Planar collision-type dependence on incident angle and on friction coefficient

Abstract

This work addresses the ‘hard collision’ approach to the solution of planar, simple non-holonomic systems undergoing a one-point collision-with-friction problem, showing that (i) there are no coherent types of collision whereby forward sliding follows sticking, unless the initial relative tangential velocity of the colliding points vanishes; and (ii) the type of collision can be determined directly, given the collision angle of incidence $\alpha$ and Coulomb’s coefficient of friction $\mu$ between the colliding points. The classic hitting rod problem is used to illustrate the $\alpha$–$\mu$ collision-type dependence. Finally, the relation between collision with friction and tangential impact problems in multibody systems is briefly discussed.

Appendix

An algorithm for collision-type identification based upon Table 1: $\text{Type}=\text{T}5$ unless

1	$ghp > 0 \cup (\alpha /r_{m} < 1 \to T1\ \mbox{or}\ 1 < \alpha /r_{m} < 1 + e{}_{e} \to T2)$
2	$p < 0 \cup (\alpha /r_{m} < 1 \to T3\ \mbox{or}\ 1 < \alpha /r_{m} < 1 + e{}_{e} \to T4)$
3	h<0→T1

An algorithm for collision-type identification based upon Fig. 3: $\text{Type}=\text{T}5$ unless

1	$\mu > |m_{nt}|/m_{tt} \cup (\alpha /r_{m} < 1 \to T1\ \mbox{or}\ 1 < \alpha /r_{m} < 1 + e{}_{e} \to T2)$
2	$m_{nt} < 0 \cup \mu < |m_{nt}|/m_{tt} \cup (\alpha /r_{m} < 1 \to T3 \ \mbox{or}\ 1 < \alpha /r_{m} < 1 + e{}_{e} \to T4)$