Mechanical modeling of the Seismic Attenuation System for AdLIGO

Transcription

1 Università degli studi di Pisa Facoltà di Scienze Matematiche, Fisiche e Naturali Scuola di Dottorato 'G.Galilei' Dottorato di Ricerca in Fisica Applicata Ciclo XX Pretesi del II anno di Dottorato Annual status report to the Graduate School Board Candidato: Dr. Valerio Boschi Relatore interno: Prof. Virginio Sannibale Relatore esterno: Prof. Diego Passuello Mechanical modeling of the Seismic Attenuation System for AdLIGO LIGO T674--R

2 Introduction. LIGO Experiment The main goal of present experimental gravitation research is the detection of the very weak interaction between gravitational waves and matter. Gravitational waves are local perturbations of space-time geometry, propagating at the speed of light, that have been predicted by General Relativity but never directly detected. General Relativity predicts that an incoming gravitational wave would induce a phase shift in the fringes of an interferometer, proportional to the length of its arms and to the wave amplitude. For this reason, in the last years, several ground based large scale interferometers have been built around the world: experiments VIRGO, in Italy, LIGO, in USA, GEO6, in Germany, TAMA, in Japan. These instruments use the well known Michelson interferometer optical scheme with kilometric Fabry-Perot arm cavities. LIGO experiment is constituted by three detectors that operate in coincidence. Two interferometers, with km and 4 km arms respectively, are located in Hanford, WA, while one 4 km interferometer, is located in Livingston, LA. The relative distance of the two sites is approximately 3 km. At low frequency, the sensitivity of ground based gravitational waves detectors is mainly limited by the seismic noise. For this reason many seismic attenuation systems have been developed in the various interferometers: VIRGO superattenuators, TAMA SAS towers, GEO6 Triple Pendulum.. HAM Seismic Attenuation System Two kinds of vacuum chambers are used in LIGO: Beam Splitter Chambers (BSC) and Horizontal Access Module chambers (HAM). BSCs are approximately 5.5 m high and hold the beam splitter and the main interferometer mirrors. Each of the two interferometers at Hanford uses ve BSC chambers for a total of ten chambers. HAMs are smaller chambers used for the Mode Cleaner and the Recycling cavity mirrors. HAM-SAS, a single stage Seismic Attenuation System dedicated to the HAM optical benches has been recently designed [] []. The system (g. ) is composed of A four leg Inverted Pendulum (IP) table for horizontal attenuation 4 Monolithic Geometric Anti-Spring (MGAS) lters dedicated to vertical attenuation A set of 8 nm resolution LVDT position sensors and 8 voice-coil actuators to be used for active attenuation In HAM-SAS the horizontal and vertical degrees of freedom are mechanically separated and orthogonal. A rectangular symmetry instead of the commonly used (Virgo, TAMA experiments) triangular symmetry is used in order to adapt the system to the HAM vacuum chamber geometry and to match the optical table symmetry and dimension. The horizontal section of HAM-SAS is composed of a base platform supporting a set of four inverted pendulums (IP) disposed on a. x.9 m diamond

3 conguration. The IPs support an intermediate rectangular platform (.9 x.7 m) containing the vertical attenuation lters. The subsystem composed of the vertical lters and the top and intermediate platforms is called spring-box. IPs are tunable mechanical oscillators widely used for their good horizontal attenuation properties. Resonance frequencies of tens of mhz have been experimentally reached in many systems. Their attenuation performance is limited by the well known center of percussion eect due the moment of inertia of their legs. For this reason, counterweights are placed below the elastic joints in order to move the leg center of rotation at the height of the hinging point of the ex joint. Using counterweights, HAM-SAS IPs are expected to reach an attenuation level of 8 db. MGAS lters are vertical oscillators based on a cantilever-blade conguration. In these mechanical devices force created by the load is mainly redirected in the horizontal direction, thanks to the geometry of the system, and it compresses the blades allowing a stable equilibrium position. Using this principle, it's possible to obtain a very low vertical elastic constant and consequently a very low resonance frequency. Each HAM-SAS vertical lter houses 8 blades, each carrying to 3 kg of load, depending on the blade width. Tuning frequencies as low as 3 mhz have been measured on several prototypes. Attenuation factor of 8 db have been experimentally reached using so called magic wands [3]. HAM-SAS Mechanical Modeling In order to evaluate HAM-SAS performance and allow the design of the control system, a three-dimensional, multiple-degrees of freedom dynamical model has been developed [4]. Let's list rst the most important approximations used in the model: Lumped system, i.e. rigid body approximation, elastic elements approximated using quadratic potentials, i.e. small oscillation regime, dissipation mechanisms accounted using viscous damping to approximate structural/hysteretic damping, system assumed to be symmetric enough to separate horizontal displacements x, y, and yaw θ z from pitch θ y roll θ x and vertical displacement z, internal modes of the mechanical structures not accounted, angular wires' stiness neglected, Using lumped elements limits the accuracy of the simulation to frequency lower than the lowest internal frequency. HAM-SAS is expected to have internal modes frequencies starting at around Hz. The reason of using viscous instead of hysteretic damping is because of the need of having a straightforward state space representation for time domain simulations studies. In the small oscillation regime, the major dierence between the two kind of damping is that the viscous damping changes the resonant 3

