Abstract
We consider a problem of a modulating mirror fixed on active supports. It is assumed that the mirror may have several defects. The problem is to find optimal locations of supports and control forces that provide the best approximation of a given shape and phase of the oscillations for a homogeneous mirror as well as a plate with defects that have definite geometric and mechanical characteristics. The model of the Kirchhoff plate is chosen to describe the mirror. Defects are modeled by small inhomogeneities with changed elastic characteristics. An iterative technique for modeling finite-size defects in the Kirchhoff plate by point quadrupoles is developed. Isolated active supports are modeled by point forces. The optimization parameters are the location of the supports and the amplitudes and phases of the forces that generate vibrations. As an optimality criterion, the minimum of the root-mean-square deviation of the waveform of the plate from the given pattern is used.
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G. Zrazhevsky, A. Golodnikov, and S. Uryasev, “Mathematical methods to find optimal control of oscillations of a hinged beam (deterministic case),” Cybern. Syst. Analysis, Vol. 55, No. 6, 1009–1026 (2019). https://doi.org/10.1007/s10559-019-00211-x.
G. Zrazhevsky and V. Zrazhevska, “Formulation and study of the problem of optimal excitation of plate oscillations,” Bulletin of Taras Shevchenko National University of Kyiv, Ser. Physics and Mathematics, Vol. 1, 62–65 (2019). https://doi.org/10.17721/1812-5409.2019/1.12.
A. M. Khludnev, “On thin inclusions in elastic bodies with defects,” Z. Angew. Math. Phys., Vol. 70, Iss. 2, Article 45 (2019). https://doi.org/10.1007/s00033-019-1091-5.
G. Zrazhevsky and V. Zrazhevska, “Usage of generalized functions formalism in modeling of defects by point singularity,” Bulletin of Taras Shevchenko National University of Kyiv, Ser. Physics and Mathematics, Vol. 1, 58–61 (2019). https://doi.org/10.17721/1812-5409.2019/1.12.
G. Zrazhevsky and V. Zrazhevska, “Modeling of finite inhomogeneities by discrete singularities,” Journal of Numerical and Applied Mathematics, Vol. 1(135), 138–143 (2021). https://doi.org/10.17721/2706-9699.2021.1.18.
G. Zrazhevsky and V. Zrazhevska, “The extension method for solving boundary value problems of theory of oscillations bodies with heterogeneity,” World Journal of Engineering Research and Technology, Vol. 6, Iss. 2, 503–514 (2020).
L. H. Donnell, Beams, Plates, and Shells, McGraw-Hill Book Company, New York (1976).
C. L. Lawson and R. J. Hanson, Solving Least Squares Problems, Prentice-Hall, Englewood Cliffs (1974).
AORDA Portfolio Safeguard (PSG). URL: http://www.aorda.com.
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*This paper presents the results obtained in the project “Application of Buffered Probability of Exceedance (BPOE) to Structural Reliability Problems” supported by the European Office of Aerospace Research and Development. Grant EOARD #16IOE094.
Translated from Kibernetyka ta Systemnyi Analiz, No. 5, September–October, 2022, pp. 37–47.
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Zrazhevsky, G., Zrazhevska, V. & Golodnikov, O. Developing a Model for a Modulating Mirror Fixed on Active Supports. Deterministic Problem*. Cybern Syst Anal 58, 702–712 (2022). https://doi.org/10.1007/s10559-022-00503-9
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DOI: https://doi.org/10.1007/s10559-022-00503-9