On the validity limits of the parametrisation method for invariant manifolds: an assessment of practical criteria for vibrating systems

TL;DR

This paper evaluates the limits of the parametrisation method for invariant manifolds in nonlinear vibrating systems, proposing practical criteria to estimate the validity range of reduced-order models.

Contribution

It introduces and assesses three practical criteria for estimating the validity range of the parametrisation method in nonlinear vibration analysis.

Findings

The invariance error criterion effectively indicates convergence limits.

Potential singularities provide upper bounds for validity range.

Convergence rules help in assessing series validity in models.

Abstract

The parametrisation method for invariant manifolds is a powerful technique for deriving reduced-order models in the context of nonlinear vibrating systems, allowing accurate computations of nonlinear normal modes. Thanks to arbitrary order asymptotic expansions, converged results are within reach and directly applicable to finite element structures. However, since it relies on a local theory and asymptotic expansions, the results are only valid up to a given amplitude, which defines the convergence radius of the approximation. The aim of this contribution is to investigate the validity limits of the approach and review the existing error estimates, with the concrete objective of proposing a practical approach to estimate the validity range during the computation, thus producing safe bounds within which the reduced-order model can be used. Three different criteria are assessed. The first one uses the error in the invariance equation as the distance to the fixed point increases. The second one is adapted from an upper bound criterion derived for normal form transforms and based on the potential singularities of the homological operator. The third one uses Cauchy and d'Alembert convergence rules for series. The criteria are tested on a number of different examples that are representative of the situations encountered when dealing with nonlinear vibrations. The Duffing equation serves as a first benchmark that allows considering conservative oscillations, forced systems at primary resonance, and superharmonic resonance. The investigations are then extended to a vibrating system with two degrees of freedom. Finally, the different criteria are assessed on a finite element beam structure, and guidelines are formulated to generalise their practical use and produce accurate and easy-to-use error bounds in the context of model order reduction for nonlinear vibrating structures.