On optimum design of a vibrating plate with respect to its thickness and eigen-frequencies

Igor Bock; Ján Lovíšek

Igor Bock Slovak University of Technology in Bratislava
Ján Lovíšek Slovak University of Technology in Bratislava

Abstract

The eigenvalue optimization problem for anisotropic plates has been dealt with. The variable thickness of a plate plays the role of a design variable. The state problem arises considering free vibrations of a plate. The demand of the lowest first eigenfrequency means the maximal first eigenvalue of the elliptic eigenvalue problem. The continuity and differentiability properties of the first eigenvalue have been examined. The existence theorem for the optimization problem has been stated and verified. The finite elements approximation has been analyzed. The shifted penalization and the method of nonsmooth optimization can be used in order to obtain numerical results.

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References

[1] I.Bock, J.Lovíšek. Optimal control of a viscoelastic plate bending with respect to a thickness. Math. Nachr., 125: 135- 151, 1986.
[2] LBock, J.Lovíšek. Optimal control problems for variational inequalities with controls in coefficients and in unilateral constraints. Application of Math., 32: 301-314, 1987.
[3] W. Findeisen, J. Szymanowski, A. Wierzbicki. Theory and numerical methods of optimization (in Polish). Polish Scientific Publisher, Warsaw, 1980.
[4] E.J. Haug, B. Rousselet. Design sensitivity analysis in structural mechanics. Eigenvalue variations. J. Structural Mech., 8: 161-186, 1980.
[5] I. Hlaváček, I. Bock, J.Lovíšek. Optimal control of a variational inequality with application to structural analysis II, III. Appl. Math. and Optim., 13: 117-139, 1985.