033904-3 Lorenzini et al. Rev.Sci.Instrum.84,033904(2013) melting and pulling apart a cylindrical silica bar in an auto- 0.18 matic way,achieving a very good control on the diameter in the body of the resulting wire.Although being cylindrical in the central region,this wire is not regular due to the pulling process that leaves at both ends two thicker tapered necks as joining regions between the thin pulled part and the thick orig- inal bar.These two necks are not symmetric due to the differ- ence in the velocity profile of the pulling process,that starts slowly and stops abruptly.As a consequence of this particu- lar profile,the effective bending length cannot be obtained by 800 900 1000 1100 1200 1300 analytical expressions,and it must be evaluated either exper Time [s] imentally or using finite element analysis.We followed both methods and we found the same result,within the experimen- FIG.5.Example of the variation of the oscillation amplitude when the verti tal errors cal position is changed.In this case,the wire head is displaced continuously with time,and the amplitude passes through a minimum. A.Test measurement a coupling between the horizontal and vertical displacement. A measurement has been carried on a commercial har- Although the vertical motion has a doubled frequency with re- monic steel cylindrical wire,with a regular diameter of spect to the wire head rotation,the induced horizontal oscilla- 0.8 mm.The wire elasticity was directly evaluated,applying tion occurs at the same frequency.Given this,the best way to increasing stresses and measuring the corresponding strain. determine the bending point position is to find the minimum The resulting value for the Young's modulus is 120+1 GPa. of the total amplitude,for instance by fitting the experimental This allows the two characteristic frequencies fB and fv.i to points with a second order curve.The uncertainty on P,can be calculated assuming a load of 5.3 kg,obtaining: be reasonably assumed of the order of the error associated with the position measurements performed with the caliper, =19.6士0.8，f1=o0=79±2H that is,0.1 mm. fB=2元 2π After that,the bending length is referred to some element We observed these frequencies in the loaded wire,and we at the wire end,which is usually supported in a clamp struc- were able to measure their values making use of shadow sen- ture. sors yielding: The setup is also able to measure the frequency of the bouncing mode fe of the wire and of the first violin mode fv.1. fB=20.23±0.02Hz,fv.1=79.73±0.04Hz using again a displacement sensor made with a split photodi- ode.The theoretical expressions for these two characteristic These numbers are in perfect agreement with the calculated frequencies are ones. Assuming the measured value of the wire Young's mod- 1ES 1 T fB= ulus,the bending length can be obtained analytically making 2元VML' fv.1= 2LV Sp use of the definition given in the Introduction.The result is A =(6.63+0.06)x 10-3 m,that has to be compared with the where M is the loaded mass and L is the wire length. measurement. III.MEASUREMENTS 0.8 Two series of bending length measurements have been 0.7 performed,with the aim of testing the setup performance.The 0.6 load Twas chosen to be 5.3 kg,very close to the working load of the wires suspending the test masses in Virgo.5 0.5 In the first measurement,a commercial harmonic steel 0.4 wire with regular circular cross section was used.Due to the 03 cylindrical shape,the position of the bending point can be an- alytically calculated,and this value was compared with the 0.2 experimental measurement.This gives a proof of the work- 01 y=795286x2-10565x+35,309 ing principle and highlights the precision that can be achieved with the setup,better than 0.1 mm. 0 In the second case,a laser pulled fused silica wire is 5.50E-03 6.00E-03 6.50E-03 7.00E-03 7.50E-03 measured.Fused silica wires"of this kind are employed for Vertical displacement [m] the construction of quasi-monolithic mirror suspensions in FIG.6.Oscillation peak to peak amplitude of the steel wire with respect advanced interferometric gravitational wave detectors.They to the vertical displacement.Experimental points are fitted with a quadratic are produced using a dedicated CO2 laser machine capable of function.The resulting bending length is A=6.6+0.1 mm Reuse of AlP Publishing content is subject to the terms at:https://publishing.aip.org/authors/rights-and-permi ssions.Download to IP:183.195.251.6 On:Fri.22 Apr 2016 00:58:16033904-3 Lorenzini et al. Rev. Sci. Instrum. 