OPTIMA 2018: Optimization and Applications pp 277-290

# Numerical Damping of Forced Oscillations of an Elastic Beams

• Andrey Atamuratov
• Igor Mikhailov
• Nikolay Taran
Conference paper
Part of the Communications in Computer and Information Science book series (CCIS, volume 974)

## 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.

## Keywords

Marquardt minimization method Oscillations damping Fixed point actuators Matrix sweep method

## References

1. 1.
Lagness, J.: Control of wave process with distributed controls supported on a subregion. SIAM J. Control Optim. 1(1), 68–85 (1983)
2. 2.
Russel, D.: Controllability and stabilization theory for linear partial differential equations. SIAM Rev. 20(5), 639–739 (1978)
3. 3.
Butkovsky, A.G.: Metody upravleniia sistemami s raspredelennymi parametrami [Methods of controlling systems with distributed parameters]. Nauka, Moscow (1975)Google Scholar
4. 4.
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)Google Scholar
5. 5.
Muravey, L.: On the suppression on membrane oscillations. In: Summaries of IUTAM Symposium “Dynamical Problems of Rigid-elastic System”, Moscow, pp. 50–51 (1990)Google Scholar
6. 6.
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)Google Scholar
7. 7.
Atamuratov, F., Mikhailov, I., Muravey, L.: The moment problem in control problems of elastic dynamic systems. Mechatron. Autom. Control 17(9), 587–598 (2016)Google Scholar
8. 8.
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)Google Scholar
9. 9.
Lions, J.L.: Optimal Control of Systems Governed by Partial Differential Equations. Springer, New York (1971)
10. 10.
Samarsky, A.A., Gulin, A.V.: Chislennye metody: Ucheb. posobie dlia vuzov [Numerical methods. Textbook for Universities]. Nauka, Moscow (1989)Google Scholar
11. 11.
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)Google Scholar

© Springer Nature Switzerland AG 2019

## Authors and Affiliations

• Andrey Atamuratov
• 1
• Igor Mikhailov
• 1
• 2
• Nikolay Taran
• 1
Email author
1. 1.Moscow Aviation Institute (National Research University)MoscowRussia
2. 2.Federal Research Center “Informatics and Control” of RASMoscowRussia