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

Abstract

The purpose of this paper is to propose an analytical method to solve natural frequencies and natural modes of a circular 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 coordinate from each local coordinate at center of all circles. Not only avoiding the computation of the principal value but also the calculation of higher-order derivatives in the plate problem can be easily determined. According to the specified boundary conditions, a coupled infinite systems of simultaneous linear algebraic equations are derived and its solution can be obtained in an analytical way. The direct searching approach is adopted to determine the natural frequency through singular value decomposition (SVD). After determining the unknown Fourier coefficients, the corresponding mode shapes are obtained by using the direct boundary integral equations for domain points. Some numerical results are presented. Moreover, the inherent problem of spurious eigenvalue using 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.