Abstract
The beam oscillations are modeled by the fourth-order hyperbolic partial differential equation. The minimized functional is the energy integral of an oscillating beam. Control is implemented via certain function appearing in the right side of the equation. It was shown that the solution of the problem exists for any given damping time, but with decreasing this time, finding the optimal control becomes more complicated. In this work, numerical damping of beam oscillations is implemented via several fixed point actuators. Computational algorithms have been developed on the basis of the matrix sweep method and the second order Marquardt minimization method. To find a good initial approximation empirical functions with a smaller number of variables are used. Examples of damping the oscillations via a different number of actuators are given. It is shown that the amplitude of the oscillations of any control functions increases with the reduction of the given damping time. Examples of damping the oscillations in the presence of constraints on control functions are given; in this case, the minimum damping time exists. The damping of oscillations is considered also in the case when different combinations of actuators are switched on at different time intervals of oscillation damping.
Supported by Russian Science Foundation, Project 17-19-01247.
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References
Lagness, J.: Control of wave process with distributed controls supported on a subregion. SIAM J. Control Optim. 1(1), 68–85 (1983)
Russel, D.: Controllability and stabilization theory for linear partial differential equations. SIAM Rev. 20(5), 639–739 (1978)
Butkovsky, A.G.: Metody upravleniia sistemami s raspredelennymi parametrami [Methods of controlling systems with distributed parameters]. Nauka, Moscow (1975)
Butkovsky, A.G.: Prilozhenie nekotorykh rezultatov teorii chisel k probleme finitnogo upravleniia i upravliaemosti v raspredelennykh sistemakh [Application of some results of number theory to the problem of finite control and controllability in distributed systems]. Proc. USSR Acad. Sci. 227(2), 309–311 (1976)
Muravey, L.: On the suppression on membrane oscillations. In: Summaries of IUTAM Symposium “Dynamical Problems of Rigid-elastic System”, Moscow, pp. 50–51 (1990)
Muravey, L.: Mathematical problems on the damp of vibration. In: Preprint of IFAC Conference “Identification and system parameter estimations”, Budapest, vol. 1, pp. 746–747 (1991)
Atamuratov, F., Mikhailov, I., Muravey, L.: The moment problem in control problems of elastic dynamic systems. Mechatron. Autom. Control 17(9), 587–598 (2016)
Atamuratov, A., Mikhailov, I., Taran, N.: Numerical damping of oscillations of beams by using multiple point actuators. Mathematical Modeling and Computational Physics: Book of Abstracts of the International Conference, p. 158. JINR, Dubna (2017)
Lions, J.L.: Optimal Control of Systems Governed by Partial Differential Equations. Springer, New York (1971)
Samarsky, A.A., Gulin, A.V.: Chislennye metody: Ucheb. posobie dlia vuzov [Numerical methods. Textbook for Universities]. Nauka, Moscow (1989)
Panteleev, A.V., Letova, T.A.: Metody optimizatsii v primerakh i zadachakh: Ucheb. Posobie [Optimization methods in examples and tasks. Textbook for Universities]. Vysshaia shkola, Moscow (2005)
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Atamuratov, A., Mikhailov, I., Taran, N. (2019). Numerical Damping of Forced Oscillations of an Elastic Beams. In: Evtushenko, Y., Jaćimović, M., Khachay, M., Kochetov, Y., Malkova, V., Posypkin, M. (eds) Optimization and Applications. OPTIMA 2018. Communications in Computer and Information Science, vol 974. Springer, Cham. https://doi.org/10.1007/978-3-030-10934-9_20
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DOI: https://doi.org/10.1007/978-3-030-10934-9_20
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