Green's function in frequency analysis of circular thin plates of variable thickness

Abstract

Free vibration analysis of homogeneous and isotropic circular thin plates with variable distribution of parameters by using Green's functions (solution to homogeneous ordinary differential equations with variable coefficients) is considered. The formula of Green's function (called the influence function) depends on the Poisson ratio and the coefficient of distribution of plate flexural rigidity, and the thickness is obtained in a closed-form. The limited independent solutions to differential Euler equations are expanded in the Neumann power series using the Volterra integral equations of the second kind. This approach allows one to obtain the analytical frequency equations as the power series rapidly convergens to exact eigenvalues for different values of the power index and different values of the Poisson ratio. The six lower natural dimensionless frequencies of axisymmetric vibration of circular plates of constant and variable thickness are calculated for different boundary conditions. The obtained results are compared with selected results presented in the literature.

Analytical Bending Solutions of Orthotropic Rectangular Thin Plates with Two Adjacent Edges Free and the Others Clamped or Simply Supported Using Finite Integral Transform Method

“…e existing solutions are very few, and the solution procedure is much more difficult which needs a thorough knowledge of mathematics and mechanics. Recently, a Green's function approach is utilized to solve the free vibration problems of circular thin plates [19][20][21]. is approach allows obtaining the analytical frequency equations as power series fast convergent to exact eigenvalues for different number of nodal diameters.…”

Investigation of the Dynamic Behaviour of Non-Uniform Thickness Circular Plates Resting on Winkler and Pasternak Foundations

“…Meanwhile, the HPM also suffers the setback of finding the embedded parameter and initial approximation of the governing equation that satisfies the given conditions. Nonetheless, several researches on a free vibration of circular plates using different methods have been presented in the literature [11][12][13][14][15][16][17][18][19][20][21][22][23][24][25]. Moreover, the reliability and flexibility of the Galerkin method of weighted residual [26] have made it more effective than any other semi-numerical method.…”

“…Figure23. Influence of density variation on deflection of circular plate for free surface of foundation.…”

Thermodynamic Response of Beams on Winkler Foundation Irradiated by Moving Laser Pulses

“…The methods of obtaining of the specific diverse Green's functions for structural members with homogeneous or nonhomogeneous material, uniform or non-uniform thickness, additional discrete elements, etc., have been reported. For example,Żur [15][16][17][18][19] presented a series of work over the free vibration analysis of thin circular plates and elastically supported functionally graded annular plates using the Green's functions. Zhao et al [20][21][22] analytically studied the vibration of a cracked Euler-Bernoulli beam induced by a heat flux or a harmonic force and that of Timoshenko beams due to a heat flux together with an external load.…”