Mathematical analysis and numerical study of true and spurious eigenequations for free vibration of plates using real-part BEM

Abstract

In this paper, an imaginary-part BEM for solving the eigenfrequencies of plates is proposed for avoiding singularity and saving half effort in computation instead of using the complex-valued BEM. By employing the imaginary-part fundamental solution, the spurious eigenequations in conjunction with the true eigenequation are obtained for free vibration of plate. To verify this finding, the circulant is adopted to analytically derive the true and spurious eigenequations in the discrete system of a circular plate. In order to obtain the eigenvalues and boundary modes at the same time, the singular value decomposition (SVD) technique is utilized. The analytical solutions are derived in the discrete system. Three cases, clamped, simply-supported and free circular plates, are demonstrated analytically and numerically to see the validity of the present method. SVD updating technique is adopted to suppress the ocurrence of the spurious eigenvalues, and a clamped plate is demonstrated analytically for the discrete system in this paper.

Method of Fundamental Solutions for Plate Vibrations in Multiply Connected Domains

“…In Fig. 3, the value λ, in which there is a cusp, is an eigenvalue since the corresponding determinant value is suddenly minified [13]. The numerical eigenvalues are in good agreement (up to 0.01) with the exact solutions as shown in Table 1.…”

“…2, and let s E = (R E , θ E ), s I = (R I , θ I ), and x = (ρ, φ) denote the interior source, exterior source, and boundary field points, respectively. To analytically derive the MFS solutions in an annular domain, it is necessary to decompose the kernels (6a) ~ (6d) into circular harmonics, which will result in the following degenerate kernels [9,13]: ( )…”

“…However, he did not circumvent the problems of spurious eigenvalues. Recently, Chen et al [13] further developed this method and utilized the SVD updating technique to avoid the occurrence of the spurious eigenvalues. However, a general method for computing eigenvalues with various boundary conditions in multiply connected domains, especially for a simple scheme free from spurious eigenvalues, is still lacking.…”

Improved non-dimensional dynamic influence function method based on tow-domain method for vibration analysis of membranes

“…2,7 Note that the proposed method is occasionally ill-posed when too many nodes are used. On the other hand, the proposed method does not consider the problem of spurious eigenvalues, which have been studied in many articles related to plate vibrations, [11][12][13][14][15][16] because the spurious eigenvalues do not appear in the NDIF method for fixed membranes.…”

Free Vibration Analysis of Multiply Connected Plates Using the Method of Fundamental Solutions

“…Chen et al have tried to solve the eigenproblem of multiply connected membrane and found that spurious eigenvalues also appear (Ref. [5]) as well as BEM (Ref. [6]).…”