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

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.

Figures (28)

• Figure 1: Evolution of the radius of convergence with the tolerance value $\varepsilon$ for the unforced and undamped Duffing oscillator using the invariance equation criterion. The continuous lines indicate results obtained by \ref{['eq:inv_eq_error']}, while the dashed lines represent the simplification introduced in \ref{['eq:inv_eq_error_ANM']}. Parameter values are fixed as $\omega = 1.5$ and $h=1$.
• Figure 2: Validity limit calculated by the series criteria for the unforced and undamped Duffing oscillator as a function of the parametrisation order. For each of the criteria, the displacement $u$, velocity $v$, and reduced dynamics $f$ power series are studied. Results obtained by extrapolation of parametrisations up to order 15 are also presented, in dashed lines. Parameter values are fixed as $\omega = 1.5$ and $h=1$.
• Figure 3: Validity limit calculated by the criterion of the singularity of the homological operator for the unforced and undamped Duffing oscillator as a function of the parametrisation order. The lines represent circles of radius equal to the minimum amplitude among points of the same order. Parameter values are fixed as $\omega = 1.5$ and $h=1$.
• Figure 4: Comparison of the radii of convergence calculated by the tested criteria, for different values of $\theta$. The points correspond to the $z_1$ coordinates in the complex plane for each combination of $\rho$ and $\theta$. For the invariance equation and singularity criteria, the expansion order is chosen as 15, while for the series criteria, it goes up to 35. Parameter values are fixed as $\omega=1.5$ and $h=1$.
• Figure 5: Backbone curves of the conservative unforced Duffing oscillator: CNF with orders $o$=5, 10 and 15 and reference solution Salas2014. Dashed horizontal lines: validity limit criteria obtained by the three methods: series convergence, singularity of the homological operator and error in the invariance equation (Inv.) with $\varepsilon=0.01$. Parameter values are fixed as $\omega = 1.5$ and $h=1$.

Theorems & Definitions (9)

• remark 1
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