Classical planar impact theory and the tip impact of a slender rod

This paper covers the topic of the planar eccentric impact of a rigid body at a point. The tip impact of a slender rod against a massive surface or barrier is used as a means to illustrate the principles and solution techniques. Two issues are addressed; the first is how the planar impact problem should be, or at least can be, formulated and solved. The second is how close classical solutions match those obtained by methods used in the field of shock and vibration. In formulating the classical impact problem, three well-defined coefficients of restitution are available, the kinematic (defined as a ratio of velocities), kinetic (defined as a ratio of impulses) and energetic (defined as a ratio of energies). Their relationship, advantages and disadvantages are discussed. The best approach to treat tangential impulses in general has not yet reached agreement. For eccentric impacts with Coulomb friction, for example, various combinations of sliding and sticking including tangential velocity reversals can occur during contact. Several approaches have been proposed and all are related. A method using the impulse ratio as a basic parameter is covered and provides solutions for arbitrarily shaped lamina and arbitrary initial conditions. Solutions of the tip impact of a long slender rod using the classical approach are found for various combinations of coefficients of friction and restitution and initial velocities. The final velocities, impulses and energy losses from these solutions are compared with solutions obtained by integrating the differential equations of motion of a rigid rod striking a viscoelastic surface. The comparisons show excellent agreement for most of the cases considered.