Beating in the problem of longitudinal impact on a thin rod

The longitudinal impact on an elastic rod generating a periodic system of longitudinal waves in the rod, is considered. For certain values of the problem parameters in the linear approximation, these waves generate parametric resonances accompanied by an infinite increase in the transverse vibrations amplitude. To obtain the finite values of the amplitudes, a quasilinear system where the influence of transverse vibrations on the longitudinal ones is taken into account was considered. Earlier, this system was solved numerically by the Bubnov—Galerkin method and the beatings accompanied by energy exchange between the longitudinal and transverse vibrations were obtained. Here an approximate analytic solution of this system based on two-scale expansions is constructed. A qualitative analysis is performed. The maximum transverse deflection depending on the loading method is estimated.