Modelling of the amplitude of transverse vibrations of a homogeneous rod upon impact against a rigid obstacle taking into account its own weight

The accurate solution of dynamic problems in a nonlinear formulation is associated with certain mathematical difficulties. The available research works devoted to the problem solution on the dynamic deflection and stability of the rods under shock wave loading have considered perfect rods with a straight axis and have not considered any external effects. The account of additional external factors influencing the state of a shock system makes this task more complicated. By using the method of the initial parameters and the wave model of the longitudinal impact, this work attempts to develop the calculation method for dynamic deflection of a homogeneous rod producing lateral vibrations under the longitudinal impact against a hard obstacle, also taking lateral loads into account. Depending on the pre-impact state of the rod, the corresponding initial parameters are set, such as initial deflection, rotation, bending moment and shear force. By using the wave model of the longitudinal impact and method of characteristics, it becomes possible to take the short-term exposure of the shock compressing load into account, after which the rod is considered as a vibratory system having the initial velocity and the deviation from the equilibrium position. In this state, the rod makes transverse shock vibrations. The present article describes the calculation method of the amplitude of transverse vibrations of the rod with a constant longitudinal stiffness under the impact against an absolutely rigid barrier. It is of a particular interest to use this method in order to calculate the system of uniform and stepped rods making lateral vibrations under the longitudinal impact. Such systems of rods are often found in present impact mechanisms, they are used as structural elements, e.g. core elements of trusses, frames, pillars, columns, piles, etc.