Discussion: “Predicting Rebounds Using Rigid-Body Dynamics” (Smith, C. E., 1991, ASME J. Appl Mech., 58, pp. 754–758)

In a recent paper, Smith (1991) presents comments related to specific items from the works of Brach (1984Brach ( , 1989)).In particular, Smith says: (1) the statement of Brach (1989) that the kinematic coefficient of restitution is bounded by 0 and 1, that is 0<e<l, is incorrect or at least unsupported, (2) Eq. ( 19) in Brach (1984) for the energy loss of a collision of two particles is incorrect, and (3) if the quantity, K, in the denominator of Smith's expression (unnumbered) for the change in kinetic energy AK were replaced by 1, Smith's results would agree with Brach's energy loss expression.The work of Stronge (1990) clearly points out that an energetic coefficient of restitution, E 2 , is bounded by 0 and 1, that is, 0<E 2 <:\.Since the kinematic coefficient, e, and energetic coefficient, E 2 , are generally different, Smith's criticism of Brach's bounds as summarized in item 1 above is justified.Equation ( 19) for the energy loss of a planar particle collision is Brach (1984) follows directly from Newton's laws of particle dynamics as applied to collisions.This is demonstrated clearly in Brach (1984) and by a different method in Brach (1991).Consequently, Smith's claim in item 2 above is itself incorrect.With reference to item 3 above, if the quantity K is replaced by 1, the appropriate expressions of Brach and Smith have the same form, however, there exists a subtle and important difference.In Smith's equation for AK, ^ is a coefficient of friction whereas in Brach's Eq. ( 19) /x is an impulse ratio.These can be equal, but only under certain conditions.Smith's equation will give a negative energy loss (which is physically unrealistic) for relatively small initial tangential contact velocities and relatively large coefficients of friction.Brach uses the impulse ratio specifically to avoid this problem.