Using Sphere Symmetry Breaking to Calculate SCHENBERG’s Antenna Quadrupolar Frequencies

Gravitational waves (GW) play an important role in the understanding of several astrophysical objects, like neutron stars and black holes. One technology used to detect them involves massive objects that vibrate as GW cross it, and the detectors built are, accordingly, of the resonant-mass type. SCHENBERG is a resonant-mass GW detector, built in Brazil, whose antenna is a spherical, 65 cm in diameter mass made of a CuAl alloy, and its quadrupole vibrational modes would be excited by GW, as predicted by general relativity. The chosen alloy can be cooled down to mK temperatures with a good mechanical quality factor. The quadrupole mode frequencies were measured at 4K, and a frequency band of about 67.5 Hz was found, but when the antenna was simulated in SolidWorks FEM software version 2010–2011 (as well as in Ansys SpaceClaimTM), the band obtained for a free sphere was different—around 30 Hz. When the holes for the suspension were included in the simulation, the same discrepancy persisted. In this work, gravity was included in the FEM simulation, and we show that the bandwidth results are even smaller. We were then able to obtain a bandwidth close to the measured one by including a small deviation from the vertical axle, as well as variations on the sphere microstructure, which are assumptions that break the symmetry of a perfect, homogeneous free sphere. We believe that the microstructure variations are due to differences in the cooling time during the sphere casting. As for a good mechanical quality factor, the sphere was not submitted to homogenization. With these additions to the FEM simulation, a reasonable frequency distribution was found, consistent with the one measured for SCHENBERG’s antenna.