Investigation of crackling noise in the vibration isolation systems of the KAGRA gravitational wave detector
Shin Kirii, Yingtao Liu, Takashi Uchiyama, Ryutaro Takahashi, Yuhang Zhao, Seiji Kawamura
IInvestigation of crackling noise in the vibration isolationsystems of the KAGRA gravitational wave detector
Shin Kirii a , Yingtao Liu a , Takashi Uchiyama a , Ryutaro Takahashi b ,Yuhang Zhao c , Seiji Kawamura a,d, ∗ a KAGRA Observatory, ICRR, The University of Tokyo, Hida, Gifu 506-1205, Japan b Gravitational Wave Science Project, National Astronomical Observatory of Japan, Mitaka.Tokyo 181-8588, Japan c Department of Astronomy, Beijing Normal University, Beijing, Beijing 100875, China d Division of Particle and Astrophysical Sciences, Nagoya University, Nagoya, Aichi,464-8602, Japan
Abstract
It is essential to investigate various types of noise in gravitational-wave tele-scopes such as KAGRA. A crackling noise is an intermittent noise, which canoccur when a material experiences stress. KAGRA could be prevented fromreaching the target sensitivity if the crackling noise appears in the geometricanti-spring filter (GAS) of the vibration isolation system. Therefore it is nec-essary to investigate the effect of crackling noise in the GAS. For this research,a crackling noise measurement system with a miniature GAS was built, andthe noise was measured when stress was intentionally added to the GAS. Thescaling law of crackling noise was also investigated using a GAS of a differentdesign. Then the upper limit of the crackling noise in KAGRA was estimated,from the results of the experiment and the derived scaling law, to be less thanthe target sensitivity of KAGRA at frequencies above 55 Hz.
Keywords: gravitational wave, vibration isolation system, crackling noise,geometric anti-spring ∗ Corresponding author
Email address: [email protected] (Seiji Kawamura)
Preprint submitted to Elsevier September 2, 2020 a r X i v : . [ phy s i c s . i n s - d e t ] S e p . Introduction Gravitational-wave astronomy has already begun. Gravitational waves area phenomenon predicted by general relativity, in which distortions of spacetimepropagate far away. In 2015, a gravitational wave was detected for the first timeby the two US detectors comprising the Laser Interferometer Gravitational-wave Observatory (LIGO)[1]. Now the Large-scale Cryogenic Gravitational-wave Telescope (KAGRA) is under construction[2]. It must reach the targetsensitivity and join the world network of gravitational wave telescopes as soonas possible for the development of gravitational-wave astronomy.KAGRA is constructed underground at Kamioka in Japan (very close tothe neutrino detector Super Kamiokande) to reduce the effect of seismic noise.KAGRA is a 3 km-long resonant sideband extraction (RSE) interferometer asshown in Fig. 1.
ETMY ETMXBSY ‐ arm X ‐ armITMY ITMXLASER Photodetector (PD)SRMPRM Figure 1: RSE interferometer used for KAGRA.
The RSE interferometer consists of a Fabry-Perot Michelson interferometer(FPMI), a power recycling mirror (PRM), and a signal recycling mirror (SRM).The FPMI consists of four mirrors as test masses (ITMX, ITMY, ETMX, andETMY) and a beam splitter (BS). The FPMI senses a difference between thetwo arm lengths to obtain the gravitational wave signals. The PRM is installed2o amplify the power of the beam incident to the FPMI for better shot noise, andthe SRM is installed to extract the gravitational wave signals from the FPMIbefore cancellation. In KAGRA, the four test-mass mirrors will be cooled downto about 20 K to reduce thermal noise.Vibration isolation is very important for improving the detection sensitivityinside the gravitational wave observation band. In KAGRA, all the mirrorsconstituting the interferometer are suspended as multistage pendulums for vi-bration isolation in all degrees of freedom. In particular, the vibration isolationsystem (VIS) for the four test-mass mirrors is crucial. It is shown in Fig. 2,and is called a Type-A suspension system. It has seven stages, low-temperaturemasses at the bottom and room temperature ones above. The test-mass mirroris suspended at the very bottom of the Type-A suspension and is cooled downto a cryogenic temperature. At the lowest end of the room-temperature part,there is a “bottom filter” (BF). . m . m Figure 2: Type-A suspension system for the four test-mass mirrors used in KAGRA.
