Abstract
In general the motion of a body takes place in a confined environment and collision of the body with the containing wall is possible. In order to predict the dynamics of a body in this condition one must know what happens in a collision. Therefore, the problem is: if one knows the precollision dynamics of the body and the properties of the body and the wall one wants to predict the postcollision dynamics. This problem is quite old and it appeared in the literature in 1668. Up to 1984 it seemed that Newton's model was enough to solve the problem. But it was found that this was not the case and a renewed interest in the problem appeared. The aim of this paper is to treat the problem of plan collisions of rigid bodies, to classify the different models found in the literature and to present a new model that is a generalization of most of these models.
Collisions; dynamics; modeling
A Brief Review and a New Treatment for Rigid Bodies Collision Models
Edson Cataldo
Universidade Federal FluminenseUFF
Departamento de Matemática Aplicada
Rua Mário Santos Braga, s/ No. Praça do Valonguinho. Centro
24020140 Niterói, RJ. Brazil
Rubens Sampaio
Pontifícia Universidade Católica do Rio de JaneiroPUCRio
Departamento de Engenharia Mecânica
Rua Marquês de São Vicente, 225 Gávea
22453900 Rio de Janeiro, RJ. Brazil
In general the motion of a body takes place in a confined environment and collision of the body with the containing wall is possible. In order to predict the dynamics of a body in this condition one must know what happens in a collision. Therefore, the problem is: if one knows the precollision dynamics of the body and the properties of the body and the wall one wants to predict the postcollision dynamics. This problem is quite old and it appeared in the literature in 1668. Up to 1984 it seemed that Newton's model was enough to solve the problem. But it was found that this was not the case and a renewed interest in the problem appeared. The aim of this paper is to treat the problem of plan collisions of rigid bodies, to classify the different models found in the literature and to present a new model that is a generalization of most of these models.
Keywords: Collisions, dynamics, modeling
Introduction
From the simplest observation, we can say that the dynamics of a body or of a system with more than one particle can be modeled properly only if collisions are taken into account. In the works of Galileu and Descartes there are references to the collision between particles, but the first published model of this problem seems to be due to John Wallis and Christopher Wren, independently, in 1668. In the same period, Mariote did some experimental work that had great repercussion. Some great scientists such as Newton, Huygens, Coriolis, Darboux, Routh, Apple, Carnot and Poisson have also treated the problem. At the beginning of this century the problem generated some discussions, as we can see in the works of Painlevé (1905) and Klein (1910). But, up to 1984, all of these works used the theory developed by Newton or by Poisson and the dificulty was to include friction in the modeling, as was pointed out by Painlevé in his famous paper "Sur les lois de frottement de glissement".
In 1984, Kane published a work, "A Dynamics Puzzle", in a journal with limited circulation, where he pointed out an apparent paradox: the application of Newton's theory with Coulomb's friction, universally accepted, in a problem of collisions of a double pendulum, conducted to generation of energy. When the tip of the pendulum collided with a fix barrier, then the energy increased. What was wrong?
In 1986, Keller presented a solution to Kane's paradox, but the solution was not easy to generalize. Keller's work was published in a journal with large circulation and arose widespread interest. In these thirteen years the interest has increased and there are some books totally dedicated to this topic, as the ones written by GlockerPfeiffer (1995), Brach (1991), Brogliato (1996) and MonteiroMarques (1993).
Brach (1989) presented a model with linear equations containing some nondimensional parameters that characterize the collision and he defined "ratio between impulses" instead of coefficient of friction. However, his consideration did not give clear solutions to the problem when one considers during the collision. Stronge (1990) suggested a coefficient of restitution relating the energy during the compression phase to the energy during the expansion phase. Smith (1991) presented a model with nonlinear equations. WangMason (1992) applied the Routh's technique (1877,1891) and compared the coefficients of restitution given by Newton and Poisson. Sabine Durand (1996) studied the dynamics of systems with unilateral restrictions and included some systems related to the collisions. Chatterjee (1997) presented new laws based in simple algorithms. He has not used many parameters and he obtained good results. Soianovici and Hermuzlu (1996) have shown the limits of validity of some rigid bodies collision models. As their main interest was in Robotics, they focused in collisions of slender bodies at low velocities. Cathérine Cholet (1998) presented a new theory of rigid bodies collisions using the basic formalism of Continuum Mechanics. Her work was based in the ideas introduced by Michel Frémond: a system formed by a set of rigid bodies is deformable because the relative positions between each pair of bodies vary. In Cholet's work the theory is discussed and she showed that it is coherent from the mathematical point of view and also experimentally validated.