84, 033904 (2013) FIG. 5. Example of the variation of the oscillation amplitude when the verti￾cal position is changed. In this case, the wire head is displaced continuously with time, and the amplitude passes through a minimum. a coupling between the horizontal and vertical displacement. Although the vertical motion has a doubled frequency with re￾spect to the wire head rotation, the induced horizontal oscilla￾tion occurs at the same frequency. Given this, the best way to determine the bending point position is to find the minimum of the total amplitude, for instance by fitting the experimental points with a second order curve. The uncertainty on Pλ can be reasonably assumed of the order of the error associated with the position measurements performed with the caliper, that is, 0.1 mm. After that, the bending length is referred to some element at the wire end, which is usually supported in a clamp struc￾ture. The setup is also able to measure the frequency of the bouncing mode fB of the wire and of the first violin mode fV,1, using again a displacement sensor made with a split photodi￾ode. The theoretical expressions for these two characteristic frequencies are fB = 1 2π ES ML, fV,1 = 1 2L T Sρ , where M is the loaded mass and L is the wire length. III. MEASUREMENTS Two series of bending length measurements have been performed, with the aim of testing the setup performance. The load T was chosen to be 5.3 kg, very close to the working load of the wires suspending the test masses in Virgo.5 In the first measurement, a commercial harmonic steel wire with regular circular cross section was used. Due to the cylindrical shape, the position of the bending point can be an￾alytically calculated, and this value was compared with the experimental measurement. This gives a proof of the work￾ing principle and highlights the precision that can be achieved with the setup, better than 0.1 mm. In the second case, a laser pulled fused silica wire is measured. Fused silica wires4 of this kind are employed for the construction of quasi-monolithic mirror suspensions in advanced interferometric gravitational wave detectors. They are produced using a dedicated CO2 laser machine capable of melting and pulling apart a cylindrical silica bar in an auto￾matic way, achieving a very good control on the diameter in the body of the resulting wire.5 Although being cylindrical in the central region, this wire is not regular due to the pulling process that leaves at both ends two thicker tapered necks as joining regions between the thin pulled part and the thick orig￾inal bar. These two necks are not symmetric due to the differ￾ence in the velocity profile of the pulling process, that starts slowly and stops abruptly. As a consequence of this particu￾lar profile, the effective bending length cannot be obtained by analytical expressions, and it must be evaluated either exper￾imentally or using finite element analysis. We followed both methods and we found the same result, within the experimen￾tal errors. A. Test measurement A measurement has been carried on a commercial har￾monic steel cylindrical wire, with a regular diameter of 0.8 mm. The wire elasticity was directly evaluated, applying increasing stresses and measuring the corresponding strain. The resulting value for the Young’s modulus is 120 ± 1 GPa. This allows the two characteristic frequencies fB and fV,1 to be calculated assuming a load of 5.3 kg, obtaining: fB = ωB 2π = 19.6 ± 0.8 Hz, fV,1 = ωV,1 2π = 79 ± 2 Hz. We observed these frequencies in the loaded wire, and we were able to measure their values making use of shadow sen￾sors yielding: fB = 20.23 ± 0.02 Hz, fV,1 = 79.73 ± 0.04 Hz. These numbers are in perfect agreement with the calculated ones. Assuming the measured value of the wire Young’s mod￾ulus, the bending length can be obtained analytically making use of the definition given in the Introduction. The result is λ = (6.63 ± 0.06) × 10−3 m, that has to be compared with the measurement. FIG. 6. Oscillation peak to peak amplitude of the steel wire with respect to the vertical displacement. Experimental points are fitted with a quadratic function. The resulting bending length is λ = 6.6 ± 0.1 mm. Reuse of AIP Publishing content is subject to the terms at: https://publishing.aip.org/authors/rights-and-permissions. Download to IP: 183.195.251.6 On: Fri, 22 Apr 2016 00:58:16