Optimal motions of an elastic rod controlled by a piezoelectric actuator

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Resumo

The longitudinal vibrations of an elastic rod controlled by normal forces in the cross section, which are uniformly distributed along the length over a selected interval, are studied. Such a system can be implemented using an actuator consisting of piezoelectric elements located along the axis of the rod. Criteria for the uncontrollability of individual vibration modes are given. A generalized solution to the initial-boundary value problem is found applying d’Alembert traveling waves, which are determined on the space-time mesh formed by characteristics. Linear combinations of the traveling wave and control functions define the sought displacements and dynamic potential in the energy space. The latter in a certain way relates the momentum density and the force in the cross section. The problem is to transfer the rod to a prescribed state in a fixed time while minimizing the norm of the control force. The optimal motion and the corresponding feedforward control law are found by reducing the original problem to a one-dimensional variational one. The example shows the control of vibrations for certain geometric parameters of the piezoelectric actuator.

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Sobre autores

G. Kostin

Ishlinsky Institute for Problems in Mechanics RAS

Autor responsável pela correspondência
Email: kostin@ipmnet.ru
Rússia, Moscow

Bibliografia

  1. Lions J.L. Optimal Control of Systems Governed by Partial Differential Equations. N.Y.: Springer-Verlag, 1971. 400 p.
  2. Бутковский А.Г. Теория оптимального управления системами с распределенными параметрами. М.: Наука, 1965. 474 c.
  3. Романов И.В., Шамаев А.С. О задаче граничного управления для системы, описываемой двумерным волновым уравнением // Изв. РАН. ТиСУ. 2019. № 1. C. 109–116.
  4. Черноусько Ф.Л. Ананьевский И.М., Решмин С.А. Методы управления нелинейными механическими системами. М.: Физматлит, 2006. 328 с.
  5. Chen G. Control and Stabilization for the Wave Equation in a Bounded Domain. II // SIAM J. Control Optim. 1981. V. 19. № 1. P. 114–122.
  6. Гавриков А.А., Костин Г.В. Изгибные колебания упругого стержня, управляемого пьезоэлектрическими силами // ПММ. 2023. Т. 87. № 5. С. 801–819.
  7. IEEE Standard on Piezoelectricity // ANSI/IEEE Std 176-1987. 1988. https://doi.org/10.1109/IEEESTD.1988.79638
  8. Kucuk I., Sadek I., Yilmaz Y. Optimal Control of a Distributed Parameter System with Applications to Beam Vibrations Using Piezoelectric Actuators // J. Franklin Inst. 2014. V. 351. № 2. P. 656–666.
  9. Kostin G.V., Saurin V.V. Dynamics of Solid Structures. Methods Using Integrodifferential Relations. Berlin: De Gruyter, 2018.
  10. Kostin G., Gavrikov A. Controllability and Optimal Control Design for an Elastic Rod Actuated by Piezoelements // IFAC-PapersOnLine. 2022. V. 55. № 16. P. 350–355. https://doi.org/10.1016/j.ifacol.2022.09.049
  11. Гавриков А.А., Костин Г.В. Оптимизация продольных движений упругого стержня с помощью периодически распределенных пьезоэлектрических сил // Изв. РАН. ТиСУ. 2023. № 6. С. 93–109.
  12. Kostin G., Gavrikov A. Modeling and Optimal Control of Longitudinal Motions for an Elastic Rod with Distributed Forces // ArXiv. 2022. arXiv:2206.06139 5. P. 1–11. https://doi.org/10.48550/arXiv.2206.06139
  13. Gavrikov A., Kostin G. Optimal LQR Control for Longitudinal Vibrations of an Elastic Rod Actuated by Distributed and Boundary Forces // Mechanisms and Machine Science. V. 125. Berlin: Springer, 2023. P. 285–295. https://doi.org/10.1007/978-3-031-15758-5_28
  14. Ho L.F. Exact Controllability of the One-dimensional Wave Equation with Locally Distributed Control // SIAM J Control Optim. 1990. V. 28. № 3. P. 733–748.
  15. Bruant I., Coffignal G., Lene F., Verge M. A Methodology for Determination of Piezoelectric Actuator and Sensor Location on Beam Structures // J. Sound and Vibration. 2001. V. 243. № 5. P. 861–882. https://doi.org/10.1006/jsvi.2000.3448
  16. Gupta V., Sharma M., Thakur N. Optimization Criteria for Optimal Placement of Piezoelectric Sensors and Actuators on a Smart Structure: A Technical Review // J. Intelligent Material Systems and Structures. 2010. V. 21. № 12. P. 1227–1243. https://doi.org/10.1177/1045389X10381659
  17. Botta F., Rossi A., Belfiore N.P. A Novel Method to Fully Suppress Single and Bi-modal Excitations Due to the Support Vibration by Means of Piezoelectric Actuators // J. Sound and Vibration. 2021. V. 510. № 13. P. 116260. https://doi.org/10.1016/j.jsv.2021.116260
  18. Тихонов А.Н. Самарский А.А. Уравнения математической физики. М.: Наука, 1977. 735 с.
  19. Михлин С.Г. Курс математической физики. М.: Наука, 1968. 576 с.
  20. Иосида К. Функциональный анализ. М.: Мир, 1968. 624 с.

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2. Fig. 1. Diagram of the rod with a control piezoelectric actuator.

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3. Fig. 2. Grid on the space-time domain D for the system parameters: x- = 0, l = 1/4, T = 7/3.

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4. Fig. 3. Optimal control u(t).

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5. Fig. 4. Optimal force generated by the piezoelectric actuator.

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6. Fig. 5. Optimal dynamic potential of the rod v(t, x).

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7. Fig. 6. Optimal displacements of the rod points w(t, x).

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8. Fig. 7. Optimal values of the objective functional J depending on the control time T for different PA locations at x- = j /4, where j = 0, 1, 2, 3.

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