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For mechanical systems subject to periodic excitation, forced response curves(FRCs) depict the relationship between the amplitude of the periodic responseand the forcing frequency. For nonlinear systems, this functional relationshipis different for different forcing amplitudes. Forced response surfaces (FRSs),which relate the response amplitude to both forcing frequency and forcingamplitude, are then required in such settings. Yet, FRSs have been rarelycomputed in the literature due to the higher numerical effort they require.Here, we use spectral submanifolds (SSMs) to construct reduced-order models(ROMs) for high-dimensional mechanical systems and then use multidimensionalmanifold continuation of fixed points of the SSM-based ROMs to efficientlyextract the FRSs. Ridges and trenches in an FRS characterize the main featuresof the forced response. We show how to extract these ridges and trenchesdirectly without computing the FRS via reduced optimization problems on theROMs. We demonstrate the effectiveness and efficiency of the proposed approachby calculating the FRSs and their ridges and trenches for a plate with a 1:1internal resonance and for a shallow shell with a 1:2 internal resonance.

Fast computation and characterization of forced response surfaces via spectral submanifolds and parameter continuation