The Bounds on the Coefficients of Restitution for the Frictional Impact of Rigid Pendulum Against a Fixed Surface

Upper bounds on Newton's, Poisson's, and energetic coefficients of normal restitution for the frictional impact of rigid pendulum against a fixed surface are derived, demonstrating that the upper bound on Newton's coefficient is smaller than 1, while the upper bound on Poisson's coefficient is greater than 1. The upper bound on the energetic coefficient of restitution, which is a geometric mean of Newton's and Poisson's coefficients of normal restitution, is equal to 1. Lower bound on all three coefficients is equal to zero. The bounds on the tangential impact coefficient, defined by the ratio of the frictional and normal impulses, are also derived. Its lower bound is negative, while its upper bound is equal to the kinetic coefficient of friction. Simplified bounds in the case of a nearly vertical impact are also derived. DOI: 10.1115/1.3172198 The determination of the rebounding velocity components of colliding bodies is an old mechanics problem, with its origin in early work by Newton and Poisson. Newton defined the coeffi- cient of normal restitution as the ratio of the relative normal ve- locities after and before the impact. In contrast to Newton's kine- matic definition, Poisson's kinetic definition is based on the ratio of the magnitudes of the normal impulses corresponding to the periods of restitution and compression. In the absence of friction frictionless impact, the Poisson definition of the coefficient of normal restitution yields the same expression, in terms of the rela- tive velocities, as does the Newton definition, which is demon- strated in standard dynamics textbooks, e.g., Ref. 1. In the pres- ence of friction, however, the two definitions are, in general, not equivalent. The simplest theory of the frictional impact is that of Whittaker 2, in which it is assumed that the frictional impulse is in the slip direction and is equal to the product of the coefficient of friction and the magnitude of the normal impulse. Kane 3 observed that this theory leads to an increase in kinetic energy upon the impact of a double pendulum with a rough horizontal surface, for some values of the coefficients of friction and normal restitution, and for some kinematic parameters of motion. Keller 4 explained this by noting that Whittaker's theory 2 applies only when the direction of sliding is constant throughout the col- lision. If there is a reversal of the slip direction during the impact process, the coefficient of the proportionality between the tangen- tial and normal impulses is different from the coefficient of kinetic friction. Keller's 4 analysis also demonstrated the advantage of using the normal impulse as an independent variable, instead of physical time, to cast and analyze the governing differential equa- tions of motion during the impact process. Stronge 5 introduced an energetic coefficient of normal restitution, whose square is equal to the negative ratio of the elastic strain energy released during restitution and the internal energy of deformation absorbed during compression phase of the impact. Numerous papers, pro- posing different models of frictional impact, were published since. A comprehensive treatment of the subject, with a historical out- line, can be found in the monographs or review articles 6-9. In this paper we revisit a classical problem of frictional impact of rigid pendulum against a fixed surface. By employing Keller's 4 method of analysis, we derive the expressions for the angular velocity in terms of the monotonically increasing normal impulse during the impact process. Three different definitions of the coef- ficient of normal restitution are used to specify the rebounding angular velocity and the total normal impulse: Newton's kine- matic, Poisson's kinetic, and Stronge's energetic definitions. It is shown that the energetic coefficient of normal restitution is a geo- metric mean of the Newton and Poisson coefficients of normal restitution. The upper bounds on all three coefficients are estab- lished, demonstrating that the upper bound on the Newton coeffi- cient is smaller than 1, while the upper bound on the Poisson coefficient is greater than 1. For the pendulum striking a rough surface elastically, without dissipation due to deformation, the Newton and Poisson coefficients are the reciprocals of each other. If, upon the impact, the pendulum sticks to the ground, there is no restitution phase of the impact, and all three coefficients of normal restitution are equal to zero, which represents their lower bound. The bounds on the tangential impact coefficient, defined by the ratio of the frictional and normal impulses, are also derived. Its lower bound is negative, while its upper bound is equal to the kinetic coefficient of friction. Simplified bounds in the case of a nearly vertical impact are also derived.