigid body dynamics while the larger body continues to suffer dispersal of energy fromthe contact region by elastic waves. In the smaller body the energy transmitted by elasticwaves is not lost - this energy is redistributed by the waves until it is approximately equalto the energy distribution required by rigid body dynamics. For the larger body, however,there may be insufficient time for some parts of the radiating stress wave to reflect fromany boundary, so that the distributions of kinetic energy and strain energy during contactare not even approximately equivalent to those required by rigid body dynamics.

Hunter (1956) examined the vibration energy in an elastic half space struck by anelastic sphere. He showed that the work done by the contact force in driving the halfspace is less than 1% of the energy absorbed, if the elastic wave speed c\ is much largerthan the normal component of relative velocity at impact [(ci/i?o)3/5 ^> 1]; i.e., most ofthe absorbed energy is represented by elastic strain energy in material near the contactpoint C. This view was corroborated by a dynamic analysis of elastic impact by Tsai(1971). Early in the compression period the radial stress near the surface is larger thanexpected from a quasistatic analysis; nevertheless, Tsai concluded that the Hertz theory isa good approximation for the contact force during impact. Essentially, very large stressesare concentrated in a small contact region by geometric effects. A spherical elastic waveexpands as it radiates away from the small contact region; a part of the expanding stresswave is continuously being reflected back towards the source due to stiffness that increases


as the surface area of the wavefront increases. Thus at locations which are far from thecontact region only very modest stresses are required to accelerate the bulk of the massfrom the initial to the separation speed. The jump in stress [a] across an elastic waveprovides an estimate of the stress required: [cr]/E ^ Vo/c\. Consequently, compact bodieswith a small contact region have most of the impact energy absorbed by deformation ofmaterial near the contact point. For collisions involving 3D deformation fields (sphericalcontact), vibrational energy can only be significant if at least one of the bodies is slender,i.e. if it has a dynamic response that is similar to that of a beam, plate or shell.

Hunter (1956) obtained his estimate of energy loss in an elastic half space by consid-ering the steady state solution for an oscillating normal force acting in a circular regionon the surface of the half space and calculating the work done by this force during halfa period of loading. That is, he approximated the contact force as being sinusoidal andobtained the work done by this force during the contact period tf = TT/COQ ^ l.O68i?o/Scwhere for bodies composed of rate-independent materials the maximum indentation 8C

occurs simultaneously with the maximum force Fc. In an elastic half space with Young'smodulus E and Poisson's ratio v the work W done during the contact period was calculatedby Hunter (1956) as

where the parameter £ = £(v) is a function of Poisson's ratio:

£(0.25) = 0.537, £(0.33) = 0.415.

The expression (6.30b) can be used to obtain the ratio between the work done duringcontact (or energy loss due to elastic waves) and the incident kinetic energy of normalrelative motion:

1 / 1 I / O _ 1 / 1 A

wmv2j2

Here the penultimate term is a geometric factor that depends on the curvature of the contactregion R~l and the shape of the colliding body, e.g. whether it is spherical, ellipsoidal, ora slender rod. For a solid colliding body that is spherical, the energy loss to elastic wavescan be expressed also as

mvf{= 3.85 x £(1 + v) (- — | ( — )

\l-2vj \coj

,3/5

(6.42a)

(6.42b)

Here it is clear that the energy losses in elastic waves are a negligibly small part of theincident kinetic energy of relative motion if a small body collides against an elastic halfspace. Hunter (1956) stated that for a steel half space struck by a high-strength steelsphere the loss ratio is 2W/mvl = l.O4(i?o/co)3/5, while for a half space made of glassit is 2W/mvl = l.27(vo/co)3/5

6.6 / Transverse Impact of Rough Elastic-Plastic Cylinders 137

6.5 Applicability of the Quasistatic Approximation

The previous example demonstrates that elastic collisions between slightly de-formable bodies have small contact areas and very brief collision periods; consequentlyvery large contact pressures develop during the collision. This raises the question ofapplicability of the quasistatic analysis.

Almost all of the mass in colliding bodies is decelerated by stresses that are transmittedfrom the contact region by elastic stress waves; the predominant mode of energy trans-mission is by dilatational waves travelling at speed c\ = [£"/(l — v/)//O;(l — v, — 2vf)]1^2.This speed depends on the bulk modulus of the material - for metals it is typically of theorder of 4 or 5 x 103 m s"1 (see Table 7.1).

The contact period for elastic collisions 2tc can be expressed in terms of the wavespeed c\. For a collision between two identical spheres this gives m = raB/2, R = RB/2and E = EB/2(l — Vg); thus the elastic contact period is given by

( 6 . 4 3 )

The time tB for an elastic wave to transit each sphere is tB = 2/?B/^I SO that the numberof wave transits n during the contact period can be expressed as

Thus collisions of compact or stocky bodies at low impact speeds (up to a few metersper second) result in contact periods that are long enough for several but not a very largenumber of transits by elastic waves.

Love (1906) proposed that the quasistatic Hertz theory applies only if the number ofelastic wave transits is very large, i.e. (c\ /vo)l/5 ^> 1. If this condition is not satisfied, Lovepresumed that a significant part of the impact energy remains in the bodies after contactceases - remaining in the form of elastic vibration energy. By considering collisionswhere one body is very large in comparison with the other it can be shown that this notionis not correct.

Rather, the quasistatic contact theory gives accurate results for collisions if the regionof significant deformation or internal energy density remains small in comparison withall dimensions of the colliding bodies; i.e., applicability depends on the geometry ofthe colliding bodies and how this affects diffusion of energy in 2D or 3D elastic wavesemanating from the contact region.

The Hertz quasistatic analysis neglects the effect of stresses distributed throughoutthe body. By considering the quasistatic analysis of stress distribution in a heavy sphere,Villaggio (1996) investigated the error introduced by assuming that all compliance islumped in a small region of negligible mass. This approximation causes a small reductionin the contact period without any significant change in the load-deflection relation.

6.6 Transverse Impact of Rough Elastic-Plastic Cylinders - Applicabilityof Energetic Coefficient of Restitution

Impact of hard metal bodies at modest speeds results in plastic deformationaround the point of initial contact, and this has been shown to be a major source of energy

s1a'-_

»§mm

ifft1II

pinstntntuiuft+tTTtnnttnTuT!T1l!|111 ilnTil

iiitiitttnt-IsBSi1w

£<

Y

s

r/g/rf cylinder

half-space

Figure 6.9. Finite element mesh for analyzing transverse impact of cylinder on elastic-plastic half space.

dissipation during impact of these bodies. For oblique impact between bodies with roughsurfaces, both normal and friction forces contribute to the local stress field. The energeticcoefficient of restitution, as defined in Chapter 3, depends on the normal component offorce only. Consequently, for those collisions where the incident velocity is large enoughto cause plastic deformation, it is important to understand whether the coefficient ofrestitution is sensitive to frictional impulse, i.e. the extent to which normal forces duringelastic-plastic impact are affected by friction.

To assess the effect of friction on the coefficient of restitution for oblique impact ofrough elastic-plastic bodies, 2D plane strain analysis of a cylinder colliding transverselyagainst an elastic-plastic half space has been performed using the finite element codeDYNA2D (see Lim and Stronge, 1998a). The finite element mesh is illustrated in Fig. 6.9.This continuum analysis of impact on a deformable body is compared with a hybridquasistatic analysis where normal forces derived for elastic-plastic local deformationare combined with elastic tangential compliance and Coulomb friction. The analyticalsolution for the normal force and indentation follows closely the pattern presented inSect. 6.1 for a colliding body with a spherical contact surface.

Consider two parallel cylinders with radii R& and RB> that are composed of materialswith Young's moduli EB and E&. Here we again employ the definitions of an effectiveradius R* and an effective elastic modulus E*\

6.6.1 Elastic Normal Compliance

For elastic deformations, an elliptic pressure distribution over the contact radiusa gives a normal contact force F per unit depth and mean contact pressure p that are


related to the contact area byp F na

and a normal indentation 8 that depends on the normal force according to

8 F ( . v1 - v ,

where f = Id /a is a characteristic depth for the deformation field. This relation is insen-sitive to the characteristic depth for f ^> 1, so that in the elastic range, indentation depthis proportional to normal contact force:

F _ 8FY 8Y

6.6.2 Yield for Plane Strain Deformation

For 2D deformation associated with indentation by a long cylinder, yield initi-ates at a mean pressure p/Y = 1.5 = pY/Y, i.e. at a pressure somewhat larger than thatrequired for yield under a spherical indenter. Once again the normal force and indentationat yield are material properties,

FY 36 Y

aY 12 Y^ = - —• (6.45)/?* n E*

Equating the work done during compression and the incident kinetic energy of normalrelative motion for a cylinder of mass m and length L, the incident normal velocity atyield vY is obtained as

2 3YRIL / 6 7 \ 3 r AA nE* 1 + v lvr = n ~K~ \ T I 41n § + 41n — - - . (6.46)

(I — vz)m \nE*J [_ 67 1 —vj

6.6.3 Elastic-Plastic Indentation

Employing Johnson's model for constrained plastic deformation, the mean pres-sure and indentation in the elastic-plastic range of deformation are expressed as

PY ^ \aY J °YThis pressure gives the normal contact force as a function of deflection,

Elastic-plastic deformation terminates and uncontained plastic deformation begins whenthe mean contact pressure equals p/Y = 2.4.

Table 6.3. Incident Normal Velocities for Yield and Uncontained Plastic Deformationin Plane Strain

Aluminum alloy,2014-T6

Stainless steel,302 cold-rolled

^ ( i n s l)

Analytical DYNA2D

1.62 1.55

0.36 0.37

Vp (m s l)

Analytical DYNA2D

18.12 17.50

8.39 7.90

6.6.4 Fully Plastic Indentation

In the range of uncontained plastic deformation, additional indentation occurswith no increase in contact pressure; the contact area merely spreads outward over a radius

a/aY = 0.625F/FY.

The indentation is related to contact area by the same relation (6.47), which gives nonormal displacement at the periphery of the contact strip. The normal contact force thatis developed by the uniform contact pressure equals

11/2

The incident impact speeds for yield vY and uncontained plastic deformation vP are listedin Table 6.3 for an aluminum alloy and a stainless steel. The normal velocity at yield for2D plane strain deformation is an order of magnitude larger than the corresponding valuefor spherical contact.

6.6.5 Analyses of Contact Forces for Oblique Impact of Rough Cylinders

For oblique impact there are tangential forces in the contact region in addition tothe normal force analyzed in the previous section. The tangential forces are assumed tobe due to dry friction that can be represented by Coulomb's law with a single coefficientthat represents both static and dynamic friction.7

In the present analysis the energy losses are assumed to be entirely due to friction plusthe work absorbed due to the hysteresis of the normal component of force. The latterfactor is represented by the energetic coefficient of restitution. Here, the coefficient ofrestitution includes the effect of energy lost to elastic waves; for transverse impact of acylinder on a half space at a normal speed equal to vY this gives a coefficient of restitutione* & 0.8. Thus for elastic impact [\v{0)\ < vY], as shown in Fig. 6.10a, the normalcomponent of relative velocity is somewhat smaller at separation than it was at incidence.The finite element calculations by DYNA2D show these same energy losses to elasticwaves irrespective of the angle of incidence. At an impact speed that is large enough to

7 While numerical simulations that incorporate separate values for static and dynamic coefficients of fric-tion can achieve greater accuracy, for purposes of interpreting phenomenological effects this distinctionmerely muddies the water.

0.2-0.4 0.6 0.8 1.0 1.2 Utime t/tc

1.6 1.8 2.0

g-0.6-0.8-

- t o o(b)

0.2 0M 0.6 0.8 1.0 1.2time t / t c

U

Figure 6.10. Normal velocity as a function of time during oblique impact on a rough halfspace with /x = 0.3: (a) incident velocity in elastic range, \v(0)/Vy \ = 0.67, and (b) incidentvelocity in range of uncontained plastic indentation, \v(0)/vy\ = 33.3.

cause uncontained plastic deformation [\v(0)/vY\ = 33.3], as shown in Fig. 6.10b, theenergy lost to plastic work causes the separation velocity to be substantially smaller thanthat for elastic impact; nevertheless, for a coefficient of friction /x = 0.3 these lossesremain insensitive to friction.

The friction forces that result from oblique impact at a range of different incidentvelocities are shown in Fig. 6.11; this figure compares analytical results which incorpo-rate elastic tangential compliance with DYNA2D finite element calculations. At smallangles of incidence, shown in Fig. 6.11a, the contact initially sticks and does not beginsliding until late in the contact period. For incident normal speeds that give elastic be-havior [\v(0)/vY | < 1] the collision terminates when the contact separates at a time that isroughly double the compression period. At larger incident speeds, which result in morehysteresis, the period of restitution is smaller than the period of compression for theserate-independent materials.

For intermediate angles of incidence the contact initially slips, but slip vanishes dur-ing the contact period. During a subsequent period of stick the direction of slip reverses;

frictionboundary

I l0 0.2 0.4 0.6 0.8 1.0 1.2 U 1.6 1.8

time t / t c

(a)

0 0.2 0.4 0.6 0.8 1.0 1.2 U 1.6 1.8 2.0time t/tc

(b)

Figure 6.11. Variation of ratio between tangential and normal force during contact periodresulting from oblique impact with /x = 0.3 - a comparison of predictions of the lineartangential compliance model with finite element calculations: (a) small angle of incidence[i>i(0)/i?3(0) = 0.176 < fir]2], exhibiting stick-slip behavior (rf- = 1.25 for cylindrical con-tact in plane strain), and (b) intermediate angle of incidence [v\ (0)/i?3(0) = 0.58], exhibitingslip-stick-slip behavior.

finally, however, as the normal contact force decreases, slip resumes. Figure 6.1 lb com-pares analytical and finite element calculations.

Other calculations at a large angle of incidence where there is gross slip in the initialdirection similarly indicate that for normal impact speeds in the range \v(0)/vY | < 33 andcoefficients of friction \i < 0.3 the effect of tangential force on the normal complianceis negligible. Hence the energetic coefficient of restitution is unaffected by friction fornormal impact speeds that are below the limit for uncontained plastic deformation.

6.6.6 Loss of Internal Energy to Elastic Waves for Planar (2D) Collisions

Calculations similar to those in Sect. 6.4.2 have been performed by Lim (1996)and Lim and Stronge (1998a), for transverse impact of a cylinder on an elastic half space.

AluminumSteel

0.000.00

0.020.03

0.160.13

0.240.23

0.270.25

6.7 / Synopsis for Spherical Elastic-Plastic Indentation 143

Table 6.4. DYNA2D Calculations of Energy Loss to Elastic Waves for Dissimilar Size ElasticCylinders That Collide at Incident Speed v0 = 1 m s~l

Irrecoverable Part of Initial Kinetic Energy, (To — 7/)/ 7b

Material RB/RB' = 1 RB/RB> = 2 RB/RB' = 4 RB/RB' = 8 RB/RB> = 16 RB/RB' -> oo

0.310.28

Transverse impact of parallel cylinders gives planar (2D) deformations8; these collisionconfigurations result in a much larger part of the initial kinetic energy of relative motionbeing removed by elastic waves than occurs for impact between spheres:

(f^f ^ (f ) (6.48)pc3

0 \ l 2 j l \ l 2 v jFor two bodies of similar size and mass the loss of energy to elastic waves is negligible,

as listed in Table 6.4.9 For bodies that are very dissimilar in size, however, the irrecoverableenergy due to elastic waves in the large body can be significant. In this case the size ofthe contact region and the strain energy stored in this region can be almost the samefor each body (see Sect. 6.4.1), so that only the mass of the small body is important fordetermining the contact duration. For the large and massive body, the waves transmittingenergy away from the contact region have little effect on the momentum in the normaldirection. It is the energy in these waves which is lost in a collision if the masses of thecolliding bodies are quite different.

6.7 Synopsis for Spherical Elastic-Plastic Indentation

Below is a synopsis of the maximum normal force Fc, maximum indentationdepth 8C and work done by the normal component of contact force at the termination ofcompression Wc for spherical indentation of elastic-perfectly plastic solids. The work Wc

deforms the bodies and decreases the kinetic energy of relative motion. Also shown arethe work done during restitution, Wr, that restores some of the kinetic energy of relativemotion, and finally the coefficient of restitution e*. All of these variables are expressedas functions of the nondimensional indentation 8/8Y.

Elastic

p < I AY, 8/8Y < 1, a2/a2Y = 8/8Y

e* = 1

8 For transverse compression of parallel cylinders the deformation field is 2D at points far from the ends.Near an end there is also an axial component of displacement, and this varies within the cross-section.

9 Analyses of energy losses in elastic cylinders with distinct sizes were performed with ABAQUS/Explicitby Sunib Seah, National University of Singapore.

Elastic-plastic:

I AY 84, a2/aj = 28/8Y - 1

A = 2.55, f - 1 5 5 ( ? « -

£ = -3090+ 1.60 ( g - )

r]

3090 + 1.60(2«c/«r - I)2 J

From either elastic-plastic or fully plastic states the final or residual indentation 8/ isgiven by

For colliding bodies with an effective mass m and a nominal initial radius of contactingsurfaces /?*, the kinetic energy % of normal relative motion is transformed to deformationenergy during compression, so that Wc = f0 = mv^/l. Irrespective of the impact speed,the specific energy absorbed during compression,

{) (652)(U {3Y) 2YRI

Problems 145

can be used to calculate the minimum nondimensional contact radius ac = ac/ay.Equation (6.52) applies to impacts of bodies with a normal effective mass m and aneffective radius of curvature /?* in the contact region. For impact of an elastic sphere onan elastic half space,

^ = 2.74Wy

PROBLEMS

/M4K\3Yj Y

6.1 Derive expression (2.8) from the sum of the kinetic energies of normal relative motionthat are dissipated during compression. For an impact speed of 3 m s"1 and data fromcollisions between ivory and ivory and between lead and lead spheres, calculate thecoefficient of restitution e* for a collision between a lead and an ivory sphere ofthe same size. What assumption does this calculation make regarding the relativeduration of the periods of contact for collisions between different pairs of identicalspheres?

6.2 A ball peen hammer of mass M = 1 kg that is made of steel (E = 210 GPa, Y —600 MPa, p = 7.8 g cm"3) strikes a soft aluminum bar (E = 70 GPa, Y = 120 MPa,p = 2.7 g cm"3); the bar has thickness h = 6 mm, while the impact surface of thehammer has a radius of 10 mm.(a) Calculate the minimum impact speed Vy for yield. For impact at this speed

describe the location where yield initiates.(b) Calculate an estimate of the coefficient of restitution e* for an impact speed

Vb = 0.1 m s"1 (a light tap).(c) Describe the final surface and subsurface deformation that you expect from the

impact specified in (b).

CHAPTER 7

Axial Impact on Slender Deformable Bodies

Alice laughed. 'There's no use trying,' she said, 'one can't believeimpossible things.' 'I daresay you haven't had much practice,' saidthe Queen. 'When I was your age, I always did it for half-an-hour aday. Why, sometimes I've believed as many as six impossible thingsbefore breakfast.'

Lewis Carroll, Through the Looking Glass (1872)

Axial impact on a deformable body results in a disturbance which initiallypropagates away from the impact site at a specific speed. This disturbance is a pulseor wave of particle displacement (and consequent stress). Wave propagation relates topropagation of a coherent pulse of stress and particle displacement through a mediumat a finite speed. Familiar manifestations of this phenomenon are the transmission ofsound through air, water waves across the surface of the sea and seismic tremors throughthe earth; thus, waves exist in gases, liquids and solids. Sources of excitation may beeither concentrated or distributed spatially, and brief or extended functions of time. Theunifying characteristic of waves is propagation of a disturbance through a medium.Properties of the medium that result in waves and determine the speeds of propagationare the density p and moduli of deformability (Young's modulus E, shear modulus G, bulkmodulus K, etc.).

7.1 Longitudinal Wave in Uniform Elastic Bar

Consider a uniform slender elastic bar of cross-sectional area A, elastic modulusE and density p; the bar contains a region with axial stress a(x, t) that is propagatingin the positive x direction as shown in Fig. 7.1. For a differential element of the barlocated at spatial coordinate x let u(x, t) be the axial displacement at any time t. Thestress pulse results in a difference in force (A da/dx)dx across the differential element;this difference gives an axial equation of motion for the element,

d2u dadtl dx

For a linear elastic material and a one dimensional state of stress, Hooke's law gives astress-strain relation,

cr = Edu/dx. (7.1)

When £ , A and p are independent of x the constitutive relation (7.1) results in a linear

146

7.1 / Longitudinal Wave in Uniform Elastic Bar 147

dx , A

Y///A

\Y///A * —r

H N U"-f6x

Figure 7.1. Differential element in bar of cross-sectional area A at successive times t andt + df, the element is being traversed by a stress pulse traveling in direction x at speed CQ.Particle displacement u(x, t) and axial stress a(x, t) vary across the element.

wave equation,

Z±=cf± C2 Edt2 udx2 p

where the wave propagation speed c0 is known as the bar velocity. Equation (7.2) is a ho-mogeneous, linear, hyperbolic partial differential equation (PDE). It is linear because thedependent variable u(x, t) and its derivatives occur in the first degree only. Consequently,if Mi and u2 are two independent solutions, the sum c\u\ + c2u2 will also be a solutionwhen c\ and c2 are arbitrary constants; i.e., the principle of superposition applies.

In contrast to many problems in wave propagation, there is a general solution to thehomogenous wave equation,

u(x, t) = \[f(x - cot) + g(x + cot)] = lj[f(rj) + *(£)] (7.3)

where / and g are arbitrary functions of their respective arguments rj = x — c^t and£ = x + cot. By differentiating, one can show that if / and g possess second derivativeswith respect to their arguments at all but a finite number of points, they satisfy Eq. (7.2).l

The functions / and g will be determined by initial conditions on the problem.The form of Eq. (7.3) suggests the term wave. The function / represents a disturbance

traveling in the direction of increasing x with speed c0. This spatial variation in stresspropagates in direction x without change in shape - no force is required to maintain thewaveform. Similarly the function g represents a disturbance traveling in the direction ofdecreasing x without change in shape.

7.1.1 Initial Conditions

An initial displacement field u(x, 0) results from a compressed or stretchedsegment of the bar. Similarly there can be an initial velocity field u(x, 0). Denoting thederivative of any function f(r]) with respect to its argument by f = df/dt], the initial

1 The solution (7.3) to the homogenous wave equation (7.2) can be obtained by introducing a transfor-mation of variables £ = x + ct,rj = x — cf.

dt2 d^d

This implies that

where g = fgfd^.

Note that the first order PDE cdu/dx + du/dt = 0 also has this wave type solution.

148 7 / Axial Impact on Slender Deformable Bodies

\ V a

Figure 7.2. Distribution of displacement amplitude u(x, t) at successive times followingsudden release at time t = 0 of an initial compressed region of length /.

conditions are generally expressed as

«(*, 0) = f [-/'(*) + g'(x)], where ^ = ^ = - c 0 / : »; = * - c0/Z Ot Of] Ot

Prescribed Displacement M(JC, 0) = h(x) and Zero Velocity M(JC, 0) = 0

The second condition is used to obtain the relationship between the functions / and g,viz. g(x) = f(x). Hence, f(x) = g(x) = h(x), so that f(rj) = h(rjX g$) = ft(§) and

u(x, t) = -[h(x - cot) + h(x + cot)]. (7.4)

Hence, waves having the initial deformed shape h(x) propagate in both directions awayfrom their initial positions. The speed of propagation is c0 = y/E/p. The amplitude ofeach wave is 1/2 the amplitude of the initial disturbance. Figure 7.2 illustrates this splitof the solution into equal parts traveling in opposite directions away from the site of aninitial disturbance.

Zero Displacement and Prescribed Velocity W(JC, 0) = hix)

uix, 0) = 0 => fix) = —gix) =>• fix)

Co ,

2

.*. f(x)=— f hix)dxCo Jo

uix,t) = / hix)dx- h(x)dx\ =2CQ \_JO JO J

— / h(x)dx (7.5)2c 0 Jx-Cot

Table 7.1. Elastic

Material

Aluminum alloyBrassCopperLeadSteelGlassGraniteLimestonePerspex

co- ( ) , c\ -\p

Wave Speeds for Several Materials

P(kg/m3)

2,7008,3008,500

11,3007,8001,8702,7002,600

L(l + v)(l

E(kN/mm2)

7095

11417.5

21055

-2v)J '

v

.34

.35

.45

.31

.22

.33

.40

Co(ms-1)

509233833662124451895300312049202260

P)

C\

(ms"1)

64604300

2200606068002900

2600

(ms-1)

310020502300700

32603250

1090

CR

(ms"1)

2970

3040

With an imposed initial velocity field h(x), the disturbance spreads from the initial dis-turbance with speed c0.

In Table 7.1 the bar wave speed c0 for uniaxial stress is shown for a variety of materials.Also shown are the wave speed c\ for uniaxial displacement (plane strain), the shear wavespeed c2 and the Rayleigh wave speed cR (surface waves). In hard materials (e.g. metals)all of these speeds are of the order of kilometers per second. The shear and Rayleigh wavespeeds are slower than the dilatational (i.e. compressive) wave speeds.

Force FQ Suddenly Applied to End of Finite Length BarA tension Fo is suddenly applied to the end of a bar 0 < x < oo. Since the disturbanceis outgoing from the source, g(i-) = 0 and

u(x, t) = f(x — cot) for x < cot.

This gives du/dx = f and du/dt = — cof'. Consequently, for a wave traveling in thepositive x direction

du/dx = -CQ1 du/dt for x < cot. (7.6)

Discontinuities in any dependent variable are indicated by a bracket [ ]. Equation (7.6)indicates that across a wavefront propagating in the positive x direction, discontinuitiesin stress and particle velocity are related by

[a] = E[du/dx] = -pcolu]. (7.7)

Similarly, if Fo is applied to the end of a bar where — oo < x < 0, the general solutionhas / = 0 (to satisfy the radiation condition) and u(x, t) = g(x + cot). Thus for a wavetraveling in the negative x direction, discontinuities in stress and particle velocity satisfy

[cr] = E[du/dx] = +pco[u]. (7.8)

Thus Eq. (7.7) suffices for both directions of travel if the wave speed c0 is given the signof the direction of propagation. Note that for a longitudinal wave, a tensile discontinuityin stress [ax] > 0 results in a jump in particle velocity [w] that is opposite in sign from

the wave speed CQ. A compressive stress wave [crx] < 0 has a jump in particle velocity inthe same direction as the motion of the wavefront.

Example 7.1 A uniform elastic rod with Young's modulus E, density p and cross-sectional area A has constant pressure po applied to the end x = 0 for a period of timer. Obtain expressions for strain energy U(x) and kinetic energy T(x) in the bar at timet = x. Compare these parts of the total mechanical energy with work W(x) done by theapplied force that acts on the end of the rod.

Solution Boundary condition:

-p0, t < x0, x < t.

Solution for stress distribution:

1 0, cot < x-A) , coit - x) < x < cot.

0, x < co(t - x)

Strain energy:

,0 - ' " — " 2E

Kinetic energy:

2 Jo - - ' • " — " 2EWork of external force:

W(x) = f [-Acr(0, t)]u(O, t)dt = P°Ac°T

Jo EPartition of internal energy:

U/W = T/W = 1/2.

7.1.2 Reflection of Stress Wave from Free End

If a constant tension Fo is suddenly applied to one end of a free elastic bar, a stresswave of magnitude Fo/A = a0 moves away from the loaded end with speed c0; behindthe wavefront (i.e. for x < cot) the stress and the particle velocity u(x, t) = —a o/pcoare constant. If the bar has length L then the wavefront approaches the free end at timet = (L/co)—. After incidence of the wave on the free end, Eq. (7.7) gives a stress o\ andparticle velocity u\ behind the reflected wavefront,

\qx - a0] = -p(-co)[ui -(-cro/pco)], LIco < t < 2L/c0. (7.9)

At the free end there is a boundary condition a(L, f) = 0 so that G\ = 0; thus for times

Arian

Highlight

Arian

Highlight

(r(O,t) c0 0(a)

Figure 7.3. Characteristic diagram for a rod with a free end at x = L that is subject totensile stress cr0 at end x = 0 during 0 < t < r. In each region between wavefronts thestress and particle velocity (a, u) are indicated.

< t < 2L/c0,

a(x, t) = <To and M(JC, t) = —ao/pco, x < 2L — c$t

a(x, t) = 0 and u(x, t) = u\ = — 2a o/pco, x > 2L — cot.

Behind the reflected wave propagating from the free end, the stress is zero to satisfy theboundary condition (BC) and the particle velocity is twice that of the incident wave.

Figure 7.3 is a characteristic diagram for stress waves in a rod of length L. The diagramillustrates the location of wavefronts emanating from the origin x = 0 at time t — 0 whena tensile stress cr0 is suddenly applied and time t — x when a 0 is suddenly removed. Theeffects of subsequent reflections from the free end x = L are also shown. In each regionbetween wavefronts the stress and particle velocity (a, w) are indicated.

An arbitrary stress wave distribution f(x —c ot) emanating from x = 0 has the followinggeneral solution during the period of time t < L/c$ before the wavefront reaches the freeend:

u(x, t) = f(x - cot), x < cot0, cot < x < L.

(7.10)

If this stress pulse is reflected from the free end, then after incidence during the periodL/CQ < t < 2L/CQ the solution will be

u(x, t) = J f(x - cot), x < 2L - cotfix - cot) + gix + cot), 2L - cot < x < L.

(7.11)

The reflected wave component g is determined by the BC at the free end, viz., the strainduix, t)/dx = f\x — cot) + g\x + cot) at x = L and at any time t must be zero:

Let £' = L + cot; hence the BC above gives g'(f') = ~f'i2L ~ £')• Consequently byextension (i.e. substitution of §' = x + cot) we have g\x + cot) = —f\2L — x — cot),

Arian

Highlight

Arian

Highlight

I *0x=2L-cot

Figure 7.4. Stress distribution a (x, t) in a rod at time t > L/c0 after reflection from freeend.

so that behind the reflected wavefront

o(x, t) = E ^ p ^ - = E{f(x - cot) - f(2L -x- cot)).OX

(7.12)

The reflected stress wave has opposite sign to the incident stress wave and apparently isidentical to a pulse with the same distribution but opposite sign that initiates at time t = 0from x = 2L in a virtual extension of the rod (see Fig. 7.4).

The reflected particle velocity wave has the same sign as the incident particle velocitywave. The displacement field at times t > L/CQ is then

u(x, 0 = f(x-cot) + f(2L-x-c0t), 2L-cot<x<Lmdt> L/c0. (7.13)

The second term above is the reflected wave from the free end.

7.1.3 Reflection from a Fixed End

Suppose a tension Fo is suddenly applied to one end of an elastic bar of lengthL while the other end is fixed. Again, a stress wave of magnitude o = a0 is generated atthe point of loading and propagates toward the fixed end at speed c0. At the fixed end werequire the particle velocity to be zero. Applying Eq. (7.7) after incidence of the wavewith the free end again gives Eq. (7.9). Together with the BC

u(L,t) = 0, t>0.

Eq. (7.9) gives stresses and particle velocity distributions during L/c0 < t < 2L/c0,

a(x, t) = ao and M(JC, t) = —ao/pco, x < 2L — c$t

a(x, t) = 2GO and u(x, t) = u\ = 0, x > 2L — erf.

At the fixed end the stress magnitude behind the reflected wave is twice the stress in theincident wave, while the particle velocity behind the reflected wave is zero.

7.1.4 Reflection and Transmission at Interface - Normal Incidence

When a wave crosses an interface between two materials, part of the wave isreflected and part transmitted. Because the wave equation is linear, after the wave crossesthe interface the stress (and particle velocity) in the first material is the sum of those of theincident and reflected waves, f + g'. The relative magnitude of reflected and transmittedcomponents of the wave will depend on the density pj and elastic modulus Ej of the two

Arian

Highlight

Arian

Highlight


PVAV p2. A2.E2

Figure 7.5. Stresses in a bar at times preceding and following the wavefront crossing adiscontinuity in material of cross-sectional area Ay. The incident, reflected and transmittedcomponents of stress are denoted by subscripts / , R and T respectively.

materials and the cross-sectional areas Aj(j = 1,2) of the rods. The relative magnitudesare determined by the ratio of impedance pc0 of the materials.

Figure 7.5 is an illustration of a wave with stress 07 and particle velocity «/ approachingan interface in a slender composite bar. At the interface in a slender bar, contact forceAjGj and particle velocity Uj are continuous.2 When the bars are unstressed before thearrival of a wave, Eq. (7.7) gives the reflection and transmission coefficients YR, YR forstress and particle velocity:

crR V -w h e r e r =

A2P2C2

GT = 2T(Al/A2)cri ~ r + 1

= -YR

(7.14a)

(7.14b)

(7.14c)

uT

AXT YT. (7.14d)

The reflection and transmission coefficients for stresses are shown in Fig. 7.6 for bars ofequal area A2 = A\ and varying impedance ratio P.

When a transient wave passes through a layered material (e.g. a composite), the suc-cession of reflections and transmissions from interfaces can be analyzed using this onedimensional model. Interlaminar tensile fractures can be formed near a surface wherecompressive forces are applied for a short duration. Tensile stresses are large if the acous-tic impedance of adjacent layers are quite different and the impedance of the surface layeris relatively small (Achenbach, Hemann and Ziegler, 1968).

7.1.5 Spall Fracture Due to Reflection of Stress Waves

At a free surface or an interface where the acoustic impedance increases in thedirection of propagation, compressive incident waves are reflected as tensile waves. Whenbehind the wavefront the stress magnitude decreases, a net tensile stress occurs in the bar

2 Discontinuities in cross-sectional area are handled by similar means in a ID theory. At the interface,the force acting on the end of the second bar is equal in magnitude and opposite in direction to the forceacting on the first bar. Consequently, the boundary condition on stress is replaced by (a/ + OR)A\ =

2

1

r\

1

(

(

1

^ *

+

2

~~—

3

— -

— —

-

— —

5

m.. ••

m.« "

6

_—^»

•i -

r7

Figure 7.6. Reflection and transmission coefficients as a function of impedance ratio F =PiCi^ilP\C\ A] for a one-dimensional stress wave in a slender composite bar having equalareas A2 = A\.

after reflection. A tensile fracture, or spall, results where the stress equals an ultimatestress.3

Consider a free bar of length L with an exponentially decreasing compressive stresssuddenly applied to one end: a(0, t) = —<T 0 exp(—t/r). 4 The BC at the origin is used toevaluate the function f(x - cot) for a radiating wave:

or(0, t) = -a0 exp(-;/r) = Ef'(0 - cot), t > 0.

Substituting £' = —cot, we obtain the function / (£ ' ) and subsequently by extensionf(x - cot) = -(GQCQT/E) exp[(x - cot)/coT]. Consequently,

OT(JC, 0 = ^ / r ( jC - COt) =

u(x,t) = -cof(x -cot) =

0,- cot)/cor], x < cot

x > cot

| (cro/pco) exp[(x - cot)/cor], x < cot| 0, x > co£.

The bar has a free end at x — L, where the incident stress wave is reflected with a changein sign. At the free end the boundary condition is

0 = cr(L, t) = E{f'(L - cot) + g\L + c001-

Hence g'(S') = -f(2L - £'), so that

-x-c0fg\x + cot)

Go (1L — X — Cot\= -^ exp 5-E \ cor J

3 In ductile metals, fracture occurs by tearing between voids that have opened from fracture initia-tion sites. The density of sites is stress-dependent. Consequently, when an ultimate plastic shearstrain is required for tearing between voids, ductile fracture is time-dependent. The apparent ulti-mate stress for failure caused by short duration impact loads is increased by this effect (Meyers andMurr, 1980).

4 This theory assumes a uniform distribution of stress and displacement across the section, i.e. onedimensional (ID) deformation. It will be a poor approximation for higher frequency components of awave which have wavelength shorter than the bar diameter.

Consequently, during the period L/co < t < 2L/c0 after the first reflection,

- cot)/cor] x < 2L - cot

- cot)/cor] - exp[(2L - x - cot)/cor] x>2L- cota(x,t) = -ao\

Figure 7.4 shows the distribution of stress at a time in the period L/c0 < t < 3L/c0

after reflection of an initially compressive stress wave. If the stress wave decreases expo-nentially behind the wavefront, during this period the largest tensile stress will be at thereflected wavefront where 2L — x = ct. This stress increases with distance between thewavefront and the free end; at any location x the maximum stress amax is given by

(7.15)

The spall thickness is determined by the ratio of maximum applied pressure —a 0 tofracture stress Of and by the rate of decay r in pressure amplitude behind the front.

