Wave concept of motion in mathematical models of the dynamics of two-dimensional media studying

Received: November 12, 2019
Revised: December 21, 2019
Accepted: December 28, 2019
Lviv Polytechnic National University
Hetman Petro Sahaidachny National Army Academy
Lviv Polytechnic National University

The methodology of the studying of dynamic processes in two-dimensional systems by mathematical models containing nonlinear equation of Klein-Gordon was developed. The methodology contains such underlying: the concept of the motion wave theory; the single - frequency fluctuations principle in nonlinear systems; the asymptotic methods of nonlinear mechanics. The aggregate content allowed describing the dynamic process for the undisturbed (linear) analogue of the mathematical model of movement. The value determining the impact of nonlinear forces on the basic parameters of the waves for the disturbed analogue is defined.

[1] M. B. Sokil, O. L. Lyashuk, and А. Р. Dovbush, “Dynamics of flexible elements of drive systems with variable contact point to the pulleys”, INMATEH Agricultural Engineering, vol. 48, no. 1, pp. 119–124, 2016.

[2] I. Gevko, O. Lyashuk, A. Djachun, and A. Dovbush, “Interpretation of the choice of conveyers with improved technological characteristics”, MOTROL. Commission of Motorization and Energetics in Agriculture, vol. 17, no. 4, pp. 107–116, 2015.

[3] O. L. Lyashuk, et al., “Mathematical model of bending vibrations of a horizontal feeder-mixer along the flow of grain mixture”, INMATEH Agricultural Engineering, vol. 55, no. 2, pр. 35–44, 2018.

[4] I. Hevko, et al., “Resonant oscillation of vertical working part of conveyer-loader”, Bulletin of the Karaganda University. Physics series, no. 2 (94), pp. 73–82, 2019.

[5] Yu. A. Mitropolskii, and B. I. Moseenkov, Asimptoticheskie resheniia uravnenii v chastnykh proizvodnykh [Asymptotic solutions of partial differential equations]. Kyiv, Ukraine: Vyshcha shkola Publ., 1976. [in Russian].

[6] N. N. Bogoliubov, and Yu. A. Mitropolskii, Asimptoticheskie metody v teorii nelineinykh kolebanii [Asymptotic methods in the theory of nonlinear oscillations]. Moscow, Russia: Nauka Publ., 1974. [in Russian].

[7] I. Fazekas, and A. Kukush, “Asymptotic properties of estimators in nonlinear functional errors-in-variables with dependent error terms”, Journal of Mathematical Sciences, vol. 92, no. 3, pp. 3890–3895, 1998.

[8] Ye. V. Kharchenko, and M. B. Sokil, “Neliniyni protsesy u seredovyshchakh, yaki kharakteryzuyutʹsya pozdovzhnim rukhom, i vplyv sposobu zakriplennya na yikh kolyvannya” [“Nonlinear processes in media characterized by longitudinal motion and the influence of the method of fixation on their oscillations”], Avtomatizacìâ virobničih procesìv u mašinobuduvannì ta priladobuduvannì [Industrial Process Automation in Engineering and Instrumentation], vol. 41, pp. 156–159, 2007. [in Ukrainian].

[9] Martyntsiv M. P., and Sokil M. B., “Odne uzahalʹnennya metodu D’Alambera dlya system, yaki kharakteryzuyutʹsya pozdovzhnim rukhom” [“One generalization of the D’Alembert method for systems characterized by longitudinal motion”], Naukovyy visnyk Derzhavnoho lisotekhnichnoho universytetu Ukrayiny [Scientific Bulletin of Ukrainian State Forestry University], vol. 13.4, pp. 64–67, 2003. [in Ukrainian].

[10] Y. A. Mitropol'skii, and B. I. Sokil, “On the application of Ateb-functions to the construction of an asymptotic solution of the perturbed nonlinear Klein-Gordon equation”, Ukrainian Mathematical Journal, no. 5 (50), pp. 754–760, 1998.

[11] B. I. Sokil, “On a method for constructing one-frequency solutions of a nonlinear wave equation”, Ukrainian Mathematical Journal, no. 6 (46), pp. 853–856, 1994.

A. Andrukhiv, B. Sokil, M. Sokil, "Wave concept of motion in mathematical models of the dynamics of two-dimensional media studying", Ukrainian Journal of Mechanical Engineering and Materials Science, vol. 5, no. 3-4, pp. 8-15, 2019.