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CrossMark REVIEW OF SCIENTIFIC INSTRUMENTS 84.033904(2013) A tool for measuring the bending length in thin wires M.Lorenzini,1.a)G.Cagnoli,12 E.Cesarini,13 G.Losurdo,1 F.Martelli,1.3 F.Piergiovanni,1.3 F.Vetrano,1.3 and A.Vicere1.3 INFN.Sezione di Firenze.50019 Sesto Fiorentino,Italy 2Laboratoire des Materiaux Avances (LMA).IN2P3/CNRS.69622 Villeurbanne,Lyon,France Universita degli Studi di Urbino'Carlo Bo.61029 Urbino.Italy (Received 15 November 2012:accepted 8 March 2013:published online 29 March 2013) Great effort is currently being put into the development and construction of the second genera- tion,advanced gravitational wave detectors,Advanced Virgo and Advanced LIGO.The develop- ment of new low thermal noise suspensions of mirrors,based on the experience gained in the pre- vious experiments,is part of this task.Quasi-monolithic suspensions with fused silica wires avoid the problem of rubbing friction introduced by steel cradle arrangements by directly welding the wires to silica blocks bonded to the mirror.Moreover,the mechanical loss level introduced by silica (10-7 in thin fused silica wires)is by far less than the one associated with steel.The low frequency dynamical behaviour of the suspension can be computed and optimized,provided that the wire bending shape under pendulum motion is known.Due to the production process,fused silica wires are thicker near the two ends(necks),so that analytical bending computations are very complicated.We developed a tool to directly measure the low frequency bending param- eters of fused silica wires,and we tested it on the wires produced for the Virgo+monolithic suspensions.The working principle and a set of test measurements are presented and explained. 2013 American Institute of Physics.[http://dx.doi.org/10.1063/1.4796095] I.INTRODUCTION The bending point is actually the intersection of two lines that are tangent to the end and to the middle part of the wire In order to obtain the static equilibrium conditions of the suspensions of GW detector test masses,-3 or to prop- (see Figure 1).Therefore,each wire end has a specific bend- erly design the position active control strategy,o one must ing length,and for cylindrical wires,the two A are equal. In the general case of a wire with a diameter that changes describe the low frequency behaviour of the suspension el- ements.Modeling of the suspension wires at low frequency along the length,since the moment of inertia also changes, can be conveniently performed by using a 3-segment model,? the best way to obtain the position of the bending points is that is,a simple model consisting of three rigid segments con- direct measurement,or,assuming a precise knowledge of the nected at the effective bending points of the wire.The only wire profile,finite element analysis.Anyway,the validity of parameters required for this model are the two bending points, the 3-segment model is unchanged. corresponding to the upper and lower ends. For a cylindrical wire having a constant thickness the model can be analytically described,by solving the beam Il.EXPERIMENTAL SETUP equation for a slightly deflected rod under a tension T in the frequency domain:8 In principle,the detection of the bending point position can be performed as follows.Consider a thin wire clamped at EIXIV(z)-TX ()pS@-X(z)=0, its top to a rotary stage,supporting a load T(refer to Figure 2). where E is the Young's modulus,I is the (constant)moment If the bending point is placed exactly along the rotation axis of the stage,when the top of the wire is rotated,the bottom of inertia of the cross section S,p is the density,w is the angu- part does not move according to the definition of the bending lar frequency,z is the coordinate along the wire,and X is the length.If instead the bending point does not correspond to the Fourier transform of the wire deflection.Notice that the Ro- rotation center,the wire bottom moves in different directions man numerals refer to the order of the derivatives with respect depending on whether the bending point is above or below the to 3.A general solution can be found in the form: center itself. X(z)=Ae-i/Be-(L-2/+C cos(pa)+D sin(p). A specific apparatus,shown in Figure 3,has been built to perform the measurement of A of a thin wire.A vertical where,at low frequency,.p≈0andi≈√EI/T is called rotary stage,mounted at the top of a rigid structure,is driven bending length and corresponds to the distance of the bending point from the wire end.One can show?that the knowledge by a motor which is controlled by a computer program;on the stage,a micrometric linear translation stage,sliding along of A fully describes the wire dynamics in the low frequency a diameter of the rotary stage,can be controlled by means limit. of a joystick.The upper end of the wire (usually placed in a clamp structure)is tightly clamped to the micrometric transla- a)Electronic mail:matteo.lorenzini@roma2.infn.it tional stage.The position of the center of rotation is found by 0034-6748/2013/84(3)/033904/5/$30.00 84,033904-1 2013 American Institute of Physics Reuse of AlP Publishing content is subject to the terms at:https://publishing.aip.org/authors/rights-and-permissions Downlo8dolP:183.195251.60:Fi.22Apr2016 00:5816

