On some applications of Hertz's theory of impact

Abstract

Statement of the Theory. — One of the premises on which the mathematical theory of the collision of elastic solids given by Hertz is based is that the strains produced in the immediate neighborhood of the region of contact are determined by the pressure subsisting at any instant between the bodies, and are practically the same as under statical conditions. This premise is valid even when the impinging bodies do not move as rigid bodies, and the impact results in part of the translational kinetic energy being transformed into energy of elastic wave-motion in the substance of the solids. Hertz's theory of impact with suitable modifications may accordingly be applied under a very wide variety of conditions. In the present paper, an attempt is made to discuss the problem of the transverse impact of a solid sphere on an infinitely extended elastic plate of finite thickness and to calculate the theoretical coefficient of restitution which is a function of the elastic constants and densities of the materials, the diameter of the sphere and the thickness of the plate, and of the velocity of impact. As the result of the impact, annular waves of flexure are set up in the plate, and the sum of the kinetic and potential energies of the wave-motion may be determined in terms of the magnitude and duration of the impulse on certain simplifying assumptions. The calculation results in a simple formula for the coefficient of restitution. Experiments. —A series of experiments carried out in the author's laboratory by Mr. A. Venkatasubbaraman has furnished a quantitative confirmation of the formula within the limits of its applicability, that is for plates not thinner than about half the diameter of the spheres. For plates much thinner than this, theory and experiment agree in indicating a zero coefficient of restitution. The formula indicates that the coefficient of restitution should increase and approach unity for greatly diminished velocities of impact, and this is also confirmed in experiment. The paper concludes with indications of some further applications and extensions of Hertz's theory of impact.