Elastic impact of a composite rod

The elastic impact of a composite rod of variable cross section is considered (Fig. !). The geometric parameters of the i-th rod (i = I, 2, ..., n) are determined by the length Zi, and the cross-sectional area Si(x), where Li-x~x ~Li; Lo= 0. The physicomechanical parameters of the i-th rod are constant over its length: a i is the velocity of the elastic-perturbation propagation in the rod material; E i is the elastic modulus of the rod material. The system moves at an initial velocity Vo and strikes a motionless barrier. The energy losses of deformation at the Junctions are not taken into account. The motion of the i-th rod is described by a differential equation in terms of the displacement function ui(x, t) O2u, 1 dSi (x) 1 O~ui (i = 1, 2, n). (1)