4 z, Yaw Optical Table x, Roll y, Pitch Spring Box Figure : Three-dimensional CAD drawing of HAM-SAS and simplied sketch of the mechanical model developed (IP counterweight bells, MGAS wands and wands' counterweights not shown). frequencies of the the modes. Practically, this turns out to be just a minor drawback because resonances are tuned to some nominal frequencies. A symmetric system such as HAM-SAS has some orthogonal degrees of freedom that simplify the implementation of the dynamic model. Asymmetries are expected to introduce coupling no more than % among degrees of freedom. Therefore, for seismic attenuation performance estimation purposes such approximation should be reasonable. Neglecting angular wires' stiness produces an underestimation of the pendular modes frequencies, which is negligible in our case vis-a-vis the wires' cross section and tensions. Figure shows a sketch of this mechanical model with rigid bodies, linear springs, and exural joints. Inverted pendulums counterweight bells and MGAS wands and wands' counterweights are not shown for sake of simplicity. More detailed sketches are shown in gures and 4. The physical parameters used in the model have been in part extracted 4

5 from HAM-SAS ocial production drawings [5], in part evaluated to match experimental data acquired on several prototypes. Masses and moments of inertia have been calculated using SolidWorks CAD from the three-dimensional drawings of the system realized by Y.Huang. In the model the entire mechanical system is sitting on a base which is used to excite the Attenuator in all the 6 degree of freedom. Internal resonances of the IP legs and other supporting structures are not included in the simulation. The mechanical model has been implemented using scripts in Maple symbolic language. Those scripts produce a state-space representation of the system that can be easily imported into Matlab and Simulink for control simulations. The way that the code has been written is such that allows to progressively introduce new features to improve the accuracy and remove degrees of freedom to check the consistency of the simulation. For example, it is possible to freeze all the degrees of freedom but the one describing the IP and verify that the model gives the expected simple response similar to a compound pendulum. This feature allowed us to increase our condence about the model.. MGAS Filter Model The MGAS lter is modeled with an equivalent system which is able to account for most of the blades' mechanical compliances, and for attenuation saturation eects. The equivalence is achieved by tuning the model with measurements. Three orthogonal springs with the proper elastic constant are connected together by one end to the payload (the optical table), and the other three ends are attached to the lter frame. Saturation eects due to the blade's distributed mass are simulated with a wand pivoting around a point rigidly connected to the lter frame. One wand's end is then attached to a counterweight and the other is free to rotate about the point connected to the payload. The tuning of the attenuation saturation is done by changing the counterweight and/or the wand's length and/or wand's central pivot point. In the actual MGAS lter wands are introduced to neutralize the attenuation saturation eects, while in the model the wands are used both to generate, end eventually to neutralize the saturations. This phenomenological model with two wands plus counterweight system, shown in gure sketch, is used to simulate the horizontal and vertical saturations of MGAS transmissibility. The transmissibility is essentially a measure of how a mechanical system respond, in the frequency domain, to a generic excitation. Mathematically, the Transmissibility T i (s) along the i-th degree of freedom of a mechanical system is dened as the ratio, dependent on the Laplace variable s = jω, T i (s) = Qi (s) Q i (s) between the Laplace transform of the vibration q i (t) of the system and the Laplace transform of the excitation q(t) i applied to the system, along the i-th degree of freedom. In the case of a simple unidimensional damped harmonic oscillator, excited with a sinusoidal displacement x sin(ωt), the transmissibility magnitude is T (s) = x x = + 4γ (ω/ω ) ( (ω/ω ) ) + 4γ (ω/ω ) 5