Example 7.2 Consider coaxial impact of two long slender bars, each of length L andcross-sectional area A, as shown if Fig. 7.7. The impact occurs at time t = 0, and theimpact interface is designated as x = 0. Bar 2 is butted against a rigid barrier at endx — L, and initially it is stationary while bar 1 is moving at initial velocity V o. Let theimpedance of bar 1 be picOi and that of bar 2 be P2C02, where Co2/co\ > 1. Obtain thedistribution of stress and particle velocity in the rod as a function of time.

0<t

(0, Vo) ,' f iJ (0.0)

2L/c02:

"2a1

/ c 0 2 < t

(0, u, 1*2, • °2

2Z./C Q1

Figure 7.7. Elastic waves from collinear impact of traveling bar against stationary barshown at successive times after impact. The wavefronts are indicated, and between themthere is a constant state of stress and particle velocity (a, u).

Solution Equations (7.7) and (7.8) for the velocity change across a wavefrontare used together with boundary and interface conditions to obtain the stress and particlevelocity distributions at successive times:

0 < t < L/co2 : o\a = a2a = —(1 + F) p2co2Vo, V = p2co2/P\Co\

L/CQI < t : crlb = 0, o2b = -2(1 + rylp2c02V0

At time 2L/CQ2 the faster wave returns to the interface x — 0, where it is partly reflectedand partly transmitted:

2L/c02 < t < 4L/(c0i + c02): o2c = - ( 3ulc = u2c =

At time (3cOi + 2CO2)£/(CQ1 + coiCo2) the two waves cross in bar 1:

4L/(c0i + c02) < t < 2L/c0i: ald = -2(1 + ry2p2c02 Vo

Notice that at the interface for time t < 2L/c0i the stress is compressive (p2c < 0)irrespective of the impedance ratio P. At larger times the stress at the interface must bechecked to ensure that it remains negative - when this condition is no longer satisfied,the interface condition changes so that the interface is free of stress.

7.2 Planar Impact of Rigid Mass against End of Elastic Bar

Let the end x = L of a uniform, straight elastic bar be struck by a rigid bodyof mass M that is moving with initial speed Vb while the end x = 0 is restrained bya viscoelastic damper as shown in Fig. 7.8. Here we assume that changes in stress arerapid so that they are transmitted by waves propagating at an elastic wave speed CQ. Fora wave traveling in the direction of decreasing (increasing) x, stress and particle velocityare related by conservation of momentum across a wavefront [Eq. (7.7)]:

[a] = pco[u], c0 < 0 (7.7a)[a] = -pco[ul c0 > 0 (7.7b)

where u(x, t) is the particle velocity at x.

MP.A.E

Figure 7.8. Impact of heavy mass against end of elastic bar. The distal end of the bar isrestrained by a viscous dashpot generating a force proportional to the rate of extension"(0, 0.

7.2 / Planar Impact of Rigid Mass against End of Elastic Bar 157

7.2.1 Boundary Condition at Impact End

At any instant the stress (and particle velocity) at each end of the bar is a sumof an incident (or incoming) and a reflected (or outgoing) wave with magnitudes 07 andoR respectively. Suppose mass M collides against end x = L with incident speed Vo.Subsequently the mass has velocity V(t). While contact persists, this velocity is the sameas the particle velocity at the impact end u(L, t). The BC is

The accelerations of the colliding mass depends on the force at the end of the bar:

MdV/dt = -[a/(L, 0 + crR(L, t)]A = pc0A[ii/(L, 0 - uR(L, t)]

= pcoA[2uj(L,t)-V(t)]

where the last equality follows from the BC above. Denote the ratio of masses of collidingmissile and bar by a = M/pAL and solve the ordinary differential equation for thevelocity V(t) of the colliding mass,

V(t) = ( ^ ) e-c°t/aL \c + T ii7(L, t')e-c»t>/aL dt'\ (7.16)

Initially V(0) = Vo and ii7(L, 0 = 0, so that C = (aL/2co)Vo and

Cot<2L. ( 7 . 1 7 )

During the first transit period, the outgoing wave from the impact end is obtained byextending this BC (i.e. replacing cot by cot + x — L):

e_(cot+x_L)/aL^ ^ < 2LpcV0

7.2.2 Boundary Condition at Dashpot End

The dashpot force is proportional to the velocity at the end w/(0, t) + UR(0, t). Ifthe constant of proportionality is /xo, equilibrium offerees gives the boundary conditionat the restrained end x = 0,

0 = [o,(0, t) + <r«(0, t)]A - A*O[«/(O, 0 + uR(0, /)]

= pc0A[u,(0, t) - uR(0, t)] - /iO[«/(O, t) + «*(0, 01- (7-19)

Let F = iio/pcoA, and rearrange to obtain the reflection coefficient YR'-

_aR(0,t)= uR(0,t) = f - lYR~ a,(0,t) ~ 11(00 r+1"

During the first transit period L/c0 < t < 3L/c0, the reflected (outgoing) wave atthe dashpot end (x = 0) is determined from an incident wave (7.19) and the reflectioncoefficient (7.20),

ii*(0, 0 = -y««/(0, t) = -Voe-^'-L)/a(L\ L/co < t < 3L/c0. (7.21)

cot/L

-2 -1 0 3 x/L

Figure 7.9. Characteristic diagram for wavefront components in extended virtual bar.Location of bar is the grey band 0 < x/L < 1.

The outgoing wave is obtained by extending this BC (i.e. replacing cot — Lbycot—x — L)\

UR(X, 0/VO = -yRe-(cot-x-L)/o(L, x < cot - L and L/c0 < t < 3L/c0.(7.22)

7.2.3 Distribution of Stress and Particle Velocity

The stress wave components obtained from continuation of the solution by suc-cessive use of boundary conditions are superposed to obtain the impact response. At anytime the solution for a region 0 < x < L is the sum of those components that are gen-erated by the initial impact and subsequent reflections. An additional component entersat alternate ends of region 0 < x < L after each time period cot/L as illustrated in thecharacteristic diagram, Fig. 7.9. For each successive wave entering the bar at the impactend, the constant of integration in Eq. (7.16) is determined by continuity of V(t):

x > L — cot

x < -L + cot

pcoVo

pcoVo-u2 - 1

x > 3L — cot

cr4

x < 3L + cot.

The collision terminates at some time tf > 2L/c 0 when cr(L, tf) — 0. Thereafter areflected wave at the free end just cancels a/(L, t).

Figure 7.10 shows the variation of stress with time at the impact end of the bar for twovalues of the colliding mass ratio, a = 1 and 4 and a fixed distal end f ^> 1. After initial

7.2 / Planar Impact of Rigid Mass against End of Elastic Bar 159

0-lL.tY

2.607

2.135\

2.974

X\\\\\

0 2 4 6 L

Figure 7.10. Stress at impact end as a function of time for two mass ratios, a = 1 and 4.

Figure 7.11. Maximum stress at fixed distal end (F ;$> 1) as function of mass ratio a:elastic wave solution (solid line), dynamic solution assuming uniform strain (dashed line)and quasistatic solution assuming uniform strain (dash-dot line).

incidence the light mass rapidly decelerates and the bodies separate at time cot/L ^ 3,while with the heavier mass, separation occurs at cot/L % 6. At the impact end of thebar there is a jump in stress each time the initial wavefront returns.

At the distal end of the bar the effect of mass ratio a on maximum stress is shown inFig. 7.11. The maximum stress calculated from the elastic wave solution is larger thanthe maximum stress obtained from a strength-of-materials analysis where at the instantthe colliding mass is brought to rest the initial kinetic energy of the colliding body isdistributed uniformly along the bar.

7.2.4 Experiments

John Hopkinson (1872) and Bertram Hopkinson (1913) performed experimentswith steel spheres striking the end of an iron wire. The wire was threaded through a hole inthe sphere, and the sphere was dropped from a height h. Within a broad range of collidingmass, the minimum height h required to break the wire was independent of the mass Mof the sphere. Moreover, the wire broke at the upper (fixed) end and not at the impactend. If the drop height was increased above a limiting value hcr, fracture occurred at theimpact end if h > hcr.

These results were explained in terms of stress waves in an elastic wire. The stress atthe impact end is

while the stress at the clamped end (f ^> 1) is twice this value during L < ct < 3L.Subsequent reflections increase the stress further. G. I. Taylor (1946) showed that withoutfracture, the maximum stress occurring at the fixed end is approximately

<W(£, 0 = pc0V0[l + Ja + 2/3}.

7.3 Impact, Local Indentation and Resultant Stress Wave

When an elastic body strikes the end of an elastic rod, both bodies suffer localindentation as described in Chapter 6. The axial force in the contact region is the sourcethat generates a stress wave in the rod - a wave that propagates away from the impact siteduring an initial period of motion.

Suppose an elastic sphere B2 traveling in direction x with an initial velocity Vb suddenlystrikes the domed end of a stationary elastic bar as shown in Fig. 7.12. Let the sphere haveradius R2 and be composed of material with density p2 and elastic modulus E2, while thedomed end of the bar has radius Ry. The bar is slender and is composed of material withdensity p\ and elastic modulus E\\ the cross-section has area A\. With these materialproperties, the mass of the solid sphere is M2 = ^np2R\, while the wave speed in thebar is c0 = y/E\/p\.

The stress wave propagating away from the domed end of the bar results in an axialdisplacement W(JC, t) = fit — x/c0). Consequently sections of the bar have particlevelocity u(x,t) and strain e(x,t) = du/dx given by

II(JC, t) = fit - x/co), six, t) = -c~lfit - x/co).

Following incidence there is local indentation 8it) that develops at the end of the rod. Therate of indentation is given by

& = Vit) - ii(0, 0 = Vit) - fit). (7.23)

At the contact surface x = 0 the positive axial contact force F(0, t) is related to both thestrain and the local indentation:

F(0, 0 = ^—^fit) = KS83/2 (7.24)Co

where E* = E\E2/iE\ + E2), R* = R\R2/iR\ + R2), and the indentation stiffness isKS = ^E*RlJ2 in accord with Eq. (6.8). It is this contact force which propagates awayfrom the contact surface as a stress wave. After substituting the expression for f into

Vit)

Figure 7.12. Coaxial impact of elastic sphere of mass M2 striking domed end of slenderelastic rod. Axial speed of sphere Vit) is retarded during collision.

7.3 / Impact, Local Indentation and Resultant Stress Wave 161

(7.22) and noting that F(0, t) — —M 2 dV/dt, the equation of motion is obtained as

fl KC^d^ Kf l KsC^d^ K^ 3/2

dt2 AXEX dt M2

Making the change of variables y = 8/R*, r = tJ KSRXJ2 /M2 and denoting the ratio of

axial rod to local indentation compliance by 2f = (3co/2A\E\)y M2KSRXJ2, we obtainthe equation of motion in terms of nondimensional variables:

y3/2 (7.25)

where y = dy/dr.To interpret the nondimensional results obtained from this analysis it is useful to note

that for a bar with circular cross-section of diameter 2a the parameters can be expressedas

(?)3 (T) and r = —„7tp2R2

while the initial conditions are

v(0) - 0, dy(0)/dx = V0(np2/E*)l/2(R2/R*)5/4

Notice that the analysis above requires that the contact radius be small in comparison withthe radius of the bar so that the Hertz stress field is representative of contact conditions.Consequently, the asymptotic limit as R*/a -> oo of the analysis above does not approachthe solution for planar axial impact of a rigid body (Sect. 7.2).

Example 7.3 A sphere of radius R2 = a collides against the flat end of a bar made ofan identical material. Find the contact force F(t) for impact speeds Vb = 1 and 10ms"1.

Solution For a solid sphere that collides against the flat end of a rod made ofthe same material,

? 2 = (2nT\R2/a)\ x = (2nTl (a / R2)c0t / a, y(0) = (2n)l/2V0/c0.With these relations, Eq. (7.24) has been used to calculate the interface force as a functionof nondimensional time r for impact speeds of Vb = 1 and 10 ms"1 . Figure 7.13 showsthat the contact duration decreases slowly with increasing impact speed. In Fig. 7.14 themaximum force is plotted as a function of the nondimensional incident velocity VQ/CQ.Also shown is the ID stress wave solution for coaxial impact of a rigid mass on the end of aslender elastic bar. For impact between elastic bodies, the local compliance in the contactregion reduces somewhat the maximum force at the interface; there is a correspondingincrease in the duration of contact. In these elastic collisions against an infinitely longbar, the part of the kinetic energy of relative motion that is lost during collision equalsthe sum of the strain plus the kinetic energy in the bar.5

5 Saint-Venant (1867) mentioned that for a collision at the end of an elastic bar, the stress wave propagatingaway from the end transports energy away from the contact region; consequently, this energy is notavailable during restitution to restore the initial kinetic energy of the colliding body.


15

Figure 7.13. Contact force F/E*Rl during impact of steel sphere on flat end of steel barwhere R2 = a. Impact speeds of VQ = 1,10 ms"1 give initial conditions dy(0)/dr =0.48 x 10-3, 4.8 x 10"3.

10 10-3

impact speed V0/c0

10-2

Figure 7.14. Maximum contact force Fmwi/E*R* for elastic impact of steel sphere on flatend of steel bar where R2 = a. The dashed line shows the ID stress wave solution forcoaxial impact of a rigid body against the end of an elastic bar; this gives a maximum force

7.4 Wave Propagation in Dispersive Systems

The wave equation has a solution which represents a pulse which propagateswithout distortion. There are variants of the wave equation which also represent distur-bances that initially propagate away from a point of initiation, but whose solutions involvedistortion of the pulse shape as it propagates. This continuous variation of the pulse shapeis known as dispersion.

Consider a longitudinal wave propagating in a bar that has continuous elastic supportas shown in Fig. 7.15; the bar has cross-sectional area A, Young's modulus E and densityp. At any point the support provides a distributed force per unit length — KU(X, t) thatis proportional to the displacement u(x, t). The equation of motion for this system is alinear Klein-Gordon equation,

d2udx2 E U ddt2'

(7.26)

7.4 / Wave Propagation in Dispersive Systems 163

/y yy y /y yy y. _K\S\\ \\ HI if

SAf JJJ1 1/ / / / / A / / / / / / / / / / /

Huix.t)Figure 7.15. Axial wave motion in a bar embedded in an elastic medium.

Figure 7.16. Variation of phase velocity cp with wavelength X in elastically embedded bar.

If the solution is both spatially and temporally harmonic, we can write u(x, t) = w0 exp[i(kx — cot)], where w0 is the amplitude, / = \f—i, k = 2TT/X is the wave number(inversely proportional to the wavelength X), and at any location co is the frequency ofoscillation. Substituting this doubly harmonic form of solution into the equation of motiongives a dispersion relation between frequency co and wave number k,

co2/cl = k2+/c/E. (7.27)

At any time the harmonic disturbance has a wavelength X. This disturbance propagateswith the bar wave speed c0, resulting in a frequency to for an observer at any spatiallocation x; i.e. kx — cot = k(x — cpt). Consequently co = kcp, and the phase velocity cp

varies with the wavelength of the disturbance:

The phase velocity is the speed of propagation of a component of the stress pulse that haswavelength X. The phase velocity increases with the wavelength as shown in Fig. 7.16.

If one considers propagating waves with wavelength that grows without bound(X -> oo), the corresponding frequency decreases to a cutoff frequency coc. Excitationsat frequencies smaller that coc do not propagate - they are dissipated:

u = u0e-ikxe-ioa, for co < coc. (7.29)

This oscillating displacement field which decays exponentially with distance from asource is known as an evanescent wave.

7.4.1 Group Velocity

A stone dropped into a pond (Fig. 7.17) causes ripples that propagate outward onthe water surface. Rayleigh observed these wave groups and the individual waves withina group. He wrote that "when a group of waves advances into still water, the velocity of

Figure 7.17. Phase and group velocities cp and cg for ripples spreading on surface ofshallow water.

the group is less than that of the individual waves of which it is composed; the wavesappear to advance through the group, dying away as they approach its anterior limit."

This behavior can be understood by examining the superposition of two simple har-monic waves of equal amplitude and slightly different wavelengths:

u = u0 [ei(kx-^ + ei(*ox-*>ot)]

^ - ^ ( A k A l k k ± ^ co =k

Letting 2 Ak = k — k0, 2 Aco = co — co$ yields the phase velocity cp and the groupvelocity cg:

cp = coll, cg = Aco/Ak ^ dco/dk. (7.30)

Noting that co = kcp gives cg = cp + kdcp/dk = cp — Xdcp/dX. In effect there is acarrier frequency co propagating with the phase velocity and an amplitude modulationterm that travels with the group velocity. Three conditions can occur:

(a) cp > cg: waves appear at the back of a group, travel to the front and disappear.This is normal dispersion.

(b) cp = cg: no dispersion and no change in pulse shape.(c) cp < cg: waves appear at the front of a group and travel to the back.

For the previous example of longitudinal waves in a bar embedded in an elastic medium,cg/co = co/Cp = (1 + K/EJC2)~1/2, which is case (a).

The wave group between two nodes has a certain energy. In a dispersive medium, energypropagates with the group velocity. Thus the flexural response of a beam to transverseimpact is a wave traveling away from the impact site; this wave has a leading element, orwavefront, that travels at a speed equal to the group velocity cg.

7.5 Transverse Wave in a Beam

7.5.1 Euler-Bernoulli Beam Equation

Flexural waves in a beam involve translation perpendicular to the axis. FromFig. 7.18 the equation of motion can be written

d2w d2Mdd= JTdx+adx

where w(x,t) is the transverse displacement, S(x, t) is the shear force, M(x, t) is thebending moment, S = dM/dx, and q is a distributed force per unit length. Ignoring any

7.5 / Transverse Wave in a Beam 165

4TTMTTYflT*fnvfTTyfTTfrrnM^^

-Ik* tfl—HT'i;M™dx

(a)

Figure 7.18. (a) Beam loaded by downward pressure q(x,t), and (b) stress resultants actingon deformed element of beam.

effects of rotational inertia of the element dx, the shear force and bending moment arerelated by 5" = dM/dx. With the moment-curvature relation M = —El d 2w/dx2 (whichassumes that plane sections remain plane) one obtains the Euler-Bernoulli equation forbeam elements,

^ = -c2k2^- + ^lH c2 = - (7.31)

dt2 ° r dx4 pA ' ° p

and the radius of gyration kr is related by k2 = I/A to the second moment of area, / , ofthe cross-section about the neutral axis.

For a free wave solution (q = 0), substitute w =wo exp i(kx — cot) into the homoge-neous Euler-Bernoulli equation. This gives a dispersion relation,

co = ±cokrk2

and a phase speed and group velocity,

Cp/co = krk, cg/co = 2krk.

Although this system exhibits normal dispersion, it is unrealistic in that both phasespeed and group velocity become large without bound as the wavelength X -> 0 (i.e.k —> oo). Experimentally it has been observed that the transverse motions of a beam havea finite maximum wave speed irrespective of the wavelength.

7.5.2 Rayleigh Beam Equation

The Euler-Bernoulli equation is based on the assumption that rotary inertiais negligible, an assumption that is valid for long wavelengths only. To consider shortwavelengths the rotary inertia of sections must be included. Rayleigh first developed theequation of motion incorporating rotary inertia for transverse waves in a beam. Consider-ing the equation of rotational acceleration due to moments acting on a differential elementof length dx gives

dM d2oSdx dx = pi dx — -

dx dt2

where the (small) rotation of the element is 0 ^ dw/dx. Therefore

ds d2M d ( d3w \ d2

dx dx2 dx V dx dt2

Figure 7.19. Element rotation 0 + y0 caused by combination of flexure and shear warping.

By substitution into (7.31) Rayleigh obtained the following equation for a uniform beam:

2 / 2 I T 2 I ^ V > /

dx2 dt2 pA(7-32)

This equation has a homogeneous solution w = wo exp i(kx — cot) with a dispersionrelation

This gives a phase speed and group velocity,

cp/c0 = krk(l + k2k2)~l/\ cg/c0 = krk(2 + k2k2)(\ + k2k2)~V2

While this analytical model provides a bounded phase speed and group velocity for shortwavelengths, the asymptotic limits are too large. The assumptions of this analysis are animprovement over the Euler-Bernoulli theory but still are not representative of transientsthat include short wavelengths.

7.5.3 Timoshenko Beam Equation

Short wavelength deformations (kr/X > 0.1) require more accurate representa-tion of the deformation field than occurs with the Rayleigh beam equation. This equationconsiders the rotation 0 due to flexure dO/dx = —M/EI, but it neglects warping ofinitially plane sections - warping caused by the shear stress distribution over the cross-section. The change of inclination caused by shear varies from zero on the top and bottomsurfaces to a maximum y0 at the neutral axis. In the deformed configuration, lines thatinitially were perpendicular to the neutral axis have a maximum rotation dw/dx = 0 + y0

at the neutral axis, as shown in Fig. 7.19.Section rotation caused by warping is overestimated by the warping rotation of the

neutral axis /o- An effective rotation from warping is y = Ey0, where the constant S isobtained on the basis of average shear stress (or strain) across the section, 0 < S < 1.Alternatively the coefficient S = y/yo can be obtained by equating the work per unitlength done by the shear force (Syo/2) to the shear strain energy per unit length. Thesealternative definitions for the Timoshenko beam coefficient S result in somewhat differentvalues, as shown in Table 7.2.

The effective shear stress across the section Gy is related to the shear force S by

S = GAEyo = GAE(dw/dx - 0)

while the bending moment is related to flexural rotation of the neutral axis by

M = -EldO/dx.

7.5 / Transverse Wave in a Beam 167

Table 7.2. Timoshenko Beam Coefficient S of VariousCross-Sections

Cross-Section

RectangleCircleIsosceles triangle

From Av. Shear Stress

0.670.750.75

From Shear Strain Energy

0.880.9

Hence for a uniform beam with a distributed load q(x, t), Eq. (7.31) for rotation of adifferential element results in

d20 d20„ _ _ , , _ . (7.33)

An independent equation representing transverse motion of a beam can be obtained as

d2w d0\ d2w( 7 - 3 4 )

Noting that dS/dx = GA E(d2w/dx2 - d0/dx) = pA d2w/dt2, Eqs. (7.33) and (7.34)can be combined to give the Timoshenko beam equation;

El d4w I / E \ d4w d2w^oXJx4 ~ ~A \ + ~GEJ dx2 dt2 + ~dt2

pi d4w

q(x,t) kzrd2q El dzq

pA S dt2 pAE dx2

Solutions to the Timoshenko beam equation are most clearly expressed from solutionsto the coupled equations for rotational displacement (7.33) and transverse displacement(7.34). Harmonic solutions representing propagating waves [q(x, t) = 0] will be of theform

w = wo exp i(kx — cot), 0 = 6$ exp i(kx — cot)

where the wave number k = 2n/X. Solutions to (7.33) and (7.34) of this form satisfy

0 = iGAEkwo - (GAE + Elk2 - plco2)00 (7.35a)

0 = (GAEk2 - pAco2)w0 + iGAEk00. (7.35b)

These equations provide an amplitude ratio Oo/wo and a characteristic equation for thefrequency co:

Go _ i(GAEk2 - pAco2) _ iGAEk~w~o ~ GAEk ~~ GAE + EIk2- pico2 (?*

0 = F ^ 4 " (p + 7 + 7^0 kW + -GA \kz A GAJ A

Recalling that co = kcp and k2 = I/A, we have the dispersion relation

0 = 4 - (*?** + f (1 + ^ 2 ) ) | | + 4. (7.37)Limiting cases for short and long wavelengths can be examined after defining a nondi-

mensional phase speed cp = CP/CQ\

(a) Short wavelengths (k -> oo):

These limits represent the longitudinal bar and effective shear wave speeds,respectively,

(b) Long wavelengths (k —>• 0):

These limits on the admissible range of frequencies represent cutoff frequenciescoc for predominately flexural and predominately shear deformation, respectively.

The mode of deformation corresponding to the cutoff frequency coc = (c0/ kr)^/GE/Ecan be obtained by substituting this limiting frequency into the characteristic equations(7.35). Letting o(e) denote small terms as k -> 0,

0 = io(e)w0 - ( G A S - plco2c)00

0 = -pA(i>2cwo + io(8)00.

If the frequency approaches the cutoff frequency ooc —> (co/k r)^GE/E, then w = 0and 0 = Oo exp(—ia> ct), where for long wavelengths both w and 0 are independent of thespatial coordinate x.

7.5.4 Comparison of Euler-Bernoulli, Rayleigh andTimoshenko Beam Dynamics

Figure 7.20 compares the phase speed cp for flexural (and shear) waves thatwas obtained from the three models of elastic beam deformation in the range of longwavelengths 0 < h/k < 2. This comparison was calculated for a beam of rectangularcross-section with depth h and Poisson ratio v = 0.3. Notice that for short wavelengths(X —> 0), the wave propagation speed has an upper limit in both the Rayleigh and Tim-oshenko theories. In the latter case, the group velocity approaches the Rayleigh surfacewave speed cR/c0 = 0.56, which is slightly less than the shear wave speed c2/co = 0.57.Throughout this range of wavelengths the Timoshenko theory is within 1% of the exactsolution given by a 2D elasticity solution. It should not be expected however, that the Tim-oshenko theory is accurate for wavelengths that are much smaller than the cross-sectiondepth. Cremer and Heckl (1973) explained why higher order shear and through-thicknessdeformation modes need to be considered if the wavelength is very small. Calculationsof the beam transient response to a suddenly applied transverse force were published byBoley (1955).

Problems 169

0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

Figure 7.20. Phase speed cp as a function of wavelength X for Euler-Bernoulli, Rayleighand Timoshenko beam theories.

PROBLEMS

7.1 A uniform elastic bar with Young's modulus E, density p and cross-sectional area Ahas pressure p{t) applied to the end x = 0 for a period of time r. The pressure p{t)satisfies p = 2p0t/r for t/z < 1/2, p = 2/?0(l - f/r) for 1/2 < r/r < 1 and p = 0for 1 < t/r. Obtain expressions for the strain energy U(r) and kinetic energy T(z)in the bar at time t = r. Compare these parts of the total mechanical energy withthe work W(r) done by the applied force that acts on the end of the rod.

7.2 Two long uniform elastic bars are moving in the same direction before they collideend on. The first bar, of length L/2, is moving with initial speed V\, and the secondbar, of length L, is moving with initial speed V2. The bars are identical in all otherrespects. Find the intensity of the stress due to longitudinal impact, and determinehow far the interface between the bars moves between start and end of contact. Ona characteristic diagram plot the position of the bar ends as a function of time.

7.3 For the previous problem show that at time t = L/ACQ the sum of the kinetic andstrain energies in the two bars is equal to the kinetic energy of the system beforethe collision. Also show that at this time the axial momentum of the system is equalto the momentum before collision. Identify the collision periods in which energyand/or momentum are conserved.

7.4 A thin aluminum alloy bar of length L has an axial velocity VQ before impact withone end of an initially stationary, equal diameter, steel bar of length 3L. The steel baris suspended by long, flexible strings which keep the axes of the two bars aligned.Denote the elastic wave speed in the aluminum and steel bars by ca and cs respectively.Plot the position of the ends of each bar for time 0 < t < 4L/ca. Show that contactbetween the rods ceases at time t = 2L/ca, and calculate the mean velocity of eachrod at this time. Determine if momentum and energy are conserved in this impact,and account for any losses. What is the impact velocity if the largest stress equalsthe yield stress in one of the bars?

7.5 A pile driver consists of a steel block, the monkey, which is dropped onto the top of avertical timber pile. The length of the pile is L, and the impact speed is Vb- Assumingthat the steel block is rigid compared with the timber, show that the initial retardationof the monkey is (coVo/aL) — g, where g is the acceleration due to gravity, M isthe mass of the monkey, a = M/pAL is the ratio of the mass of the monkey to themass of the pile, and CQ is the elastic bar wave speed for the timber. Assuming thefoot of the pile to be rigidly fixed, prove that the retardation of the monkey just aftertime 2L/co from impact is given by

2c0V0 /coVo \ , 0 . .— 7 - + — r ~ 8 exp(-2/a).

7.6 A stress wave is traveling away from the end x = 0 of an elastic bar of length L andcross-sectional area A. Beginning at time t = 0, the end of the rod is driven axiallyat a velocity V(t) = Voe~at.(a) Find an expression for the particle velocity in the bar.(b) The end of the rod at x = L has an axial damper that provides a damping force

F{t) = -inou(L, t). Show that the amplitude UR of the wavefront reflected fromthe damped end can be expressed as

w* _ 1 - fip/pcpAVo 1 +

For what value of the damping constant /JLO is the wave completely absorbed sothere is no reflection?

(c) Obtain an expression for the particle distribution for times t > L/c\. Sketch thisparticle velocity distribution at time t = 3L/2c\. For any location x give anupper limit for the time t when this expression is applicable.

7.7 A uniform, slender elastic bar of length L, cross-sectional area A, mass M andelastic wave speed CQ is moving initially in an axial direction with speed 2Vb- Attime t = 0 one end of the moving bar collides against the end of an identical bar.The second bar is initially stationary and the bars are oriented coaxially. A heavyparticle of mass M/a rests against the free end of the stationary bar at x = L, wherespatial coordinate x is measured from the interface and directed parallel to the initialvelocity. Axial motions are unconstrained.(a) Produce sketches of the distribution of particle velocity u(x, t) and stress cr(x, t)

in the bars at time t = L/2CQ. On this sketch indicate the amplitude of anydiscontinuities. The sketches should be labeled so that positive values are upward.

(b) Calculate the speed of the heavy particle, V{t)\ show that this speed equalsV_ _ r 0, t < L/cVo ~ { 2 - 2exp[a(l - ct/L)], L/c < t < 3L/c.

(c) Sketch the distribution of particle velocity and stress in the bars at time t =3L/2c; on this sketch indicate the amplitudes of any discontinuities.

(d) Find the first time when stress vanishes at the interface between the rods.7.8 Two coaxial uniform, slender elastic bars are joined by a heavy particle with mass

M'. Each bar has length L, cross-sectional area A, mass density p and elastic wavespeed c. At time t = —L/c an axial stress GQ is suddenly applied at a free end of onebar, where x = 0; i.e., <T(0, t) = ao,t > —L/c, The interface between the bars cansupport tension, and axial motions are unconstrained.(a) For the stress wave incoming to the interface, the incident stress o\ is related to

the incident particle velocity u\ by o\ = —pcui. Write similar equations that

Problems 111

7.9

relate reflected and transmitted stresses aR, aT to the reflected and transmittedparticle velocities UR,UT-

(b) For a wave incident on the interface, express the continuity conditions acrossthe interface in terms of a mass ratio OL — M''/pAL. These equations relateincident, reflected and transmitted stresses and particle velocities across theinterface.

(c) Obtain the reflection and transmission coefficients YR = aR/aI and yT = GT/GIfor waves incident on the interface, and show that they can be expressed asfunctions of ct /a L.

(d) For mass ratio a = 1, sketch the stress distribution along the rod at time t =0.5(L/c), i.e. after the wavefront has passed the interface. Indicate on yoursketch the amplitude of stresses at the interface.

A long freight train has n identical wagons (freight cars), each of mass M and length /across the buffers. The buffer springs between two wagons are of combined stiffnessS. The couplings put some initial compression in the springs,(a) By considering the forces acting on a wagon and/or by analogy with a uniform

bar, show that waves of tension and compression travel down the train with avelocity

(b) If the driver suddenly applies the brakes to give the locomotive a severe uniformdeceleration - V for a time nl/4c, suggest when and where is there a possibilityof a coupling breaking in tension. Take the locomotive to be massive comparedwith the train of wagons.

Dispersive Waves7.10 (a) A ID system has a dispersion relation a> = co{k). Two waves of equal amplitude

but slightly different frequencies travel along the system, in the same direction.Show that the envelope of the resulting wave pattern travels (approximately) atthe group velocity.

(b) Consider a finite section of the same system, of length L. Make a plausible guessfor the approximate form of standing waves (vibration modes) in the system. Formodes well up the modal series for 'his system, what is the approximate intervalbetween the wave numbers of successive modes? Hence show that the spacing (inhertz) between adjacent mode frequencies is given approximately by A / ^ cg/2L,where cg is the group velocity.

7.11 A simple model for the vertical vibration of a railway rail on its bed of ballast is theaccompanying drawing. An Euler-Bernoulli beam of flexural rigidity El and massper unit length m is supported on an elastic foundation, which exerts a restoring forceSy(x, t) per unit length when the displacement is y(x, t). The equation of motion is

thus

d4y d2y

(a) Obtain the dispersion relation. For each (real) value of frequency &>, describe thepossible propagating and/or evanescent waves on the rail.

(b) Calculate the phase velocity and the group velocity for the traveling waves, andplot both (on the same axes) as a function of frequency.

(c) A train wheel rolls along the rail and applies a broadband input force to the rail(arising from roughness of the rail surface and wheel tread). Describe (withoutcalculations) the qualitative response pattern of the rail.

7.12 For axial displacement u(x, t) in an elastic circular rod of radius a made of ma-terial with density p, Young's modulus E, and Poisson's ratio v, Love derived anapproximate expression for the effect of lateral inertia on a longitudinal wave ofaxial displacement,

d2u pv2a2 d4u _ _2d2u

Find the dispersion relation by considering a solution of the form u(x, t) = A exp[i(cot — kx)]. Calculate the phase velocity cp and the group velocity cg, and plot agraph of c = CP/CQ as a function of the radius to wavelength ratio k = vak/y/l inthe range 0 < % < 5.(a) Use your graph to delimit the circumstances where the elementary theory of

longitudinal wave propagation in a bar is satisfactory.(b) Compare your solution with the 2D Pochhammer-Chree solution for longitudi-

nal waves in an elastic rod, and estimate the maximum value of k where Love'stheory is satisfactory.

7.13 Two semi-infinite Euler-Bernoulli beams are connected together end to end by apinned joint. Both beams have flexural rigidity El and mass per unit length m. Whatare the appropriate boundary conditions for the displacement and its derivatives inthe vicinity of the joint? Hence calculate the reflection and transmission coefficientsfor a flexural traveling wave that is incident on the joint from the left. Show also thatthe amplitudes of the evanescent waves on either side of the joint are equal; do thisby finding reflection and transmission ratios, YER and YET, for evanescent waves.

CHAPTER 8

Impact on Assemblies of Rigid Elements

Physics is popularly deemed unnecessary for the astronomer, buttruly it is in the highest degree relevant to the purpose of this branch ofphilosophy, and cannot indeed, be dispensed with by the astronomer.For astronomers should not have absolute freedom to think up any-thing they please without reason; on the contrary, you should givecausas probabiles for your hypotheses which you propose as thetrue cause of the appearances, and thus establish in advance theprinciples of your astronomy in a higher science, namely physics ormetaphysics.

J. Kepler, Epitome Astronomiae Copernicanae, transl. N. Jardine,The Birth of History and Philosophy of Science, CUP (1984)

Impact against a mechanism composed of nearly rigid bodies is a feature of manypractical machines. These systems may include mechanisms where the relative velocitiesat joints are initially zero and finally must vanish or they can be another type of systemsuch as a gear train or an agglomerate of unconnected bodies where at each contact thenormal component of terminal relative velocity must be separating. These two classes ofmultibody impact problems - mechanisms and separate bodies that are touching - aredistinguished from analyses of two-body impacts by the addition of constraint equationsthat describe limitations on relative motion at each point of contact between bodies.These constraint equations express the linkage between separate elements. Books ondynamics typically analyze the impulsive response of systems composed of rigid bodieslinked by frictionless or nondissipative pinned joints. In these systems, pairs of bodiesare connected by a joint or hinge that imposes a constraint on the relative velocity at thepoint of connection. A common assumption is that during impact the relative velocity atjoints remains negligible; this assumption applies if the compliance of the joints is smallin comparison with the compliance at the point of external impact. Ordinarily, however,there is no reason for a point of external impact to be more compliant than all other pointsof contact within the system.

Any connection between bodies can be represented by a constraint equation - anequation that expresses the limitation on relative motion provided by the physical link. Oneclass of constraint equations represents pinned joints; these bilateral constraint equationsrequire that at the yth point of contact the relative velocity vanishes at initiation andtermination of impact: y/(0) = Vj(p/) = 0. A second class of constraint equationsrepresents contact where there is no physical connection; these unilateral constraint

173

174 8 / Impact on Assemblies of Rigid Elements

equations require that at separation the normal component of relative velocity at eachpoint of contact is positive. The latter form of constraint arises in systems where bodiesare in initial contact but not physically linked.

