Simulation of oscillations and study of resonant properties of a rope of variable length lying on an elastic foundation, taking into account dissipation

Currently, matters of dependability in the design of technical systems with moving boundaries require an increasingly complete consideration of underlying dynamic phenomena.

Aim. The aim of the study is to develop a mathematical model and an approximate analytical method for studying the transverse vibrations and resonant properties of a viscoelastic rope of variable length lying on an elastic foundation, taking into account energy dissipation. The relevance of the work is due to the widespread use of technical systems with moving boundaries (lifting mechanisms, flexible transmissions, railway contact networks, rail tracks, belt conveyors, drill strings, etc.), for which dynamic loads and resonance are dangerous. The existing methods do not allow for a complete consideration of a system of factors, i.e., changes in the object’s length, resistance of the medium, elastic properties of the foundation and internal friction.

Methods. To solve the problem, the Kantorovich–Galerkin method, effective for systems with moving boundaries, was applied. The original boundary value problem for a partial differential equation was reduced to a system of ordinary differential equations. The solution procedure included the transition to dimensionless variables, the selection of coordinate functions in the form of eigenmodes and the application of the Galerkin procedure. The small parameter method was used to analyze non–stationary processes. In the considered model, the drag force of the rope movement is assumed to be proportional to the velocity, and the bending rigidity of the structure is also taken into account.

Results. Calculation expressions are presented for the amplitude of oscillations corresponding to the n–th dynamic mode. Particular attention is paid to the study of the phenomena of steady–state resonance and passage through resonance. The solution covers the most common case in practice of the action of external disturbances on the moving boundary of the system. It is established that the amplitude significantly depends on the velocity of the boundary, dissipation parameters and the rigidity of the foundation. The conditions for steady–state resonance are determined for a certain ratio of the frequency of the external influence and the natural frequency of the system. The phenomenon of passage through resonance is studied. The resulting analytical expressions were verified by comparison with known special cases, confirming the method’s validity with an error of up to 5% for the fundamental modes.

Conclusions. The resulting analytical expressions for the oscillation amplitude, steady–state resonance conditions, and resonance passage parameters enable the formulation of a number of practical recommendations for design engineers aimed at increasing the dependability and durability of technical systems with moving boundaries and preventing resonant failures in variable–length systems. Key applied problems solved using this model include fatigue life assessment, residual life prediction, and emergency prevention. Consideration of dissipation and an elastic foundation is critical for assessing resonant properties. To prevent resonance, it is recommended to optimize the boundary velocity, use materials with increased friction or dampers, and increase the foundation rigidity. The results have practical significance for improving the dependability of systems with moving boundaries. Research prospects are related to taking into account nonlinear effects and non–harmonic influences.

1. Vesnitskiy A.I., Potapov A.I. [Transverse vibrations of ropes in mine hoists]. Dinamika sistem. Gorkovsky universitet 1975;7:84–89. (in Russ.)

2. Kolosov L.V. [Longitudinal and transverse vibrations of the rope string of a hoisting unit]. Minerals and Mining Engineering 1981;3:83–86. (in Russ.)

3. Litvinov V.L. [Study of passage through resonance of a viscoelastic rope of variable length in lifting mechanisms]. In: Proceedings of Aerospace engineering, high technologies and innovations 2008: XI All–Russian science and technology conference. Perm; 2008. (in Russ.)

4. Zhu W.D., Chen Y. Theoretical and experimental investigation of elevator cable dynamics and control. Trans. ASME. J. Vibr. and Acoust. 2006;1:66-78.

5. Samarin Yu.P., Anisimov V.N. [Forced transverse vibrations of a flexible link during acceleration]. BMSTU Journal of mechanical engineering 1986;12:17–21. (in Russ.)

6. Ryue J., Thompson D. Decay rates of propagating waves in railway tracks at high frequencies. J. Sound and Vibr. 2009;4-5:955-976.

7. Lei Xiao–yan. Effect of abrupt changes in the rigidity of the railway bed base on its vibration under a moving load. Journal of Vibration Engineering 2006;2:195–199.

8. Mulukhov K.K. [Features of Dynamic Calculation of Belt–Wheel Conveyors]. Proceedings of the North Caucasus State Technological University 2000;7:266–269. (in Russ.)

9. Samarin Yu.P. On Wave Phenomena in Domains with Moving Boundaries. Volzhskiy matematichesky sbornik. Kuibyshev 1967;5:337–340. (in Russ.)

10. Vesnitsky A.I. Waves in Systems with Moving Boundaries and Loads. Moscow: Fizmatlit; 2001. (in Russ.)

11. Anisimov V.N., Litvinov V.L., Korpen I.V. On a Method for Obtaining an Analytical Solution of the Wave Equation Describing Oscillations of Systems with Moving Boundaries. Bulletin of Samara State Technical University. Series: Physical and Mathematical Sciences 2012;3(28):145–151. DOI: 10.14498/vsgtu1079. (in Russ.)

