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TECHNICAL BRIEFS

# Finite Duration Impacts With External Forces

[+] Author and Article Information
D. Dane Quinn

Department of Mechanical Engineering, The University of Akron, Akron, OH 44325-3903quinn@uakron.edu

Each of these examples leads to an equivalent equation of motion for the contact deformation during the collision.

J. Appl. Mech 72(5), 778-784 (Sep 30, 2004) (7 pages) doi:10.1115/1.1875552 History: Received March 29, 2004; Revised September 30, 2004

## Abstract

This work considers the effect of external forces during finite duration collisions using an incremental model of impact. The deformation of the “rigid” body is modeled through an elastic element and the time interval over which contact occurs is of finite duration. Moreover, the work done by the external forces is nonzero during the collision. This model allows for an equivalent coefficient of restitution $e$ to be identified. In the presence of a constant external force the coefficient of restitution depends not only on the system parameters, but the initial relative velocity at the point of impact. For external forces which tend to bring the colliding bodies together, the colliding bodies remain in contact for sufficiently small impact velocities $(e=0)$ while for larger incoming speeds, the coefficient of restitution is positive. This state dependent restitution arises from the coupling of external forces to the collision model, and is not seen in more familiar models of impact. Finally, based on the results of the experimental system and the incremental model, the standard algebraic model of restitution is modified to include these finite duration effects.

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Copyright © 2005 by American Society of Mechanical Engineers
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## Figures

Figure 1

Experimental setup

Figure 2

Representative impact data (voltage versus time)

Figure 3

Experimental coefficient of restitution. Each panel is the compilation of ten individual tests

Figure 4

Incremental model

Figure 5

Equivalent Newtonian restitution (F0=0)

Figure 6

Equivalent coefficient of restitution. The value of e∞ is shown by the dashed line.

Figure 7

Critical ν below which the coefficient of restitution vanishes

Figure 8

Critical ν versus e∞

Figure 9

Duration of the collision tℓ

Figure 10

Equivalent Newtonian restitution for the nonlinear contact model of Hunt and Crossley (δ=0.10). The measured coefficient for F0=0 is shown for reference as the dashed line.

Figure 11

Experimental critical cutoff velocity. The solid curve is Eq. 9 with σ=7.83×10−3s.

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