Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques

The paper states and summarizes the quadratic normal-form mapping for conservative second-order systems, giving explicit second-order coefficients âij, b̂ij, ĉij, α̂ij, β̂ij, γ̂ij in terms of the resolvents [(ωi±ωj)^2 M − K]^{-1}G(φi, φj), with ĉij = 0, α̂ij = 0, β̂ij = 0, and γ̂ij = ((ωj+ωi)/ωj)Ψ̂(P)ij + ((ωj−ωi)/ωj)Ψ̂(N)ij, exactly as in the candidate solution. These formulas appear explicitly in Eqs. (47)–(48) of the review (derived from [214, 298]), alongside the same near-identity mapping X, Y in terms of R, S (Eqs. (45)–(46)) and rely on the same non-resonance (no second-order internal resonance) assumption for invertibility of (ωi±ωj)^2M − K. The symmetry of the quadratic force tensor G (hence G(φi, φj) = G(φj, φi)) is also assumed in the paper and used by the model. Therefore, the model’s derivation and final expressions coincide with the paper’s stated results and underlying assumptions, notationally and conceptually.