LIGO-P040042-A-D ARTICLE IN PRESS
Nuclear Instruments and Methods in Physics Research A 556 (2006) 616–623 www.elsevier.com/locate/nima
Mechanical design of a single-axis monolithic accelerometer for advanced seismic attenuation systems Alessandro Bertolinia,b,, Riccardo DeSalvob, Francesco Fidecaroa, Mario Francesconia, Szabolcs Markac, Virginio Sannibaleb, Duccio Simonettia, Akiteru Takamorib,d, Hareem Tariqb a Dipartimento di Fisica dell’Universita` di Pisa and INFM, Largo Pontecorvo 2, I-56127 Pisa, Italy LIGO Project, California Institute of Technology, 1200 E. California Blvd., Pasadena, CA 91125, USA c Department of Physics, Columbia University, 538 W. 120th St., New York, NY 10027, USA d Earthquake Research Institute, University of Tokyo, 1-1-1 Yayoi, Bunkyo-Ku, Tokyo 113-0032, Japan
b
Received 17 August 2005; received in revised form 30 October 2005; accepted 30 October 2005 Available online 28 November 2005
Abstract The design and mechanics for a new very-low noise low frequency horizontal accelerometer is presented. The sensor has been designed to be integrated in an advanced seismic isolation system for interferometric gravitational wave detectors. The motion of a small monolithic folded-pendulum (FP) is monitored by a high resolution capacitance displacement sensor; a feedback force actuator keeps the mass at the equilibrium position. The feedback signal is proportional to the ground acceleration in the frequency range 0–150 Hz. The very high mechanical quality factor, Q ’ 3000 at a resonant frequency of 0.5 Hz, reduces the Brownian motion of the proof mass of the accelerometer below the resolution of the displacement sensor. This scheme enables the accelerometer to detect the inertial displacement of a platform with a root-mean-square noise less than 1 nm, integrated over the frequency band from 0.01 to 150 Hz. The FP geometry, combined with the monolithic design, allows the accelerometer to be extremely directional. A vertical–horizontal coupling ranging better than 103 has been achieved. A detailed account of the design and construction of the accelerometer is reported here. The instrument is fully ultra-high vacuum compatible and has been tested and approved for integration in seismic attenuation system of japanese TAMA 300 gravitational wave detector. The monolithic design also makes the accelerometer suitable for cryogenic operation. r 2005 Elsevier B.V. All rights reserved. PACS: 06.30.G Keywords: Accelerometer; Flexure; Capacitance sensor
1. Introduction 1.1. Vibration isolation in gravitational wave detectors Gravitational wave detectors currently in operation or under construction [1,2] are expected to detect events that modify the distance between masses a few kilometers apart by an amplitude up to 1018 m, within a frequency range of Corresponding author. Present address: DESY, Notkestrasse 85, D-22607 Hamburg, Germany. Tel.: +49 040 3998 1769; fax: +49 040 3998 333. E-mail address:
[email protected] (A. Bertolini).
