Analysis of free oscillations of round thin plates of variable thickness with a point support - Trapezon, Kirill; Trapezon, Alexandr; Orlov, Anatolii (Eastern-European Journal of Enterprise Technologies, 2020)
This paper reports the derived general analytical solution to the IV-order differential equation with variable coefficients for the problem on free axisymmetric oscillations of a circular plate of variable thickness. The plate thickness changes along the radius ρ in line with the parabolic law h=H0(1–µρ)2. When building a solution, the synthesis of the factorization method and the symmetry method was used. The factorization method has enabled us to represent the solution to the original IV-order equation as the sum of the solutions to the two respectively constructed II-order equations. The method of symmetry has produced precise solutions to these two equations. The problem on a point fixation of the plate has been considered as a boundary case of the problem on the rigid fixation of the inner contour of a circular plate whose ρ→0. To this end, the general solution has been transformed into the form that pre-meets the conditions on a rigid point support. The result of such a transformation is a simpler solution, with only two permanent integration variables instead of four. As a result, the frequency equation for a plate under any conditions on the outer contour is significantly simplified because it is derived from the second-order determinant. The frequency equation for a plate with a point support and with a free edge at µ=1.39127, which corresponds to the ratio of the limit thicknesses equal to 10.8, yielded the first five eigenvalues λi (i=1÷5). The oscillation shapes have been constructed as a graphic illustration for λi (i=1÷3). The numerical values of amplitude ratios have been given, as well as coordinates (relative radii) of the oscillation antinodes and nodal circles for each of the five oscillation shapes (i=1÷5). The derived numerical values of the oscillation parameters could in practice be used to initially identify the type of an oscillatory system and its possible characteristics for the case when a plate is fixed inside the inner contour of the small diameter. The criterial ratio of the fastening contour diameter to the plate diameter could serve the same purpose. If this ratio is equal to or less than 0.2, then it is permissible to assume that it is a point-type fastening. In this case, it is possible to calculate the circular plate oscillations with its internal contour fixed using the algorithm set out for a plate with a point support