December  2018, 10(4): 411-417. doi: 10.3934/jgm.2018015

On motions without falling of an inverted pendulum with dry friction

Steklov Mathematical Institute of the Russian Academy of Sciences, Moscow, Russia

* Corresponding author: Ivan Polekhin

Received  March 2017 Revised  September 2018 Published  November 2018

Fund Project: This work was supported by the Russian Science Foundation under Grant No. 14-50-00005.

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.

Citation: Ivan Polekhin. On motions without falling of an inverted pendulum with dry friction. Journal of Geometric Mechanics, 2018, 10 (4) : 411-417. doi: 10.3934/jgm.2018015
References:
[1]

B. Bardin and 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 (1995), 879-886.  doi: 10.1016/0021-8928(95)00121-2.  Google Scholar

[2]

S. V. Bolotin and 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 (2015), 894-901.  doi: 10.4213/im8413.  Google Scholar

[3]

E. I. Butikov, On the dynamic stabilization of an inverted pendulum, American Journal of Physics, 69 (2001), 755-768.   Google Scholar

[4]

A. F. Filippov, Differential Equations with Discontinuous Righthand Sides, Mathematics and its Applications (Soviet Series), 18. Kluwer Academic Publishers Group, Dordrecht, 1988. doi: 10.1007/978-94-015-7793-9.  Google Scholar

[5]

A. P. Ivanov, Bifurcations in systems with friction: Basic models and methods, Regular and Chaotic Dynamics, 14 (2009), 656-672.  doi: 10.1134/S1560354709060045.  Google Scholar

[6]

P. Kapitsa, Pendulum with vibrating axis of suspension (in Russian), Uspekhi fizicheskich nauk, 44 (1954), 7-20.   Google Scholar

[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 (2014), 465-472.   Google Scholar

[8]

I. Polekhin, Forced oscillations of a massive point on a compact surface with a boundary, Nonlinear Analysis: Theory, Methods & Applications, 128 (2015), 100-105.  doi: 10.1016/j.na.2015.07.022.  Google Scholar

[9]

I. Polekhin, On forced oscillations in groups of interacting nonlinear systems, Nonlinear Analysis: Theory, Methods & Applications, 135 (2016), 120-128.  doi: 10.1016/j.na.2016.01.021.  Google Scholar

[10]

I. Polekhin, A topological view on forced oscillations and control of an inverted pendulum, Systems Control Lett., 113 (2018), 31-35.  doi: 10.1016/j.sysconle.2018.01.005.  Google Scholar

[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, (2017), 329-335.   Google Scholar

[12]

V. Popov, Contact Mechanics and Friction: Physical Principles and Applications, Springer Science & Business Media., 2010. Google Scholar

[13]

R. Reissig, G. Sansone and R. Conti, Qualitative Theorie nichtlinearer Differentialgleichungen, Edizioni Cremonese, 1963.  Google Scholar

[14]

A. Seyranian and A. Seyranian, The stability of an inverted pendulum with a vibrating suspension point, Journal of Applied Mathematics and Mechanics, 70 (2006), 754-761.  doi: 10.1016/j.jappmathmech.2006.11.009.  Google Scholar

[15]

R. Srzednicki, On periodic solutions in the Whitney's inverted pendulum problem, arXiv preprint, arXiv:1709.08254, 2017. Google Scholar

[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 (1947), 279-313 (1948).   Google Scholar

show all references

References:
[1]

B. Bardin and 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 (1995), 879-886.  doi: 10.1016/0021-8928(95)00121-2.  Google Scholar

[2]

S. V. Bolotin and 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 (2015), 894-901.  doi: 10.4213/im8413.  Google Scholar

[3]

E. I. Butikov, On the dynamic stabilization of an inverted pendulum, American Journal of Physics, 69 (2001), 755-768.   Google Scholar

[4]

A. F. Filippov, Differential Equations with Discontinuous Righthand Sides, Mathematics and its Applications (Soviet Series), 18. Kluwer Academic Publishers Group, Dordrecht, 1988. doi: 10.1007/978-94-015-7793-9.  Google Scholar

[5]

A. P. Ivanov, Bifurcations in systems with friction: Basic models and methods, Regular and Chaotic Dynamics, 14 (2009), 656-672.  doi: 10.1134/S1560354709060045.  Google Scholar

[6]

P. Kapitsa, Pendulum with vibrating axis of suspension (in Russian), Uspekhi fizicheskich nauk, 44 (1954), 7-20.   Google Scholar

[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 (2014), 465-472.   Google Scholar

[8]

I. Polekhin, Forced oscillations of a massive point on a compact surface with a boundary, Nonlinear Analysis: Theory, Methods & Applications, 128 (2015), 100-105.  doi: 10.1016/j.na.2015.07.022.  Google Scholar

[9]

I. Polekhin, On forced oscillations in groups of interacting nonlinear systems, Nonlinear Analysis: Theory, Methods & Applications, 135 (2016), 120-128.  doi: 10.1016/j.na.2016.01.021.  Google Scholar

[10]

I. Polekhin, A topological view on forced oscillations and control of an inverted pendulum, Systems Control Lett., 113 (2018), 31-35.  doi: 10.1016/j.sysconle.2018.01.005.  Google Scholar

[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, (2017), 329-335.   Google Scholar

[12]

V. Popov, Contact Mechanics and Friction: Physical Principles and Applications, Springer Science & Business Media., 2010. Google Scholar

[13]

R. Reissig, G. Sansone and R. Conti, Qualitative Theorie nichtlinearer Differentialgleichungen, Edizioni Cremonese, 1963.  Google Scholar

[14]

A. Seyranian and A. Seyranian, The stability of an inverted pendulum with a vibrating suspension point, Journal of Applied Mathematics and Mechanics, 70 (2006), 754-761.  doi: 10.1016/j.jappmathmech.2006.11.009.  Google Scholar

[15]

R. Srzednicki, On periodic solutions in the Whitney's inverted pendulum problem, arXiv preprint, arXiv:1709.08254, 2017. Google Scholar

[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 (1947), 279-313 (1948).   Google Scholar

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