Green’s function approach to frequency analysis of thin circular plates

Abstract

Abstract.The free vibration analysis of homogeneous and isotropic circular thin plates by using the Green's functions is considered. The formulae for construction of the influence function for all nodal diameters are presented in a closed form. The limited independent solutions of differential Euler equations were expanded in the Neumann power series using the method of successive approximation. This approach allows to obtain the analytical frequency equations as power series rapidly convergent to exact eigenvalues for different number of nodal diameters.

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

“…Here, the same term t is chosen for m and n, with their upper limit taken as M � (2t − 1)/2 and N � (2t − 1)/2. By substituting the above constant solutions into equation (19), we finally get the analytical bending solutions for plate with two adjacent edges free and the other two edges clamped. e bending moment along the clamped edge can be easily found by the following equation:…”

“…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.…”

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.…”