On the Timoshenko Beam Vibrating under an Obstacle Condition

The dynamic impact problem for the Timoshenko beam against a rigid frictionless obstacle is studied. The unknown reaction is modeled as a positive measure with support contained in the contact set and acting on the centroid of the beam in the vertical direction. Three independent invariant quantities of energy type for the free beam are derived. These quantities turn out to be useful in the description of the impact lines, the crucial assumption being the conservation of the local energies in order to model the perfectly elastic impact. The problem of extending the solution after the first influence line is considered. The strict hyperbolicity of the system leads to a free-boundary problem similar to a previous one studied by L. Amerio for theimpact of two strings with different propagation velocities. In a “generic” case, a necessaryand sufficient condition for the solvability of the free-boundary problem is provided.