# zbMATH — the first resource for mathematics

Mathematical methods to find optimal control of oscillations of a hinged beam (deterministic case). (English. Russian original) Zbl 07190727
Cybern. Syst. Anal. 55, No. 6, 1009-1026 (2019); translation from Kibern. Sist. Anal. 2019, No. 6, 145-164 (2019).
Summary: We consider several problem statements for the optimal controlled excitation of oscillations of a hinged beam. Oscillations occur under the influence of several external periodic forces. In the simplest statement, it is assumed that the structure of the beam is homogeneous. In a more complex formulation, inhomogeneities (defects) on the beam are allowed. The goal of controlling the oscillations of the beam is to provide a predetermined shape and a predetermined pointwise phase of oscillations in a given frequency range. The problem is to determine the number of forces and their characteristics (application, amplitude, and phase of oscillations), which provide the desired waveform with a given accuracy. With the help of analytical mathematical methods, the problems in question are reduced to simpler multiextremum problems of minimizing basic functionals, which can be numerically solved using the multifunctional package AORDA PSG.

##### MSC:
 49J20 Existence theories for optimal control problems involving partial differential equations 49M99 Numerical methods in optimal control 74K10 Rods (beams, columns, shafts, arches, rings, etc.) 74K30 Junctions
##### Keywords:
vibrations; waveform; optimal actuation
##### Software:
FEAPpv; Portfolio Safeguard; PSG
Full Text:
##### References:
 [1] G. M. Zrazhevsky, “Determination of the optimal parameters of the beam waveform actuation,” Bulletin of Taras Shevchenko National University of Kyiv, Ser. Physics & Matematics, Issue 3 (2013), pp. 138-141. URL: http://nbuv.gov.ua/UJRN/VKNU_fiz_mat_2013_3_34. [2] L. H. Donnell, Beams, Plates, and Shells, McGraw-Hill Book Co., New York (1976). [3] S. Timoshenko and S.Woinowsky-Krieger, Theory of Plates and Shells, McGraw-Hill Book Co., New York (1959). · Zbl 0114.40801 [4] O. C. Zienkiewicz and R. L. Taylor, The Finite Element Method for Solid and Structural Mechanics, Butterworth-Heinemann, Oxford (2005). · Zbl 1084.74001 [5] N. I. Akhiezer, Lectures on the Approximation Theory [in Russian], Nauka, Moscow (1965). [6] AORDA Portfolio Safeguard (PSG). URL: http://www.aorda.com/html/PSG_Help_HTML/index.html?bpoe.htm.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.