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Energies
Volume 18
Issue 23
10.3390/en18236206
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This is an early access version, the complete PDF, HTML, and XML versions will be available soon.
Article

The Mathematical Modelling of Nonlinear String Oscillations in an Isotropic Viscoelastic Medium Using the Example of a Long Power Line

by
Andriy Chaban
1,2,3,
Petro Pukach
4,
Tomasz Perzyński
1,
Andrzej Szafraniec
1,*,
Vitaliy Levoniuk
2,
Aleksander Dydycz
1,* and
Szymon Arkanowicz
5
1
Faculty of Transport, Electrical Engineering and Computer Science, Casimir Pulaski Radom University, 26-600 Radom, Poland
2
Department of Electrical Systems, Stepan Gzhytskyi National University of Veterinary Medicine and Biotechnologies of Lviv, 79010 Lviv, Ukraine
3
Institute of Power Engineering and Control Systems, Lviv Polytechnic National University, 79000 Lviv, Ukraine
4
Institute of Applied Mathematics and Fundamental Sciences, Lviv Polytechnic National University, 79000 Lviv, Ukraine
5
Faculty of Electrical Engineering, Czestochowa University of Technology, 42-201 Czestochowa, Poland
*
Authors to whom correspondence should be addressed.
Energies 2025, 18(23), 6206; https://doi.org/10.3390/en18236206 (registering DOI)
Submission received: 29 October 2025 / Revised: 24 November 2025 / Accepted: 25 November 2025 / Published: 26 November 2025
(This article belongs to the Special Issue Modeling, Simulation and Optimization of Power Systems: 2nd Edition)
Versions Notes

Abstract

In this study, a nonlinear mathematical model of a thin string oscillating in an isotropic viscoelastic medium is developed. The model addresses external and internal mechanical energy dissipation in the string using components described by nonlinear exponential functions. The differential state equation of the studied item is based on a modified Hamilton–Ostrogradsky integral variation principle. The principle is modified by expanding the Lagrangian with two additional components: one addressing the external and internal mechanical energy dissipation in a system and the other addressing the energy of external non-potential forces acting on a system. To substantiate the existence and uniqueness of the solution to a mixed initial boundary problem, the general theory of nonlinear differential equations is applied. A long, single power line is used as an example; its elements oscillate between two support points of the wire. The computer simulation results for the nonlinear vibrations of the object are presented and analysed.
Keywords: conductor oscillation; mathematical modelling; viscoelastic medium; string vibration; Hamilton–Ostrogradsky principle conductor oscillation; mathematical modelling; viscoelastic medium; string vibration; Hamilton–Ostrogradsky principle

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MDPI and ACS Style

Chaban, A.; Pukach, P.; Perzyński, T.; Szafraniec, A.; Levoniuk, V.; Dydycz, A.; Arkanowicz, S. The Mathematical Modelling of Nonlinear String Oscillations in an Isotropic Viscoelastic Medium Using the Example of a Long Power Line. Energies 2025, 18, 6206. https://doi.org/10.3390/en18236206

AMA Style

Chaban A, Pukach P, Perzyński T, Szafraniec A, Levoniuk V, Dydycz A, Arkanowicz S. The Mathematical Modelling of Nonlinear String Oscillations in an Isotropic Viscoelastic Medium Using the Example of a Long Power Line. Energies. 2025; 18(23):6206. https://doi.org/10.3390/en18236206

Chicago/Turabian Style

Chaban, Andriy, Petro Pukach, Tomasz Perzyński, Andrzej Szafraniec, Vitaliy Levoniuk, Aleksander Dydycz, and Szymon Arkanowicz. 2025. "The Mathematical Modelling of Nonlinear String Oscillations in an Isotropic Viscoelastic Medium Using the Example of a Long Power Line" Energies 18, no. 23: 6206. https://doi.org/10.3390/en18236206

APA Style

Chaban, A., Pukach, P., Perzyński, T., Szafraniec, A., Levoniuk, V., Dydycz, A., & Arkanowicz, S. (2025). The Mathematical Modelling of Nonlinear String Oscillations in an Isotropic Viscoelastic Medium Using the Example of a Long Power Line. Energies, 18(23), 6206. https://doi.org/10.3390/en18236206

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