Analytical and numerical studies of free vibrations of plate by imaginary-part BEM formulations

In this paper, the spurious solution of characteristic equation for the eigenproblems of a circular plate is studied in the continuous and discrete systems. Since any two boundary integral equations in the plate formulation (four equations) can be chosen, 6(C24) options are considered instead of only two approaches (single-layer and double-layer methods, or singular and hypersingular equations). The occurring mechanism of the spurious solution for the circular plate in the imaginary-part formulations is studied analytically. For the continuous system, degenerate kernels for the fundamental solution and the Fourier series expansion for the boundary density are employed to derive the true and spurious solutions of characteristic equation analytically. For the discrete system, the degenerate kernels for the fundamental solution and circulants resulting from the circular boundary are employed to determine the spurious solution of characteristic equation. It is found that true solution for characteristic equation depends on the specified boundary condition while spurious one is embedded in each formulation. Also, we provide two methods (singular value decomposition updating technique and the Burton and Miller method) to suppress the occurrence of the spurious eigenvalues. A general-purpose program of imaginary-part boundary element method was developed for general cases. Several examples including circular and noncircular cases were demonstrated to check the validity of the formulation.