Technical Observations on Nonlinear Force Assembly and Harmonic Balance in an Elastic Pipeline Metastructure

This technical discussion note examines the mathematical formulation presented in the article by G. Zhang, X. Fang, D. Yu, and W. Lacarbonara, “Combining nonlinear resonators and nonlinear clamps for broadband vibration mitigation in elastic metastructures,” published in the Journal of Sound and Vibration, Volume 647 (2027), Article 120186, available online on 10 October 2026. Three mathematical observations are investigated: A dimensional inconsistency in the inertial coefficient appearing in Equation (1), concerning the pipe mass contribution and the distinction between volumetric and linear mass density. An algebraic inconsistency between cubic restoring forces defined through relative displacements and the constant-matrix representation of componentwise cubic displacement terms in Equations (3)–(5). A corrected incidence-matrix formulation is provided, together with an explicit algebraic counterexample demonstrating the necessity of mixed displacement monomials. An incorrect mixed sine–cosine monomial in the first-order harmonic-balance formulation of Equation (6). The correct fundamental harmonic projections are derived and corresponding reduced equations are presented. The note includes analytical verification, numerical cross-checks, and references to established literature on nonlinear oscillations, structural dynamics, and contact mechanics. The reported 27.3% bandgap widening is confirmed arithmetically from the published frequency intervals. The observations concern the printed mathematical equations and do not establish that the authors' numerical implementation or experimental conclusions are incorrect. The purpose of this note is to encourage constructive scientific clarification and improve the mathematical reproducibility of nonlinear elastic metastructure models.