Advancing asymptotic approaches to studying the longitudinal and torsional oscillations of a moving beam

Authors

DOI:

https://doi.org/10.15587/1729-4061.2022.257439

Keywords:

nonlinear oscillations, asymptotic method, elastic beam, longitudinal oscillations, torsional oscillations

Abstract

This paper analyzes the influence of kinetic and physical-mechanical parameters of systems on the characteristics of dynamic processes in moving one-dimensional nonlinear-elastic systems. Improved convenient calculation formulas have been derived that describe the laws of changing the amplitude-frequency characteristics of systems for both a non-resonant case and a resonant one. An important issue of studying the influence of the speed of movement of elements of mechanisms on the oscillations of one-dimensional nonlinear-elastic systems has not been considered in detail until now in the scientific literature. This issue relates to the vibrations of shafts in gears, pipe strings when drilling oil and gas wells, the oscillations of turbine blades and rotating turbine discs, the longitudinal vibrations of the beam as an element of structures. The main reason for this in the analytical study of dynamic processes were the shortcomings of the mathematical apparatus for solving the corresponding nonlinear differential equations that describe the laws of motion of those systems.

It was found that in the case of longitudinal oscillations in the moving beam with an increase in the longitudinal speed of the medium to 10 m/s, the amplitude of the oscillation also increases by 13.5 %. However, when the longitudinal velocity of the beam is 5 m/s, the amplitude will increase by only 3 %. It is established that with the growth of the amplitude, the frequency of longitudinal oscillations decreases sharply, and if the system moves at a higher speed, for example, 20 m/s, it reduces the frequency of oscillation by about 13 %.

The results reported here make it possible to assess the effect of kinetic and physical-mechanical parameters on the frequency and amplitude of oscillations. The research that involved the asymptotic method makes it possible to predict resonant phenomena and obtain engineering solutions to improve the efficiency of technological equipment.

