Efficient Model Reduction and Prediction of Superharmonic Resonances in Frictional and Hysteretic Systems

The paper explicitly defines Fnq,broad and Fkq,sup,n and derives the nth-harmonic rearrangement (its Eqs. (8)–(11)) used for VPRNM, and poses the orthogonality (phase) constraint (its Eq. (12)); the candidate reproduces this derivation for (i) and, in (ii), adds a mathematically explicit IFT/IVT argument for existence and robustness of a phase-resonant root and a no-resonance implication when the modally filtered broadband forcing vanishes. These extensions are consistent with the paper’s modeling assumptions (including neglecting Fkq,sup,n for the constraint) and its discussion of the modal filter and residual sign change, but are not proved there; hence both are correct, with the model providing a complementary proof-level treatment rather than contradicting the paper’s statements. See the paper’s definitions and equations for Fnq,broad and Fkq,sup,n and their rearrangement (), the VPRNM constraint and its interpretation (), the modal-filtered residual behavior (), and the stated need for nonzero modal broadband forcing for a superharmonic resonance ().