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IssuesArchive of Issues2021-2pp.220-229

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Bityurin A.A., "Mathematical Modeling of the Amplitude of Transverse Vibrations of Homogeneous Rods under Longitudinal Impact," Mech. Solids. 56 (2), 220-229 (2021)
Year 2021 Volume 56 Number 2 Pages 220-229
DOI 10.3103/S0025654421020047
Title Mathematical Modeling of the Amplitude of Transverse Vibrations of Homogeneous Rods under Longitudinal Impact
Author(s) Bityurin A.A. (Ul'yanovsk State Technical University, Ulĺyanovsk, 432027 Russia, sntk_2015@mail.ru)
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 ôzone of maximum deflectionsö, namely, the interval of change in the preimpact 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.
Keywords transverse vibrations, deflection, longitudinal impact, wave model, method of initial parameters, pre-impact velocity, rod
Received 01 January 2020Revised 14 January 2020Accepted 17 January 2020
Link to Fulltext https://link.springer.com/article/10.3103/S0025654421020047
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