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TECHNICAL PAPERS

# On the Normal Component of Centralized Frictionless Collision Sequences

[+] Author and Article Information
Pieter J. Mosterman

The MathWorks, Inc., 3 Apple Hill Dr., Natick, MA 01760

Notation is derived from (3). Contact points will be superscripted, whereas reference to bodies around contact points and their variables are subscripted.

During continuous behavior, $v1,Ei=v1,Ai$ and $v2,Ei=v2,Ai$, as the left and right limit values of a point on a continuous $(C0)$ curve have to be equal.

Note that $gi$ is not indexed by either $E$ or $A$ as it is a $C0$ variable.

Alternatively, to achieve the same effect, the clause can be replaced by $Ik−1P=∅$.

The MATLAB code of the collision models is available upon request.

J. Appl. Mech 74(5), 908-915 (Jan 04, 2001) (8 pages) doi:10.1115/1.2712237 History: Received December 05, 2000; Revised January 04, 2001

## Abstract

A typical assumption for rigid body collisions with multiple impact points is that all collisions occur simultaneously and are synchronized in their compression/expansion behavior, a useful assumption given the microscopic time scale at which collisions occur. In the case in which collisions are dependent upon one another, however, there is interaction between and within compression and expansion phases. Instead of treating the collisions as separate consecutive impacts or by activating all constraints at the same time, a rule is presented that orders the collisions as a sequence of interacting events at a point in time to handle the normal component of the collisions.

###### FIGURES IN THIS ARTICLE
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Copyright © 2007 by American Society of Mechanical Engineers
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## Figures

Figure 1

A sequence of dependent collisions

Figure 2

Collision between two bodies and a floor

Figure 3

Resulting velocities for Newton’s collision rule: (a) As a function of ϵ12; (b) as a function of ϵ12 with m3 large; (c) as a function of ϵ23

Figure 4

Resulting velocities for Poission’s collision rule: (a) As a function of ϵ12; (b) as a function of ϵ12 with m3 large; (c) as a function of ϵ23

Figure 5

Different elasticity leads to differences in relative velocity

Figure 6

Resulting velocities for the new collision rule: (a) As a function of ϵ12; (b) as a function of ϵ12 with m3 large; (c) as a function of ϵ23

Figure 7

A sequence of four colliding bodies

Figure 8

Resulting velocities for the new collision rule: (a) As a function of ϵ12; (b) as a function of ϵ23; (c) as a function of ϵ34

Figure 9

Final velocities for m2=0.1 and 0≤ϵ23≤1

## Errata

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