REVIEW OF SCIENTIFIC INSTRUMENTS 84, 033904 (2013) A tool for measuring the bending length in thin wires M. Lorenzini,1,a) G. Cagnoli,1,2 E. Cesarini,1,3 G. Losurdo,1 F. Martelli,1,3 F. Piergiovanni,1,3 F. Vetrano,1,3 and A. Viceré1,3 1INFN, Sezione di Firenze, 50019 Sesto Fiorentino, Italy 2Laboratoire des Matériaux Avancés (LMA), IN2P3/CNRS, 69622 Villeurbanne, Lyon, France 3Università degli Studi di Urbino ‘Carlo Bo’, 61029 Urbino, Italy (Received 15 November 2012; accepted 8 March 2013; published online 29 March 2013) Great effort is currently being put into the development and construction of the second generation, advanced gravitational wave detectors, Advanced Virgo and Advanced LIGO. The development of new low thermal noise suspensions of mirrors, based on the experience gained in the previous experiments, is part of this task. Quasi-monolithic suspensions with fused silica wires avoid the problem of rubbing friction introduced by steel cradle arrangements by directly welding the wires to silica blocks bonded to the mirror. Moreover, the mechanical loss level introduced by silica (φfs ∼ 10−7 in thin fused silica wires) is by far less than the one associated with steel. The low frequency dynamical behaviour of the suspension can be computed and optimized, provided that the wire bending shape under pendulum motion is known. Due to the production process, fused silica wires are thicker near the two ends (necks), so that analytical bending computations are very complicated. We developed a tool to directly measure the low frequency bending parameters of fused silica wires, and we tested it on the wires produced for the Virgo+ monolithic suspensions. The working principle and a set of test measurements are presented and explained. © 2013 American Institute of Physics. [http://dx.doi.org/10.1063/1.4796095] I. INTRODUCTION In order to obtain the static equilibrium conditions of the suspensions of GW detector test masses,1–3 or to properly design the position active control strategy,6 one must describe the low frequency behaviour of the suspension elements. Modeling of the suspension wires at low frequency can be conveniently performed by using a 3-segment model,7 that is, a simple model consisting of three rigid segments connected at the effective bending points of the wire. The only parameters required for this model are the two bending points, corresponding to the upper and lower ends. For a cylindrical wire having a constant thickness the model can be analytically described, by solving the beam equation for a slightly deflected rod under a tension T in the frequency domain:8 EIXIV (z) − T XII (z) − ρSω2 X(z) = 0, where E is the Young’s modulus, I is the (constant) moment of inertia of the cross section S, ρ is the density, ω is the angular frequency, z is the coordinate along the wire, and X is the Fourier transform of the wire deflection. Notice that the Roman numerals refer to the order of the derivatives with respect to z. A general solution can be found in the form: X(z) = Ae−z/λ + Be−(L−z)/λ + C cos(pλ) + D sin(pλ), where, at low frequency, p ≈ 0 and λ ≈ √EI/T is called bending length and corresponds to the distance of the bending point from the wire end. One can show7 that the knowledge of λ fully describes the wire dynamics in the low frequency limit. a)Electronic mail: matteo.lorenzini@roma2.infn.it The bending point is actually the intersection of two lines that are tangent to the end and to the middle part of the wire (see Figure 1). Therefore, each wire end has a specific bending length, and for cylindrical wires, the two λ are equal. In the general case of a wire with a diameter that changes along the length, since the moment of inertia I also changes, the best way to obtain the position of the bending points is direct measurement, or, assuming a precise knowledge of the wire profile, finite element analysis. Anyway, the validity of the 3-segment model is unchanged. II. EXPERIMENTAL SETUP In principle, the detection of the bending point position can be performed as follows. Consider a thin wire clamped at its top to a rotary stage, supporting a load T (refer to Figure 2). If the bending point is placed exactly along the rotation axis of the stage, when the top of the wire is rotated, the bottom part does not move according to the definition of the bending length. If instead the bending point does not correspond to the rotation center, the wire bottom moves in different directions depending on whether the bending point is above or below the center itself. A specific apparatus, shown in Figure 3, has been built to perform the measurement of λ of a thin wire. A vertical rotary stage, mounted at the top of a rigid structure, is driven by a motor which is controlled by a computer program; on the stage, a micrometric linear translation stage, sliding along a diameter of the rotary stage, can be controlled by means of a joystick. The upper end of the wire (usually placed in a clamp structure) is tightly clamped to the micrometric translational stage. The position of the center of rotation is found by 0034-6748/2013/84(3)/033904/5/$30.00 © 2013 American Institute of Physics 84, 033904-1 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

