Free Vibration of Tapered, Damaged, and Functionally Graded Beams with Elastic Supports: A Chebyshev Collocation Study

Abstract

This paper presents a comprehensive investigation of the free vibration characteristics of beams with variable cross-sections and elastically restrained boundaries using a high-precision Chebyshev collocation method. The methodology employs Chebyshev–Gauss–Lobatto collocation points with cosine transformation and direct analytical differentiation formulas to reduce the governing differential equations to algebraic eigenvalue problems. Three distinct structural configurations are analyzed: (i) linearly tapered beams with flexible ends representing non-ideal structural connections, (ii) beams with exponentially varying properties and damaged boundaries for structural health monitoring applications, and (iii) functionally graded beams with general elastic constraints relevant to advanced material systems. The implementation utilizes multiprecision arithmetic to ensure numerical stability and reliable eigenvalue separation. Extensive validation against three independent analytical and numerical solutions demonstrates exceptional agreement. The method requires only 20–25 collocation points compared to 40–60 nodes typically needed in finite element or differential quadrature methods, while maintaining spectral accuracy. Natural frequencies accurate to six decimal places are presented for various boundary conditions, different taper, varying damage parameters, and different material gradation indices .The results provide valuable benchmark data for structural design optimization, damage detection algorithms, and validation of commercial finite element software.