For both classes of constraints, the action due to an external impact propagates awayfrom the impact point. If the point of external impact is much more compliant than othercontact points, it is satisfactory to assume that the reaction at each point of constraintacts simultaneously with the contact force at the point of external impact. For all otherdistributions of contact compliance, however, it is necessary to return to discrete modelingof the local compliance of each contact region in order to follow the chain reactionthat transmits energy through the compliant elements of the system. Energy propagatesoutward from the point of external impact and travels through the connecting joints toadjacent bodies.

Irrespective of the form of constraint equations, convenient methods for analyzing thedynamics of multibody systems employ generalized coordinates which are associated withLagrange's equations of motion. Lagrangian methods are advantageous for multibody sys-tems because they eliminate any requirement to consider reaction forces arising at the con-straints. This chapter employs Lagrangian dynamics to analyze the behavior of systems ofcompact colliding bodies - compact or stocky bodies that have small structural compliancein comparison with the compliance at points of connection between bodies in the system.

8.1 Impact on a System of Rigid Bodies Connected by NoncompliantBilateral Constraints

Analysis of the dynamics of a mechanism composed of an assembly of rigidbodies that are connected by frictionless pinned joints (nondissipative, noncompliantbilateral constraints) is based on the following assumptions: (a) the time of contact isbrief, so that there is no change in configuration during impact; and (b) the reactionsat points of contact occur simultaneously. The latter assumption is appropriate if thecompliance of each joint and body within the system is small in comparison with thecontact compliance at any site of external impact.

8.1.1 Generalized Impulse and Equations of Motion

Let S be a set of n particles with the jth particle located at a position vectorTj, j = 1 , . . . ,«; each particle is subject to external impulse p7 applied at the instant ofimpact. The velocity V7 of the jth particle is the rate of change of the position vector,Yj(Pj) = drj/dt; the particle velocity is a function of the impulse p , . Suppose the particlevelocities are subject to constraints that are represented by 3n — m holonomic constraintequations; e.g., let there be a fixed distance between the jth and &th particles, so that(Vfc — \j) • (rk — Tj) = 0. Also there can be m — m nonintegrable (nonholonomic)

m

constraint equations ^asrqr + bs, s = 1 , . . . , m — m. Then the particle velocities canbe expressed in terms of generalized speeds qr, r = 1 , . . . , m, and time t\ i.e., V/ =Y/(tf i> <?2, • • •, #m, t), where m is the minimum number of variables required to definethe constrained motion of the system. The number m of generalized speeds qr equals thenumber of degrees of freedom of the system.

Virtual displacements 5r;- are a displacement field that is compatible with the dis-placement constraints. Similarly virtual velocities S\j are compatible with the velocity

8.1 / Impact on a System of Rigid Bodies . . . 175

constraints of the system. The virtual velocity of the yth particle can be expressed interms of generalized speeds qr as1

During impact on a system of rigid bodies the virtual differential of work 8(dW) done byexternal forces dpj/dt is used to define a differential of generalized impulse dUr:

8(dW) = J2T,J Jy = l r=\ ^r r=l

where dUr = ^dpj • (d\j/dqr).2 It is important to recognize that generalized active

forces Fr = ^(d^j/dt) • (d\j/dqr) are the only forces that contribute to the differentialof generalized impulse; other forces, which do not contribute, include any equal butopposite forces of interaction at rigid (i.e. noncompliant) constraints and external bodyforces or pressures which remain constant during impact.

A system of n particles connected by 3n — m velocity constraints has a kinetic energyT which is a scalar that varies with the applied impulse. This kinetic energy can beexpressed either in a global coordinate system or as a function of the generalized speeds:

-i n -i

T = -J2 Mj(Vj • \j) = -j=\ r=\ s=\

where My is the mass of the j th particle. The inertia matrix mrs for the constrained systemwith generalized speeds qr can be obtained from the expression for the kinetic energy ofthe constrained motion.

For a system subject to velocity constraints, the equations of motion in terms ofgeneralized speeds qr are obtained directly from the kinetic energy T and the differentialof generalized impulse dl\r.

For a system subject to a differential of generalized impulse dTlr and a set of3n — mvelocity constraints, the equations of motion in terms of m independent generalizedspeeds qr are obtained as

d^-=dUr. (8.4)dqr

PROOF For external impulse p7 acting on a set of n particles, the differential formof Newton's second law is expressed as

= d(M\) =

1 If the velocity of the j th particle is expressed as V, = \j{q\, qi, . . . , qm ,t), note the distinction betweenvirtual velocity 8\j and the differential of velocity d\j = f^Li {(d\j/dqr)Sqr + (d\j/dt)dt).

2 This formulation requires that the differential of impulse at each compliant contact dpj be includedin calculating the generalized impulse in a mechanism consisting of rigid bodies linked by compliantconstraints.


where the system of particles has mass M = J2 Mj and the center of mass of the sys-

tern has velocity V. Multiplying each side of the equation by the set of virtual velocitycoefficients d\j/dqr, we obtain

JT{ Jdqr J \dqrj j ^ ' ' \dqr) f^Vj \dqr

Since d\j/dqr depends solely on the configuration of the system and this does not changeduring the instant of impact, the differential of these virtual velocity coefficients mustvanish [d{d\j/dqr) = 0], giving

d = dUr. q.e.d.dqr

The expression d(dT/dqr) is termed a differential of generalized momentum?; it can beexpressed in terms of generalized speeds as d(dT/dqr) = mrs dqs.

Example 8.1 Six identical uniform rigid bars form a regular hexagon with frictionlesspinned joints connecting adjacent bars at each corner. The hexagon lies on a smoothhorizontal plane surface and is initially stationary. A transverse impulse pn acts at pointA located at the center of one bar; the impulse acts in a direction normal to the bar andtangent to the plane. For this impulsive load, find the speed of the center of mass of themechanism V and the speed VA of the loaded side immediately after the impulse; thusshow that VA/V = 20/11.

Solution Let each bar have mass Mj = M and length Lj=L, and let 0 be theangular speed of each side bar (symmetry reduces the problem to one with two degrees offreedom, so there are only two independent generalized speeds, q\ = V and q2 = L0).From (1.11) we obtain that the total kinetic energy T for this assembly of rigid bars isthe sum of the kinetic energies of the bars: T = \ J2(Mj% • V/ + Wj • I7 • u)j)\ forsymmetrical motion this hexagonal mechanism has a kinetic energy

T = ^M(6V2 + ^(L6>)2).

This symmetrical problem has two degrees of freedom. The impulse pn is applied atpoint A; this point has a velocity that can be expressed as YA = (V + L0/2)n.

Virtual velocity coefficients:

avA d\A 1= n, ^dV d(LO) 2

Generalized impulses:

1 PYl\ = pn - n = /?, n 2 = pn • - n = —.

3 For an impulsive load it is not useful to define a Lagrangian function L = T — U which incorporatesthe potential energy U of conservative active forces. These forces do no work during the instant thatan impulse is applied, so they do not affect Eq. (8.4).


Equations of motion:

d^=dUx =» V=l-M-lpoq\ 6

dq2 11Speed of point A after application of impulse:

:. VA/V = 20/11.

8.1.2 Equations of Motion Transformed to Normal and Tangential Coordinates

In the preceding example a specified impulse was applied to the system. Fortwo or more bodies that come into contact with a relative velocity at some contact point,additional relations are required to describe the properties of the contact region. In orderto apply laws of impact and friction that are related to normal or tangential componentsof relative velocity at the contact point, the equations of motion must be transformed tocomponents in the normal and tangential directions.

For a system of rigid bodies that are linked by nondissipative, noncompliant joints, letthe velocity at contact point C be separated into a component Ve in the tangent plane anda component V3n3 normal to the tangent plane; i.e., Vc = {Ve, V3}7'. For a differentialimpulse dp = {dpe, dp}T applied at C the equations of relative motion can be expressedas

Batlle (1993, 1996) has shown that

where [m] = mrs are generalized inertia coefficients and d\c/dqr are virtual velocitycoefficients for the system.

Impact is initiated when two systems come together with a relative velocity v =ve + v3n3 = Vc — Vc between the contact point C and the coincident point C'. Ifthere is slip at C (i.e., n3 x v ^ 0), then for dry friction with a coefficient of friction/x, Coulomb's law can be used to relate the tangential components of the contact forcedpe to the normal force dp; i.e., dpe = —\isdp, s = [(n3 x v) x n3]/|(n3 x v) x n3|.In this case there is a set of n -f- n' generalized speeds for the combined systems. Henceduring slip, the equations of relative motion at a point of external impact can be expressedas

dvjdp\_\h-n[b]s

Figure 8.1. Rigid compound pendulum at angle of inclination 6 when it strikes against aninelastic half space where the energetic coefficient of restitution is e* and the coefficient offriction is //.

Here the terms on the right side that come from [N] are inertia properties of the systemfor a point of external impact C.

The next two examples consider systems that collide against a massive half space; i.e.,the body B' has an indefinitely large mass. In this case M'~x ->• 0, so that the mass M'does not appear in the equations of relative motion for the contact point C.

Example 8.2 A compound pendulum with mass M pivots around a frictionless pin O,and the tip strikes a rough half space at contact point C. The position of C relative to O isr c = —r\Tk\ — r3n3, where the unit vector ni is parallel to the tangent plane and n3 is thecommon normal direction as shown in Fig. 8.1. The pendulum has radius of gyration kr

for O and at the contact point the energetic coefficient of restitution is e*. Find the ratioof terminal to incident angular velocities Of/Oo assuming that tangential compliance isnegligible and that friction at C is represented by Coulomb's law with a coefficient offriction \±.

Solution

Kinetic energy:

Velocity of C:

Vc = Vo + u; x r c = ——ii! —n 3.kr kr

The differential of impulse dp = dp xni +dp?> n3 at contact point C satisfies the Amontons-Coulomb law:

dp\ = ix dp 1—/xdp < dpi < fidp\ for I Vc • ni = 0 \ , where dp = dp3.

dpi = -ix dp)

The differential of the generalized impulse is

ndp3

, Vc m <0, Vc-n! > 0 *

The equations of motion are obtained from (8.4) after recognizing that the slip reverses indirection at impulse pc simultaneously with the transition from compression to restitution.After integration, one obtains the following generalized speed as a function of normalimpulse p:

ftfi(0) - (krM)-\rx + iir3)p, p < pc

~ \-{krM)-\rx - fir3)p, pc < p < p/

Notice that slip reversal requires \i < fi = rx/r3\ otherwise the pendulum sticks in thecompressed configuration.

Compression impulse pc from n3 • \c(Pc) = 0:

pc=kr(rl+iir3ylMql(0).

Work of normal impulse during compression W3(pc):

Work of normal impulse during restitution W3(pf) — W3(pc):

Energetic coefficient of restitution:

2 = W3(pf) - W3(pc) = r\ -— - 1

W3(pc) rf + firxr3 [_p*

Terminal impulse pf as function of angle of inclination 0 = cot-1(r3/ri):

Pf , , \r\ + \JLrxr3 \ 1 + fi cot 0— = 1 + e* -\ = 1 + e* — .pc y r\- iirxr3 y 1 - /x cot 0

Ratio of final to initial angular speed:

qi(Pf) _ /I - co t 0e00 qi(0) V l+cot(9

Figure 8.2 illustrates the ratio of angular speeds as a function of the eccentricity angle 0of the pendulum at impact and the energetic coefficient of restitution e*. The result showsthe effect of energy dissipated by friction even if the bodies are elastic. At small anglesof eccentricity 0 the work done by friction is large in comparison with the work done

1.0-

•o

cI2

- /

7 L

JM =

/

0

0.1

A = 0.

1

5

/ r / 6 7C/3

angle of inclination, 0

Figure 8.2. Rebound of compound pendulum as a function of configuration angle 0 forcoefficients of friction {i = 0.1 and 0.5.

by the normal component of contact force. Also for small 0 there is no rebound (i.e., thecontact sticks) if friction is sufficiently large (/A > /x = tan 0) (Stronge, 1991).

A more complex problem of impact at the tip of a double compound pendulum wasproposed by Kane and Levinson (1985). The double pendulum is swinging when the tipstrikes against a rough inelastic half space. This problem has generated renewed interestin analytical methods for representing impact with friction.

Example 8.3 Two identical uniform rods OB and BC are joined at ends B by a friction-less joint in order to form a double pendulum; the other end of OB is suspended from africtionless hinge at O as shown in Fig. 8.3. When the free end C of rod BC strikes againsta rough half space, the rods have angles of inclination from vertical denoted by 0\ and 02

and angular speeds 0\ and 02 respectively. Denote the coefficient of friction between Cand the half space by /z, and the energetic coefficient of restitution at the same locationby e*. Assume the motion is planar.

(a) Find an expression for the critical coefficient of friction jl that prevents slipreversal.

(b) For ii < jl find expressions for the angular speeds at separation.

Solution After defining generalized speeds q \energy of the system can be expressed as

= L0i, qi = L02/2, the kinetic

T =ML2,

M . .2

+ 022

• 2 ^

30x02 cos(<92 -

8.1 / Impact on a System of Rigid Bodies , 181

*• u, //AS

Figure 8.3. Double pendulum colliding against an inelastic half space where the energeticcoefficient of restitution is e* and the coefficient of friction is \x.

so that the generalized momenta are8T _ Mdqi 3dT Af .

^2 COS(02 - 01)]

qi cos(02 - 00].(a)

The velocity of the contact point Vc can be written, with s6t = sin 9t and cOi = cos Oi,

Vc = 0i(-Lc0ini + Lj0in3) +0 2 ( -Lc0 2 ni(b)

The differential of generalized impulse dUi for increment of impulsedp = dpi ni + dp3 n3 is

dnx = -dpicOi +dp3s0u dU2 = -2dpic62 + 2dp3s02.

Initially the slip is in direction ni, so Coulomb's law gives dpi = —\xdp 3 = —\xdpand (8.4) results in the equations of motion

c{e2 - 00\ = _3_ \licBi + s0x \ \

dq2/dp\ M l2/xc02 + 2^02 j \\bi\b2

After solving for the differentials and then integrating with initial conditions ^/(0), weobtain

ftfil = (<7i(0)J J_U2J l92(0)J ^ A cos(02 - (c)

where at impulse

- 3b2c(02 - 0i)]c0i + 2[4Z?2 - 3&ic(02 - 1

the tangential speed vanishes (Vc • iM = q\c9\ + 2q2c92 = 0), while at impulse

-[qi(0)s9i+2q2(0)s92]APc = [4*, - 3b2c(92 - 81)]s9l + 2[4b2 - 3bxc(92 -

(d)

(e)

Table 8.1. Results for Double Pendulum Striking Rough Half Space

Coeff. ofFriction,

0.00.20.50.7

Initial Velocity(rads"1)

em *2(0)

-0.1 -0.2-0.1 -0.2-0.1 -0.2-0.1 -0.2

Rel. Impulse(slip - 0),Ps/Pc

1.321.301.291.28

Rel. Impulse(separation),Pf/Pc

1.501.521.561.58

Final Velocity(rads"1)

0i(Pf)

-0.230-0.214-0.199-0.193

02(Pf)

+0.292+0.259+0.223+0.210

Final Dir.of Slip,

_

--Stick

Final NormalVelocity,V3(p/)/V3(0)

-0.50-0.42-0.33-0.28

Final Kin.Energy,

0.7070.6210.5520.527

Configuration gives a coefficient of friction for stick of /x = 0.62. Coefficient of restitution, e* = 0.5

the normal component of relative velocity vanishes (Vc • 113 = qis0\ + 2q2s02 = 0).If the contact point C slides in the positive direction, during compression the normalcomponent of impulse does work W^{pc) equal to

W3(pc) = {qiMsOx + 2q2(0)s02}pc2 s4-\

- 3b2c(02 - 0l)]s6l + 2[4b2 - 3blC(02 - e^OJ^

After initial sliding is brought to a halt, if sliding resumes, it occurs in direction — ni, sothat coefficients b\ and b2 transform to b\ and b2 where

= M I -

For impulse applied after slip is halted (ps i)c(6>2 - 0x)^ ~ Sc202 + 2c20x - 6c6xc62c(02 - 6X) ' ( g )

For some specific initial values,4 0\ =TT/9, 02 = TT/6, 0\ = — 0.1 rad s~\ 02 = — 0.2rad s"1, e* =0 .5 , Table 8.1 contains results for this double pendulum obtained with anenergetic coefficient of restitution at C.

8.2 Impact on a System of Rigid Bodies Connected byCompliant Constraints

Unilateral constraints require that at separation each pair of contact points havea normal component of relative velocity that is nonnegative, i.e. that finally the contactpoints move apart or remain in contact - they cannot interpenetrate. This constraint isappropriate to problems of granular flow where a colliding body may strike against a

4 For this case of a noncollinear configuration with friction, alternative definitions of the coefficient ofrestitution result in calculated values of T//TQ > 1.0.

8.2 / Impact on a System of Rigid Bodies Connected by Compliant Constraints 183

cluster of separate bodies or else against a body that is at rest against a wall. In each casethe bodies have no specified relative velocity during contact; at each contact point thereis a constraint on the relative velocity at separation only.

8.2.1 Comparison of Results from Alternative Analytical Approximationsfor Multibody Systems with Unilateral Constraints

Impulse-momentum relations have been a direct and effective route for analyz-ing the process of impact between two rigid bodies (Keller, 1986; Stronge, 1990). Formultibody systems, however, use of impulse-momentum relations implicitly incorporatesone of two assumptions - either (a) impulsive reactions occur simultaneously or (b) theyoccur sequentially. Alternative formulations based on sequential impacts have been pro-posed by Johnson (1976), Han and Gilmore (1993) and Adams (1997); on simultaneousimpacts, by Glockner and Pfeiffer (1995) and by Pereira and Nikravesh (1996). Here wewill demonstrate that neither the simultaneous nor the sequential approach gives resultsin agreement with a simple experiment on a multibody system where elastic bodies arelinked by compliant unilateral constraints.

When two bodies collide, contact pressures develop in a small area around the point ofinitial contact. These pressures are compatible with the local deformations of the bodies,and they are just sufficient to prevent interpenetration of the bodies. If the bodies arecomposed of sufficiently hard, rate-independent materials, the resultant force F(t) thatacts on each body has a component normal to the common tangent plane F(t) = F • n thatincreases while the colliding bodies outside the local contact region have a relative velocityof approach during a period of compression. During a subsequent period of restitutionthe colliding bodies are driven apart by strain energy stored in the contact region duringcompression. For these materials the transition from compression to restitution occurswhen the normal relative velocity across the contact region vanishes and the normalcomponent of contact force is a maximum (Fig. 2.2).

Hard bodies have deformations that remain localized around a small contact area, sothat the contact compliance is very small and the period of impact is extremely brief; theseconditions are necessary for changes in inertia properties during impact to be negligible.Despite the brief period of contact, it is important to recognize that during impact, changesin relative velocity across each contact region are a continuous function of either timeor impulse. The continuous nature of changes in relative velocity across the contactregions can be modeled most effectively by introducing an infinitesimal deformableparticle between each pair of contact points; in multicontact problems, although thecontact compliances are very small, they are not negligible. In particular, they are notnegligible in comparison with the compliance at other connections to adjacent bodies.With the artifice of an infinitesimal deformable particle separating each pair of contactpoints, bodies B and B' have coincident contact points C and C with normal components ofvelocity Vc(t) and V^it) respectively. These velocities vary continuously during impact asa function either of the time t or of the normal component of impulse, p(t) = / J n-F(0 dt.

The following comparisons of different approximations for impact relations have beenselected to bring out the most important features representing contact interactions duringmultibody impact. In that spirit we analyze the dynamic response to axial impact on aperiodic series of collinear bodies that are initially at rest and touching.

Figure 8.4. Direct impact on a uniform collinear system of n identical rigid bodies withunilateral constraints at n — 1 contact points.

Simultaneous Reactions at Multiple, Noncompliant Points of ContactConsider direct impact on a set of n identical bodies in a collinear configuration witha unique common normal direction x as shown in Fig. 8.4. Newton's second law ofmotion provides the relation for differential changes of velocity for body j in terms of theimpulse acting at each contact point. Let the normal component of velocity of body j beV), and denote the contact point between body j and body j + 1 as C/, consistent withthe multibody system shown in Fig. 8.4. After recognizing that the two bodies touchingat unilateral constraint Cj have equal but opposite impulses pj, and denoting the normalcomponent of relative velocity at contact point Cj by Vj = V)+i — Vj• = n • (V/+i — V,),the differential equations for the relative velocities across this series of contact points canbe written as (see Wittenburg, 1977)

= M~l$>jkdpk, j = 1, 2 , . . . , n - 1 (8.8)

where the inverse of the inertia matrix, <t>y£, generally depends on the directions ofcommon normals as well as the inertia properties of each body. For the present case ofparallel collinear collisions between identical bodies one obtains

2 - 1 0 - - 0 "1 2 - 1 - . . 00 - 1 2 - . . 0

0 0 0 - - - 2

(8.9)

The narrow bandwidth of this matrix results from each body being subject to a pair ofcollinear impulses only; in systems containing branches or loops the bandwidth is muchlarger.

Equation (8.8) can be integrated to obtain the changes in relative velocity as a functionof the set of impulses /?#:

(8.10)

To analyze collisions between two bodies, a characteristic normal impulse for compres-sion pc was defined as the impulse at the transition from compression to restitution, sothat v(pc) = 0. For multicontact point collisions the equivalent idea requires that eachcontact point have a simultaneous time of transition from compression to restitution. Thisgives

Pj(tc) = - (8.11)


where the inertia matrix

" n - 1 n-2 n-3 -" 1n - 2 2(n - 1) • • • 2

1 n-3

1 2 ••• n - \

The ratio between the terminal impulse Pj(tf) and the normal impulse Pj(tc) for com-pression at the yth contact point is given by the local coefficient of restitution

= pj(tf) - Pj(tc)

Pj(tc)

At different contacts this coefficient can differ, since it depends on material properties,the impact configuration, incident relative velocities, etc.

With incident relative velocities 17/(0) Eq. (8.10), (8.11) and (8.12) give

vj(p) = vj(0) - (1 + ekWjkQjvtf) (8.13)

e.g., the initial conditions Vj(0) = {—Vo, 0, 0,• • •, 0} r give terminal relative velocities

Vi(Pf)/v0 = -1 + 2n~\n - 1)(1 + ex) - n~\n - 2)(1 + e2)v2(Pf)/v0 = 0- n~\n - 1)(1 + e{) + 2n~\n - 2)(1 + e2) - ( ^ ) (1 + e3).

For example, if all collisions are elastic (ey = 1), these initial conditions yield a distribu-tion of terminal velocity Vj(pf)/V0 = 2n~l{\ - n/2, 1 , 1 , . . . , l } r .

Simple experiments on a periodic collinear system of identical elastic spheres in initialcontact (Newton's cradle) suggest that this solution is not representative of the dynamicsof multibody collisions (Johnson, 1976).5

Sequential Reactions at Multiple, Noncompliant Points of ContactIn multibody systems with compliant unilateral constraints an alternative method ofrelating the reaction impulses is to assume that they act sequentially, usually in order ofincreasing distance from the site of an external impact. This assumption is unequivocal aslong as the series of bodies is linked solely by single points of contact, but for any body incontact with three or more bodies (i.e. at a branch point), some other ad hoc assumptionregarding the order of impulses is required.

For the simple collinear system of identical bodies with equal masses Mt = M de-scribed in Fig. 8.4, each body in turn is accelerated by impulse from the preceding mov-ing body while the speed of the striking body is retarded. If all contacts are elastic andthe collision has incident velocity V/(0) = {Vo, 0, 0 , . . . , 0} r , this interaction hypothesisyields a set of terminal velocities Vj(pf)/ Vo = {0, 0, 0 , . . . , l } r . This result agrees withcrude observations of Newton's cradle. The question is whether the sequential method isgenerally applicable.

5 The name "Newton's cradle" seems to have evolved from Kerwin's (1972) description of a "Newtonmomentum-conservation apparatus". Analyses of impact against collinear series of spheres, however,began much earlier (for references see Chapman, 1960).

\ /

Figure 8.5. Typical elements in a uniform collinear system with bilinear springs represent-ing local elastic compliance at points of contact.

Note that both simultaneous and sequential solutions above satisfy conservation ofmomentum and conservation of energy; Le.,formultibody elastic collisions, conservationof momentum and energy are not sufficient to uniquely determine the solution.

A third, more detailed approach employed by Cundall and Strack (1979) and by Stronge(1994b) abandons impulse-momentum relations and develops contact relations directlyin terms of interaction forces. This approach obtains changes in relative velocity at contactpoints as continuous functions of time (or relative displacement), but has the disadvantageof requiring additional information about the contact geometry and material propertiesin order to represent the compliance of each local contact region.

Force-Acceleration Relations for Multibody Systems with Compliant Points of ContactThe compliance of hard bodies with small areas of contact results from local deformationaround the contact area. Irrespective of whether the bodies are elastic or inelastic, themagnitude of contact stresses rapidly decreases with increasing distance from the contactregion; thus strain energy is localized in a small neighborhood around contact area. Thesmall size of the region of large strain causes the compliance of this region to be smalland consequently the contact period to be very brief.

This small local compliance at each contact point can be modeled as either an elasticor an inelastic spring of infinitesimal length oriented normal to the surface. Outside thecontact region the bodies are assumed to be rigid. For this periodic collinear system,let the element spacing be L, the normal contact stiffness at each interface be K andthe infinitesimal displacement of the y'th element be u};, as shown in Fig. 8.5. A typicalelement has an equation of motion

j j j for Uj < Uj-\, OL>1 = K/MJ=K/M. (8 .14)

For a wave of slowly varying amplitude the displacement varies according to

where the wave number k is related to the wavelength X by k = 2TT/X, and / = V^T.Substitution into (8.14) gives an expression for the dispersion relation [see (7.27)]

co/coo = ±2 sin(kL/2), co/co0 < 2. (8.15)

Individual wavelength components of the wave propagate with phase velocity

co sin(kL/2) co0k . (TTL\CP = T= ±c°o j T n = ± sin — (8.16)k kL/2 n \ k J


rcut-offwave number

wave number 7T

Figure 8.6. Phase velocity cp and group velocity cg as functions of wave number k for aperiodic system of equal masses separated by bilinear compliant elements.

while energy propagates at the group velocity

dcoCg = = ±(JOQL COS (T) (8.17)

Figure 8.6 illustrates that this system has a cutoff wave number k = n/L, which gives alower bound for wavelength A. of a propagating disturbance: propagation occurs only ifk > 2L. This minimum compares with a value of roughly 3L measured from photoelasticpatterns in a collinear series of thin disks driven axially by a small explosive charge (Singh,Shukla and Zervas, 1996).

The limiting wave speed is obtained for waves with vanishingly small wave number k =0; this gives a group velocity cg = COQL that can be calculated for spherical elastic elementswith individual masses Mj = (4n/3)pR3. If the linear spring constant K represents contactstiffness, the work done during indentation to the yield limit 8Y, Eqs. (6.8) and (6.10a)for Hertzian compliance give an equivalent linear contact stiffness K that depends on theyield pressure #yF, viz. K = (4n/5)$YYR*. Thus in a collinear series of identical elasticspheres, the limiting wave speed is

For metal spheres, this speed is of the order of 1/10 the elastic wave speed of the medium.

High Frequency Waves, UJ >Depending on size and constitution of the colliding bodies, the contact force F{t) gen-erated by the external impact may contain frequency components larger than the cutofffrequency. In this system these high frequency components cannot propagate. What hap-pens to the energy contained in high frequency components of the collision force?

Consider a complex wave number k = kR + ik\, where kR = n/L gives continuitywith lower frequencies at the cutoff frequency co = 2COQ. The wave solution becomes

Uj = Ce-kljLei{nj-wt\ i = v ^ j = element number.

This steady state solution is a standing wave (nonpropagating) that decays exponentiallyin amplitude with increasing distance from the end, x = 0. The solution is termed anevanescent wave; each element has an oscillatory motion with displacement that is outof phase by n with the motion of adjacent elements.


-0.2

Figure 8.7. Velocity distribution at successive times in a collinear periodic system of rigidbodies separated by bilinear compliant elements; initially one end is struck by a singlecolliding element moving at speed Vo in the coaxial direction.

8.2.2 Numerical Simulation and Discussion of Multibody Impact

Figure 8.7 illustrates the spatial velocity distribution at successive times in a periodicarray of identical elements connected by springs which are linear in compression only; thebilinear springs provide no resistance to extension. As an initial disturbance propagatesthrough the system, it slowly decreases in amplitude and the pulse width broadens; i.e.,the wave is dispersive. Less obvious is the fact that each element near the impact endfinally separates from its neighbor with a small negative velocity - the amplitude ofthese residual velocities decreases almost exponentially with increasing distance fromthe external impact point. In Fig. 8.7 the curve with negative velocity represents thisresidual velocity distribution.

In a more limited collinear system of six touching spheres, the temporal variation invelocity for each element illustrated in Fig. 8.8a clearly shows both dispersion and theexponentially decreasing residual negative velocity. If the bilinear springs of the systemin Fig. 8.8a are replaced by nonlinear springs where the compressive force-deflectionrelation Fj = KS83/2 is obtained from the Hertz contact law, then the velocity distribu-tion as a function of time is shown in Fig. 8.8b. In comparison with similar results forbilinear springs, the nonlinear compliance relation results in a smaller rate of decrease inpeak amplitude and a smaller amplitude of residual negative velocity. Nevertheless, boththe bilinear and the nonlinear compliance relations result in dispersion and evanescence(Stronge, 1999).

Figure 8.9 illustrates the effect of two colliding spheres striking a chain of four identicalspheres with Hertzian compliance relations. Here waves of change in relative velocitypropagate in both directions away from the off center point of impact. If the chain is struckcoaxially by a pair of spheres with twice the modulus of the spheres in the chain, the finalvelocity distribution is more widely spread than that anticipated from experiments witha Newton cradle.

In systems of bodies connected by unilateral constraints, the forces are a reactionto relative displacements between adjacent bodies, and they develop from the relativedisplacements as a result of small local compression. Any analysis of the effect of theseforces generally requires consideration of infinitesimally small relative displacementswhich develop across contact regions. In general, analyses of multibody impact need toconsider local displacements at contact points although global displacements during the


*-0.1314

-0.2,

Figure 8.8. Variation in velocity for each ball in a collinear elastic system where a singleball strikes a set of five identical balls: (a) Linear interaction compliance and (b) Hertznonlinear interaction compliance.

i i i i i

time

Figure 8.9. Variation in velocity for each ball in collinear set of four balls struck collinearlyby two identical balls traveling at an initial speed Vo. Speeds calculated using Hertz nonlinearinteraction compliance.

contact period may be sufficiently small so that the configuration can be considered to betime-invariant.

The present formulation of equations of motion is for systems of rigid bodies connectedby compliant constraints; this analytical simulation uses two distinct scales for the effectof displacements. Infinitesimal displacements generate interaction forces at compliantconstraints, so it is necessary that they be included in order to represent the interactionforces that prevent interference. These infinitesimal displacements, however, have noeffect on the inertia or the kinetic energy T of the system. Thus the equations of motion(8.4) do not have terms arising from changes in the impact configuration during contact.

V^(t) V2(tJ V*(t)

Figure 8.10. Set of three spheres in collinear configuration; spheres are separated by bi-linear springs with differing contact stiffness.

8.2.3 Spatial Gradation of Normal Contact Stiffness KJ = %K

In an assembly of rigid bodies where there are multiple deformable points of contact,the simultaneity of reaction forces depends on the distributions of inertia and contactcompliance. By considering the dynamic response of a small system of three collinearspheres that are initially in contact at two contact points, the effect of gradation of contactstiffness will be investigated.

Consider the set of three identical spheres numbered 1-3 with equal masses Mj = Mas shown in Fig. 8.10. Let the contact points be numbered 1, 2 successively, and atthe contact points denote the normal relative displacement between bodies (indentationpositive) by fy = Uj — Uj+\. For a bilinear contact stiffness, the normal contact force Fj atthe jth contact point is Fj = -Kj8j if fy > 0 and Fj = 0 iffy < 0. Let KJ = XJ~1K> a n c i

recall that equal but opposite reaction forces act on adjacent bodies to derive equationsof relative motion for indentation fy at contact points 1 and 2,

during the period when S\ > 0, 82 > 0. These equations of relative motion give modalfrequencies cot for the first and second modes,

coi/coo = [(1 + x) ± (1 - x + X2)1/2]1 /2 (8.19)

Figure 8.11 shows these frequencies as a function of the stiffness gradient x and illustratesthat the ratio of characteristic frequencies coi/a)2 is a minimum near x = 1-

For an unstressed initial state Si(0) = 82(0) — 0 the modal solution for indentationscan be expressed as

\8x(t)\ \AX A2

where the /th eigenvector equals {1 x ~l ^ / } -If the first body has initial speed Vb whenit collides against two stationary balls, the initial conditions <5i(0) = Vo, 82(0) = 0 givethe constants

= 1 - —-) —\ £22/ a>\ M ?rAi' (8-21)

This solution has been used to evaluate the times of maximum compression at contactpoints 1 and 2 as shown in Fig. 8.12. For x < 1 (decreasing stiffness with increasing x)the times of maximum compression differ by more than a factor of 2; thus decreasing


1 2 3stiffness gradient

Figure 8.11. Characteristic frequencies for system of three masses as a function of stiffnessgradient x m a system with springs linear in compression.

oU)(/)Q)

I28Xo

«*-o

1 "

U2 max

J\ max

0 1 2 3stiffness gradient

Figure 8.12. Time of maximum compression at contacts Q and C2 as a function of stiffnessgradient x m a system of two spheres hit in the axial direction by a third sphere.

contact stiffness can be approximated as a sequence of individual collisions between pairsof bodies. On the other hand, for x > 3 (increasing stiffness with increasing x) the timesof maximum compression asymptotically approach the same limit; this case is closelyapproximated by simultaneous collisions at all points of contact.

The final velocities of the three spheres as functions of the stiffness ratio x are shownin Fig. 8.13. When x is very small or very large these terminal velocity distributionsapproach the results of sequential or simultaneous collision approximations, respectively.For uniform stiffness (x = 1), however, the velocity distributions are between these twolimits.

Example 8.4 A stack of two spherical balls has a common initial velocity — Von whencontact point Q on ball Bi strikes a massive barrier. The impact configuration is collinear

sequentialj impact

simultaneousimpact f

0.2 0.5 1 2stiffness ratio X

10

Figure 8.13. Terminal velocities after collinear impact of a sphere against two identicalspheres. Heavy lines are for spring stiffness given by Hertz contact law, F = KS83/2. Lightlines are for linear stiffness, F = KS.

V2(t)

(a)

Figure 8.14. Normal collinear impact of a quiescent stack of two dissimilar balls againstan immovable barrier: (a) initial velocities and (b) analytical model for compliant bodieswith bilinear contact stiffness.

as shown in Fig. 8.14. The mass of ball Bi is M, and that of ball B2 is M/a. Assumethat at contact point Ci ball Bi and the barrier are separated by an elastic element whosespring constant equals tc if the relative displacement is compressive (8\ < 0); similarly, atcontact C2 between balls Bi and B2 assume that there is an elastic element whose springconstant equals xK if the relative displacement at C2 is compressive (82 < 0). At bothcontact points the spring constant K = 0 if the elastic element is extended (8j > 0). Thebodies are assumed to be rigid outside of these elastic elements. Find the ratios V+/ Vb offinal to initial velocities for each of the balls as functions of the mass ratio a for stiflFnessratios x = 1/6, 1/2, 1.

SolutionLet the rates of change of relative displacements be defined as

The equations of relative motion for initial period where 8\ < 0, 82 < 0 are

1 K

The determinant of coefficients gives the natural (modal) frequencies,

2

1/2"

coo U - "A — r • v a + x+<*x)2,The normal mode solution satisfying initial conditions 8\{0) = 82(0) = 0 is

^i(f) = A\ sina>\t + A2 sin&>2^ ^1 = X~ (1 ~~ 1/^0)

^ ( 0 = Ai^2i sin&>i + A2^2 sin^2/, ^2 = X~ (l ~" ^2/^0)*The initial conditions 5i(0) = — VQ, 82(0) = 0 give

The relative displacements and velocities can be expressed as

f cooSi/Vo 1 _ —coo/coi |"l —CO1Q1/CO2&2] isi\co082/QiVoj ~ 1 - J2i/£22 L1 - ^ 1 / ^ 2 J {si

(O0S1/V0 ) = - 1 [1 - f iJ 1 - £2!/£22 L1 - 1

Double contact ceases at time ts when contact separates at either Q or C2 (S\ < 0 or<52 < 0).