033904-2 Lorenzini et al. Rev.Sci.Instrum.84,033904(2013) MICROMETRICAL MOTOR Bending length ROTARY STAGE CLAMP FIBRE SHADOW SENSOR FIG.1.Sketch of the definition of bending length A. inserting in the hollow shaft of the rotary stage a steel bar on which a cut is machined,precisely indicating the rotation axis. A load was hung from the bottom of the wire;since the bending position depends on the load,T must be chosen prop- erly,according to the purpose of the measurement,and de- clared together with the results. COIL SUPPORT Once suspended,the loaded mass starts swinging at the pendulum frequency.Since the swing can be so large to com- FIG.3.Picture of the apparatus for the measurement of the bending point pletely mask out the displacement induced by the rotating position.The coil actuator under the load was used for exciting the bouncing stage,pendulum motion has to be effectively reduced.A mode of the loaded mass. rough damping of the swing was obtained using a PVC bar fixed to the load mass and inserted into a rubber pipe(see sition.The oscillating displacement(Figure 5)is recorded by Figure 4)which was in frictional contact with the bench the apparatus was placed on.The movements of the bottom part the sensor and its amplitude is obtained,while the distance of the wire are detected using a split photodiode forming a among the clamped end of the wire and the rotation axis is shadow sensor.A low pass filter was used to further reduce measured using a caliper.After that,the head is displaced by the signal from the pendulum motion. a small amount along the micrometric translational stage and When the rotary stage motor is driven with a periodic the resulting oscillation amplitude is again acquired.By re- signal,a small oscillation of the wire head occurs.The pe- peating this procedure,the profile of the amplitude versus the riod of this motion is chosen to be large compared to that of spatial coordinate along the wire is determined.The bending the pendulum swing.At the beginning,the bending point will point is thus found between the two subsequent measurement not be placed,in general,along the rotation axis:thus,the positions whose recorded amplitude is at the minimum. bottom part of the wire starts moving around a centered po- In all realistic situations,the amplitude of the oscillation never becomes zero,not even when the bending point is per- fectly aligned with the rotation axis.A residual oscillation is ROTARY STAGE always present,probably due to the presence of the rubber damper.When the wire head is rotating,even in correspon- CLAMP dence of the bending point,while the horizontal displacement is made zero,the load is vertically displaced.The rubber pipe, being in contact with the supporting table,is responsible for FIBRE FIBRE LOAD MAGNET LOAD eertealretio C ROD side he FIG.2.Scheme of the measurement method.The wire is clamped on a rotary stage and supports a load which is free to move in the horizontal direction. RUBBER PIPE When the bending point stays exactly on the axis of the rotary stage,the load does not move when the stage is rotated.The bending length can be evaluated FIG.4.Schematic view (left)and picture (right)of the rubber pipe used to as the distance between the clamp and the rotation axis. damp the pendulum swing. Reuse of AlP 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

033904-2 Lorenzini et al. Rev. Sci. Instrum. 84, 033904 (2013) FIG. 1. Sketch of the definition of bending length λ. inserting in the hollow shaft of the rotary stage a steel bar on which a cut is machined, precisely indicating the rotation axis. A load was hung from the bottom of the wire; since the bending position depends on the load, T must be chosen properly, according to the purpose of the measurement, and declared together with the results. Once suspended, the loaded mass starts swinging at the pendulum frequency. Since the swing can be so large to completely mask out the displacement induced by the rotating stage, pendulum motion has to be effectively reduced. A rough damping of the swing was obtained using a PVC bar fixed to the load mass and inserted into a rubber pipe (see Figure 4) which was in frictional contact with the bench the apparatus was placed on. The movements of the bottom part of the wire are detected using a split photodiode forming a shadow sensor.9 A low pass filter was used to further reduce the signal from the pendulum motion. When the rotary stage motor is driven with a periodic signal, a small oscillation of the wire head occurs. The period of this motion is chosen to be large compared to that of the pendulum swing. At the beginning, the bending point will not be placed, in general, along the rotation axis: thus, the bottom part of the wire starts moving around a centered poFIG. 