531.3

nonlinear dynamics, oscillations, chaos, spatial problem, double spherical pendulum, Lagrange equation of the 2nd kind, mathematical model, numerical experiment.

The article considers the spatial motion of a mechanical system where a heavy beam of a given mass and dimensions is suspended at one end by a weightless inextensible cable to a trolley, which can move along horizontal guides without resistance. The system has five degrees of freedom. Based on the apparatus of analytical mechanics and Lagrange equations, a mathematical model of the considered mechanical system in the form of a system of five nonlinear differential equations of the second order is obtained. The mathematical model is implemented in the form of a computer program that allows you to determine the coordinates (positions) of the beam at any time, build the trajectory of the center of mass, determine the kinematic characteristics of the movement, calculate the cable tension and determine its extreme value. Based on the numerical experiment, graphs and phase trajectories of these parameters are constructed, including the 3D trajectory of the center of mass of the beam. The system can show quite complex dynamics depending on the initial conditions, as evidenced by the results of numerical calculations. Under certain conditions, chaotic behavior of the system is possible. Having a mathematical model and a calculation program, it is possible to conduct further studies of the system under consideration, revealing the positions of stable and unstable equilibrium, modes of self-oscillations, revealing areas of periodic and chaotic modes, bifurcations, and so on.

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11. Wang J., Wang S., Chen H.,Niu A., Jin G. Dynamic Modeling and Analysis of the Telescopic Sleeve
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17. Loveykin V., Lymar P. Dynamic analysis of movement of carriage hoisting crane with a displaced center of mass cargo for grips. Bulletin of TNTU. Ternopil: TNTU. 2014. Vol.73. No. 1. P. 102–109.
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1. Vekeryk V. ta in. Teoretychna mekhanika. Chastyna druha. Dynamika. Navchal'nyy posibnyk. Ivano-Frankivs'k: Fakel, 2002. Р. 342. [In Ukrainian].
2. Zhang M., Zhang Y., Cheng X. Finite-Time Trajectory Tracking Control for Overhead Crane
   Systems Subject to Unknown Disturbances. IEEE, Vol. 7. 2019. Digital Object Identifier 10.1109/ACCESS.2019.2911538.
3. Niu D., Zhu Y., Chen X., ets. An anti-sway positioning control method via load generalized position tracking with disturbance observer. Measurement and Control. 53 (9–10). 2020. P. 2101–2110.
4. L. Ramli, Z. Mohamed, M. Efe, I. M. Lazim, H.I. Jaafar. Efficient swing control of an overhead crane with simultaneous payload hoisting and external disturbances/ Elsevir. Mechanical Systems and Signal Processing. 135 (2020). 106326
5. M. J. Maghsoudi, H. Nacer, M. O. Tokhi, Z. Mohamed. A Novel Approach in S-Shaped Input Design for Higher Vibration Reduction. Journal of Applications of Modelling and Simulation, 2018.
6. Loveykin V. S., Chovnyuk Yu.V., Serdyuchenko Yu.Yu., Dikteruk M. H. Analiz kolyvan' vantazhu na zovnishniy pidvistsi helikoptera pry yoho horyzontal'nomu pol'oti z postiynoyu shvydkistyu. Vestnyk KhNADU. Vol. 57. 2012. Р. 234–238. [In Ukrainian].
7. Podlesny S. Dynamics of a spherical pendulum on a nonlinear elastic suspension under the action of a variable side aerodynamic load. Visnyk TNTU. Tern.: TNTU, 2020. Vol. 98. No. 2. P. 49–58.
8. Korytov M. S., Shcherbakov V. S., Tytenko V. V., Belyakov V. E. Model' sferycheskoho mayatnyka s podvyzhnoy tochkoy podvesa v zadache prostranstvennoho peremeshchenyya hruza hruzopodemnyh kranom pry ohranychenyy kolebanyy. Dynamyka system, mekhanyzmov y mashyn. 2019. Tom 7. No. 1. P. 104–110. URL: https://doi.org/10.25206/2310-9793-7-1-104-110. [In Russian].
9. Perig A. V., Stadnik A. N., Deriglazov A. I., Podlesny S. V. “3 DOF spherical pendulum oscillations with a uniform slewing pivot center and a small angle assumption”. Shock and Vibration. Vol. 2014. Article ID 203709. 32 p. URL: https://www.researchgate.net/publication/265385700.https://doi.org/10.1155/2014/ 203709.
10. Zhang M., Ma X., Rong X., Song R., Tian X., Li Y. An enhanced coupling nonlinear tracking controller for underactuated 3d overhead crane systems. Asian Journal of Control. Vol. 20. No. 5. 2018. P. 1839–1854.
11. Wang J., Wang S., Chen H.,Niu A., Jin G. Dynamic Modeling and Analysis of the Telescopic Sleeve
12. Antiswing Device for Shipboard Cranes. Hindawi. Mathematical Problems in Engineering. Vol. 2021. Article ID 6685816. 15 p. URL: https://doi.org/10.1155/2021/6685816
13. H. C. Cho and K. S. Lee. Adaptive control and stability analysis of nonlinear crane systems with perturbation. Springer. Journal of Mechanical Science and Technology. 22. 2008. P. 1091–1098.
14. Kutsenko L.M., Adashevs'ka I.Yu. Heometrychne modelyuvannya kolyvan' bahatolankovykh mayatnykiv. Monohrafiya. X.: NTU “KhPI”, 2008. 176 р. [In Ukrainian].
15. Miyamoto К. Long Term Simulations of the Double Pendulum by Keeping the Value of Hamiltonian Constant. 8th EUROSIM Congress on Modelling and Simulation, 2013. Р. 130–135.
16. Wu Y., Sun N., Liang X., Fang Y., Xin X. A Robust Control Approach for Double-Pendulum Overhead Cranes With Unknown Disturbances. 2019 IEEE 4th International Conference on Advanced Robotics and Mechatronics (ICARM). P. 510–515.
17. Loveykin V., Lymar P. Dynamic analysis of movement of carriage hoisting crane with a displaced center of mass cargo for grips. Bulletin of TNTU. Ternopil: TNTU. 2014. Vol.73. No. 1. P. 102–109.
18. O’Connor W., Habibi H. Gantry crane control of a double-pendulum, distributed-mass load, using mechanical wave concept. Mech. Sci. 4. 2013. Р. 251–261.