Nomenclature
_{N }= Relative Normal Velocity _{T} = Relative Tangential Velocity
e_{b} = BeghinBoulanger's coefficient of normal restitution
_{N} = Relative Angular Velocitye_{m} , e_{mC}e e_{mE} = Coefficients of moment
e_{n} = Newton's coefficient of normal restitution
e_{t }= Coefficient of tangential restitution
I = Impulse caused by the reaction in the contact
I_{N} = Normal Impulse
I_{T }= Tangential Impulse
I_{q} = Angular Impulse
J = Moment of inertia related to the center of mass
m = Mass
[M] = Mass Matrix
[ML] = Local Mass Matrix
q = Vector of Generalized Coordinates
Q_{i} = Contribution of the External Generalized Forces
r = Generalized Collision Force
r_{i}Generalized Force due to the Reaction in the Contact
R = Reaction Force in Collision
t = Transposition Symbol
T = Kinetic Energy
u_{N = }Unitary Vector of the Normal Direction
u_{r} = Unitary Vector of the Tangential Direction
[W] = Matrix relating r with R
W_{N }, W_{T} and W_{q}= Columns of the Matrix [W]
= Vector Related to the Rotation
index A = Indicates Precollision
index C = Indicates end of the Compression Phase
index E = Indicates end of the Expansion Phase or Postcollision
m = Coefficient of Friction
m_{0} = Coefficient of Statics Friction
m_{s }= Coefficient of Critical Friction
n = Coefficient of Tangential Restitution
(t , N) = Collision Frame
Motion Equations
The collision is plan and it is modeled as instantaneous. Let the generalized position of the system in the instant t be defined by q = (q_{1},q_{2},...,q_{n})^{t}. We consider the contact between two bodies C_{1} and C_{2} and let R be the force of reaction exerted by C_{1} on C_{2}. Then we write R = (R_{N} R_{T} )^{t}.
The dynamics of the system is given by the Lagrangean equations :
with Q_{i} the contribution of the external generalized forces, r_{i} the generalized force due to the reaction in the contact and T the kinetic energy of the system. We should observe that r_{i} is only present when there is contact, otherwise it is null.
Considering only a planar situation, we have n parameters of position and two reactions in the contact (R_{N} and R_{T} ) also unknowns. Then, we need, not only the n equations obtained from Lagrange's equations but also two equations more given by the collision laws that will be discussed later.
We consider P_{1} and P_{2} the points of C_{1} and C_{2}, respectively, that will be in contact in the collision. We denote by D the vector that represents the relative displacement between the two bodies and by the vector that represents the relative velocity between the bodies, as shown in the Fig.1.
In the point of contact we represent the impulses in the normal and tangential directions by I_{N} and I_{T} . We use u_{n} and u_{t} the unitary vectors of the normal direction (given by n) and tangential direction (given by t ) in a frame which we will call collision frame, shown in Fig.1.
Evaluating the relative velocity between the contact points we have,
Introducing the notations,
We consider, then, W_{T} the column vector in which the components are and W_{N} the column vector in which the components are
We can write the normal (
_{N}) and tangential (_{T}) components of as
or, we can write,
The generalized force r can be written in terms of [W] and of R as
Integrating Eq.1 in the interval (te , t+e ,), with t the instant of collision, we have,
We observe that is bounded. We consider Q_{i} continuous and bounded and we make e ® 0, then
We use the index E to represent the right limit and A to represent the left limit.
We know that r = [W] R.
Then,
We write the impulse I caused by the reaction R as
Then,
We denote as the vector in which the components are , so that we can write
But,
Our problem is to find and I given [M], [W] and . Then, there are n equations and we want to find n +2 unknowns. Therefore, we need two more equations. These two equations are given by the restitution laws discussed later.
In some cases we can consider also an impulse of moment denoted by I_{q} . Some models consider that in the collision appear not only the normal and the tangential impulse but also the impulse of moment. In this case, the equation will be given by
In this case, we have n equations to find n+3 unknowns. We need three more equations. These three equations will be given by the restitution laws.
We construct a collision model when we join the n equations that describe the motion of the system with the equations given by the restitution laws.