2. Scheme of the measurement method. The wire is clamped on a rotary stage and supports a load which is free to move in the horizontal direction. When the bending point stays exactly on the axis of the rotary stage, the load does not move when the stage is rotated. The bending length can be evaluated as the distance between the clamp and the rotation axis. FIG. 3. Picture of the apparatus for the measurement of the bending point position. The coil actuator under the load was used for exciting the bouncing mode of the loaded mass. sition. The oscillating displacement (Figure 5) is recorded by the sensor and its amplitude is obtained, while the distance among the clamped end of the wire and the rotation axis is measured using a caliper. After that, the head is displaced by a small amount along the micrometric translational stage and the resulting oscillation amplitude is again acquired. By repeating this procedure, the profile of the amplitude versus the spatial coordinate along the wire is determined. The bending point is thus found between the two subsequent measurement positions whose recorded amplitude is at the minimum. In all realistic situations, the amplitude of the oscillation never becomes zero, not even when the bending point is perfectly aligned with the rotation axis. A residual oscillation is always present, probably due to the presence of the rubber damper. When the wire head is rotating, even in correspondence of the bending point, while the horizontal displacement is made zero, the load is vertically displaced. The rubber pipe, being in contact with the supporting table, is responsible for FIG. 4. Schematic view (left) and picture (right) of the rubber pipe used to damp the pendulum swing. 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

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 00:58:16

033904-3 Lorenzini et al. Rev. Sci. Instrum. 84, 033904 (2013) FIG. 5. Example of the variation of the oscillation amplitude when the vertical 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 respect to the wire head rotation, the induced horizontal oscillation 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 structure. 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 photodiode. 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 analytically calculated, and this value was compared with the experimental measurement. This gives a proof of the working 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 automatic 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 original bar. These two necks are not symmetric due to the difference in the velocity profile of the pulling process, that starts slowly and stops abruptly. As a consequence of this particular profile, the effective bending length cannot be obtained by analytical expressions, and it must be evaluated either experimentally or using finite element analysis. We followed both methods and we found the same result, within the experimental errors. A. Test measurement A measurement has been carried on a commercial harmonic 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 sensors 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 modulus, 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

033904-4 Lorenzini et al. Rev.Sci.Instrum.84,033904(2013) The bending point was measured as described in Sec.II. In this case,the wire head is directly clamped to the micro- metric translational stage.The measurement is thus referred to the true end of the wire (a quite different arrangement is used for silica wires,for reasons that will be explained in the fol- lowing).The recorded peak to peak amplitude of the swinging motion under the applied rotation of the upper end,as a func- tion of the vertical displacement,is shown in Figure 6.The ex- perimental points have been fitted with a parabolic curve and the position of the minimum of the curve has been obtained. The resulting bending length,referred to the clamped end of the wire,is =6.6+0.1 mm.This value is in agreement within experimental error with the one previously obtained by making use of the definition of. B.Measurement on silica wires The geometry of a typical silica wire produced for the quasi-monolithic assembly in Virgo+is shown in Figure 7. Two silica parts,called cone and anchor,are welded to the ends of the silica bar before the pulling process.5 This particu- lar design choice is explained by the following considerations. A well known problem in handling silica parts is that even a FIG.8.Sketch of the steel cone holder blocking one end of the wire. very soft touch produces a mechanical damaging,a crack,that causes an abrupt decrease in the breaking strength,eventually leading to a suspension failure.Therefore,one has to always avoid any contact with the thin wires.To handle them during the production and measurement process,the wire ends are welded to chunky parts that in turn are placed in dedicated steel clamping structures,as shown in Figure 8 for the silica cone.