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

Abstract

For general nonlinear mechanical systems, we derive closed-form, reduced-order models up to cubic order based on rigorous invariant manifold results. For conservative systems, the reduction is based on Lyapunov Subcenter Manifold (LSM) theory, whereas for damped-forced systems, we use Spectral Submanifold (SSM) theory. To evaluate our explicit formulas for the reduced model, no coordinate changes are required beyond an initial linear one. The reduced-order models we derive are simple and depend only on physical and modal parameters, allowing us to extract fundamental characteristics, such as backbone curves and forcedresponse curves, of multi-degree-of-freedom mechanical systems. To numerically verify the accuracy of the reduced models, we test the reduction formulas on several mechanical systems, including a higher-dimensional nonlinear Timoshenko beam.

Comparison of nonlinear mappings for reduced-order modelling of vibrating structures: normal form theory and quadratic manifold method with modal derivatives

“…A main result of our investigations is that the results predicted by the QM approach with MDs converge to those provided by the normal form approach, only in the case where a slow/fast assumption between master and slave coordinates holds. This result is fully in the line of general theorems provided in [17,60] and thus further illustrates the general findings given in these papers where a more general framework including damping is given, together with an exact result that do not rely on asymptotic expansion. A first quantification of the limit value for the slow/fast assumption to hold has been provided, based on the predicted values for the type of nonlinearity, showing that a small gap is needed: ω s > 4ω p , thus justifying a posteriori the good results found by previously published papers using this method.…”

“…[12,33,34]). However, recent developments show that the coefficients can be directly computed, for the case of spectral submanifold [60], or for the case of normal form in a non-intrusive manner [62], so that this limitation does not hold anymore.…”

“…Enforcing the invariant property is key in a perspective of reduced-order modelling, since it is the only way to ensure that the trajectories of the ROM will also exist for the full system. Elaborating on this idea, NNMs have been used in the perspective of model-order reduction using either centre manifold theorem [39,46], normal form theory [52,54,57], or spectral submanifolds [9,16,40,60].…”

“…These simplifications have been used in numerous recent papers in order to explain why a number of methods for producing ROMs are able to predict very good results in this case, see, e.g. [12,19,60,61]. In order to propose a simple two-dof system mimicking these particular rela-tionships, the von Kármán model for slender beams is used and simplified to two vibration modes, one flexural and one longitudinal, in order to produce the simplified system, from which the coefficients can be related to meaningful quantities of the beam and in particular to its slenderness.…”

Reduced order models for geometrically nonlinear structures: Assessment of implicit condensation in comparison with invariant manifold approach

“…However, recent developments overcome this limitation, see e.g. [54] for a direct approach using spectral submanifolds (with general third-order formula equivalent to the ones given with the invariant manifold method proposed by Shaw and Pierre), and [55,56] for a direct method based on normal form.…”