COMPARATIVE ANALYSIS OF CHEBYSHEV PSEUDOSPECTRAL METHODS FOR FREE VIBRATION OF UNIFORM, TAPERED, AND FUNCTIONALLY GRADED RODS

Abstract

This paper presents a comprehensive comparative analysis of three Chebyshev pseudospectral methods—Gauss (CG), Radau (CGR), and Lobatto (CGL)—for analyzing free longitudinal vibrations of rods with varying geometric and material properties. The study employs a novel trigonometric transformation approach that enables analytic differentiation, eliminating the need for numerical differentiation matrices. The methodology encompasses uniform rods, linearly tapered rods, and functionally graded (FG) rods under various boundary conditions including fixed-fixed (F-F), fixed-free (F-Fr), and free-free (Fr-Fr) configurations. A systematic investigation of accuracy, convergence rates, and computational efficiency is conducted for each method across different rod types. The trigonometric formulation provides several advantages: derivatives are computed analytically via closed-form expressions, the method is naturally well-conditioned for smooth functions achieving spectral accuracy, and collocation points automatically concentrate near boundaries. Results demonstrate that all three methods achieve exponential convergence with relative errors below  for fundamental frequencies using - collocation points. The CGL method emerges as the most practical choice for engineering applications due to its natural boundary condition implementation and superior robustness, while CG and CGR methods offer marginally higher accuracy per degree of freedom. The study provides detailed recommendations for method selection based on problem complexity, accuracy requirements, and computational constraints.