Direct and Inverse Spectral Problems in the Theory of Oscillations of Elastic Plates with Additional Point Interactions
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DIRECT AND INVERSE SPECTRAL PROBLEMS IN THE THEORY OF OSCILLATIONS OF ELASTIC PLATES WITH ADDITIONAL POINT INTERACTIONS N. F. Valeev and E. A. Nazirova
UDC 517.4; 519.71
Abstract. This paper is devoted to a new statement and the study of direct and inverse spectral problems for small linear oscillations of orthotropic plates that carry concentrated masses at a finite set of points, which, in turn, are connected to a stationary base by elastic springs with known stiffness coefficients. Keywords and phrases: inverse spectral problem, differential operator with distribution coefficients, theory of oscillations of elastic plates and shells. AMS Subject Classification: 47E05, 34E05, 34L05
1. Introduction. Problems of linear oscillations of elastic plates and shells with concentrated masses and additional bracings arise in various branches of mechanics and engineering. A plate or a shell often acts as a supporting surface to which certain structural elements with relatively small sizes are attached. If a device operates under dynamical impacts, then the frequencies and modes of oscillations of a plate or a shell depend substantially on the elasticity coefficients and point masses of bracings. In this situation, problems of controlling the frequency-resonance characteristics of a device by available bracing parameters naturally arise. Namely, choosing the location, rigidity, and mass of bracings, we can model eigenfrequencies and normal modes of oscillations. We also note a similar problem of identification (or diagnostics) of parameters of additional constraints by a finite set of known eigenvalues. Problems of this type can be considered as inverse spectral problems with a finite set of spectral data consisting of a certain number of eigenvalues. In this case, the formulation and methods of investigation substantially differ from those for classical inverse spectral problems. In our opinion, it is appropriate to study such problems as multi-parameter spectral problems using appropriate methods and approaches (see [4, 7]). In this paper, we consider direct and inverse spectral problems for small linear oscillations of an orthotropic plate Ω, which carries concentrated masses mj at points Mj = Mj (xj , yj ) attached to a rigid base with elastic supports (springs) with force constants kj , j = 1, . . . , n, in a new formulation. The essence of the inverse spectral problem consists of determining force constants and concentrated masses by 2n known eigenfrequencies of a constrained plate. A mathematical model proposed is based on the bending model of a homogeneous plate under the assumption of the validity of the Kirchhoff–Love hypotheses . We consider boundary conditions of the following types: a free end, a fixed end, and a hinged support. These well-known boundary conditions are given by systems of boundary differential operators. It is convenient to describe the interaction of a plate with point masses and springs by using the Dirac δ-function; in this case, the degree of idealization of the model remains the same as in the transitio
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