Null-field integral equation approach for free vibration analysis of circular plates with multiple circular holes

Abstract

In this paper, a semi-analytical approach for the eigenproblem of circular plates with multiple circular holes is presented. Natural frequencies and modes are determined by employing the null-field integral formulation in conjunction with degenerate kernels, tensor rotation and Fourier series. In the proposed approach, all kernel functions are expanded into degenerate (separable) forms and all boundary densities are represented by using Fourier series. By uniformly collocating points on the real boundary and taking finite terms of Fourier series, a linear algebraic system can be constructed. The direct searching approach is adopted to determine the natural frequency through the singular value decomposition (SVD). After determining the unknown Fourier coefficients, the corresponding mode shape is obtained by using the boundary integral equations for domain points. The result of the annular plate, as a special case, is compared with the analytical solution to verify the validity of the present method. For the cases of circular plates with an eccentric hole or multiple circular holes, eigensolutions obtained by the present method are compared well with those of the existing approximate analytical method or finite element method (ABAQUS). Besides, the effect of eccentricity of the hole on the natural frequency and mode is also considered. Moreover, the inherent problem of spurious eigenvalue using the integral formulation is investigated and the SVD updating technique is adopted to suppress the occurrence of spurious eigenvalues. Excellent accuracy, fast rate of convergence and high computational efficiency are the main features of the present method thanks to the semi-analytical procedure.

Free vibration analysis of a circular plate with multiple circular holes by using the addition theorem and direct BIEM

“…The concept of this technique is to provide sufficient constrains to overcome the rank deficiency of the system. Since four null field integral equations for the plate formulation are provided [10], there are 6 (C (4,2)) options for choosing any two equations to solve the eigenproblem. The U formulation in section 3 uses the first and the second equations.…”

“…Recently some researchers intended to extend an annular plate [7,8] to the plate with an eccentric hole. Lee et al [9,10] proposed a semi-analytical approach to solve the free vibration analysis of a circular plate with multiple holes and pointed out some insufficient accurate results in [8] after careful comparisons.…”

“…One is the improper integral in the boundary integral equation; the other is that both field point and source point are located on different circular boundaries. These problems have been treated by using the degenerate kernel and tensor transformation [9,10], respectively. But tensor transformation accompanied with the higher order derivative increases the complexity of computation and then affect the accuracy of its solution [9].…”

“…But tensor transformation accompanied with the higher order derivative increases the complexity of computation and then affect the accuracy of its solution [9]. In addition, the collocation method in [9,10] belongs to point-matching approach instead of analytical derivation. It also increases the effort of computation since boundary nodes for collocation are required.…”

“…Some numerical computations are presented. Moreover, the SVD updating technique [10] is employed to suppress the appearance of spurious eigenvalues.…”