An Improved Fourier–Ritz Method for Analyzing In-Plane Free Vibration of Sectorial Plates

Abstract

A modified Fourier–Ritz approach is developed in this study to analyze the free in-plane vibration of orthotropic annular sector plates with general boundary conditions. In this approach, two auxiliary sine functions are added to the standard Fourier cosine series to obtain a robust function set. The introduction of a logarithmic radial variable simplifies the expressions of total energy and the Lagrangian function. The improved Fourier expansion based on the new variable eliminates all the potential discontinuities of the original displacement function and its derivatives in the entire domain and effectively improves the convergence of the results. The radial and circumferential displacements are formulated with the modified Fourier series expansion, and the arbitrary boundary conditions are simulated by the artificial boundary spring technique. The number of terms in the truncated Fourier series and the appropriate value of the boundary spring retraining stiffness are discussed. The developed Ritz procedure is used to obtain accurate solution with adequately smooth displacement field in the entire solution domain. Numerical examples involving plates with various boundary conditions demonstrate the robustness, precision, and versatility of this method. The method developed here is found to be computationally economic compared with the previous method that does not adopt the logarithmic radial variable.