Optimal motions of an elastic rod controlled by a piezoelectric actuator

Capa

Citar

Texto integral

Acesso aberto Acesso aberto
Acesso é fechado Acesso está concedido
Acesso é fechado Somente assinantes

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.

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.

Arquivos suplementares

Arquivos suplementares
Ação
1. JATS XML
Baixar

Declaração de direitos autorais © Russian Academy of Sciences, 2024