利用Fourier級數求解具裂縫矩形板 之自由振動

Abstract

本研究係利用傅立葉餘弦級數展開求解具裂縫矩形薄板之自由振動問題；以多項式作為輔助函數，修正傅立葉餘弦級數在逐項微分所面臨到之問題。由無裂縫矩形板出發，延伸至當薄板考慮邊緣裂縫或是內部裂縫時的振動分析。當考慮邊緣裂縫時，將矩形薄板分割成四個部分的矩形區域；若考慮的為內部裂縫，則分割成六個部分的矩形區域；其中將裂縫視為無剪力及彎矩之自由端，搭配邊界條件、相鄰區域共邊處之連續關係及薄板之自由振動控制方程式，以求解無因次化自由振動頻率。在收斂性分析中，隨著級數展開項次之增加，自然振動頻率由下限單調收斂至真值。此外，亦以不同裂縫長度，裂縫位置，長寬比之結果與文獻值進行比較。最後再以配點法求解具邊緣裂縫板之振動問題，並與傅立葉餘弦級數展開所得之結果進行比較；配點法之結果不若Fourier級數之解析解優。

The existence of a crack yields the exact closed-form solutions for vibrations of cracked rectangular to be intractable, if they exist. This work presents analytical solutions for vibrations of horizontally or vertically cracked rectangular plates having various boundary conditions. The solutions are constructed by using Fourier cosine series combining with domain decomposition. A rectangular plate is divided into four and six rectangular sub-domains for the plate with a side crack and an internal crack, respectively. Fourier series solutions satisfying the governing equations for vibrations of a plate based on the classical plate theory are first established for each sub-domain. The solutions for each sub-domain are related to each other by satisfying the continuity conditions along the interconnection boundaries between two adjacent sub-domains. Finally, the boundary conditions of the cracked plate are enforced on the solutions. Comprehensive convergence studies are performed for intact plates and cracked plates with various boundary conditions and comparisons between the present results of natural frequencies and the published one are also made to validate the correctness of the proposed solutions. The convergence studies indicate that the present solutions provide lower bounds to the exact values of frequencies. The present solutions are further applied to determine the first five frequencies of rectangular plates with side cracks and internal cracks having various crack lengths and locations. The results for SSSS, CFFF, FSFS and FFFF boundary conditions are tabulated, some of which are first shown in literature. Finally, seeks the free vibration for the rectangular thin plates with cracks by point allocation, and compare the result with Fourier cosine series.