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«RIGID BODY COLLISIONS: SOME GENERAL CONSIDERATIONS, NEW COLLISION LAWS, AND SOME EXPERIMENTAL DATA Anindya Chatterjee January 6, 1997 Abstract This ...»

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RIGID BODY COLLISIONS: SOME GENERAL

CONSIDERATIONS, NEW COLLISION LAWS, AND SOME

EXPERIMENTAL DATA

Anindya Chatterjee

January 6, 1997

Abstract

This thesis attempts to present a uni ed view of the subject of rigid body collisions. This includes

discussion of basic assumptions, fundamental and reasonable constraints on collision laws, a survey

of commonly used laws, some new collision laws, a brief discussion of non-rigid body collisions in the context of rigid-body collisions, and some new experimental data, interpreted in the context of the previous theoretical considerations.

Contents 1 Introduction 8

1.1 Why Collisions are Hard to Model, and Simplistic Models are Popular 9 ::::::::

1.2 Brief Review of Existing Approaches : : : : : : : : : : : : : : : : : : : 11 ::::::::

1.3 Contribution of this Thesis : : : : : : : : : : : : : : : : : : : : : : : : 12 ::::::::

1.4 Outline of Remainder of this Thesis : : : : : : : : : : : : : : : : : : : 13 ::::::::

2 Preliminaries 16

2.1 Collision Laws for Rigid Bodies and Ideal Mechanisms : : 16 :::::::::::::::

2.1.1 Collisions : : : : : : : : : : : : : : : : : : : : : : : 16 :::::::::::::::

2.1.2 De nition of a Collision Law : : : : : : : : : : : : 16 :::::::::::::::

2.1.3 Desirable Properties in a Collision Law : : : : : : 17 :::::::::::::::

2.2 The Usual Assumptions of Rigid Body Collision Modeling 18 :::::::::::::::

2.3 Impulse-Momentum Relations the Local Mass Matrix : : 25 :::::::::::::::

2.3.1 The Local Mass Matrix for Some Special Cases : : 28 :::::::::::::::

–  –  –

7.4 78 :::::::::::::

–  –  –

10.1 In nitesimal perturbations can break up simultaneous impacts into sequences of single impacts : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 102

10.2 A simultaneous impact occurs when collisional contact at one point causes impulsive constraint forces at a pre-existing sustained contact : : : : : : : : : : : : : : : : : : : 103

10.3 Physical realization of arbitrary mass matrices : : : : : : : : : : : : : : : : : : : : : 110

10.4 Region in impulse space covered by Ivanov's restitution parameter, for values between 0 and 1 : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 113

11.1 Composite axisymmetric puck : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 117

11.2 Two-stage collision of simpli ed puck model : : : : : : : : : : : : : : : : : : : : : : : 118

11.3 Axisymmetric delrin pucks : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 120

11.4 Normal restitution for pucks 1 and 2 identical, regular circular pucks : : : : : : : : 122

11.5 Normal restitution for puck 3 circular puck with hole : : : : : : : : : : : : : : : : : 122

11.6 Normal restitution for pucks 4 and 5 identical circular pucks with holes : : : : : : : 123

11.7 Normal restitution for pucks 6 and 7 identical circular pucks with attached disks : : 123

11.8 Normal restitution for puck 8 circular puck with attached disk : : : : : : : : : : : : 124

11.9 Normal restitution for puck 9 circular puck with attached disk : : : : : : : : : : : : 124

11.10Post-collision tangential velocity VfT for pucks 1 and 2 identical, regular circular pucks : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 125

11.11Post-collision tangential velocity VfT for puck 3 circular puck with hole : : : : : : : 126

11.12Post-collision tangential velocity VfT for pucks 4 and 5 identical circular pucks with holes : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 126

11.13Post-collision tangential velocity VfT for pucks 6 and 7 identical circular pucks with attached disks : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 127

11.14Post-collision tangential velocity VfT for puck 8 circular puck with attached disk : : 127