In order to solve the problem we use a strategy that consists in defining a process called virtual process. It is not related to time. We show a scheme in the Fig.2 to ilustrate this idea.
The Local Matrix Mass
Instead of writing the equations in terms of we can use . The vector D was shown in Fig.1 and it is important because it monitors when the collision occurs.
We can write
Then,
But,
Then,
So,
when [
_{L}] is invertible and []^{1}_{L} =[M_{L}]. We call [M_{L}] the local matrix mass.Compression Phase and Expansion Phase
In order to describe some of the collision models we will think, formally, that the change between the precollision velocity to the postcollision velocity occurs in two phases: the compression phase and the expansion phase. The virtual process will be composed by these two phases as it is shown schematically in Fig.3.
The Restitution Laws
As we have already said, to solve a collision problem we need the n equations given by the Lagrange formalism and two more equations (or three, if we consider the impulse of moment). These equations are given by the restitution laws that will be divided in restitutions laws in the normal direction and restitution laws in the tangential direction. These equations are constitutive and are not basic laws. They depend on materials and processes.
Restitution Laws in the Normal Direction
In the normal direction, the restitution laws mostly used are those given by Newton and by Poisson. As ilustration we will also comment about the restitution law given by BeginBoulanger, since it is based in a different principle. Newton is a purely kinematical law, Poisson's is dynamical and BeginBoulanger uses energy consideration. Each one of these laws define a coefficient of restitution, used in the models, that indicates the behavior postcollision in the normal direction.
The coefficient of restitution given by Newton considers the normal relative velocities pre and postcollision. The coefficient of restitution given by Poisson considers the normal impulses in the compression phase and in the expansion phase and the coefficient of restitution given by BeghinBoulanger considers the kinetic energy in the compression phase and in the expansion phase.
Coefficient of Restitution given by Newton
The coefficient of restitution given by Newton, denoted by e_{n}, is defined as the ratio between the normal relative velocity postcollision (_{N} ) and the normal relative velocity precollision (_{N}). We can write,
This coefficient of restitution takes into consideration only the kinematics of the system in the collision.
Coefficient of Restitution given by Poisson
The coefficient of restitution given by Poisson, denoted by e_{np} , is defined as the ratio between the normal impulse in the expansion phase (I_{NE} ) and the normal impulse in the compression phase (I_{NC}) .
This coefficient of restitution takes into consideration the dynamics of the system in the virtual process of collision.
Coefficient of Restitution given by BeghinBoulanger
The coefficient of restitution given by BeginBoulanger, denoted by e_{b} , is related with the loss of kinetic energy during the collision. This coefficient shows the relation between the kinetic energy in the expansion phase and the kinetic energy in the compression phase.
Then,
This coefficient of restitution considers the exchange of energy during the virtual process of collision.
It is easy to show that when friction is not considered all of these coefficients of restitution are equivalent [3].
Restitution Laws in the Tangential Direction
In the tangential direction, the first law to be considered is the case of perfect collision; that is, when the tangential impulse is null; I_{T} = 0 . This is the case when we do not consider friction in the collision.
When we consider friction, the mostly used law is Coulomb's law. In reality, we use a modification of Coulomb's law, which is expressed in terms of impulses (and not in terms of forces).
We write
and
A Classification of the Collision Models
We want to discuss some collision models used in the literature and to do this we can classify them in four groups.
First Group
The first group does not consider the compression phase; that is, we use only the A and E indexes. This group also does not consider the impulse of moment.
We have,
Second Group
This group does not consider the compression phase (as the first group) but it considers the impulse of moment.
The equations used are
and
Third Group
This group considers the compression phase (index C) and the expansion phase (index E). This group does not consider the impulse of moment.
The equations used are
and
Fourth Group
The equations used in this group form the most general case. This group considers the compression and expansion phases and it also considers the impulse of moment.
The equations are given by
and
The Collision Models
In order to characterize a collision model we need the equations given by one of the four groups present in the last section and the restitutions laws. In this section we want to present some rigid bodies collision models. We also present a new model: CS model [3]. We show in Tab.1 the collision models that will be discussed, the group of equations used by them and the restitutions laws used for each one.
