The Factorial Fundamental Frequency Explicit Formulas for Homogenous Rectangular Thin Plates with Four Free Edges

Abstract Purpose This study aims to investigate fundamental modal solutions for rectangular thin plates with four free edges: (1) to provide a novel criterion for fundamental mode identification, and (2) to derive high accurate explicit formulas for calculating the fundamental frequency. Method To achieve the study objectives, an investigation into the fundamental mode space was conducted, varying structural and material parameters including the plate’s dimensions (a, b), thickness (h), and Poisson’s ratio (ν). A novel, explicit criterion for fundamental mode identification was formulated, incorporating a six sigma tolerance tailored for specific aspect ratios. Ordinary least squares (OLS) multiple linear regression methodology was integrated within a dedicated process to identify optimal factor combinations for factorial frequencies based on 56 features. This methodology allowed for the derivation of factorial frequency formulas as a function of a, b, h, and ν, providing explicit expressions for calculating the fundamental frequency. Conclusion The study concludes that the global fundamental mode shapes of rectangular thin plates with four free edges are primarily governed by the aspect ratio a/b, Poisson’s ratio ν, and the thickness-induced shearing effect h on both dimensions. The derived explicit formulas for calculating the fundamental frequency exhibit exceptional accuracy, with a maximum error of −0.217% compared to extensive finite element analysis results, surpassing current state-of-the-art accuracy. This work presents physically meaningful, explicit formulations and introduces a novel perspective in plate and shell research, particularly relevant in scenarios with adjacent free edges and similar global modes.