Fourier series solutions for vibrations of a rectangular plate with a straight through crack

Chiung-Shiann Huang*, Y. J. Lin

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

18 Scopus citations


The existence of a crack in a rectangular plate makes the exact closed-form solutions for the vibrations of the plate intractable, if they even exist. This work presents analytical solutions for vibrations of horizontally or vertically cracked rectangular plates under various boundary conditions. The solutions are constructed by combining Fourier cosine series with domain decomposition. A rectangular plate with a side crack or an internal crack is divided into four or six rectangular sub-domains, respectively. The series solutions that satisfy the governing equations for the vibrations of a plate are firstly established for each sub-domain based on the classical plate theory. The solutions for each sub-domain are related to each other by satisfying the continuity conditions along the interconnection boundaries between the sub-domains. Finally, the boundary conditions of the cracked plate are imposed on the solutions. Comprehensive convergence studies are performed for intact plates and cracked plates with various boundary conditions, and the presented natural frequencies are compared with the published ones to confirm the correctness of the proposed solutions. Not like a typical energy method, which overpredicts true frequencies if the used shape functions do not exactly satisfy the governing equations for the problem under consideration, the convergence studies reveal that the present solutions converge from the lower bounds to the exact frequencies as the number of series terms increases. The present solutions are further applied to determine the first five frequencies of vibration of rectangular plates with side cracks and internal cracks of various lengths and locations. The results obtained under SSSS, CFFF, FSFS and FFFF boundary conditions are tabulated, and some of these are presented here for the first time.

Original languageEnglish
Pages (from-to)10389-10403
Number of pages15
JournalApplied Mathematical Modelling
Issue number23-24
StatePublished - 1 Dec 2016


  • Cracked rectangular plate
  • Fourier series solution
  • Lower bound solution
  • Vibration


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