The equation of motion after initial separation at Q , i.e. for t > ts and 5i < 0, is

82 + o)2^2 = 0, cb2 = (1 + a)x/c/M

with transition conditions ^ = 52(^), <5i5 = <5i(/5) and 82s = 82(ts), so that

82 = (82s/cb) sin o)(£ — f5) + 25 cos ct)(/ — O

<52 = 525 cos<5>(? — ts) — d)8 2s sino>(r — ts)

giving velocities

Vi = V{(ts) - (1 + a)~\82 - 82s\ V2 = V2(ts) + a(l + a)~\82 - 82s).These equations apply unless there is a second hit at contact point Cj, i.e. 8\ (t) = 0, t > ts,where

= (t- ts)8ls + (1 + a ) " 1 ^ * - t s - a)'1

-(1 + arAlternatively, the equation of motion after initial separation at C2, i.e. for t > ts and

82 < 0, is

8{ + a>l&i = 0

with transition conditions 8\s = 8i(ts), 8\s = 8i(ts) and 82s = 82(ts), so that

= (8is/coo)smcoo(t - ts) + 8U cos a)0(t - ts)

= S\s cosa)0(t - ts) - co08Xs sin^o(^ - ts)

giving velocities

Vx = Sis cosco0(t - ts) - Q)08xs sin<wo(* - ts), V2 = V2(ts).

A second hit occurs at contact point C2 if S2(t) = 0, t > ts, where

h = if - ts)V2s - co~l8u sinco0(t - ts) + Sis[l - cosco0(t - ts)].

Figure 8.15 illustrates these results for three values of the spring constant ratio x« Ineach case the curves can be compared with the result of rigid body assumptions of (a) si-multaneous impact V+/ Vo = V^~/ Vb = — 1 or (b) sequential impact. The final velocitiesresulting from an assumption of sequential rigid body impact are shown as the light lines on

oo

oooooooooo

•M o

JOOOOOOOOOOOOOOOOOOOOO

X=1

0 1 2 U 6 8mass ratio

oooo first separation at C2

•_+_+ second hit at Ci+++++ second hit at C2

x-x-x third hit at Ci

Figure 8.15. Final velocities of compliant spheres Bj and B2 with masses M and M/otrespectively, as functions of mass ratio a, for three ratios of local elastic stiffness. In eachcase multiple hits occur between Bi and B2 if ot < 1. The graph for rapidly decreasingstiffness (/ = 1/6) has light lines that show the result of assuming that at contact pointsbetween rigid bodies the collisions occur sequentially.

the graph for x = 1/6. If the mass ratio a > 1 (i.e. M2 < Mi), the assumption of simultane-ous impact provides a rough approximation of the elastic result if x > 1 (seeHarter, 1971).On the other hand, sequential rigid body impact is a good approximation if the stiffnessdecreases rapidly (x < 1/6) with increasing contact number starting from the impact site.

At small mass ratios a < 1 (i.e. M\ < M2) after initial separation at contact C2, bodyBi rebounds from second strikes against both the massive barrier and body B2. As adecreases towards zero the number of impacts between the bodies increases withoutbound - in the limit as a approaches zero, body Bi can be thought of as a small elasticparticle oscillating between the surfaces of two immovable bodies.

Eccentric Impact with Compliance at Multiple Points of ContactThe previous examples considered multibody impact in a collinear impact configuration.The following problem considers the impact of a rigid compound pendulum against anelastic half space when there is compliance at the pivot as well as at the contact with thehalf space; in this case the configuration is eccentric. Effects of friction are assumed tobe negligible.

Example 8.5 A compound pendulum pivots around a frictionless pivot Q before col-liding with an elastic half space at contact point C2. The pendulum has mass M anda radius of gyration kr about the center of mass. Denote the position vectors from thecenter of mass G to Ci and C2 by r' and r, respectively, as illustrated in Fig. 8.16. Thependulum has rotational velocity #n2, while the translational velocity of the center ofmass is denoted by V = V\U\ + V3113. Contact Ci is constrained to move parallel to thetangent plane. At d and C2 there are contact forces ¥[ = F{n\ + F3II3 and F2 = F3II3if friction is negligible. These forces arise from linear compliance at each contact; let arepresentative spring constant at Ci be denoted by K, and at C2 let the spring constant beXK in compression and 0 in extension. Find the ratio of incident to rebound speeds for thenormal component of relative velocity at C2 as a function of the stiffness ratio x • Assumethe displacements are small so that inertia properties do not change during contact.

C2

Figure 8.16. Collision of compound pendulum having linear elastic compliance at pivotQ as well as bilinear compliance at impact point C2.


Solution

Velocity constraint on vertical motion at Ci:

V3 = r[0.

Definition of generalized speeds that satisfy the constraint:

q\ = V\, q2 = kr0.

Velocities at contact points d and C2:

91 + ^kr

V2 = ( + f q 2 ) i + ^kr J • kr

Differentials of generalized impulse, dUj = (d Yi/dqj)-Ff dt+(dY2/dqj)'Fdt:

dU2 = J-F[dt+ 1A

l F3dt. (a)

Kinetic energy:

Generalized momenta:

dT dT ( r[2

Relations (a) and (b) give

At each contact a component of relative velocity is defined for each component ofactive force (positive for separation):

&i = - V i • iij = -qx - ^q2kr

r'-n ( d )

82 = V2 • n3 = ——-q 2.kr

The contact forces related to relative displacements are

82 > 0 ( ,( C )

Substituting (d) and (e) into (c) results in

\ S i \ , 2

where bij are configuration parameters, b\\ = 1 +r^/(rf2 + k2), b\2 = r^(r[ —k2), b22 = (rj — r\)2/{rf2 + k2) Equation (f) has characteristic frequencies

a>i I b\\ + xîcoo [ 2

Modal solution from substitution into (f) results in

8\ = A\ sinco\t + A2 sina)2t + A3 cosa)\t + A4 cos co2t

82 = A\Q\ sincoit + A2^2 smco2t + A3Q1 cosco\t + A^2cosco2t

with 2i = ( x f t n ) " 1 ^ ! — ^1/^0) ' ^2 = (Xî2)~1(^n —^2/^0)- When impact initiatesat time ts, the transition conditions6 8\(ts) = 8\s, 82(ts) = 0, <51 (^) = 8\s, 82(ts) = 82sgive relative displacements and relative velocities at Q and C2,

81 = (Q2 - ^ I ) " 1 {coîQrfis - 82s) sin^Cr - ts) - co

— ts) + 8is[Q2coscoi(t — ts) — Qi cosco2(t — ts)]}

82 = (Q2 - Q{)-1 [Q)^1Q1(Q2SIS ~ S2s)sincox(t - ts)

l — 82s) sin co2(t — ts)

- ts) - cosco2(t - ts)]}

is - 82s)cosco{(t - ts)

- 82s)cosco2(t - ts) - 8is[coiQ2 sinco\(t - ts)

— ts)]}

82 = (£22 - Q{yl {Qi(Q2Sls - 82s)cosco{(t - ts)

- 82s)cosco2(t - ts)

cosco\(t — ts) — a>2 cosco2(t — ts)]}.

The preceding equations apply so long as 82 < 0.Initial contact at C2 ceases at time tr when the state of the system is 8 \{tr) = 8\r, 82(tr) =

0, 8\(tr) = 8\r, 82(tr) = 82r. The subsequent motion with F3 = 0 must be checked todetermine if there is a second impact. With F3 = 0 the system reduces to a single DOF witha characteristic frequency cl) = &>o\/î2- Together with the initial conditions this gives7

81 = 8\r coscl)(t — tr) + (b~l8\r sma)(t — tr)

82 = (t-tr)b^l[bl28lr+bu82r]

\l\8\r\\ - coscb(t - tr)] - db~l8irsmco{t - tr))

— tr) + 8\r cos co(t — tr)

82 = b^[bi28ir +bn82r] +bi2b^{a>8ir sincb(t - tr) - 8ircosco(t - tr)}.6 This general form of initial conditions is required when there are a sequence of impacts at C2; for the

first impact, ts = 0, <$i(0) = 0, 5i(0) = 0.7 The equations of motion presented here have assumed any changes in configuration are infinitesimal.

These solutions consider velocity changes during impact only and not during the subsequent phase ofmotion involving rotation of the pendulum.


0.1 0.5 1 2stiffness gradient, X

10

Figure 8.17. Ratio of rebound to incident normal relative velocity — V3 (tf)/ V3 (0) at impactpoint C2 as a function of stiffness ratio x f° r a compound pendulum with 6 = n/4.

For a specific example of a compound pendulum composed of a slender uniform barand assuming that the bar is at an angle 0 from n3 when it collides with a massive elasticbody at C2, one obtains the inertia coefficients b\\ = 1 + (3 cos2 0)/(l + 3 sin2 0), bn =(6 sin 0 cos 0)/(l + 3 sin2 6>), and b22 = (12 sin2 0)/(l + 3 sin2 0).

The normal relative velocity at C2 as a function of stiffness ratio x is illustrated inFig. 8.17 for 0 = n/4. For stiffness ratios that are either very large or small comparedwith unity, the magnitudes of the incident and the rebound speed are almost the same.For ratios near unity, however, the impact forces have transformed a substantial part ofthe energy into a transverse translational mode; e.g., for x = 0-78 at separation we haveS(tf) = 0, since the rotational velocity 9{tf) = 0. In this elastic system no energy isdissipated at either contact point, but an eccentric impact configuration together with acompliant support can transform the mode of motion and thereby transfer energy intoadjacent contacts.

The previous example demonstrates that if contact compliance is considered and theimpact configuration is noncollinear, the energetic and kinematic coefficients of restitutionare not equivalent, irrespective of friction.

8.2.4 Applicability of Simultaneous Impact Assumption

In a system where rigid bodies are linked by several compliant contacts, energypropagates away from any point of external impact as a dispersive wave. This wave travelsat a speed cg that depends on inertia, the contact arrangement and the local compliance ateach contact. Since the points of contact are ordinarily much more compliant than the bodybetween them, it is contact compliance that determines the speed of propagation for hardbodies that are compact (stocky) in shape. Calculations based on assuming that externalimpact at one contact point results in a sequence of independent collisions at successivecontacts can give a good approximation only if the system has a series arrangement and thespeed of propagation cg decreases with increasing distance from the impact point. Calcu-lations based on simultaneity of collisions give a good approximation only if the speed ofpropagation cg has a substantial increasing gradient with distance from the contact point.

Problems 199

PROBLEMS

8.1 For a uniform slender rigid bar of length L and mass M with velocity V,- at the i thend, i = l,2, derive the following expression for the kinetic energy: T = (M/6) x(Vi • Vi + Vi • V2 + V2 • V2) (Bahar, 1994).

8.2 A set of three uniform bars of equal length L and mass M are joined end to endby two frictionless joints. The bars are collinear and at rest with the ends numberedsequentially 1, 2, 3, 4 beginning from one end. The bars are stationary until timet = 0 when an impulse p acts in the transverse direction at point 1.

r kVl kV4

(a) Identify the number of degrees of freedom, and show that the equations of motioncan be expressed as

2100

1410

0141

0012

dv\dV2dvsdV4

6dp000

(Note that axial velocity u\ is an ignorable coordinate - it is a constant.)(b) Solve for the transverse velocities vf immediately following the impulse, and

show that your solution gives a final translational momentum for the systemequal to p.

(c) Suppose a sphere with mass M' = aM is traveling in a direction transverse tothe bar and moving with speed Vo'. If the sphere strikes the side of the linkageat end 1 and the coefficient of restitution is e*, find the terminal impulse pf atseparation.

8.3 Two uniform slender rigid bars, each with mass M and length L, are connectedby a frictionless joint at one end of each bar. The bars are mutually perpendicularand lying on a smooth level table when an axial impulse p is applied to the endof one bar. Show that immediately thereafter, the other bar has an angular velocity00$ = 2p/ML.

8.4 Two uniform slender rigid rods, each with length L and mass M, are connected by africtionless joint at an end of each bar. Initially the assembly lies at rest on a smoothsurface with an angle 6 between the axes of the rods. An impulse p acts at the endof one rod in a direction inclined by an angle 0 to the axis.

kV\

u\

(a) Identify the number of degrees of freedom, and show that the kinetic energy canbe expressed as T = (M/6)[6u2

l+vf + ViV2+4vj + 3(v2c0-uis0)Ld-\-L202l(b) Write equations of motion for the response.

8.5 Let AC be a uniform rigid rod with mass M and length 2L; the rod has a uniformtransverse velocity Vbn3 when end C strikes against an inelastic stop at a normalangle of obliquity. The stop has coefficient of restitution e*. For the free end of therod A find the velocity v£ immediately after impact. Show that the ratio of final toinitial kinetic energy equals Tf/ To = (3 + el)/4.

8.6 A uniform rigid bar of mass M and length 3L lies centered across two elastic railsthat are separated by distance L before a sphere with mass M/a strikes the bartransversely at a distance k from the center. The center of the bar has radius ofgyration kr. Suppose at contact point C\ between the bar and a rail the elasticstiffness is K, while at C\, where the sphere strikes the rail, the stiffness is xK-Obtain coefficients b\\, b\2, b22 if the equations for relative displacement at C\ andC2 are expressed as

bn b22X J \S2\8.7 In example 8.3 of a double compound pendulum striking an inelastic half space

(Sect. 8.1.2), for a coefficient of friction /x = 0.2 and coefficient of restitution e* = 0.5,obtain the impulse for compression pc, the work done by the normal contact forceduring compression W?,(pc) and the total work of both friction and normal force dur-ing contact W\(pf), W^{pf). The coefficient of friction fi = 0.2 is relatively small;explain why in this case such a large part of the initial kinetic energy To is dissipatedby friction during contact.

8.8 A collinear stack of two elastic balls B i and B2 with masses M and M/a respectivelyis illustrated in Fig. 8.14(a) an instant before Bi strikes against an elastic half spaceat contact point C\. Before impact the balls have a common velocity — Vbn. Assumesequential impacts, first at C\ and secondly at C2.(a) Show that the final velocity of the balls can be expressed as

V+ a - 3 V+ 3a - 1Vo a + 1' Vo a + 1

V,+ -a + 3 V9+ 3a - 1

3 <a

<aff

Sketch these relations, and explain why the sign of V^/ Vo changes at a = 3.(b) Consider a < 1, and obtain relations for the velocities similar to those above.

Also find the range of values of mass ratio a wherein these relations are valid.8.9 Perform an experiment, dropping a stack of two balls onto a hard surface, and

make rough measurements of the maximum rebound height of each ball. Do thisexperiment with two identical balls (tennis balls or basketballs) and then with twodissimilar balls (tennis ball and basketball), first with one on the top and then theother. Explain the results, and comment on the applicability of the assumption ofsequential impacts. Some of these questions have been addressed by Kerwin (1972),Newby (1984) and Spradley (1987).

CHAPTER 9

Collision against Flexible Structures

We are all agreed that your theory is crazy. The question which di-vides us is whether it is crazy enough to have a chance of beingcorrect. My own feeling is that it is not crazy enough.

Niels Bohr, alleged comment at close of seminarwhere Pauli had presented a new theory

Problems of impact against a flexible structure or of a collision in a system thatcontains flexible structures will involve dynamic deformation of the structure in additionto local deformation of the contact region. In comparison with impact between compactbodies, global deformations in the structure generally prolong the contact period, reducethe maximum contact force and transfer significant energy into structural vibrations.Structural deformations also can result in the more complex interactions that are associatedwith multiple degree of freedom dynamic systems, e.g. repetitive impacts or chatteringat contacts of the type introduced in Chapter 8.

9.1 Free Vibration of Slender Elastic Bodies

9.1.1 Free Vibration of a Uniform Beam

Transverse displacements w(x, t) of a beam with mass per unit length pA(x)and cross-sectional bending stiffness EI(x) are represented by an equation of motion1

If the beam is uniform, this linear, fourth order partial differential equation can be ex-pressed as

d2w 2d4w 2 Eldt2

It has a separable solution

w(x,t) = X(x)(A\ cosa)0t + A2 si

1 This equation for a beam neglects rotary inertia and the effect of transverse shear; thus, it applies onlyfor wavelengths that are long in comparison with cross-sectional dimensions of the beam.

201

202 9 / Collision against Flexible Structures

where &>0 is the angular frequency of free vibration. Substitution into (9.1) gives

o = xdx4 y2

This fourth order equation has a solution with repeated roots. The solution can be ex-pressed in terms of two pairs of trigonometric functions that are respectively symmetricand antisymmetric,

X(x) = C\(coskx + cosh/:*) + C2(coskx — cosh &JC)+ sinhA:^) + C^sinkx — sinh &jt) (9.2)

where k = V<Wx is the wave number.

9.1.2 Eigenfunctions of a Uniform Beam with Clamped Ends

Solutions of the form (9.2) with coefficients that satisfy the boundary conditionsfor the beam are known as eigenfunctions; these depend on the end constraints (boundaryconditions) appropriate to a particular case. For a beam of length L with clamped (i.e.encastre) ends the boundary conditions are

X(0) = X(L) = 0, dX(0)/dx = dX(L)/dx = 0. (9.3)

With clamped boundary conditions (9.3), the constants C\, C2, C3, C4 are related by

C2 cos kL — cosh kL sin kL — sinh kLC\ = C3 = 0, — = = .

C4 sin kL + sinh kL cos kL — cosh kLThe last equality gives an equation for the eigenvalue k,

cos kL cosh kL= 1. (9.4)

The first five eigenvalues are listed in Table 9.1, while the associated mode shapesare sketched in Fig. 9.1b. The eigenvalues are closely approximated by kjL % (j +l/2);r, 7 = 1,2, . . . , 00.

The jth eigenfunction (i.e. natural mode of vibration) for a uniform bar with both endsclamped can be expressed as the sum of an antisymmetric and a symmetric function of kjX,

. . sinkjX — sinhkjX cosk,x — coshk;xXj = Cj (sin kj L - sinh kjL)] J J J J

sin kj L — sinh kj L cos kj L — cosh kj L(9.5)

(a) (b)

Figure 9.1. (a) Differential element used to derive equation of motion, and (b) first, second,and third mode shapes for a beam clamped at both ends.

9.1 / Free Vibration of Slender Elastic Bodies 203

where the constant Cj can be chosen in order that eigenfunctions are normalized; i.e.

1= f XjXjdx. (9.6a)Jo

Alternatively, the constant Cj can be chosen such that the maximum amplitude Xj |max isunity,

l=XjUx. (9.6b)

The circular frequency COJ for the y'th mode is obtained from the principle of virtual work,which gives2

CO2 CL rl2Y rl2Y CL

°± \ iAllAldx^ I XjXjdx = k4j.

Y2 Jo dx2 dx2 Jo J J J(9.7)

For the linear equation (9.1), the transverse displacements are obtained by superpositionof the response in independent modes,

oo

w(x, t) = ^2,Xj{A\j cos oo jt + A2j- sincojt) (9.8)7=0

where the coefficients A\j, A2j are obtained from the initial conditions.

9.1.3 Rayleigh-Ritz Mode Approximation

When only a few modes are required to represent the structural response, a setof shape functions Xj(x) that approximate the mode shapes are sufficiently accurate togive a good estimate of modal frequencies.

For a structural element with uniform properties, the functions employed to approxi-mate the deflected shape must be continuous functions that satisfy the natural boundaryconditions. Furthermore, it simplifies the analysis if the set of functions which approx-imate distinct modes are orthogonal so that the frequency equations are decoupled. Letthe approximation be expressed as

oo

w(x, t) = ] P AjXj sin(cojt - (pj) (9.9)j=o

with phase angle <pj and amplitude Aj. This gives a partial kinetic energy 7) for the jthapproximate mode,

x l fL A 2, w [AjCDjCOS(CDjt-<t)j)]2 fL 2

Tj = - pAwfa, t)dx = — L-L -1 J-— / pAX](x)dx.^ Jo ^ Jo

The partial kinetic energy associated with the jth mode is a maximum when the modedeflection is a maximum,

z Jo

L

pAX2(x)dx. (9.10)

2 The eigenfunctions are orthogonal; i.e., 0 = /QL XjXk dx, j / k. Hence 0 = /QZ\dA[Xj/fdx4)Xk dx =f0

L(d2Xj/dx2)(d2Xk/dx2)dx if j ±k.


Table 9.1. Modal Eigenvalues and Frequencies for Uniform Barwith Clamped Ends

Modeno.

12345

Wave number

kjL(rad)

4.7307.853

10.99614.13717.279

C2/C4

-0.98-1.00-1.00-1.00-1.00

Modal

Exact

22.3761.67

120.91199.85298.56

Frequency

RayleighApprox.

22.8078.98---

L2COJ/Y

Single DOFApprox.

22.80----

Similarly, associated with mode Xj there is a potential Uj(t), and for a beam in bending,this part of the potential energy is written as

where curvature d2 w /dx2 is related to the bending moment M(x ,t)byM = EI(d2w /dx2).The maximum value of this potential occurs when the modal displacement is maximum,

( 9 - n )

In a conservative system the maximum values of kinetic and potential energies are equal;i.e. for any mode, Uj |max = Tj |max. This gives an equation for the modal frequency COJ,

H'Example 9.1 A uniform beam of length L, with Young's modulus E and cross-sectionalsecond moment of area Is has both ends clamped. This beam satisfies the equation of mo-tion (9.1) and boundary conditions (9.3). Estimate the frequencies of the first symmetricand antisymmetric modes.

Solution Mode approximations (not normalized):

luxX\ — 1 — cos

2nx 1 ATZXX2 = sin sin ,

symmetric

antisymmetric.

It is worth noting that in addition to satisfying the boundary conditions, these functions

9.1 / Free Vibration of Slender Elastic Bodies 205

are orthogonal. Substitution of the mode approximations into (9.12) and integration gives

L2cox/y = 22.80

L2a)2/y = 78.98.

In Table 9.1 these approximations are compared with exact values. With these approxi-mating shape functions, Rayleigh's method gives errors of 2% for the frequency of thelowest mode and 28% for the frequency of the second mode. In general the approximatefrequency is always in excess of the exact value because the mode approximation rep-resents a variant of the actual deformation mode shape - a variant that is constrained tosatisfy the specified displacement function.

9.1.4 Single Degree of Freedom Approximation

If the mass of the colliding body, M', is small in comparison with the mass ofthe structure, M, a single degree of freedom analysis for the dynamic response can give auseful estimate of the structural response to impact. The single degree of freedom approxi-mation essentially is an approximation of the mode shape X(x). In this approximation, thestructure displacement w(x, t) is assumed to be a separable function, w(x, t) = qo(t)X(x).For the dynamic model shown in Fig. 9.2, the equivalent spring stiffness k is obtainedfrom the strain energy at maximum deflection with the proviso that the shape functionX(x) satisfies the normalization condition (9.6b). For example, if the structure is a uniformbeam in bending, this gives

X\2

(9.13)

By normalizing the mode shape such that the maximum displacement is unity; i.e.^Uax = 1, the equivalent stiffness is obtained as

(9.14a)

Figure 9.2. Single degree of freedom model of a compact, perfectly plastic body with massM' striking a flexible system represented by effective mass M and equivalent stiffness ic.


Similarly, an equivalent mass M is obtained from the maximum kinetic energy in themode:

CL

M= / pAX2dx. (9.14b)Jo

These parameters give an approximate circular frequency cb2 = k/M.

Example 9.2 A uniform beam of length L and mass M = pAL is clamped at both ends.For the mode approximation X = 0.5[l — cos(27r;c/L)] find the equivalent mass M andthe modal frequency cb for free vibrations of the beam.

Solution Let the transverse displacement w(x, t) = qo(t)X(x) = O.5qo(t)[l —COS(27TJC/L)]. The kinetic energy T of the mode approximation is obtained as

T=l- I pA[q0X(x)]2dx=^-Mq2.2 Jo 16

Since T = 0.5Mql and M = pAL, this gives an equivalent mass M = 3M/S for a uniformclamped-clamped beam. The maximum potential energy U for this mode approximationis

The equivalent single degree of freedom system (Fig. 9.2) has an equivalent spring stiff-ness K obtained from U = 0.5/c^; hence, k = 2JT4EI/L3. This gives a frequency cb forthe mode approximation,

Since for this example the mode approximation is geometrically similar to the deformationprofile used in the Rayleigh mode approximation (Ex. 9.1), the frequencies obtained bythe single degree of freedom and the Rayleigh method are identical.

9.2 Transverse Impact on an Elastic Beam

For transverse impact on a flexible structure, the analytical method to be em-ployed depends on the mass of the colliding body in comparison with the mass of thestructure, the location of the impact in relation to boundaries or other points where dis-placements of the structure must satisfy boundary conditions, and the deformability ofthe contact region in comparison with the structural compliance. These factors determinethe number of structural modes which significantly affect the structural response andconsequently the duration of the period of contact. If the structure is very stiff in com-parison with the contact region, the results of Chapter 5 are little affected by structuraldeformations. On the other hand, if the structure is compliant in comparison with thecontact region, structural vibrations play a major part in the response to impact.

9.2 / Transverse Impact on an Elastic Beam 207

9.2.1 Forced Vibration of a Uniform Beam

For a uniform beam in bending that is acted on by a distributed transverse forcef(x,t) the partial differential equation for transverse displacement w(x, t) is written as

Sw + w-*?-A separable solution w(x, t) = q(t)X(x) can be obtained in terms of the eigenfunctionsXj(x) obtained for free vibration where f(x, t) — 0. For forced vibration, the principleof virtual work and a set of orthogonal eigenfunctions Xj gives

Recalling from (9.7) that /0L (d2Xj/dx2) dx = k* /QL X2 dx, the previous equation yields

an ordinary differential equation for the time-dependent function qj(t) associated witheach eigenfunction. If in addition the force acts at point XQ only - i.e., f(x, t) = F(t)8(xo),where 8(x) is the Dirac delta function - then the differential equation becomes

, ? n 4. dlq> - F{t) XJ(Xo) ,?-vH* ro 17^CO -q j ~T~ — — 7 , CO j — y K i. \zf.vl)

df pA f0LX2j(x)dx

This differential equation is solved by convolution of the impulse response function. Fora transient force applied to an initially quiescent structure, it has a solution

= Xj(x0)fiF(t')sina>j(t-t')df ( 9 i g )

Finally, the partial displacements from the eigenfunctions are summed to obtain thedistribution of bar displacement in response to the applied force F(t),

^ XjXj(xo) /o F(t')sincoj(t - t')dtf

w(x, t) = J J - Jo — - - / Q 1O^

9.2.2 Impact of a Perfectly Plastic Missile

If a ID structure (e.g. a beam) is struck transversely by a perfectly plastic missileat a point located at x$ and local deformation (indentation) in the contact region is negli-gible, then during the contact period the missile and the point of contact on the beam willundergo the same acceleration. Let the missile with mass M' initially be traveling in adirection normal to the surface of the structure with speed VQ. After initiation of contact,the contact force F(t) that acts on the beam is related to the rate of change of momentumof the missile, F = —M fdV'/dt. If at the contact point the local indentation of bothmissile and beam are negligible, then the transverse acceleration of the missile equalsthe transverse acceleration of the contact point on the beam, F = —M'X(xo)d 2q/dt2.Consequently (9.17) can be written as

( 9 . 2 0 )

tfxjdx

which gives qj = Aj sin cbjt where cbj is a modified modal frequency that takes intoaccount the effect of additional mass from the missile resting on the beam during contact,

For the beam with an attached mass M' the equation of motion for modal response has asolution

q} = Aj sin(Q)jt + 0,) .

The beam displacement and velocity of this modified system can be expressed as

00

jCbj cos(a)jt + 4>j) Xj(x). (9.22b)7 = 1

Initial conditions are used to obtain the coefficients for amplitude Aj and phase angle4>j. Suppose an initially quiescent body is struck by a rigid missile of mass M' and beforeit strikes the missile is traveling with speed VQ\ this gives initial conditions that can be rep-resented by w(x, 0) = 0, w(x, 0) = M' VQ8(XO). Multiplying each side of the expressionsabove by the mode shape Xj and then integrating over the volume of the structure gives

fL _ fL

0 = Aj sin</)j, / w(x, 0)Xj(x)dx = AjCbj coscpj / X)dx.Jo Jo

The coefficients that satisfy conservation of linear momentum at impact are as follows:

&7-Xjdx

Example 9.3 A uniform elastic beam of length L, cross-sectional area A, elastic modulusE, and second moment of area / is clamped at both ends. The beam is initially at restbefore it is struck by a rigid missile of mass M' that is traveling with speed VQ in a directionperpendicular to the axis of the beam. Using the two mode Rayleigh-Ritz approximationobtained in Ex. 9.1, find the response of the beam to impact for impact points xo/L = 0.25and 0.5; in each case obtain an estimate of the time of separation.

Solution Mode approximations obtained in Ex. 9.1:

2nxX\ = 1 — cos , symmetric

2TTX 1 4nxA 2 = sin sin , antisymmetric.

LJ A Li

Recall from Ex. 9.1 that o)\ = 22.8y/L2 and co2 = 78.98y/L2, so that the modified modal


Table 9.2. Modal Frequencies and Amplitudes as Functions of Mass Ratio M'/Mfor a Uniform Clamped-Clamped Beam Hit at XQ/L by a Rigid Missile

xo/L = 0.25 xo/L = 0.5

L2(b\M'/M y

AXL2 L2Q)2 L2cb\ L2cb2

YVo'

0.11.0

10.0

22.117.78.2

0.0030.0380.808

75.355.823.8

0.0010.0180.420

21.414.96.0

0.0060.0892.211

79.079.079.0

000

frequencies obtained from (9.21) are

22.8y/L2

2MfXl(x0)1 + 3M

O)2 =78.98y/L2

1 + MThe amplitude coefficients Aj are obtained as

AXL2 2M'Xl(x0) I , 2M'Xx(xo)3M 1 + 22.8

A2L2 M'X2(x0) I , M'X2(x0)'I H — xM V " M 78.98

Amplitudes calculated for various mass ratios M'/M are listed in Table 9.2.

Unless the colliding mass is very large in comparison with the mass of the structure,the analysis in this section gives too long a period of contact and thus underestimatesthe maximum contact force. Usually the structural compliance will be larger than thecompliance at the contact point, so that the period of contact is controlled by local contactrather than structural compliance.

9.2.3 Effect of Local Compliance in Structural Response to Impact

In general the compliance of the structure prolongs the contact duration andreduces the maximum force in any impact. The magnitude of these effects depends onthe difference between the contact period for local deformations and the period of thefundamental mode of structural vibration. To illustrate these interactions, consider auniform elastic beam with length L and cross-sectional second moment of area / that iscomposed of material with Young's modulus E. The beam is struck by a colliding missileof mass M' that is initially traveling in a direction transverse to the axis of the beam. Themissile is collinear. At the impact point x0 there is a contact force F(t), and this resultsin flexural deflection of the beam w(x0, t), for which Eq. (9.19) gives

,0 =X2(x0) /J F(t')sinojj(t - t')dt'~^~ fQ

LX2(x)dx '

Figure 9.3. Local indentation 8(t) = u(t) — w(x$, t) during collision between a sphere andan initially stationary beam or plate. The displacements of contact points on the collidingmass u(t) and beam w(jto, t) do not include displacement due to local deformation.

Simultaneously, an equal but opposite contact force is decelerating the colliding body.Here we consider that the contact force acts in a direction normal to the initial contactsurface and has a line of action passing through the center of mass of the colliding body;consequently, the colliding body is decelerated from an impact speed Vo, but it suffers norotation. During contact the center of mass of the colliding body undergoes a displacementu(t) where

u(t)=Vot-^- I (t-tf)F{tf)dtr.fJo

The combined local indentation of missile and beam is represented by 8(t) = u(t) — w(t)as shown in Fig. 9.3. This local indentation is related to the contact force through aforce-indentation relation that depends on the geometry of the contacting surfaces andthe magnitude of the indentation:

8=K~

8 = K~lF, 8<8Y,

linear approximation (9.24a)

cylindrical elastic contact surfaces (9.24b)

spherical elastic contact surfaces. (9.24c)

With any particular force-indentation relation, compatibility of displacements at xo givesan equation for the contact force F(t) acting on the beam,

= *V-^7 f\t-tf)F(tr)dtr

-EI

X2j(x0) /0' F(f') sin(Oj(t - t')dt'

^ 7 f0LX*dx '

(9.25)

This integral equation for the contact force F{t) was obtained by Timoshenko (1913) whoused quadrature to obtain a numerical solution to the equation for particular cases of spher-ical contact; i.e., he discretized the integrals and solved for F at each successive time step.

Lennertz (1937) developed an approximate solution to the equation based on thestructural response of the lowest mode of vibration only. This method was improved andextended by E.H. Lee (1940). The method gives reasonable results if the contact period(Eq. 6.31) is short in comparison with the modal period of vibration 2n/co\.

Lee's method assumes that during contact the reaction force is a pulse that variessinusoidally with time. The pulse has period Q and amplitude Fc that are obtained fromstructural response of the compliant system, i.e. F(t) & Fc sin£2r, 0 < t < TT/Q. Aftersubstitution into (9.26) and integration, the indentation 8(t) is obtained as

Fc Fc [ Qsma)\t — co\\\(o) [ Q2-co

Ml(x0) / 2 2= pA /Jo

where M\(xo) is an equivalent mass for a beam that is hit at impact point JC0.3 Because thecontact period n/ Q is short in comparison with the period of the fundamental mode of vi-bration, the function sin co\ t can be approximated by sin a)\t^a)\t while the colliding mis-sile is in contact. After substituting this approximation into the equation above, we have

At this point the force-indentation relation for a particular contact geometry and inden-tation depth 8 is substituted into the equation and the functions of time t and sin £2t areequated separately.

Linear compliance, 6We have

M'Q2 (1 -K

M'V0 D /

= K

f a)Qn 2 -

2-cojCO2

FcQ~l

where mass ratio a — M'/M\. These expressions give

£ll OKI

" J( 9 . 2 7 a )

M' I

Fc = KVO/Q. (9.27b)

(The expression for Q2 contains a choice of sign as + ; this choice can be explained byconsidering the limit as co\ —> 0.)

Hertz compliance, 6 = *S~2 / 3F2/3.To solve the nonlinear equation resulting from beam dynamics with Hertz contact com-pliance, Lee (1940) adopted an approximation for the sinusoidal function, (sin £2t)2/3 ^

3 Note that if the impact point approaches a fixed boundary where the displacement vanishes, the equiv-alent mass becomes large without limit.

1.093 sin £2t. For the period of contact this approximation gives an impulse equal to thatof the Hertzian pulse. With this approximation, Eq. (9.26) can be rearranged to give

l.093M'Q2

This equality gives frequency Q and force amplitude Fc,

i/2 ( 9 2 8 )

( }1.306M*

For a spherical contact between elastic bodies, the stiffness KS = ^E*RlJ2 [Eq. (6.8)].If the period of structural response is much longer than the contact period £2 ^> o)\ theexpressions above give B ^ 1 + a and

where TQ is the incident kinetic energy of relative motion. While the approximationto the trigonometric function (sin Qt)2/3 can be improved by a method of successiveapproximation, in most cases this effort is not justified.For either linear or Hertz compliance, at the impact point x0 the displacement of the beamw(xo, t) and that of the center of mass of the sphere u(t) are given by

w(x0, t) = — c I — sin^r - sin tit \ (9.30)(ft2 \)M() \(O\ J

= V0t - -z^-ri^t - sin at)

Vot f sin Qt}c + t<tf. (9.31)1 +

Example 9.4 A simply supported steel beam of length L = 153.5 mm, depth h = 10 mmand width b = 10 mm is struck at midspan by a steel sphere of radius R' = 10 mm with anincident velocity of 0.01 m s"1. Compare the maximum force Fc at the interface with theforce obtained from a similar impact against an elastic half space, and use a single degreeof freedom analysis to estimate the dynamic deflection w(xo, t) of the impact point onthe beam.

Solution

Material properties: E = 220 GPa, p = 7900 kg m~3.Section properties: El = 183.3 N m, pA = 0.78N s2m"2, / = 15.23 m2s"1.Beam properties: mass M = 0.120 kg, mode equivalent mass M\ = Af/2 =

0.060 kg, frequency of fundamental mode a>\ = 6381 rad s"1.Sphere properties: mass M' — 0.033 kg, mass ratio a = Mf/M\ = 0.55.Contact conditions: stiffness KS = 29.3 x 109 N m~3/2, frequency Q = 27.07 x

103 rad s"1, contact force Fc = 5.76 N, contact period tf = 0.116 x 10~3 s.


