Identification of properties of the inhomogeneous viscoelastic circular plate

Abstract

In the present research, we construct a model of significantly inhomogeneous viscoelastic circular plate clamped along its contour. We consider the steady vibrations of the plate caused by a load distributed on a circumference of a certain radius at the plate's surface. To take into account the damping effect of viscoelastic materials, we use the standard model of viscoelastic body based on the theory of complex modules. To simulate the plate's deformation behavior, we use the Kirchhoff-Love hypothesis. With the use of the variational principle of Hamilton-Ostrogradskii, we derive the wave equation and the boundary conditions for the plate. Two inverse problems on the identification of the instant and long modules included in the complex modulus are formulated. The first inverse problem uses the additional data on the given function of frequency flexure of the plate. The second one uses the measured function values at a single point for some definite set of frequencies. In the second section by methods for solving formulated direct and inverse problems are described. The first inverse problem is linear, and it is solved using the Galerkin method; a specific set of basis functions is selected for that. To solve the second problem which is significantly nonlinear and ill-posed, we develop a special iterative procedure based on the linearization method combining the use of the Galerkin method to solve direct problems at its every step and the solution of the systems of Fredholm's integral equations of the 1st kind; to regularize the latter, we employed the Tikhonov regularization method. The third section of proposed of approaches are illustrated by a representative set of computational experiments where both monotonic and non-monotonic functions are reconstructed. The error does not exceed 6% in all of the experiments conducted indicating that the proposed approaches are efficient enough in solving such problems.