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Optimization of Longitudinal Motions of an Elastic Rod Using Periodically Distributed Piezoelectric Forces

  • SYSTEMS WITH DISTRIBUTED PARAMETERS
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Journal of Computer and Systems Sciences International Aims and scope

Abstract

The longitudinal vibrations of an elastic rod controlled by a distributed force, which is applied to individual sections of the rod, are studied. It is assumed that the force varies in space in a piecewise constant manner. Such a mechanical system can be implemented using piezoactuators attached along the rod. The dynamics of the system is determined from the solution of the variational problem following the method of integrodifferential relations. The variational problem is solved analytically. To do this, traveling waves of the d’Alembert type are introduced on the space-time mesh, which determine continuous displacements and a dynamic potential. The latter relates the momentum density and stresses. A control problem is posed under the condition of the weighted minimization of the vibrational energy stored by the rod at the terminal time instant, and the mean potential energy generated by the control actions. The extremal motion and the corresponding control law are found explicitly by solving the Euler–Lagrange equations. As an example, the control capabilities for certain configurations of piezoelectric elements are studied.

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REFERENCES

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

    Book  Google Scholar 

  2. A. G. Butkovskii, Theory of Optimal Control of Systems with Distributed Parameters (Nauka, Moscow, 1965) [in Russian].

    Google Scholar 

  3. 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 (1), 105–112 (2019).

    Article  MathSciNet  Google Scholar 

  4. F. L. Chernous’ko, I. M. Anan’evskii, and S. A. Reshmin, Control Methods for Nonlinear Mechanical Systems (Fizmatlit, Moscow, 2006) [in Russian].

    Google Scholar 

  5. G. Chen, “Control and stabilization for the wave equation in a bounded domain. II,” SIAM J. Control Optim. 19 (1), 114–122 (1981).

    Article  MathSciNet  Google Scholar 

  6. I. Kucuk, I. Sadek, and Y. Yilmaz, “Optimal control of a distributed parameter system with applications to beam vibrations using piezoelectric actuators,” J. Franklin Inst. 351 (2), 656–666 (2014).

    Article  MathSciNet  Google Scholar 

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

    Google Scholar 

  8. A. A. Gavrikov and G. V. Kostin, “Optimal control of longitudinal motion of an elastic rod using boundary forces,” J. Comput. Syst. Sci. Int. 60 (5), 740–755 (2021).

    Article  MathSciNet  Google Scholar 

  9. G. Kostin and A. Gavrikov, “Energy-optimal control by boundary forces for longitudinal vibrations of an elastic rod,” in Lecture Notes in Mechanical Engineering: Proceedings of the XLIX International Summer School-Conference “Advanced Problems in Mechanics”, 2021, St. Petersburg, Russia (Springer, 2023).

  10. G. Kostin and A. Gavrikov, “Controllability and optimal control design for an elastic rod actuated by piezoelements,” IFAC-PapersOnLine 55 (16), 350–355 (2022). https://doi.org/10.1016/j.ifacol.2022.09.049

    Article  Google Scholar 

  11. G. Kostin and A. Gavrikov, “Optimal motions of an elastic structure under finite-dimensional distributed control,” 2023. https://arxiv.org/pdf/2304.05765. https://doi.org/10.48550/arXiv.2304.05765.

  12. G. Kostin and A. Gavrikov, “Optimal Motion of an Elastic Rod Controlled by Piezoelectric Actuators and Boundary Forces,” in 16th Int. Conf. “Stability and Oscillations of Nonlinear Control Systems” (Pyatnitskii Conference) (STAB) (IEEE, Moscow, 2022), pp. 1–4. https://doi.org/10.1109/STAB54858.2022.9807484.

  13. G. Kostin and A. Gavrikov, “Modeling and optimal control of longitudinal motions for an elastic rod with distributed forces,” 2022. https://arxiv.org/pdf/2206.06139. https://doi.org/10.48550/arXiv.2206.06139.

  14. A. Gavrikov and G. Kostin, “Optimal LQR control for longitudinal vibrations of an elastic rod actuated by distributed and boundary forces,” in Mechanisms and Machine Science (Springer, Berlin, 2023), Vol. 125, pp. 285–295. https://doi.org/10.1007/978-3-031-15758-5_28

    Book  Google Scholar 

  15. L. F. Ho, “Exact controllability of the one-dimensional wave equation with locally distributed control,” SIAM J Control Optim. 28 (3), 733–748 (1990).

    Article  MathSciNet  Google Scholar 

  16. I. Bruant, G. Coffignal, F. Lene, and M. Verge, “A Methodology for determination of piezoelectric actuator and sensor location on beam structures,” J. Sound Vib. 243 (5), 861–882 (2001). https://doi.org/10.1006/jsvi.2000.3448

    Article  Google Scholar 

  17. V. Gupta, M. Sharma, and N. Thakur, “Optimization criteria for optimal placement of piezoelectric sensors and actuators on a smart structure: A technical review,” J. Intell. Mat. Syst. Struct. 21 (12), 1227–1243 (2010). https://doi.org/10.1177/1045389X10381659

    Article  Google Scholar 

  18. F. Botta, A. Rossi, and N. P. Belfiore, “A novel method to fully suppress single and bimodal excitations due to the support vibration by means of piezoelectric actuators,” J. Sound Vib. 510 (13), 116260 (2021). https://doi.org/10.1016/j.jsv.2021.116260

  19. A. N. Tikhonov and A. A. Samarskii, Equations of Mathematical Physics (Nauka, Moscow, 1977) [in Russian].

    Google Scholar 

  20. S. G. Mikhlin, Course of Mathematical Physics (Nauka, Moscow, 1968) [in Russian].

    Google Scholar 

  21. K. Yosida, Functional Analysis (Springer, Berlin, 1968; Mir, Moscow, 1968).

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Funding

This work was supported by the Russian Science Foundation, project no. 21-11-00151.

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Authors and Affiliations

  1. Ishlinsky Institute for Problems in Mechanics RAS (IPMech RAS), 119526, Moscow, Russia

    A. A. Gavrikov & G. V. Kostin

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  1. A. A. Gavrikov
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  2. G. V. Kostin
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Correspondence to G. V. Kostin.

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Gavrikov, A.A., Kostin, G.V. Optimization of Longitudinal Motions of an Elastic Rod Using Periodically Distributed Piezoelectric Forces. J. Comput. Syst. Sci. Int. 62, 800–816 (2023). https://doi.org/10.1134/S1064230723050064

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