Skip to main content
SpringerLink
Log in

Control for minimizing vibrations in systems with compliant elements

  • Control Systems of Moving Objects
  • Published:
Journal of Computer and Systems Sciences International Aims and scope

Abstract

A controlled mechanical system containing a compliant element is investigated. Such an element can be, for example, a compliant platform where the operated object (plant) is installed or an elastic gear that connects an engine with this movable object. The control (force or torque) is bounded in magnitude. Only the first (lowest) resonance frequency is taken into account. The system under examination has two degrees of freedom. The linear mathematical model of this system has a double zero eigenvalue and two complex eigenvalues. A control law for this system is designed that steers the plant from the given initial state to a given terminal state in finite time (assuming that there is no damping). The control is divided into intervals on which it varies linearly depending on time or is constant. The time intervals on which the control varies are equal to the period of the natural vibrations of the system. If there is no damping, this makes it possible to completely avoid vibrations on the time intervals where the control is constant.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. I. S. Dobrynina and F. L. Chernous’ko, “Bounded Control of a Fourth-Order Linear System,” Izv. Ross. Akad. Nauk, Teor. Sist. Upr., No. 4, 108–115 (1994).

  2. F. L. Chernous’ko and I. S. Dobrynina, “Constrained Control in a Mechanical System with Two Degrees of Freedom,” IUTAM Symp. on Optimization of Mechanical Systems, Ed. by D. Bestle and W. Schiehlen (Kluwer Acad. Publ, Stuttgart, 1995), pp. 57–64.

    Google Scholar 

  3. J.-J. Slotine and Li Weiping, Applied Nonlinear Control (Prentice-Hall, New Jersey, 1991).

    MATH  Google Scholar 

  4. V. G. Boltyanskii, Mathematical Methods in Optimal Control (Nauka, Moscow, 1969) [in Russian].

    Google Scholar 

  5. L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze, and E. F. Mishchenko, The Mathematical Theory of Optimal Processes (Fizmatlit; Interscience, New York, 1962).

    MATH  Google Scholar 

  6. Y. Aoustin and A. Formal’sky, “On the Synthesis of a Nominal Trajectory for Control Law of a One-Link Flexible Arm,” Int. J. Robotics Research 16(1), 36–46 (1997).

    Article  Google Scholar 

  7. A. De Luca and B. Siciliano, “Trajectory Control of a Non-Linear One-Link Flexible Arm,” Int. J. on Control 50(5), 1699–1715 (1989).

    Article  MATH  Google Scholar 

  8. L. D. Akulenko, “Quasistationary finite-interval control of the motion of hybrid oscillatory systems,” J. Appl. Math. Mech. 55, 145–152 (1991).

    Article  MathSciNet  MATH  Google Scholar 

  9. L. A. Shpigler, A. B. Gudzenko, and L. V. Gannel, “Restricting Elastic Vibrations of Manipulator Gears Using an Electrical Drive,” Stanki Instrument, No. 3, 9–11 (1990).

  10. F. Gravot, Y. Hirano, and S. Yoshizava, “Generation of “Optimal” Speed Profile for Motion Planning,” in Proc. of the Intelligent Robots and Systems, San-Digo, 2007.

  11. P. H. Meckl, “Optimized S-Curve Motion Profiles for Minimum Residual Vibration,” in Proc. of the American Control Conf., Philadelphia, 1998.

Download references

Author information

Authors and Affiliations

  1. Anorad (Rockwell Automation), Tel-Aviv, Israel

    L. V. Gannel

  2. Institute of Mechanics, Lomonosov Moscow State University, Moscow, Russia

    A. M. Formal’skii

Authors
  1. L. V. Gannel
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. M. Formal’skii
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. M. Formal’skii.

Additional information

Original Russian Text © L.V. Gannel, A.M. Formal’skii, 2013, published in Izvestiya Akademii Nauk. Teoriya i Sistemy Upravleniya, 2013, No. 1, pp. 122–134.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Gannel, L.V., Formal’skii, A.M. Control for minimizing vibrations in systems with compliant elements. J. Comput. Syst. Sci. Int. 52, 117–128 (2013). https://doi.org/10.1134/S1064230713010061

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S1064230713010061

Keywords

Search

Navigation