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.

[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.

[3]

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

[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.

[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.

[6]

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

[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.

[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.

[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.

[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.

[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.

[12]

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

[13]

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

[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.

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

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.

[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.

[3]

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

[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.

[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.

[6]

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

[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.

[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.

[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.

[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.

[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.

[12]

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

[13]

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

[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.

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

[1]

Mari Paz Calvo, Jesus M. Sanz-Serna. Carrying an inverted pendulum on a bumpy road. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 429-438. doi: 10.3934/dcdsb.2010.14.429

[2]

Giuseppe Maria Coclite, Angelo Favini, Gisèle Ruiz Goldstein, Jerome A. Goldstein, Silvia Romanelli. Continuous dependence in hyperbolic problems with Wentzell boundary conditions. Communications on Pure & Applied Analysis, 2014, 13 (1) : 419-433. doi: 10.3934/cpaa.2014.13.419

[3]

Jianquan Li, Yiqun Li, Yali Yang. Epidemic characteristics of two classic models and the dependence on the initial conditions. Mathematical Biosciences & Engineering, 2016, 13 (5) : 999-1010. doi: 10.3934/mbe.2016027

[4]

Leonid Shaikhet. Improved condition for stabilization of controlled inverted pendulum under stochastic perturbations. Discrete & Continuous Dynamical Systems - A, 2009, 24 (4) : 1335-1343. doi: 10.3934/dcds.2009.24.1335

[5]

Yubai Liu, Xueshan Gao, Fuquan Dai. Implementation of Mamdami fuzzy control on a multi-DOF two-wheel inverted pendulum robot. Discrete & Continuous Dynamical Systems - S, 2015, 8 (6) : 1251-1266. doi: 10.3934/dcdss.2015.8.1251

[6]

Xianwei Chen, Zhujun Jing, Xiangling Fu. Chaos control in a pendulum system with excitations. Discrete & Continuous Dynamical Systems - B, 2015, 20 (2) : 373-383. doi: 10.3934/dcdsb.2015.20.373

[7]

Liejune Shiau, Roland Glowinski. Operator splitting method for friction constrained dynamical systems. Conference Publications, 2005, 2005 (Special) : 806-815. doi: 10.3934/proc.2005.2005.806

[8]

J. Ángel Cid, Pedro J. Torres. On the existence and stability of periodic solutions for pendulum-like equations with friction and $\phi$-Laplacian. Discrete & Continuous Dynamical Systems - A, 2013, 33 (1) : 141-152. doi: 10.3934/dcds.2013.33.141

[9]

Yalin Zhang, Guoliang Shi. Continuous dependence of the transmission eigenvalues in one dimension. Inverse Problems & Imaging, 2015, 9 (1) : 273-287. doi: 10.3934/ipi.2015.9.273

[10]

Jiří Benedikt. Continuous dependence of eigenvalues of $p$-biharmonic problems on $p$. Communications on Pure & Applied Analysis, 2013, 12 (3) : 1469-1486. doi: 10.3934/cpaa.2013.12.1469

[11]

Rafał Kamocki, Marek Majewski. On the continuous dependence of solutions to a fractional Dirichlet problem. The case of saddle points. Discrete & Continuous Dynamical Systems - B, 2014, 19 (8) : 2557-2568. doi: 10.3934/dcdsb.2014.19.2557

[12]

Pavel Krejčí, Thomas Roche. Lipschitz continuous data dependence of sweeping processes in BV spaces. Discrete & Continuous Dynamical Systems - B, 2011, 15 (3) : 637-650. doi: 10.3934/dcdsb.2011.15.637

[13]

Ramon Quintanilla. Structural stability and continuous dependence of solutions of thermoelasticity of type III. Discrete & Continuous Dynamical Systems - B, 2001, 1 (4) : 463-470. doi: 10.3934/dcdsb.2001.1.463

[14]

B.G. Fitzpatrick, M.A. Jeffris. On continuous dependence under approximation for groundwater flow models with distributed and pointwise observations. Discrete & Continuous Dynamical Systems - A, 1996, 2 (1) : 141-149. doi: 10.3934/dcds.1996.2.141

[15]

P.E. Kloeden, Pedro Marín-Rubio. Equi-Attraction and the continuous dependence of attractors on time delays. Discrete & Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 581-593. doi: 10.3934/dcdsb.2008.9.581

[16]

Paola Goatin, Philippe G. LeFloch. $L^1$ continuous dependence for the Euler equations of compressible fluids dynamics. Communications on Pure & Applied Analysis, 2003, 2 (1) : 107-137. doi: 10.3934/cpaa.2003.2.107

[17]

Giselle A. Monteiro, Milan Tvrdý. Generalized linear differential equations in a Banach space: Continuous dependence on a parameter. Discrete & Continuous Dynamical Systems - A, 2013, 33 (1) : 283-303. doi: 10.3934/dcds.2013.33.283

[18]

Margarita Arias, Juan Campos, Cristina Marcelli. Fastness and continuous dependence in front propagation in Fisher-KPP equations. Discrete & Continuous Dynamical Systems - B, 2009, 11 (1) : 11-30. doi: 10.3934/dcdsb.2009.11.11

[19]

Luisa Malaguti, Cristina Marcelli, Serena Matucci. Continuous dependence in front propagation of convective reaction-diffusion equations. Communications on Pure & Applied Analysis, 2010, 9 (4) : 1083-1098. doi: 10.3934/cpaa.2010.9.1083

[20]

Hong Cai, Zhong Tan, Qiuju Xu. Time periodic solutions to Navier-Stokes-Korteweg system with friction. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 611-629. doi: 10.3934/dcds.2016.36.611

2017 Impact Factor: 0.561

Metrics

  • PDF downloads (34)
  • HTML views (86)
  • Cited by (0)

Other articles
by authors

[Back to Top]