Abstract—

The solution of dynamic problems on longitudinal impact of a system of homogeneous rods in a nonlinear formulation presents, as is known, significant mathematical difficulties. The existing approaches have a rather limited application, mainly due to the significant approximation of the solution of the problem. In addition, such approaches involve an extensive mathematical apparatus, which makes it difficult to use them in engineering calculations. In practice, it is of considerable interest to model the amplitude of transverse vibrations of a system of homogeneous rods having different lengths and thicknesses under longitudinal impact. The solution of the problem posed is complicated by the chaotic nature of the interference pattern of longitudinal waves when they pass through the impact cross-section of the rods, which is the boundary of the homogeneous sections for the rod system under consideration. In this regard, deformations and longitudinal forces along the length of the rods change rapidly over time. This study deals with the mathematical modeling of the amplitude of transverse vibrations of physically and geometrically homogeneous rods, one of which is stationary and interacts with an absolutely rigid obstacle, and the another, moving with a pre-impact speed, collides with the first rod. In the first case, the collision is considered by taking into account the eccentricity in the impact cross-section of the rods. In the second case, the initial curvature of one of the rods is taken into account. Modeling is carried out using the method of initial parameters and a wave model of longitudinal impact. The results of modeling the maximum deflection are obtained for various values of the initial curvature and eccentricity depending on the pre-impact velocity. The analysis of the results revealed the so-called “zone of maximum deflections”, namely, the interval of change in the pre-impact velocity, at which the maximum amplitude of transverse vibrations of homogeneous rods is observed. Additionally, a number of regularities are noted that have been identified when comparing the results for both cases. The relevance of using this calculation method in solving dynamic problems arising in the design of engineering structures with dynamically loaded rod systems is pointed out.

中文翻译：

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纵向冲击作用下均质杆横向振动幅度的数学建模

摘要-

众所周知，以非线性公式表示的动态问题对均质杆系统纵向冲击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，这主要是由于问题解决方案的近似值。另外，这样的方法涉及广泛的数学装置，这使得难以在工程计算中使用它们。在实践中，对在纵向冲击下具有不同长度和厚度的均质杆系统的横向振动的幅度进行建模非常有意义。当纵波穿过杆的冲击横截面时，由于纵波干涉图的混沌性质，所造成的问题的解决变得复杂。这是所考虑的杆系统的均质截面的边界。在这方面，沿着杆的长度的变形和纵向力随时间快速变化。这项研究涉及物理上和几何上均一的棒的横向振动幅度的数学模型，其中一个是固定的，并且与绝对刚性的障碍物相互作用，而另一个以预撞击的速度运动，与第一个棒发生碰撞。 。在第一种情况下，通过考虑杆的冲击横截面中的偏心量来考虑碰撞。在第二种情况下，考虑其中一根杆的初始曲率。使用初始参数的方法和纵向冲击波模型进行建模。对最大挠度建模的结果是根据冲击前速度对初始曲率和偏心率的各种值获得的。对结果的分析揭示了所谓的“最大挠度区”，即预冲击速度的变化间隔，在该间隔下可以观察到均匀杆的横向振动的最大幅度。此外，在比较这两种情况的结果时，注意到了许多规律性。指出了使用这种计算方法解决动态加载杆系统工程结构设计中出现的动态问题的相关性。对结果的分析揭示了所谓的“最大挠度区”，即预冲击速度的变化间隔，在该间隔下可以观察到均匀杆的横向振动的最大幅度。此外，在比较这两种情况的结果时，注意到了许多规律性。指出了使用这种计算方法解决动态加载杆系统工程结构设计中出现的动态问题的相关性。对结果的分析揭示了所谓的“最大挠度区”，即预冲击速度的变化间隔，在该间隔下可以观察到均匀杆的横向振动的最大幅度。此外，在比较这两种情况的结果时，注意到了许多规律性。指出了使用这种计算方法解决动态加载杆系统工程结构设计中出现的动态问题的相关性。