PERIODIC ATEB-FUNCTIONS AND THE VAN DER POL METHOD FOR CONSTRUCTING SOLUTIONS OF TWO-DIMENSIONAL NONLINEAR OSCILLATIONS MODELS OF ELASTIC BODIES

Yaroslav Romanchuk; Mariia Sokil; Leonid Polishchuk

doi:10.35784/iapgos.6377

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Published: Sep 30, 2024

DOI: https://doi.org/10.35784/iapgos.6377

Issue Vol. 14 No. 3 (2024)

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DOI

https://doi.org/10.35784/iapgos.6377

Authors

Yaroslav Romanchuk

romanchuky@ukr.net

Hetman Petro Sahaidachny National Army Academy, Ukraine

Mariia Sokil

mariia.b.sokil@lpnu.ua

Lviv Polytechnic National University, Ukraine

https://orcid.org/0000-0003-3352-2131

Leonid Polishchuk

leo.polishchuk@gmail.com

Vinnytsia National Technical University, Ukraine

https://orcid.org/0000-0002-5916-2413

Abstract

In the process of operation, the simplest elements (hereinafter elastic bodies) of machines and mechanisms under the influence of external and internal factors carry out complex oscillations ‒ a combination of longitudinal, bending and torsion combinations in various combinations. In general, mathematical models of the process of such complex phenomena in elastic bodies, even for one-dimensional calculation models, are boundary value problems for systems of partial differential equations. A two-dimensional mathematical model of oscillatory processes in a nonlinear elastic body is considered. A method of constructing an analytical solution of the corresponding boundary-value problems for nonlinear partial differential equations is proposed, which is based on the use of Ateba functions, the Van der Pol method, ideas of asymptotic integration, and the principle of single-frequency oscillations. For "undisturbed" analogues of the model equations, single-frequency solutions were obtained in an explicit form, and for "perturbed" ‒ analytical dependences of the basic parameters of the oscillation process on a small perturbation. The dependence of the main frequency of oscillations on the amplitude and non-linearity parameter of elastic properties in the case of single-frequency oscillations of "unperturbed motion" is established. An asymptotic approximation of the solution of the autonomous "perturbed" problem is constructed. Graphs of changes in amplitude and frequency of oscillations depending on the values of the system parameters are given.

Keywords:

oscillations, nonlinear elastic bodies, two-dimensional mathematical model

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Romanchuk, Y., Sokil, M., & Polishchuk, L. (2024). PERIODIC ATEB-FUNCTIONS AND THE VAN DER POL METHOD FOR CONSTRUCTING SOLUTIONS OF TWO-DIMENSIONAL NONLINEAR OSCILLATIONS MODELS OF ELASTIC BODIES. Informatyka, Automatyka, Pomiary W Gospodarce I Ochronie Środowiska, 14(3), 15–20. https://doi.org/10.35784/iapgos.6377

PERIODIC ATEB-FUNCTIONS AND THE VAN DER POL METHOD FOR CONSTRUCTING SOLUTIONS OF TWO-DIMENSIONAL NONLINEAR OSCILLATIONS MODELS OF ELASTIC BODIES

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