ISSN 0869-6632 (Print)
ISSN 2542-1905 (Online)

#### For citation:

Savchin V. M., Trinh P. Variational approach to the construction of discrete mathematical model of the pendulum motion with vibrating suspension with friction. Izvestiya VUZ. Applied Nonlinear Dynamics, 2022, vol. 30, iss. 4, pp. 411-423. DOI: 10.18500/0869-6632-2022-30-4-411-423, EDN: BPZEKD

Full text PDF(Ru):
Full text PDF(En):
Language:
Russian
Article type:
Article
UDC:
531.011
EDN:

# Variational approach to the construction of discrete mathematical model of the pendulum motion with vibrating suspension with friction

Autors:
Trinh Phuoc Toan, RUDN University
Abstract:

The main purpose of this work is, first, a construction of the indirect Hamilton’s variational principle for the problem of motion of a pendulum with a vibration suspension with friction, oscillating along a straight line making a small angle with the vertical line. Second, the construction on its basis of the difference scheme. Third, to carry out its investigation by methods of numerical analysis. Methods. The problem of motion of the indicated pendulum is considering as a particular case of the given boundary problem for a nonlinear second order differential equations. For the solution of problem of its variational formulation there is used the criterion of potentiality of operators — the symmetry of the Gateaux derivative of nonlinear operator of the given problem. This criterion is also used for the construction of variational multiplier and the corresponding Hamilton’s variational principle. On its basis there is constructed and investigated a discrete analog of the given boundary problem and a problem of motion of the pendulum. Results. It is proved that the operator of the given boundary problem is not potential with respect to the classical bilinear form. There is found a variational multiplier and constructed the corresponding indirect Hamilton’s variational principle. On its basis there is obtained a discrete analog of the given boundary problem and its solution is found. As particular cases one can deduce from that the corresponding results for the problem of motion of the pendulum. There are performed numerical experiments, establishing the dependence of solutions of the problem of motion of the pendulum on the change of parameters. Conclusion. There is worked out a variational approach to the construction of two difference schemes for the problem of a pendulum with a suspension with friction, oscillating along a straight line making a small angle with the vertical line. There are presented results of numerical simulation under different parameters of the problem. Numerical results show that under sufficiently small amplitude and sufficiently big frequency of the oscillations of the point of suspension the pendulum realizes a periodical motion.

Key words:
Acknowledgments:
This paper has been supported by the RUDN University Strategic Academic Leadership Program — 2030
Reference:
1. Kapitsa PL. The dynamic stability of a pendulum for an oscillating point of suspension. Journal of Experimental and Theoretical Physics. 1951;21(5):588–597 (in Russian).
2. Kapitsa PL. A pendulum with a vibrating suspension. Physics-Uspekhi. 1951;44(5):7–20 (in Russian). DOI: 10.3367/UFNr.0044.195105b.0007.
3. Bogolyubov NN. Perturbation theory in nonlinear mechanics. Proceedings of the Institute of Structural Mechanics of the Academy of Sciences of the Ukrainian SSR. 1950;14:9–34 (in Russian).
4. Bogatov EM, Mukhin RR. The averaging method, a pendulum with a vibrating suspension: N.N. Bogolyubov, A. Stephenson, P.L. Kapitza and others. Izvestiya VUZ. Applied Nonlinear Dynamics. 2017;25(5):69–87 (in Russian). DOI: 10.18500/0869-6632-2017-25-5-69-87.
5. Butikov EI. The rigid pendulum — an antique but evergreen physical model. European Journal of Physics. 1999;20(6):429–441. DOI: 10.1088/0143-0807/20/6/308.
6. Samarskii AA. The Theory of Difference Schemes. Boca Raton: CRC Press; 2001. 786 p. DOI: 10.1201/9780203908518.
7. Goloviznin VM, Samarskii AA, Favorskii AP. A variational approach to constructing finite difference mathematical models in hydrodynamics. Proceedings of the Academy of Sciences of the USSR. 1977;235(6):1285–1288 (in Russian).
8. Filippov VM, Savchin VM, Shorokhov SG. Variational principles for nonpotential operators. Journal of Mathematical Sciences. 1994;68(3):275–398. DOI: 10.1007/BF01252319.
9. Savchin VM. Mathematical Methods of Mechanics of Infinite-Dimensional Non-Potential Systems. Moscow: Peoples’ Friendship University Publishing; 1991. 237 p. (in Russian).
10. Demidenko GV, Dulepova AV. On stability of the inverted pendulum motion with a vibrating suspension point. Journal of Applied and Industrial Mathematics. 2018;12(4):607–618. DOI: 10.1134/S1990478918040026.
11. Demidenko GV, Dulepova AV. On periodic solutions of one second-order differential equation. Modern Mathematics. Fundamental Directions. 2021;67(3):535–548 (in Russian). DOI: 10.22363/2413-3639-2021-67-3-535-548.