Simultaneous identification of mechanical properties of functionally graded plates under the Kirchhoff and Tymoshenko models

Abstract

The paper presents models of deformation of functional-gradient round plates in the framework of the hypotheses of Kirchhoff and Timoshenko. Based on the equations of oscillations and boundary conditions obtained earlier using the Hamilton - Ostrogradsky variational principle, the formulations of problems in a cylindrical coordinate system are written, taking into account the variability of the functions of cylindrical stiffness and density along the radial coordinate. The plates was considered rigidly clamped along the edge. The case of steady-state vibrations caused by a load applied to the surface was considered. A scheme for solving direct problems calculating vibrations of plates based on the Galerkin method is constructed. An analysis of the influence of the functions of cylindrical stiffness and density on the amplitude-frequency characteristics (AFC, acoustic response) was carried out, which revealed that both functions significantly affect the AFC, and the greatest influence is observed in the vicinity of resonant frequencies. The results of the analysis made it possible to formulate new inverse problems of simultaneous identification of the functions of cylindrical stiffness and density of an inhomogeneous circular plate using additional information about the acoustic response for both hypotheses. To solve them, a special projection technique was built, based on the expansion of unknown functions of mechanical characteristics, as well as dynamic quantities (functions of deflection and angle of rotation of the normal) in terms of some systems of linearly independent functions that satisfy the boundary conditions. The coefficients of these expansions are determined from the solution of special systems of linear and nonlinear equations obtained from the formulated weak statements of both problems. As a result, it is possible to identify the desired characteristics in the given classes of functions. The identification results are illustrated with a set of computational experiments for various functions.