Geometric Anti-Spring filters (GAS) are used for vertical vibration isolation3n KAGRA. They are used in many parts of the Type-A suspension, with thelowest of them in the BF. A GAS consists of several metallic blades that areevenly pressed toward a keystone, as shown in Fig. 3 to generate anti-springforce. As a result, the resonant frequency of the vertical spring is reduced, andbetter isolation is obtained while the load capacity is barely affected[3].
KeystoneBlade
Figure 3: General type of the GAS filters used in the suspension of KAGRA. The diameterof the disk frame is, for example, 0.7 m for the bottom filter.
Although the GAS is more compact than just a blade spring that has thesame resonant frequency, a large stress is applied to the metal blades of theGAS. In this state, the crackling could occur in the metal blades of the GASin response to a change of the blades’ strain caused by vertical seismic motion.The crackling is an unsteady and impulsive phenomenon that the internal strainin solid changes discontinuously when stress is externally applied[4]. Cracklingis known to occur in various crystals and materials, and the crackling is alsoobserved for metal crystals. Crackling in metal is already observed, for example,in Nickel[5]. If crackling frequently occurs in the GAS blades, the sensitivityof KAGRA could be impaired. Incidentally, the crackling noise occurring in asimple blade spring has been investigated by experiment at California Instituteof Technology, and it was found that the crackling noise would not impair theLIGO target sensitivity[6][7][8]. 4 . Objectives and strategies
The purpose of this investigation is to estimate the effect of crackling noisethat could occur in KAGRA. For that purpose, we employed the following strat-egy. First, we design a crackling-noise measurement system (CNMS) consistingof a specially-arranged GAS and an interferometric measurement system in sucha way that the crackling noise in the CNMS reveals itself as much as possiblecompared with the real vibration isolation system for KAGRA. This way, wecan estimate the effect of crackling noise for KAGRA in better sensitivity. Thenwe convert the crackling noise (or an upper limit) obtained in the CNMS intothe estimated crackling noise using a conversion ratio. The conversion ratioincludes some obvious geometric factors and a scaling law of the GAS. To im-plement this strategy in the CNMS, we employed the following schemes:(1) In the CNMS, the vertical motion of the mirror, which is along the beam axis,is measured, while in KAGRA, the horizontal motion of the test-mass mirrorsalong the beam axis is measured. In KAGRA, the crackling noise of the GASshakes the test-mass mirrors in the vertical direction, which is converted intothe horizontal motion of the mirrors because of the ground tilt (intentionallyimplemented for drainage of the spring water) and the curvature of the Earth.Therefore in this sense, the CNMS shows larger crackling noise compared withKAGRA.(2) The mirror is suspended as a single pendulum from the keystone of the GASin the CNMS, while in KAGRA, the test-mass mirror is suspended as a triplependulum from the keystone of the GAS in the BF. The transfer function ofthe vertical motion from the keystone to the mirror depends on the number of apendulum used to suspend the mirror; it is isolated more with more pendulums.Therefore in this sense, the CNMS shows larger crackling noise compared withKAGRA.(3) A compactly-arranged GAS system is used in the CNMS, while in KAGRA,a regular-size GAS is used. It is anticipated that the smaller GAS would producecrackling noise of larger amplitude because the smaller GAS usually provides5he higher resonant frequency (see section 4, 3-i) for more details). Thereforethe CNMS shows larger crackling noise compared with KAGRA. On the otherhand, crackling noise is expected to occur less frequently in the smaller GAS(see section 4, 3-ii) for more details). Therefore, it is not obvious if the CNMSshows larger crackling noise compared with KAGRA at this point. Nevertheless,we decided to use the smaller GAS because of the easiness of the manufacturingand handling of the GAS.(4) The GAS system in the CNMS is shaken intentionally with large sinusoidalsignals, while in KAGRA, the GAS is shaken only seismically. In this sense, theCNMS shows larger crackling noise compared with KAGRA.The scaling law of the crackling noise for different sizes of GASes will betheoretically analyzed in this paper. Although it is necessary to verify theresult of the analyzed scaling law by experiment, in this paper, we assume thatthe obtained scaling law is correct to estimate the crackling noise for KAGRA.There are two kinds of crackling noise. One is what occurs in response to achange in stress, and the other is what occurs, even if there is no stress change.In this paper, we estimate only the former one because it is anticipated to belarger than the latter. We should also note that we assume that the cracklingnoise frequently occurs enough so that each piece of the crackling noise cannotbe separated, and all the crackling noise increases the power spectrum of thenoise. We do not consider infrequently-occurring crackling noise because dataanalysis with two detectors can reject such infrequently-occurring noise.