(The particular shapes of cones and anchors are opti- mized for the integration in the quasi-monolithic assembly.) The difference in profile and clamping parts at the two ends of the wire is reflected by two different values for the bending length,that can be marked as Ac (cone)and A(an- chor).For the sake of this article,we will consider only the measurement of Ac,but the case of the anchor is completely similar.Considering again Figure 8,we notice that hereon the bending length value is referred to the back plane surface of the cone holder. 0,14 0,12 0.1 0.08 0.06 0.04 0.02 y=157420x2-10687x+181,42 3,30E-02 3.34E-02 338E-02 3.42E-02 3,46E02 3,50E-02 Vertical displacement [m] FIG.7.Scheme of a typical silica wire produced to be assembled in the Virgo+quasi-monolithic suspension. FIG.9.Oscillation peak to peak amplitude of the silica wire with respect to the vertical displacement.Experimental points are fitted with a quadratic function.The resulting bending length is =33.90.1 mm. Reuse of AlP 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

033904-4 Lorenzini et al. Rev. Sci. Instrum. 84, 033904 (2013) The bending point was measured as described in Sec. II. In this case, the wire head is directly clamped to the micrometric translational stage. The measurement is thus referred to the true end of the wire (a quite different arrangement is used for silica wires, for reasons that will be explained in the following). The recorded peak to peak amplitude of the swinging motion under the applied rotation of the upper end, as a function of the vertical displacement, is shown in Figure 6. The experimental points have been fitted with a parabolic curve and the position of the minimum of the curve has been obtained. The resulting bending length, referred to the clamped end of the wire, is λ = 6.6 ± 0.1 mm. This value is in agreement within experimental error with the one previously obtained by making use of the definition of λ. B. Measurement on silica wires The geometry of a typical silica wire produced for the quasi-monolithic assembly in Virgo+ is shown in Figure 7. Two silica parts, called cone and anchor, are welded to the ends of the silica bar before the pulling process.5 This particular design choice is explained by the following considerations. A well known problem in handling silica parts is that even a FIG. 7. Scheme of a typical silica wire produced to be assembled in the Virgo+ quasi-monolithic suspension. FIG. 8. Sketch of the steel cone holder blocking one end of the wire. very soft touch produces a mechanical damaging, a crack, that causes an abrupt decrease in the breaking strength, eventually leading to a suspension failure. Therefore, one has to always avoid any contact with the thin wires. To handle them during the production and measurement process, the wire ends are welded to chunky parts that in turn are placed in dedicated steel clamping structures, as shown in Figure 8 for the silica cone. (The particular shapes of cones and anchors are optimized for the integration in the quasi-monolithic assembly.) The difference in profile and clamping parts at the two ends of the wire is reflected by two different values for the bending length, that can be marked as λC (cone) and λA (anchor). For the sake of this article, we will consider only the measurement of λC, but the case of the anchor is completely similar. Considering again Figure 8, we notice that hereon the bending length value is referred to the back plane surface of the cone holder. FIG. 9. Oscillation peak to peak amplitude of the silica wire with respect to the vertical displacement. Experimental points are fitted with a quadratic function. The resulting bending length is λ = 33.9 ± 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

033904-5 Lorenzini et al. Rev.Sci.Instrum.84,033904(2013) x10 specifying a different moment of inertia for each mesh ele- ment.Finally,deflection at each point along the wire is ob- -fibre displacement tained by the equation of motion written in terms of mass and -linear fit stiffness matrices.The wire bending is computed by imposing a static value for the deflection at the free end of the wire. It must be underlined that this method is in general much more complicated than the direct measurement of A.mainly due to the acquisition of the wire profile(a profiling machine has been developed in Glasgow,that allows automated mea- surement of the diameter along the wire's length;this is nec- essary for the estimation of the thermoelastic noisel0).In the present case,the profile has been obtained by a camera picture of the wire head,processed with Matlab graphic tools.The output of the simulation tool is shown in Figure 10;the result- 25 30 35 40 ing bending length is Ac=33.85+0.05 mm.This test con- Position along the wire [mm] firms the validity of the obtained measurement of the bending point. FIG.10.Simulated bending shape of the silica wire (blue line)and linear fit after the bending region(red line).The bending point,as from definition,is located at the crossing among the red line and the zero-deformation axis,that is.