Vestnik КRAUNC. Fiz.-Mat. Nauki. 2026. vol. 54. no. 1. P. 124 – 140. ISSN 2079-6641

PHYSICS
https://doi.org/10.26117/2079-6641-2026-54-1-124-140
Research Article
Full text in Russian
MSC 74–00, 35Q74, 74H20, 74J05

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Interaction of a Rod With an External Medium According to Modified Winkler Models

A. Begmatov, N. T. Mamatova^{\ast}

National University of Uzbekistan after named Mirzo Ulugbek, 100174, Tashkent, Uzbekistan, str. University, 4.

Abstract. This paper investigates the interaction of a semi-infinite rod with a stationary and moving external medium under dynamic loading applied to the rod end. The interaction with the surrounding medium is described using two modified Winkler-type models. It is assumed that at small values of displacement or deformation the interaction obeys the Coulomb–Amontons dry friction law. When the displacement or deformation exceeds a certain critical value, the friction law changes and the interaction is described by the Winkler model, where the tangential stress is proportional to deformation. A mathematical formulation of the dynamic problem for a semi-infinite rod subjected to exponential dynamic loading is presented. The approximate solution of the governing equations is obtained using the method of integral relations. The calculations are performed using the Maple-18 computational system. Numerical results are presented in the form of graphs describing the evolution of velocity, displacement and deformation over time, as well as tables characterizing the transition time from Coulomb friction to Winkler-type interaction. The influence of the friction parameters, the velocity of the external medium and the loading characteristics on the deformation process and the transition to equilibrium is analyzed in detail.

Key words: Coulomb-Amontons dry friction, Winkler model, rod-environment interaction, method of integral relations, free boundary, wave propagation, dynamic loading.

Received: 11.03.2026; Revised: 18.03.2026; Accepted: 25.03.2026; First online: 29.03.2026

For citation. Begmatov A., Mamatova N. T. Interaction of a rod with an external medium according to modified Winkler models. Vestnik KRAUNC. Fiz.-mat. nauki. 2026, 54: 1, 124-140. EDN: JABPKQ. https://doi.org/10.26117/2079-6641-2026-54-1-124-140.

Funding. No funding was received.

Competing interests. There are no conflicts of interest regarding authorship and publication.

Contribution and Responsibility. All authors contributed to this article. Authors are solely responsible for providing the final version of the article in print. The final version of the manuscript was approved by all authors.

^{\ast}Correspondence: E-mail: nigmamatova@yandex.ru

The content is published under the terms of the Creative Commons Attribution 4.0 International License

© Begmatov A., Mamatova N. T., 2026

© Institute of Cosmophysical Research and Radio Wave Propagation, 2026 (original layout, design,

compilation)

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Information about the authors

Begmatov Abduvali – D. Sci. (Phys. & Math.), Professor of the Department of Mechanics and Mathematical Modeling of the National
University of Uzbekistan named after Mirzo Ulugbek, Tashkent, Uzbekistan, ORCID 0009-0005-8819-6646.

Mamatova Nigora Tuxtabevna – Ph. D. (Phys. & Math.), Associate Professor of the Department of Mechanics and Mathematical Modeling of the National University of Uzbekistan named after Mirzo Ulugbek, ORCID 0000-0001-7161-3187.

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