Vibration analyses of cracked rectangular plates via Fourier series
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.
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