Green's function for frequency analysis of thin annular plates with nonlinear variable thickness

• Finding general forms of Green's function for uniform and non-uniform circular and annular plates.
• Obtaining solutions for plates with discrete elements and without discrete elements.
• Presenting ratios of basic frequencies of uniform and hyperbolic annular plates.

This study considers the free vibration analysis of homogeneous and isotropic annular thin plates with variable distributions of parameters by using the properties of Green's function and Neumann series. The general forms of Green's function depending on the Poisson ratio and the coefficient of distribution for the plate's flexural rigidity and thickness are obtained in closed-form. The fundamental solutions of differential Euler equations are expanded in the Neumann power series using the method of successive approximation based on the properties of integral equations. This approach allows us to obtain the nonlinear frequency equations as a power series that converges rapidly to exact eigenvalues for different power index values and Poisson ratio values. The Neumann power series can then be used to solve the boundary value problem for the free vibration of circular and annular plates with discrete elements, such as an additional mass or ring elastic support. Numerical solutions of the characteristic equations are presented for annular plates with constant and hyperbolic varying thickness, as well as different boundary conditions. The results obtained are compared with selected results from previous studies.