入=33.85±0.05mm. IV.CONCLUSIONS This work completely demonstrates the efficiency of our First of all,the characteristic frequencies of the wire have setup in determining experimentally the position of the bend- been determined,obtaining ing points in wires with varying diameter.The setup further fB=5.735±0.006，fv.1=451.30±0.2Hz. allows the measurement of the frequencies of the bouncing mode and of the first violin mode,such that a full characteri- An estimation of the expected values for these frequencies can zation of the wire can be simply made;the latter is needed to be given,defining an effective length L of the wire equal to properly install the test mass suspension system and to design the distance among the two bending points.Assuming usual the position control strategy.The described method has been parameters values for the produced silica wire,such as the successfully employed during the installation of the quasi- measured length L=670.0+0.3 mm,the measured diameter monolithic payload in Virgo+. d =140.0+0.5 um,the Young's modulus E 72.5 GPa as retrieved in literature,we find ACKNOWLEDGMENTS fB=5.64±0.20，fv,1=462±17Hz. Part of the presented experimental work has been funded The main contribution to the error in the estimation comes by the EGO consortium in Cascina(PD),which also provided from the diameter uncertainty.These numbers are in agree- the employed facilities and spaces. ment with the measured ones within experimental error. To allow the bending point measurement,the cone holder IThe relevant documentation on interferometric gravitational wave de- tectors can be found in the Advanced Virgo Technical Design Report has been clamped to the micrometric translational stage.The available at http://wwwcascina.virgo.infn.it/advirgo/docs.html and in Ad- recorded peak to peak amplitude of the swinging motion un- vanced LIGO Reference Design available at https://dcc.ligo.org/cgi-bin/ der the applied rotation of the upper end,as a function of the DocDB/ShowDocument?docid=m060056. vertical displacement,is shown in Figure 9.Again,the exper- M.Lorenzini,"Suspension thermal noise issues for advanced GW inter- ferometric detectors."Ph.D.dissertation (Universita di Firenze,2008).see imental points have been fitted with a parabolic curve and the https://gwic.ligo.org/thesisprize/2007/Lorenzini_Thesis.pdf. position of the minimum of the curve has been obtained.The 3F.Piergiovanni,M.Punturo,and P.Puppo."The thermal noise of the resulting bending length is入c=33.9±0.1mm. Virgo+and Virgo Advanced Last Stage Suspension (the PPP effect).," A different approach to determine the bending point is to Virgo Note:VIR-0015E-09,2009. P.Amico,L.Bosi,L.Carbone,L.Gammaitoni,M.Punturo,F.Travasso, use finite element analysis.Once the profile of the wire has and H.Vocca.Class.Quantum Grav.19,1669 (2002). been recorded,this task can be performed making use of a 5M.Lorenzini on behalf of Virgo Collaboration,Class.Quantum Grav.27. dedicated simulation Matlab code developed by Piergiovanni 084021(2010) et al.?This code is based on a one-dimensional mesh along 6F.Acernese er al.Astropart.Phys.20.617-628(2004). 7F.Piergiovanni.M.Lorenzini,G.Cagnoli,E.Campagna,E.Cesarini,G. the wire length.Within each element of the mesh,the wire de- Losurdo,F.Martelli,F.Vetrano,and A.Vicere,J.Phys.Conf.Ser.228. flection is expressed using Hermite cubics interpolation func- 012017(2010). tions.The coefficients of these functions are obtained by im- BL.D.Landau and E.M.Lifshitz,Theory of Elasticiry,2nd ed.(Pergamon Press,Oxford,1970). posing the boundary conditions,then the mass and stiffness 9G.Cagnoli,L.Gammaitoni,J.Kovalik,F.Marchesoni,and M.Punturo, matrices are computed by the definition of kinetic and poten- Phys.Lett.A213,245(1996). tial energy.The variable profile of the wire is accounted for 10A.Cumming et al,Rev.Sci.Instrum.82.044502(2011). Reuse of AlP 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

