Longitudinal-transverse bending of an elastic rod under short-term axial action

The paper considers the problem of dynamic longitudinal–transverse bending of an elastic rod with initial curvatures under short-term axial action. The motivation of the research is the current problems related to the analysis of the stability and strength of structural elements under shock loads, applicable, in particular, to the problems of launching aircraft, pile foundations and vehicles. The main attention is paid to two special cases: the effect of a short-term longitudinal impact of constant force and the application of an axial force that increases linearly over time. The problem statement specifically takes into account the fact that the time range of the load application is half the time of wave propagation along the length of the rod, which makes it possible to correctly simulate wave effects. Unlike classical approximate models, in which the propagation of a longitudinal wave is assumed to be instantaneous and the voltage is assumed to be constant along the rod, equations are used in this work taking into account the finite velocity of propagation of longitudinal waves. To numerically study the dynamics of small transverse deviations, eigenfunction decomposition is applied and a system of ordinary differential equations for Fourier amplitudes is solved. The influence of various shapes of the initial bend is analyzed in detail, as well as changes in the maximum oscillation amplitudes for various parameters of the model. A comparative analysis of the behavior of the rod for simple and complex initial forms of curvature is carried out, and the occurrence of irregularities close to those observed in physical experiments is shown. The results obtained confirm the importance of taking into account wave effects in short-term dynamic tasks and expand the scope of the developed models.