Mechanics of Solids

, Volume 47, Issue 3, pp 285–297 | Cite as

Stability and instability of controlled motions of a two-mass pendulum of variable length

  • V. S. AslanovEmail author
  • S. P. Bezglasnyi


The problem of parametric control of plane motions of a two-mass pendulum (swing) is considered. The swing model is a weightless rod with two lumped masses one of which is fixed on the rod and the other slides along it within bounded limits. The control is the distance from the suspension point to the moving point. The proposed control law of swing excitation and damping consists in continuously varying the pendulumsuspension length depending on the phase state. The stability of various controlled motions, including the motions near the upper and lower equilibria, is studied. The Lyapunov functions that prove the asymptotic stability and instability of the pendulum lower position in the respective cases of the pendulum damping and excitation are constructed for the proposed control law. The influence of the viscous friction forces on the pendulum stable motions and the onset of stagnation regions in the case of its excitation is analyzed. The theoretical results are confirmed by graphical representation of the numerical results.


two-mass pendulum controlled system asymptotic stability Lyapunov function 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    T. G. Strizhak, Methods for Studying ‘Pendulum’-Type Dynamical Systems (Nauka, Alma-Ata, 1981) [in Russian].Google Scholar
  2. 2.
    K. Magnus, Vibrations. Introduction to the Study of Oscillatory Systems (Mir, Moscow, 1982) [in Russian].Google Scholar
  3. 3.
    S. L. Chechurin, Parametric Vibrations and Stability of PeriodicMotion (Izd-vo LGU, Leningrad, 1983) [in Russian].Google Scholar
  4. 4.
    L. D. Akulenko, Asymptotic Methods of Optimal Control (Nauka, Moscow, 1987) [in Russian].zbMATHGoogle Scholar
  5. 5.
    A. P. Markeev, Theoretical Mechanics (CheRo, Moscow, 1999) [in Russian].Google Scholar
  6. 6.
    Yu. F. Golubev, Foundations of Theoretical Mechanics (Izd-vo MGU, Moscow, 2000) [in Russian].Google Scholar
  7. 7.
    A. P. Seyranian, “The Swing: Parametric Resonance,” Prikl. Mat. Mekh. 68(5), 847–856 (2004) [J. Appl. Math. Mech. (Engl. Transl.) 68 (5), 757–764 (2004)].MathSciNetGoogle Scholar
  8. 8.
    A. A. Zevin and L. A. Filonenko, “A Qualitative Investigation of the Vibrations of a Pendulum with a Periodically Varying Length and a Mathematical Model of a Swing,” Prikl. Mat. Mekh. 71(6), 989–1003 (2007) [J. Appl. Math. Mech. (Engl. Transl.) 71 (6), 892–904 (2007)].MathSciNetzbMATHGoogle Scholar
  9. 9.
    L. D. Akulenko and S. V. Nesterov, “The Stability of the Equilibrium of a Pendulum of Variable Length,” Prikl. Mat. Mekh. 73(6), 893–901 (2009) [J. Appl. Math. Mech. (Engl. Transl.) 73 (6), 642–647 (2009)].MathSciNetGoogle Scholar
  10. 10.
    L. D. Akulenko, “Parametric Control of Vibrations and Rotations of a Compound Pendulum (a Swing),” Prikl. Mat. Mekh. 57(2), 82–91 (1993) [J. Appl. Math. Mech. (Engl. Transl.) 57 (2), 301–310 (1993)].MathSciNetGoogle Scholar
  11. 11.
    E. K. Lavrovskii and A. M. Formal’skii, “Optimal Control of the Pumping and Damping of a Swing,” Prikl. Mat. Mekh. 57(2), 92–101 (1993) [J. Appl.Math. Mech. (Engl. Transl.) 57 (2), 311–320 (1993)].MathSciNetGoogle Scholar
  12. 12.
    V. M. Volosov and B. I. Morgunov, Averaging Method in the Theory of Nonlinear Oscillatory Systems (Izd-vo MGU, Moscow, 1971) [in Russian].Google Scholar
  13. 13.
    L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamlrelidze, and E. F. Mishchenko, Mathematical Theory of Optimal Processes (Nauka, Moscow, 1969) [in Russian].Google Scholar
  14. 14.
    I. G. Malkin, Theory of Motion Stability (Nauka, Moscow, 1966) [in Russian].zbMATHGoogle Scholar

Copyright information

© Allerton Press, Inc. 2012

Authors and Affiliations

  1. 1.Korolyov Samara State Aerospace UniversitySamaraRussia

Personalised recommendations