Development of a method for computer simulation of a swinging spring load movement path

Authors

• Leonid Kutsenko National University of Civil Defense of Ukraine Chernyshevska str., 94, Kharkiv, Ukraine, 61023, Ukraine
• Oleg Semkiv National University of Civil Defense of Ukraine Chernyshevska str., 94, Kharkiv, Ukraine, 61023, Ukraine
• Andrii Kalynovskyi National University of Civil Defense of Ukraine Chernyshevska str., 94, Kharkiv, Ukraine, 61023, Ukraine
• Leonid Zapolskiy Ukrainian Civil Protection Research Institute Rybalska str., 18, Kyiv, Ukraine, 01011, Ukraine
• Olga Shoman National Technical University "Kharkiv Polytechnic Institute" Kyrpychova str., 2, Kharkiv, Ukraine, 61002, Ukraine
• Gennadii Virchenko National Technical University of Ukraine «Igor Sikorsky Kyiv Polytechnic Institute» Peremohy ave., 37, Kyiv, Ukraine, 03056, Ukraine
• Viacheslav Martynov Kyiv National University of Сonstruction and Architecture Povitroflotskyi ave., 31, Kyiv, Ukraine, 03037, Ukraine
• Maxim Zhuravskij National University of Civil Defense of Ukraine Chernyshevska str., 94, Kharkiv, Ukraine, 61023, Ukraine
• Volodymyr Danylenko National Technical University "Kharkiv Polytechnic Institute" Kyrpychova str., 2, Kharkiv, Ukraine, 61002, Ukraine
• Nelli Ismailova Military Academy Fontanska doroha str., 10, OdesSa, Ukraine, 65009, Ukraine

Keywords:

pendulum oscillations, periodic paths of movement, swinging spring, Lagrange equation of the second kind

Abstract

Studies of geometric modeling of non-chaotic periodic paths of movement of loads attached to a variety of mathematical pendulums were continued. Pendulum oscillations in a vertical plane of a suspended weightless spring which maintains straightness of its axis were considered. In literature, this type of pendulum is called a swinging spring. The sought path of the load of the swinging spring was modeled with the help of a computer using values of the load weight, stiffness of the spring and its length without load. In addition, initial values of oscillation of the swinging spring were used: initial angle of deviation of the spring axis from the vertical, initial rate of change of this angle as well as initial parameter of the spring elongation and initial rate of elongation change. Calculations were performed using Lagrange equation of the second kind. Variants of finding conditionally periodic paths of movement of a point load attached to a swinging spring with a movable fixing point were considered.

Relevance of the topic was determined by necessity of study and improvement of new technological schemes of mechanical devices which include springs, in particular, the study of conditions of detuning from chaotic oscillations of the elements of mechanical structures and determination of rational values of parameters to ensure periodic paths of their oscillation.

A method for finding values of a set of parameters for providing a nonchaotic periodic path of a point load attached to a swinging spring was presented. The idea of this method was explained by the example of finding a periodic path of the second load of the double pendulum.

Variants of calculations for obtaining periodic paths of load movement for the following set parameters were given:

‒ length of the spring without load and its stiffness at an unknown value of the load weight;

‒ length of the spring without load and the value of the load weight at unknown spring stiffness;

‒ value of the load weight and stiffness of the spring at an unknown length of the spring without load.

As an example, determination of the values of a set of parameters to provide a non-chaotic, conditionally periodic path of movement of a point load attached to a swinging spring with a movable attachment point was considered.

Phase paths of functions of generalized coordinates (values of angles of deflection of the swinging spring axis from the vertical and extension of the spring) were constructed with the help of which it is possible to estimate ranges of these values and rates of their variation.

The results can be used as a paradigm for studying nonlinear coupled systems as well as in calculating variants of mechanical devices where springs affect oscillation of their elements when it is necessary to detune from chaotic movements of loads in the technologies using mechanical devices and provide periodic paths of their movement

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

Oleg Semkiv, National University of Civil Defense of Ukraine Chernyshevska str., 94, Kharkiv, Ukraine, 61023

Doctor of Technical Sciences, Vice-Rector

Department of Prevention Activities and Monitoring

Andrii Kalynovskyi, National University of Civil Defense of Ukraine Chernyshevska str., 94, Kharkiv, Ukraine, 61023

PhD, Associate Professor

Department of Engineering and Rescue Technology

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

PhD, Senior Researcher

Department of Scientific and Organizational

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

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

Doctor of Technical Sciences, Associate Professor

Department of Descriptive Geometry, Engineering and Computer Graphics

Viacheslav Martynov, Kyiv National University of Сonstruction and Architecture Povitroflotskyi ave., 31, Kyiv, Ukraine, 03037

Doctor of Technical Sciences, Associate Professor

Department of Architectural Constructions

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

PhD

Department of Educational and Methodical

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

Associate Professor

Department of Geometrical Modeling and Computer Graphics

Nelli Ismailova, Military Academy Fontanska doroha str., 10, OdesSa, Ukraine, 65009

Doctor of Technical Sciences, Associate ProfessorDepartment of Engineering Mechanics

References

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

Vlasov, V. N. Velichayshaya Revolyuciya v Mekhanike 4. Available at: http://www.trinitas.ru/rus/doc/0016/001d/2114-vls.pdf

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.

