The solution of dynamic problems of the longitudinal impact of a homogeneous rod in a nonlinear formulation presents, notoriously, significant mathematical difficulties. Existing approaches have a rather limited application mainly due to the significant approximation of the solution of the problem. In addition, such approaches affect an extensive mathematical apparatus, which complicates their use in engineering calculations. The task of solving the problem of the longitudinal impact of a geometrically heterogeneous (stepped) rod remains far from the final solution to this day. The solution of this problem is complicated by the randomness of the interference pattern of longitudinal waves during their transition through the boundaries of homogeneous sections of a stepped rod, and therefore the deformations and longitudinal forces along the length of the rod quickly change in time. In this work, using the initial parameter method and the wave model of longitudinal impact, an attempt is made to develop a methodology for calculating the dynamic deflection of a stepped rod that performs transverse vibrations during a longitudinal impact on a rigid barrier. The value of the maximum deflection of a stepped rod with the initial curvature of one of its sections is modeled. The results of modeling the maximum deflection for various values of the initial curvature and the pre-impact speed of the rod are obtained. During the processing of simulation results, the so-called “Zone of maximum deflections” is the interval of variation of the pre-shock velocity, at which the maximum amplitude of the transverse vibrations of the curved section of the stepped rod is observed. The possibility of calculating the amplitudes of the transverse vibrations of the rods and rod elements under the most various fixing schemes and the various nature of the pre-shock state is noted. The relevance of applying this technique in dynamic calculations of various impact mechanisms, as well as in engineering calculations of rod systems for various purposes, is emphasized.

中文翻译：

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刚性杆撞击时具有初始曲率的阶梯杆的最大挠度建模

众所周知，非线性公式中均质杆纵向冲击的动力学问题的解决方案存在很大的数学困难。现有方法的应用相当有限，这主要是由于该问题的解决方案非常近似。另外，这些方法影响广泛的数学仪器，这使其在工程计算中的使用复杂化。迄今为止，解决几何异质（阶梯式）杆的纵向冲击问题的任务还远远没有最终解决方案。在纵波穿过阶梯状杆的均匀截面的边界过渡时，纵波干涉图的随机性使该问题的解决变得复杂，因此，沿杆长度方向的变形和纵向力会迅速变化。在这项工作中，使用初始参数方法和纵向冲击波模型，试图开发一种计算阶梯杆的动态挠度的方法，该阶梯杆在对刚性屏障的纵向碰撞过程中会产生横向振动。对阶梯杆的截面之一的初始曲率的最大挠度值进行建模。获得了对杆的初始曲率和预冲击速度的各种值的最大挠度进行建模的结果。在模拟结果的处理过程中，所谓的“最大挠度区”是指冲击前速度的变化间隔，在该处观察到阶梯杆的弯曲部分的横向振动的最大振幅。指出了在最多种固定方案和震荡前状态的各种性质下计算杆和杆元件的横向振动幅度的可能性。强调了在各种冲击机构的动态计算以及出于各种目的的杆系统的工程计算中应​​用该技术的相关性。