A two-dimensional low-frequency vibration attenuator using X-pendulums 3
D. Tatsumi , Mark A. Barton
y
, T. Uchiyama and K. Kuroda
Institute for Cosmic Ray Research, University of Tokyo, Tanashi, Tokyo 188, Japan
Abstract We have designed and constructed an improved two dimensional X-pendulum vibrationisolation table. It achieved a lower resonant frequency (7 s) than previous prototypes, and the e ects of many parasitic elastic resonances have been reduced by careful balancing, leading to much improved vibration isolation around a few Hertz.
1 Introduction To detect gravitational waves using an interferometric detector, mirrors should be isolated from seismic motion. The seismic vibration is a particular problem at low frequency, because the mirror is suspended with pendulum with a resonant frequency around 1 Hz. The Japanese interferometric gravitational wave detector, TAMA300, will have a vibrational attenuation system consisting of three stages: a stack, the two-dimensional X-pendulum attenuator presented here, and a double pendulum suspension system. The stack and the double pendulum suspension are only e ective at relatively high frequencies, so the X-pendulum system will be used to improve the isolation at low frequency (less than 10 Hz). Since the stack and the double pendulum suspension systems have several resonances around a few Hz, we need to attenuate the amplitude of horizontal vibration by an order of magnitude (020 dB) to let the interferometer achieve its maximum sensitivity. The simplest way to do this would be to construct a very long period simple pendulum. However such a pendulum needs a tall supporting frame and this needs to be placed in a vacuum chamber (the TAMA chambers are 1 m in diameter and not quite 2 m in height). Our X-pendulum system is designed to have a suitably long period but is much more compact vertically and easily meets the space requirements [1, 2]. The basic X-pendulum is seen in Fig. 1. When a center of mass of the load is set just below the critical point [1], we can obtain a long period as a simple pendulum.
3 Electronic y
mail:
[email protected]
Present address: California Institute of Technology, M/S 51-33, Pasadena, California 91125
1
L1 upper plate
X-wires
LG
Hv lower plate
L2
Hc
H
critical point
M mass
stem
Figure 1: Schematic view of the basic X-pendulum. Potential curve at each center-of-mass position is also shown with dashed line. This group has previously reported results for two versions of prototype two-dimensional Xpendulum systems, which showed that the concept was sound [3, 4]. We successfully suspended a table so that it had a long period in both horizontal directions, but was moderately sti in other degrees of freedom and was simple to assemble and adjust. As shown in Fig. 2, a combined unit of two X-pendulums becomes a basis of the two-dimensional vibrational attenuator. The upper part is the same as the basic X-pendulum except the stem which is replaced by the V-shaped suspension wires [3]. The lower one is made by turning the upper one upside down and then rotating horizontally by 90 degrees. The V-shaped middle wires allow pure horizontal motion of two degree of freedom. The load table is suspended by four units of the above combined X-pendulums. 2
Upper X-wires
Support
Normal X-pendulum
z Intermediate wires
y x Inverted X-pendulum
Lower X-wires
Bottom plate
Figure 2: Wire con guration of a combined unit of X-pendulums. The upper part is the same as the basic X-pendulum except the stem which is replaced by the V-shaped suspension wires. The lower one is made by turning the upper one upside down and then rotating by 90 degree. The V-shaped middle wires allow pure horizontal motion of two degree of freedom. Unfortunately the vibration isolation performance of the previous prototype was disappointing, due to a number of resonances from elastic modes of the moving parts [5, 6]. In the next section, we describe the performance of the previous prototype and how we modify it. Finally we present the test result of the modi ed attenuator.
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2 Modi cation of the previous prototype The predecessor to the current design [4] was tested on a shaker table while tuned to 0.25 Hz and achieved a useful amount of horizontal vibration isolation, in particular, 20 dB at 1 Hz. The height of the system was only about 40 cm. However, since little e ort had been made to balance the moving parts, the transfer function contained a number of very large peaks due to elastic resonances, as can be seen in Fig. 3 and listed in Table 1.