0168-9002/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.nima.2005.10.117
between few Hz to few kHz (far enough from the low frequency fluctuations of the gravity field of the Earth’s crust). The Earth crust moves randomly and continuously with amplitudes sometime exceeding 106 m, depending on the site; thus the required vibration isolation level is formidable. Moreover the differential residual motion between the mirrors must be kept well below 1013 m, to prevent the saturation of phase sensors (their typical dynamic range is seven decades) used in extracting the output signal of the antenna. An effective solution for the seismic attenuation is represented by the superattenuator (SA) chains first introduced by VIRGO [3]. Advanced nearly fully passive isolators have also been developed by
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ACIGA group [4], while active attenuation solutions have been developed by the Advanced LIGO seismic isolation team [5]. 1.2. SAS: Seismic Attenuation Systems Recently a further development of VIRGO’s SA, named SAS (Seismic Attenuation System) has been designed by a Caltech, University of Tokyo, University of Pisa collaboration for TAMA GW interferometric detector [6]. A view of a fully instrumented prototype of a TAMA–SAS seismic attenuation chain is presented in Fig. 1. The tower consists of a 1.3-m tall ultra-low frequency inverted pendulum (IP), tuned below 30 mHz, for the horizontal degrees of ^ pre-isolation. The IP absorbs the ^ y^ and f, freedom, x, effects of the microseismic peak and provides precise positioning [7]. Furthermore its geometry, stiff in the vertical, pitch and roll degrees of freedom, effectively decouples the horizontal degree of freedom (d.o.f.) from the vertical ones. a very-low frequency (tuned around 100 mHz) passive filter, the Filter 0 (F0), for the vertical pre-isolation. The design is based on monolithic geometrical anti-spring (MGAS) concept [8]. a four stages chain of passive filters for in-band attenuation and a two-level hierarchical mirror positioning system that maintains the mirror within the required 1013 m from the dark fringe [9], without introducing actuation or Brownian noise in the frequency band of interest. Even if strongly shielded by the pre-attenuator, the chain resonances, located between 0.1 and 3 Hz, would be excited by the residual seismic motion, swinging the mirror well
Coil Magnet Actuator
Horizontal Accelero meter Vertical Filter LVDT Position Sensor
617
beyond the dynamic range of the mirror positioning system. The extraction of the energy from these resonances, as well as from the IP’s normal modes, is achieved actively via low noise non-contacting voice-coil actuators [10]. The feedback signal for the actuators is obtained by combining the signal of position sensors (LVDT) [11] and specifically designed high sensitivity accelerometers; these accelerometers are the subject of this paper. All the passive filters are connected in series to the IP-Filter 0 platform. The collective modes of the chain recoil against the IP-Filter 0 and are detectable by the accelerometers. These modes are electronically controlled (inertial damping), by means of a suitable feedback applied to the top of the chain. The main tilt modes of the passive filters are naturally coupled to the horizontal modes and are effectively damped by the horizontal accelerometers. Sets of three accelerometers, LVDT’s and actuators are arranged in a pinwheel configuration around the F0 to control the yaw d.o.f. As the horizontal and vertical isolation is dealt with at different stages, the horizontal accelerometers sit on a platform that, in the vertical direction, is short circuited to ground seismic motion. It is then important that the horizontal accelerometer be highly insensitive to vertical excitation. A digital Multiple-In-Multiple-Out (MIMO) control system transforms the signal from the sensors and generates the damping forces down to the lowest frequencies (mHz) in each individual d.o.f. Since the accelerometers do not sense position, at frequencies below 10 mHz, the positioning of the IP is controlled by a mix of signals from auxiliary LVDT position sensors and the interferometer error signal itself, acting through the MIMO control system. All active isolation is limited to 3–4 Hz, one third of the lower end of the frequency band of interest (10 Hz), to avoid reintroduction of the up-conversion noise generated by sensor’s, actuator’s and feedback loop’s non-linearities. The vertical d.o.f. is, independently, similarly treated. A mirror suspension point stability of 0:2 mm rms, well inside the mirror positioning system dynamic range, has already been obtained in the TAMA–SAS prototype installed at University of Tokyo [12]. 1.3. Advanced inertial sensors for SAS attenuators
Inverted Pendulum Leg
Vertical Filter Monolithic GAS Suspension Platform Mini-GAS
Coil Holder Intermediate Mass
Magnetic Damper Coaxial Recoil Mass
Mirror Suspension Mirror
Fig. 1. Schematic view of a fully instrumented TAMA–SAS attenuation tower; the two-level mirror control system is emphasized.