0.1 0.2time A(ms)

0.3

Figure 9.4. Transverse impact of steel sphere, with radius R' = 0.01 m, at midspan of sim-ply supported steel beam, b = h = 0.01 m, L = 0.1535 m, which gives mass ratio a = 0.55.Incident speed of sphere, 0.01 m s"1, yields solely elastic deformation. Contact force F(t),beam displacement w(x0, t) and sphere displacement u{t) for single degree of freedom ap-proximation (dashed curves) are compared with a numerical solution that includes higherorder modes of deformation (solid curves: numerical solution from Timoshenko, 1913).

The maximum indentation 8C of the half space is obtained from equating the work duringcompression to the change in kinetic energy, f*c KS83/2 d8 = M'VQ/2. Hence

= 0.456 x 10"6m4/c,

so that the maximum contact force is

and the contact period is

tf(n) = 2.94— = 0.135 x 10~ 3 s [Eq. (6.31)].

Figure 9.4 compares the contact force during impact obtained from this single degree offreedom modal approximation with the displacements obtained from a numerical solutionof Eq. (9.26) (see Timoshenko, 1913).4 The numerical solution includes the effect ofhigher order modes. This comparison shows that the fundamental mode approximationgives an increased magnitude and decreased period for the pulse of contact force; these areconsequences of excessive beam stiffness. The higher order modal deformations (whichare neglected by the single mode solution) become more important as slendemess ratioL/h increases and as the impact point becomes more off-center.

4 Lee, Hamilton and Sullivan (1983) developed a higher order lumped parameter method and used com-parison with this example problem to demonstrate that their calculated result converges to Timoshenko'snumerical solution.


1.0time t, \ms)

1.5 2.0

Figure 9.5. Transverse impact of steel sphere, with radius R' — 0.02 m, at midspan ofsimply supported steel beam, b = h= 0.01 m, L = 0.307 m, which gives mass ratio a = 2.2.Incident speed of sphere, 0.01 ras"1. The contact force F(t) converges to the numericalsolution as the number of degrees of freedom increases. Modal approximations used inthese calculations are as follows: mode 1, dotted curves; modes 1 and 3, dashed curves;modes 1, 3, 5 and 7, solid curves (Lee, Hamilton and Sullivan, 1983).

During collision of an elastic sphere against a slender elastic beam, the maximumcontact force is substantially reduced in comparison with the maximum force duringcollision of the same sphere against an elastic half space. The force reduction is due tothe larger compliance of the beam; this effect increases as the beam compliance becomeslarger in comparison with compliance of the contact region.

For mass ratio a > 1, the response is complicated by multiple strikes occurring at theimpact point before separation at time tf. Figure 9.5 illustrates the variation of contactforce for a case of multiple impact in a system that is but a slight modification of thatin Ex. 9.4, i.e. direct impact in a transverse direction by an elastic sphere on a slendersimply supported beam. In this case the mass ratio has been increased to a = 2.2. Thedotted and dashed curves were calculated using only low order modes; this set of curvesindicate convergence to the continuum solution (crosses in Fig. 9.5) as the number ofmodes in the approximate solution is increased.

For a light mass striking a slender simply supported beam (a = 0.205) at midspan,Fig. 9.6 compares the maximum force calculated from Eq. (9.29) with experimentalmeasurements by Schwieger (1970). Both the ball and beam are steel, and the impact speedranges from 0 to 1.5 m s"1. For a light mass, in order to obtain an accurate approximationfor the contact force it is necessary to consider the effect of local indentation, e.g. theapproximation by Lee.

9.2.4 Impact on Flexible Structures - Local or Global Response?

A light mass striking a stiff and heavy structure results in little motion of thestructure during impact. The response to impact by a light mass is accurately represented


2 0 -

impact speed,

Figure 9.6. Maximum force F from central impact of mass M' = 0.885 kg striking steelbeam of length L = 0.86 m, depth h = 0.0051 m and width b = 0.00254 m. For this impactthe mass ratio a = 0.205, while the stiffness KS =24.2 x 109 Nm"3 / 2. The curve is theanalytical approximation, while crosses indicate experimental data by Schwieger (1970).

by the Hertzian local indentation model of Chapter 5, since during the contact periodthere is little motion of the structure; consequently the contact period is brief, the contactforce is large, and there is little energy lost to structural vibrations.

On the other hand, a heavy missile striking a slender beam or plate results in substantialstructural deformations that limit the contact force; consequently, the indentation is smalland the inertia of the missile is the dominant factor, so that the response is quasistaticas described in Sect. 9.2.2. Quasistatic structural response gives a relatively long contactperiod (roughly half the period of the lowest mode of structural vibration), a reducedcontact force, energy loss to structural vibrations and a likelihood of multiple impact.

For inertia and stiffness parameters that are between these limits, an analysis is requiredthat incorporates both local indentation and structural deformation, i.e. an analysis suchas that in Sect. 9.2.3. What, however, is sufficiently light and stiff or heavy and compliantin order for one approximate method or another to be acceptable?

Christoforou and Yigit (1998) used elastic-plastic contact relations to analyze impactof spherical missiles against beams and rectangular plates having a range of stiffness. Thecharacterizing feature of their linearized analysis was a nondimensional contact force

F(r) = F(0/M'V0'Q0

where £2o is the frequency for local indentation of an elastic half space (linearized stiffness)

Q20 = Ks8l

Y/2/Mf.

For a light mass striking a stiff and heavy structure, the response is local, so the Hertziancontact relation gives the correct contact force, viz. the maximum nondimensional forceFc — 1. At the other end of the spectrum, for a heavy mass striking a light and compliantstructure the quasistatic approximation essentially can be represented as a body of mass M'striking a pair of springs arranged in series. This quasistatic limit gives a maximum nondi-mensional force which depends on the ratio K = K/K of the structural stiffness ic to the

il/2 ^linearized stiffness of the local contact region, K — itcs8lY

/2 = ^ * (for a spherical


indentor); viz.vl/2-m

For impact on thin plates and cylinders, Swanson (1992) has shown that the error of thisapproximation is less than 5% if the colliding body is sufficiently heavy so that the massratio a > 10.

To determine the effect of impact on systems with intermediate mass or stiffnessratios, Christoforou and Yigit (1998) investigated the impact response of infinitely longbeams and plates of thickness h. These solutions for unbounded systems are valid onlyso long as the contact terminates before shear waves emanating from the impact can bereflected from boundaries and return to the impact site. The Christoforou-Yigit solutionfor infinitely large structures depends on an additional nondimensional parameter - thevibration energy loss factor £. This factor represents the energy transformed to elasticvibrations of the beam or plate during the contact period; it depends on the mass of thecolliding body, M'\ the mass per unit area of the plate, ph (or for a beam pbh); the localcontact stiffness K; and the beam or plate bending stiffness Dn = (h3/l2)[E/(l — v2)].The vibration energy loss factor acts like a damping ratio; for a beam or plate it has thefollowing representations:

Uniform beam:

\/2M'K y/4

)KphDuJUniform plate:

M'K ^'2K

1 /~T6\16 \phDn,

Figure 9.7 distinguishes domains for local and quasistatic behavior based on the maxi-mum value of the nondimensional contact force Fc. In this figure the curves for impacton bodies of infinite extent are obtained from the infinite elastic beam or plate solutions,while if the vibration energy loss factor is large (£ > 1), one has the branches shown forimpact on beams or plates with finite length and width. The solutions for finite size struc-tural elements asymptotically approach quasistatic solutions as the loss factor becomeslarge, ? » 1.

For impact near a boundary where displacements are constrained, the quasistatic so-lution for a heavy mass on a flexible structure results in an increasing frequency andcontact force as the boundary is approached. Figure 9.8 illustrates that near a boundarythe apparent coefficient of restitution (based on assuming that at separation the platevelocity is negligible) approaches unity as the energy transformed into elastic vibrationsdiminishes. The width of the region wherein the coefficient of restitution (or energy ab-sorbed by elastic vibrations) is affected by boundary conditions is roughly equal to halfthe distance traversed by shear waves during the contact period (Sondergaard, Chaney andBrennen, 1990).

In general the effect of a resilient structure is to prolong the contact period and reducethe maximum contact force in comparison with the Hertz solution for an elastic halfspace. For elastic-plastic solids, the compliance of the structure effectively increases thenormal impact speed for yield. Thus the energetic coefficient of restitution depends on the

Problems 111

ooXa

local \ transition1.0-

0.8

0.6

0.4

\ global

0 . 2 -

110"2 10"1 10° 101

vibration energy loss factor,102

Figure 9.7. Maximum contact force divided by contact force for half space, F c , as afunction of both the vibration energy loss factor f and the stiffness ratio K: solid curves,uniform beam; dashed curves, uniform plate (Christoforou and Yigit, 1998).

11.10 mm

20 40 60distance from support, x0

80(mm)

Figure 9.8. Coefficient of restitution e for impact of different size solid spheres near aclamped support on a 12.7 mm thick Lucite plate. Impact speed Vo' = 3.5 ms"1 of a steelsphere with radius R' as specified (Sondergaard, Chaney and Brennen, 1990).

compliance of the structure at the point of impact in addition to the material properties,impact speed and relative mass of the colliding bodies.

PROBLEMS

9.1 A uniform beam of length L, cross-sectional area A, and second moment of area / iscomposed of material with density p and Young's modulus E. The beam is simplysupported. Find the mode shapes and modal frequencies.

9.2 For a uniform beam with simple supports, use the Rayleigh-Ritz method with sinu-soidal shape functions to obtain estimates of the lowest symmetric and antisymmetricmodes. Compare the modal frequency approximations with exact values obtained inProblem 9.1.

9.3 The simply supported beam in Problem 9.1 is struck at midspan by a rigid missileof mass M'. Initially the beam is at rest while the missile is moving transverse tothe axis at speed VJJ. For eigenfunctions Xj(x) = sinkjL, kj = jn/L, find theequivalent mass Mj, the equivalent stiffness ic and the amplitude Aj of response ofthe beam as a function of the mass ratio M/Mf.

9.4 Suppose the beam in Problem 9.3 has a uniform cross-section 10 mm x 10 mmand length L = 153.5 mm and is composed of steel (p = 7.9 x 103 kg m~3, E =210 GPa). Show that the modal frequencies of free vibration are COJ/2TT ^ j2x 103 Hz.Find the amplitude of vibration if the stationary beam is struck transversely atmidspan by a 20 mm diameter steel sphere traveling with an initial speed 10 mm s"1

and the impact is perfectly plastic.9.5 A slender elastic beam is struck transversely at low speed by a spherical elastic

missile.(a) Assuming the contact period is short in comparison with the fundamental period

of vibration, use the terminal missile velocity u(tf) to show that for elasticimpact on a flexible body, the single mode approximation gives a coefficient ofrestitution e* = (1 — a)/(I +a). Explain the physical significance of a coefficiente* < 0 if the mass ratio a > 1.

(b) Obtain an expression for the displacement of the sphere (and the contact pointon the beam) at termination of contact (a < 1).

CHAPTER 10

Propagating Transformations of Statein Self-Organizing Systems

Molecules far from equilibrium have far reaching sensitivity whereasthose near equilibrium are sensitive to local effects only,

Ilya Prigogine, Cambridge Lecture, 1995

A ball that falls in a gravitational field before colliding against a flat surfacewill rebound from the surface with a loss of energy that depends on the coefficient ofrestitution. If the ball is free, it will continue bouncing on the surface in a series ofcollisions; these arise because in each collision the ball is partly elastic and during theperiod between collisions the ball is attracted towards the surface by gravity. In Chapter 2it was shown that an inelastic ball (0 < e* < 1) which is bouncing on a level surface in agravitational field has both a period of time between collisions and a bounce height thatasymptotically approach zero as the number of collisions increases. In other words, withincreasing time this dissipative system asymptotically approaches a stable attractor- theequilibrium configuration where the ball is resting on the level surface.

Some other systems can experience energy input during each cycle of impact andflight; consequently these systems exhibit more complex behavior. For example, a pencilhas a regular hexagonal cross-section with six vertices. If the pencil rolls down a plane,the mean translational speed of the axis asymptotically approaches a steady mean speedof rolling where the kinetic energy dissipated by the collision of a vertex against theplane equals the loss in gravitational potential energy as the pencil rolls from one flatside to the next. Sequential toppling of dominoes is another system where a gravitationalpotential drives a series of dissipative collisions. Here again there is a natural speed ofpropagation (toppling) where the energy dissipated by each collision equals the changein gravitational potential as the wavefront moves forward one domino in a uniform set.The equations representing sequential toppling will be shown to be directly analogousto those for a rolling pencil; i.e., there is an intrinsic speed of toppling that depends onthe domino spacing and size but is independent of the initial conditions. A third systeminvolving a sequence of collisions is a ball bouncing on a vibrating table. Here, however,there are excitation frequencies where steady bouncing develops and other frequencieswhere the bounce period is chaotic. The key to classifying these alternative behaviors isto identify the steady state solutions, i.e. the solution attractors.

In each of these systems, the release of energy from some source is triggered by activityat a wavefront. Typically the potential energy that can be released is uniformly distributed,so that the rate of energy added to the system increases linearly with speed of propagation.

219

220 10 / Propagating Transformations of State in Self-Organizing Systems

These systems also have some source of energy loss or dissipation. In any system whichexhibits an intrinsic wave speed for propagation of activity, the dissipation is always a non-linear function of speed of propagation; e.g., for domino toppling the energy dissipationrate from collisions between dominoes depends on the cube of the speed of propagation.Hence these systems always satisfy a kinetic or evolution equation of the form

rate of change of active energy= — (rate of change of potential energy)

— (rate of change of dissipation).

In the case of a mechanical system such as domino toppling or progressive collapse ofwarehouse racking, the active energy is kinetic energy. For nerve signals in a neuronsystem, the active energy is an electrical potential (voltage), while for propagation of aninfectious disease the active energy is infection.

10.1 Systems with Single Attractor

10.1.1 Ball Bouncing down a Flight of Regularly Spaced Steps

A ball of mass M is dropped from height ho onto the top of a flight of regularlyspaced steps. Gravity produces a steady downward force — Mg acting on the ball. The ballfalls onto the top step with a vertical relative velocity — Vo, then rebounds and continuesto bounce just once on each successive step in the set. The path of the ball is illustratedin Fig. 10.1. At each impact the coefficient of restitution is e*.

In this representation of a bouncing ball, each level step is an equilibrium configurationfor the ball. At each bounce the ball loses a part 1 — e\ of the kinetic energy that it possessedjust before impact; after each collision it falls to the next step down, thereby gaining anamount of kinetic energy equal to the negative of the loss in potential —Mgb, where b isthe height of each step. If the energy gain equals the energy dissipated, then every cycleis identical and the ball bounces steadily down the flight of stairs.

Height of Fall for Steady Bouncing, h*If the ball bounces steadily down the stairs, then at each bounce the ball attains the sameheight above the next step as the initial drop height h0; i.e., h* = h0. After each collision

Figure 10.1. Inelastic ball bouncing down a flight of steps with one bounce per step.

10.1 / Systems with Single Attractor 221

the ball that was dropped from an initial height h* attains a maximum height h*el abovethe step that it bounced from; hence if the height of fall for each successive step is identical,

This gives a drop height for steady bouncing,

K = 1b ~ 1 - el' (10.1)

Evolution in Bounce Height for General Initial ConditionsOrdinarily the initial drop height is not equal to the height for steady bouncing (/**), sothat the height of each bounce differs as the ball bounces down the stairs. To obtain anexpression for the evolution of drop height with number of bounces, consider the cycleof fall, impact and rebound for any step. Let hi denote the height of fall onto the ith step.Then the height of fall for any two successive steps is

- el).We note that

dhi/di ^ hi+\ — hi.

The difference equation has a solution

The constant of integration B is evaluated from the initial condition ht = h0 for / = 0;thus B = h0 — h* and the height of fall hi onto the ith step is given by

hi =h* + (ho - h*)ef. (10.2)

Equation (10.2) represents bouncing where the bounce height asymptotically ap-proaches the height for steady bouncing independently of whether the initial drop heightho is larger or smaller than the height h* for steady bouncing. This asymptotic approachis illustrated in Fig. 10.2. The system is stable, and with increasing time (or number of

3.0i

o>2.0

%O.Q

1.0

= 0.775

- 0

0 1 2 3 4 5 6 7 8step number

Figure 10.2. Change in bounce height as ball progresses down steps for coefficients ofrestitution e* = 0.6 and 0.775 and initial drop heights ho/b = 0.5 and 3.0.

bounces) the solution asymptotically approaches the single attractor where the bounceheight above each step equals h*.

Example 10.1 A sphere is dropped onto a level anvil from an initial height h0.(a) Find the rebound height hi after / bounces if each impact is represented by a

coefficient of restitution £*,(b) Find the time tf when bouncing ceases.Neglect the effect of air drag on the sphere.

Solution After falling a distance ho in a gravitational field with intensity g, therelative speed at incidence t?0 = v(0) can be obtained from conservation of energy,

v2 = 2gh0.

The coefficient of restitution then gives the separation speed Vf = v{tf) for this impactand subsequently the rebound height h\ for the first bounce:

v) = e\v\ = 2e\ghohi = elh0.

After the /th bounce,

hi = e\hi-\ = e2jho. (a)

Thus the rebound height is reduced to ht = ah0 after / = 0.5 In a bounces, i.e., after /bounces the maximum height is reduced to ht/ho = e2!.

The time required to complete / bounces is

t = & 1 - 1 - >+2T 4 0 11 V * I '* UTFor a coefficient of restitution in the range 0 < e* < 1, the time when bouncing terminatescan be obtained by taking the limit of this expression as / -> oo. Letting x = e* andnoting that (1 — x)~l = 1 + x + x2 -\ gives a terminal time

1+e* &tf= lmU = / . (b)

10.2 Systems with Two Attractors

10.2.1 Prismatic Cylinder Rolling down a Rough Inclined Plane

A regular prismatic cylinder with an angle 2*I> between the sides can be in equi-librium with any side resting against a rough inclined plane if the angle of inclination 0is such that 0 < ^ and the coefficient of friction JJL > tan 0. Suppose a pencil (i.e. a reg-ular prismatic cylinder with hexagonal cross-section) rolls down a rough plane withoutbouncing from the surface. The hexagonal cylinder passes through a series of possi-ble equilibrium configurations, each with a smaller potential energy than the precedingequilibrium configuration - in this respect the rolling prismatic cylinder is similar to the

10.2 / Systems with Two Attractors 223

Figure 10.3. Hexagonal prism rolling down a rough inclined plane. The velocity of thecenter of mass is shown an instant before impact at vertex Q+i.

sphere bouncing down regularly spaced steps. Both the bouncing ball and the rollingprismatic cylinder exhibit steady motion if the kinetic energy gained during each cycleof motion equals the energy dissipated by each collision and other dissipative processes.

Consider a prismatic cylinder with a polygonal cross-section having N equal sides,where N > 4, as shown in Fig. 10.3. Between adjacent vertices a regular polygon has acentral angle 2*I>, where *I> = n/N. Let the sides of the cylinder be slightly convex, sothat when each side collides with the plane, the reaction impulse acts at the corner Ci+\.Let the prismatic cylinder of mass M have a radius a from the center G to each vertexC[. Hence the cylinder has a polar moment of inertia / for the center of mass G, where

Ma2

I = - — ( 2 + cos 2vl>). (10.3)6

From the parallel axis shift theorem, the polar moment of inertia for any vertex can beobtained as

IC=I + Ma2.

Equations of MotionAssume that friction is sufficiently large that there is no sliding during each collision. Asthe cylinder rotates about a vertex from one side to the next in a gravitational field withintensity g, the decrease in the potential energy equals Mga sin *I> sin 0. This decrease inpotential energy increases the kinetic energy of the cylinder during the interval ti+\ — tt

(i.e. during the period of time between the collision at vertex Q and the collision at vertexQ+i). During this interval the cylinder rolls about the vertex Q , so the only active forceis the conservative force of gravity. Hence for this period, conservation of energy givesthe change in angular velocity cofc) as

/c^2+i(-) - /c^ 2(+) = ^Mga sin *I> sin 6

where the angular speed of the prism just after the collision at vertex Q is denoted bycoi(+) = co(ti+), and the angular speed just before the same collision by &>;(—) = co(ti —).

At this point it is useful to introduce a parameter representing inertia and the activeforces during the rolling phase of motion. Suppose the prismatic cylinder were pendu-luming about vertex Q . For small angular deflections 0 the equation for free oscillationsin the gravitational field gives

0 = 0 + o)2g(/>, co2

g = Mga/Ic.

With this definition of a natural frequency cog the previous equation for conservation ofenergy during rotation about the ith corner can be expressed as

co2+l(-) - co2(+) = 4co2

g sin V s in0. (10.4)

The other part of each cycle of rotation is the collision of the next vertex Q+i with theinclined plane at time ti+\. During the instant of collision the active force is simply theimpulsive reaction at Q+i and there is no impulsive couple, so the moment of momentumabout C|+i is conserved1:

(/ + Afa2)<w/+i(+) = (/ + Ma2cos2^)coM(-).

Hence the ratio ;+1(+) depends on the geometry and inertia properties:

£»,•+!(-) t + Ma2 8 + cos2vJ/w = = T = (10.5)

coi+\(+) / + M a 2 cos 2*1/ 2 + 7cos2vI/

where 1. Combining (10.4) and (10.5), the ratio of speeds just after two successiveimpacts gives an iteration equation for evolution of the rolling speed,

Q)f(+) <P2 [ CO2 J

Steady State and Transient SolutionsA steady rolling speed2 &>*(+) implies that a)i+\(+) = &>/(+) = a>*(+). Substituting thiscondition into (10.6), we obtain

co\ <p2 - 1

Apparently rolling persists at the steady speed if the initial conditions are consistent withthis motion. In general however the rolling cylinder will have an initial speed &>o(+)which is not the same as the steady state solution. The evolution of the rolling speed canbe obtained by replacing the parameter co2 in the iteration equation (10.6) with expression(10.7):

? - <P~2 {<»?(+) + (<P2 ~ 1

1 Before the (/ + l)th impact the velocity of the center of mass G is perpendicular to line GQ as shownin Fig. 10.3; therefore at the instant f,-+i (—) the velocity of the center of mass is not perpendicular tolineGQ+i.

2 Of course the speed of rolling varies during each cycle, but in steady rolling the variation in speed isperiodic with N cycles per revolution. Thus for steady rolling at the instant just after each collision,the angular speed is <w*(+).

U 6 8 10corner number

12 U 16

Figure 10.4. Evolution of rolling speed from initial states co0(+) that are either larger orsmaller than the speed of steady rolling, &>*(+). Solid lines are for N = 6 (cp = 1.545), whiledashed lines are for N = 12 (o(+) this iteration equation has a solution

^ 2 ( + ) = a>l(+) + <p-2i {co2(+) - a>l(+)), i = 0, 1, 2 , . . . (10.8)

Here it is apparent that rolling is a stable process where the speed asymptotically ap-proaches the steady rolling speed from either above or below, as shown in Fig. 10.4; i.e.,the steady rolling speed is an attractor.

Minimum Initial Speed for Rolling, u;mjn(+)If the inclination of the plane is sufficiently large (0 > *!>), the process of rolling is self-starting from an initial condition coo(-\-) = 0. For somewhat smaller angles of inclination,rolling still asymptotically approaches the steady state if the initial speed is larger than aminimum value that is necessary to initiate rolling, &>min(+). The minimum initial speedfor continuous rolling is obtained from the smallest kinetic energy which is sufficient tobring the center of mass to a position vertically above the contact point,

This gives

- cos(* - 0)1

- cos(vj/ - 0)](cp2 -2 sin vj/ sin 0

(10.9)

Minimum Angle of Inclination for Rolling, 0cr

If the angle of inclination of the plane is too small 0 < 0CT < ^ the steady state rollingspeed is inaccessible. For small angles of inclination where &>o(+) > &>*(+) an initialrolling speed &>o(+) > &>min(+) will slow at each successive impact until &>*(+) < &>/(+)< &>min(+)- At this point the cylinder will no longer have sufficient energy to continuerolling but instead will rock back and forth, with decreasing amplitude of rotation, be-tween two vertices on the same face as shown in Fig. 10.5. The inclination 0CT where thesteady state rolling becomes inaccessible is given by the condition &>min(+) = &>*(+);thus it is a root of

0 = 2 sin V sin<9cr - (cp2 - - 0CT)] (10.10)

^0.75

$ 0.5CL

% 0.25

I 0

/V = 6

15° 30° 45° 60° 75°inclination of plane

90°

Figure 10.5. Attractors &>;(+) = 0 and &>*(+) for rolling of a hexagonal prism (N = 6) asfunctions of the angle of inclination of the rough plane. There are two attractors for anglesof inclination in the range 6cr < 9 < a\ in this range the attractor which is approached byct>/(+) depends on the initial rolling speed (Abeyaratne, 1989).

Hence if the angle of inclination 6 is in the range 0 < 0 < *I>, the stationary state &>; (+) = 0is an attractor. The state &>;(+) = &>*(+) is an attractor for 0 > 0CT. These two regionsoverlap and there are two attractors for 0CT < 0 < *I>. In Fig. 10.5 this behavior is illustratedfor a hexagonal prism. Suppose the prism (e.g. a pencil) is resting on a rough plane andthe angle of inclination of the plane is slowly increased. The hexagonal prism does notbegin to roll until the angle of inclination equals *I>, but then the rolling speed rapidlyaccelerates to the steady rolling speed &>*(+). Now assume that the angle of the planedecreases slowly. Steady rolling continues until the angle of inclination equals #cr, whererolling ceases and the hexagonal prism oscillates to rest.

Table 10.1 lists the minimum inclination for rolling and the steady state rolling speedsfor prismatic cylinders with an increasing number of sides. The minimum number of sidesfor rolling is four, as was noted by Abeyaratne (1989). For a triangular prism (N = 3),the moment of momentum equation (10.5) shows that when any vertex strikes the plane,the direction of rotation reverses rather than the cylinder rolling onto the next side.

Energy Dissipated by CollisionsIn this system the loss of energy is solely due to the perfectly plastic collisions whichoccur as each vertex strikes the plane. From the equation preceding (10.8) the change of

Table 10.1. Minimum Inclination for Rolling (0cr)and Steady Rolling Speed to*(+)for PrismaticCylinders with n Sides

N

4569

90180

(deg)

4536302021

*

0.4340.8871.2011.8086.588

24.91

kinetic energy T per cycle is obtained as

i = dc*2(+)/di = fi>2

where T has been multiplied by 2 / / c . Likewise the change of a similarly modifiedpotential energy per cycle, dUt(+)/di, equals

dUt/di = -4co2 sin V sinO = -(cp2 - l)a>l(+)

where co2 is a parameter and the second equality comes from (10.7). Since the rate ofcollision is proportional to the angular speed cot(+), the rate of change of kinetic energyand rate of change of potential energy with time can be expressed as

dt

(10.11)

With these expressions the rate of change of dissipation Dt can be expressed as

dDt/dt = -dTi/dt - dUt/dt

= (1 - Cp~2) [(<p2 - l)< (10.12)

Figure 10.6 compares the rate of energy dissipation dDt/dt with the rate that the lossof potential energy adds kinetic energy to the system, —dUi/dt. If the rolling speed isbelow the intrinsic speed for rolling [&>/(+) < &>*(+)], the rate of decrease of potentialenergy is larger than the rate of dissipation, so the speed of rolling increases. On the otherhand, if &>;(+) > &>*(+), the rate of dissipation exceeds the rate of decrease of potentialenergy, so the kinetic energy and the rolling speed decrease. Whether the rolling speedis above or below the intrinsic speed, with increasing time the system asymptoticallyapproaches the intrinsic speed of rolling &>*(+) in a stable manner. This indicates that

rolling speed

Figure 10.6. Rate of energy dissipation by collisions (dDi/dt) and rate of decrease ofpotential energy (—dUi/dt) are equal when the rolling speed &>;(+) equals the intrinsicspeed for rolling, &>*(+).

228 10 / Propagating Transformations of State in Self Organizing Systems

the speed &>*(+) is an attractor. Since the rate of dissipation and the rate of change ofpotential energy also are equal for &>;(+) = 0, this too is an attractor.

10.2.2 The Domino Effect - Independent Interaction Theory

The domino effect describes a wave of impacts and toppling that propagatesthrough a periodic array of regularly spaced elements where each element is marginallymetastable in the initial state. The effect is readily observed in an array of slender blocks(dominoes) that initially stand on end with a regular spacing between faces of adjacentelements; the blocks stand in a gravitational field with the faces vertical and parallel. Inthis array, toppling one element can initiate a sequence of collisions where each topplingblock knocks over its neighbor; an impact that imparts sufficient kinetic energy to knockover the first block is sufficient to knock over the entire array in a wave of destabilizingcollisions.

The term "domino effect" has been used to describe disparate phenomena such assequential collapse of periodic frame structures, blowdown of trees in ice-laden forestsand propagation of neuron firing in a synapse-coupled neural network. The effect is adiscrete counterpart of propagation phenomena in exothermic chemical reactions that arerepresented by reaction-diffusion equations (Nicholas and Prigogine, 1977; Bimping-Bota et al., 1977). In the case of dominoes however, the energy loss at the wavefront isdue to dissipation by successive collisions rather than diffusion.

Toppling and Collision of DominoesConsider a regularly spaced array of uniform slender blocks that initially stand on end on arough level surface as shown in Fig. 10.7. Each block has mass M, length L and thicknessh, and they are spaced at distance k + h along the surface. Each block is initially vertical,so the center of mass is displaced by a small angle *I>0 = arctan(/z/L) from the verticalline passing through any edge in contact with the supporting surface. If the block rotatesabout the edge, the potential energy increases until the rotation angle equals ^o- At thisangle of rotation the block becomes unstable and for even larger angles of rotation thepotential energy begins to decrease. Since there is no sliding on the supporting surface,an unstable domino topples into a collision with its neighbor after rotating through anangle ^ ; thus at collision there is a total rotation *I>0 + ^ I = arcsin(A,/L). With thepresent assumptions, the collision is the only coupling between neighboring elements ina domino array.

Figure 10.7. Geometry of toppling in a regularly spaced array of dominoes.

At the head of the group of toppled dominoes, the domino at the wavefront is rotatingwith angular speed co(t). Let the angular speed of this leading domino be denoted by(o(ti+) = coi(+) at the instant just after impact against the ith domino and co(ti+i—) =(t>i+\(—) at the instant just before impact against the (/ + l)th domino. Following theimpact of the (i — l) th domino against the /th domino, the latter has an initial kineticenergy 7] (—) = (ML2/6)cof(—) sec 2 *I>o- During the subsequent motion of the / th dominoin the gravitational field there is a change in potential energy AUt = f/£-+1(—) — f/ ,(+) sothat for the rotation phase of each cycle the principle of conservation of total mechanicalenergy gives an energy ratio

AU; 2(O2g8 (cos ^ i - cos vl>o) (10.13)

where cog = [(3g cos *I>0)/(2L)]1/2 is the natural frequency of small oscillations for theblock rotating as a pendulum about the supporting edge.3 This oscillation frequency forpenduluming is the same idea used in the example of a rolling pencil to obtain a parameterthat represents the ratio of active force to inertia for the prescribed motion.

Conservation of energy and this change in potential energy during toppling give forthe ratio of angular speed at impact to initial angular speed

G ) , - + I ( - ) M ( + ) = [1 - AI///7K+)]1/2. (10.14)

The collision against each block occurs at distance § above the supporting plane, where§/L = cos(*I>o + ^ I ) - The impact against the (/ + l)th domino generates an impulse pi+\normal to the surface and if the dominoes are rough, there is also a component of impulsetangent to the surface M/?;+I , where /JL is the coefficient of Coulomb friction. The frictionalimpulse acts in a direction opposed to sliding. The following analysis assumes that slidingcontinues throughout the contact region; this is a valid assumption for colliding bodiesif the angle of incidence is substantially larger than the angle of friction (Maw, Barberand Fawcett, 1981). The analysis also assumes that during each collision the sliding iscontinuously in the same direction; the case of slip reversal during collision occurs onlyif both the coefficient of friction and the coefficient of restitution are large (Stronge,1987).

The normal impulse pt that initiates toppling in the ith domino can be expressed asPi = 27}(+)/(£ — /z/z)&>;(+). For block / — 1 the component of impulse normal to thelongitudinal axis is /?;(§ + fiX)/L. Then for each block the change in angular velocity isrelated to the moment of the impulse about the supporting edge. The changes in angularvelocity are also related to the work done in changing the kinetic energy of relative motionof the bodies by the coefficient of restitution e*. Thus we obtain the ratio of angular speedfor block / — 1 at the instant before impact, &>;(—), to the initial angular speed of theneighbor, o>/(+), as

= <p-l(l+e*), 2. Multiplying this expression with (10.14) and

3 Like the natural frequency defined following (10.3), this characteristic frequency depends on gravityand inertia properties of the system for rotation about a particular instantaneous center - the supportingedge of the domino.

noting that &>;+i (+)/&>;(+) = coi+\{—)/cOi(—) gives an iteration equation for evolution ofthe angular speed of the domino at the wavefront,

This expression is analogous to (10.6) in the example of a rolling prism. (Recall thatthe analysis of the rolling prism in Sect. 10.2.1 assumed perfectly plastic collisions, i.e.e* = 0.)

As toppling propagates from one domino to the next, each collision dissipates energy.The loss of energy per collision can be evaluated from theorem (3.21):

A7}+1

This result can be obtained from the difference in kinetic energies at the inception andtermination of each collision.

Steady Speed of TopplingEquation (10.16) has a steady state solution where each successive domino has identicalmotion when it is at the wavefront; i.e., the interval between impacts is constant. At theintrinsic speed of propagation the initial angular speed of every domino is the same; hencesetting (10.16) equal to unity after substituting from (10.13) gives the initial angular speedat the intrinsic or steady speed for propagation co*, where

r.T?T- <•<><?2-(i + e*)2 J

Since 2 and e* < 1, there is a real intrinsic speed if *I>i > *I>0; (i.e. k/h > \fl cosThus a real intrinsic speed of propagation can exist only if the domino spacing is largerthan a minimum. For slender dominoes this minimum spacing is slightly smaller than thedomino thickness.

Transient Solution for Approach to Steady StateToppling of an array of dominoes is initiated by some external impulse which is sufficientto topple one element. Ordinarily the impulse does not initiate toppling at the intrinsicspeed. If X/h > >/2cos \J/0 and toppling is initiated at a speed that is smaller than theintrinsic speed [&>;(+) < &>*], then the speed of propagation will steadily increase towardsthe intrinsic speed; likewise, if the initial speed is larger than the intrinsic speed the speedof propagation will steadily decrease towards the intrinsic speed. An equation describingthe approach to the intrinsic speed to* can be obtained directly from a linear recurrenceequation resulting from Eqs. (10.13), (10.16) and (10.17):

*>?+i(+) = o>l + <p-\\ + e*)2h2(+) - col].For an initial angular speed co\(+) of the first block, this equation has a solution

*,2(+) = col + K ( + ) " col] l<P~lV + e*)f~2- (10.18)Because cp > 2, the factor (1 + e*)/cp < 1, so that for a sufficiently large number / oftoppled dominoes the initial angular speed of each successive domino asymptoticallyapproaches the intrinsic angular speed for steady toppling.

A//? = 2.89

2 A 6 8 10 12 ttspatial coordinate, x/[\+h)

Figure 10.8. Intercollision period during toppling of dominoes with array spacings k/hof (a) 2.89, (b) 3.89 and (c) 4.51. Each test had an initial speed ct>o/co* that was lower than,about the same as or higher than the natural speed of propagation.

Figure 10.8 shows measurements of the period between successive collisions duringpropagation of toppling in domino arrays with three different spacing. In these experi-ments, toppling was initiated by imparting a specified impulse to the end domino. The datapoints for the interval of time between successive collisions show that at each spacing theintercollision period rapidly settles to a roughly constant value representative of steadypropagation. That steady propagation speed increases with domino spacing. If topplingis not initiated at the steady speed, it closely approaches the steady speed by the timethat 5-10 dominoes have been toppled; thus, at the spacing described in Fig. 10.8 steadypropagation is quite stable.

Closely Spaced Dominoes, \£i < \I>oFor closely spaced dominoes (k/h < \fl cos ^ 0 ) it is clear from (10.17) that real steadywave speeds do not exist. To determine the evolution of angular speeds as the wavefrontpasses through the array of dominoes, first consider the minimum initial speed for topplingof an individual block, com[n, where

If the initial speed of the /th block is &>;(+) = Pcom[n where P > 1 then a ratio of initialangular speeds between neighboring blocks will be

*>/(+) 1 + 21-cos^o

^ I J 1/2Jj ' (10.19)

For closely spaced dominoes where cos *I>0 < cos ^ i this expression shows that &>;< cot (+); i.e., the wave of collisions continually slows until there is insufficient momentumto topple the next domino.

