Analytical Free Vibration Solutions of Rectangular Thin Plates Subjected to Three Edges Rotationally-Restrained and One Edge Free

Due to the challenges in mathematical solving, accurate analytical solutions for rectangular thin plates are mainly limited to these plates subject to simple Lévy-type boundary conditions (BCs). Conversely, plates with more complicated BCs are more frequently encountered in practical engineering applications. This study employs the finite integral transform method to analyze the free vibration behavior of isotropic rectangular thin plates with three edges rotationally-restrained and one edge free, which is difficult for other methods. Primarily, the method adopted eliminates the necessity for predetermining trial function for plate deflection, offering simplicity and generality for plate mechanical problems. Furthermore, the approach establishes one way to allow one solving linear algebraic equations instead of handling complex boundary value problems of higher-order vibration partial differential equation (PDE), resulting precise and rapidly converging analytical solutions. Finally, new accurate free vibration studies for plates with four non-Lévy-type BCs and three non-classical BCs are presented via altering the introduced rotating fixed coefficients. The results obtained are contrasted with the numerical simulations from ABAQUS software and existing literature works, confirming the accuracy of this method and providing benchmark results for further exploration of free vibration problems of rectangular plates.