6 Optical Table Wand Counterweight η Wand ξ Counterweight Spring Box Figure : MGAS lter model sketch. The third spring and the third wand+counterweight are not shown because are orthogonal to the drawing. Pivot points are shown with black dots. where ω is the oscillator resonance frequency and γ is the resonance damping factor. Figure 3 shows a comparison between a model result and measurements of the vertical transmissibility of a MGAS blade with a single counterweight and wand [3]. The model has been tuned in order to match the resonance and the notch frequencies of the measurement. A signicant systematic discrepancy at low frequency (below. Hz) of approximately a 5 db is clearly visible. The large uncertainty of the accelerometers calibration at low frequency used to take the measurement can partially explain such dierence. Discrepancies above 5Hz are mainly due to the internal resonances of the supporting frame, the shaker, and the blade support designed for lter tuning studies.. Inverted pendulum Leg Model The inverted pendulum dynamics have been modeled with an ideal exural joint connected to a leg with a counterweight as shown in gure 4. The transmissibility saturation has been tuned to about -6dB, which is a conservative number considering previous measurement with the HAM-SAS IP legs. IP Resonant frequency was tuned to 3 mhz, which is a frequency routinely obtained by Virgo superattenuators and also obtained by SAS prototypes..3 Ground Transmissibilities The two state-space models generated by the Maple scripts have a total of 3 inputs and 8 outputs. The inputs are constituted by 7 force/torque actuators F i placed on the system and 6 force/torque actuators F i, one for every degree of freedom i, placed on an ideal innite mass base platform, used to shake the 6

7 4 MGAS Model vs. GAS+wand Measurements Data set Data set Model Frequency [Hz] Figure 3: Comparison of modeled and experimental vertical transmissibility of an MGAS lter with assembled with a wand with a counterweight. Experimental data were taken in two separate measurements [3]: green asterisks Dataset #, blue asterisks Dataset #. system, that represents the ground. The outputs are composed by the 6 degrees of freedom of the optical table, of the spring box, q i, and of the base, q i, centers of mass. The frames are oriented as shown in gure. Transmissibilities from ground to the optical table center of mass and from ground to spring box center of mass for all the six degrees of freedom are shown in gures 5 and 6. To compute such transmissibilities we rst calculate the transfer functions from generalized forces/torques F i applied to the shaker base, to the optical table and spring box degrees of freedom q i. Then the transmissibilities have been obtained applying the proper transfer function ratios to remove the dependency on the generalized forces: T i (s) = Qi (s) F i (s) F i (s) Q i (s) for the generic ith degree of freedom. Because internal modes are not included in the mechanical models, this simulations are valid up to approximately 5 Hz. Above that frequencies discrepancies are expected to be large especially around the internal mode frequencies due to eect of distributed masses. The transmissibility generated by the model are Horizontal x/x transmissibility. The blue and green curves of gure 5 upper plot are the transmissibilities of the horizontal degree of freedom x. The resonance at 3 mhz with very low quality factor corresponds to the inverted pendulum mode frequency. The.Hz resonance is due to the 7

8 Counterweight Bell Spring Box θ i Hollow ϕ i Leg x i Little Pendulum Flexural Joint Figure 4: Inverted pendulum mechanical model sketch. Pivoting point are shown with black dots. The connection between the spring box and the little pendulum is not clear from the sketch. horizontal stiness of the MGAS blades, which has been experimentally measured to be about.54 Hz using a MGAS lter prototype. Finally, the resonance at 3 Hz is due to the short pendulums which connect the inverted pendulum legs to the MGAS frame. The presence of this resonance is well known and has been conrmed by ANSYS FEM simulations [6]. Saturation above Hz has been tuned using the inverted pendulum counter-weight to about -65 db, which is considered a reasonable value after the past Virgo results and measurements on HAM-SAS legs. Longitudinal y/y transmissibility. Because of the symmetry of the system, the transmissibility curves for the longitudinal degree of freedom y (red and cyan curves of gure 5 upper plot) are essentially the same as the x degree of freedom. Minor dierences come from the moments of inertia and lever arms which are dier by less than % in the two orthogonal directions. Yaw θ z /θ () z transmissibility. The blue and green curves of gure 5 lower plot show the transmissibility for the angular degree of freedom θ z (yaw). Vertical z/z transmissibility. The blue and green traces of gure 6 upper plot are magnitude and phase of the transmissibility of the vertical degree of freedom z. The vertical resonance has been tuned at mhz and the saturation has been set to about -8dB. Those values have been 8

9 Transmissibilities x and y Direction Magnitude (db) 5 5 x Spring Box x Optical Table y Spring Box y Optical Table Frequency (Hz) Transmissibilities θ z Direction Magnitude (db) 5 5 θ z Spring Box θ z Optical Table Frequency (Hz) Figure 5: Transmissibilities for the horizontal (x, y) and (θ z ) degrees of freedom for a system with IPs tuned at 3 mhz. experimentally obtained for a MGAS with a payload of about the same of the HAM-SAS system. This saturation level can be reached using a properly tuned wand and counterweight. The major advantage in lowering the saturation levels is to reduce the magnitude of the internal resonances in the transmissibility curves. The quality factor for the vertical resonance was set to about 3. Roll θ x /θ () x transmissibility. The blue and green traces of gure 6 lower plot are magnitude and phase of the transmissibility of the angular degree of freedom θ x (roll). As expected this transfer function is similar to the vertical transfer function but with dierent resonant frequency( this resonance depends on the moment of inertia of the payload and not just on the mass ) Pitch θ y /θ y () transmissibility. The red and cyan traces of gure 6 lower plot are magnitude and phase of the transmissibility of the angular degree of freedom θ y (pitch). The dierent between the previous transmissibility is mainly due to the dierent moment of inertia about the x and y axis..4 Attenuation performance The seismic noise reference spectra of Hanford (LHO) and Livingston (LLO) sites, used to estimate the attenuation of the horizontal and vertical seismic spectra, are based on AdLIGO Seismic Isolation requirements design document 9

