Exact solutions for axisymmetric flexural free vibrations of inhomogeneous circular Mindlin plates with variable thickness

Abstract

Circular plates with radially varying thickness, stiffness, and density are widely used for the structural optimization in engineering. The axisymmetric flexural free vibration of such plates, governed by coupled differential equations with variable coefficients by use of the Mindlin plate theory, is very difficult to be studied analytically. In this paper, a novel analytical method is proposed to reduce such governing equations for circular plates to a pair of uncoupled and easily solvable differential equations of the Sturm-Liouville type. There are two important parameters in the reduced equations. One describes the radial variations of the translational inertia and flexural rigidity with the consideration of the effect of Poisson's ratio. The other reflects the comprehensive effect of the rotatory inertia and shear deformation. The Heun-type equations, recently well-known in physics, are introduced here to solve the flexural free vibration of circular plates analytically, and two basic differential formulae for the local Heun-type functions are discovered for the first time, which will be of great value in enriching the theory of Heun-type differential equations.

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

“…The reason, of course, is not the lack of interest in the possibility of solving the problem but apparently is the difficulties of a mathematical nature, in the absence of a correct solution method. The works cited above [14,15] are indicative in this respect. These studies apply a "new approach" to achieve the goals stated, the essence of which is to treat absolutely arbitrarily the elastic and physical constants of a material, considering these constants to be variables.…”

“…Our analysis of current publications [1,2,[9][10][11][12][13][14][15] leads to the conclusion that solving the problem about bending oscillations of circular plates is in high demand for research-ers and specialists in applied acoustics and mechanics. This judgment is supported by the presence of a huge number of technical applications of circular plates and various ways to solve the relevant eigenvalue problem.…”

“…This judgment is supported by the presence of a huge number of technical applications of circular plates and various ways to solve the relevant eigenvalue problem. However, most of the methods and algorithms considered in [1,2,[9][10][11][12][13][14][15] are specific and focus on iterative numerical approaches. In addition, the results reported in the cited papers typically refer to plates of constant thickness.…”

“…In article [15], the authors considered a plate with variable thickness, stiffness, and density using the parameter p. However, it should be noted that the authors artificially create in their study a function of changing the E elasticity module, density ρ, thickness h along the radius of the plate: E=E 0 (1+p(r/R) 2 ) n+0, 5 ; ρ=ρ 0 (1+p(r/R) 2 ) n-0,5 ; h=h 0 (1+p(r/R) 2 ) 0,5 . These parameters are selected in such a way that they correspond to the original equation of the oscillation shapes.…”

“…Paper [14] introduces a "material's coefficient", variable along the radius of the plate, instead of the Poisson coefficient. Work [15] introduced the elasticity module E(r) and the density ρ(r), variable along the radius. As a result of this "method", work [15], for example, "converts" the constant thickness of the plate into a variable and a simple problem turns into a problem about the oscillations of a plate of variable thickness.…”