Solving a paradox in impacts dynamics of continuous systems

In this work, as a representative of continuous systems impacting with a body, we consider the problem of a ball bouncing on a beam, focusing our attention on the infinite dimensionality of the system. A modal expansion of the equation of motion is used, and the solution during the contact phase is obtained in closed form. It is shown that the time duration of the impact can be shrunk to zero, i.e. the impact can be considered instantaneous in time. Then, the distribution of the impact contact force in space is addressed. We prove that it cannot be considered as a Dirac delta function, i.e. localized in space, since this leads to an inconsistency, caused by the divergence of the series involved in the problem and entailing an unphysical situation, that has not been previously highlighted (to the best of authors' knowledge). The conclusion is that in continuous bodies the impact cannot be simultaneously istantaneous and localized. This marks a strong difference with the finite dimension (discrete) counterpart, in which an impulsive contact force in both space and time is possible and is commonly used. A solution is proposed to overcome this apparent paradox and a simple impact model is developed and illustrated with some examples.