In this work we present the horizontal accelerometer designed to instrument SAS systems inertial damping. The particulars of its mechanical design and machining, and the expected resolution performances are described. These accelerometers show enhanced resolution and strongly reduced cross-talk levels with respect to similar currently designed instruments like VIRGO sensors [13]. Apart from the linear sensitivity to tilt motion in the sensing direction, the response is almost purely uniaxial with a cross-coupling ranging around 104 , with respect to all the other 4 d.o.f. The closed-loop bandwidth of the instrument is 150 Hz and care has been taken to reduce phase rotations in the acceleration response ensuring easy use of the accelerometer as a sensor in feedback loop. The dynamic range, defined as the peak acceleration over the acceleration noise
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A. Bertolini et al. / Nuclear Instruments and Methods in Physics Research A 556 (2006) 616–623
spectral density is larger thanpffiffiffiffiffiffi 160 ffi dB, with a spectral sensitivity better than 108 m= Hz between pffiffiffiffiffiffiffi 10 mHz and 0.5 Hz, while a noise floor of 1012 m= Hz is ought to be reached above 1 Hz. The corresponding rms noise integrated over all the frequencies above 0.01 Hz is predicted to be less than 1 nm. Although designed mainly for resonance damping, the high resolution of this instrument allows for active reduction of the residual background seismic motion by implementing a more aggressive control strategy. A further reduction of the chain residual motion allows for less powerful and less noisy mirror positioning actuators thus allowing enhanced sensitivity of the interferometer. 2. The accelerometer mechanics The accelerometer mechanics were designed taking in consideration the following criteria: The performance of an accelerometer is closely related to its resonant frequency o0 ; a softer mechanics allows an increase in the low frequency resolution proportional to o20 . With capacitive position sensors, a natural period of a few seconds is needed to achieve sub-nanometer accuracy around the microseismic peak (150 mHz) [14]. A high-Q mechanics prevents hysteresis (it corresponds to low friction and low creep of the elastic elements) and reduces non-linearities in the device output. A high quality factor is also mandatory in order to limit the amplitude of the Brownian motion of the test mass which is one of the main sources of noise in very low frequency devices. A MIMO control strategy for the IP requires true singleaxis accelerometers; a basic constraint in designing the mechanics is to make the instrument as insensitive as possible to all the other four orthogonal d.o.f. Geometrical cross-talks inside the mechanics of the accelerometer degrade its performance, by transforming accelerations in transversal directions into movements along the main axis of the sensor. Using an accelerometer in a feedback loop, any sensitivity to unwanted d.o.f. is particularly deleterious. While the in-axis signal is used to control a specific d.o.f., any sensed noise from an orthogonal direction is injected into the same feedback channel with the gain of the feedback loop. With an accelerometer totally insensitive to orthogonal directions one can depress the seismic perturbation down to the sensitivity of the accelerometer itself by simply increasing the gain. If the accelerometer is coupled to other directions with a sensitivity b, the suppression of the noise in the d.o.f. of interest equals the re-injection of noise from another d.o.f. (assuming the same level of initial seismic perturbation in all d.o.f.) with the loop gain G ¼ 1=b. Hence it is clear that the insensitivity of an accelerometer to unwanted d.o.f. is just as important as its sensitivity to the d.o.f. of the interest. We have designed the horizontal accelerometer as a small folded pendulum (FP). The FP geometry, also named
‘‘Watt-linkage’’ [15], along with a monolithic design, allows us to satisfy all the requirements while preserving a compact, relatively rugged, all-metal structure. The FP was proposed in 1993 by D. G. Blair et al. [16], to provide ultra-low frequency vibration isolation for a laser interferometer gravitational wave detector and to be used as a long-period (1-min) seismometers for studying the dynamics of ocean’s shore waves [17]. A FP is essentially a mass M suspended on one end by a simple pendulum and supported on the other by an IP working antagonistically. In this configuration the straight pendulum positive gravity restoring force is balanced by the negative IP one. The only dissipation in this design is given by the flexures connecting the different parts of the system. The kinematics of the FP can be described by a rough model illustrated in Fig. 2a, where a simple pendulum, with mass M 1 and length L1 , and an IP, with mass M 2 and length L2 , are connected by a massless rigid beam. The pendula arms are also massless and rigid, with suitable flex joints. In the small angle approximation, the gravitational restoring force is given by M1 M2 F ¼ gx (1) L1 L2
L1
M1
M2
L2
(a) xs
θ
ma1,J1 xc mp2 mp1
xp
lp
l
ma2,J2
(b) Fig. 2. (a) Gravitational spring and anti-spring principle. (b) Folded pendulum model including mass and inertia of the arms.
