Mathematical analysis and numerical study of the true and spurious eigenequations for free vibration of annular plate using the BEM

Abstract

In this paper, the complex-valued BEM for solving the eigenfrequencies of the annular plates is proposed. By employing the complex-valued BEM, the spurious eugenvalues in conjunction with the true eigenvalues are ontained for free vibration of the multiply-connected plate. We analytically and numerically examine the occurance of the spurious eigenvalues in the continuous and discrete systems of an annular plate. For the continuous system, the degenerate kernels for thefundamental solution and the Fourier series expansion for the boundary density are employed to derive the true and spurios eigenequations analytically. The circulant is adopted to analytically derive the true and spurious eigenequation in the discrete system. It is found that the spurious eigenvalues parasitizing in the multiply-connected plate depend on the associated true eigenvalues of the simply-connected plate with a radius b which is the inner circle of the annular domain. Three methods (SVD updating technique, the Burton & Miller method and the CHIEF method) are adopted to suppress the occurence of the spurious eigenvalues, and a clamped-clamped annular plate is demonstrated analytically for the discrete system in this paper. Serval examples were demonstrated to check the validity of the formulation. 本文以邊界元素法求解同心圓板之特徵值問題。在多連通問題使用複數行邊界元素法在求解過程中所伴隨而來的假根問題為此文章之討論重點。為解析假根產生之機制，本文在連續系統中採用退化核及傅立葉級數來進行數學推導，離散系統中使用循環矩陣來說明假根之產生，同時以固力端邊界條件之同心圓板為例來解析證明。經解析發現在多連通問題所產生之假根為對應內徑b之單連通圓板問題之真根。本文中並提出三種方法(奇異值分法之補充式技巧，Burton & Miller方法及CHIEF方法)來克服假根之產生，最後本文藉由不同的數值算例，來驗證上述的理論推導之正確性。