ExperimentsA series of high speed photographs of a wave of collisions in a widely spaced arrayof dominoes (k/h = 2.89) are shown in Fig. 10.9. These photographs cover the periodin which the toppling wavefront progresses a distance of about 1.5 domino spacingsalong the array. The size of the dominoes and measured values of their characteristicparameters are given in Table 10.2. For each spacing the theoretical value for speed ofthe wavefront has been compared with the measured value; the theoretical value wasobtained by integrating the angular speed [obtained from (10.14)] with respect to timeto determine the period between collisions if toppling is initiated at the intrinsic steadywave speed &>;(+). The table indicates satisfactory agreement only if the spacing islarge.

For dominoes with a spacing k/ h < 4, the high speed photographs show that this singlecollision model is not satisfactory because there are multiple collisions. For more closelyspaced dominoes multiple collisions are even more common, as shown in Fig. 10.10 fora spacing k/h = 1.7. Multiple collisions on the domino leading the toppling group drivethe system at a larger speed of propagation than would occur if there were solely singlecollisions.

Table 10.2. Domino Toppling Experiment

Exp. Exp. Theor.Steady Steady Steady

Spacing Initial Collision Collision Collision Collision WaveRatio Angle Angle Height Period Speed Speedk/h *I>o ^ 1 £/£ Ar V* V*

(deg) (deg) (ms) (ms"1) (ms"1)

2.893.894.51

10.310.310.3

21.334.644.6

0.8510.7080.573

28.438.347.8

1.040.970.87

0.650.800.86

Domino dimensions: L = 41.78 mm, h = 7.58 mm, w = 21.90 mm.Natural frequency: co = 18.95 rad s"1.Coefficient of restitution: e* = 0.846 ± 0.03 (edge impact on domino surface). Inthis experiment the coefficient of restitution was measured for an edge of oneplastic domino colliding with the face of a similar domino at a speed in the range0.5-1.0 ms-1.Coefficient of friction: \x = 0.176^QQ4 (edge sliding on domino surface).


t = 0.0 ms

t = 6.2 ms

t = 13.9 ms

t = 20.0 ms

t = 26.2 ms

t = 32.3 ms

t = 38.5 ms

t =46.2 ms

t = 52.4 ms

t =60.1 ms

Figure 10.9. Frames from high speed film of toppling in dominoes with moderate spacing, X/h2.89.

234 10 / Propagating Transformations of State in Self Organizing Systems

t zz 0.0 ms 1111t - 6.75 ms

t n 13.50 ms

t = 20.25 ms

t = 27.00 ms

t ~ 33.75 ms

=40.50 ms

t = 47.25 ms

Figure 10.10. Frames from high speed film of domino toppling, X/h = 1.7.


10.2.3 Domino Toppling - Successive Destabilization by Cooperative Neighbors

The following analysis considers domino toppling caused by a group of neigh-boring elements that interact as they topple. The toppled dominoes behind the wavefrontare assumed to each lean against their neighbor, and they maintain contact during topplingrather than just striking once and then separating. The dominoes all lean in the same direc-tion towards a leading element at the wavefront. Ahead of the wavefront, all dominoes arestationary and vertical, while behind the wavefront, the dominoes all rotate in the samedirection. Toppling of the next undisturbed element by the cooperative group has an in-trinsic speed of propagation whenever the weight of the leaning dominoes is sufficient todestabilize an undisturbed element at the wavefront. The natural speed for toppling by acooperative group is larger than the natural speed for toppling by single collisions unlessthe loss of energy at each collision is small and the spacing between elements is large.

Collision and Toppling by Cooperative NeighborsConsider a uniformly spaced row of identical slender blocks (dominoes) that initiallystand on end on a rough level surface in a gravitational field. Each block has length L,thickness h, width w and mass M and is spaced a distance X from its nearest neighbors asshown in Fig. 10.11. In this analysis the toppling blocks are numbered backwards fromthe block at the wavefront i = 1, so that the first block is the leading block in the topplinggroup. Before toppling, the long edges of each block are vertical; thus the center of massis displaced from vertical by a small angle *I>0 = arctan(A/L). The friction betweeneach block and the rough level surface is assumed to be large enough to prevent anysliding as the block pivots about the edge; hence a typical block simply rotates about thesupporting edge until it collides with the next block. At any instant the / th block behind thewavefront has rotated through an angle 0/. When the leading domino has rotated throughan angle 0\ = arcsin(A./L), the top edge collides against the face of the next domino. Letthe angular speed of this leading domino an instant before the collision against the Othblock be 0 i ( - ) = wi(- ) .

Several features of toppling by united action of a cooperative group are illuminatedby an analysis which considers the following hypothetical interaction conditions. Thisanalysis assumes:

(a) there are an indefinitely large number of toppling dominoes behind the collisionwavefront;

(b) each domino behind the wavefront leans forward against its neighbor;

Figure 10.11. Geometry for toppling by cooperative neighbors.


(c) at the sliding contact between each pair of neighboring dominoes, friction isnegligible;

(d) the coefficient of restitution is negligible, so that after a collision the collidingblocks remain in sliding contact.

With these hypotheses, the rotations of two neighboring dominoes behind the wavefrontare related by the kinematic condition that sliding contact is maintained. Thus,

L sin(0/+i - Oi) = (k + h)cos 0t - h.

In order to obtain the ratio of angular speeds at any instant, the above equation can bedifferentiated with respect to time to give

SinOi (10.20)£ 1 () .Oi cot V L J cos(0i+l - Ot)

Hence the angular speed &>;(#;) of each block can be determined from a function thatdepends solely on the angular speed of the first block. Let &>i(0) = oo\(+) be the angularspeed of the first block at the instant when it begins to move. At this instant the kineticenergy T of the group of toppling dominoes can be expressed as T = $T\, where anenergy ratio # is calculated from (10.20) as

and T\ = (ML2/6)co\(+) sec2 *I>o is the kinetic energy of the first domino an instant afterthe impact that sets it in motion.

If a block is far behind the wavefront, its rotation 0t approaches an angle 0 = arccos( 1 +k/h) where each block lies flat against its neighbor.4 As the wavefront moves forward oneelement in the array, the change in potential energy of the group of toppling dominoes,—AU, equals the change in potential energy of a single block rotating from 0 = 0 to0 = 0. A ratio of the change of potential energy to the kinetic energy of the leadingdomino gives

-AU 2co2

= 0 8 [cos vj/o - cos(# - *I>o)] (10.21)

where cog = [(3g cos *I>0)/(2L)]1/2 is the natural frequency of a block penduluming freelyabout a supporting edge.

If there are no energy losses due to friction, the sum of the kinetic and potentialenergies is constant during the toppling phase of motion that precedes the next collision.Consequently the leading element in the group of toppling dominoes has an angular speedat collision, co\(—), that is related to the initial angular speed of this element a)\(+):

4 The rotation angle approaches this bound in the limit as the number of toppling dominoes becomesindefinitely large.

At impact against the next stationary domino, an impulse is imparted to the face of theOth domino and an equal but opposite impulse acts on a top edge of the first domino. Thisimpulse initiates motion of the Oth domino at an angular speed#o(+) = &>o(+)- With nofriction between sliding blocks, the entire energy dissipation per cycle, Z)o, occurs at thecollision,

A)

Hence as the collision wavefront travels through the array, the change of total mechanicalenergy per cycle gives a ratio of initial angular speeds of adjacent elements equal to

, 1/2

(10.23)

Intrinsic Speed of Propagation for Toppling by a Cooperative GroupFrom Eq. (10.23) it is evident that the wave of destabilizing collisions moves from oneelement to the next at a speed that depends on the initial angular speed of the leadingblock and the change in this speed during the toppling that precedes the next collision.A natural or intrinsic speed of propagation for successive toppling implies that there isa speed where every element has an identical motion but this is displaced in time by acommon period between collisions; i.e., &>o(+) = &>i(+)- The initial angular speed forsteady propagation, &>*(+), is obtained from (10.21) and (10.23) with this condition ofidentical motion for every element:

= {2(1 - tf-^fcos ^o - cos(£ - ^ 0 ) ]} 1 / 2 . (10.24)

There is a real natural speed of propagation whenever the limiting angle of rotation0 > 2^o- This condition is satisfied if the spacing k/h> 2[(L/h)2 — I ] " 1 . For typicaldominoes with L/h^5 the required spacing is very small. This minimum spacing forsteady propagation is required for potential energy to be supplied to the system by thetoppling of each domino. If the spacing is very small, so that 0 < 2^ 0 , an initial distur-bance imparted at one end propagates into the array with a speed that decreases steadilyuntil it vanishes after a finite number of dominoes have been displaced.

Logistic Map for Transient Phase of PropagationGenerally toppling is initiated at a speed other than the steady speed for toppling. Supposethat after toppling has been initiated, the leading domino in the toppling group has aninitial angular speed co\(+) = /?&>*(+). Then the angular speed ratio (10.23) can be usedto obtain the initial speed for the neighbor that is next at the wavefront,

<wo(+)A»i(+) = [1 - »-ld ~ /T2)]1 / 2 . (10.25)

This ratio is illustrated in Fig. 10.12 in the form of a logistic map (Collet and Eckmann,1980). Such maps are sometimes used by ecologists for analyzing the dynamic effectof interactions on the population of successive generations of animals (May, 1986). Atypical analysis for k/h = 2 and a ratio of initial to intrinsic angular speeds f$ = 0.7 isindicated by the broken line in the figure. Successive dominoes have angular speed ratios/3 = 0.7, 0.82, 0.89, etc.; the initial angular speed of successive dominoes monotonouslyapproaches the intrinsic speed &>*(+). This map shows stable behavior for dominoes since


incident speed,

Figure 10.12. Logistic map for change in initial angular speed between successive colli-sions (h/L = 0.12). The broken line indicates a typical trajectory for an initial speed lessthan the intrinsic speed co*. This map shows a stable approach to the intrinsic speed frominitial speeds either above or below.

as sequential toppling progresses, the wavefront travels at a speed that asymptoticallyapproaches the intrinsic speed for initial conditions either above or below this naturalspeed of propagation.

Stable behavior which results in an asymptotic approach to the steady speed is alsoevident from the finite difference equation for initial angular speed of successive blocks.Equations (10.21), (10.23) and (10.24) give a change of kinetic energy per cycle, To - Tu

equal to the loss in potential energy minus the dissipation per cycle,

Average Translational Speed of WavefrontThe translational wavefront speed V(t) varies as each successive domino is toppled;nevertheless the average speed V* can be calculated by integrating the motion to obtainthe period between collisions and dividing this period into the distance traversed. Thusfor toppling that initiates at the intrinsic angular speed &>*(+),

fJO

arcsin(A./A) — i

where1/2

Figure 10.13 compares the intrinsic speed for steady toppling by a cooperative groupwith that calculated for single collisions between neighbors. For almost every spacingthe cooperative group gives a larger change in potential per cycle so that at any spacingit has a larger natural speed.


Table 10.3.

Spacingk/h

0.781.02.03.04.05.06.0

Material

Tufnol (T)Perspex (P)

Steady Speed of Toppling for Two Sets of Dominoes

(Exp. Speed)/Tufnol

1.55-1.571.471.421.361.23

L h(mm) (mm)

80.0 9.680.0 9.9

Perspex

1.491.501.511.451.40-1.20w(mm)

50.050.0

(Theor. Speed)/ jgL,

1.991.911.681.591.521.461.42

^o e* /xdyn(rad)

0.12 0.62 0.150.12 0.55 0.25

2.0

2 1.0Q.in

^ 0.5

cooperative group theory

single collision theory

0 1 2 3 U 5spacing \/h

Figure 10.13. Experimental measurements of nondimensional intrinsic speed of propaga-tion compared with theory over a range of domino spacings. Experiments were made onthree sets of dominoes made from different materials: Opalene, Tufnol, Perspex.

Toppling ExperimentsIn addition to the experiments using Opalene dominoes that were described in Table 10.2,toppling experiments were performed with sets of thin rectangular blocks manufacturedfrom Perspex and Tufnol. These blocks had better uniformity and much smaller edgeradii than commercially available dominoes. Dimensions of these blocks, a coefficient ofrestitutions* and a coefficient of sliding friction /x are listed in Table 10.3. The coefficientsof restitution and friction were measured in separate experiments that closely simulatedcontact conditions during a collision at the intrinsic speed.

Toppling of each set was initiated by a low speed collision at the center of percussionof the domino at the end of the array. Usually there was an initial phase of propagationwhere the speed approached the steady speed for toppling - typically this required thewavefront to pass through 6-15 blocks. Thereafter the speed of the wavefront settleddown to a small but regular variation about a constant value. Figure 10.13 compares thetheoretical nondimensional speed of propagation V*/v/g£ with measurements obtained

from the three sets of dominoes. These nondimensional speeds are almost the same asthat obtained for Opalene dominoes (Stronge, 1987) and slightly larger than the speedmeasured by McLachlan et al. (1983).

Frames from a segment of high speed film taken of the toppling process (Fig. 10.10)indicate that successive toppling never settles down to a single type of interaction betweenevery pair of dominoes; when the dominoes are closely spaced, there are some pairsof blocks near the wavefront with sliding contact and others with multiple collisions.For more widely spaced dominoes there are some single collisions and other multiplecollisions. The agreement with theory is best for an intermediate range of spacing wherethe cooperative neighbor hypotheses are most representative. If the spacing is small(X/ h < 2), there is a relatively large impulsive reaction at the pivot point on the collidingblock when it is at the wavefront. Friction does not prevent closely spaced blocks fromsliding on the supporting surface, and this sliding dissipates energy. There is friction alsowhere the corner of one block slides down the face of its neighbor. These sources offriction dissipate energy that is not accounted for, so that for small spacing the theoryunderestimates the intrinsic speed of propagation.

Domino toppling is a nondispersive wave that has an intrinsic speed of propagation. Inthis wave, particle velocity and pulse shape are properties of the system. Other systemshaving similar characteristics also can propagate changes of state where destabilizationreduces the potential energy. Figure 10.14 shows photographs of sequential collapsein heavily loaded warehouse racking. In heavily loaded periodic frame structures andicy forests the source of distributed potential energy that drives progressive collapse isgravity, whereas neuron firing is driven by an electrical potential. The release of part of thepotential energy in a neuron is chemically triggered by incoming action potential pulses.In all of these marginally stable systems, a wave of destabilization has a natural speed ofpropagation where the rate of change of energy per cycle vanishes (Scott, 1975).

10.2.4 Wavefront Stability for Multidimensional Domino Effects

The domino effect is a solitary wave of reaction in a periodic dissipative systemwith two stable states of potential energy U. The reaction transforms potential energy intoactive energy T that propagates from sequentially triggered initiation sites into unreactedmedium; this propagated energy triggers subsequent reactions at neighboring sites. Thetraveling reaction is self-sustaining if active energy supplied by each reaction is sufficientto trigger reactions at nearest neighbors.

In domino toppling the active energy is kinetic energy, and this is transported byconvection; in this case the propagating phase transformation can be represented by eitherdifferential or difference forms of the Bloch equation (Harris and Stodolsky, 1981; Leggettet al., 1987). Similar traveling reactions occur in other periodic, dissipative systems; e.g.involuntary spasms in striated muscle tissue propagate from one sacromere to the nextwhen contraction triggers the release of Ca2+ ions in the adjoining sacromere (Regirir,1989).

Domino toppling is powered by loss in gravitational potential U and transported byconvection; the transportable or active energy is the kinetic energy T. The energy dis-sipation D is predominantly related to collisions at the reaction wavefront; dissipationincreases as the cube of wavefront speed. Since the distribution of energy in the leading


oCO

in

rH

II-P

o

114J

Figure 10.14. Sequence of photographs showing progressive collapse of heavily loadedwarehouse racking. Unbraced, 3 m high cold-rolled steel racks are collapsing at a loadF = 0.6Fc, where the buckling load Fc = 5.6 kN per leg (Bajoria, 1986).

group of toppling dominoes is independent of the wavefront speed,

dT/dt = -dU/dt - dD/dt. (10.26)

For planar (ID) reaction waves in a homogeneous medium, domino toppling has a intrinsicor natural speed of propagation where reaction rate equals dissipation rate. For cylindricaland spherical wavefronts (2D and 3D) or radius r, however, reaction and dissipationrates increase as rj, where the dimensionality of the wavefront equals j + 1 (i.e., j =0, 1, 2 for ID, 2D, 3D). Consequently, radiating 2D and 3D reaction wavefronts naturallypropagate more slowly than the ID intrinsic wave speed. As the wavefront radius becomesindefinitely large, the speed of propagation asymptotically approaches the intrinsic speedfor ID propagation. In contrast, converging cylindrical or spherical wavefronts propagate


2 .5 -

20 10• distance from centre r/(>+/?)

Figure 10.15. Speed of radiating and converging cylindrical wavefronts as function ofradius r for domino spacings Xjh — X and 4.

Figure 10.16. Cylindrical wavefront of 2D domino toppling shown at successive times.An initially irregular wavefront becomes more circular (smoother) as it propagates.

at a speed that increases as the radius decreases. For 2D domino toppling, these trendsare illustrated in Fig. 10.15.

For a radiating wavefront with uneven radius, the perturbations with a smaller radiusof curvature progress more slowly than adjacent parts which have a larger radius ofcurvature. This stabilizes or smoothes the reaction wavefront as the reaction progresses.This smoothing is illustrated in Fig. 10.16.

10.3 Approach to Chaos - an Unbounded Increase in Numberof Attractors

10.3.1 Periodic Vibro-impact of Single Degree of Freedom Systems

Vibrating systems that strike a barrier during a cycle of motion are inherentlynonlinear. If such a nonlinear system is subjected to harmonic excitation, it can exhibit

10.3 / Approach to Chaos - An Unbounded Increase in Number of Attractors 243

Figure 10.17. Forced vibration of single degree of freedom system with unilateral dis-placement constraint u <UQ.

a rich variety of periodic responses depending of the excitation force and frequency,the clearance of the barrier and the coefficient of restitution. Such problems have beeninvestigated both analytically and with the aid of numerical simulations by Holmes (1982),Shaw and Holmes (1983), Shaw (1985), Thompson and Stewart (1986) and Gontier andToulemonde (1997). In this system the period of the response can be the same as thatof the excitation (period 1 orbits), twice as long (period 2 orbits), three times as long(period 3 orbits), etc. In each case the number of impacts in one period is indicated bythe numeral, but these impacts are not necessarily regularly spaced in time during theperiod. The approach to chaos is marked by a very rapid doubling of the period of thestable response with small increases in excitation frequency.

The single degree of freedom oscillator shown in Fig. 10.17 is subjected to a sinusoidalbase displacement C sin Cii. The displacement u(t) of the system is limited by a stop thatprovides a unilateral constraint, u < UQ. When the mass collides against the stop, thechange in velocity is given by a coefficient of restitution e*. This vibro-impact system hasan equation of motion for the two phases of motion,

d2u/dt2 + co2u = C sin Qt,

du(ti+)/dt = -e

u < u0

u(tt) = u0

where GO is the natural frequency of this undamped oscillator and e* is a coefficient ofrestitution which applies to each impact against the stop. For this system one can definenondimensional variables

u = = Q/co, C = C/u0co2, t = cot.

In terms of these nondimensional variables the equations of motion can be expressed astwo first order ordinary differential equations in the dependent variables for displacementu and velocity it,

du/dt = u

du/dt + u = C sin Qt, u < 1

(10.27a)

(10.27b)

(10.27c)

For 0 < e* < 1 this is a nonconservative oscillatory system with energy dissipated atevery impact.

10.3.2 Period 1 Orbits

On the /th cycle let the impact velocity be denoted by i?,- = {«(*,-—), 1, tx•,—}. Thesubsequent motion can be expressed as

( C \

u(vi91) = I 1 sin Qti I cos(£ — ti)

e*«, + D cosiit,jsm{f t,)+ Dsi( C \

ii(Vj, t) = — I 1 sin Sltj I sin(r — f,

sm Qt (10.28a)

— ( ^ *w/ H / x CQcos(r — ti) -\ cos Qt (10.28b)

where D = 1 — £22. From (10.28a) the time ti+\ of the (/ + l)th impact can be obtained,and this is subsequently used in (10.28b) to determine the impact velocity vi+\. ThusVi+X = </)(Vi).

10.3.3 Poincare Section and Return Map

A Poincare section E is defined as a plane at a constant value of u in the solutionspace. This plane can be chosen as the impact surface u = 1, and on it the solutionsof (10.28a) and (10.28b) appear as points where the solution pierces the surface (seeFig. 10.18). The section E is designated as

E = : u = 1, u > 0}.

A point (w/, tt) maps to (—e*ui, ti) due to the impact rule and then follows a free flightmotion to w/+i, U+\. A Poincare map P is a rule that transforms points on a section Einto other points on the section; i.e.

P : E E, or P : (iif, 4) (10.29)

Figure 10.18. Solution trajectory in phase space (u,u,t) and points / = 1, 2, 3 where thesolution pierces the Poincare section E.

10.3 / Approach to Chaos - An Unbounded Increase in Number of Attractors 245

If before the initial conditions are repeated there are j impacts in an orbit, the impactvelocity at the end of the j th impact is denoted by Vj:

Vj = Pj(v0), Pj = PP P (j times) (10.30)

where in a set the initial impact velocity is denoted as v0 = (wo, to). Hence a j-impactperiodic motion is described by

pj(vo) = v0.

This orbit has period t\ — t0 = 2jn/ Q.

10.3.4 Stability of Orbit and Bifurcation

When a periodic orbit exists, it can be either stable or unstable; i.e., neighboringorbits may be either attracted or repelled from the periodic orbit. The stability is deter-mined by considering a small perturbation v = v0 -f dv from a periodic orbit i?0. At theend of a period this gives, to first order,

P(v) = P(v0 + dv) ?*vo + (DP)dv.

The derivative DP is the Jacobian of the Poincare map,

f0 dti/diio 1'/o oiii/ olio J

and its elements are calculated using implicit differentiation:

DP = P\_dui/dt0

DP= *- a- a, a- a- a,aui aui ati oui oui ati

.dto dti dto diiQ dti diiQ.

(10.31)

For the impact oscillator this Jacobian has a determinate and trace:

det(Z)F) = eluo/iix

_ 2e*u0 cos(2j7t/ Q) + (1 + e*)[l - C sin(^^0)] sin(2jn/ Q)

The eigenvalues X\ and X2 of DP are roots of the characteristic equation,

X2 - tr(DP) X + det(DP) = 0. (10.32)

Since DP is real, the eigenvalues are either real or a complex conjugate pair.If both eigenvalues are inside the unit circle, the periodic orbit is stable. On the other

hand, if one eigenvalue is outside the unit circle, the orbit is unstable. Bifurcation occurswhen an eigenvalue crosses the unit circle. For periodic orbits ii0 = u\ of the vibro-impactsystem, the determinant det(Z)P) < 1, so that the single period orbit becomes unstableonly if kt = ± 1 (Budd and Lee, 1996; Shaw and Holmes, 1983). If X = 1, the orbitbecomes unstable because of a saddle point bifurcation. Gontier and Toulemonde (1997)show that this occurs when C <CC, which implies that the exciting force is insufficient to

1.6 1.8 2 2.2 2M 2.6 2.8 3 3.2 3Mexcitation frequency, co

Figure 10.19. Frequency response of impact speed Vj for a sphere vibrating against a stop.For (M0 = 0, e* = 0.8) chaotic motion occurs at frequencies a little larger and smaller than

give an amplitude large enough for impact. If X = — 1, the orbit undergoes a subharmonic,or flip, bifurcation.

Figure 10.19 shows the frequency response of the single degree of freedom oscillatorwith a unilateral displacement constraint. For the specified parameters (w0 = 0, e* = 0.8)the velocity of impact Vj has been plotted for 100 successive impacts. The frequencyresponse exhibits resonance at co ^ 2, 4 , . . . , and there is a period-doubling bifurcation atco & 2.65. In the ranges 2.7 < co < 2.95 and 3.05 < co < 3.35 the response seems chaoticand of small amplitude. A small neighborhood around co = 3.0 exhibits stable behaviorwith two impacts per period. Additional detals are available in Budd and Lee (1996).

PROBLEMS

10.1 A particle dropped onto a hard level surface has a coefficient of restitution e*. Thesurface is then inclined at angle 0 from horizontal, and the particle is dropped from aheight ho above the first impact point. Find the period of the ith bounce, and show thatthis can be expressed as tt = 2el(2ho/g)l/2. Hence find expressions for the time tfwhen bouncing ceases and the distance along the plane, Xf, where bouncing ceases.Explain the motion of the particle for t > tf.

10.2 Let the staircase in Sect. 10.1.1 have a thin vertical wall of height h^ at the edge ofeach step of height b, hb < b. Find the initial drop height h* to initiate bouncingwhich continues down the flight of stairs.

10.3 For a pencil (i.e. a hexagonal prism with N = 6) rolling at the steady speed &>*(+),find an expression for the time period between successive collisions. Hence obtainan expression for the mean translational speed of steady rolling as a function of theradius a.

10.4 Consider a prism with N sides that is rolling down a plane inclined at angle 0from horizontal. Find the asymptotic steady state rolling speed as the number ofsides becomes indefinitely large (N -> oo), and compare this solution with that for

Problems 247

a circular cylinder. (Rolling of a circular cylinder involves no energy loss and henceno source of energy dissipation.)

10.5 For a regular prism with TV sides that is rolling down a rough plane, find that toprevent sliding the coefficient of friction // must satisfy

M>tan" , v-r-zzr, yHence find the limiting coefficient of friction for TV = 6 and N = 9. Note that thislimit is independent of the angle of inclination 0.

APPENDIX A

Role of Impact in the Development ofMechanics During the Seventeenth andEighteenth Centuries

He that will not apply new remedies must expect new evils, for timeis the greatest innovator.

Francis Bacon

Before publication of Newton's Principia (1687), dynamics was an empiricalscience; i.e., it consisted of propositions that described observed behavior without anyexplanation for the forces that caused motion. For example, Kepler's laws are kinematicrelations that describe orbital motion. Kepler (1571-1630) discovered these relationsby laboriously fitting various possibilities to the voluminous measurements of planetarymotion that had been recorded by Tycho Brahe. Likewise Galileo stated propositionsdescribing the motion of freely falling bodies. The propositions are based on relatingtransit times for different drop heights to clear ideas of distance, time and translationalvelocity.1 These savants, however, possessed only the vaguest notion of force.

While Galileo recognized that there must be some extended cause for acceleration orretardation, he did not realize that a uniform acceleration was a consequence of a steadyforce; he recognized that there was a cause for acceleration but was not able to relatecause and effect. Indeed, it is difficult to imagine how there could be progress in thisdirection before the creation of calculus.

At the time of Galileo, the topics at the forefront of dynamics were percussion, projec-tile ballistics and celestial mechanics. Each of these topics had technological importancefor warfare, industrial development or navigation. Percussion in particular was concernedwith the terminal ballistics of musket balls as well as the effect of a forging hammer ona workpiece. There were many examples of the powerful effects of percussion, and itwas comparatively simple to measure the apparently instantaneous changes in velocitythat resulted from impact. At that time demonstrations of impact phenomena focused oncollisions of shot fired from cannon rather than the work done by a blacksmith's hammer.

Galileo Galilei (1564-1642)La Dynamique est la science des forces accelratrices or retardatrices,et des mouvements varis qu'elles doivent produire. Cette science estdue entirement aux modernes, et Galile est celui qui en a jet les pre-miers fondemens.

Lagrange, Mec. Anal. I. 221.

1 The concept of velocity at an instant of time is a cornerstone of kinematics that was identified byscholars of Merton College, Oxford, during the fourteenth century (Truesdell, 1968, p. 30).

248

Appendix A / Role of Impact in the Development of Mechanics 249

On the basis of a treatise on the center of gravity of solid bodies, Galileo secureda position as lecturer of mathematics at the University of Pisa; he was then 24 years of age.At Pisa he conducted the famous experiments on uniformly accelerated motion in whichspheres of different weights were simultaneously dropped from the Tower onto a sound-ing board. Similar experiments had been performed since the time of Philoponus (sixthcentury A.D.),2 but Galileo improved on these by simultaneously measuring the time offall from different heights by means of a water clock (clepsydra); these experiments pro-vided data that he used to draw correct deductions regarding the rate of change of distancetraversed by a falling (uniformly accelerated) body. In year 1591 Galileo accepted theprofessorship at Padua. He soon became known for inventions such as the thermometerand the construction of telescopes. With his telescopes he discovered sunspots and foursatellites of Jupiter - this was the first discovery of bodies orbiting another planet. He alsowas renowned for his experiments and teaching in mechanics and Ptolemaic astronomy.His teaching in celestial mechanics, however, was considered heretical by the Church,and in 1615 the Holy Office in Rome condemned him for claiming that the sun was thecenter of the world.3 In that trial Galileo was forbidden to "hold, teach or defend" histheories, and he retained his freedom only by promising to obey. His next book, Dialogosopra i due massimi sistemi dei monda Tolemaico e Copernicano {Dialogue Concerningthe Two Chief World Systems - Ptolemaic and Copernican) was published at Florence in1632. The book took the form of a dialogue between three characters - Salviati, an expertwho usually expresses Galileo's view; Sagredo, an intelligent layman; and Simplicio, aslow-witted student. In order to obtain the inquisitor's imprimatur for publication, Galileowas required to include an argument against the Copernican system. He rashly put thisinto the mouth of Simplicio. Subsequently he was condemned to prison for having floutedthe Inquisitor's order; the Vatican's Inquisitors declared Galileo's writings "an absurd andfalse proposition, that the Sun is at the center of the world and does not move from eastto west, and the earth moves and is not the center of the world."4 In order to gain releasefrom prison he was compelled to publicly recant and accept that the Earth was stationary.5

To circumvent the Church ban, Galileo's subsequent work had to be published outsidethe reach of Rome. His next book was a more general exposition on mechanics. Dis-corsi e Dimostrazioni Matematiche interno due Nuove Scienze, Mecanica & MovimentiLocali (Dialogue Concerning Two New Sciences) was published in Ley den in 1638. Thisremarkable book on statics and dynamics presented a systematic development of me-chanics organized into four chapters, or Days. It seems there was a draft for a fifth Daydealing with the forza della percossa, but the printer was rushed, so the fifth Day neverappeared.6

In the Two New Sciences, Galileo often gropes for terminology to describe the phenom-ena that he observed. Lacking any idea of mass, he frequently uses velocity and momentum

2 M.R. Cohen and I.E. Drabkin, A Source Book in Greek Science, Harvard Univ. Press (1948).3 It was not until 15 March 1616 that the Congress of Cardinal Inquisitors officially condemned the

writings of Copernicus.4 Finocchiaro (1989).5 In 1823 the Vatican quietly authorized publication of a book by G. Settele, a priest, which acknowledged

that Galileo was correct, but it was not until 1992 that the Pope admitted that the church had doneGalileo an injustice.

6 Galileo's dialogue on percussion appears in Drake's translation (Galileo, 1638b, pp. 281-303). Thismaterial was not printed in the original edition.

250 Appendix A / Role of Impact in the Development of Mechanics

synonymously (velocitatem, impetum seu momentum)? Nevertheless the book containsa clear statement of the principle of inertia (Newton's first law of motion)8: " . . . anyvelocity once imparted to a moving body will be rigidly maintained as long as externalcauses of acceleration or retardation are removed, " [243].9

To examine uniformly accelerated motion, Galileo simultaneously dropped different-sized spheres from the Tower of Pisa. In 1604 he wrongly believed that velocity increasedin proportion to distance. By 1638, however, he correctly described uniformly acceleratedmotion as having velocity that increases in proportion to time, and distance traversed thatis proportional to time squared [209].

In regard to impulsive forces in collisions, Galileo states that "the impetus of collision"depends on the relative velocity (velocit delpercuziente); i.e., it depends on the differencebetween the velocities of the colliding bodies [291]. This idea of relative velocity wouldlater be successfully picked up by Wallis and Wren and most thoroughly by Huyghensin papers they published in 1668-1669. Galileo's illustration indicates that he recognizesthat a significant effect of impact - the impetus of collision - is directly proportional tothe normal component of relative velocity:

To what has hitherto been said concerning momenta, blows or shocks of projectiles, wemust add another very important consideration; to determine the force or energy of theshock (forzsa ed energia della percossa) it is not sufficient to consider only the speedof the projectiles, but we must also take into account the nature and condition of thetarget which, in no small degree, determines the efficiency of the blow. First of all it iswell known that the target suffers violence from the speed {velocit) of the projectile inproportion as it partly or entirely stops the motion [291].

Moreover it is to be observed that the account of yielding in the target depends notonly upon the quality of the material, as regards hardness, whether it be iron, lead, wool,etc., but also upon the angle of incidence. If the angle of incidence is such that the shotstrikes at a right angle, the momentum imparted by the blow {impeto del colp) will be amaximum; but if the motion be oblique, that is to say slanting, the blow will be weaker;and more and more so in proportion to the obliquity; for, no matter how hard the materialof the target thus situation, the entire momentum {impeto e moto) of the shot will notbe spent and stopped; the projectile will slide by and will, to some extent, continue itsmotion along the surface of the opposing body [292].6

Despite these observations, Galileo did not decompose the incident velocity into nor-mal and tangential components, nor did he employ the principle of inertia in any analyticalmanner.

A contemporary of Galileo, Professor Marcus Marci, wrote the treatise De ProportioneMotus (published in Prague, 1639) which includes some observations on impact. He wrote

7 According to the Oxford English Dictionary, impeto seu momento meant both moving power andmovement (Jouguet, 1871, Vol. I, p. 106; ref. Dugas, 1957, p. 140). Jouguet observed that Galileosometimes uses impeto to mean "velocity acquired by a body in a given time" and sometimes "thedistance travelled in a certain time as a body accelerates from rest"; i.e., he confuses momentum andkinetic energy acquired during uniform acceleration.

8 The idea of impetus, or momentum, that maintains an existing state of motion was conceived by JohnBuridan, rector of the University of Paris, about 1327. Buridan proposed that a moving body has animpetus that is proportional to the velocity and heaviness of the body.

9 Numbers in square brackets are Galileo's section numbers. This quotation is from the rather freetranslation by Crew and de Salvio in 1914 (Galilei, 1638a); a later translation by Drake in 1976(Galilei, 1638b) is more literal.


Figure A.I. Illustration from De Proportione Motus by Marcus Marci. The illustration ofimpact of a cannon ball against an identical sphere depicts their positions at equal intervalsof time before and after collision. The second illustration shows a cue stick striking a billiardball e, which in its turn will strike a set of tightly packed balls; this is reminiscent of thepresent day "Newton cradle." (Illustration from Mach, 1883.)

that in an elastic collision between two identical bodies, if before the collision one bodymoves with speed v while the other is stationary, then after the collision the reverse istrue - the body that was moving is now stationary while the second body moves awaywith speed v. Marci's illustration of this observation is illustrated as Fig. A.I. It showsthe positions of the balls at equal intervals of time before and after a collinear collisionon the table top.

Rene Descartes (1596-1650)

Descartes is known as a philosopher, mathematician and physicist; the rectilinearor Cartesian coordinate system was named after him. He is remembered most for teach-ing an aesthetically satisfying system of world order governed by a minimum numberof universal laws of nature. He looked for scientific explanation based on a few funda-mental principles. Descartes's greatest failing was to rely solely on reasoning and neglectexperience.

Conservation of Quantity of Motion

Independently of Galileo, Descartes also addressed the motion of bodies in theabsence of active forces. In 1829 he wrote to Father Mersenne, "I suppose that the motionthat is once impressed on a body remains there forever if it is not destroyed by somemeans. In other words, that something which has started to move in a vacuum will moveindefinitely and with the same velocity." Later, in Principia Philosophiae (1644), he madethe following laws a centerpiece of his mechanics:

The law of nature: that everything whatever, - so far as depends on it, - always perseveres

in the same state; and thus whatever once moves always continues to move.

The second law of nature: that all motion by itself alone is rectilinear


DFigure A.2. Descartes's analysis of a sphere A reflecting from a "hard" plane surface CBEduring elastic impact. The illustration uses a vector diagram for velocities which conservemomentum in the tangential direction and have a normal component of velocity at separationwhich is the negative of that at incidence.

Of these the first [law] is that everything whatever, insofar as it is simple and indivisible,remains, so far as depends on it, always in the same state, nor ever changes [in its state]save by external causes. (Transl. in Cohen and Drabkin, 1948, p. 184.)

Here motion or quantity of motion equals the product of the weight of a body and itsvelocity; i.e., this is a precursor to or early form of the principle of inertia. Descartes doesnot say what can cause velocity to change or how it changes when the body is acted on byan "external cause". At that time the word force was used quite loosely. By it Descartesusually meant the work required to lift a weight to some height (Dugas, 1957, p. 156).

