GENERALIZED IMPULSE AND MOMENTUM APPLIED TO MULTIBODY IMPACT WITH FRICTION^*

Abstract

A mechanism composed of rigid bodies joined by ideal nondissipative pinned joints has a configuration that can be described in terms of generalized coordinates q _i and time t; the system has a kinetic energy . The generalized momentum of the system is defined as a vector, . If the system is subject to a set of time-dependent external forces F _j that act at locations r _j , these forces result in differential impulses dp _j = F _j dt. The generalized impulse vector Π_ι associated with these impulses has a differential d, where are velocities of the points at which the impulses are applied. If the forces act impulsively (i.e., there is a negligibly small period of force application), the differentials of generalized momentum and generalized impulse are equal,

and if the forces are proportional during the “period” of application, then integration of this equation gives a change of state .

When applied to impact between systems of hard bodies for which there is friction and slip that changes direction during contact, the differential relation is required. If the direction of slip is constant, however, it is more convenient to use the integrated form of this generalized impulse-momentum relation. In either case, at the point of external impact, the terminal impulse is obtained from the energetic coefficient of restitution.

^*Communicated by B. Gilmore.

Acknowledgments

Notes

^*Communicated by B. Gilmore.

^*For e _* = 1, μ = 0, and using the coordinate system of the present paper, Pereira and Nikravesh [[7]] give a terminal velocity V ⁽⁺⁾ _C = L(− 1.7n ₁ + 0.416n ₂), which results from rad/s, rad/s. Although these coefficients of friction and restitution represent a conservative system, their solution has substantial energy loss (T ₀ − T _f )/T ₀ = 0.80 that is unaccountable.