Modal analysis for elastic structures featuring shock nonlinearities

by E. H. Moussi, EDF R&D / AMA

This work is a deliverable of the project "Advanced Numerical Methods in Mechanics", in partnership with the Laboratory of Mechanics and Acoustics of Marseilles (CNRS).

Engineers in structural dynamics often heavily rely on modal analysis to reduce model order. These methods are limited to the analysis of elastic structures therefore non-linearities cannot be taken into account. Nevertheless, the operator DYNA_TRAN_MODAL in Code_Aster enables transient modal analysis with the ability to take into account localized non-linearities (point-wise contact and friction) but going back and forth between "modal" and "physical space" coordinates makes the calculation costly.

In recent years, a theory of the non-linear modes, able to directly capture these non-linearities in the spectral analysis, has emerged. Within the work thesis of El Hadi Moussi [1], these new products of research have been implemented in Code_Aster.
Two operators dedicated to non-linear modal calculation were introduced:

• MODE_NON_LINE dedicated to the calculation of non-linear modes, e.g. sets of periodic solutions. The algorithm featured is a combination of the harmonic balance method and asymptotic numerical method (ANM);
• REST_MODE_NONL that allows to retrieve a specific periodic solution from the set of periodic solutions.

In addition, a specific operator (CALC_STABILITE) enables to determine the stability or instability of a periodic solution.

So far, these tools are limited to the treatment of point-wise contact without damping. Extension towards non-conservative cases is planned in 2014.

In the pictures below, the non-linear modal behavior of a U-shaped tube is analysed using the tools described above. One can observe the interaction between two mode shapes which do not lie in the same plane.

Figure 1: U-shaped tube with spacer plate	Figure 2: Frequency-Energy Plot (FEP) representing the first non-linear mode
Figure 3: Deformed shape of the periodic solution (1) in Figure 2	Figure 4: Deformed shape of the periodic solution (2) in Figure 2	Figure 5: Deformed shape of the periodic solution (3 ) in Figure 2

[1] E.H. Moussi, Analyse de structures vibrantes dotées de non-linéarités localisées à jeu à l’aide des modes non-linéaires. Thèse de doctorat de l’université d’Aix-Marseille, 2013.