A Chebyshev--Ritz Spectral Framework for Nonlinear Vibration of CNT-Reinforced Composite Beams

Abstract: This study develops a spectral Ritz formulation for the nonlinear free vibration analysis of carbon nanotube-reinforced composite (CNTRC) beams. Boundary-adapted Chebyshev basis functions are constructed to exactly satisfy clamped and simply supported boundary conditions. The governing equations incorporate von~K\'{a}rm\'{a}n geometric nonlinearity, while the effective material properties for both uniform and functionally graded (FG) CNT distributions are evaluated using a modified rule of mixtures. Discretization via the Chebyshev-Ritz approach produces a reduced-order model exhibiting exponential convergence; for basis sizes $N \geq 12$, the fundamental frequency error remains below $0.1\%$ relative to published benchmarks. Computational results demonstrate substantial efficiency gains, with the spectral approach requiring significantly less time than high-fidelity finite element discretizations of comparable accuracy. Parametric studies reveal that the fundamental frequency increases with CNT volume fraction and is sensitive to the interfacial load-transfer efficiency parameter $\eta_E$. Selected FG patterns are shown to enhance stiffness relative to uniformly distributed CNTs. Validation against established numerical benchmarks yields relative differences of only a few percent. The current limitation of the method is its reliance on the Euler-Bernoulli beam assumption, which neglects transverse shear deformation and damping; addressing these effects is proposed for future work. All numerical data and scripts are provided as supplementary material to ensure reproducibility.