10 Transmissibilities x and y Direction 5 Magnitude (db) 5 5 z Spring Box z Optical Table Frequency (Hz) 5 Transmissibilities θ z Direction Magnitude (db) 5 5 θ x Spring Box θ x Optical Table θ Spring Box y θ Optical Table y Frequency (Hz) Figure 6: Transmissibilities for the vertical (z), pitch and roll (θ x, θ y ) degrees of freedom for a system with MGAS tuned at mhz with 8 db attenuation factor. [7]. The horizontal and vertical ground noise are considered equal and expressed, in the frequency range mhz< f <4 Hz, as a polynomial expansion in log space: log x g (f) = p (log f) n + p (log f) n p n log f + p n+ where x g (f) is the displacement spectral density. The LHO and LLO reference spectra have been merged and extrapolated below mhz using the USGS New Low Noise Model developed by J.Peterson [8]. The ground tilt noise was generated in the mhz - 4 Hz band using the Rayleigh waves propagation model [9]. In this model the ground tilt noise spectrum is proportional to the vertical component of Rayleigh waves through the relation: θ g (ω) = ω c S v where θ g (ω) is expressed in rad/ Hz, S v is the vertical seismic motion and c is the local speed of seismic waves. This is probably an underestimation of the actual LIGO sites angular noise especially because of asymmetries and of the internal modes of the HAM supporting structure at low frequencies which introduce phase delays in the noise propagation. Direct measurements are needed to estimate the amount of angular noise. The requirements, contained in document [7], set a displacement noise limit, for the Power Recycling and Mode Cleaner optics, of x 7 m/ Hz in.-. Hz band, x 3 m/ Hz at Hz and 3x 4 m/ Hz above Hz.

11 The spectra obtained ltering LHO and LLO combined seismic noise with the HAM-SAS model are: x/x power spectrum. Figure 7 shows the reference spectrum, the design requirement spectrum and two predicted attenuated spectra for the horizontal degree of freedom. The blue curve is obtained considering the inverted pendulum resonant frequency tuned at mhz and the red curve with the IP tuned at 3mHz. The red curve shows that requirements cannot be met in the.8-.5 Hz interval. There are essentially three possible solutions to this problem that can be combined together. The rst solution is to tune the notch produced by the inverted pendulums counterweight at the MGAS resonant frequency to add passive attenuation in correspondence of that MGAS horizontal resonance. The second solution is to tune the inverted pendulum resonance frequency at mhz as shown in the red curve. Finally the third solution is to use the active damping control to reduce the height of. Hz resonance. y/y power spectrum. The ltered seismic noise spectrum for the horizontal degree of freedom y is not reported because is essentially the same as the x horizontal degree of freedom. In fact the only dierences comes from a slight dierence on the two horizontal transmissibilities. θ z /θ () z power spectrum. Figure 8 shows the ltered and unltered power spectral density of the angular degree of freedom θ z (yaw). Requirements are met everywhere with a quite large safety margin. As in the other angular degrees of freedom, the angular noise roughly translates into position noise expressed in meters. As previously mentioned, it is important to notice that the angular noise spectrum could be underestimated. z/z power spectrum. Figure 9 shows the ltered and unltered power spectral density of the vertical degree of freedom z. Using a mhz MGAS resonance frequency, even with a -8dB saturation, experimentally reached with magic wands, requirements cannot be met. However this problem does not compromise HAM-SAS performance because HAM chambers requirements dened in [7] are probably too stringent. A recent study [] performed by P. Fritschel, one of the authors of the original seismic requirement document [7], has shown that the Mode Cleaner noise limit at Hz can be increased of - orders of magnitude with no problem for AdLIGO target sensitivity. We have also to notice that considering the horizontal and vertical components of seismic noise to be equal causes an overestimation of the real displacement noise along vertical direction. θ x /θ x () and θ y /θ y () power spectra. Figure shows the ltered and un- ltered power spectral density of the angular degree of freedom θ x (roll). Requirements are met everywhere with a quite large safety margin. Because the arm lever of the triple pendulum supporting frame is about m long, the angular noise translates directly into meters. It is again important to mention that the ground angular noise spectrum could be underestimated.

