Invariant-manifold-based model reduction for geometrically exact beam dynamics

This paper presents an invariant-manifold (IM)-based reduction approach for geometrically exact beams undergoing finite rigid-body motions and elastic deformations. For a free beam or beam assembly, direct application of standard IM reduction is hindered by infinite system equilibria and secular growth in physical coordinates. To address this challenge, the floating frame of reference (FFR) method is employed to decompose the beam kinematics into a fictitious rigid-body motion and elastic deformation relative to the FFR. Furthermore, externally applied loads are recast as ordinary differential equations for an excitation vector. These steps render the system dynamics independent of the FFR motion tensor and the time variable, yielding an autonomous system with all parameters bounded in the vicinity of the state-space origin. Polynomial parameterization of the target invariant manifold is enabled using the FFR velocity, a set of dominant modal coordinates, and the excitation vector. A hierarchy of cohomological equations is formulated to compute the manifold mapping and the reduced dynamics. An efficient solution strategy is proposed to solve the high-dimensional cohomological equations, where higher-order problems are separated into resonant and non-resonant parts: the resonant part, associated with small-scale singular linear systems, is solved using least-squares minimization, while the non-resonant part is solved sequentially through a hierarchy of linear systems. Numerical results show that the proposed IM-based reduction yields highly accurate predictions of the nonlinear dynamic response of geometrically exact beams using only third–order polynomial expansions of the manifold and reduced dynamics. Moreover, the method achieves substantial computational savings, as dynamic simulation requires integrating only the low-dimensional reduced-order equations rather than the full state dynamics.