Nonlinear mapping of non-conservative forces for reduced-order modelling

Non-intrusive or indirect reduced-order modelling strategies, such as the implicit condensation and expansion method, are applicable to geometrically nonlinear structures modelled using commercial finite-element (FE) software. Traditionally, the non-conservative forces acting on the structure are reduced via a linear projection onto the space spanned by the reduced modeshapes. As such, only the forces acting directly on these reduced modes can be captured, while any energy gained or dissipated by the statically condensed modes is neglected. This can lead to significant inaccuracies in the reduced-order model (ROM) predictions, which is demonstrated here using a 2-degrees-of-freedom (DOF) oscillator, and an FE model of a cantilever beam. It is shown that the non-conservative forces acting on the statically condensed modes can be captured using a nonlinear mapping of the physical DOFs into the reduced coordinates. This introduces additional terms in the reduced equations of motion, which we describe as force compensation. Excellent agreement is observed between the forced response curves of the full-order models and those of our proposed ROMs, both for the oscillator as well as the cantilever beam under different external excitation conditions (i.e. a constant-direction force and a follower force).