Elastodynamics and resonances in elliptical geometry

Stéphane Ancey (1) , Emmanuelle Bazzali (1) , Paul Gabrielli (1) , Michaël Mercier (1)

1 SPE - Sciences pour l'environnement

The resonant modes of two-dimensional elastic elliptical objects are studied from a modal formalism by emphasizing the role of the symmetries of the objects. More precisely, as the symmetry is broken in the transition from the circular disc to the elliptical one, splitting up of resonances and level crossings are observed. From the mathematical point of view, this observation can be explained by the broken invariance of the continuous symmetry group O(2) associated to the circular disc. The elliptical disc is however invariant under the finite group C 2v and the resonances are classified and associated with a given irreducible representation of this group. The main difficulty stands in the application of the group theory in elastodynamics where the vectorial formalism is used to express the physical quantities (elastic displacement and stress) involved in the boundary conditions. However, this method significantly simplifies the numerical treatment of the problem which is uncoupled over the four irreducible representations of C 2v. This provides a full classification of the resonances. They are tagged and tracked as the eccentricity of the elliptical disc increases. Then, the splitting up of resonances, which occurs in the transition from the circular disc to the elliptic one, is emphasized. The computation of vibrational normal modes displacement also highlights mode splittings. A physical interpretation of resonances in terms of geometrical paths is provided.

Domaines: Acoustique [physics.class-ph]

Stéphane Ancey, Emmanuelle Bazzali, Paul Gabrielli, Michaël Mercier. Elastodynamics and resonances in elliptical geometry. Journal of Physics A General Physics (1968-1972), 2013, pp.435204. ⟨10.1088/1751-8113/46/43/435204⟩. ⟨hal-01150359⟩