Analytical solution to the problem about free oscillations of a rigidly clamped circular plate of variable thickness

Abstract

This paper reports an analytical solution to one of the problems related to applied mechanics and acoustics, which tackles the analysis of free axisymmetric bending oscillations of a circular plate of variable thickness. A plate rigidly-fixed along the contour has been considered, whose thickness changes by parabola h(ρ)=H 0 (1+µρ) 2 . For the initial assessment of the effect exerted by coefficient μ on the results, the solutions at μ=0 and some μ≠0 have been investigated. The differential equation of the shapes of a variable-thickness plate's natural oscillations, set by the h(ρ) function, has been solved by a combination of factorization and symmetry methods. First, a problem on the oscillations of a rigidly-fixed plate of the constant thickness (μ=0), in which h(1)/h(0)=η=1, was solved. The result was the computed natural frequencies (numbers λ i at i=1...6), the constructed oscillation shapes, as well as the determined coordinates of the nodes and antinodes of oscillations. Next, a problem was considered about the oscillations of a variable-thickness plate at η=2, which corresponds to μ=0.4142. Owing to the symmetry method, an analytical solution and a frequency equation for η=2 were obtained when the contour is rigidly clamped. Similarly to η=1, the natural frequencies were calculated, the oscillation shapes were constructed, and the coordinates of nodes and antinodes of oscillations were determined. Mutual comparison of frequencies (numbers λ i ) shows that the natural frequencies at η=2 for i=1...6 increase significantly by (28...19.9) % compared to the case when η=1. The increase in frequencies is a consequence of the increase in the bending rigidity of the plate at η=2 because, in this case, the thickness in the center of both plates remains unchanged, and is equal to h=H 0 . The reported graphic dependences of oscillation shapes make it possible to compare visually patterns in the distribution of nodes and antinodes for cases when η=1 and η=2. Using the estimation formulae derived from known ratios enabled the construction of the normalized diagrams of the radial σ r and tangential σ θ normal stresses at η=1 and η=2. Mutual comparison of stresses based on the magnitude and distribution character has been performed. Specifically, there was noted a more favorable distribution of radial stresses at η=2 in terms of strength and an increase in technical resource