Постановка та дослідження задачі про оптимальне збудження коливань пластини

Abstract

A model problem of harmonic oscillations of a hinged plate, that is is under the influence of a certain number of point concentrated forces, is considered. The plate model is considered to satisfy Kirchhoff's conditions. The main task of the consideration is to determine the optimal characteristics of excitation - the number of forces, coordinates of their application, amplitudes and phases. The optimality criterion is constructed as the standard deviation of the complex deflections from a given profile function. With the given excitation characteristics, the problem of determining the vibrations is solved in the form of a superposition of the Green functions with singularities at the points of application of forces. The Green function is constructed as a Fourier series by a circular coordinate. By using Parseval equality in L2, the objective function of the optimization problem is represented as a combination of linear and Hermitian forms with respect to complex amplitudes of forces whose matrices are nonlinear (and not convex) dependent on the coordinates of singular points. A complete study of the objective function is performed. Sufficient conditions are determined for reducing the dimension of the control space by analytical determination of the amplitudes of forces. Expressions were obtained to calculate the gradients of the objective function by angular and radial coordinates. A partial case of grouping of excitation forces on concentric circles is considered, that leads to the degeneration of the problem.