033904-5 Lorenzini et al. Rev. Sci. Instrum. 84, 033904 (2013) FIG. 10. Simulated bending shape of the silica wire (blue line) and linear fit after the bending region (red line). The bending point, as from definition, is located at the crossing among the red line and the zero-deformation axis, that is, λ = 33.85 ± 0.05 mm. First of all, the characteristic frequencies of the wire have been determined, obtaining fB = 5.735 ± 0.006, fV,1 = 451.30 ± 0.2 Hz. An estimation of the expected values for these frequencies can be given, defining an effective length L of the wire equal to the distance among the two bending points. Assuming usual parameters values for the produced silica wire, such as the measured length L = 670.0 ± 0.3 mm, the measured diameter d = 140.0 ± 0.5μm, the Young’s modulus E = 72.5 GPa as retrieved in literature, we find fB = 5.64 ± 0.20, fV,1 = 462 ± 17 Hz. The main contribution to the error in the estimation comes from the diameter uncertainty. These numbers are in agreement with the measured ones within experimental error. To allow the bending point measurement, the cone holder has been clamped to the micrometric translational stage. The recorded peak to peak amplitude of the swinging motion under the applied rotation of the upper end, as a function of the vertical displacement, is shown in Figure 9. Again, the experimental points have been fitted with a parabolic curve and the position of the minimum of the curve has been obtained. The resulting bending length is λC = 33.9 ± 0.1 mm. A different approach to determine the bending point is to use finite element analysis. Once the profile of the wire has been recorded, this task can be performed making use of a dedicated simulation Matlab code developed by Piergiovanni et al.7 This code is based on a one-dimensional mesh along the wire length. Within each element of the mesh, the wire de- flection is expressed using Hermite cubics interpolation functions. The coefficients of these functions are obtained by imposing the boundary conditions, then the mass and stiffness matrices are computed by the definition of kinetic and potential energy. The variable profile of the wire is accounted for specifying a different moment of inertia for each mesh element. Finally, deflection at each point along the wire is obtained by the equation of motion written in terms of mass and stiffness matrices. The wire bending is computed by imposing a static value for the deflection at the free end of the wire. It must be underlined that this method is in general much more complicated than the direct measurement of λ, mainly due to the acquisition of the wire profile (a profiling machine has been developed in Glasgow, that allows automated measurement of the diameter along the wire’s length; this is necessary for the estimation of the thermoelastic noise10). In the present case, the profile has been obtained by a camera picture of the wire head, processed with Matlab graphic tools. The output of the simulation tool is shown in Figure 10; the resulting bending length is λC = 33.85 ± 0.05 mm. This test con- firms the validity of the obtained measurement of the bending point. IV. CONCLUSIONS This work completely demonstrates the efficiency of our setup in determining experimentally the position of the bending points in wires with varying diameter. The setup further allows the measurement of the frequencies of the bouncing mode and of the first violin mode, such that a full characterization of the wire can be simply made; the latter is needed to properly install the test mass suspension system and to design the position control strategy. The described method has been successfully employed during the installation of the quasimonolithic payload in Virgo+. ACKNOWLEDGMENTS Part of the presented experimental work has been funded by the EGO consortium in Cascina (PI), which also provided the employed facilities and spaces. 1The relevant documentation on interferometric gravitational wave detectors can be found in the Advanced Virgo Technical Design Report available at http://wwwcascina.virgo.infn.it/advirgo/docs.html and in Advanced LIGO Reference Design available at https://dcc.ligo.org/cgi-bin/ DocDB/ShowDocument?docid=m060056. 2M. Lorenzini, “Suspension thermal noise issues for advanced GW interferometric detectors,” Ph.D. dissertation (Università di Firenze, 2008), see https://gwic.ligo.org/thesisprize/2007/Lorenzini_Thesis.pdf. 3F. Piergiovanni, M. Punturo, and P. Puppo, “The thermal noise of the Virgo+ and Virgo Advanced Last Stage Suspension (the PPP effect),” Virgo Note: VIR-0015E-09, 2009. 4P. Amico, L. Bosi, L. Carbone, L. Gammaitoni, M. Punturo, F. Travasso, and H. Vocca, Class. Quantum Grav. 19, 1669 (2002). 5M. Lorenzini on behalf of Virgo Collaboration, Class. Quantum Grav. 27, 084021 (2010) 6F. Acernese et al., Astropart. Phys. 20, 617–628 (2004). 7F. Piergiovanni, M. Lorenzini, G. Cagnoli, E. Campagna, E. Cesarini, G. Losurdo, F. Martelli, F. Vetrano, and A. Viceré, J. Phys. Conf. Ser. 228, 012017 (2010). 8L. D. Landau and E. M. Lifshitz, Theory of Elasticity, 2nd ed. (Pergamon Press, Oxford, 1970). 9G. Cagnoli, L. Gammaitoni, J. Kovalik, F. Marchesoni, and M. Punturo, Phys. Lett. A 213, 245 (1996). 10A. Cumming et al., Rev. Sci. Instrum. 82, 044502 (2011). 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

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上海交通大学：《探索物质微观世界》课程教学资源（补充材料）引力波_A tool for measuring the bending length in thin wires