Skip to main content
SpringerLink
Log in

Modeling and Optimizing the Turn of a Loaded Elastic Rod Controlled by an Electric Drive

  • CONTROL IN SYSTEMS WITH DISTRIBUTED PARAMETERS
  • Published:
Journal of Computer and Systems Sciences International Aims and scope

Abstract

In this study, we consider a flat rotation of an electric rectilinear rod controlled by an electric drive, loaded at the free end by a point mass. The problem is to optimally control the voltage in the motor’s winding, which transfers the system to the final state with the damping of the elastic vibrations. A generalized formulation of the equations of the rod’s state and a method for approximating unknown displacements, momentum density, and bending moment are proposed. The procedure for minimizing the objective functional and regularizing the numerical error is described.

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

Fig. 1.
Fig. 2.
Fig. 3.
Fig. 4.
Fig. 5.

Similar content being viewed by others

REFERENCES

  1. J. L. Lions, Optimal Control of Systems Governed by Partial Differential Equations (Springer, New York, 1971).

    Book  Google Scholar 

  2. N. U. Ahmed and K. L. Teo, Optimal Control of Distributed Parameter Systems (North-Holland, New York, 1981).

    MATH  Google Scholar 

  3. A. G. Butkovsky, Distributed Control Systems (Elsevier, New York, 1969).

    Google Scholar 

  4. W. Krabs, Optimal Control of Undamped Linear Vibrations (Heldermann, Lemgo, 1995).

    MATH  Google Scholar 

  5. M. Gugat, “Optimal control of networked hyperbolic systems: evaluation of derivatives,” Adv. Modell. Optimiz. 7, 9–37 (2005).

    MathSciNet  MATH  Google Scholar 

  6. J. E. Lagnese, G. Leugering, and E. J. P. G. Schmidt, Modeling, Analysis and Control of Dynamic Elastic Multi-Link Structures (Birkhauser, Boston, 1984).

    MATH  Google Scholar 

  7. G. Leugering, “A domain decomposition of optimal control problems for dynamic networks of elastic strings,” Comput. Optimiz. Appl. 16, 5–29 (2000).

    Article  MathSciNet  Google Scholar 

  8. S. P. Banks, State-Space and Frequency-Domain Methods in the Control of Distributed Parameter Systems (Peregrinus, London, 1983).

    MATH  Google Scholar 

  9. R. F. Curtain and H. J. Zwart, An Introduction to Infinite-Dimensional Linear Systems Theory (Springer, New York, 1995).

    Book  Google Scholar 

  10. F. L. Chernousko, “Control of elastic systems by bounded distributed forces,” Appl. Math. Comput. 78, 103–110 (1996).

    MathSciNet  MATH  Google Scholar 

  11. M. Gerdts, G. Greif, and H. J. Pesch, “Numerical optimal control of the wave equation: optimal boundary control of a string to rest infinite time,” Math. Comput. Simul. 79, 1020–1032 (2008).

    Article  Google Scholar 

  12. A. I. Ovseevich and A. K. Fedorov, “Asymptotically optimal control for a simplest distributed system,” Dokl. Math. 95, 194–197 (2017).

    Article  Google Scholar 

  13. I. V. Romanov and A. S. Shamaev, “On the problem of precise control of the system obeying the delay string equation,” Autom. Remote Control 74, 1810 (2013).

    Article  MathSciNet  Google Scholar 

  14. I. V. Romanov and A. S. Shamaev, “On a boundary controllability problem for a system governed by the two-dimensional wave equation,” J. Comput. Syst. Sci. Int. 58, 105 (2019).

    Article  Google Scholar 

  15. R. W. Lewis, P. Nithiarasu, and K. N. Seetharamu, Fundamentals of the Finite Element Method for Heat and Fluid Flow (Wiley, Chichester, 2004).

    Book  Google Scholar 

  16. M. J. Balas, “Finite-dimensional control of distributed parameter systems by Galerkin approximation of infinite dimensional controllers,” J. Math. Anal. Appl. 114, 17–36 (1986).

    Article  MathSciNet  Google Scholar 

  17. P. D. Christofides, Nonlinear and Robust Control of PDE Systems: Methods and Applications to Transport-Reaction Processes (Birkhauser, Boston, 2001).

    Book  Google Scholar 

  18. T. Belytschko, Y. Y. Lu, and L. Gu, “Element-free Galerkin method,” Int. J. Numer. Methods Eng. 37, 229–256 (1994).

    Article  MathSciNet  Google Scholar 

  19. S. N. Atluri and T. Zhu, “A new meshless local Petrov-Galerkin (MLPG) approach in computational mechanics,” Comput. Mech. 22, 117–127 (1998).

    Article  MathSciNet  Google Scholar 

  20. P. B. Bochev and M. D. Gunzburger, Least-Squares Finite Element Methods (Springer, New York, 2009).

    MATH  Google Scholar 

  21. G. V. Kostin and V. V. Saurin, Dynamics of Solid Structures.Methods Using Integrodifferential Relations (De Gruyter, Berlin, 2018).

    MATH  Google Scholar 

  22. G. V. Kostin, “Modelling and optimization of controlled longitudinal motions for an elastic rod based on the Ritz method,” in Proceedings of the 2018 14th International Conference on Stability and Oscillations of Nonlinear Control Systems STAB (Pyatnitskiy’s Conference) (IEEE, Moscow, 2018), pp. 1–4. https://doi.org/10.1109/STAB.2018.8408369

  23. L. G. Donnell, Beams, Plates and Shells (McGraw-Hill, New York, 1976).

    MATH  Google Scholar 

  24. G. V. Kostin, “Dynamics of controlled rotations of a loaded elastic link in a handling system with an electromechanical drive,” Izv. Akad. Nauk SSSR, Tekh. Kibern., No. 3, 146–153 (1990).

  25. M. G. Chilikin, V. I. Klyuchev, and A. S. Sandler, Theory of Automated Electric Drives (Energiya, Moscow, 1979) [in Russian].

    Google Scholar 

  26. M. Giaquinta and S. Hildebrandt, Calculus of Variations. I (Springer, Berlin, 2004).

    Book  Google Scholar 

Download references

Funding

The work was supported by the Russian Science Foundation, project no. 16-11-10343.

Author information

Authors and Affiliations

  1. Ishlinsky Institute for Problems in Mechanics, Russian Academy of Sciences, 119526, Moscow, Russia

    G. V. Kostin

Authors
  1. G. V. Kostin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to G. V. Kostin.

Additional information

Translated by A. Ivanov

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kostin, G.V. Modeling and Optimizing the Turn of a Loaded Elastic Rod Controlled by an Electric Drive. J. Comput. Syst. Sci. Int. 59, 504–517 (2020). https://doi.org/10.1134/S1064230720040103

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S1064230720040103

Search

Navigation