Mathematical analysis of the true and spurious eigensolutions for free vibration of plate using real-part BEM

Abstract

In this paper, a real-part BEM for solving the eigenfrequencies of plates is proposed for saving half effort in computation instead of using the complex-valued BEM. By employing the real-part fundamental solution, the spurious eigensolutions in conjunction with the true eigensolution are obtained for free vibration of plate. To verify this finding, the circulant is adopted to analytically derive the true and spurious eigenequation 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. For the continuous system, mathematical analysis for the spurious eigensolution was done by using the degenerate kernel and Fourier series. Good agreement among the analytical solutions (continuous and discrete systems) is made. The clamped circular plate is demonstrated analytically and numerically to see the validity of the present method. 本文以實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。使用實部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。為證明假根產生之機制，本文在連續系統中採用退化核及富利葉級數來進行數學推導，於離散系統中利用退化核及循環矩陣來做一解析之動作，並使用奇異值分解法來同時獲得特徵頻率之邊界模態。本中並以一固端圓板為例來說明由本文所提出之方法不論是在連續或是離散系統中，均能得到相吻合之結果，以驗證此方法之正確性。