Harmonic thickness vibrations of inhomogeneous elastic layers with curved boundaries

Abstract

The thickness vibrations of elastic inhomogeneous bodies of different geometry under dynamic harmonic loading are studied. The dependence of the amplitude-frequency characteristics of homogeneous and inhomogeneous bodies on excitation frequency is analyzed in detail. The frequency spectra for plane, cylindrical, and spherical layers are determined Introduction. Lame's static and dynamic problems for elastic homogeneous (and piecewise-homogeneous) cylinders and spheres have exact analytic solutions [1, etc.]. However, engineering practice deals with continuously inhomogeneous layers (plane, cylindrical, spherical, etc.). Such inhomogeneities are caused by various structural, technological, operational, climatic, and other factors. This issue for cylindrical bodies was discussed in [4][5][6][8][9][10][11]. To study the effect of various radial inhomogeneities on the frequency and amplitude-frequency characteristics (AFCs) of elastic layers with curved boundaries undergoing harmonic vibrations, we need not only efficient mathematical and computational algorithms for solving boundary-value problems of elasticity, but also a certain strategy for physical analysis of the solution found.Here we present a unified approach to the analysis of the harmonic vibrations of a plate, a cylinder, and a sphere with transverse anisotropy and inhomogeneity. We will use the developed algorithm to compare the AFCs and natural frequencies of inhomogeneous bodies of different geometry.1. Problem Statement. To study the ADCs of inhomogeneous elastic plane, cylindrical, and spherical layers undergoing thickness vibrations, we start with the equations of motion

Comparative analysis of the electroelastic thickness vibrations of layers with curved boundaries

“…With more complicated transformations, the Hamiltonian formalism was generalized to the equations of elasticity and electroelasticity in cylindrical coordinates [10, 11, etc. ] and spherical coordinates (for centrosymmetric problems) [4].…”