A Method of Calculating Energy Losses During Impact

Abstract

Abstract When an elastic sphere collides with another perfectly elastic body, part of the initial kinetic energy is lost in starting elastic waves in the two bodies. The energy thus dissipated can be calculated by previous analytical methods only when it is a small fraction of the initial kinetic energy. This paper develops an approximate analytical method for this calculation which is applicable even when the greater part of the energy is dissipated. Thus in a particular case where the energy dissipated is 90 per cent of the original kinetic energy, an error of only 0.7 per cent is made. The new element in this analysis is the introduction of a “normalized” interaction force which is necessarily quite insensitive to one’s ignorance of the exact interaction force. The powerfulness of the method is illustrated by a complete survey of the problem of impact of spheres with beams fixed at each end. Graphs are constructed showing the variation of the coefficient of restitution with the length of the beam and the mass of the sphere. Impacts with multiple blows are excluded. It is found that when the mass of the sphere is kept constant, the coefficient of restitution has a minimum for a certain optimal beam length, and is independent of the beam length for values greater than twice the optimal length. The coefficient is a minimum when the period of the fundamental mode of vibration is approximately equal to 2.4 times the time of contact. The method is also applied to the impact of spheres with large thin plates. The semiempirical formula of Raman is derived. Although this analytical method cannot be applied to impacts with multiple blows, it nevertheless gives the conditions for such multiple blows.