Analytical free vibration solutions of rectangular edge-cracked plates by the finite integral transform method

Analytical free vibration solutions of rectangular edge-cracked thin plates are obtained by the finite integral transform (FIT) method. While the previous studies focus on the mechanical behaviors of intact plates, this study establishes an FIT-based new scheme for analyzing the free vibration characteristics of rectangular thin plates with an edge crack. The plates may have any combined conditions of simply supported, clamped, and free boundaries. In specific solution scheme, an edge-cracked plate is decomposed into four subdomains, and a double cosine FIT is imposed on the fourth-order partial differential governing equations of each subdomain, which yields the relationship between the transformed modal deflections and naturally defined unknowns. By substituting the exterior boundary conditions of the plate, continuity conditions across the subdomains, and free conditions at the crack into the inverse transforms, the natural frequencies and unknowns are successively determined via the established homogeneous linear algebraic equations. The modal deflections are provided by further substituting the obtained unknowns into the inverse transforms. Comprehensive benchmark results for the edge-cracked plates under representative boundary conditions are shown. The results are well verified by other solution methods. A parametric study quantitatively reveals the effects of boundary conditions and crack length on the natural frequencies of edge-cracked plates. Due to the rigorous but easy-to-implement mathe-matical derivations, this study presents a novel solid way for exploring analytical solutions of vibration problems.

Automated summary

Analytical free vibration solutions of rectangular edge-cracked thin plates are obtained using the finite integral transform (FIT) method. A new scheme based on FIT is established to analyze the free vibration characteristics of these plates with different boundary conditions. The study decomposes the edge-cracked plate into four subdomains and applies double cosine FIT to obtain the relationship between the transformed modal deflections and unknowns. The natural frequencies and unknowns are determined by solving the established algebraic equations, and the modal deflections are obtained by substituting the unknowns into the inverse transforms. Comprehensive benchmark results and a parametric study are presented to validate the effectiveness of the method.