3. Experimental setup
In the CNMS, a Michelson interferometer shown in Fig. 4 is used. Each armof the Michelson interferometer has a ”GAS unit.” It is a combination of anEnd Mirror (EM) and a Folding Mirror (FM) for folding the horizontal beamto the vertical direction. The EM is attached to the lower part of the testmass, which is suspended by wire from the keystone of the GAS to isolate theEM from seismic motion both in a horizontal and vertical motion. The FM is6lso suspended in the same manner independently under the EM suspension.The beam splitter (BS) is also suspended as a double pendulum for horizontalvibration isolation (the optical axis is horizontal.) All the mirrors and the beamsplitter are isolated from seismic noise as much as possible so that the cracklingnoise appearing in the GAS can be measured precisely. The interferometer iscontrolled to be maintained mid-fringe at the photodetector by feeding back theinterference signal to the magnet-coil actuator of the BS. Large sinusoidal forceis applied to the two keystones commonly by a magnet-coil actuator to inducethe crackling noise. It should be noted that since the lengths of the two armschange commonly, the feedback signal does not contain this large signal at all inprinciple. However, the crackling noise, which occurs independently in the twoarms, could appear in the feedback signal. This feedback signal is measured toanalyze the noise performance of the interferometer.
EMTest massFM
Figure 4: Schematic diagram of the CNMS for measuring the crackling noise. . Conversion ratio In this section, let us consider the conversion ratio, with which the cracklingnoise obtained in the CNMS should be converted to the estimated cracklingnoise for KAGRA.The following shows the details of the conversion ratio. (1) The verticalmotion of the test-mass mirror in KAGRA is converted to the motion of themirror along the optical axis because of the ground tilt and the curvature of theEarth. The ground tilt of the KAGRA arm tunnel is 1/300, and the differencebetween the optical axis and the horizontal axis due to the curvature of theEarth is 0 . × − . Therefore, the maximum ratio between the optical andhorizontal axis is 3 . × − . This is the conversion ratio with which the motionof the mirror along the optical axis should be estimated from the vertical motionof the mirror for KAGRA. (2) The transfer function of the vertical motion fromthe keystone to the test-mass mirror for KAGRA is calculated by SUMCON,which is a mechanical suspension modeling tool in Mathematica. The obtainedtransfer function, H ( f ), is shown in Fig. 5. It has three resonant peaks, andthe frequency dependence above the frequency of the third peak is f − . This isconsistent with the features of the triple pendulum below the lowest GAS bladefor KAGRA. It should be noted that the transfer function has some pairs of adip and a peak, which come from resonances in the auxiliary degrees of freedom.The transfer function of vertical motion from the keystone to the test massof GAS for the CNMS is calculated as a simple spring which has one resonantpeak by SUMCON. The obtained transfer function, | h ( f ) | , is shown in Fig. 6.To estimate the motion of the mirror for KAGRA from the measured motionof the mirror in the CNMS, the ratio between the two transfer functions ( H ( f ) h ( f ) )should be used.(3) Here we consider the scaling law of the size of GASes in terms of the twoaspects of the crackling noise: the amplitude of the noise and the occurrencefrequency of the noise. Then finally we combine both effects to derive the scalinglaw in terms of the power spectrum of the crackling noise.8 igure 5: Transfer function, H ( f ), of the vertical motion from the keystone of the GAS at thetop of the CNMS to the mirror on the test mass.Figure 6: Transfer function, h ( f ), of vertical motion from the key stone to the test mass ofGAS at the top of CNMS.
