Abstract:
The problem on the normal impact of an elastic sphere upon an elastic Timoshenko beam of thin-walled monosymmetrical open section is considered. The process of impact is accompanied by the dynamic flexure and torsion of the beam, resulting in the propagation of plane flexural, shear, and torsional waves of strong discontinuity along the beam axis. Behind the wavefronts the solution is constructed in terms of one-term ray expansions. During the impact the sphere moves under the action of the contact force which is determined due to the Hertz's theory, but the contact region moves under the action of the contact force, and the bending-torsional moment and transverse forces, which are subjected to the lateral surfaces of the contact region and are determined using one-term ray expansions. The joint consideration of the equations of sphere and the contact region motion leads to the Abel equation of second kind in the rate of change of the value of the sphere and beam drawing together. The solution to the Abel equation written in the form of a series in terms of this value allows one to determine all characteristics of the sphere and beam shock interaction.