True and spurious eigensolutions of an elliptical membrane by using the non-dimensional dynamic influence function method

Abstract

In this paper, we employ the non-dimensional dynamic influence function (NDIF) method to solve the free vibration problem of an elliptical membrane. It is found that the spurious eigensolutions appear in the Dirichlet problem by using the double-layer potential approach. Besides, the spurious eigensolutions also occur in the Neumann problem if the single-layer potential approach is utilized. Owing to the appearance of spurious eigensolutions accompanied with true eigensolutions, singular value decomposition (SVD) updating techniques are employed to extract true and spurious eigenvalues. Since the circulant property in the discrete system is broken, the analytical prediction for the spurious solution is achieved by using the indirect boundary integral formulation. To analytically study the eigenproblems containing the elliptical boundaries, the fundamental solution is expanded into a degenerate kernel by using the elliptical coordinates and the unknown coefficients are expanded by using the eigenfunction expansion. True and spurious eigenvalues are simultaneously found to be the zeros of the modified Mathieu functions of the first kind for the Dirichlet problem when using the single-layer potential formulation, while both true and spurious eigenvalues appear to be the zeros of the derivative of modified Mathieu function for the Neumann problem by using the double-layer potential formulation. By choosing only the imaginary-part kernel in the indirect boundary integral equation method (BIEM) to solve the eigenproblem of an elliptical membrane, spurious eigensolutions also appear at the same position with those of NDIF since boundary distribution can be lumped. The NDIF method can be seen as a special case of the indirect BIEM by lumping the boundary distribution. Both the analytical study and the numerical experiments match well with the same true and spurious solutions. 本文利用無因次動力影響函數法求解橢圓薄膜的自由振動問題。我們發現使用雙層勢能法求解 Dirichlet 問題時會有假根的產生。此外，當使用單層勢能法求解Neumann 問題時也會有假根的產生。由於假根伴隨著真根的出現，因此我們使用奇異值分解補充技巧來分別萃取真假根。因為在橢圓型邊界的離散系統下循環矩陣的特性被破壞，所以我們使用間接邊界積分方程法來解析地探討預測假根的產生機制。為了解析研究 橢圓形邊界的真假根問題，將其閉合型的基本解在橢圓座標下展開成分離核的形式，而未知的密度函數則使用特徵函數展開。使用單層勢能法求解 Dirichlet 問題時，真根及假根同時相對應於第一類修正型 Mathieu函數的零值，同樣地使用雙層勢能法求解 Neumann 問題時，真根及假根皆對應於一階微分修正型 Mathieu函數的零值。當使用間接邊界積分方程法的虛部核函數與使用無因次動力影響函數法求解橢圓薄膜特徵值問題時，假根出現的位置是相同的，因為連續分佈的密度函數可被分段局部集中，所以無因次動力影響函數法可視為間接邊界積分方程法的特例。最後，解析預測和數值實驗所得到的真假根皆ㄧ致吻合。