Synthesis and classification of periodic motion trajectories of the swinging spring load

Authors

DOI:

https://doi.org/10.15587/1729-4061.2019.161769

Keywords:

pendular oscillations, path of motion, swinging spring, Lagrange second-degree equation.

Abstract

The study of possibilities of geometric modeling of non-chaotic periodic paths of motion of a load of a swinging spring and its variants has been continued. In literature, a swinging spring is considered as a kind of mathematical pendulum which consists of a point load attached to a massless spring. The second end of the spring is fixed motionless. Pendular oscillations of the spring in a vertical plane are considered in conditions of maintaining straightness of its axis. The searched path of the spring load was modeled using Lagrange second-degree equations.

Urgency of the topic is determined by the need to study conditions of dissociation from chaotic oscillations of elements of mechanical structures including springs, namely definition of rational parameter values to provide periodic paths of their oscillations. Swinging springs can be used as mechanical illustrations in the study of complex technological processes of dynamic systems when nonlinearly coupled oscillatory components of the system exchange energy with each other.

The obtained results make it possible to add periodic curves as «parameters» in a graphic form to the list of numerical parameters of the swinging spring. That is, to determine numerical values of the parameters that would ensure existence of a predetermined form of the periodic path of motion of the spring load. An example of calculation of the load mass was considered based on the known stiffness of the spring, its length without load, initial conditions of initialization of oscillations as well as (attention!) the form of periodic path of this load. Periodic paths of the load motion for the swinging spring modifications (such as suspension to the movable carriage whose axis coincides with the mathematical pendulum) and two swinging springs with a common moving load and with different mounting points were obtained.

The obtained results are illustrated by computer animation of oscillations of corresponding swinging springs and their varieties.

The results can be used as a paradigm for studying nonlinear coupled systems as well as for calculation of variants of mechanical devices where springs influence oscillation of their elements and in cases when it is necessary to separate from chaotic motions of loads and provide periodic paths of their motion in technologies using mechanical devices

Author Biographies

Leonid Kutsenko, National University of Civil Defense of Ukraine Chernyshevska str., 94, Kharkiv, Ukraine, 61023

Doctor of Technical Sciences, Professor

Department of Engineering and Rescue Technology

Volodymyr Vanin, National Technical University of Ukraine «Igor Sikorsky Kyiv Polytechnic Institute» Peremohy ave., 37, Kyiv, Ukraine, 03056

Doctor of Technical Sciences, Professor

Department of Descriptive Geometry, Engineering and Computer Graphics

Olga Shoman, National Technical University "Kharkiv Polytechnic Institute" Kyrpychova str., 2, Kharkiv, Ukraine, 61002

Doctor of Technical Sciences, Professor, Head of Department

Department of Geometrical Modeling and Computer Graphics

Leonid Zapolskiy, Ukrainian Civil Protection Research Institute Rybalska str., 18, Kyiv, Ukraine, 01011

PhD, Senior Researcher

Department of Scientific and Organizational

Petro Yablonskyi, National Technical University of Ukraine «Igor Sikorsky Kyiv Polytechnic Institute» Peremohy ave., 37, Kyiv, Ukraine, 03056

PhD, Associate Professor

Department of Descriptive Geometry, Engineering and Computer Graphics

Serhii Vasyliev, National University of Civil Defense of Ukraine Chernyshevska str., 94, Kharkiv, Ukraine, 61023

PhD, Associate Professor

Department of Engineering and Rescue Machinery

Volodymyr Danylenko, National Technical University "Kharkiv Polytechnic Institute" Kyrpychova str., 2, Kharkiv, Ukraine, 61002

Associate Professor

Department of Geometrical Modeling and Computer Graphics

Elena Sukharkova, Ukrainian State University of Railway Transport Feierbakh sq., 7, Kharkiv, Ukraine, 61050

Assistant

Department of Descriptive Geometry and Computer Graphics

Svitlana Rudenko, National University of Civil Defense of Ukraine Chernyshevska str., 94, Kharkiv, Ukraine, 61023

PhD

Department of Engineering and Rescue Machinery

Maxim Zhuravskij, National University of Civil Defense of Ukraine Chernyshevska str., 94, Kharkiv, Ukraine, 61023