12 6 4 LLO+LHO Displacement Noise Spectrum Filtered Seimsic Noise (3 mhz) Filtered Seimsic Noise ( mhz) AdLIGO Requirements LHO+LLO Seismic Noise Displacement Noise [nm/sqrt(hz)] Frequency [Hz] Figure 7: Seismic noise square-root power spectral densities of the horizontal (x) degree of freedom: gold curve required AdLIGO noise performance, green curve LHO+LLO seismic noise model, blue and red curve ltered seismic noise for a system with IPs tuned at 3 mhz and mhz respectively..5 Eect of asymmetric parameters A preliminary study on the eects of the introduction of asymmetric parameters in the model have been performed. This has done in order to help to determine the precision required for HAM-SAS construction and assembly. Two cases have been considered: Asymmetric IP leg lenghts: Two of the four IP legs have lenghts dierent (- mm< l < mm) from the design value i.e. the intermediate and top platforms are inclined by an angle θ x or θ y respect to the ground. This case has to be considered because it's dicult to manufacture metallic surfaces of the dimension required for the top and intermediate platforms (.9 x.7 m) with an high level of planarity. The transmissibilities and the ltered seismic spectral densities for the horizontal and yaw degrees of freedom are shown in gures and respectively. As expected, the asymmetry introduces a coupling between the horizontal and angular directions. This causes the IP tuning frequency along the excited direction to change and to mix with the main resonance frequency of the non-excited degree of freedom. Since, as shown in g. 7, the height of. Hz resonance depends strongly on the IP tuning frequency, when the system is excited along x, the Hz region is over the requirements in the case of positive l and under them in the case of negative l. When the system

13 LLO+LHO Angular Noise Spectrum ( θ z ) Filtered Seismic Noise AdLIGO Requirements LHO+LLO Seismic Noise Angular Noise [nrad/sqrt(hz)] Frequency [Hz] Figure 8: Seismic noise square-root power spectral densities of the yaw (θ z ) degree of freedom: gold curve required AdLIGO noise performance, green curve LHO+LLO seismic noise model, blue and red curve ltered seismic noise for a system with IPs tuned at 3 mhz. is excited along θ z instead, we can see that AdLIGO requirements are met with a good margin even with an asymmetric system. As before, we have to notice that ground angular seismic noise spectrum could be underestimated. Asymmetric MGAS elastic constants: Two of the four MGAS springs have elastic constants dierent (.k < k <.k ) from the design value k. Since MGAS are systems in a quasi-equilibrium state, a precise tuning of the spring box is very dicult. For this reason, in the real system, a % error in the value of the elastic constants is expected. The transmissibilities for the vertical, pitch and roll degrees of freedom are shown in gures 3 and 4 respectively. In order to amplify the eects of asymmetry, the quality factors of all resonances are set to innity. Looking at the 6-4 mhz region we can see, as in the asymmetric IP table case, the presence of coupling between the vertical and angular degrees of freedom. However, unlike the previous case, the eects are very small and become completely negligible when we use the expected resonance quality factors. This result is expected since for an ideal spring the resonance frequency f k. These results have been presented to the August 6 LIGO Scientic Collaboration Meeting. 3

14 6 4 LLO+LHO Displacement Noise Spectrum (z) Filtered Seismic Noise AdLIGO Requirements LHO+LLO Seismic Noise Displacement Noise [nm/sqrt(hz)] Frequency [Hz] Figure 9: Seismic noise square-root power spectral densities of the yaw ( z) degree of freedom: gold curve required AdLIGO noise performance, green curve LHO+LLO seismic noise model, blue and red curve ltered seismic noise for a system with MGAS tuned at mhz with 8 db attenuation factor..6 Mode Cleaner suspension Every HAM vacuum chamber is equipped with a so called Triple suspension. This seismic attenuation device, based on the Triple Pendulum design developed for GEO6 - the German-British gravitational wave detector -, is made of a chain of three pendulums and is capable of providing 9 db of horizontal attenuation and 5 db of vertical attenuation at Hz. The system (g. 5) is composed by three bodies, called top, intermediate, and test mass that weight approximately 3 kg each and provide three stages of passive horizontal attenuation. The T-shaped top mass is suspended from the top of an external cage through two wires connected to two maraging steel blades. The blades are used in order to provide vertical attenuation. The intermediate mass is connected to the top mass through four wires and four maraging blades while the test mass is connected to the intermediate with four silica bers. Both intermediate and test masses are made of fused silica. The triple pendulum has also 4 sensors and actuators, 6 on the top mass and 4 on both the intermediate and test masses, called OSEMs (Optical Sensor and ElectroMagnetic actuator). The top mass sensors and actuators are used for 6-degrees of freedom active damping of the structure resonances. The triple pendulum external cage is supported by the optical table, that will be placed on HAM-SAS top platform. In this way the motion of the test 4

