ON THE VIBRATIONS OF INHOMOGENEOUS PIEZO DISC

Abstract

In the framework of the model of coupled electroelasticity of inhomogeneous bodies, the problem of steady-state oscillations of a thin piezodisc with inhomogeneous properties is considered, in particular, in the presence of radial polarization. The necessary simplifications are made within the framework of traditional hypotheses, the formulated boundary-value problem is reduced to a canonical system of first-order differential equations with respect to dimensionless components of radial displacement and radial stress with corresponding boundary conditions. The direct problem of oscillations of an inhomogeneous disk is solved numerically based on the shooting method by numerically analyzing auxiliary Cauchy problems. The analysis of the amplitude-frequency characteristics and resonance frequencies depending on various laws of variation of the inhomogeneous properties of the piezodisc is performed, which in the presented model are characterized by two functions, one of which characterizes the change in the elastic modulus, the second changes in the piezomodule. The inverse problem is formulated in the first statement, in which the laws of variation of the piezodisc heterogeneity (two functions) are restored from the values of the functions characterizing the radial displacement and stress, known in a finite set of points. The results of computational experiments on solving the inverse problem in the first formulation are presented, various aspects of reconstruction are discussed. The second formulation of the inverse problem is formulated to determine the piezoelectric characteristics of the disk, where a function that describes the laws of change in the elastic characteristics of the disk and the amplitude-frequency characteristic is considered known. To solve the inverse problem, in this formulation, the Fredholm integral equation of the first kind with a smooth kernel is formulated. The results of numerical experiments on solving the Fredholm integral equation of the first kind using the Tikhonov regularizing method are presented, various aspects of reconstruction are discussed.