11.15Post-collision tangential velocity VfT for puck 9 circular puck with attached disk : : 128

11.16Angles and : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 128

11.17tan vs. tan for pucks 1 and 2 identical, regular circular pucks : : : : : : : : : : : 129

11.18tan vs. tan for puck 3 circular puck with hole : : : : : : : : : : : : : : : : : : : : 129

11.19tan vs. tan for pucks 4 and 5 identical circular pucks with holes : : : : : : : : : : 130

11.20tan vs. tan for pucks 6 and 7 identical circular pucks with attached disks : : : : 130

11.21tan vs. tan for puck 8 circular puck with attached disk : : : : : : : : : : : : : : : 131

11.22tan vs. tan for puck 9 circular puck with attached disk : : : : : : : : : : : : : : : 131

11.23Impulse ratio for pucks 1 and 2 identical, regular circular pucks : : : : : : : : : : : 132

11.24Impulse ratio for puck 3 circular puck with hole : : : : : : : : : : : : : : : : : : : : 133

11.25Impulse ratio for pucks 4 and 5 identical circular pucks with holes : : : : : : : : : : 133





11.26Impulse ratio for pucks 6 and 7 identical circular pucks with attached disks : : : : : 134

11.27Impulse ratio for puck 8 circular puck with attached disk : : : : : : : : : : : : : : : 134

11.28Impulse ratio for puck 9 circular puck with attached disk : : : : : : : : : : : : : : : 135

11.29Impulse ratio for all pucks : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 136

11.30Impulse ratio predicted by some collision laws : : : : : : : : : : : : : : : : : : : : : : 136

11.31Non-axisymmetric Delrin puck : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 137

11.32Due to symmetry in the puck, may be assumed to be nonnegative between 0 and =2 : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 138

11.33Ratio of tangential to normal impulse vs. contact point location and incidence angle : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 139

11.34Sample of data points: vs. : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 139

11.35Coe cient of normal restitution vs. contact point location : : : : : : : : : : : : : : 140

11.36The angle  between the transmitted impulse vector and the position vector from the contact point to the center of mass : : : : : : : : : : : : : : : : : : : : : : : : : : 141

11.37Coe cient of normal restitution vs. angle  between impulse vector and position vector from contact point to center of mass : : : : : : : : : : : : : : : : : : : : : : : 141

11.38Ratio of tangential to normal impulses observed for both sticking and sliding collisions, vs. incidence angle : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 143

11.39Impulse ratio vs. incidence angle, showing sticking and sliding points separately : : 143

11.40Ratio of tangential to normal impulses observed for both sticking and sliding collisions, vs. contact point angle : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 144

11.41Impulse ratio for puck no. 2, for two velocities : : : : : : : : : : : : : : : : : : : : : : 145

11.42Load-displacement graph of Delrin puck loaded along diameter between at steel plates : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 147 Chapter 1 Introduction The objective of this thesis is to present a uni ed view of the subject of rigid body collisions.

This includes discussion of basic assumptions, fundamental and reasonable constraints on collision laws, a survey of commonly used laws, some new collision laws, a brief discussion of non-rigid body collisions in the context of rigid-body collisions, and some new experimental data.

When two objects collide, understanding and modeling the resulting mechanical interaction is far from a purely academic exercise. In the physical world, at human length scales, one of the primary modes of interaction between bodies is through contact, including collisional contact.

In dynamic models of mechanical systems, an extremely popular and useful idealization of a solid object is as a rigid body. The world of ideal rigid bodies has clear and well de ned rules for how objects move under the action of forces and moments, as well as how constraints like rolling, sliding or pivoting a ect the motions of systems of objects. Yet, in a world of ideal rigid bodies where objects are allowed to collide, it is neither widely known nor completely understood how general, sensible rules for collisional interaction between ideal rigid objects might be constructed how good the underlying assumptions are behind the rules that are available and how meaningful, physically, the predictions of these or any other rules really are in a world of not-truly-rigid bodies.