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and the resonant frequency is s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi M1 M2 g þg o0 ¼ L1 L2 ðM 1 þ M 2 Þ
(2)
where g represents the effect of cumulative flex-joint’s stiffness. As o0 depends essentially only on the position of the centre of mass (CM) and on the length of the two pendula, an arbitrarily low resonant frequency can be achieved, as long as M1 M2 gðM 1 þ M 2 Þ (3) X g L1 L2 otherwise the system collapses. This scheme is representative of a real FP if equivalent masses and lengths are used for taking in account the distribution of the weight on the two pendula arms and their moment of inertia. A more accurate description of the FP behaviour is given by the Lagrangian model which also includes the effect of the arm’s masses and moment of inertia. See Ref. [18] for the mathematical details. This model (see Fig. 2b) considers two vertical beams with the same total length l and masses ma1 and ma2 , respectively, J 1 and J 2 are their moment of inertia. The oscillating mass M is shared depending on its CM position with respect the two pendula locations. Two equivalent masses mp1 and mp2 , located near the hinge points, account for the distribution of M; l p is the distance between the pivot point of the pendulum arm and the hinging point of the proof mass; l p is also equal for the two arms. The oscillating mass moment of inertia can be disregarded because, within the first-order approximation, it does not rotate. Let k be the cumulative angular spring constant due to flexure’s stiffness, and let each arm be approximated by a rigid beam with J ¼ ml 2 =12. By solving the equations of motion, the resulting FP’s resonant frequency is o20 ¼
ð1=2Þðma1 ma2 Þðl=l p Þ þ ðmp1 mp2 Þ þ ðk=gl p Þ g lp ðma1 þ ma2 Þðl 2 =3l 2p Þ þ ðmp1 þ mp2 Þ
(4)
while the pendulum equivalent mass M e is M e ¼ ðma1 þ ma2 Þ
l2 þ ðmp1 þ mp2 Þ. 3l 2p
(5)
In this picture we are still neglecting the dissipation that will be later discussed in detail. The transfer function (TF) between the frame motion xs and the pendulum mass motion xp is given by xp o20 Ao2 ¼ 2 . xs o0 o2
(6)
Where the factor A scales as the ratio between the arm mass and the pendulum mass, and it originates from the fact that the two arms of the FP are made of rigid beams (they are physical pendula). In this way the position of the hinges with respect to the p arm’s ffiffiffiffi CM affects TF. Above a critical frequency f c ¼ o0 = A, if one shakes the FP frame, each pendulum arm tends to rotate around its own center
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of percussion (c.o.p.) which remains still, while the arm’s free end countershakes by an amount that pffiffiffiffi is a function of A. As a result the TF flattens above o0 = A [19]. Unlike the case in which the FP is used as a vibration isolator, the c.o.p affects only marginally the accelerometer operation, because the feedback force is proportional to the mass displacement with respect to the support, that is ðxp xs Þ. This quantity is modified by the c.o.p only by a multiplicative factor close to unity (as long as A51). Several materials were considered for their high strength, low creep and low friction characteristics and finally the accelerometers were made out of aluminium alloy 7075-T6 and copper–beryllium (C-17200). For standard inertial damping application, aluminium is preferred because of its good performances while preserving low cost of machining and material. Copper–Beryllium allows the best performance because of lower hysteresis. Both materials show good thermal conductivity and immunity from static and dynamic magnetic fields. The final shape of the accelerometer is achieved by a combination of traditional machining and electric-discharge-machining (EDM). The monolithic design eliminates shear effects at the contact surfaces between separate mechanical parts that can generate hysteresis and dissipation; in this way, it also preserves the high mechanical quality factor of the material. The use of a single material also limits the instrument’s thermal sensitivity. Co-machining of all the joints effectively eliminates all directional cross-talks due to misalignments between the components in the assembly, typical of a composite structure. A side view and a schematics of the monolithic horizontal accelerometer are shown in Figs. 3 and 4. The final form of the accelerometer is obtained by machining a 140 134 40 mm block of metal. Prior to machining, the block has been inspected by X-rays to reject material containing defects and, for Cu–Be, precipitation hardened to obtain the desired mechanical