Exact model reduction by a slow–fast decomposition of nonlinear mechanical systems

Abstract

We derive conditions under which a general nonlinear mechanical system can be exactly reduced to a lower-dimensional model that involves only the most flexible degrees of freedom. This Slow-Fast Decomposition (SFD) enslaves exponentially fast the stiff degrees of freedom to the flexible ones as all oscillations converge to the reduced model defined on a slow manifold. We obtain an expression for the domain boundary beyond which the reduced model ceases to be relevant due to a generic loss of stability of the slow manifold. We also find that near equilibria, the SFD gives a mathematical justification for two modal-reduction methods used in structural dynamics: static condensation and modal derivatives. These formal reduction procedures, however, are also found to return incorrect results when the SFD conditions do not hold. We illustrate all these results on mechanical examples.

Exact nonlinear model reduction for a von Kármán beam: Slow-fast decomposition and spectral submanifolds

“…As shown by Haller and Ponsioen [3], assumptions (A1)-(A3) guarantee that the critical manifold M 0 (τ ) perturbs into a nearby attracting slow manifold M (τ ) for > 0 small enough. On this slow manifold, the discretized beam system (4) admits an exact reduced order model given bÿ…”

“…For general finite-dimensional mechanical systems characterized by such a dicotomy of time scales, Haller and Ponsioen [3] deduced conditions under which positions and velocities in the fast degrees of freedom (y,ẏ) can be expressed as a graph over their slow counterparts (x,ẋ), resulting in a globally exact model reduction. If these conditions for a Slow-Fast Decomposition (SFD) are satisfied, then all trajectories of the full system (close enough to the slow manifold in the phase space) synchronize with the reduced model trajectories at rates faster than those within the slow manifold.…”

“…Most classic reduction techniques, coming from a intuive and heuristic standpoint, do not provide an a priori estimate of their accuracy or even validity for a given system. To this effect, Haller and Ponsioen [3] proposed requirements for mathematically justifiable and robust model reduction in a non-linear mechanical system. These requirements ensure not only that the lower dimensional attracting structure (manifold) in the phase space is robust under perturbation, but also that the full system trajectories are attracted to it and synchronize with the reduced model trajectories at rates that are faster than typical rates within the manifold.…”

“…Recent advances in nonlinear dynamics enable such an exact model-reduction using the slow-fast decomposition (SFD) [3] and spectral submanifold (SSM) based reduction [4]. SFD is a general procedure to identify if a mechanical system exhibits a global partitioning of degrees of freedom into slow (flexible) and fast (stiff) components such that the fast variables can be enslaved to the slow ones.…”

Explicit third-order model reduction formulas for general nonlinear mechanical systems

“…While such a manifold can formally be sought, it generally does not actually exist in the configuration space but in the phase space. Accordingly, modal derivatives only give an accurate reduced-order model in slow-fast systems in which spectral submanifolds have a weak dependence on the velocities (see [12]).…”