Exact Model Reduction by a Slow-Fast Decomposition of Nonlinear Mechanical Systems

TL;DR

This paper presents a method for exactly reducing nonlinear mechanical systems to lower dimensions using a slow-fast decomposition, providing conditions for validity and connecting to existing modal reduction techniques.

Contribution

It introduces a formal framework for exact model reduction via slow-fast decomposition, clarifies its domain of validity, and relates it to classical modal reduction methods.

Findings

Exact reduction is possible under specific conditions.

The domain boundary for reduction validity is characterized.

Connections to static condensation and modal derivatives are established.

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.