Vibrations in engineering and technology

Dmytriv Vasyl - Doctor of Engineering Sciences, Professor, Head of Department of Desing Machine and Automotive Engineering, Lviv Polytechnic National University, 79013, 12 Stepan Bandera Str., Lviv, Ukraine, е-mail: Vasyl.T.Dmytriv@lpnu.ua,  https://orcid.org/0000-0001-9361-6418

Berehuliak Stepan - recipient of the PhD degree of Department of Desing Machine and Automotive Engineering, Lviv Polytechnic National University, 79013, 12 Stepan Bandera Str., Lviv, Ukraine, е-mail: Stepan.T.Berehuliak@lpnu.ua,  https://orcid.org/0009-0005-6597-9752

Mechanical systems that combine structural elements with elastic and viscous properties and operate under dynamic and impulse loads require accurate mathematical models to predict their oscillatory behaviour. The purpose of this study was to develop a mathematical model of an elastic-damper system based on the Kelvin-Voigt body, aimed at describing viscoelastic material behaviour and analysing the influence of stiffness and damping parameters on the dynamic response of the system. The research applied analytical methods for solving differential equations, analysis of characteristic equations for both real and complex roots, as well as algebraic transformations for constructing general and particular solutions. As a result, an analytical solution was obtained for determining the dynamic response of the masses under impulse loading. The proposed model took into account variable stiffness, damping, and mass parameters, and distinguishes between regimes with real and complex eigenvalues. It was established that increasing the damping coefficient reduces amplitude and accelerates oscillation decay, while increasing mass extends the decay duration. The influence of pulse duration and repetition frequency on the oscillation profile was investigated. The simulation results confirmed the accuracy of the analytical expressions and enable the modelling of transient processes under various conditions. Compared to purely numerical approaches, the proposed method provides broader opportunities for system analysis and control. The developed model can be applied in the study of technical systems with arbitrary configurations of elastic-viscous element connections and varying numbers of components, making it possible to optimise the design of systems that utilise oscillations, vibrations, damping.

1. Huilai, S., Ruichuan, L., Jikang, X., Funing, X., Bo, Z., & Xinyuan, D. (2021). Fractional modeling and characteristic analysis of hydro-pneumatic suspension for construction vehicles. Processes, 9(8), article number 1414. doi: 10.3390/pr9081414.

2. Lanets, O.S., Dmytriv, V.T., Borovets, V.M., Derevenko, I.A., & Horodetskyy, I.M. (2020). Analytical model of the two-mass above resonance system of the eccentric-pendulum type vibration table. International Journal of Applied Mechanics and Engineering, 25(4), 116-129. doi: 10.2478/ijame-2020-0053.

3. Bonfanti, A., Kaplan, J.L., Charras, G., & Kabla, A. (2020). Fractional viscoelastic models for power-law materials. ARXIV. doi: 10.48550/arXiv.2003.07834.

4. Wu, Y., Xuan, H., Wu, C., & Mi, D. (2022). Effect of under-platform dampers on the forced vibration of high-speed rotating blades. Journal of Mechanical Science and Technology, 36(8), 3837-3850. doi: 10.1007/s12206-022-0707-4.

5. Zhou, R., Zhang, B., & Li, Z. (2022). Dynamic modeling and computer simulation analysis of the air spring suspension. Journal of Mechanical Science and Technology, 36(4), 1719-1727. doi: 10.1007/s12206-022-0308-2.

6. Leniowski, R., & Wronski, M. (2022). Vibration analysis and modelling of light-weight robot arms. Vibrations in Physical Systems, 33(2), article number 2022220. doi: 10.21008/j.0860-6897.2022.2.20.

7. Satpute, N., Iwaniec, M., Narina, R., & Satpute, S. (2020). Energy harvesting shock absorber with linear generator and mechanical motion amplification. Vibrations in Physical Systems, 31, article number 2020104. doi: 10.21008/j.0860-6897.2020.1.04.

8. Dmytriv, V.T., Dmytriv, I.V., Lavryk, Y.M., Krasnytsia, B., Bubela, T.Z., Yatsuk, V.O. (2017). Study of the pressure regulator work with a spring-damper system applied to milking machine. INMATEH - Agricultural Engineering, 52(2), 61-67

9. Jugulkar, L.M., Singh, S., & Sawant, S.M. (2016). Analysis of suspension with variable stiffness and variable damping force for automotive applications. Advances in Mechanical Engineering, 8(5). doi: 10.1177/1687814016646695.

10. Madeira, R.H., & Coda, H.B. (2016). Kelvin viscoelasticity and lagrange multipliers applied to the simulation of nonlinear structural vibration control. Latin American Journal of Solids and Structures, 13, 964-991. doi: 10.1590/1679-78252624.

11. Monsia, M.D., & Kpomahou, Y.J.F. (2014). Simulating nonlinear oscillations of viscoelastically damped mechanical systems. Engineering, Technology & Applied Science Research, 4(6), 714-723. doi: 10.48084/etasr.518.

12. Herisanu, N., Marinca, B., & Marinca, V. (2022). Dynamics of the vibro-impact nonlinear damped and forced oscillator under the influence of the electromagnetic actuation. Mathematics, 10(18), article number 3301. doi: 10.3390/math10183301.

13. Gomez-Aguilar, J.F., Yépez-Martínez, H., Calderón-Ramón, C., Cruz-Orduña, I., Escobar-Jiménez, R.F., & Olivares-Peregrino, V.H. (2015). Modeling of a mass-spring-damper system by fractional derivatives with and without a singular kernel. Entropy, 17(9), 6289-6303. doi: 10.3390/e17096289.

14. Hamedi, B., Shrikanthan, S., & Taheri, S. (2024). The reduced-order modeling approach for a double-damper concept: A comparison with a single damper for comfort analysis. Vibration, 7(3), 644-661. doi: 10.3390/vibration7030034

15. Stankovic, B., & Atanackovic, T.M. (2002). Dynamics of a rod made of generalized kelvin-voigt visco-elastic material. Journal of Mathematical Analysis and Applications, 268, 550-563. doi: 10.1006/jmaa.2001.7816.

16. Rossikhin, Y.A., & Shitikova, M.V. (2012). Analysis of damped vibrations of thin bodies embedded into a fractional derivative viscoelastic medium. Journal of Mechanical Behavior of Materials, 21(5-6), 155-159. doi: 10.1515/jmbm-2013-0002.

17. Tychkov, D., & Bondarenko, M. (2023). Mathematical model of static and quasi-static electric fields of ultra-low power under a polyelectrode cylindrical transducer. Bulletin of Cherkasy State Technological University, 28(1), 13-22. doi: 10.24025/2306-4412.1.2023.273684.

18. Bobyleva, T., & Shamaev, A. (2020). Problem of damping oscillations of a mechanical system with integral memory. IOP Conference Series: Materials Science and Engineering, 869(2), article number 022011. doi: 10.1088/1757-899X/869/2/022011.

19. Zhang, J., Chen, S., Wu, Z., Feng, J., & Fan, Y. (2014). Research on damping noncoincidence and its influence factors of multibranch hydropneumatic suspension. Advances in Mechanical Engineering, 7(2). doi: 10.1155/2014/362086.