properties. The movable parts of the accelerometer are literally carved into the material by means of wire EDM. A thin 250 mm wide cut is used to separate the two pendulum arms and the test mass from theframe, and to machine the 50 mm thick tensional flexures, four on each side of the accelerometer, that connect the test mass to the structure. The arms are 71.5 mm long and spaced by 102 mm and the equivalent suspended mass is 0.83 kg for the aluminium accelerometer (2.48 kg for the Cu–Be version). The simple pendulum arm, linked to the accelerometer frame through the left-top flexure, is identifiable on the left side of the drawing. The IP leg, connected to the frame through the right-bottom link, is located on the right side. The IP links are mounted in a S-shape structure to allow for the use of a tensional flex-joint in the IP leg. The details of the FP arms are shown in Fig. 5. Both of the legs of the FP have suitable openings in order to reduce their masses and moment of inertia without giving up rigidity. The three big pins sticking out above and below the accelerometer body (Fig. 3), help lock the instrument mass with respect to its outer frame for transportation. At the
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front and the back of the accelerometer (Fig. 6), two deep wells are milled to house the capacitance sensor and the voice-coil actuator. A small tuning ballast mass could be also provided for the fine adjustment of the CM between the legs, and the Fp resonant frequency. The accelerometer body rests on two wedges that allow for levelling. 2.1. Insensitivity to transversal d.o.f.
Fig. 3. Side view of the accelerometer.
1
7
5
7
3
6
2
The entire machining, milling and EDM, are done with the full accuracy of numerically controlled machines which guarantee the equality of the two pendula length, their relative positioning and alignment, as well as the parallelism and equality of the eight flexures, within a few microns. This local and overall accuracy in mechanics generates a high level of insensitivity to unwanted d.o.f. that would be difficult to achieve in a composite structure. This geometry, the position and the aspect ratio of the flexures (short, thin and wide) make the accelerometer extremely rigid in vertical direction (z axis), strongly suppressing the sensitivity to that d.o.f. Hence a resonant frequency higher than 1 kHz can be estimated for the first vertical normal mode. The mechanical coupling between the y direction and the sensitive x axis is reduced by at least a factor 1500 (estimated by the ratio of angular spring constants). Additionally, the position sensor and the force actuator are designed coaxially with respect to the load’s CM, and are built to be insensitive or ineffective in all other directions to enhance the directionality of the instrument. Use of exactly the same flex-joints on both straight and IP legs and the wide spacing between the two planes of flexures, make the instrument quite insensitive to torsional (yaw) and transverse tilt (yy ) accelerations. The mass of each leg has been accurately distributed in order to have the CM well-aligned vertically with the flexures, because a misaligned leg would produce levelling (tilt) error of the instrument and a linear sensitivity to vertical acceleration. 2.2. Flexures and stress evaluation
6
4
7
5
Fig. 4. Accelerometer details: (1) simple pendulum arm; (2) inverted pendulum arm; (3) test mass; (4) tuning mass; (5) suspension points; (6) test mass–pendulum links; (7) transport locks.
The most critical and delicate part of the FP are the flexjoints which support the load against the gravity and allow the movement of the test mass. The accelerometer has four couples of coaxial flexures, one for each junction between the test mass and the frame of the instrument. Each couple of joints has been constructed by first drilling two throughholes, 4-mm in diameter, into the block and obtaining a long ‘‘thick’’ joint. Then the pair of joints are separated by opening a 30-mm long 6-mm wide slot on the front of the instrument’s frame. Two of these slots are visible in Fig. 6. This preliminary machining yields pairs of links, each 1mm thick at the bottleneck and 5-mm wide. The thickness of these links is then reduced to 50 mm by wire EDM. This thickness is close to the limit accuracy of the EDM machine used. Thinner flexures could be obtained with better machines. After the machining the surface of the joints has been electropolished to remove microcracks and
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Fig. 5. Details of the two pendulum arms: (right) the simple pendulum; (left) the inverted one.
