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 00:5816REVIEW 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 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 (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 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 In order to obtain the static equilibrium conditions of the suspensions of GW detector test masses,1–3 or to prop￾erly design the position active control strategy,6 one must describe the low frequency behaviour of the suspension el￾ements. 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 con￾nected 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 angu￾lar frequency, z is the coordinate along the wire, and X is the Fourier transform of the wire deflection. Notice that the Ro￾man 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 bend￾ing 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 transla￾tional 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