Free Vibration Analysis of Thin Rectangular Plates with Mixed Point Support Conditions Using the Rayleigh–Ritz Method

This study addresses the semi-analytical solution of vibration modes for thin rectangular plates with mixed point support conditions, a scenario that has not been comprehensively explored in existing research. The research aims to expand the range of point support conditions to a collection of three types, including simply supported points, clamped points, and sliding-supported points, and to develop and validate a solution process within the Rayleigh–Ritz method for the vibration of thin rectangular plates with mixed point support conditions that incorporate the three types. The three types of point support conditions are expressed in the form of constraint equations on the mode shape function of a plate, and are incorporated into the Rayleigh–Ritz method with the employment of Lagrange multipliers. Semi-analytical solutions are obtained for plates with specified support conditions, and the accuracy of the approach is validated against finite element method (FEM) analysis results. The present method successfully determines the vibration modes for plates with mixed point support conditions. Specific examples include a plate with four clamped corners and a plate with mixed support at corner points. Results of the solution show good agreement with FEM analysis, with the error of the natural frequencies at around 3% or lower, and the mode shapes show a high degree of similarity. Further investigation reveals that higher-order basis functions improve accuracy, with Legendre or Chebyshev polynomials outperforming trigonometric functions in achieving precise results with fewer basis function orders. The findings supplement the existing research on rectangular plates with clamped points, and enable the semi-analytical solution of plates with mixed point support conditions involving sliding degrees of freedom at points, which facilitates the vibration analysis of such structures.