Author Biographies

Andrii Slipchuk, Lviv Polytechnic National University

PhD, Associate Professor

Department of Robotics and Integrated Mechanical Engineering Technologies

Petro Pukach, Lviv Polytechnic National University

Doctor of Technical Sciences, Professor

Department of Computational Mathematics and Programming

Myroslava Vovk, Lviv Polytechnic National University

PhD, Associate Professor

Department of Mathematics

Olha Slyusarchuk, Lviv Polytechnic National University

PhD, Associate Professor

Department of Mathematics

References

  1. Andrukhiv, A., Sokil, B., Sokil, M. (2018). Resonant phenomena of elastic bodies that perform bending and torsion vibrations. Ukrainian Journal of Mechanical Engineering and Materials Science, 4 (1), 65–73. doi: https://doi.org/10.23939/ujmems2018.01.065
  2. Humbert, S. C., Gensini, F., Andreini, A., Paschereit, C. O., Orchini, A. (2021). Nonlinear analysis of self-sustained oscillations in an annular combustor model with electroacoustic feedback. Proceedings of the Combustion Institute, 38 (4), 6085–6093. doi: https://doi.org/10.1016/j.proci.2020.06.154
  3. Haris, A., Alevras, P., Mohammadpour, M., Theodossiades, S., O’ Mahony, M. (2020). Design and validation of a nonlinear vibration absorber to attenuate torsional oscillations of propulsion systems. Nonlinear Dynamics, 100 (1), 33–49. doi: https://doi.org/10.1007/s11071-020-05502-z
  4. Pipin, V. V., Kosovichev, A. G. (2020). Torsional Oscillations in Dynamo Models with Fluctuations and Potential for Helioseismic Predictions of the Solar Cycles. The Astrophysical Journal, 900 (1), 26. doi: https://doi.org/10.3847/1538-4357/aba4ad
  5. Barbosa, J. M. de O., Fărăgău, A. B., van Dalen, K. N., Steenbergen, M. J. M. (2022). Modelling ballast via a non-linear lattice to assess its compaction behaviour at railway transition zones. Journal of Sound and Vibration, 530, 116942. doi: https://doi.org/10.1016/j.jsv.2022.116942
  6. Hatami, M., Ganji, D. D., Sheikholeslami, M. (2017). Differential Transformation Method for Mechanical Engineering Problems. Academic Press. Available at: https://www.sciencedirect.com/book/9780128051900/differential-transformation-method-for-mechanical-engineering-problems
  7. Lavrenyuk, S. P., Pukach, P. Y. (2007). Mixed problem for a nonlinear hyperbolic equation in a domain unbounded with respect to space variables. Ukrainian Mathematical Journal, 59 (11), 1708–1718. doi: https://doi.org/10.1007/s11253-008-0020-0
  8. Chen, G., Deng, F., Yang, Y. (2021). Practical finite-time stability of switched nonlinear time-varying systems based on initial state-dependent dwell time methods. Nonlinear Analysis: Hybrid Systems, 41, 101031. doi: https://doi.org/10.1016/j.nahs.2021.101031
  9. Bayat, M., Pakar, I., Domairry, G. (2012). Recent developments of some asymptotic methods and their applications for nonlinear vibration equations in engineering problems: A review. Latin American Journal of Solids and Structures, 9 (2), 1–93. doi: https://doi.org/10.1590/s1679-78252012000200003
  10. Sokil, B. I., Pukach, P. Y., Sokil, M. B., Vovk, M. I. (2020). Advanced asymptotic approaches and perturbation theory methods in the study of the mathematical model of single-frequency oscillations of a nonlinear elastic body. Mathematical Modeling and Computing, 7 (2), 269–277. doi: https://doi.org/10.23939/mmc2020.02.269
  11. Andrukhiv, A., Sokil, B., Sokil, M. (2018). Asymptotic method in investigation of complex nonlinear oscillations of elastic bodies. Ukrainian Journal of Mechanical Engineering and Materials Science, 4 (2), 58–67. doi: https://doi.org/10.23939/ujmems2018.02.058
  12. Andrukhiv, A., Sokil, M., Fedushko, S., Syerov, Y., Kalambet, Y., Peracek, T. (2020). Methodology for Increasing the Efficiency of Dynamic Process Calculations in Elastic Elements of Complex Engineering Constructions. Electronics, 10 (1), 40. doi: https://doi.org/10.3390/electronics10010040
  13. Le van, A. (2017). Nonlinear Theory of Elastic Plates. ISTE Press – Elsevier. Available at: https://www.sciencedirect.com/book/9781785482274/nonlinear-theory-of-elastic-plates
  14. Hashiguchi, K. (2020). Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity. Elsevier. doi: https://doi.org/10.1016/c2018-0-05398-0
  15. Jauregui, J. C. (2015). Parameter Identification and Monitoring of Mechanical Systems Under Nonlinear Vibration. Woodhead Publishing. doi: https://doi.org/10.1016/c2013-0-16479-3
  16. Kazhaev, V. V., Semerikova, N. P. (2019). Self-modulation of quasi-harmonic bending waves in rods. Bulletin of Science and Technical Development, 13–19. doi: https://doi.org/10.18411/vntr2019-142-2
  17. Babenko, A. Ye., Boronko, O. O., Lavrenko, Ya. I., Trubachev, S. I. (2020). Kolyvannia nekonservatyvnykh mekhanichnykh system. Kyiv: Nats. tekhn. un-t Ukrainy «KPI imeni Ihoria Sikorskoho», 153. Available at: https://ela.kpi.ua/handle/123456789/38187?locale=uk
  18. Sokil, B. I., Khytriak, O. I. (2009). Vibrations of drive systems flexible elements and methods of determining their optimal nonlinear characteristics based on the laws of motion. Military Technical Collection, 2, 9–12. doi: https://doi.org/10.33577/2312-4458.2.2009.9-12
  19. Chen, L.-Q. (2005). Analysis and Control of Transverse Vibrations of Axially Moving Strings. Applied Mechanics Reviews, 58 (2), 91–116. doi: https://doi.org/10.1115/1.1849169
  20. Marynowski, K., Kapitaniak, T. (2014). Dynamics of axially moving continua. International Journal of Mechanical Sciences, 81, 26–41. doi: https://doi.org/10.1016/j.ijmecsci.2014.01.017
  21. Nazarov, V. E., Kiyashko, S. B. (2020). Stationary and Self-Similar Waves in a Rod with Bimodular Nonlinearity, Dissipation, and Dispersion. Technical Physics, 65 (1), 7–13. doi: https://doi.org/10.1134/s106378422001020x
  22. Pukach, P., Beregova, H., Slipchuk, A., Pukach, Y., Hlynskyi, Y. (2020). Asymptotic Approaches to Study the Mathematical Models of Nonlinear Oscillations of Movable 1D Bodies. 2020 IEEE 15th International Conference on Computer Sciences and Information Technologies (CSIT). doi: https://doi.org/10.1109/csit49958.2020.9321908

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Published

2022-06-30

How to Cite

Slipchuk, A., Pukach, P., Vovk, M., & Slyusarchuk, O. (2022). Advancing asymptotic approaches to studying the longitudinal and torsional oscillations of a moving beam . Eastern-European Journal of Enterprise Technologies, 3(7(117), 31–39. https://doi.org/10.15587/1729-4061.2022.257439

Issue

Vol. 3 No. 7(117) (2022): Applied mechanics

Section

Applied mechanics

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Copyright (c) 2022 Andrii Slipchuk, Petro Pukach, Myroslava Vovk, Olha Slyusarchuk

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