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transfer function
1 10 10 10 10 10
-1 -2 -3 -4 -5
10
-1
1
10
f (Hz) Figure 3: Transfer function of the previous prototype X-pendulum system. Its resonant frequency was about 0.25 Hz and it attenuated horizontal vibration by 20 dB at 1 Hz. However, some undesirable parasite resonances were seen. To reduce these e ects, we made a number of modi cations to the shapes of the moving parts. The rst was to arrange that the center of mass of the load as a whole was at the same height as the X wire attachment points. Since the double pendulum had a rigid safety cage which was heavy, the center of mass extended a considerable distance below the X wire attachment points. To eliminates the coupling from horizontal elastic forces to pitch and roll of the load, considerable counterweighting was required. (The suspended parts of the double pendulum stage are not included when calculating the center of mass, since they do not participate in pitching or rolling.) In fact the counterweight had to be placed above the support table and was connected to the load table by thick rods passing through the support table. 4
The X plate was completely redesigned. The wire attachment points (both for X and intermediate wires) were brought to the same height as the center of mass to reduce the e ect of pitching and rolling modes (just as for the load table). The previous design had been particularly bad in this respect because (i) it used three X wires to reduce the number of parts, thus reducing the sti ness in roll, (ii) the X wire clamps were mounted well above the X plate to eliminate interference with the clamps in the adjustment mechanism, (iii) the intermediate wire clamps were mounted on the opposite side of the X plate to avoid interference with the central X wire clamp. However experience with the earlier prototype had shown that we did not need independent adjustments for the two outer X wires. Therefore we arrived at a very simple design as follows: (i) we made the `X' vertically asymmetrical (L1 > L2 ) to separate the X clamps horizontally, (ii) we reverted to a four wire design with all X wire clamps at or near the edges, freeing up space for the intermediate wire clamps in the center, (iii) we further reduced the number of adjustment mechanisms, to just two, in the form of wide lever plates with one double screw actuator in the center and two X wire clamps at the edges. Apart from the tool steel inserts used as clamp jaws, the new design was made from a single piece of aluminium. We also implemented the high-frequency optimization proposed in [4]. If the ratio of moment of inertia to mass of the X plate is optimized, then the natural motion of the isolated X plate when driven horizontally through the X wires is to pitch about the hinge line at the far end of the intermediate wires. This minimizes the force transmitted to the load at high frequencies. Finally, we substituted exure strips made of carbon steel and of rectangular cross-section for the X wires (which had been ordinary piano wire). This was done to enhance the mechanical quality factor (Q) of the system. For periods above around 10 s, most of the restoring force in the pendulum modes is provided by the bending elasticity ke of the X wires. A thin strip gives a much smaller ke value for the same cross-sectional area (i.e., for the same breaking strength) and thus a higher Q (Details of it will be found in Ref. [7]). The spring constant ke is give by
p
ke = TW EW IW ;
(1)
where TW , EW and IW are the tension, Young's modulus and the moment of area of an X wire. In the case of squared cross section wire, IW is expressed by the thickness a and the width b as
IW =
ab3 12
:
(2)
We chose a = 2:0 mm and b = 0:3 mm, which gives a cross-sectional area as large as that of 1 mm diameter wire.
3 The third prototype test results and Discussion We constructed the new version of the two-dimensional horizontal attenuator as shown in Fig. 4 and tested it on a shaker table.