9) First, we consider the scaling law for the amplitude of the crackling noise. Wecalculate how the crackling event occurring at an arbitrary location of the GASblade is transmitted to the vertical displacement of the keystone. We treat theblade as a beam with one end fixed to a wall and a mass M attached to theother end, as shown in Fig. 7.
𝐹 = 𝐹 sin (Ω𝑡)𝜉 𝑀𝑙
Figure 7: Schematic model of the crackling event occurring in the GAS blade.
We regard each crackling event as an external force applied to the beam atits local position. First the vertical displacement of the beam at a horizontalposition x from the wall, w(x, t), when an external force ( F = F sin(Ω t ))is applied to the beam at the horizontal position of x = ξ in the downwarddirection is the following[9]: w ( x, t ) = F ∞ (cid:88) i =0 W i ( x ) W i ( ξ ) ω i − Ω (sin (Ω t ) − Ω ω i sin ( ω i t )) , (4.1)where i is the resonance mode number, W i is the mode shape of the i th modeat the beam, and ω i is the resonant frequency of the i th mode. Here it shouldbe noted that F is not constant, it depends on the configuration of the beam.If we neglect modes of higher order than the 0th, we can rewrite Eq. (4.1) asthe following: w ( x, t ) = F W ( x ) W ( ξ ) ω − Ω (sin (Ω t ) − Ω ω sin ( ω t )) . (4.2)10ccording to [9], the 0th mode shape W ( x ) is given by W ( x ) = C [(cos (1 . x/l ) − cosh (1 . x/l )) (4.3) − . . x/l ) − sinh (1 . x/l ))] . When we put m ( x ) = (cos (1 . x/l ) − cosh (1 . x/l )) − . . x/l ) − sinh (1 . x/l )),the vertical position of the mass, w(x=l, t) is derived from the normalization of W as follows, w ( x = l, t ) = F M m ( ξ ) m ( l ) 1 ω − Ω (sin (Ω t ) − Ω ω sin ( ω t )) . (4.4)See [9] for details of this calculation. In the cace that the external force is veryfast (Ω (cid:29) ω ) and t ∈ [0 π ], we can regard the external force as an impact (forexample crackling noise). From this condition and Eq. (4.4) at t = π , we canwrite w ( x = l, t = π
2Ω ) = − F M m ( ξ ) m ( l ) 1 − π + O ( | ω Ω | )Ω . (4.5)The maximum of this quantity is F M Ω . Now let us compare two such systemsof different size and mass. Since we can reasonably assume that Ω does notdepend on the size and mass of the system, the ratio of the vertical motion ofthe mass between the two systems (A and B) is the following. w A w B = M B M A F A F B . (4.6)Here, F , M, and ω : F = kδy ( δy is vertical displacement of the keystonewhat depends on the type of material.), and k = M ω . These lead to thus F A F B = M A ω A M B ω B . Thus, finally, w A w B is expressed by the following simple equation. w A w B = ω A ω B . (4.7)This equation represents a scaling law of the amplitude of the vertical motion ofthe keystone caused by crackling events with a GAS system of a different size.ii) Here we consider the occurrence frequency of the crackling. We believewe can safely assume that the occurrence frequency of crackling ( N ) of the11rackling is proportional to an absolute change in the blade’s strain and keystonedisplacement caused by the external stress, and the total number of the crystallattice. It indicates that the occurrence frequency ( N ) is N = (cid:37)V ω ext π δγ, (4.8)where ω ext is a frequency of external stress, δγ is a difference between themaximum shear strain and minimum one, V is a volume of the blade, and (cid:37) is a proportionality constant, depending on the number density of the crystallattices ( (cid:37) is common for the same material). The shear strain of the blade δγ is also proportional to the vertical displacement of the keystone. Thus we canwrite δγ = χδy . χ is a proportionality constant, which can be calculated by thematerial dynamics. We assume that the shape of the blade is a trapezoid asshown in Fig. 8. ℎ" $ % Figure 8: Dimension of a blade used for a GAS.
