×

zbMATH — the first resource for mathematics

On motions without falling of an inverted pendulum with dry friction. (English) Zbl 1411.70030
Summary: An inverted planar pendulum with horizontally moving pivot point is considered. It is assumed that the law of motion of the pivot point is given and the pendulum is moving in the presence of dry friction. Sufficient conditions for the existence of solutions along which the pendulum never falls below the horizon are presented. The proof is based on the fact that solutions of the corresponding differential inclusion are right-unique and continuously depend on initial conditions, which is also shown in the paper.
MSC:
70K40 Forced motions for nonlinear problems in mechanics
37B55 Topological dynamics of nonautonomous systems
34A60 Ordinary differential inclusions
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] B. Bardin; A. Markeyev, The stability of the equilibrium of a pendulum for vertical oscillations of the point of suspension, Journal of Applied Mathematics and Mechanics, 59, 879-886, (1995) · Zbl 0900.70281
[2] S. V. Bolotin; V. V. Kozlov, Calculus of variations in the large, existence of trajectories in a domain with boundary, and Whitney inverted pendulum problem, Izvestiya: Mathematics, 79, 894-901, (2015) · Zbl 1367.37053
[3] E. I. Butikov, On the dynamic stabilization of an inverted pendulum, American Journal of Physics, 69, 755-768, (2001)
[4] A. F. Filippov, Differential Equations with Discontinuous Righthand Sides, Mathematics and its Applications (Soviet Series), 18. Kluwer Academic Publishers Group, Dordrecht, 1988.
[5] A. P. Ivanov, Bifurcations in systems with friction: Basic models and methods, Regular and Chaotic Dynamics, 14, 656-672, (2009) · Zbl 1229.70072
[6] P. Kapitsa, Pendulum with vibrating axis of suspension (in Russian), Uspekhi fizicheskich nauk, 44, 7-20, (1954)
[7] I. Y. Polekhin, Examples of topological approach to the problem of inverted pendulum with moving pivot point (in Russian), Nelineinaya Dinamika [Russian Journal of Nonlinear Dynamics], 10, 465-472, (2014) · Zbl 1353.70051
[8] I. Polekhin, Forced oscillations of a massive point on a compact surface with a boundary, Nonlinear Analysis: Theory, Methods & Applications, 128, 100-105, (2015) · Zbl 1372.70023
[9] I. Polekhin, On forced oscillations in groups of interacting nonlinear systems, Nonlinear Analysis: Theory, Methods & Applications, 135, 120-128, (2016) · Zbl 1353.70050
[10] I. Polekhin, A topological view on forced oscillations and control of an inverted pendulum, Systems Control Lett., 113, 31-35, (2018) · Zbl 1386.93259
[11] I. Polekhin, On topological obstructions to global stabilization of an inverted pendulum, Geometric Science of Information. GSI 2017, Lecture Notes in Comput. Sci., 10589, Springer, Cham, 329-335, (2017) · Zbl 1427.70048
[12] V. Popov, Contact Mechanics and Friction: Physical Principles and Applications, Springer Science & Business Media., 2010. · Zbl 1193.74001
[13] R. Reissig, G. Sansone and R. Conti, Qualitative Theorie nichtlinearer Differentialgleichungen, Edizioni Cremonese, 1963.
[14] A. Seyranian; A. Seyranian, The stability of an inverted pendulum with a vibrating suspension point, Journal of Applied Mathematics and Mechanics, 70, 754-761, (2006)
[15] R. Srzednicki, On periodic solutions in the Whitney’s inverted pendulum problem, arXiv preprint, arXiv:1709.08254, 2017.
[16] T. Ważewski, Sur un principe topologique de l’examen de l’allure asymptotique des intégrales des équations différentielles ordinaires, Annales De La Societe Polonaise De Mathematique, 20, 279-313 (1948), (1947) · Zbl 0032.35001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.