Advertisement

Automation and Remote Control

, Volume 80, Issue 11, pp 1996–2004 | Cite as

Stabilizing the Oscillations of a Controlled Mechanical System

  • V. N. TkhaiEmail author
Nonlinear Systems
  • 16 Downloads

Abstract

A mechanical system subjected to the action of positional forces and small smooth control is considered. It is assumed that in the absence of control, the system may have a family of single-frequency oscillations. A universal control—a nonlinear force that implements and simultaneously stabilizes a cycle in the system—is found. An example is given.

Keywords

mechanical system universal small smooth control natural stabilization 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Pontryagin, L.S., On Dynamic Systems That are Close to Hamiltonian Systems, Zh. Eksper. Teor. Fiz., 1934, vol. 4, no. 9, pp. 883–885.Google Scholar
  2. 2.
    Klimina, L.A., Iterative Method of Construction of a Bifurcation Diagram of Autorotation Motions for a System with One Degree of Freedom, AIP Conf. Proc., 2018, vol. 1959, no. 030011, pp. 30011–1–030011–5. doi: 10.1063/1.5034591CrossRefGoogle Scholar
  3. 3.
    Patidar, V., Sharma, A., and Purohit, G., Dynamical Behaviour of Parametrically Driven Duffing and Externally Driven Helmholtz-Duffing Oscillators under Nonlinear Dissipation, Nonlin. Dynam., 2016, vol. 83, nos. 1-2, pp. 375–388.MathSciNetCrossRefGoogle Scholar
  4. 4.
    Zaitsev, S., Shtempluck, O., and Gottlieb, E.B., Nonlinear Damping in a Micromechanical Oscillator, Nonlin. Dynam., 2016, vol. 67, no. 1, pp. 859–883.CrossRefGoogle Scholar
  5. 5.
    Shiriaev, A., Perram, J.W., and Canudas-de-Wit, C., Constructive Tool for Orbital Stabilization of Underactuated Nonlinear Systems: Virtual Constraints Approach, IEEE Transact. Autom. Control, 2005, vol. 50, no. 8, pp. 1164–1176.MathSciNetCrossRefGoogle Scholar
  6. 6.
    Tkhai, V.N., The Behaviour of the Period of Symmetrical Periodic Motions, J. Appl. Math. Mech., 2012, vol. 76, no. 4, pp. 446–450.MathSciNetCrossRefGoogle Scholar
  7. 7.
    Malkin, I.G., Nekotorye zadachi teorii nelineinykh kolebanii (Some Problems in the Theory of Nonlinear Oscillations), Moscow: Gostekhizdat, 1956.Google Scholar
  8. 8.
    Tkhai, V.N., Stabilizing the Oscillations of an Autonomous System, Autom. Remote Control, 2016, vol. 77, no. 6, pp. 972–979.MathSciNetCrossRefGoogle Scholar
  9. 9.
    Tkhai, V.N., The Stability of Regular Grioli Precessions, J. Appl. Math. Mech., 2000, vol. 64, no. 5, pp. 811–819.MathSciNetCrossRefGoogle Scholar
  10. 10.
    Tkhai, V.N., Dissipation in the Vicinity of an Oscillation of the Mechanical System, AIP Conf. Proc., 2018, vol. 1959, no. 030022, pp. 030022–1–030022–5.CrossRefGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2019

Authors and Affiliations

  1. 1.Trapeznikov Institute of Control SciencesRussian Academy of SciencesMoscowRussia

Personalised recommendations