Flexural vibrations of clamped-free rhombic plates with corner stress singularities. Part I: review of research

In this paper (part I) an ample review of the published literature of skew plate vibrations summarizes well over 100 references as background, bringing forth a need for accurate solutions incorporating stress singularity-based methodologies for analyzing this applied mechanics problem. Such an accurate method is presented in a part II companion paper for analysis of flexural vibrations of rhombic plates with all combinations of clamped and free edge conditions. The prime focus therein is that the analysis explicitly considers the bending stress singularities that occur in the two opposite, clamped-free corners with obtuse angles of the rhombic plates. The strength of these singularities increases significantly as the obtuse angles at the corners increase. Frequency relations summarized herein from this review of Mindlin and Reddy skew plate vibrations research enable investigators to obtain accurate 3-4 significant digit upper bounds on exact solutions of shear deformable skew plates, including the effects of stress singularities, given analogous Kirchhoff skew (rhombic) plate frequency solutions, and thereby eliminating the need to solve the complicated equations of shear deformation theories including stress singularity effects published elsewhere.