Published on March 5, 2009
Finite element model of the Virgo MirrorsPaola Puppo : Finite element model of the Virgo MirrorsPaola Puppo Developed a FEM of the Virgo mirrors NI-WI, NE-WE, and BS including all the details of the design; The agreement with the measured frequencies is very good for for all the mirror models; Using this model we can understand why we are not able to see the splitting of first two axysimmetric modes in the test masses; We can also study how the equivalent mass of the modes depends on the laser beam size and centering on the mirror; Slide 2: Input Mirrors made of SUPRASIL 312 Thickness: 96.8 mm Diameter: 350 mm Output Mirrors made of HERASIL 1 Thickness: 95.7 mm Diameter: 350 mm FEM Model Magnets Spacers Markers Modeled the internal modes of the Virgo mirrors; The markers, the spacers and the magnets are modeled following the specifications of the Virgo design; Lateral cuts allowing to perform the silica bonding of spacers; Mirror suspended to C85 wires having a a diameter of 0.2 mm; Specifications r = 2.203 g/cm3 s = 0.165 ; Y = 7.3 1010 N/m2 Slide 3: Poisson ratio s Thickness d Young Modulus Y We computed the internal modes for several values of the s, Y and d; We found a function which best fits the simulated data; We based ourselves on the issues of the elasticity theory in the case of circular shells and cylindric bars; Slide 4: (1,0) Mode m1 = 9.456 10-4 Hz/dyne1/2 m2 = -3.8211 10-5 Hz cm/dyne1/2 m3 = 7.4364 10-7 Hz cm2/dyne1/2 m4 = -5.3039 10-9 Hz cm3/dyne1/2 A10 = 0.241 B10 = 0.8057 (0,2) Mode m1 = 6.9070 10-4 Hz/dyne1/2 m2 = -2.1871 10-5 Hz cm/dyne1/2 m3 = 2.6667 10-7 Hz cm2/dyne1/2 m4 = ---- A02 = 0.886 B02 = 0.0415 (0,3) Mode m1 = 1.5543 10-3 Hz/dyne1/2 m2 = -6.8829 10-5 Hz cm/dyne1/2 m3 = 7.4227 10-7 Hz cm2/dyne1/2 m4 = 1.9751 10-8 Hz cm3/dyne1/2 A03 = 0.841 B03 = 0.0347 Results of the fit Notation: the mode (l,m) has l nodal circumferences and m nodal diameters Summary of the measurements : Summary of the measurements From C2-C3 Using the fitted equation and the internal frequencies values of the North and West cavities that were measured during the runs C2 and C3, we have derived the parameters and the thicknesses of NI, WI, NE and WE mirrors. Slide 6: Results for test masses Simulation Measured Modes splitting NI: (3917.2 ± 0.5) Hz (NI/WI) 3912.6 Hz-3916.7 Hz WI: (3916.0 ± 0.5) Hz Modes splitting NE: (3883.0 ± 0.5) Hz (NE/WE) 3882.4 Hz-3882.6 Hz WE: (3884.2 ± 0.5) Hz (0,2 Mode) The mode splitting is mainly due to the mirror lateral cuts and the lateral magnets and spacers. Slide 7: Simulation Measured NI/WI: 5584.9 Hz NI: (5585.7 ± 0.5) Hz WI: (5583.5 ± 0.5) Hz NE/WE: 5546.1 Hz NE: (5543.2 ± 0.5) Hz WE: (5545.6 ± 0.5) Hz (1,0 Mode) Y = (7.32 ± 0.02) 1010 N/m2 s = (0.164 ± 0.003) dInput=(9.68 ± 0.01) cm dEnd =(9.57 ± 0.01) cm The measured data are in agreement with the model parameters: Slide 8: Simulation North/West Input 7595.3 Hz-7602.6 Hz North/West End 7551 Hz-7558 Hz These modes were not observed. (0,3 Mode) Slide 9: During C2 and C3 we were not able to excite the (0,2) doublets on the test masses Why? .... some remaks about the axisymmetric modes (0,2).... 3882.4 Hz 3882.6 Hz End Mirror Simulation of the mirror harmonic response to a force exerted by the up-down coils of the reference mass; like it was done during C2 and C3 runs. The force we apply selects only one of the (0,2) modes.... Slide 10: ... This is the mode we can excite with two coils: the lower frequency mode...... Slide 11: To see the other (0,2) mode we should use four coils exerting different forces...... but maybe it is enough the small difference in the diagonalization of the driving matrix...... F2 F2 F1 F1 F1 ¹ F2 Slide 12: Measured Beam Splitter Modes on C3 During C3 we have observed the Beam Splitter modes (0,2): (5346.9 ± 0.5) Hz (5355.2 ± 0.5) Hz Finite element model of BS : Finite element model of BS Specifications made of SUPRASIL 311 Thickness: 55.0 mm Diameter: 230 mm No lateral cuts. Magnets and markers are all placed on the BS front side. r = 2.203 g/cm3 s = 0.165 ; Y = 7.27 1010 N/m2 Slide 14: Measured frequencies n02 (1) = (5346.9 ± 0.5) Hz n02 (2) = (5355.2 ± 0.5) Hz Simulation n02 (1) = 5344 Hz n02 (2) = 5355.2 Hz In this case the splitting of the modes depends on the lateral magnets and spacers (0,2 Mode) Slide 15: (1,0 Mode) This mode was not observed. Equivalent mass, beam size and beam position : Equivalent mass, beam size and beam position The interferometer output can detect the mirror displacement because it is sensitive to the position of its center; Moreover we have to consider the beam size on the mirror; The observed coordinate zobs is the average of the mirror displacement over an area defined by the beam size ws... ...weighted by the beam gaussian intensity... ...then it depends on the beam size ws... If the beam is not centered on the mirror, zobs depends also on ro The equivalent mass is defined as: Slide 17: Using the results of the FE Model we have evaluated the equivalent masses for the Input and Output mirrors for the (0,1) mode (5585 Hz/5546 Hz). Input Mirror: ws = 2 cm Output Mirror: ws = 5.5 cm Equivalent mass of the (0,1) mode @ 5 kHz ro (cm) If ro = 2 cm Meq increases of 7% Meq Input = 5.6 kg Meq Output = 6.7 kg Slide 18: Using the results of the FE Model we have evaluated the equivalent masses for the Input and Output mirrors for the (2,0) mode. Input Mirror: ws = 2 cm Output Mirror: ws = 5.5 cm Equivalent mass of the (2,0) mode @ 3.9 kHz Meq (ws,0) > Mmirror If ro = 1 cm as the beam moves from the mirror center, the equivalent mass decreases.