Here the material dynamics gives (see [10] for the details) δy = 6 αW l b h E (1 − ν ) , (4.9)where W is force added to the blade tip and E is a modulus of the longitudinalelasticity of the blade and ν is Poisson’s ratio of the blade material (In this caseit is maraging steel). α is a correction factor for shape. It is given by α = α ( β ) = − β + 3 β ( − log β )(1 − β ) , (4.10)where β = b b . Using Hooke’s law and assuming that the applied force P gives12 uniform stress to each part of the blade, we obtain δγ = b h (1 − ν )(1 − ν )3 αl V δy. (4.11)Thus we obtain χ = b h (1 − ν )(1 − ν )3 αl V . Therefore the scaling law of the occurrencefrequency of the crackling between GAS A and GAS B is N A N B = b A h A α B l B b B h B α A l A ω ext ; A δy A ω ext ; B δy B j A j B , (4.12)where j A and j B are the number of blades in each GAS.Now that we derived the scaling law of the amplitude and occurrence fre-quency of the crackling noise. To estimate the scaling law of the power spectrumof the crackling noise, we assume that the crackling occurs very frequently sothat the power spectrum of the crackling noise is proportional to the squareroot of the occurrence frequency. Thus the scaling law of the power spectrum P A P B is given by P A P B = w A w B (cid:113) N A N B . From Eq. (4.7) and Eq. (4.12), we obtain P A P B = ω A ω B (cid:115) b A h A α B l B b B h B α A l A ω ext ; A δy A ; seis ω ext ; B δy B ; seis j A j B . (4.13)(4) From the above explanations altogether, the conversion function ( c ( f ))from the power spectrum of the vertical displacement of the mirror in the CNMSdue to the crackling noise to the estimated power spectrum for KAGRA in termsof strain due to the crackling noise is the following c ( f ) = 3 . × − (cid:12)(cid:12)(cid:12)(cid:12) H ( f ) h ( f ) (cid:12)(cid:12)(cid:12)(cid:12) P KAGRA P CNMS / . (4.14)Note that the conversion function has 3000 (arm length) in denominator toconvert the displacement sensitivity to the strain sensitivity. Table 1 shows thecomparison of the mechanical parameters of the Type-A GAS for KAGRA andthe small GAS for CNMS.
