Vibrations of a beam between stops: Collision events and energy balance properties

Abstract

<p style='text-indent:20px;'>We consider the model problem of an elastic beam vibrating between two stops. More precisely the beam is clamped at its left end while its right end may undergo contact and collision events with two stops. We model the interaction between the beam and the stops either with Signorini complementarity conditions when the stops are perfectly rigid or with a normal compliance contact law allowing some penetration within the stops and given by a linear relationship between the shear stress and the penetration at some positive power <inline-formula><tex-math id="M1">\begin{document}$ \beta $\end{document}</tex-math></inline-formula> when contact occurs. <p style='text-indent:20px;'>Motivated by computational issues we study the evolution of the energy functional defined as the sum of the kinetic energy and the potential energy of elastic deformation of the beam. When contact is modelled with a normal compliance law we prove an energy conservation property. Then we interpret the relationship between the shear stress and the penetration in case of contact as a penalization of the non-penetration condition. We show that the solutions of the penalized problems converge to a <i>strong solution</i> of the problem with Signorini conditions as defined in [<xref ref-type="bibr" rid="b26">26</xref>] and we prove that the limit satisfies an energy conservation property through instantaneaous collision events.