2018, no. 1

### OPTIMIZATION OF VIBRATION FREQUENCIES OF THE ELASTIC PLATE IN AN IDEAL FLUID

#### Kostikov Yu.A., Pavlov V.Y., Romanenkov A.M.

In this paper, we consider the problem of optimizing the vibration frequency of the elastic plate, which is completely immersed in the ideal liquid. The model of small oscillations of a plate in this case is an integrodifferential equation with the appropriate boundary conditions. This problem is considered as a model example in the book Banichuk N. B. . The peculiarity of this problem is a special kind of functional eqution, which satisfies the function of deflection of the plate from the equilibrium position. Due to the fact that the integral operator is self-adjoint, it was possible to obtain the necessary extremum conditions, on the basis of which an original numerical algorithm for optimizing the oscillation frequencies was developed. The search for the solution of the optimization problem is based on the method of gradient design, this work has been obtained exact formulas for finding the projection of the gradient.
Using the method of hydrodynamic potentials the problem of plate oscillation was reduced to the problem of beam oscillation. In work different ways of fixing of a beam on the ends were considered: hinged and rigid. The case of rigid restrictions imposed on the thickness of the plate was also investigated. For all considered methods of fixing numerical calculations which results are presented on the corresponding schedules were carried out.
The paper investigates the problem of optimizing the frequencies of an elastic plate oscillating in an ideal fluid. The formulation of the corresponding problem of hydroelasticity is given. By methods of the theory of functions of complex variable the solution of an external hydrodynamic problem is received and the forces acting from liquid on oscillating plate are defined. With the help of ideas and methods proposed in [1, 5], an integro-differential equation describing one-dimensional oscillations of a plate in a liquid is obtained. A formal mathematical formulation and study of the optimization problem are given. The numerical algorithm for determining the optimal forms and the results of calculations on the PC.

Keywords: vibrations of an elastic plate, the gradient projection.

UDC: 519.6

PACS: 04.50.Kd

DOI: 10.17238/issn2226-8812.2018.1.82-91