12. Anisimov V.N., Litvinov V.L. Resonance Properties of Mechanical Objects with Moving Boundaries: a monograph. Samara State Technical University. Samara; 2009. (in Russ.)

13. Lezhneva A.A. Flexural Oscillations of a Variable– Length Beam. Bulletin of the USSR Academy of Sciences. Solid Body Mechanics 1970;1:159–161. (in Russ.)

14. Anisimov V.N., Litvinov V.L. Analysis of the Influence of Boundary Movement in the Study of Resonance Properties of Damped Systems. Bulletin of Samara State Technical University. Series: Physical and Mathematical Sciences 2009;2(19). DOI: 10.14498/vsgtu1079. (in Russ.)

15. Anisimov V.N., Litvinov V.L. Calculation of Natural Frequencies of Transverse Oscillations of a Viscoelastic Rope Moving in the Longitudinal Direction and Having Flexural Rigidity. In: Mathematical Modeling and Boundary Value Problems: Proceedings of the Fifth All–Russian Scientific Conference with International Participation. Part 1: Mathematical Models of Mechanics, Strength, and Reliability of Structural Elements. Samara: SamSTU; 2008. (in Russ.)

16. Litvinov V.L., Litvinova K.V. Application of the Kantorovich–Galerkin method for the analysis of resonant characteristics of systems with damping. Teoreticheskaia i Matematicheskaia Fizika 2025;224(1):95–104. DOI: 10.4213/tmf10930. (in Russ.)

17. Litvinov V.L. Solution of boundary value problems with moving boundaries using an approximate method for constructing solutions of integro–differential equations. Trudy Instituta Matematiki i Mekhaniki UrO RAN 2020;26(2):188–199. DOI: 10.21538/0134-4889-2020-26-2-188-199. (in Russ.)

18. Litvinov V.L., Litvinova K.V. An approximate method for solving boundary value problems with movable boundaries by reduction to integro–differential equations. Zhurnal Vychislitel’noi Ma-tematiki i Matematicheskoi Fiziki 2022;62(6):977–986. DOI: 10.31857/S0044466922060126. (in Russ.)

19. Litvinov V.L., Shamolin M.V. On an asymptotic method for solving homogeneous integro–differential equations describing oscillations of objects with moving boundaries. Sibirskii Zhurnal Industrial’noi Matematiki 2025;28(2):93–107. DOI: 10.33048/SIBJIM.2025.28.203. (in Russ.)

20. Savin G.N. On the basic equations of a mine hoisting rope. Prikladnaia Mekhanika 1955;1(1):15–24. (in Russ.)

21. Litvinov V.L., Anisimov V.N. Mathematical modeling and investigation of the resonant properties of mechanical objects with a changing boundary: a monograph. Samaral; SamGTUl; 2020. (in Russ.)

22. Litvinov V.L., Litvinova K.V. On one inverse method for solving problems of oscillations of me-chanical systems with moving boundaries. Vestnik Moskovskogo Universiteta. Seriia 1: Matematika. Mekhanika 2024;3:53–59. DOI: 10.55959/MSU0579-9368-1-65-3-8. (in Russ.)

23. Selivanova N.Yu., Shamolin M.V. Local solvability of a one-phase problem with free boundary. Journal of Mathematical Sciences 2013;189(2):274-283. DOI: 10.1007/s10958-013-1209-9 EDN: RFJYVD. (in Russ.)

24. Selivanova N.Yu., Shamolin M.V. Studying the interphase zone in a certain singular–limit problem. Journal of Mathematical Sciences 2013;189(2):284–293. DOI: 10.1007/s10958-013-1210-5. (in Russ.)

25. Erofeev V.I., Leont’eva A.V. Quasiharmonic longitudinal wave propagating in a Mindlin–Hermann rod embedded in a nonlinearly elastic medium. Teoreticheskaia i Matematicheskaia Fizika 2022;211(2):216–235. DOI: 10.4213/tmf10253. (in Russ.)

26. Erofeev V.I., Lisenkova E.E. General relations for waves propagating in one–dimensional elastic systems. In: Mathematical Methods in Mechanics: materials of the international conference. To the 90th anniversary of academician A.G. Kulikovsky. Moscow, MIAN; 2023. Pp. 26–27. (in Russ.)

27. Semenov A.L., Litvinov V.L., Shamolin M.V. Investigation of the influence of boundary motion on the oscillatory and resonant properties of mechanical systems of variable length. Computational Mathematics and Information Technologies 2025;9(2):34–43. DOI: 10.23947/2587-8999-2025-9-2-34-43. (in Russ.)