Mathematical analysis and numerical study of the true and spurious eigenequations for free vibration of plate using an imaginary-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. 本文以虛部邊界元素法求解一固定圓板之特徵頻率問題以節省數值運算時間與避免奇異性。使用虛部邊界元素法在求解板自由振動過程中所伴隨而來的真假特徵方程為此文章之討論重點。為證明假根產生之機制，本文在於離散系統中利用退化核及循環矩陣來探討解析一圓形板的真假特徵方程。文中以一圓板（固定端，簡支撐及自由邊界條件）為例在離散系統中來說明，並以驗證此方法之正確性。最後則提出採用「奇異值分解法-補充式」之技巧來克服假根之產生，並以一固定端圓板為例在離散系統下說明並解析。