Impact of Bodies

In his book Dioptrics, Descartes relied on an analogy between reflections of lightrays and the rebound of a ball that collides with a surface at an angle of incidence. He useda spatial diagram (Fig. A.2) to illustrate the laws of reflection. He considered a ball thatcollides with a plane surface CBE; the ball moves steadily from point A until it collideswith the surface at B and then rebounds. Descartes explicitly neglects "the heaviness, thesize and the shape" of the ball, and supposed the earth to be "perfectly hard and flat". Heasserts that the ball is reflected on meeting the earth and its "determination to tend to B"is modified "without there being any other alteration of the force of its motion than this".

He states that "the determination to move towards some direction, like the movement,to be divided into all the parts of which it can be imagined that it is composed". The ball isthus driven by two "determinations"; one makes it descend, and the other makes it travelparallel to the surface. Impact with the surface disturbs the first but has no effect on thesecond, in accord with the conservation of the quantity of motion in the second direction.

Since collisions obviously modified the quantity of motion, the impact event hadsubstantial importance in Principia Philosophiae. Descartes proposed seven rules fordirect impact of elastic bodies:10

(1) If two equal bodies impinge on one another with equal velocity, they recoil, eachwith its own velocity.

10 Dugas (1957, p. 162).


(2) If one of the two is greater than the other, and the velocities equal, the lesseralone will recoil, and both will move in the same direction with the velocity theypossessed before impact.

(3) If two equal bodies impinge on one another with unequal velocities, the slowerwill be carried along in such a way that their common velocity will be equal tohalf the sum of the velocities they possessed before impact.

(4) If one of the two bodies is at rest and another impinges on it, this latter will recoilwithout communicating any motion to it.

(5) If a body at rest is impinged on by a greater body, it will be carried along andboth will move in the same direction with a velocity which will be to that of theimpinging body as the mass of the latter is to the sum of the masses of each body.

(6) If a body C is at rest and is hit by an equal body B, the latter will push C alongand, at the same time, C will reflect B. If B has a velocity 4 it gives a velocity 1to C and itself moves backwards with velocity 3.

(7) (A seventh rule related velocities of two unequal bodies initially traveling in thesame direction.)

The greatest importance of these rules for subsequent development of this subject wasthat many were easily shown to be wrong; it was apparent that they did not representthe results of simple experiments. This stimulated further examination of impact byHuyghens, Newton, Wallis and Wren. Today only one of these rules, (3), can be acceptedas having any validity.

Descartes himself apparently suspected that some of these rules were not in accordwith experimental evidence. He dismissed his critics, however, with a disdainful commentthat his rules applied to ideal collisions and hence they were not entirely representativeof anything that could be measured.

John Wallis (1616-1703)

Descartes's speculations led to 10 laws of nature; most are laws of impact forelastic particles. Many of his impact laws are vague, and most are wrong. Having inmind the mistaken concepts proposed by Descartes, The Royal Society, in 1668, initi-ated an investigation into the laws relating to the collision of bodies. Wallis, Wren andHuyghens replied with papers published in 1668 and 1669. Wallis dealt with perfectlyplastic collisions, while Huyghens and Wren dealt with elastic collisions.

In 1668 Wallis was the Savilian Professor of Geometry at Oxford and a foundingmember of The Royal Society of London. He had been a scholar at Emmanuel College,Cambridge, and then was ordained in the Church of England and later appointed to thechair in Oxford by Oliver Cromwell in recognition of his discovery of a method of de-ciphering codes during the civil war. He wrote an important book on algebra (ArithmeticaInfinitorum) and later a treatise on mechanics (Mechanica sive Tractatus de Motu, 1670);the historian Duhem calls the latter "the most complete and the most systematic [bookon mechanics] which had been written since the time of Stevin". In this treatise Wallisextended the idea of force to include forces other than gravity; moreover his propositionsexpressed for the first time a relation between force and momentum in bodies beingaccelerated from rest (although he calculated momentum as the product of velocity andweight rather than mass).


Wallis's Royal Society paper dealt with impact of what he called perfectly hard bodies',he explained that this meant that during the compression phase of impact they did notstore energy in elastic deformations that subsequently would drive the bodies apart duringrestitution. He distinguished these collisions from those of soft or elastic bodies where" . . . part of the contact force is expended in deforming the bodies during compression".Thus Wallis considers inelastic impact where the bodies have a common velocity whenimpact terminates; he does not however seem to recognize that the reaction that changesthe momentum of each of the colliding bodies during compression also changes the kineticenergy of the system - the kinetic energy lost during compression goes into work donein deforming the bodies, whether or not that work is reversible.

For impact of hard or perfectly inelastic bodies Wallis wrote, "If a body in motioncollides with a body at rest, and the latter is such that it is not moving nor prevented frommoving by any external cause, after the impact the two bodies will go together with avelocity which is given by the following calculation. Divide the momentum furnished bythe product of the weight and the velocity of the body which is moving by weight ofthe two bodies taken together. You will have the velocity after impact." For two bodiesmoving collinearly in the same direction, he let body B with weight W and velocity i?'be struck from behind by body A with weight W and velocity v; then if the bodies arehard, he said they had a common final velocity,

Wv + W'v'w + w '

In Mechanica sive Tractatus de Motu, Wallis dealt also with elastic impact of softbodies, where he commented on how the reaction force deforms the body during com-pression. He noted that this elastic deformation causes the same change in momentumduring restitution as had occurred previously during compression. Further, he suggests thecase of partly elastic collisions where the velocity changes during restitution fall betweenthe elastic and inelastic limits.

Christopher Wren (1632-1723)

Wren had a very distinguished career in science before turning increasinglytowards architecture after the age of 30. Today he is most widely known for planningthe rebuilding of London after the Great Fire of 1666, and as the architect of St. Paul'sCathedral (London), Trinity Library (Cambridge), the Sheldonian Theatre (Oxford) andmany other churches and public buildings. He started out, however, as Fellow in mathe-matics and astronomy at Gresham College, London, from 1657 to 1661, before becomingSavilian Professor of Astronomy, 1661-1673. Wren and Hooke (an Oxford don), wereamong the group of mathematicians and natural philosophers who met regularly in roomsat Gresham College, London. A merger of these meetings and those of the Oxford exper-imental club resulted in the founding of The Royal Society of London in 1660. Wren'sscientific work brought him the presidency of The Royal Society from 1680 to 1682. Itwas as a leading mechanician and founding member of the society that he contributed apaper on elastic collisions that was read before The Royal Society and published (togetherwith Wallis's paper) in the Proceedings of November-December 1668.

Like the subsequent paper by Huyghens, Wren's considered a collision between twoelastic bodies (spheres) with different weights. He began from a concept of "proper


velocity" which, for any body, is inversely proportional to weight. He regarded impactof two bodies, each traveling at its proper velocity, as equivalent to a balance oscillatingabout its center of gravity; i.e., he saw collisions from the viewpoint of the law of the lever.Wren expressed this analogy in diagrams illustrating the correct relations he developedfor the speeds at separation, given any incident speeds. The greatest virtue of Wren'spaper, however, was that it described experiments that he used to develop his ideas andvalidate the results. As Newton wrote in the scholium following the laws of motion in hisPrincipia, "Wren proved the truth of these rules before the Royal Society by means of anexperiment with pendulums." Wren's experiment involved two pendula of equal lengthbut different weights that are released from different heights - the heights were chosensuch that each pendulum bob rebounded to the same proportion of its initial release height.He demonstrated that two elastic bodies approaching each other with velocities that areinversely proportional to the weight of the other body result in simple reversal of velocityof both bodies when they collide. While Wren came up with essentially the same ruleof impact as Huyghens, he (unlike Huyghens) did not present the general principles thatwere used to derive these results. As was his wont, this 1 ^ page paper was the only thingthat Wren ever wrote on this subject.

Christian Huyghens, 1629-1695

Huyghens was born in Holland but spent much of his life in Paris. He was an em-inent mathematician and physicist who is remembered for his concepts in physical opticsand the wave theory of light. Using his knowledge of optics, he constructed telescopeswith improved lenses that allowed him to discern that the planet Saturn was surroundedby an annular disk rather than a pair of moons. In 1655 he discovered Titan - the firstmoon of Saturn to be identified.

In 1652-1657 Huyghens was working on an improved description of motion andtreatment of impact.11 From 1652 he distinguished between momentum ("quantity ofmovement") and kinetic energy, but did not understand the relationship between thesevariables. While Galileo is credited with suggesting that if no external forces act ona system, then translational momentum is conserved, Huyghens used this concept andconservation of total mechanical energy (conservatio vis ascendentis), without pointingout that these concepts had any special significance (Mach, 1919).1213 (SubsequentlyLiebnitz named the latter principle as vis viva or living force - it is presently known as theconservation of energy.) About 1659 Huyghens distinguished between mass and weight.14

Most of these investigations did not have as wide an influence as they might have had ifHuyghens had not been so reticent to publish. His posthumous book on impact, Tractus deMotu Corporum exPercussione, was written almost entirely before 1659 but not publisheduntil 1703. By that time Newton's Principia had already been in print for 15 years.

1 ] Correspondence with Schooten, Gutschoven and Slusius (see Bell, 1953, p. 110).12 In Huyghens's application the term conservatio vis ascendentis meant that in a conservative system

(e.g., a pendulum swinging in a gravitational field), any displacement of a body in a direction parallelto gravity resulted in work that changed the kinetic energy. This work was just sufficient to raise theswinging pendulum back to the height from which it was first released.

13 Regarding previous statements of a idea of conservation of translational momentum, see footnote 8.14 Crew (1935) remarks that this is probably the earliest suggestion of such a distinction. (See "De Vi

Centrifuga", Oeuvres completes de Huyghens, Dutch Scientific Society, Vol. XVI, p. 266.)


Figure A.3. Drawing from De Motu Corporum ex Corporum ex Percussione illustratingimpact of bobs of two pendula supported in a moving boat as viewed from a stationaryreference frame (the man on the bank).

Huyghens was widely recognized for his mathematical skills and physical intuition.He was an early member of The Royal Society of London and a founding memberof the French Academy. In 1668 he was a late addition to the group of three savantscommissioned by The Royal Society to clarify the phenomena of impact - because oftheir methods, Newton referred to Huyghens, Wallis and Wren as the three geometers.Huyghens's paper on the subject was published in March 1669, and it treated impact ofelastic particles. While each of the three authors produced a correct solution, Huyghens'spaper had the major distinction of recognizing that all changes in velocity during collisiondepend on the relative velocity at the instant when the bodies first come into contact. Hefreely uses translating coordinate systems in order to express velocity changes relative to acoordinate system moving in the normal direction with the center of gravity of the system,as shown in Fig. A.3. Regarding conservation of momentum in a collision between twoelastic particles, he says " . . . the common center of gravity of the bodies advances alwaysequally towards the same side in a straight line before and after impact". In arriving athis propositions, Huyghens uses two concepts:

(i) He explicitly gives a translational velocity to the entire system such that thecenter of gravity of the system is stationary both before and after the collision.

(ii) He considers that for equal but opposite efforts applied to the two bodies, relativeto the coordinate system moving with the center of gravity, each body has a speedof approach that is inversely proportional to its weight; i.e., he gives the bodiesequal but opposite initial quantities of motion relative to the center of gravity.

The latter proposition is similar to but more general than the idea behind Wren'sdemonstration experiment. Here, for any initial velocity of each of the colliding bodies,Huyghens calculates the speed of the steadily moving reference frame such that thedifference in velocities satisfies proposition (ii).


MA=3MB

Figure A.4. Marriotte's illustration of direct collision of balls A and B with unequal massthat approach each other along a line. The horizontal axis represents time. The positions ofthe balls are indicated at equal intervals of time before and after impact at E. The speed ofthe center of mass (shown as a dashed line MEP) is constant.

Although this book is about impact, it is well to acknowledge that in mechanics,Huyghens is best known as the father (if not the inventor)15 of the pendulum clock. Theclocks which Huyghens invented were intended for use at sea - mainly for determinationof longitude. In this connection he went beyond the dynamics of particles and providedan expression to calculate the moment of inertia of solid bodies (by summation).16 Inhis magnum opus on the pendulum clock, Horologium Oscillatorium (1673), he namedand provided the first correct explanation for centrifugal force. Subsequently Newtonacknowledged with chagrin, "What Mr. Huyghens has published since about centrifugalforce I suppose he had before me."

Edme Mariotte (1620-1684)

Mariotte was a Roman Catholic abbe who was best known for experimentation.While not in the first rank of contributors who developed an understanding of principlesdescribing impact, he solved a number of problems of elastic collisions in a mannersimilar to Huyghens and wrote a useful book, Traite de la Percussion ou chocq des Corps(Paris, 1673), that gives a clear presentation of the geometric method of analysis in vogueat that time.17 Mariotte's analysis was based on a principle of invariability of the quantityof motion (conservation of translational momentum of the system) which clearly resolvedthe velocities of colliding spheres into components normal and tangential to the plane ofthe common tangent at the contact point.

Figure A.4 is an example of a collision at a normal angle of incidence between twoelastic balls, A and B, where the masses are unequal: MA = 3MB. The vertical axes onthe left and right give the positions of the balls at instants of time r before and after

15 Galileo had designed a clock escapement based on the pendulum, but it is unlikely that this was everconstructed.

16 Wallis introduced him to this idea in a letter written 1 January 1659, where he described his methodof finding a moment of weight about a certain axis as imagining the body separated into segments andforming a series in which each term was the weight of a segment multiplied by a distance. To find thedistance from the axis to the center of gravity Wallis divided the sum of the moments of weight of theindividual segments about the axis by the total weight of the body (Bell, 1953, p. 125).

17 Mariotte redefined Wallis's confusing terminology for elastic and inelastic impact. He used the moreaccurate descriptions^fe;c/&/e and resilient for elastic collisions and flexible and not resilient for inelasticcollisions.


the impact at E, while the horizontal axis represents time. The two balls approach eachother at a speed proportional to length AB, and after impact they separate at a speedproportional to length JL. A dashed line MEP represents the motion of the center of massduring this period; i.e., lengths AM and MB are inversely proportional to the massesof the adjacent balls, A and B, respectively. Hence Mariotte determines the separationvelocities by considering that any changes in velocity depend only on velocity differencesrelative to a reference frame moving with the center of mass; for elastic collisions (theonly case he considers) he simply reflects the normal component of incident velocityabout the vertical line through E, where changes in velocity are relative to the steadilymoving center of mass of the system. It is likely that Mariotte followed Huyghens inrecognizing that changes in velocity during collision are independent of the referenceframe of the observer.

Marriotte's diagram (Fig. A.5) represents two examples of oblique impact of a movingball A against a stationary ball B. Here again the lines represent velocity vectors for eachball before and after impact. Following collision, ball B moves in the direction of thecommon normal with speed proportional to BE, while ball A has velocity proportional tobF. The center of mass of the system has a postcollision velocity bG given by conserva-tion of quantity of motion, so that the speeds are related by bG/Ab = MA/(MA + MB).Mariotte illustrates cases of (i) balls of equal mass (MA = MB) and (ii) balls of unequalmass (MA > MB). In these examples he follows a Cartesian principle by decompos-ing velocities into components normal and tangential to the common tangent plane atthe point of contact. In each case Ab is the incident velocity of ball A with compo-nents AD and Db in the tangential and normal directions respectively. Moreover, in bothcases the vertical (tangential) component of velocity for A is the same before and aftercollision; i.e., bF = AD, or LF = AD. Since the collision is elastic, the relative normalvelocity after impact equals that before impact. Hence Mariotte obtains the postcollisionnormal component for each ball by separating the incident relative velocity Db into twoparts that are inversely proportional to the masses; he then says that following collisionthis same proportionality applies to the speeds relative to that of the center of mass;i.e.,

IL _ DH _ MB

IE ~ Hb ~ M A '

Thus he says that for case (ii) triangles FEL = AbD and GEI = CbH. Once again thisproportionality expresses the conservation of quantity of motion.

Isaac Newton (1642-1727)

Newton came from the farming village of Woolsthorpe near Grantham, England.His family were moderately well off, but uneducated, landowners. His father died beforehis birth, and his mother remarried an elderly vicar who wanted nothing to do with theboy. Young Isaac was raised by his grandmother, who taught him to read and write. In1661 he entered Trinity College, Cambridge as a sizar (i.e., he earned most of the moneyfor his expenses by performing menial tasks for Fellows, Fellow commoners and affluentstudent pensioners). After three years he became a scholar, and in 1665 he took his B.A.;that same year, because of the Plague, he returned home for two years to reflect, study anddevelop his ideas on mathematics, optics and mechanics. This enforced period of solitarystudy was most fruitful - he worked on many of those concepts that would later become


B

(ii) MA

b

-a*

H i

Figure A.5. Oblique elastic impact of moving ball A with stationary ball B that has (i) equalmass or (ii) unequal mass, MA = 2MB . In this figure Marriotte uses arrows to represent theincident and separation velocities of both A and B. At incidence the normal component ofrelative velocity is Db. At separation, ball A has velocity bF, ball B has velocity bE, and thecenter of mass of the system has velocity bG.

the more important contributions in the Principia. Following his return to Cambridgein 1667, he was elected a Fellow of his college and subsequently succeeded his mentor,Barrow, as Lucasian Professor of Mathematics. He published his Theory of Light in 1671,shortly before being elected a Fellow of The Royal Society. The Principia was publishedin 1687. In later years his extraordinary contribution to science was recognized by hiscountry when he became Master of the Mint and President of The Royal Society. A mostthorough and entertaining biography of Newton was written by Westfall (1980).

Leaving aside the individual developments that Newton expressed so clearly in thePrincipia, his major contribution was a new approach to problems - he sought to explaincauses of phenomena rather than merely describing them. He was a master of synthesis;e.g., the law of universal gravitation was synthesized as a consequence of the third lawof motion. Moreover, he continually tested hypotheses against experimental evidence.Nowhere is use of experiments to validate theory more clearly demonstrated than in histreatment of impact.

Newton's third law of motion states: To every action there is always an equal andopposed reaction. In a scholium, or example, illustrating this law, Newton analyzedan experiment verifying that the law applies to collisions also. He provided a drawing(Fig. A.6) describing his experiment which involved two spherical bodies A and B hung atthe end of 10 foot long cords, CA and DB. His description of his experimental techniquedemonstrates concern for eliminating experimental errors; he says "we are to have dueregard as well to the resistance of the air as to the elastic force of the concurring bodies."18

18 Excerpts from the Principia are from Cajori's 1934 revision of an original translation by Motte (1729)(see Newton, I.).


EG CD F H

Figure A.6. Illustration from the Principia of Newton's experiment where pendulum bobswith different weights collide. Incident and separation speeds were calculated from mea-sured heights of descent and reascent by Galileo's expression of proportionality betweensquare of speed and height of descent. Letters R, S,T, V, etc. were used to correct themeasurements in order to allow for air resistance.

In order to correct for air resistance, the free-swinging path EAF of body A is subdividedas follows: if body A, when released from R, returns to V after one free swing, thenthe translational speed of A at the bottom of its swing (impact point) will be the sameas if it had been released in vacuo from 5, where RS: RV = 3:8. Similarly supposethat, after reflection from body B, A comes to place s. Then if a free-swinging pendulumA returns to r when released from v, the corrected height after the collision is t ratherthan the measurement s. After this demonstration of his care, Newton goes on to say;"Thus trying the thing with pendulums of 10 feet, in unequal as well as equal bodies, andmaking the bodies to concur after a descent through large spaces, as of 8, 12, or 16 feet,I found always, without an error of 3 inches, that when the bodies concurred togetherdirectly, equal changes towards the contrary parts were produced in their motions, and,of consequence, that the action and reaction were always equal."

At this point Newton goes on to describe his results from impact experiments withbodies composed of various materials. This may have been the first time the materialused in experiments was actually named; neither Wren nor Mariotte had provided thisinformation. Of course they considered perfectly elastic impacts only. Newton is carefulto distinguish his results from those for elastic collisions. He writes,

. . . I must add, that the experiments we have been describing, by no means dependingupon that quality of hardness, do succeed as well in soft as in hard bodies. For if the ruleis to be tried in bodies not perfectly hard we are only to diminish the reflection in such acertain proportion as the quantity of the elastic force requires. By the theory of Wren andHuyghens, bodies absolutely hard return one from another with the same velocity withwhich they meet. But this may be affirmed with more certainty of bodies perfectly elastic.In bodies imperfectly elastic the velocity of the return is to be diminished together withthe elastic force; because that force (except when the parts of bodies are bruised by theirimpact, or suffer some such extension as happens under the strokes of a hammer) is (asfar as I can perceive) certain and determined, and makes the bodies to return one from theother with a relative velocity, which is in a given ratio to that relative velocity with whichthey met. This I tried in balls of wool, made up tightly, and measuring their reflection, Idetermined the quantity of their elastic force; and then, according to this force, estimatedthe reflections that ought to happen in other cases of impact. And with this computationother experiments made afterwards did accordingly agree; the balls always receding onefrom the other with a relative velocity, which was to the relative velocity with which theymet as about 5 to 9. Balls of steel returned with almost the same velocity; those of corkwith a velocity something less; but in balls of glass the proportion was as about 15 to 16.


And thus the third Law, so far as it regards percussions and reflections, is proved by atheory exactly agreeing with experience (Motte).19

Thus Newton uses the principle of conservation of translational momentum, but in additionthe third law, to find that irrespective of hardness of the colliding bodies, the changes invelocity are always in proportion to the relative velocity at incidence. He finds that theproportion of this relative velocity which is recovered depends on the bodies - i.e., hisproportionality is essentially a material property. Here it seems that he was too enamoredof large pendulum swings, which could more easily be measured, and missed the fact thathis conclusion about this proportionality being independent of relative speed at incidenceis logically inconsistent in the limit of indefinitely small incident speeds. Nevertheless, werecognize that in comparison with previous writing on our topic, these passages represent atremendous leap in style, thoroughness and clarity in regard to relating cause and effect.20

Leonhard Euler (1707-1783)

Euler was a student at the University of Basel, where he was tutored by JohnBernoulli, professor of mathematics. Truesdell (1968) quotes from a 1720 letter fromBernoulli's son, Daniel: "Mr. Euler . . . of Basel is a student of my father who will dohim much honour." In 1727 Euler was appointed professor of physics of the Academy ofSciences in St. Petersburg by Catherine I, a patron of science. He went there, joining hisfriend, Daniel Bernoulli who had taken up the chair of mathematics two years previously.In year 1732, D. Bernoulli moved to Berlin to take up chairs, first in anatomy and laterin physics. Euler was elected to fill the vacant chair in mathematics in St. Petersburg.He remained there until 1741, when Frederick the Great called him to Berlin. In 1766he returned to St. Petersburg at the request of Catherine the Great, and he soon wentcompletely blind, but nonetheless continued to produce important papers. He almostcertainly is the most prolific writer of mathematical papers of any period. His work isnow readily accessible through Opera Omnia, edited by Charles Blanc and published bySocietatis Scientiarum Naturalium Helveticae.

The first investigation of dynamics of collisions between rigid bodies as distinct fromparticle collisions is due to Euler in a paper submitted to the Academy of St. Petersburg in1737.21 A root of this work goes back to Euler's mentor, John (Johan or Jean) Bernoulli,

19 Newton's reference to steel is not specific, but steel was produced as early as 1000 B.C. by heatingiron objects in contact with charcoal. This process of surface carbonization was initially used by theChalbyes in Northern Anatolia to improve the edge-holding properties of iron tools (Wertime, T.A.and Muhly, J.D., Coming of Age of Iron, Yale Univ. Press, 1980).

20 Newton could achieve clarity because he firmly grasped the concepts. Barbour (1989, p. 678) remarksthat Kepler recognized that dynamic behavior of individual bodies must be characterised by a quantitywhich measures its resistance to applied force; he introduced the term "laziness", or inertia, for thisquantity. It was Newton who took the step of replacing "the concept of laziness with respect to motionby laziness with respect to change in motion" in order to arrive at inertial mass. Previously Huyghenshad expressed a similar idea (see footnote 12).

21 Herivel describes an unpublished manuscript, predating publication of the Principia, where Newtonanalyzes planar impact of two elastic bodies of arbitrary shape. He does not, however, give a method ofcalculating the radius of the "equator of reflected circulation" (i.e. the radius of gyration). Herivel hasshown how Newton's variables can be defined so that the changes in angular and translational velocityof each body are obtained as functions of the radii of the equators of reflected circulation - this givesthe correct result, although Newton's method is obscure.


the Professor of Mathematics at Basel and father of Daniel Bernoulli, who is frequentlycredited with formulating in 1703 the concept of moment of inertia for a body.22

Despite the invention of calculus by Leibnitz and Newton, the problem of mechanicsof impact or percussion remained of central importance. The Royal Academy of Sciencesin Paris biannually awarded a prize for the most outstanding paper; in 1724 and 1726 theprizes were for papers on percussion. Colin Maclaurin, professor of mathematics at theUniversity of Aberdeen, was awarded the prize in 1724 (over John Bernoulli) for his paperentitled Demonstration des lois du choc des corps. He expressed that the interaction forceson colliding bodies are equal in magnitude but opposed in direction (Newton's third law)and used the physical construct of an elastic spring between the contact points in orderto obtain changes in velocity during compression and restitution phases of collision. Thespring represented a small compliant region - implicitly the tangential compliance wasassumed to be negligible.23 Maclaurin says that he won the prize for clarifying that duringa collision the changes in velocity depend on relative velocity only - but omits saying thatthis point had previously been made by Huyghens. The recipient of the next Academyprize was Pere Maziere in 1726. That year the runner-up was again John Bernoulli, whoalso had his paper published. Bernoulli's paper, Discours sur les loix de la communicationdu mouvement, related velocities before and after collision by means of conservation ofenergy; consequently, his analysis was unintentionally limited to elastic collisions. Thefact that Bernoulli did not comment on or otherwise acknowledge this limitation broughtforth jeers of derision from Maclaurin24 and a pointed criticism from Robins.25 Despitethese setbacks, J. Bernoulli was an indefatigable campaigner, and during his lifetime hewon 10 French Academy prizes.

Euler submitted his paper on planar collisions between two rigid bodies, De com-municatione motus in collisione corporum sese non directe percutinentium, in 1737; itwas published in the Proceedings of the Scientific Academy of St. Petersburg (1744). Inthis communication he considers planar collisions between two bodies which can rotateas well as translate; the rotational inertia of each body is represented by a moment ofinertia about the center of mass - a parameter he terms the " . . . sitque horum factorumaggregatum = S". In this problem there is negligible friction, so the interaction force atthe contact point is perpendicular to the common tangent plane. The bodies have centersof mass that are not on the line of action of the contact force; i.e., the impact configurationis noncollinear. Euler described his analytical method with Fig. A.7.

Euler considers limiting cases of perfectly elastic and perfectly inelastic collisions.His analysis supposes that at initial contact (incidence) the two bodies are separatedby an infinitesimal elastic element - an artifice originally introduced by Maclaurin torepresent the normal compliance of an infinitesimally small deforming region. With thisartifice each body has a contact point. Although these points are coincident, there is arelative velocity between them that varies during the collision. The elastic element is

22 Earlier Wallis and Huyghens had expressed the moment of inertia of a body about an axis of rotationas a property that they calculated by summation over all parts of the body.

23 Whereas Galileo envisioned elastic deformations of colliding bodies as homogeneous, so that whencompressed together, spherical bodies became spheroidal, Euler recognized that stresses decreasedrapidly with increasing distance from the contact area, so that only a small part of the body around thecontact area is significantly deformed.

24 Maclaurin (1748, p. 192).25 Robins, B., The Present State of the Republic of Letters, May 1728.


(a)

P P

Figure A.7. Euler's illustration of eccentric collision of a sphere B moving at speed fiBagainst stationary elliptical body with center of gravity at A. After an elastic collision therelative velocity between the contact points turns out to be equal but opposite to what it wasat incidence; i.e., Cg = fiB. The normal velocity at separation is composed of translationalvelocities Cb and yg of the centers of mass plus the translational velocity Gy at C on bodyA due to the product of the angular speed of A and the perpendicular distance AG to theline of action. Line segments Cb and yg have lengths that are inversely proportional to themasses the corresponding bodies.

oriented in a direction n normal to the common tangent plane at the contact point, so theinteraction force F, which results from compressing this compliant element, acts solelyin the direction of the normal to the common tangent plane, F = Fnn. Euler was thefirst to consider interaction forces with a line of action that does not pass through bothcenters of mass, a consideration that becomes important if the shapes of the collidingbodies are not spherical. The changes in relative velocity are due to changes in bothtranslational velocity of each center of mass and rotational velocity of each body. Eulerassumes that the collision period is divided into an initial compression period, where thenormal component of relative velocity vn = v • n between the contact points is decreasingas the contact points approach each other, and a subsequent period of restitution, wherethis component of relative velocity is increasing as the contact points separate.26 Thetransition between these periods occurs when the relative velocity at the point of contacthas a normal component that vanishes.

Using a time-dependent analysis, Euler obtained the relative velocity at the contactpoint as a function of work done on the deformable spring by the contact force F. His elasticanalysis assumes the contact force is reversible, so in an elastic collision fo

f Fnvn dt = 0.He relates the work done by the contact force to an expression proportional to changesin the kinetic energy of the system. Euler's determination of the final or separation con-ditions on the basis of the ratio of kinetic energy restored to the system by contact forceduring restitution in comparison with the energy absorbed during compression is a majordeparture from Newton's kinematic law of restitution. Euler simply used this ratio forthese limiting cases without commenting on the distinction from the kinematic law of

26 The idea of dividing the impact into separate periods of compression and restitution seems to haveoriginated with Nicolas de Malebranche (1675-1712), a Parisian priest.


restitution. After one takes the step of modeling the collision with an elastic elementbetween the contact points, it seems natural to use the energy stored and subsequentlyreleased by this element to define the terminal conditions at separation.

In 1744 Benjamin Robins wrote New Principles of Gunnery, which among other thingsdescribed the use of a ballistic pendulum to measure the momentum of a cannon ball atvarious distances from the muzzle of a gun. With these measurements he established thesignificance of air drag in calculating the trajectory of projectiles if the muzzle velocity islarger than 130 ms"1. Also he discussed the aerodynamic lift and consequent curvatureof the path of a projectile which is spinning about a transverse axis - what is now termedthe Magnus effect. Euler annotated and corrected some technical errors in Robins's book;his annotated version (in German) is five times the length of the original. Nevertheless,W. Johnson (1986) suggests that Euler's version contains little that is original. The an-notated version was translated into English by Hugh Brown (published as The TruePrinciples of Gunnery, London, 1777), and into French by J.L. Lombard (1783).

Historical References

Barbour, J.B. (1989) Absolute or Relative Motion? Vol. 1: Discovery of Dynamics. Cam-bridge Univ. Press.

Bell, E.T. (1953) Men of Mathematics, Vol. I, Penguin.Blanc, C. (1957-58) Opera Omnia of E. Euler, Societatis Scientiarum Naturalium Hel-

vetica. Teubner.Cajori, E (1934) transl. of Principia Mathematica by I. Newton, 1687, Univ. of California

Press, Berkeley.Cohen, M.R. and Drabkin, I.E. (1948) A Source Book in Greek Science, Harvard Univ.

Press.Coriolis, G. (1835) Theorie Mathematique des Ejfets du Jeu de Billiard, Carilian-Joeury,

Paris.Crew, H. (1935) The Rise of Modern Physics, Bailliere, Tindall and Cox, London.Dugas, R. (1957) History of Mechanics (transl. by J.R. Maddox), Routledge & Kegan

Paul, London.Duhem, P. (1903) UEvolution de la Mecanique, Joanin, ParisEuler, L. (1737) "De communicatione motus in collisione corporum sese non directe per-

cutientium," Comment. Acad. Sci. Petropolitanae 9,50-76 (1744); in CommentationesMechanicae (ed. C. Blanc), Societatis Scientiarum Naturalium Helveticae.

Finocchiaro, M.A. (1989) Galileo Affair: A Documentary History, Univ. of CaliforniaPress, Berkeley.

Galilei, Galileo (1632) Dialogo sopra i due massini Sistemi del Mondo, Tolemaico eCopernicano (Dialogue Concerning the Two Chief World Systems: Ptolomaic andCopernican) Florence.

Galilei, Galileo (1638a), Dialogues Concerning Two New Sciences (transl. by H. Crewand A. de Salvio), Macmillan (1914).

Galilei, Galileo (1638b), Dialogues Concerning Two New Sciences (transl. by S. Drake),Univ. of Wisconsin Press (1974).

Herivel, J. (1965) The Background to Newton's Principia, Oxford Univ. Press.Huyghens, C. (1703) De Moto Corporum ex Corporum ex Percussione in Ovevres Com-

pletes, Vol. 16 (1929) (English transl. by R. J. Blackwell, Isis 68, 574-97, 1977).


Johnson, W. (1986) "Benjamin Robins: new principles of gunnery", Int. J. Impact Engng.4, 205-219.

Johnson, W. (1992), "Benjamin Robins (18th century founder of scientific ballistics):European dimensions and past and future perceptions", Int. J. Impact Engng. 12, 293-324.

Jouguet, E. (1871) Lectures de Mecanique, Johnson Reprint Corp., New York (1966).Mach, E. (1883) Die Meckanik in Ihrer Enwicklung Historisch-Kritisch Dargestellt (The

Science of Mechanics, transl. by T. J. McCormack) Open Court, La Salle, 111. (1960).Mach, E. (1919), The Science of Mechanics (transl. by T.J. McCormack), 224.Maclaurin, C. (1748) An Account of Newton's Philosophical Discoveries, Patrick

Murdoch.Malebranche, N. (1960) Oeuvres Completes de Malebranche: Pieces Jointes, Ecrits

Divers. Des Lois du Mouvement (ed. Pierre Costabel), Vrin, Paris.Marci, M. (1639) De Proportione Motus, Prague.Newton, I. (1687) Mathematical Principles of Natural Philosophy and his System of

the World (transl. by A. Motte; revised and annotated by F. Cajori, 1934), Univ. ofCalifornia Press, Berkeley.

Parsons, W.B. (1939) Engineers and Engineering in the Renaissance, MIT Press.Robins, B. (1742) New Principles of Gunnery, J. Nourse, London; Richmond Publ.,

Richmond, UK (1972).Truesdell, C. (1968) Essays in the History of Mechanics, Springer-Verlag.Wertime, T.A. and Muhly, J.D. (1980) Coming of Age of Iron, Yale Univ. Press.Westfall, R.S. (1980) Never at Rest, Cambridge Univ. Press.Wilson, J. (1761) Mathematical Tracts of the Late Benjamin Robins, Esq., J. Nourse,

London.

APPENDIX B

Glossary of Terms

This model will be a simplification and an idealization, and conse-quently a falsification. It is to be hoped that the features retained fordiscussion are those of greatest importance in the present state ofknowledge.

A. M. Turing, 1952

angle of incidence angle between the direction of the incident relative velocity of the contactpoints and the common normal direction. Direct or normal collisions have zero angle ofincidence, whereas oblique collisions have a nonzero angle of incidence.

angle of rebound angle between the direction of the relative velocity of the contact pointsat separation and the common normal direction.

attractor steady state solution that is approached asymptotically with increasing time if thesystem has small dissipation.

coefficient of friction upper limit on ratio of tangential to normal force at contact.coefficient of stick geometric parameter specifying ratio of tangential to normal force for

stick.collinear (or central) impact configuration colliding bodies oriented so that each center

of mass is on common normal line passing through the point of initial contact.common normal direction normal to common tangent plane that passes through contact

point C.common tangent plane If at least one of the bodies has a topologically smooth surface at

the contact point, this is the plane that is tangent to the surface at the point of initial contact.Usually both bodies have smooth surfaces around their respective points of contact, so theyhave a common tangent plane.

compression phase of collision part of the contact period in which the normal componentof relative velocity between contact points is negative, i.e., the contact region is beingcompressed.

configuration description of position and orientation of each body in a dynamic system.conforming bodies two bodies with contact surfaces where the curvature of one is the

negative of the curvature of the other so that they initially come into contact over an arearather than at a point.

conforming contact surfaces surfaces which conform or touch over a finite area.constitutive relation equation relating stress at a point to strain or local deformation.