PhD

Department of Educational and Methodical

References

  1. De Sousa, M. C., Marcus, F. A., Caldas, I. L., Viana, R. L. (2018). Energy distribution in intrinsically coupled systems: The spring pendulum paradigm. Physica A: Statistical Mechanics and its Applications, 509, 1110–1119. doi: https://doi.org/10.1016/j.physa.2018.06.089
  2. De Sousa, M. C. de S., Marcus, F. A. M., Caldas, I. L. C. (2016). Energy distribution in a spring pendulum. Proceedings of the 6th International Conference on Nonlinear Science and Complexity. doi: https://doi.org/10.20906/cps/nsc2016-0022
  3. De Sousa, M. C., Marcus, F. A., Caldas, I. L., Viana, R. L. Energy Distribution in Spring Pendulums. Available at: https://www.researchgate.net/publication/316187700
  4. Buldakova, D. A., Kiryushin, A. V. (2015). Model of the shaking spring pendulum in the history of physics and equipment. Elektronnoe nauchnoe izdanie «Uchenye zametki TOGU», 6 (2), 238–243. Available at: http://pnu.edu.ru/media/ejournal/articles-2015/TGU_6_104.pdf
  5. France, W. N., Levadou, M., Treakle, T. W., Paulling, J. R., Michel, R. K., Moore, C. (2001). An Investigation of Head-Sea Parametric Rolling and its Influence on Container Lashing Systems. SNAME Annual Meeting 2001 Presentation, 24.
  6. Zhang, P., Ren, L., Li, H., Jia, Z., Jiang, T. (2015). Control of Wind-Induced Vibration of Transmission Tower-Line System by Using a Spring Pendulum. Mathematical Problems in Engineering, 2015, 1–10. doi: https://doi.org/10.1155/2015/671632
  7. Christensen, J. (2004). An improved calculation of the mass for the resonant spring pendulum. American Journal of Physics, 72 (6), 818–828. doi: https://doi.org/10.1119/1.1677269
  8. Bayly, P. V., Virgin, L. N. (1993). An Empirical Study of the Stability of Periodic Motion in the Forced Spring-Pendulum. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 443 (1918), 391–408. doi: https://doi.org/10.1098/rspa.1993.0152
  9. Li-Juan, Z., Hua-Biao, Z., Xin-Ye, L. (2018). Periodic solution and its stability of spring pendulum with horizontal base motion. Acta Physica Sinica, 67 (24). doi: http://doi.org/10.7498/aps.67.20181676
  10. Klimenko, A. A., Mihlin, Yu. V. (2009). Nelineynaya dinamika pruzhinnogo mayatnika. Dinamicheskie sistemy, 27, 51–65.
  11. Cross, R. (2017). Experimental investigation of an elastic pendulum. European Journal of Physics, 38 (6), 065004. doi: https://doi.org/10.1088/1361-6404/aa8649
  12. Petrov, A. G. (2015). O vynuzhdennyh kolebaniyah kachayushcheysya pruzhiny pri rezonanse. Doklady Akademii nauk, 464 (5), 553–557.
  13. Petrov, A. G., Shunderyuk, M. M. (2010). O nelineynyh kolebaniyah tyazheloy material'noy tochki na pruzhine. Mekhanika tverdogo tela, 2, 27–40.
  14. Lynch, P., Houghton, C. (2004). Pulsation and precession of the resonant swinging spring. Physica D: Nonlinear Phenomena, 190 (1-2), 38–62. doi: https://doi.org/10.1016/j.physd.2003.09.043
  15. Aldoshin, G. T., Yakovlev, S. P. (2012). Dinamika kachayushcheysya pruzhiny s podvizhnym podvesom. Vestnik Sankt-Peterburgskogo universiteta. Seriya 1. Matematika. Mekhanika. Astronomiya, 4, 45–52.
  16. Awrejcewicz, J., Sendkowski, D., Kazmierczak, M. (2006). Geometrical approach to the swinging pendulum dynamics. Computers & Structures, 84 (24-25), 1577–1583. doi: https://doi.org/10.1016/j.compstruc.2006.01.003
  17. Zaripov, M. N., Petrov, A. G. (2004). Nelineynye kolebaniya kachayushcheysya pruzhiny. Dokl. RAN, 399 (3), 347–352.
  18. Dullin, H., Giacobbe, A., Cushman, R. (2004). Monodromy in the resonant swing spring. Physica D: Nonlinear Phenomena, 190 (1-2), 15–37. doi: https://doi.org/10.1016/j.physd.2003.10.004
  19. Gendelman, O. V. (2001). Transition of Energy to a Nonlinear Localized Mode in a Highly Asymmetric System of Two Oscillators. Normal Modes and Localization in Nonlinear Systems, 237–253. doi: https://doi.org/10.1007/978-94-017-2452-4_13
  20. Petrov, A. G. (2006). Nelineynye kolebaniya kachayushcheysya pruzhiny pri rezonanse. Izvestiya Rossiyskoy Akademii Nauk. Mekhanika Tverdogo Tela, 5, 18–28.
