531.3

oscillations, nonlinear dynamics, phase portrait, pendulum, mathematical model, Lagrange equations of the second kind, d'Alembert principle, numerical experiment.

A mechanical system, where the load in the form of material point is suspended on inextensible thread screwed on the rotating cylindrical drum, but the drum is connected to the boom rotating around fixed horizontal axis is considered. Using the Lagrange equation of the second kind, a mathematical model of the motion of the mechanical system is obtained. The system has three degrees of freedom, two of which are cylindrical. The investigation of the system motion is carried out using computer technology. As a result, the dependences of linear and angular coordinates and velocities in time at different values of the output data for two main modes of the system operation, namely – under the conditions of lifting and lowering the load are obtained. Appropriate graphs are constructed, including the trajectories of the cargo motion. The mathematical model takes into account nonlinearities of the system and allows you to find the amount of tension of the hoisting rope at any time. The analysis showed that vertical oscillations occur twice as fast as horizontal ones. The phase portrait of the generalized coordinate (angle of the rope with the vertical axis) is the focus, which is untwisted when lifting due to nonlinearity in the system, and when the load moves down, the focus, which twists and approaches the mathematical pendulum is obtained. The obtained results can be used in modeling of controlled pendulum motions for different mechanical systems. The methodology and program are recommended to the students and graduate students in terms of learning the principles of construction and analysis of complex nonlinear dynamical systems.

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1.   Podlesny S. (2020) Dynamics of a spherical pendulum on a nonlinear elastic suspension under the action of a variable side aerodynamic load.  Scientific Journal of TNTU (Tern.). Vol. 98. No. 2. Р. 49–58.
2.   Freundlich J., Sado D. Dynamics of a coupled mechanical system containing a spherical pendulum and a fractional damper. Published 2020. Physics. Meccanica. 55:2541–2553. URL: https://doi.org/10.1007/ s11012-020-01203-4(0123456789().,-volV)( 01234567.
3.   Zinko R. Maiatnyk zminnoi dovzhyny. Elektronnyi resurs. URL: http://www.zinko.lviv.ua/index.php? artid=1472900553.
4.   Olshanskyi S. V. Nestatsyonarnыe kolebanyia ostsylliatora peremennoi massы s uchёtom viazkoho trenyia. Vibratsii v tekhnitsi ta tekhnolohiiakh. No. 3 (75). 2014. Р. 18–27.
5.   Loveikin V., Lymar P. Dynamichnyi analiz peremishchennia vizka vantazhopidiomnoho krana zi zmishchenym tsentrom mas vantazhu vidnosno zakhvatu. Visnyk TNTU. 2014. Tom 73. No. 1. Р. 102–109.
6.   Perig A. V., Stadnik A. N., Kostikov A. A., Podlesny S. V. Research into 2D dynamics and control of small oscillations of a cross-beam during transportation by two overhead cranes. Shock and Vibration. 2017. URL: http://downloads.hindawi.com/journals/sv/2017/9605657.pdf.
7.   Loveikin V. S., Romasevych Yu. O., Stekhno O. V. Optymizatsiia rezhymu rukhu mekhanizmu zminy vylotu vantazhu bashtovoho krana z horyzontalnoiu striloiu. Mashynobuduvannia. 2017. No. 20. Р. 11–18.
8.   Podoliak O. S., Bolybik M. O. Matematychne modeliuvannia sumisnoho rukhu mekhanizmiv pidiomu, povorotu i zminy vylotu krana DEK-251. Mashynobuduvannia. 2017. No. 19. Р. 61–67.
9.   Palamarchuk D. A. Yssledovanye dynamyky dvyzhenyia strelovoi systemы krana pry avtomatycheskom upravlenyy mekhanyzmom yzmenenyia vыleta. Visnyk Natsionalnoho universytetu vodnoho hospodarstva ta pryrodokorystuvannia. Vypusk 3 (67). 2014 r. Seriia “Tekhnichni nauky”. P. 361–370.
10. Bulatov L. A., Bertiaev V. D., Kyreeva A. E. Yssledovanye dvyzhenyia oborotnoho matematycheskoho maiatnyka s yestyia TulHU. Tekhnycheskye nauky. 2010. Vol. 2. Ch. 1. P. 11–18.