15 LLO+LHO Angular Noise Spectrum (Roll and Pitch) Filtered Seismic Noise θ x Filtered Seismic Noise θ y AdLIGO Requirements LHO+LLO Seismic Noise Angular Noise [nrad/sqrt(hz)] Frequency [Hz] Figure : Seismic noise square-root power spectral densities of the roll (θ x ) and pitch (θ y ) degrees of freedom: gold curve required AdLIGO noise performance, green curve LHO+LLO seismic noise model, blue and red curve ltered seismic noise for a system with MGAS tuned at mhz with 8 db attenuation factor. mass will be attenuated by both HAM-SAS and the Triple suspension. It's therefore important to study the combined attenuation of the two systems. For this reason, in order to allow a preliminary study of the combined suspensions system, an independent model for the horizontal degrees of freedom of the Triple suspension has been developed. The model, shown in the sketch of gure 5, uses the same approach and code structure of the HAM-SAS simulation. Similarly to what we have done before, an ideal innite mass platform, placed at the suspension point of the pendulum chain, is used to shake the system. The generated state-space representation has 8 inputs and 9 outputs. The horizontal transmissibilities obtained from the model are shown in the upper two plots of gure 6. Thanks to the modular structure of the code, a single coordinate transformation allows the connection of the Triple Pendulum to HAM-SAS system. In gure 6 are shown the transmissibilities of the Triple Suspension combined with HAM-SAS in the horizontal (x) and yaw (θ z ) degrees of freedom for the Spring Box, Optical Table, Triple Pendulum suspension point (MC_SP_X), Triple Pendulum top (MC_X[]), intermediate (MC_X[]) and test (MC_X[3]) mass. We can see the triple pendulum resonance frequencies, in Hz region. and the three levels of attenuation introduced by three masses at higher frequencies. 5

16 Inverted Pendulum Table: Spring Box Transmissibilities Horizontal Direction (X) 4 6 Symmetric legs +/3 mm 8 +/3 mm + mm /3 mm /3 mm mm Inverted Pendulum Table: Optical Table Transmissibilities Horizontal Direction (X) 5 Displacement Noise [nm/sqrt(hz)] LLO+LHO Displacement Noise Spectrum (x) Symmetric Legs +/3 mm +/3 mm + mm /3 mm /3 mm mm LHO+LLO Seismic Noise AdLIGO Requirements Figure : Eect of asymmetric IP leg lenghts in the horizontal (x) transmissibility. The 3 mhz IP resonance frequency changes, due to the expected coupling between rotational and translational degrees of freedom. In the lower plot are shown the eects of asymmetry on the seismic attenuation performance. 6

17 Inverted Pendulum Table: Spring Box Transmissibilities Yaw θ z Symmetric legs +/3 mm +/3 mm + mm /3 mm /3 mm mm 3 3 Inverted Pendulum Table: Optical Table Transmissibilities Yaw θ z LLO+LHO Angular Noise Spectrum ( θ z ) Angular Noise [nrad/sqrt(hz)] 4 6 Symmetric Legs +/3 mm +/3 mm + mm /3 mm /3 mm mm AdLIGO Requirements LHO+LLO Seismic Noise 8 3 Figure : Eect of asymmetric IP leg lenghts in the yaw (θ z ) transmissibility. The 3 mhz IP resonance frequency changes, due to the expected coupling between rotational and translational degrees of freedom. In the lower plot are shown the eects of asymmetry on the seismic attenuation performance. 7

18 MGAS Table: Optical Table Transmissibilities Vertical Direction (Z) 5 5 Symmetric K + % + % % % MGAS Table: Optical Table Transmissibilities Vertical Direction (Z) Detail Figure 3: Eect of asymmetric MGAS elastic constants in the vertical ( z) transmissibility. In the lower plot, detail of the region 6 mhz< f <4 mhz. The quality factors of all resonances are set to innity in order to amplify the coupling eects. 3 HAM-SAS LVDT Driver characterization and optimization As already mentioned, eight LVDTs are used for position control in HAM-SAS. Linear Variable Dierential Transformers (LVDTs) are displacement sensors constituted by a primary and secondary windings. The primary winding is fed with an audio frequency (usually in the range - khz) sinusoidal signal. The secondary winding is composed by two coils wound in opposite directions. When the primary winding is displaced of an amount x, a current with the same frequency of the primary signal and modulated in amplitude proportionally to x is induced in the secondary winding. A mixer is then required to demodulate the secondary signal and produce a DC output proportional to x. A VME LVDT driver board will be used in HAM-SAS control. The LVDT board specications are: 8 independent channels, rst channel set as master in master-slave trigger congurations. One single-ended output per channel for signal monitoring One single-ended input, for external oscillator operation One single-ended output, for board synchronization 8

