Modeling the resonance of a swinging spring based on the synthesis of a motion trajectory of its load

Authors

DOI:

https://doi.org/10.15587/1729-4061.2019.168909

Keywords:

swinging spring, a swinging spring resonance, pendulum oscillations, a load motion’s trajectories.

Abstract

The paper reports a technique for building the resonance trajectories of the motion of a swinging spring load. A swinging spring is the kind of a mathematical pendulum consisting of a point load attached to a weightless spring. The other end of the spring is fixed immovably. We have considered the pendulum-like spring oscillations in a vertical plane provided its axis straightness is maintained. Calculations have been performed based on the solutions to a system of differential equations with components that include values for the frequency values of vertical and horizontal displacements of a point on a spring.

The relevance of the subject is predetermined by the necessity to study the technological processes of dynamic systems when the nonlinearly connected oscillatory components of the system exchange energy. Using a swinging spring phenomenon illustrates the exchange of energies between the transverse (pendulum) and longitudinal (spring) oscillations. In this case, we also take into consideration the influence of the initial conditions for initiating oscillations. Of particular importance is to study the resonance state of a swinging spring when the frequency of longitudinal oscillations differs by a multiple number of times from the frequency of transverse oscillations. In addition to a common «classic» case (resonance 2:1), there is a need to consider cases with different values for the frequency ratio. The result is the derived geometric shapes of the motion trajectory of a swinging spring load that correspond to the patterns in the state of its resonance.

The results obtained in the current paper make it possible, by using a computer, to synthesize the motion trajectory of a swinging spring load that would match the assigned frequency ratio of longitudinal and transverse oscillations. For this purpose, in addition to basic parameters (a load’s mass, rigidity of the spring, its length in a no-load state), we added the initial values for the parameters during oscillation initiation. Specifically, the «starting» coordinates for a load position, and the initial load motion velocities in the direction of the coordinate axes. We have considered examples of building a load motion’s trajectories for cases of resonances the type of 2:1, 7:3; 9:4; and 11:2. The results obtained are illustrated by the computerized animations of oscillations of appropriate swinging springs for different cases of resonance.

The results could be used as a paradigm in order to study the nonlinear connected systems, as well as in the calculation of variants for mechanical devices where springs affect the oscillation of their elements. Additionally, for cases when the technology of using mechanical devices necessitates abandoning the chaotic movements of loads in order to ensure the periodic trajectories of their displacements.

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

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

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

PhD, Senior Researcher

Department of Scientific and Organizational

Natalia Hrytsyna, Kharkiv National Automobile and Highway University Yaroslava Mudroho str., 25, Kharkiv, Ukraine, 61002

PhD, Associate Professor

Department of Engineering and Computer Graphics

Sergii Nazarenko, National University of Civil Defense of Ukraine Chernyshevska str., 94, Kharkiv, Ukraine, 61023

PhD

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

Elizaveta Sivak, National Technical University "Kharkiv Polytechnic Institute" Kyrpychova str., 2, Kharkiv, Ukraine, 61002

PhD, Associate Professor

Department of Geometrical Modeling and Computer Graphics

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

Assistant

Department of Fire Tactics and Rescue Operations

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Published

2019-05-29

How to Cite

Kutsenko, L., Vanin, V., Shoman, O., Yablonskyi, P., Zapolskiy, L., Hrytsyna, N., Nazarenko, S., Danylenko, V., Sivak, E., & Shevchenko, S. (2019). Modeling the resonance of a swinging spring based on the synthesis of a motion trajectory of its load. Eastern-European Journal of Enterprise Technologies, 3(7 (99), 53–64. https://doi.org/10.15587/1729-4061.2019.168909

Issue

Vol. 3 No. 7 (99) (2019): Applied mechanics

Section

Applied mechanics

License

Copyright (c) 2019 Leonid Kutsenko, Volodymyr Vanin, Olga Shoman, Petro Yablonskyi, Leonid Zapolskiy, Natalia Hrytsyna, Sergii Nazarenko, Volodymyr Danylenko, Elizaveta Sivak, Serhii Shevchenko

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