Model: Newton without Friction
This model considers only the coefficient of restitution given by Newton. It uses the equations from the first group and as we do not consider friction, the impulse in the tangencial direction is null. Then, the equations are given by
and also
Model: Brach
Brach's model is based in the equations from the second group. The parameters used are nondimensional ratios between physical measures. In the normal direction the model uses the coefficient of restitution given by Newton. In the tangential direction Brach uses a coefficient given by the ratio between the impulses in the tangential and in the normal direction, denoted also by m , although this is not the coefficient of friction given by Coulomb. This model considers a coefficient of moment denoted by e_{m}.
This coefficient of moment is given by
The equations used are those from the fourth group.
The restitution law in the normal direction is given by
and in the tangential direction given by
Model: Kane
This model considers the coefficient of normal restitution given by Newton and it considers the coefficient of friction given by Coulomb's law (modified).
The equations used are from the first group. The restitution law used in the normal direction is
_{NE} = e_{n}_{NA }and in the tangential direction it is given by
m_{0} is the static coefficient of friction and m is the kinetics coefficient of friction.
This model was used by Kane to solve a problem with collisions of a double pendulum. He observed increasing of energy. Of course, if not carefully used, is a bad model. Its importance is for showing what might happen if one is not careful.
Model: GlockerPfeiffer  first case
This model uses the coefficient of restitution given by Poisson and it considers the virtual process of collision composed by two phases: compression and expansion. In the tangential direction it uses Coulomb's law (modified). The equations used are from the third group. The GlockerPfeiffer's model will be divided in two cases, because in the expansion phase we have two differente possibilities. Although the first case is a particular case of the second case, we will separate them for didatic reasons.
This model considers the restitution law in the normal direction given by
In the tangential direction, in the compression phase, it considers
In the expansion phase it uses
Model: GlockerPfeiffer  second case
When we discussed the GlockerPfeiffer's model in the previous section, we considered the same restitution law in the tangential direction in the expansion phase and in the compression phase. But, GlockerPfeiffer also proposed a model (another one) taking in consideration what they called the reversible portions of the tangential impulse that can occur, for example, when we analyze the superball phenomenon, discussed in Pfeiffer, F. and Glocker, C. (1996) and also in Cataldo, E. (1999). These effects are taken in consideration, by the authors, with the translation of the tangential characteristic by a determined quantity, which we denote by 2I_{TS}. To introduce this quantity we need to consider new parameters for the tangential restitution.
The restitution laws used in the tangential direction are divided in two cases and given by
Case 1: I_{TC }³0 and I_{TS}³ 0
We use
Case 2: ITC £ 0 and ITS £ 0
We use
The quantity I_{TS} is given by
n and e_{t} are additional parameters.
Model: WangMason  Based in the Routh's Technique
The Routh's method is a graphic technique to analyze plane collisions with friction. Wang and Mason (1992) used the Routh's method, the Coulomb's law and the coefficient of restitution given by Newton or by Poisson to predict the impulse in the collision. The method used by Wang and Mason consists in analysing the values of the tangential and normal impulses, I_{T} and I_{N} , constructing a virtual process of collision.
Model: Smith
This model uses the coefficient of restitution given by Newton. The restitution law used in the tangential direction uses an average of the tangential components of the relative velocities in the instants pre and postcollision. The equations used are from the first group.
In the tangential direction it uses
and in the normal direction
Model: CS
This model, proposed by E. Cataldo and R. Sampaio [3][4] tries to generalize some of the models presented. The Tab.2 shows the number of parameters used for each model.
This model considers the equations from the fourth group, it uses the coefficient of restitution given by Poisson, Coulomb's law (modified) to the tangential direction and two coefficients of moment.
The equations used are described in the following
Compression Phase
We use
with .
Expansion Phase
We use,
The model uses the coefficient of restitution given by Poisson.
Particular Cases
We show some particular cases obtained from the CS model.
We have Newton's model (without friction).
We have the GlockerPfeiffer's model (first case).
We have a case similar to the Brach's model.
We have the GlockerPfeiffer's model.
Conclusions
Using our framework, we presented some of the models found in the literature. This allowed a systematic presentation and a critical comparison of the models studied. We proposed a classification of these models according to the motion equations used by each of them. Moreover, we proposed a more general model that serves to unify different theories: The CS model. This model was constructed in such way to avoid the problem of increasing of energy, such as founded by Kane (1984).
Manuscript received: March 2000. Technical Editor: Átila P. S. Freire.
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Publication Dates

Publication in this collection
28 Sept 2001 
Date of issue
2001
History

Received
Mar 2000