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

Abstract

A longitudinal impact on a thin elastic rod generating a periodic system of longitudinal waves is addressed. Linear problem is considered and for certain values of parameters these waves are shown to generate parametric resonance accompanied by unbounded increase in the amplitude of transverse oscillations. In order to obtain the finite values of amplitudes we consider a quasi-linear system that takes into account the influence of transverse vibrations on the axial ones. In the earlier research this system was numerically solved by the Bubnov—Galerkin approach which resulted in beats displaying the exchange of energy between the axial and transverse oscillations. This paper presents an approximate analytic solution to this systems based on the two-scale expansions is constructed. A qualitative analysis is carried out, too. We obtained an estimate of the maximum transverse deflection depending upon the method of loading. Both the short-term and long-term axial loading are analysed. Intensive transverse vibrations are found out to appear in the case of a suddenly applied axial pulse with the magnitude below the critical Euler force. Refs 17. Figs 7. Tables 1.