On the nonlinear stochastic dynamics of a continuous system with discrete attached elements

The paper states the eigenpair structure and transcendental equation for the fixed–mass–spring bar (equations (19)–(21)) and the orthogonality properties, but does not derive them; it cites a handbook for the result. It also qualitatively asserts that as the discrete–continuous mass ratio m* increases, the natural frequencies decrease, and observes PSD peaks near natural frequencies under white-noise forcing, all supported by Monte Carlo evidence, but without rigorous proofs. The model’s solution supplies the missing derivations: the dynamic boundary condition, the transcendental equation cot λ + (kL/(AE))(1/λ) − (m/(ρAL))λ = 0, M/K-orthogonality, strict monotonicity of λn(m*) via the implicit function theorem, precise large-mass limits (λ1 → 0, λn → (n−1)π for n≥2), and a closed-form PSD expression in the linear case that explains the observed spectral peaks and their shift with m*. These steps are consistent with, but stronger than, the paper’s statements and numerics. Key items cross-check cleanly against the paper’s own weak form and stochastic forcing set-up (Eqs. (5)–(13), (19)–(21), (25)–(26), Sec. 4.3), yet the paper does not present the proofs. Therefore, the paper is incomplete relative to the question’s demands, while the model’s solution is correct and complete for the linear analysis it undertakes. See the paper’s eigenvalue statement and weak form (including M and K) and its PSD observations for corroboration .