Optimization of Longitudinal Motions of an Elastic Rod Using Periodically Distributed Piezoelectric Forces
- Авторлар: Gavrikov A.A.1, Kostin G.V.1
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Мекемелер:
- Ishlinsky Institute for Problems in Mechanics RAS (IPMech RAS), 119526, Moscow, Russia
- Шығарылым: № 6 (2023)
- Беттер: 93-109
- Бөлім: CONTROL OF SYSTEMS WITH DISTRIBUTED PARAMETERS
- URL: https://journals.eco-vector.com/0002-3388/article/view/676453
- DOI: https://doi.org/10.31857/S0002338823050062
- EDN: https://elibrary.ru/OHEOEA
- ID: 676453
Дәйексөз келтіру
Аннотация
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.
Авторлар туралы
A. Gavrikov
Ishlinsky Institute for Problems in Mechanics RAS (IPMech RAS), 119526, Moscow, Russia
Email: kostin@ipmnet.ru
Россия, Москва
G. Kostin
Ishlinsky Institute for Problems in Mechanics RAS (IPMech RAS), 119526, Moscow, Russia
Хат алмасуға жауапты Автор.
Email: kostin@ipmnet.ru
Россия, Москва
Әдебиет тізімі
- Lions J.L. Optimal Control of Systems Governed by Partial Differential Equations. N.Y.: Springer-Verlag, 1971. 400 p.
- Бутковский А.Г. Теория оптимального управления системами с распределенными параметрами. М.: Наука, 1965. 474 с.
- Романов И.В., Шамаев А.С. О задаче граничного управления для системы, описываемой двумерным волновым уравнением // Изв. РАН. ТиСУ. 2019. № 1. С. 109–116.
- Черноусько Ф.Л., Ананьевский И.М., Решмин С.А. Методы управления нелинейными механическими системами. М.: Физматлит, 2006. 328 с.
- 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.
- 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.
- Kostin G.V., Saurin V.V. Dynamics of Solid Structures. Methods Using Integrodifferential Relations. Berlin: De Gruyter, 2018.
- Гавриков А.А., Костин Г.В. Оптимальное управление продольным движением упругого стержня с помощью граничных сил // Изв. РАН. ТиСУ. 2021. № 5. С. 74–90.
- Kostin G., Gavrikov A. Energy-Optimal Control by Boundary Forces for Longitudinal Vibrations of an Elastic Rod // Lecture Notes in Mechanical Engineering Advanced Problems in Mechanics III. Springer, 2023.
- 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
- Kostin G., Gavrikov A. Optimal Motions of an Elastic Structure Under Finite-dimensional Distributed Control // ArXiv. 2023. arXiv:2304.05765. P. 1–17. https://doi.org/10.48550/arXiv.2304.05765.
- Kostin G., Gavrikov A. Optimal Motion of an Elastic Rod Controlled by Piezoelectric Actuators and Boundary Forces // 16th Intern. Conf. “Stability and Oscillations of Nonlinear Control Systems” (Pyatnitskiy’s Conference) (STAB). M.: IEEE, 2022. P. 1–4. https://doi.org/10.1109/STAB54858.2022.9807484.
- 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.
- 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
- 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.
- 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
- 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
- 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
- Тихонов А.Н., Самарский А.А. Уравнения математической физики. М.: Наука, 1977. 735 с.
- Михлин С.Г. Курс математической физики. М.: Наука, 1968. 576 с.
- Иосида К. Функциональный анализ. М.: Мир, 1968. 624 с.
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