Impact of a cone on a thin elastic membrane

The problem of computation of parameters of motion of a thin elastic membrane under the Impact of a rigid body, was considered by various authors (see [1] together with bibliography) without, however, yielding a rational solution. This paper presents a full qualitative analysis of solution of this problem for the case of normal impact of a circular cone moving with constant velocity on an infinite elastic membrane of constant thickness. Although this is the simplest case, it is important, insofar as it brings to light the characteristic features of the problem. In the “membrane” approximation the thickness of the layer is found to be an unessential parameter, therefore the problem, as postulated by us, is self-similar and its solution is reducible to the problem for ordinary differential equations.