Parametric resonances in the problem of longitudinal impact on a thin rod

Аннотация

The longitudinal impact on a thin elastic rod, which generates a periodic system of longitudinal waves in it, is considered. At definite values of the parameters of the problem in the linear approximation, these waves induce parametric resonances, which are accompanied by an unlimited increase in the amplitude of the transverse vibrations. To obtain finite values of the amplitudes, a quasilinear system is considered in which the effect of the transverse vibrations on the longitudinal vibrations is taken into account. This system was previously solved using the Bubnov–Galerkin method and beats accompanied by energy transfer between the transverse and longitudinal vibrations were obtained. In this work, an approximate analytical solution of the system has been derived that is based on double-scale expansions. A qualitative analysis of this solution has been carried out. An estimate of the maximum transverse bending has been obtained for various methods of loading. Both shortand long-term pulses have been considered. It has been shown that, in the case of a spontaneously applied long-term pulse that is lower than the Euler critical load, intensive transverse vibrations can occur.