Analysis of free oscillations of circular plates with variable thickness based on the symmetry method

Abstract

Based on symmetry and factorization methods, a general analytical solution to the fourth-order differential equation has been derived for a problem on the free axisymmetric oscillations of a circular plate of variable thickness. The law of the thickness change is the concave parabola h=H0(1–μρ)2, where µ is a constant coefficient that determines the degree of plate concaveness. The solution has been given by the Bessel functions of zero and the first order of the actual and imaginary argument. A circular ring plate has been considered whose inner contour is rigidly fixed and whose outer edge is free, for three values of the µ coefficient. We have determined the first three natural values for the problem (frequency numbers) and their natural functions (oscillation shapes). It has been shown that the natural frequencies of the first three shapes of oscillations decrease, with the increase in concaveness (increase in µ), to varying degrees, determined by the number of the frequency number λi (i=1, 2, 3). At µ=1.21417 and µ=1.39127, the frequencies decrease, compared to the case of µ=0.5985, by (1;1.3) %, (17.6;24) %, (22.85;30.35) %, respectively, for λ1, λ2, λ3. One can see a significant drop in frequency on the higher shapes of oscillations (λ2, λ3) and a slight drop in the basic shape (λ1). We have established the values and coordinates of extreme deflections (antinodes of oscillations) and the indicative coordinates of the nodal cross-sections. The reported numerical parameters, along with the frequency indicators, are a means of identifying the oscillational properties of a plate when it is studied in practice. We have built the graphic dependences for radial σr and tangential σθ cyclical stresses at the basic shape for each of the three variants of the concaveness of a parabolic plate. It has been established that the increase in the ratio of edge thickness, that is, concaveness, leads to an increase in σr in the cross-sections outside the end constraint. These stresses, which operate far from the free edge, for example at the end constraint or the area of the maximum σθ, are greater than σθ in varying degrees. Because of this, these stresses pose a major threat in terms of the cyclical strength of the plate when σr reaches destructive values. We have pointed to the possibility to provide, by increasing the concaveness of the parabolic plate, the optimal ratio between the value of σr at the end constraint and σr operating away from the fastening. This ratio, approximately equal to 1, is ensured at µ=1.39127 considered in the current work.