266

Appendix B / Glossary of Terms 267

constraint equation kinematic equation that specifies a velocity, relative velocity or rangeof admissible velocities at point of intersection between two bodies.

contact area area around initial point of contact where surfaces of colliding bodies arecoincident.

contact force stress resultant of pressure and tangential surface tractions acting in contactarea.

deformation relative displacement between two points on the same body that is due toextension and distortion; obtained by integrating strains along a line at a particular instantof time.

direct collision collision throughout which the relative velocity of contact points is in thenormal direction. It is frictionless, since no sliding occurs in the contact region.

dispersion spreading of a pulse with time caused by dependence of speed of propagationon wavelength.

dissipated energy kinetic energy transformed during collision to plastic work, viscouslosses, residual vibrations of the bodies, etc.

eccentric (or noncollinear) impact configuration center of mass of at least one body noton the line of common normal that passes through the contact point.

effective mass m = (M#l + M^",1)"1, a composite mass term in the equation of relativemotion for a pair of colliding bodies.

elastic having a reversible constitutive relation; stress uniquely related to strain.grazing incidence angle of incidence approaching tangent to contact surface.incidence time of initial contact of colliding bodies.incident relative velocity at contact point difference between the velocities of coincident

points of contact when contact initiates.inelastic having an irreversible constitutive relation so that a cycle of loading and unloading

exhibits hysteresis.interference overlap between surfaces of two bodies.intrinsic speed natural speed that is characteristic of system.jam self-locking of sliding in eccentric impact with large coefficient of friction.noncollinear configuration orientation of colliding bodies such that the center of mass is

eccentric (not on the common normal passing through the point of initial contact).normal direction unit vector perpendicular to the common tangent plane.oblique collision one in which at incidence the relative velocity between the points of

contact has a component that is tangential to the common tangent plane, so that the angleof incidence is nonnormal.

reflection coefficient at an interface, the ratio of amplitude of reflected to incident wave.restitution phase of contact part of the contact period in which the normal component of

relative velocity between the contact points is positive; the period in which the bodies aremoving apart but have not lost contact.

rough surface one such that friction related to surface roughness opposes sliding at pointof contact.

separation instant when contact ceases.

268 Appendix B / Glossary of Terms

separation relative velocity final difference at separation between the velocities of thepoints of contact on the two bodies.

sliding tangential component of relative velocity between two coincident contact points.slip slidingsmooth surface frictionless surface; i.e., tangential component of contact force is zero.state of stress complete set of stress components at a point.stick motion in which tangential component of relative velocity remains zero.topologically smooth surface region of surface with continuous curvature in every direc-

tion. "Smooth" is also used to mean frictionless; hence the cumbersome adjective "topo-logically" is used to distinguish between these two meanings.

total mechanical energy sum of kinetic and potential energies (including energy of internaldeformation).

traction vector describing the force per unit area acting at a point on a surface.transmission coefficient at an interface, the amplitude ratio of reflected to transmitted wave.

Answers to Some Problems

Genius begins great works, labor alone finishes them.

Joseph Joubert

1.1 Vf/V0 = M/(M + MO, (To - Tf)l To = M'/{M + M')^^ niiii —2nin 2 0

1.3 p = (5, 4, 0)/3; V = (5, 2, 0)/6; I = -2n 2ni 4n2n2 03 L 0 0 5n3n3_

h = (5M/3)(0, 0, 1)1.4 (a) co+/co- = 0.5; (b) co+/co_ = 0; V = VB+ = Leo.; (c) (71 - r + ) / 7 1 = 3/41.5 co+/co^ = {2 + 1 cos 2a)/(8 + cos 2a)

2.2 V'f = 9.1 m s"1, Vf = 44.7 m s~l

2.3 V3{pf)max/V0 = 4/(1 + a)2, \ima^0[V3{pf)/ Vo] = 42.42.5 M/Mr = cos 2^ + (sin 29)/ tan 0'2.6 6> = 1.176 rad., Vlf/V0 = 0.934, y2//V0 = 0.2532.9 M 7 M = 4/3, Vf = (TTC/M)1'2

2.10 Largest possible length

3.4 V, = 0, V3 = 5Vb/7; x3B = 5V*/49pg3.5 (9B = 30°,6>^(30°)%33.7°;

2^fi = cos~1(5/9) « 56° =• m a x = ^ ( 2 8 ° ) « 33.7°

3.8 (b) /x = 2V3/21; (c) W3(/?c) = V3MV*/S3.11 (b)-^/23.12 (a) AT = 0.5(1 - el)MR2co2

0[2h/R - (h/R)2](b) /too > {2gh/[l - (1 -

6.1 Calculation assumes the period of contact is the same for each pair of like spheres. In acollision between dissimilar spheres the body with a smaller contact period (for impactagainst an identical sphere) has larger influence than that predicted by this theory.

6.2 (a) vY = 0.64 x 10~3 m s"1; (b) e* = 0.33;(c) imperceptible surface indentation.

7.1 W = Ap2cx/6E, U/W = T/W = 1/27.2 Interface a=0 for t>L/c0, but contact lost at t = 2L/c0 when displacement

£ (£±*) + £ V, = 4(3 V, + V2)

269

270 Answers to Some Problems

8.3

8.4

8.58.68.7

8.8

Va(2L/ca)/V0 = -0.493, Vs(2L/ca)/V0 = 0.17, momentum conserved;K.E. + residual strain energy = initial K.E., VY — 24.1 m s"1

Deceleration of monkey, % = ^ + ( ^ - g) exp ( ^ )(a) w = Vb^"a(/"x/co), * < cot; u = 0, JC > cor; (c) u = Voe~o('~*/co\ * < cor - 2L;

7.4

7.57.6

7.8 (b) uT = UR + iii, —(aL/c)dcr T/dt = oT — (aR + a7);(c) YR ~ exp(-2cr/aL), yT = 1 - exp(-2ct/aL)

7.9 Tension behind locomotive after t = 2nL/c7.10 (a) u(x, t) = 2u0ei(hc~(bt) cos[A£(x - cgt)]; (b) Ak = n/L7.12 cg/cQ = (1 + F)"3 / 2 ; (a) jt > 5; (b) k < 2 => A. < 0.7

8.2 (b) ) r = (6/?/45M)(+26, - 7 , +2, - l

= (M/6) [4M2 + u{u3 + M2 + v2 + i?,^ + 4r;|]12 0 0 - 3 ^ "00

-3s9

18

3c0

0 6dp

,2 = 1 + XL/2k2r,b22 =

pc = 0.1021ML, , W3(pf) = -0.0051 ML2;effect of friction large because initial speed of slip is large in comparison the normal com-ponent of incident relative velocity at C.(a) If 1 < a < 3 then V\/ Vo < 0 after initial impact between balls at B2, so second impactisatCj.(b) Vf/Vo = (-a2 + 10a - 5)/(a + I)2,V+/ Vo = (-5or2 + 10or - l)/(a + I)2, 0.528 < a < 1;V+/ Vo = - ( - « 2 H- 10a - 5)/(a + I)2,V2

+/Vo = (-5a2 + 10a - l)/(a + I)2,0.333 < a < 0.528.Two impacts between balls at B2 if 0.333 < a < 1.0.

sin(kjX/L),kj = J7i, j = 1, 2, 3, . . . ; L2a)j/y = TT2, An2, 9ic2, . . .Frequencies the same, since mode shapes are identicalMj = M/2, Kj = J4TT4EI/2L\ coj = j27t2y/L2; AjCbj = 2(M'/M)V^cbj = (Dj/^l + M'/M, j = 1, 3, 5, . . .hint: M = M/2 = 60 g, M' = 32.8 g(a) a > 1 results in multiple hits (contact chattering).

9.19.29.3

9.49.5

10.1 r7 =t > tf => rolling

1 0 - 2 - = - i + ^ ,el>hb/K

References

In which a thousand trifles are recounted, as nonsensical as they arenecessary to the understanding of this great history.

Cervantes, Don Quixote, 1605

Abeyaratne, R. (1989) "Motion of a prism rolling down an inclined plane", Int. J. Mech. Engng. Educ.17,53-61.

Achenbach, J., Hemann, J. H. and Ziegler, F. (1968) "Tensile failure of interface bonds in a compositebody subjected to compressive loads", AIAA J. 6, 2040-2043.

Adams, G.G. (1997) "Imperfectly constrained planar impact - a coefficient of restitution model", Int.J. Impact Engng. 19(8), 693-701.

Adams, G.G. and Tran, D.N. (1993) "The coefficient of restitution for a planar two-body eccentricimpact", ASME J. Appl. Mech. 60, 1058-1060.

Andrews, J.P. (1931) "Experiments on impact", Proc. Phys. Soc. Lond. 43, 8-17.Bahar, L.Y. (1994) "On use of quasi-velocities in impulsive motion", Int. J. Engng. Sci. 32, 1669-

1686.Bajoria, K.M. (1986) "Three dimensional progressive collapse of warehouse racking", PhD Disser-

tation, University Engineering Dept., Cambridge Univ., Cambridge, UK.Batlle, J.A. (1993) "On Newton's and Poisson's rules of percussive dynamics", ASME J. Appl. Mech.

60,376-381.Batlle, J.A. (1996) "The sliding velocity flow of rough collisions in multibody systems", ASME J.

Appl Mech. 63, 168-172.Batlle, J.A. (1998) "The jam process in 3D rough collisions", ASME J. Appl. Mech. 63(3), 804-810.Bilbao, A., Campos, J. and Bastero, C. (1989) "On the planar impact of an elastic body with a rough

surface", Int. J. Mech. Engng. Educ. 17, 205-210.Bimping-Bota, E.K., Nitzan, A., Ortoleva, P. and Ross, J. (1997) "Cooperative instability phenomena

in arrays of catalytic sites", J. Chem. Phys. 66, 3650-3658.Boley, B.A. (1955) "Some solutions of the Timoshenko beam equations", ASME J. Appl. Mech. 22,

579-586.Bouligand, G. (1959) Mecanique Rationnelle, 477-483, Vuibert, Paris.Brach, R.M. (1989) "Tangential restitution in collisions", Computational Techniques for Contact

Impact, Penetration and Perforation of Solids, ASME AMD 103 (ed. L.E. Schwer, N.J. Salamonand W.K. Liu), 1-7.

Brach, R.M. (1993) "Classical planar impact theory and the tip impact of a slender rod", Int. J. ImpactEngng. 13, 21-33.

Brogliato, B. (1996) Nonsmooth Impact Mechanics: Models, Dynamics and Control, LNCIS 220,Springer-Verlag, Heidelberg.

Budd, C.J. and Lee, A.G. (1996) "Impact orbits of periodically forced impact oscillators", Trans. Roy.Soc. Lond. 452, 2719-2750.

271

272 References

Calladine, C.R. and Heyman, J. (1962) "Mechanics of the game of croquet", Engineering 193, 861—863.

Chang, L. (1996) "Efficient calculation of the load and coefficient of restitution of impact betwen twoelastic bodes with a liquid lubricant", ASME J. Appl. Mech. 63, 347-352.

Chang, W.R. and Ling, F.F. (1992) "Normal impact model of rough surfaces", ASME J. Appl. Mech.114, 439-447.

Chapman, S. (1960) "Misconception concerning the dynamics of the impact ball apparatus", Am. J.Phys. 28, 705-711.

Chatterjee, A. (1997) "Rigid body collisions: some general considerations, new collision laws andsome experimental data", PhD Dissertation, Cornell Univ., Ithaca, NY.

Christoforou, A.P. and Yigit, A.S. (1998) "Effect of flexibility on low velocity impact response", J.Sound & Vibration 217(3), 563-578.

Cochran, A.J. and Farrally, M.R., eds. (1994) Science and Golf II, E&FN Spon, London.Collett, P. and Eckmann, J.P. (1980) "Iterated maps on the interval as dynamical systems", Progress

in Physics, Vol. I, Birkhauser, Boston.Cremer, L. and Heckl, M. (1973) Structure Borne Sound (transl. E.E. Ungar), Springer-Verlag,

New York.Cundall, PA. and Strack, O.D.L. (1979) "A discrete numerical model for granular assemblies",

Geotechnique 29, 47-65.Daish, C.B. (1981) The Physics of Ball Games, Hodder and Stoughton, London.Davies, R.M. (1949) "The determination of static and dynamic yield stresses using a steel ball", Proc.

Royal Soc. Lond. A 197, 416-421.Deresiewicz, H. (1968) "A note on Hertz's theory of impact", Acta Mech. 6, 110.Engel, PA. (1976) Impact Wear of Materials, Elsevier Science, New York.Evans, A.G., Gulden, M.E. and Rosenblatt, M. (1978) "Impact damage in brittle materials in the

elastic-plastic response regime", Proc. Roy. Soc. Lond. A 361, 343-365.Follansbee, PS. and Sinclair, G.B. (1984) "Quasi-static normal indentation of an elastic-plastic half-

space by a rigid sphere", Int. J. Solids Struct. 20, 81.Glockner, C. and Pfeiffer, F. (1995) "Multiple impacts with friction in rigid multibody systems",

Advances in Nonlinear Dynamics (ed. A.K. Bajaj and S.W. Shaw), Kluwer Academic.Goldsmith, W. (1960) Impact, the Theory and Physical Behaviour of Colliding Solids, Edward Arnold,

London.Goldsmith, W. and Lyman, PT. (1960) "Penetration of hard steel spheres into plane metal surfaces",

ASMEJ. Appl. Mech. 27, 717-725.Gontier, C. and Toulemonde, C. (1997) "Approach to the periodic and chaotic behaviour of the impact

oscillator by a continuation method", Eur. J. Mech. A/Solids 16(1), 141-163.Greenwood, J.A. (1996) "Contact of rough surfaces", Solid-Solid Interactions: Proceedings 1st Royal

Society-Unilever Indo-UK Forum in Materials Science and Engineering (ed. M.J. Adams, S.K.Biswas and B.J. Briscoe), Imperial College Press, London, 41-53.

Haake, S.J. (1991) "Impact of golf balls on natural turf: a physical model of impact", Solid Mechanics/V(ed. A.R.S. Ponter and A.C.F. Cocks) Elsevier, London, UK, 72-89.

Haake, S.J. (1996) The Engineering of Sport, Balkema, Rotterdam.Han, I. and Gilmore, B.J. (1993) "Multi-body impact motion with friction - analysis, simulation and

experimental verification", ASME J. Mech. Design 115, 412-422.Hardy, C. Baronet, C.N. and Tordion, G.V. (1971) "Elastoplastic indentation of a half-space by a rigid

sphere", /. Num. Methods Engng. 3, 451.Harris, R.A. and Stodolsky, L. (1981) "On time dependence of optical activity", /. Chem. Phys. 74(4),

2145-2155.Harter, W.G. (1971) "Velocity amplification in collision experiments involving Superballs", Amer. J.

Phys. 39, 656-663.Hertz, H. (1882) "Uber die Beriihrung fester elastischer Korper (On the contact of elastic solids)," J.

ReineAngew. Math. 92, 156-171.Hill, R. (1950) A Mathematical Theory of Plasticity, Oxford Univ. Press.

References 273

Hogue, C. and Newland, D. (1994) "Efficient computer simulation of moving granular particles",Powder Tech. 78,51-66.

Holmes, P.J. (1982) "The dynamics of repeated impacts with a sinusoidally vibrating table", J. Sound& Vibration 84(2), 173-189.

Hopkinson, B. (1913) "Method of measuring the pressure produced in the detonation of high explosivesor by the impact of bullets", Proc. Roy. Soc. Lond. A 89, 411-413.

Hopkinson, J. (1872) "On rupture of iron wire by a blow", Proc. Manchester Lit. Phil. Soc. XI, 40-45.

Horak, Z. (1948) "Impact of rough ball spinning round its vertical diameter onto a horizontal plane",Vyskoka skola strojniho a elektrotechnickeho inzenyrstvi pfi Ceskem vysokem uceni technickemv Praze {Trans. Fac. Mech. Elec. Engng. Tech. Univ. Prague).

Horak, Z. and Pacakova (1961) "Theory of spinning impact of imperfectly elastic bodies", Czech. J.Phys. B 11, 46-65.

Hunt, K.H. and Crossley, F.R.E. (1975) "Coefficient of restitution interpreted as damping in vibro-impact", ASME J. Appli. Mech. 97, 440-445.

Hunter, S.C. (1956) "Energy absorbed by elastic waves during impact", J. Mech. Phys. Solids 5,162-171.

Ivanov, A.P. (1995) "On multiple impact", /. Appl. Math. Mech. 59, 930-946 (transl. of Prik. Mat.Mekh. 59, 930-946).

Ivanov, A.P. (1997a) "The problem of constrained impact," /. Appl. Math. Mech. 61, 341-353 (transl.of Prik. Mat. Mekh. 61, 355-368).

Ivanov, A.P. (1997b) "A Kelvin theorem and partial work of impulsive forces", ASMEJ. Appl. Mech.64, 438-440.

Johnson, K.L. (1968) "Experimental determination of contact stresses between plastically deformedcylinders and spheres", Engineering Plasticity (ed. J. Heyman and F. Leckie), Cambridge Univ.Press, 341-362.

Johnson, K.L. (1970) "The correlation of indentation experiments", J. Mech. Phys. Solids 18, 115-126.

Johnson, K.L. (1983) "The bounce of 'Superball' ", Int. J. Mech. Engng. Educ. I l l , 57-63.Johnson, K.L. (1985) Contact Mechanics, 361-366, Cambridge Univ. Press.Johnson, S.H. and Liebermann, B.B. (1994) "An analytical model for ball barrier impact - II. Oblique

impact", Science and Golf II (ed. A.J. Cochran and M.R. Farrally), E&FN Spon, London, 315-320.Johnson, S.H. and Liebermann, B.B. (1996) "Normal impact models for golf balls", The Engineering

of Sport (ed. S. Haake), Balkema, Rotterdam.Johnson, W. (1972) Impact Strength of Materials, 303, Edward Arnold, London.Johnson, W. (1976) " 'Simple' linear impact," Int. J. Mech. Engng. Educ. 4, 167-181.Kane, T. and Levinson, D.A. (1985) Dynamics: Theory and Application, McGraw-Hill, New York.Keller, J.B. (1986) "Impact with friction", ASMEJ. Appl. Mech. 53, 1-4.Kelvin, W.T. and Tait, P.G. (1867) Treatise on Natural Philosophy, Vol. 1, Part 1, Clarendon, Oxford,

UK.Kerwin, J.D. (1972) "Velocity, momentum and energy transmissions in chain collisions", Amer. J.

Phys. 40, 1152-1157.Ko, PL. (1985) The significance of shear and normal force components on tube wear due to fretting

and periodic impacting, Wear 106, 261-281.Kozlov, V.V. and Treshchev, D.V. (1991) Billiards: a Genetic Introduction to the Dynamics of Systems

with Impacts, Transl. Math. Monographs, Amer. Math. Soc.Lamba, H. and Budd, C.J. (1994) "Scaling Lyapunov exponents at non-smooth bifurcations", Phys.

/tev. £50, 84-91.Lankarani, H.M. and Nikravesh, RE. (1990) "A contact force model with hysteresis damping for

impact analysis of multibody systems", ASME J. Mech. Design 112, 360-376.Lee, E.H. (1940) "Impact of a mass striking a beam", ASME J. Appl. Mech. 62, A129-A140.Lee, Y., Hamilton, J.F. and Sullivan, J.W. (1983) "The lumped parameter method for elastic impact

problems", ASMEJ. Appl. Mech. 50, 823-827.

274 References

Leggett, A.J., Chakravarty, S., Dorsey, A.T., Fisher, M, Garg, A. and Zwerger, W. (1987) "Dynamicsof the dissipative two-state system", Rev. Mod. Phys. 59(1), 1-86.

Lennertz, L. (1937) "Beitrag zur Frage nach der Wirkung eines Querstosses auf einer Stab," Ing. -Arch.8, 37-45.

Lewis, A.D. and Rogers, R.J. (1988) "Experimental and numerical study of forces during obliqueimpact", J. Sound and Vibration 125(3), 403-412.

Lewis, A.D. and Rogers, R.J. (1990) "Further numerical studies of oblique impact", J. Sound andVibration 141(3), 507-510.

Lifshitz, J.M. and Kolsky, H. (1964) "Some experiments of anelastic rebound", J. Mech. Phys. Solids12, 3 5 ^ 3 .

Lim, C.T. (1996) "Energy losses in normal collision of 'rigid' bodies", Proceedings of 2nd Interna-tional Symposium on Impact Engineering, Beijing, 129-136.

Lim, C.T. and Stronge, WJ. (1994) "Frictional torque and compliance in collinear elastic collisions",Int. J. Mech. Sci. 36, 911-930.

Lim, C.T. and Stronge, W.J. (1998a) "Normal elastic-plastic impact in plane strain", Math. Comp.Modelling 28, 323-340.

Lim, C.T. and Stronge, W.J. (1998b) "Oblique elastic-plastic impact between rough cylinders in planestrain", Int. J. Engrg. Sci. 37, 97-122.

Lotstedt, P. (1981) "Coulomb friction in two-dimensional rigid body system", Z Angew. Math. Mech.61, 605-616.

Love, A.E.H. (1906) A Treatise on the Mathematical Theory of Elasticity, 2nd ed. Cambridge Univ.Press.

Mac Sithigh, G.P. (1996) "Three-dimensional rigid-body impact with friction", ASMEJ. Appl. Mech.58, 754-758.

Marghitu, D.B. and Hurmuzlu, Y. (1996) "Nonlinear dynamics of elastic rod with frictional impact",J. Nonlinear Dynamics 10, 187-201.

Maw, N., Barber, J.R. and Fawcett, J.N. (1976) "The oblique impact of elastic spheres", Wear 38,101-114.

Maw, N., Barber, J.R. and Fawcett, J.N. (1981) "The role of elastic tangential compliance in obliqueimpact", ASME J. hub. Tech. 103(74), 74-80.

May, R.M. (1986) "When 2 and 2 do not make 4 - nonlinear phenomena in ecology", The CroonianLecture - 1985, Proc. Roy. Soc. Lond. B 228, 241-266.

McLachlan, B.J., Beupre, G., Cox, A.B. and Gore, L. (1983) "Falling dominoes", SIAM Rev. 25,403-404.

Meyers, M.A. and Murr, L.E. (1980) Shock Waves and High-Strain-Rate Phenomena in Metals,Plenum, New York.

Miller, G.F. and Pursey, H. (1956) "On the partition of energy between elastic waves in a semi-infinitesolid", Proc. Roy. Soc. Lond. A 225, 55-69.

Mindlin, R.D. and Deresiewicz, H. (1953) "Elastic spheres in contact under varying oblique forces",ASMEJ. Appl. Mech. 75, 327-344.

Mok, C.H. and Duffy, J. (1965) "Dynamic stress-strain relation of metals as determined from impacttests with a hard ball", Int. J. Mech. Sci. 7, 355-371.

Morin, R.A. (1845) Mecanique, Libraire Scientifique et Industrielle de A. Leroux, Paris.Mulhearn, T.O. (1959) "Deformation of metals by Vickers-type pyramidal indenters", /. Mech. Phys.

Solids 7, 85-96.Newby, N.D. (1984) "Excitation of a composite structure by collisions", Amer. J. Phys. 52, 745-748.Nicholas, G. and Prigogine, I. (1977) Self-organization in Nonequilibrium Systems, Wiley, New York.Painleve, P. (1895) Lecons sur le Frottement, Gauthier-Villars, Paris.Pereira, M.S. and Nikravesh, P. (1996) "Impact dynamics of multibody systems with frictional contact

using joint coordinates and canonical equations of motion", Nonlinear Dynamics 9, 53-71.Poisson, S.D. (1811) Traite de Mecanique, Courier, Paris (Engl. trans, by Rev. H. H. Harte, Longman

& Co., London 1817).

References 275

Regirir, S.A. (1989) "Active media with discrete sources and jumping waves", Nonlinear Waves inActive Media (ed. J. Engelbrecht) Springer-Verlag, Heidelberg, 176-184.

Routh, EJ. (1905) Dynamics of a System of Rigid Bodies, 7th ed. Part 1, McMillan, London.Saint-Venant, B. (1867) "Memoire sur le choc lognitudinal de deux barres elastiques de grosseurs et

de matieres semblables ou differentes, et sur la proportion de leur force vive qui est perde pourla translation ulterieure; et gene'ralement sur le mouvement longitudinal d'un systeme de deux ouplusieurs elastiques", J. de Math. (Liouville) 12, 237-277.

Schwieger, H. (1970) "Central deflection of a transversely struck beam", Exp. Meek 10(4), 166-169.Scott, A.C. (1975) "The electrophysics of a nerve fibre", Rev. Mod. Phys. 47, 487-533.Shaw, S.W. (1985) "The dynamics of a harmonically excited system having rigid amplitude con-

straints", ASMEJ. Appl. Mech. 52, 453-458.Shaw, S.W. and Holmes, P.J. (1983) "A periodically forced piecewise linear oscillator", J. Sound and

Vibration 90(1), 129-155.Simon, R. (1967) "Development of a mathematical tool for evaluating golf club performance", pre-

sented at ASME Design Engineering Congress, New York.Singh, R., Shukla, A. and Zervas, H. (1996) "Explosively generated pulse propagation through par-

ticles containing natural cracks", Mech. Materials 23, 255-270.Smith, C.E. (1991) "Predicting rebounds using rigid-body dynamics", ASME J. Appl Mech. 58,

754-758.Sondergaard, R., Chaney, K. and Brennen, C.E. (1990) "Measurements of solid spheres bouncing off

flat plates", ASMEJ. Appl. Mech. 57, 694-699.Spradley, J.L. (1987) "Velocity amplification in vertical collisions", Amer. J. Phys. 55, 183-184.Stoianovici, D. and Hurmuzlu, Y. (1996), "A critical study of the applicability of rigid-body collision

theory", ASMEJ. Appl. Mech. 63, 307-316.Stronge, W.J. (1987) "The domino effect - a wave of destabilizing collisions in a periodic array",

Proc. Roy. Soc. Lond. A 409, 199-208.Stronge, WJ. (1990) "Rigid body collisions with friction", Proc. Roy. Soc. Lond. A 431, 169-181.Stronge, WJ. (1991) "Friction in collisions: resolution of a paradox", / Appl. Phys. 69(2),

610-612.Stronge, W.J. (1992) "Energy dissipated in planar collision", ASMEJ. Appl. Mech. 59, 681-682.Stronge, W.J. (1993) "Two-dimensional rigid-body collisions with friction-discussion", ASME J.

Appl. Mech. 60, 564-566.Stronge, W.J. (1994a) "Swerve during three-dimensional impact of rough rigid bodies", ASME J.

Appl. Mech. 61,605-611.Stronge, W.J. (1994b) "Planar impact of rough compliant bodies", Int. J. Impact Engng. 15,435-450.Stronge, W.J. (1995) "Theoretical coefficient of restitution for planar impact of rough elastoplas-

tic bodies", Impact, Waves and Fracture, ASME AMD-205 (ed. R.C. Batra, A.K. Mai and G.P.MacSithigh), 351-362.

Stronge, W.J. (1999) "Mechanics of impact for compliant multi-body systems", IUTAM Symposiumon Unilateral Multibody Dynamics (ed. C. Glockner and F. Pfeiffer), Munich, Aug. 1998.

Sundararajan, G. (1990) "The energy absorbed during the oblique impact of a hard ball against ductiletarget materials", Int. J. Impact Engng. 9, 343-358.

Swanson, S.R. (1992) "Limits of quasi-static solutions in impact of composite structures", CompositesEngng. 2(4), 261-267.

Tabor, D. (1948) "A simple theory of static and dynamic hardness", Proc. Roy. Soc. Lond. A 192,247-274.

Tabor, D., 1951, The Hardness of Metals, 128-138, Oxford Univ. Press.Taylor, G.I. (1946) "The testing of materials at high rates of loading", J. Inst. Civ. Engng. 26, 486-

518.Thompson, J.M.T. and Stewart, H.B. (1986) Nonlinear Dynamics and Chaos, Wiley, Chichester, UK.Thomson, W and Tait, RG. (1879) Treatise on Natural Philosophy, Vol. I, Part I. Cambridge Univ.

Press.

276 References

Timoshenko, S. (1913) "Zur Frage der Wirkung eines Stosses auf einer Balken", Z. Math. Phys. 62,198-209.

Timoshenko, S. and Goodier, N. (1970) Theory of Elasticity, 3rd edition, McGraw-Hill, New York.Tsai, Y.M. (1971) "Dynamic contact stresses produced by impact of an axisymmetrical projectile on

an elastic half-space", Int. J. Solids Structures 7, 543-558.Ujihashi, S. (1994) "Measurement of dynamic characteristics of golf balls and identification of their

mechanical models", Science and Golf II (ed. A.J. Cochran and M.R. Farrally), E&FN Spon,London.

Villaggio, P. (1996) "Rebound of an elastic sphere against a rigid wall", ASME J. Appl. Mech. 63,259-263.

Wagstaff, J.E.P. (1924) "Experiments on the duration of impacts, mainly of bars with rounded ends,in elucidation of elastic theory", Proc. Roy. Soc. Lond. A 21, 544-570.

Walton, O.R. (1992) "Granular solids flow project", Rept. UCID-20297- 88-1, Lawrence LivermoreNational Laboratory.

Wang, Y. and Mason, M.T. (1992) "Two-dimensional rigid-body collisions with friction", ASME J.Appl. Mech. 59, 635-642.

Wittenburg, J. (1977) Dynamics of System of Rigid Bodies, Teubner, Stuttgart.

Index

ABAQUS, 143Amontons-Coulomb law, 40, 178angle of incidence, 3, 266angle of rebound, 266area, contact, 1, 7, 117, 186attractor, 219, 266axial impact, 9, 146-72

bar velocity, 147baseball, 33-4beam equation

Euler-Bernoulli, 164-5, 171modal, 201-2Rayleigh, 165-6Timoshenko, 166-8

bending, 5, 164, 201Bernoulli, J., 261bilateral constraint, 173billiard ball, 31,61,251bounce, 112-5,220boundary conditions, 206, 216bulk modulus, 146, 149

chaos, 242characteristic diagram, 151, 169collapse, 241collinear (central) configuration, 2, 155, 186, 266collision, 1common normal direction, 35, 266common tangent plane, 64, 266compliance

linear, 87, 96structural, 209tangential, 47, 93-114

compliant contact, 8, 117-^5, 182-98compression period, 23configuration, 2, 266

eccentric (non-collinear), 2, 35-85, 267conforming bodies, 266conforming contact surfaces, 117constitutive relation, 120constraint equation, 173, 175, 266contact point, 2, 21

convection, 240cooperative group, 235couple, 8, 224

damage number, 138damping, viscous, 87-93, 156deformation, 1, 267

elastic, 117, 138elastic-plastic, 120-3, 139fully plastic (uncontained), 123- , 140incompressible, 121

deformable particle, 22, 63density, 149, 162Descartes, R., 251-3dilatational wave speed, 149Dirac delta function, 207direct impact, 2, 21-30, 267discrete modelling, 95dispersion equation, 163, 186displacement

axial, 146radial, 121relative, 24

dissipated energy, 109-112, 267distal surface, 9, 158domino toppling, 228^1DYNA2D, 138

eccentric configuration (see configuration)eccentricity, 61eigenfunctions, 202elastic compliance, 3, 126elastic modulus, effective, 118elastic-plastic compliance, 96energy, conservation of, 186, 223energy loss factor, 216energy ratio, 236Euler, L., 261-4evanescent wave, 163, 187

finite element, 138flexible bodies, 9flexural wave, 164-9

277

278 Index

forcecontact, 3, 11, 96, 99, 162, 195, 210-2tangential, 107yield, 119

frequency, natural, 97, 224damped, 88two degree of freedom, 193

friction, 64friction, Amontons-Coulomb law, 40, 67friction

coefficient, 40, 266critical, 75evolution of sliding, 42force, 64

Galilei, G., 248-51generalized

coordinates, 174impulse, 175, 180momentum, 176, 196speeds, 174, 180

golf ball, 13, 19,62,93gravity, 39, 223, 240grazing incidence, 55, 267gross slip, 46group velocity, 163^ , 187gyration, radius of, 35, 37

hard bodies, 5, 21, 149, 186Hertz

contact force, 192pressure distribution, 117

hodograph, 74, 78Hunter, 135Huyghens, C , 33, 255-7

impact configuration, 35impedance ratio, 153impulse, 22, 24impulse

compression, 25, 44, 130halting slip, 49, 56normal, 24, 129, 184tangential, 41, 129terminal, 46-7, 49

impulsive load, 176incidence, time of, 2incident relative velocity at contact, 3,

25indentation

elastic-plastic, 120, 139local, 160, 209maximum, 130normal, 97, 118,210,213yield, 119

indicial notation, 19inelastic, 47, 119

inertiacenter of, 30dyadic, 15, 19, 65inverse for contact point, 37, 59moment of, 15, 65, 222-3

interference, 1, 118, 182intrinsic speed, 219, 227, 230, 231,

267inverse of inertia matrix, 66, 77isoclinic, 69, 74

jam, 42, 52Johnson, KL, 112, 120

kinetic energy, 16, 227, 236initial, 127normal relative motion, 25, 29, 212rotational, 16translational, 16, 204

Kronecker delta, 66

Lagrangian function, 176laws of motion, Newton, 12, 175logistic map, 237longitudinal waves, 146-64

Marriotte, E., 257-8mass, center of, 14mass, effective, 13, 127, 135, 206mass ratio, 157, 170, 211mechanical energy, total, 150, 239microslip, 94modal frequencies, 191mode, 202, 204moment of momentum, 15, 17, 224momentum, 11-12motion, equation of, 36

neural network, 228Newton, I., 11-12, 175, 248, 258-61normal direction, 21normal impulse for compression, 25notation, 19

obliquity, angle of, 3, 267orbit stability, 245orthogonality, 203, 205

parallel axis theorem, 223particle, 11

displacement, 148kinetics, 11-14velocity, 152, 155

particle impact, 4pendulum

compound, 178-180, 195spherical, 79-85

Index 279

percussion, center of, 34, 61,239

periodcompression, 2, 24-5contact, 13, 19, 127, 132restitution, 4, 24-5

permutation tensor, 19, 36phase velocity, 163, 169, 186planar motion, 41plastic deformation, perfectly, 97, 207plate, 9Poincare' section, 244point of contact, 2Poisson's ratio, 136potential energy, 176, 204, 223, 229,

236pressure

Hertz distribution, 117mean, 118

propagation, 146-170, 228-42properties, material, 130, 145, 149

radius, contact, 118radius of curvature, effectiverate-dependent, 86rate-independent, 64Rayleigh-Ritz method, 203-5, 218Rayleigh wave speed, 149reaction-diffusion equation, 228reaction wave, 241rebound angle, 57-8rebound height, 222reference frame, 12reflection coefficient, 153relative motion, equations of, 129relative velocity, 12, 22, 35restitution coefficient

effective, 134energetic, 27, 47, 131kinematic, 28kinetic, 28

restitution period, 23rigid body impact, 5rolling speed, 224rotary inertia, 165rough surface, 41

saddle point, 245second moment of area, 165separation impulse, 267separation relative velocity, 24, 28, 49,

53separatrix, 75shear stress, 120,201simple harmonic motion (SHM), 97sliding

unidirectional, 46

slip, 40-4, 48-61, 64,100-12

slip-reversal, 43, 49, 179,229

slip-stick, 43, 51, 101-3slip trajectory, 74smooth surface, 21, 37, 268spall fracture, 153speed, sliding, 67stick, 41, 43, 268stick, coefficient of, 40, 68, 180-2,

266stiffness

contact, 118, 126, 160, 186, 190,212

equivalent, 187,205,215linear, 96, 187,215ratio, 96, 112

strain, 146strain energy, 24, 150stress

components, 120effective, 120state, 120-1yield, 119, 139

Superball, 112swerve, 63

tangent plane, 2, 64tangential compliance, 111tennis ball, 19,200Timoshenko beam theory {see beam)topologically smooth surface, 21traction, 268transient dynamics, 230transmission coefficient, 153, 268transverse displacement, 5, 164Tresca yield criterion, 119

unilateral constraint, 173

vector decomposition, 18velocity, relative, 12, 22velocity, yield, 1,128, 130, 139vibroimpact, 243virtual power, 175viscoelastic compliance

hybrid, 91Kelvin-Voight, 87linear, 87Maxwell, 87nonlinear, 89

von Mises yield criterion, 119

Wallis, J., 253-4wave equation, 147, 149wave number, 163, 187, 203

280 Index

wave number, cutoff, 187wavefront, 156, 229, 236

stability, 240wave speed, 149wear, impact, 105work

contact force, 26, 150partial by component of force, 45yield, 119

Wren, C , 254-5

yieldindentation, 119stress, dynamic, 130stress, static, 130velocity, 1-2,128work, 119

Young's modulus, 118, 130, 136,149, 162

Impact Mechanics

Documents

Transcript of Impact Mechanics