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Figure 4: Schematic view of new X-pendulum system with double suspension system for the mirror. An isolated plate is suspended by four units. Its behavior is identical to that of a plate which is suspended by four wires at each corner. Figure 5 shows the measured transfer function of the new attenuator. This data consists of ve frequency regions; e.g. 0:08 0 0:2 Hz, 0:2 0 0:5 Hz, 0:5 0 5 Hz, 2 0 20 Hz and greater than 20 Hz. In the region of less than 0.5 Hz, data were taken by an optical displacement sensor which is made of a LED-photodiode pair. In the higher frequency region, a PZT accelerometer was used. Since the vibration above 30 Hz was disturbed by acoustic noise, this region is not shown here. The frequency of the swing was 0.14 Hz and was lower than that of the second prototype (0.25 Hz). As with the second prototype, some parasitic resonances were found but with lower amplitudes. These are summarized in Table 1. The lowest frequency peak of parasitic resonances is the pitching of the isolated plate. The second peak is a coupling e ect with the vertical motion of the load table, because the displacement sensor is contaminated by uninteresting motion. The
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resonant frequency of vertical motion, !v , is given by
!v2 =
2Nw Ew Aw sin2 0
Ml
;
(3)
where Nw is the number of X wires, Ew is Young's modulus, Aw is the cross-sectional area, 0 is the angle from the horizontal to the wire, and l is the length of the X wire. Since the X wires have much higher vertical compliance than that of the intermediate wires because of the small value of 0 , we neglected the e ects of the intermediate wires. The calculated value is 5.3 Hz, which is consistent with the measured value. The third peak was a resonance of the table supporting the whole attenuation system. Comparing these resonances with those in Fig. 3, we see that all of them have clearly been reduced.
10
transfer function
1 10 10 10 10 10
-1 -2 -3 -4 -5
10
-1
1
10
f (Hz) Figure 5: Transfer function of the new X-pendulum. This measurement was done by using discretely swept sinusoidal vibration to obtain higher S/N ratio of each data point (shown as ). The solid curve represents the ideal transfer function only de ned by its resonant frequency and its Q value, which were experimentally obtained. The data consists of ve frequency regions; e.g. 0:08 0 0:2 Hz, 0:2 0 0:5 Hz, 0:5 0 5 Hz, 2 0 20 Hz and greater than 20 Hz.
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Prototype (Hz) New (Hz) Horizontal mode 0.25 0.14 Yawing mode 2.7 1.5 Pitching mode 6.0 2.0 Vertical mode 7.5 5.0 Support Table 8.0 12.0 Tipping mode 20.0 | Table 1: Resonance frequencies: Horizontal mode means a basic horizontal motion as a swing. Yawing and pitching modes are rotational motion in horizontal and vertical plane, respectively. Vertical mode is mainly due to elasticity of the X-wires. Support table indicates a resonant frequency of the support table for X-pendulum attenuation system. Finally, tipping mode is due to a combination of pitching with some yawing motion of the middle X-plates.
4 Conclusion We have tested modi cations of the two-dimensional X-pendulum vibrational attenuator and found that they substantially improved the transfer function, and reduced the e ects of some parasitic resonances. We have achieved the period of seven seconds in this system. It was more longer than that of the previous prototype. And the attenuation at 1 Hz was more than 20 dB. These results show that this new two-dimensional attenuator using X-pendulums is one of the most powerful tools for improving lock stability and sensitivity at low frequency for laser interferometric gravity wave detectors.
5 Acknowledgements The authors wish to thank Dr. Araya for helping us to operate a shaker table at Earthquake Research Institute (ERI), University of Tokyo, and thank students of Kuroda group in ICRR for helping our experiments.
References [1] M. A. Barton and K. Kuroda, Rev. Sci. Instrum. 65, 3775 (1994). [2] N. Kanda, M. A. Barton and K. Kuroda, Rev. Sci. Instrum. 65, 3780 (1994). [3] M. A. Barton, N. Kanda and K. Kuroda, Rev. Sci. Instrum. 67, 3994 (1996). [4] M. A. Barton, T. Uchiyama et al., submitted to Rev. Sci. Instrum., 1998. 8
[5] M. A. Barton, talk at the meeting of the Astronomical Society of Japan, 1996. [6] M. A. Barton, talk at the meeting of the Physical Society of Japan, Oct 9, 1996, Saga university. [7] K. Yuki et al. Phys. Lett. A 223 (1996) 149-154.
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