In the classical treatises on rigid body dynamics, the treatment of collisions is practically always restricted to one of two basic approaches: an incremental approach see Routh 52  and an algebraic approach see Whittaker 71 . One or the other of these two approaches is frequently adopted as the rational basis for describing collisions, with little discussion of either the accuracy of these, or the possible validity of other approaches. More recent, specialized texts also present treatments of the subject that are restricted in that neither the weaknesses of the procedures they present nor the strengths of other possible, general approaches are discussed at any depth. The subject of rigid body dynamics is incomplete at present, due to the lack of breadth in available rigid body collision models. There is a need for a variety of collision models within the structure of rigid body mechanics, for use in important modern applications like robotics, dynamics of machines or other mechanical systems with intermittent impacts, and multibody dynamics in general.

Collisions between sti solid objects are characterized by complicated nonlinear deformations occuring in the colliding bodies, at least in the vicinity of the contact region and by complicated surface interactions between the bodies in the contact region. Consequently, simplistic and possibly inaccurate approaches are often used in collision modeling. The problems in modeling collisions are further discussed in Section 1.1 below.

A brief review of existing approaches to collision modeling is presented in Section 1.2. These approaches are discussed again in greater detail at various appropriate places in this thesis.

The contribution of this thesis is outlined brie y in Section 1.3.

An outline of the remainder of this thesis is then presented in Section 1.4.

1.1 Why Collisions are Hard to Model, and Simplistic Models are Popular There is a hierarchy in the models available for describing various phenomena in mechanics. The laws of linear and angular momentum balance are strictly and precisely true, for essentially all engineering purposes. The accuracy of these balance laws is greater than that of most measuring devices. In predictions of translational and rotational motions of sti, solid objects under the action of known forces and moments, rigid body mechanics can be very accurate in many cases, and may be ranked second. Models of how bodies deform in response to forces, while they can be very good, are not quite so accurate and therefore rank lower. For example, a model of a real material as linearly elastic or linearly viscous can have accuracies down to a fraction of a percent. Thus, such models are very good, yet not as accurate as rigid body mechanics can sometimes be. Many nonlinear material behaviors can only be modeled to accuracies of several percent, and should be ranked even lower. In this category are models for friction between solid bodies, models for fracture and models for nonlinear material response such as elastoplasticity. Of the constitutive laws needed for modeling material motion, the laws for collisions are amongst the least accurate.

Collisions are di cult to understand and model because they involve many interacting phenomena, each one di cult to model accurately even by itself. The collisional behavior of a given body is not determined by that body alone. When a body collides with something else, the outcome is not determined solely by the properties of the interior of the body say its shape, mass distribution, material properties : : :  and or the properties of its surface say frictional properties or local surface shape. The properties of both colliding bodies a ect the outcome of the collision.

If one colliding body happens to be touching a third body at the time of the collision, that other interaction is important, too. The real outcome of a collision could, in principle, be computed if all the relevant mechanical interactions could be captured by the mathematical model. However, the nal accuracy of the predicted outcome would probably be about the same as that of the least accurate component of the model material or contact behavior, or even less.

In order to accurately model the mechanical interactions in a collision, we need to know what nonlinear constitutive law to use for the material, and also to know detailed small scale geometrical characteristics of the contacting surfaces. This information is not usually available to any great precision. In modeling collisions, even if we are prepared to spend the time and e ort required for a careful numerical solution of a complicated nonlinear problem say, a nite element solution, we might still expect low accuracy due to incomplete information about constitutive laws as well as boundary conditions. In other words, the di culty in trying to model collisions accurately arises

at several levels:

1. Constitutive laws for essential phenomena like friction, fracture, and nonlinear deformation are not known accurately.

2. If they are known accurately, they still require detailed geometric information, information about ambient conditions that a ect the contact behavior of the bodies, and various initial conditions that are not known accurately due to lack of su cient data.



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