Splitting up resonances of elastic elliptical disc

Abstract

The elastodynamic resonances of two-dimensional elliptical objects are studied from a modal formalism by emphasizing the role of the symmetries as the circumference is deformed from circular to elliptical geometry. More precisely, as the symmetry is broken in the transition from the circular disc to the ellipse, resonance splittings and level crossings are observed. This observation can be mathematically explained by the broken invariance of the continuous O(2) symmetry group of the circular disc. However, the ellipse remains invariant under the finite C2v group. The main difficulty comes from the application of the group theory in elastodynamics since the vectorial formalism is used to express the physical quantities involved in the boundary conditions. This method significantly simplifies numerical computations and provides a full classification of the resonances. The vibrational normal modes are computed. We focus on the resonance splittings in the transition from the circular disc to the elliptical one. Then, the resonances are tagged and tracked as the eccentricity of the ellipse increases. A series of experiment on three-dimensional objects are also carried out to emphasize the physical effects described above, although no quantitative comparison can be done between theory and experiment. We expect that those effects in 2D appear also in 3D when the sphere is deformed to the spheroid.