What is it about?
We consider an eigenvalue problem for the biharmonic operator that describes the transverse vibrations of the plate. Under the imposed boundary conditions, the eigenvalues of this operator are indeed eigenfrequencies of the clamped plate. The domain of the plate is taken variable and the domain functional, involving an eigenfrequency, is studied. A new formula for an eigenfrequency is proved, the first variation of the functional with respect to the domain is calculated, and the necessary condition for an optimal shape is derived. New explicit formulas are obtained for the eigenfrequency in the optimal domain in some particular cases.
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Why is it important?
We consider the eigenfrequency of the plate under transverse vibration as a functional of its domain, investigate its properties on the form of the plate, formulate and solve shape optimization problem for the functional involving eigenfrequency. These problems have important practical applications since plates are important elements of many technical solutions.
Perspectives
We suggested a new method to investigate some type of shape optimization problems. And in this paper we applied this method to investigation of such problem for the eigenfrequency of the clamped plate under transverse vibrations. Hope the results will be interesting for the specialists.
Prof. Yusif Gasimov
Azerbaijan University
Read the Original
This page is a summary of: Shape optimization for the eigenfrequency of the plate, Georgian Mathematical Journal, January 2017, De Gruyter,
DOI: 10.1515/gmj-2017-0005.
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