Eigensolutions of a circular flexural plate with multiple circular holes by using the direct BIEM and addition theorem

The purpose of this paper is to present an analytical formulation to describe the free vibration of a circular flexural plate with multiple circular holes by using the null field integral formulation, the addition theorem and complex Fourier series. Owing to the addition theorem, all kernel functions are represented in the degenerate form and further transformed into the same polar coordinates centered at one of circles, where the boundary conditions are specified. Thus, not only the computation of the principal value for integrals is avoided but also the calculation of higher-order derivatives in the flexural plate problem can be easily determined. By matching the specified boundary conditions, a coupled infinite system of simultaneous linear algebraic equations is derived as an analytical model for the title problem. According to the direct searching approach, natural frequencies are numerically determined through the singular value decomposition (SVD) in the truncated finite system. After determining the unknown Fourier coefficients, the corresponding mode shapes are obtained by using the direct boundary integral formulations for the domain points. Several numerical results are presented. In addition, 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 advantages of the present method thanks to its analytical features.