Detecting internal resonances during model reduction

Model order reduction of geometrically nonlinear dynamic structures is often achieved via a static condensation procedure, whereby high-frequency modes are assumed to be quasi-statically coupled to a small set of lower-frequency modes, which form the reduction basis. This approach is mathematically justifiable for structures characterised by slow/fast dynamics, such as thin plates and slender beams, and has been shown to provide highly accurate results. Nevertheless, selecting the reduction basis without a priori knowledge of the full-order dynamics is a challenging task; retaining redundant modes will lead to computationally suboptimal reduced-order models (ROMs), whilst omitting dynamically significant modes will lead to inaccurate results, and important features such as internal resonances may not be captured. In this work, we demonstrate how the error associated with static condensation can be efficiently approximated during model reduction. This approximate error can then be used as the basis of a method for predicting when dynamic modal interactions will occur, which will guide the reduction basis selection process. Equivalently, this may serve as a tool for verifying the accuracy of ROMs without the need for full-order simulations. The proposed method is demonstrated using a simple oscillator, and a finite element model of a clamped-clamped beam.