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.

翻译：本研究提出了一种用于碳纳米管增强复合材料梁非线性自由振动分析的谱里兹公式。构建了边界适配的切比雪夫基函数，以精确满足固支和简支边界条件。控制方程引入了冯·卡门几何非线性，同时采用修正的混合律评估了均匀分布和功能梯度分布碳纳米管的等效材料属性。通过切比雪夫-里兹方法进行离散化，得到了呈现指数收敛性的降阶模型；当基函数规模 $N \geq 12$ 时，相对于已发表的基准解，基频误差保持在 $0.1\%$ 以下。计算结果表明效率显著提升，谱方法所需时间远少于达到同等精度的高保真有限元离散化。参数化研究表明，基频随碳纳米管体积分数的增加而提高，并对界面载荷传递效率参数 $\eta_E$ 敏感。研究表明，选定的功能梯度分布模式相较于均匀分布的碳纳米管能增强刚度。与已有数值基准的验证对比显示相对差异仅为几个百分点。该方法目前的局限性在于其依赖于欧拉-伯努利梁假设，忽略了横向剪切变形和阻尼；解决这些效应被提议作为未来的工作。所有数值数据和脚本均作为补充材料提供，以确保可复现性。