Abstract

Proceeding from the general theory of steady-state vibrations of inhomogeneous prestressed bodies, in the present work the problem of bending vibrations of circular and annular inhomogeneous plates is considered within the framework of Timoshenko's hypotheses, taking into account the viscoelastic (rheological) properties of the material. The material rheology is described by the three-parameter viscoelastic Zener type model (also known as the Standard Linear Solid model) employing instantaneous and long-term constitutive moduli, as well as the relaxation time. For the formulation of the governing equations the Volterra correspondence principle and the concept of complex modules were used. For the both types of plates, a method is proposed for solving the corresponding direct (forward) problems for determining the vibrations using a weak formulation, based on the Galerkin method, and taking into account that the functions involved are complex-valued.

The proposed method is verified by a comparison of the results of calculating the plate deflection with the analytical solution in the case of homogeneous prestressed plates. The influence of the prestress level on the amplitude-frequency characteristics is analyzed in order to identify the most effective modes of acoustic sounding.

Furthermore, a new formulation of the inverse problem is proposed to identify the prestress in inhomogeneous viscoelastic plates using the information on the acoustic response of the plate. To solve the formulated inverse problem, a modification of the previously developed special projection approach is used, whose applicability is illustrated by a set of numerical experiments. The influence of input data noise on the prestress identification accuracy is also analyzed.

Citation details of the article

Journal: International Journal of Applied Mathematics
Journal ISSN (Print): ISSN 1311-1728
Journal ISSN (Electronic): ISSN 1314-8060
Volume: 36
Issue: 4
Year: 2023

DOI: 10.12732/ijam.v36i4.1

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