19 MGAS Table: Optical Table Transmissibilities Roll ( θ x ) 5 5 Symmetric K + % + % 5 % % MGAS Table: Optical Table Transmissibilities Roll ( θ x ) Detail MGS Table: Optical Table Transmissibilities Pitch ( θ y ) Symmetric K + % + % % % MGAS Table: Optical Table Transmissibilities Pitch ( θ y ) 6 4 Figure 4: Eect of asymmetric MGAS elastic constants in the pitch and roll (θ x, θ y ) transmissibilities. In the lower plots, detail of the region 6 mhz< f <4 mhz. The resonances quality factors are set to innity in order to amplify the coupling eects. 9

20 D t D t D b h d H h D b D t3 H d H 3 h 3 d 3 = h 3 D b3 Figure 5: The Triple Suspension used in the Mode Cleaner HAM chamber and a sketch of the mechanical model developed.

21 Mode Cleaner Suspension: Transmissibilities Horizontal Direction (X) 5 5 Upper Mass Intermediate Mass Test Mass Mode Cleaner Suspension: Transmissibilities Yaw ( θ z ) Inverted Table Pendulum + GASF Spring box: Transmissibilities Horizontal Direction (X) Spring Box Optical Table MC_SP_X MC_X[] MC_X[] MC_X[3] 3 3 Inverted Table Pendulum + GASF Spring box: Transmissibilities Yaw θ( z ) Figure 6: Upper two plots: Transmissibilities of the Triple Suspension in the horizontal (x) and yaw (θ z ) degrees of freedom for the top (MC_X[]), intermediate (MC_X[]) and test (MC_X[3]) mass. Lower two plots: Transmissibilities of the Triple Suspension combined with HAM-SAS in the horizontal (x) and yaw (θ z ) degrees of freedom for the Spring Box, Optical Table, Triple Pendulum suspension point (MC_SP_X), Triple Pendulum top (MC_X[]), intermediate (MC_X[]) and test (MC_X[3]) mass. The quality factors of all resonances are set to innity.

22 Master-slave/asynchronous operation selectable through onboard jumpers. External/internal oscillator operation selectable through onboard jumpers. ± Vpp primary output voltage ±5V - ±8 V Supply operating voltage. 3 4-pin connectors for LVDT primary winding excitations, LVDT secondary winding readbacks, ADC The circuit, shown in gure 7, is based on the Analog Devices Universal LVDT Signal Conditioner AD698 chip []. The component feature are Tunable Internal oscillator from Hz to khz Double channel demodulator: two synchronous demodulator channels are used are used to detect primary and secondary amplitude. The component divides the output of the secondary by the amplitude of the primary and multiplies by a scale factor in order to improve temperature performance and stability. In this way a typical oset drift of 5 ppm/ C and a typical gain drift of ppm/ C are reached. Tunable low pass lter for each demodulator Amplifying stage at the output A phase compensation network is used to add a phase lead or lag to one of the modulator channels in order to compensate for the LVDT primary to secondary phase shift. A low noise instrumentation amplier, INA7, is used for LVDT secondary readbacks dierential input. Specically designed for audio signal amplication, this component has a voltage noise of.4 nv/ Hz at khz and a THD of.4% at khz for a gain factor. The gain can be adjusted through an external potentiometer. A wide-band fully dierential amplier, THS43, is used for primary winding excitation output. Several measurements have been done in order characterize the performance of the three versions of the board. An experimental setup, composed by a 5 µm resolution Line Tool micropositioner xed on an optical table and rigidly connected to the LVDT primary winding, has been used. Several custom made Horizontal LVDT prototypes have been realized in order to determine the optimal ratio between the radii of primary and secondary windings. LVDT spectral density noise measurements (gure 8) has been done after centering the LVDT primary coil to get zero signal output. Several independent measurements have been performed to cover dierent frequency ranges. Calibration measurements have shown a low level of nonlinearity (less than % of the range). Residual displacement noise of nm/ Hz has been measured for both LVDTs. Crosstalks of % between the horizontal and longitudinal and between horizontal and vertical degrees of freedom have been obtained. The results obtained in an optimized conguration are summarized in the following table: Horizontal LVDT Vertical LVDT Nonlinearity.88%.6% Sensitivity 6.49 V/mm 7.85 V/mm Range mm mm Displacement Noise nm/ Hz nm/ Hz

