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

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.