Fig. 7. SEM micrograph of a 50 mm thick flexures.
restoring force of the joints. In our approximation the resonant frequency of the system reduces to sffiffiffiffiffiffiffiffiffiffiffi k . (7) o0 ’ M e l 2p Modelling the flexures by using the Tseytlin [20] formula for thin circular notch hinges the angular stiffness is Fig. 6. Three quarter view of the accelerometer. The well on the front face houses the position sensor between the frame and the moving mass. An identical structure is machined on the opposite side for the feedback actuator.
the molten layer (about 5 mm thick) of material produced by the EDM sparks. EDM-reaming of all four couples of joints in a single step gives the best possible parallelism and positioning. The SEM micrograph of a monolithic Cu–Be 50 mm flexure is shown in Fig. 7. The accelerometer was designed with the load almost exactly balanced on the two arms; by using 50 mm thick flexures we expect that the instrument’s behaviour would be dominated by the
k¼
Eat2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 16 1 þ 1 þ 0:215ð2R=tÞ
(8)
where a is the sum of the width of all the joints, t is the thickness at the centre, R is the radius of curvature and E is the Young’s modulus of the material. With the design parameters of a ¼ 40 mm, t ¼ 50 mm and R ¼ 5 mm, we have measured a resonant frequency of 540 mHz in aluminium accelerometers, 440 mHz in Cu–Be device. Both the experimental data, obtained by measuring the freedecay oscillation of the pendulum, are in good agreement with the model. With such a high angular stiffness, the tuning mass turns out to be almost ineffective; therefore
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negative gravitational balancing or a modified aspect ratio (by elliptical EDM cut) of the joints is necessary to achieve lower resonant frequencies. In particular, in elliptically shaped flexures, the stiffness scales down as 1=e, where e is the ratio of the major to minor axes [21]; additional benefits of this configuration should be a reduced stress and a strain distributed over a larger length. The flexures are all in tension with a tensile stress sT ¼ Mg=at; for the aluminium accelerometers we have M ¼ 0:225 Kg yielding sT ¼ 9 MPa. Statically the joints are very far away from the 550 MPa of the material’s elastic limit. On the contrary much higher stresses are generated by bending: a tilt y of the frame generate a bending torque t ¼ Mgl p y, where l p is the pendula length. The corresponding stress sB is
One can find a formally identical expression for an IP; but in that case, to preserve the stability of the IP, k has to be greater than Mgl. For an FP, k can be made small, within the limitations given by the yield strength of the material and the machining capabilities. In principle large QFP values are achievable. The dissipation mechanisms also set the fundamental limit on the detection of the inertial displacement; the corresponding fluctuating Brownian force shakes the test mass simulating a frame acceleration whose spectral density is sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4K B Tfo20 abr ðoÞ ¼ . (11) M eo
tt lpy ¼ 6sT . (9) 2I t Where I is the cross-section moment of inertia of the flexure. In the accelerometer, a range-limiter screw keeps the allowable tilt within 700 mrad, limiting sB to 55 MPa, which is still safe for the instrument. During a normal operation, the feedback action of the accelerometer actuator holds the load at the equilibrium position, limiting the stress to the static load level.
