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

The paper claims exponential convergence of the normalized nonlinear frequency and O(N^2) complexity for the nonlinear force, and it presents the separable form of the von Kármán term and empirical semilog slopes supporting exponential decay. However, it does not supply a rigorous proof of exponential convergence; it relies on standard spectral heuristics and numerical evidence (e.g., ln ε ≈ −0.82 N in Fig. 3) rather than a theorem. The model supplies a coherent proof outline under explicit analytic-regularity and normal-hyperbolicity assumptions, and it correctly identifies the nonlinear force as N(q) = −4 α (q^T C q)(B q) with Θ(N^2) evaluation. Notably, the paper has a typographical omission in its compact matrix formula (its Eq. (25) is missing the final q), though the preceding component-wise derivation is correct. There is also a normalization inconsistency: early sections define f̂ = f_nl/f_ref, whereas the validation section uses f̂ = f_nl/f_lin(VCNT=0, BC). These are minor issues; substantively, the model’s argument fills the paper’s theoretical gap and is correct given its stated assumptions. Key supporting locations in the paper: boundary-adapted Chebyshev basis (ϕ_j = (1−ξ^2)^2 T_{j−1}), the separable nonlinear form with B and C, the O(N^2) complexity claim, and the observed exponential convergence slope ln ε ≈ −0.82 N are all present in the PDF. See the paper’s basis and Galerkin setup, the nonlinear term derivation and cost reduction, the explicit O(N^2) complexity statement, and the convergence figure and text (citations: , , , ).