  21. The Spring Pendulum (Optional) Available at: http://homepage.math.uiowa.edu/~stroyan/CTLC3rdEd/ProjectsOldCD/estroyan/cd/46/index.htm
  22. Broucke, R., Baxa, P. A. (1973). Periodic solutions of a spring-pendulum system. Celestial Mechanics, 8 (2), 261–267. doi: https://doi.org/10.1007/bf01231426
  23. Hitzl, D. L. (1975). The swinging spring – Invariant curves formed by quasi-periodic solutions. III. Astron and Astrophys, 41 (2), 187–198.
  24. Hitzl, D. L. (1975). The swinging spring – Families of periodic solutions and their stability. I. Astronomy and Astrophysics, 40 (1-2), 147–159.
  25. Simulation of Nonlinear Spring Pendulum. Available at: https://nl.mathworks.com/matlabcentral/fileexchange/33168-springpendulum?s_tid=srchtitle
  26. El péndulo elástico. Available at: http://www.sc.ehu.es/sbweb/fisica3/oscilaciones/pendulo_elastico/pendulo_elastico.html
  27. Van der Weele, J. P., de Kleine, E. (1996). The order – chaos – order sequence in the spring pendulum. Physica A: Statistical Mechanics and its Applications, 228 (1-4), 245–272. doi: https://doi.org/10.1016/0378-4371(95)00426-2
  28. Gavin, H. P. (2014). Generalized Coordinates, Lagrange’s Equations, and Constraints. CEE 541. Structural Dynamics.
  29. Ganis, L. (2013). The Swinging Spring: Regular and Chaotic Motion. 2013. Available at: http://depts.washington.edu/amath/wordpress/wp-content/uploads/2014/01/leah_ganis_pres.pdf
  30. Spring Pendulum. Available at: http://www.cfm.brown.edu/people/dobrush/am34/Mathematica/ch3/pendulum.html
  31. Semkiv, O. (2015). The method of determining special load oscillation trajectories of the 2D-spring pendulum. Vestnik Har'kovskogo nacional'nogo avtomobil'no-dorozhnogo universiteta, 71, 36–44.
  32. Semkiv, O., Shoman, O., Sukharkova, E., Zhurilo, A., Fedchenko, H. (2017). Development of projection technique for determining the non-chaotic oscillation trajectories in the conservative pendulum systems. Eastern-European Journal of Enterprise Technologies, 2 (4 (86)), 48–57. doi: https://doi.org/10.15587/1729-4061.2017.95764
  33. Kutsenko, L., Semkiv, O., Asotskyi, V., Zapolskiy, L., Shoman, O., Ismailova, N. et. al. (2018). Geometric modeling of the unfolding of a rod structure in the form of a double spherical pendulum in weightlessness. Eastern-European Journal of Enterprise Technologies, 4 (7 (94)), 13–24. doi: https://doi.org/10.15587/1729-4061.2018.139595
  34. Kutsenko, L., Semkiv, O., Kalynovskyi, A., Zapolskiy, L., Shoman, O., Virchenko, G. et. al. (2019). Development of a method for computer simulation of a swinging spring load movement path. Eastern-European Journal of Enterprise Technologies. 2019. Vol. 1, Issue 7 (97). P. 60–73. doi: https://doi.org/10.15587/1729-4061.2019.154191
  35. Kutsenko, L. M., Piksasov, M. M., Zapolskyi, L. L. (2018). Iliustratsiyi do statti “Heometrychne modeliuvannia periodychnoi traiektoriyi vantazhu khytnoi pruzhyny”. Available at: http://repositsc.nuczu.edu.ua/handle/123456789/7637
  36. Kutsenko, L. M., Piksasov, M. M., Vasyliev, S. V. (2019). Iliustratsiyi do statti "Klasyfikatsiya elementiv simi periodychnykh traiektoriy rukhu vantazhu khytnoi pruzhyny". Available at: http://repositsc.nuczu.edu.ua/handle/123456789/8658

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Published

2019-04-02

How to Cite

Kutsenko, L., Vanin, V., Shoman, O., Zapolskiy, L., Yablonskyi, P., Vasyliev, S., Danylenko, V., Sukharkova, E., Rudenko, S., & Zhuravskij, M. (2019). Synthesis and classification of periodic motion trajectories of the swinging spring load. Eastern-European Journal of Enterprise Technologies, 2(7 (98), 26–37. https://doi.org/10.15587/1729-4061.2019.161769

Issue

Vol. 2 No. 7 (98) (2019): Applied mechanics

Section

Applied mechanics

License

Copyright (c) 2019 Leonid Kutsenko, Volodymyr Vanin, Olga Shoman, Leonid Zapolskiy, Petro Yablonskyi, Serhii Vasyliev, Volodymyr Danylenko, Elena Sukharkova, Svitlana Rudenko, Maxim Zhuravskij

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