We have measured the quality factor of an aluminium FP accelerometer with a resonant frequency of 0.54 Hz. The results are shown in Fig. 8 as a function of the pressure. In these measurements the accelerometer mechanics was excited at its own resonance well above the environmental noise level and the decay time was evaluated. The quality factor, between 0.5 mbar and the atmospheric pressure, is limited to about 7, due to the constrained air flow between the closely spaced surfaces of the EDM cut and capacitance sensor plates. Below 0.5 mbar the quality factor increases quickly and saturates to about 3000 at 102 mbar where it becomes independent of the pressure. At the low-pressure threshold (0.5 mbar) the mean-free path of the gas molecules becomes greater than the smaller gap present in the accelerometer, that is the 250 mm wide EDM cut. In the saturation region only the internal friction of the flexures damps the motion of the test mass, though as the FP is almost perfectly balanced by design, Eq. (13) reduces
2.3. Mechanical quality factor One of the main advantages of the FP (apart from its compactness) is that both its main restoring force and the anti-spring force are gravitational and, in principle, does not require any storage of elastic energy. In reality some energy will be stored in the flex-joints; infinitely soft flexures would make a perfect FP. The trick of making a good FP is to make flex-joints which are as soft as possible without overstressing the material, because the stress would set in motion a large set of dislocations and introduce hysteresis. The FP accelerometer works under high vacuum, hence, excluding the gas damping (air viscosity and turbulent flow), the only dissipative effects are the structural ones. The damping forces are localized in the flexures which allow the movement of the test mass, while the friction at the junctions between the clamps and the flexures (stick-and-slip) has been eliminated by the monolithic design. We also took care of eliminating the residual stiffness and losses due to the wiring; no negative effects have been observed in the quality factor and in the resonant frequency from the use of a 50 mm diameter Cu–Be wire used to connect the moving coil of the feedback actuator. Eddy-current damping due to the actuator magnetic field losses, has been strongly reduced by using non-conducting materials (peek) for the coil assembly. Representing the material losses with the lossangle f [22], nearly frequency-independent, the quality factor of the FP is given by 2
QFP ¼
1 M e l p o20 . f k
(10)
104
103
Quality Factor
sB ¼
102
101
100
10-6
10-4
10-2 100 Pressure (mbar)
102
104
Fig. 8. Quality factor of the aluminium FP accelerometer as a function of the gas pressure; Q saturates to the internal friction limit around 102 mbar.
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linear inertial displacements of about 1 nm integrated over the same frequency band. The details about the readout system and the expected performances will be given in a companion paper [23]. Monolithic FP accelerometers have been approved to be used in the advanced seismic attenuation system of Japanese TAMA-300 gravitational wave detector.
10
Amplitude (a.u.)
623
5
0
-5
Acknowledgements -10 0
50
100
150 200 Time (sec)
250
300
Fig. 9. Example of a decay time curve at 106 mbar; the decay time is about 2300 s corresponding to Q ’ 3000.
to QFP ’ 1=f and a loss-angle f ’ 3 104 can be estimated for the aluminium alloy 7076, from our data (see an example of high vacuum decay curve in Fig. 9). At present we have not done any systematic investigation of copper–beryllium accelerometers because all of them are in service at University of Tokyo; the results will be published later. For the aluminium mechanics we can estimate, from Eq. (11), a Brownian noise level equivalent to a rms ground displacement of 0.5 nm, integrated from 0.01 to 10 Hz. Copper–beryllium accelerometer are expected to perform at least 3–4 times better, because of the larger mass, the lower resonant frequency and the higher quality factor. 3. Conclusions A new design accelerometer was fully realized with numerically controlled machining for best alignment of the mechanics. Four couples of widely spaced, parallel, flexjoints connecting the pendulum components allow strong reduction of spurious coupling between the sensitive axis and the other 4 d.o.f.; sensitivity to these d.o.f. have been limited to factors around 104 . This feature makes the accelerometer particularly suitable to be used as sensor in multiaxis active seismic isolation systems. The monolithic design also allows to eliminate stick-and-slip friction usually localized at the junctions between flexures and clamps. Very high quality factors, Q ’ 3000 at 0.5 Hz, have been achieved. The corresponding Brownian noise under these conditions is expected to generate a noise level of less than 0.5 nm rms integrated over all the frequencies above 0.01 Hz. The instrument, equipped with a new remotely conditioned capacitance readout system, is able to detect
The LIGO Observatories are a collaboration of the California Institute of Technology and the Massachussets Institute of Technology and are funded by the National Science Foundation under Cooperative Agreement PHY 9210038. The LIGO Laboratory operates under Cooperative Agreement PHY 0107417. This paper has been assigned the number LIGO-P040043-00-D. We would like to thank Brian Lantz, Stanford University, for useful comments. We would like to thank Antonio Bassotti e Figli company in Pistoia, Italy for the EDM machining of the monolithic mechanics.
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