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 $ \beta $ when contact occurs.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 strong solution of the problem with Signorini conditions as defined in [26] and we prove that the limit satisfies an energy conservation property through instantaneaous collision events.

中文翻译：

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停止点之间的光束振动：碰撞事件和能量平衡特性

我们考虑弹性梁在两个挡块之间振动的模型问题。更准确地说，光束在其左端被夹持，而其右端可能受到两个挡块的接触和碰撞。当挡块是完全刚性的时，我们可以使用Signorini互补条件模拟梁与挡块之间的相互作用，或者使用正常的顺应性接触定律，允许在挡块内有一定的穿透力，并由剪应力和在某些正值处的穿透力之间的线性关系给出受计算问题的推动，我们研究了能量函数的演变，该函数定义为梁的动能与弹性变形的势能之和。当使用正常的合规法对接触进行建模时，我们证明了一种节能特性。然后，我们将接触情况下的切应力和穿透力之间的关系解释为对非穿透条件的惩罚。我们证明了惩罚性问题的解决方案收敛于[]中定义的Signorini条件问题的强大解决方案26]，我们证明了该极限通过瞬时碰撞事件满足了节能特性。