Lynch, P. (2001). The swinging spring: a simple model for atmospheric balance. Large-Scale Atmosphere-Ocean Dynamics: Vol. II: Geometric Methods and Models. Cambridge University Press, Cambridge, 50.

Aldoshin, G. T., Yakovlev, S. P. (2015). Analiticheskaya model' kolebaniy molekuly uglekislogo gaza. Rezonans Fermi. Izv. RAN. MTT, 1, 42–53.

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

Castillo-Rivera, S., Tomas-Rodriguez, M. (2017). Helicopter flap/lag energy exchange study. Nonlinear Dynamics, 88 (4), 2933–2946. doi: https://doi.org/10.1007/s11071-017-3422-4

Bogdanov, K. Yu. (1993). Hishchnik i zhertva. Kvant, 2. Available at: http://kvant.mccme.ru/1993/02/hishchnik_i_zhertva.htm

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

Aldoshin, G. T. (2009). Zamechaniya k metodu linearizacii nelineynyh uravneniy s dvumya stepenyami svobody. V sb. «Matematika, informatika, estestvoznanie v ekonomike i obshchestve». Trudy mezhdunarodnoy nauchno-prakticheskoy konferencii. Vol. 1. Moscow: MFYUF.

Bubnovich, E. V., Moldaganapova, A. G. K voprosu ob issledovanii rezonansov pri vynuzhdennyh vzaimosvyazannyh kolebaniyah gibkoy niti. Available at: http://portal.kazntu.kz/files/publicate/%20Молдаганапова%20.pdf

Petrov, A. G. (2015). O vynuzhdennyh kolebaniyah kachayushcheysya pruzhiny pri rezonanse. Doklady Akademii nauk, 464 (5), 553–557. doi: https://doi.org/10.7868/s0869565215290113

Petrov, A. G., Shunderyuk, M. M. (2010). O nelineynyh kolebaniyah tyazheloy material'noy tochki na pruzhine. Izv. RAN. MTT, 2, 27–40.

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

Duka, B., Duka, R. (2018). On the elastic pendulum, parametric resonance and “pumping” swings. European Journal of Physics. 2018. doi: https://doi.org/10.1088/1361-6404/aaf146

Breitenberger, E., Mueller, R. D. (1981). The elastic pendulum: A nonlinear paradigm. Journal of Mathematical Physics, 22 (6), 1196–1210. doi: https://doi.org/10.1063/1.525030

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

Ryland, G., Meirovitch, L. (1977). Stability boundaries of a swinging spring with oscillating support. Journal of Sound and Vibration, 51 (4), 547–560. doi: https://doi.org/10.1016/s0022-460x(77)80051-5

Holm, D. D., Lynch, P. (2002). Stepwise Precession of the Resonant Swinging Spring. SIAM Journal on Applied Dynamical Systems, 1 (1), 44–64. doi: https://doi.org/10.1137/s1111111101388571

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

Klimenko, A. A., Mihlin, Yu. V. (2009). Nelineynaya dinamika pruzhinnogo mayatnika. Dinamicheskie sistemy, 27, 51–65.

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

Hitzl, D. L. (1975). The swinging spring invariant curves formed by quasi-periodic solution. III. Astron and Astrophys, 41 (2), 187–198.

Modelirovanie dvizheniya dvoynogo mayatnika v Dekartovoy sisteme koordinat. Available at: https://www.wolfram.com/mathematica/new-in-9/advanced-hybrid-and-differential-algebraic-equations/double-pendulum.html

The Spring Pendulum (Optional). Available at: http://homepage.math.uiowa.edu/~stroyan/CTLC3rdEd/ProjectsOldCD/estroyan/cd/46/index.htm

Gavin, H. P. (2014). Generalized Coordinates, Lagrange’s Equations, and Constraints. CEE 541. Structural Dynamics. Department of Civil and Environmental Engineering Duke University, 23.

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

File:Spring pendulum.gif. Available at: https://en.wikipedia.org/wiki/File:Spring_pendulum.gif

Aldoshin, G. T., Yakovlev, S. P. (2012). Dynamics of a swinging spring with moving support. Vestnik Sankt-Peterburgskogo universiteta. Seriya 1. Matematika. Mekhanika. Astronomiya, 4, 45–52.

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

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

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

2019-01-14

How to Cite

Kutsenko, L., Semkiv, O., Kalynovskyi, A., Zapolskiy, L., Shoman, O., Virchenko, G., Martynov, V., Zhuravskij, M., Danylenko, V., & Ismailova, N. (2019). Development of a method for computer simulation of a swinging spring load movement path. Eastern-European Journal of Enterprise Technologies, 1(7 (97), 60–73. https://doi.org/10.15587/1729-4061.2019.154191

Section

Applied mechanics