Most studies in the field of optimization problems in the mechanics of the deformable solid body are aimed at finding the optimal shape of structural elements, which most often leads to the search for the optimal distributions of geometric characteristics for the considered objects in order to ensure reducing the objects? weight and improve their strength characteristics. At the same time, with the advent of modern materials with variable physical and mechanical properties, new relevant branches arose in the optimization problems, as well as the inverse problems on the reconstruction of non-uniform variable characteristics. In this study, the problem of steady-state oscillations of a prestressed inhomogeneous round elastic plate in an axisymmetric formulation is considered. Variational and weak problem statements are presented. The case of free oscillations of a plate simply supported along the contour is investigated; it is shown that the eigenvalue problem is self-adjoint and quite definite. The problem of finding the optimal distribution of the plates? elastic modulus and the corresponding oscillation shape in order to maximize the first natural frequency is considered. An isoperimetric condition for the variable stiffness function is proposed. Based on the Rayleigh energy ratio, the minimum principle for the first eigenvalue and the corresponding eigenfunctionis formulated. On the basis of the variational principle, using the constructed Rayleigh ratio, the optimality condition is formulated, and an explicit representation is obtained for the functions of the optimal oscillation shape and optimal stiffness, taking into account the initial radial stretching or compression of the plate. The range of possible values for the initial stress is determined. The results of solving the problem via constructing optimal inhomogeneity laws for the stiffness function in the class of smooth functions in the considered range of changes in initial stresses are obtained and analyzed.
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