5. Experiment and results
We conducted the experiment at the KAGRA site so that we can estimate thecrackling noise for KAGRA precisely. In the CNMS, a sinusoidal signal at 0.1 Hz13 able 1: Parameters of the Type-A GAS for KAGRA and the small GAS for CNMS. “N”indicates number. b [mm] b [mm] l [mm] h [mm] N of blades N of GASes α Type-A GAS 80 13 275 2.4 5 4 1.4Small GAS 20 5.0 70 0.30 3 2 1.3was applied to the coil-magnet actuator for the keystone to serve as pseudo-seismic motion. The amplitude was 1 . × − m. We consider the dominantseismic motion of the keystone of the GAS filter in the BF in a KAGRA type-A suspension to be the vibration propagated to the BF at the first resonancefrequency of its GAS. Its amplitude is 5 . × − m, and its frequency is 0.5 Hz.The amplitude of CNMS’s keystone motion is much bigger than KAGRA’s. Itmakes the crackling effect bigger in CNMS than KAGRA’s.Fig. 9 is the noise power spectrum of the CNMS when the keystone was notexcited and excited at a frequency of 0.1 Hz. According to this figure, the noisefloors of the two measurements are almost equal at broad frequencies above10Hz. This indicates that the effect of the incessantly-occurring crackling noisecaused by the intentional stress applied to the GAS blades is not significant.Therefore, we can safely state that the noise level in the CNMS sets the upperlimit of the crackling noise.We estimated the upper limit of the strain spectrum of the crackling noise forKAGRA, using the sensitivity of the CNMS obtained in the experiment and theconversion function Eq. (4.14). The upper limit of the crackling noise effect forKAGRA is shown in Fig. 10, together with the latest estimation of sensitivitylimit based on the current design for KAGRA (which hereafter we simply callthe KAGRA target sensitivity). 14 igure 9: Power spectrum of the vertical displacement in the CNMS. The red curve is thepower spectrum with excitation (0.1Hz) and the blue one is without excitation.
6. Discussions and conclusions
If the scaling law obtained in this paper is correct, according to Fig. 10, wefound that the effect of the crackling does not impair the target sensitivity ofKAGRA above 55 Hz. This result indicates that the crackling noise in KAGRAdoes not have a significant effect on the observation of gravitational waves fromcoalescences of a binary black hole (BBH) whose masses are a few tens of thesolar mass, and coalescences of a binary neutron star (BNS), which were detectedby LIGO. More precisely, if the observation is not available under 55 Hz, theobservation range for gravitational waves from 30 M (cid:12) BBH is reduced only by15 % from 1.3 Gpc to 1.1 Gpc, and the one from the BNS is reduced only by 7 %from 158 Mpc to 147 Mpc in KAGRA. Finally, we should repeat that this resultwas obtained with the assumption of the scaling law described in this paper,which should be verified by the experiment in the next step.15 igure 10: Upper limit of the crackling noise effect in KAGRA and the KAGRA targetsensitivity. eferencesReferences [1] B. P. Abbott et al. [LIGO Scientific and Virgo Collaborations], “Observationof Gravitational Waves from a Binary Black Hole Merger”, Phys. Rev. Lett. (2016) 061102[2] T. Akutsu, et al., “First cryogenic test operation of underground km-scalegravitational-wave observatory KAGRA”, Class. Quantum Grav. (2019)165008[3] G. Cella, et al., “Monolithic geometric anti-spring blades”, Nucl. Instrum.Meth. A (2005) 502[4] James P. Sethna, Karin A. Dahmen and Christopher R. Myers, “Cracklingnoise”, Nature (2001) 242[5] D. M. Dimiduk et al., “Scale-Free Intermittent Flow in Crystal Plasticity”,Science, (2006) 1188[6] Xiaoyue Ni, “Probing Microplastic Deformation in Metallic Materi-als”, Dissertation (Ph. D.), California Institute of Technology (2018),doi:10.7907/F38W-6N47[7] G. Vajente, et al., “An instrument to measure mechanical up-conversionphenomena in metals in the elastic regime”, Rev. Sci. Instrum, (2016)065107[8] G. Vajente, “Crackling noise in advanced gravitational wave detectors: Amodel of the steel cantilevers used in the test mass suspensions”, Phys. Rev.D (2017) 022003[9] Singiresu S. Rao, “Vibration of continuous systems”, Wiley (2007)[10] Hemendra Singh Shekhawat and Hong Zhou, “Analysis and Design of Can-tilever Springs”, IJERT, (2015) 87717 cknowledgmentcknowledgment