23 C nf D +4 SW SYNC LO/DX 3 J SYNC IN 6-4 U SSM4 3 9 C8 C9 C3 uf uf Mylar 3.3nF Mylar U AD698 C OUTFILT EXC FEEDBACK 3 EXC BFILT C7 uf Mylar BFILT AFILT AFILT FREQ FREQ 8 +VCC uf Mylar U3 OP7 6 R6 R J SIGNAL MONITOR D C SYNC LO/DX U5 SSM SW SYNC INT/EXT C4 C Mylar C5 C Mylar RV RT 5T R5 RS 3 -BIN 4 +BIN 7 +ACOMP C6 C Mylar 8 -ACOMP +AIN 6 4 SIGREF 5 -AIN -VCC C nf Mylar SIGOUT 3 LEV 5 LEV 6 C4 nf RV4 K 5T RV3 5K 5T -4 TP U4 SSM4 8 7 ADC CONN - ADC CONN + C R9 39R C5 uf +4 C nf B EXC CONN + EXC CONN - R R R R +4 U6 THS R 39R +4 8 R7 39R R8 39R TP4 R4 K C3-4 uf TP3 6 U INA7 TP + 3 GAIN RV GAIN 8 R 5T - C nf TP R K R K R3 K LVDT IN + LVDT IN - B A J SYNC OUT R R 6 U SSM4 3 Title -4 LVDT DRIVER CANALI Sezione Canale - Clock int/ext A -4 Size Document Number Rev MW6 A Tuesday, July 8, 6 Date: Sheet of 3 Conclusion Figure 7: Schematics of LVDT Board master channel We developed a three-dimensional multiple degrees of freedom model for HAM- SAS mechanical structure. Passive attenuation performance has been studied and compared with AdLIGO requirements. Preliminary studies of the eects of asymmetries in model parameters have been performed. We developed a model for the horizontal degrees of freedom of the complete HAM suspension (Triple Pendulum and HAM-SAS). The next step will be the design of HAM-SAS control system. Since the quality factors of the system resonances are very low, a position DC control will be implemented. Several strategies are possible. One possibility is the use of multivariable LQG technique, that eliminates the need of sensors/actuators diagonalization process. Another solution is the use of quasi-stable oscillators. This is a very simple and promising alternative that can be applied whenever tunable mechanical oscillators are used. Tunable mechanical oscillators can become unstable or better, very close to an unstable equilibrium. Under this condition, a very weak force is necessary to keep the oscillator steady (stable), and no oscillator resonances will practically show up; therefore, there is no need to damp the harmonic resonances, and just a DC active control with a quite easy control law is needed. The side benet of such strategy, which is not marginal at all, is that the passive attenuation is maximized thanks to the lowest as practical as possible low pass lter cut o frequency. 3

24 Horizontal LVDT Prototype H: Residual displ. Noise Displacement [nm rms /sqrt(hz)] 3 Frequency [Hz] New Vertical LVDT v. Board: Residual displ. Noise Displacement [nm rms /sqrt(hz)] 3 Frequency [Hz] Figure 8: Square root power spectral densities of horizontal and vertical LVDT displacement noise. 4

25 HAM-SAS models can be improved and extended in several ways. For example, the complete elimination of symmetries and the introduction of the system intrinsic nonlinearities will allow a complete study of the space of the parameters. Once the fabrication of HAM-SAS prototype will conclude, characterization measurements will be compared and integrated in the models using system identication techniques. References [] A. Bertolini et al., Design and prototype tests of a Seismic Attenuation System for the Ad-LIGO Output Mode Cleaner, Class. Quantum Grav. 3 (6) S-S8 [] A. Bertolini et al., LIGO Output Mode Cleaner HAM Seismic Attenuation System, LIGO Internal Note T637- (6) [3] A. Stochino, Performance Improvement of the Geometric Anti-Spring (GAS) Seismic Filter for Gravitational Wave Detectors, LIGO Internal Note T539- -D (5) [4] V. Boschi, V. Sannibale, R. DeSalvo, AdLIGO HAM-SAS Mechanical Model with Lumped Elements, LIGO Internal Note T637- (6) [5] LIGO Documents D5-D545 (6) [6] I. Taurasi, Inverted Pendulum Studies for Seismic Attenuation, LIGO Internal Note T648 (6) [7] P. Fritschel et al., Seismic Isolation Subsystem Design Requirements Document, LIGO Internal Note E9933--D () [8] J. Peterson, Observations and Modeling of Seismic Background Noise, USGS Report 93-3 (993) [9] A.Takamori, Low Frequency Seismic Isolation for Gravitational Wave Detectors, PhD Thesis, LIGO DCC P349 (3) [] P. Fritschel, HAM Seismic Isolation Requirements, T675--D (6) [] Analog Devices, AD698 Datasheet (995) 5