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Response of a folded pendulum to tilt tides
31 May 1999
Physics Letters A 256 Ž1999. 132–140
Response of a folded pendulum to tilt tides Shuhua Fan a , Yunxia Cai a , Shuchao Wu a , Jun Luo a
a,b
, Houtse Hsu
b
Department of Physics, Huazhong UniÕersity of Science & Technology, Wuhan 430074, People’s Republic of China b Institute of Geodesy and Geophysics, Academia Sinica, Wuhan 430077, People’s Republic of China Received 9 June 1998; received in revised form 30 March 1999; accepted 5 April 1999 Communicated by P.R. Holland
Abstract A new application of the folded pendulum as a tiltmeter is proposed and its response to the crustal vibrations is studied. Theoretical analysis shows that the folded pendulum is very sensitive to the low-frequency components of the crustal tilt vibrations. Preliminary experimental results show that the resolving power of the folded pendulum to the tilt tides is in the order of about 3.5 = 10y9 rad. q 1999 Elsevier Science B.V. All rights reserved. PACS: 93.85.tq; 91.10.tq Keywords: Folded pendulum; Tiltmeter; Solid earth tide
1. Introduction Because of the tidal force, the earth crustal elastic deformation will occur and the solid earth tides will be produced. It is significant to measure the earth tides because it not only is an important way to understand a number of geophysical phenomena and the internal structure of the earth, but is also very valuable in the research on great natural disasters, such as the earthquakes and volcanic eruptions. The solid earth tides include gravity tides, strain tides and tilt tides. A superconducting gravimeter with the precision of several nGal is the most sensitive gravimeter for measuring the gravity tides w1x. The strain tides can be measured by the rod strainmeters w2–4x, wire strainmeters w5,6x, borehole strainmeters w7–11x and laser strainmeters w12–14x. The tilt tides can be measured by both short-base and long-base tiltmeters w15x. The short-base tiltmeters include the horizontal pendulums w16–19x, simple pendulums w20,21x, diamagnetic tiltmeters w22x and bubble tiltmeters w23,24x. The long-base tiltmeters are the liquid tiltmeters w25–27x. For the measurement of the tilt tides, many long-base tiltmeters such as water-tube tiltmeters have been used at stations. They all use the surface of a liquid to define the horizontal plane and then measure tilt variations by monitoring the motion of this surface. The sensitivity of a liquid tiltmeter, in general speaking, is dependent on the horizontal distance between its two containers. The longer the distance is, the higher its sensitivity is. So the liquid tiltmeter works well at detecting the average value of the tilts in a large scale. Unfortunately, some influences other than tilts, for example, the density variations caused by temperature differences, the surface tension acting on the walls of the liquid containers, the symmetry of the containers and transducers and so on, could make the liquid surface move as 0375-9601r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 Ž 9 9 . 0 0 2 2 3 - 6
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well. Therefore extra careful consideration should be given to these influences. A typical short-base tiltmeter is the horizontal pendulum, whose design is that the bob swings at the end of an arm attached to the frame in such a way that the bob moves along a nearly horizontal circle. This is an astatic system: the restoring force is kept so small that slight external forces can cause large motions. If the angle of inclination of the arm is small, the horizontal pendulum will magnify the tilts considerably. But, a small tilt perpendicular to the direction of sensitivity can alter the sensitivity of the horizontal pendulum significantly, so the frequent calibration is needed. At present, the folded pendulum ŽFP. is studied in laboratory mainly for vibration isolation for the laser interferometric gravitational waves detector and much progress has been achieved recently w28,29x. In this article, we proposed to use a FP as a sensitive tiltmeter to measure the earth tilt tides due to its following advantages. First, it is very sensitive to the low-frequency components of the crustal tilt vibrations, namely tilt tides. Second, it is insensitive, in contrast with a horizontal pendulum, to the tilts perpendicular to the direction of its sensitivity due to the flat linkage foils. Finally, its dimension is about half a meter or less so that it can be installed in a limited space as a usual short-base tiltmeter. The preliminary experimental result shows that the resolving power of a FP tiltmeter can reach about 3.5 = 10y9 rad.
2. Principle and design of the FP A FP, as proposed by Blair et al. w28x, consists of a positive pendulum and an inverse pendulum as shown in Fig. 1. The positive pendulum works as an ordinary simple pendulum. That is, whenever it deviates from the equilibrium position, the gravity force acting on it will tend to pull it back to its equilibrium position. On the contrary, the inverse pendulum will move further away from the equilibrium position once it deviates from there. The positive pendulum and the inverse one are connected by a horizontal platform to form a FP. Thus the negative elastic coefficient of the inverse pendulum partially counterbalances the positive one of the positive pendulum. The total elastic coefficient of the system can then be greatly reduced and the FP could have a quite low natural frequency. Of course, the longer the period is, the worse the stability of the system is and the more difficult to adjust it. The practical FP used in our experiment consists of an 18 cm long positive pendulum and a 22 cm long inverse one, which are connected by a horizontal platform whose dimensions are 30 cm = 20 cm = 1.5 cm. These movable components and the fixed frame and base are all made of aluminum. They are linked to each other with flat beryllium bronze foils, which are 2.5 mm wide and 50 mm thick. The separation between the frame and the platform is measured with a precise infrared displacement transducer. The infrared emitter and the photodiode ŽED. are fixed on the frame and the light shield is fixed on the platform. So the separation between the frame and the platform changes with the movement of the crust, which results in the changes of the output
Fig. 1. The schematic diagram of the FP.
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Fig. 2. The layout of the effect of the crustal vibrations on the FP and the coordinate system.
voltage of the photodiode. The output signal is amplified and transmitted both to an X–Y chart recorder in analog record and to a computer in digital record.
3. Response to the crustal vibration The effect of the crustal vibrations on the FP and the coordinate system can be graphically shown in Fig. 2: AC, BD and CD represent respectively the positive pendulum, the inverse pendulum and the horizontal platform, whose lengths and masses are also shown in the figure. AO’ and BO’ represent the frame and the base of the FP, respectively. X g , Yg and u indicate the horizontal and vertical components of the crustal vibrations and the angle of the crustal tilt vibrations, respectively. According to the coordinate system shown in Fig. 2, we can obtain the generalized Lagrangian equation of the FP system in linear approximation if we consider X g , Yg and u as the first order quantities as follows:
w¨ q 2 bw q v 02 w q Au¨q Bu q CX¨g s 0, .
Ž 1.
where As
m1 Ž l 1 q 3l 2 . q 2 m 2 l 2 q 6 m 3 l 2
v 02 s
2 Ž m1 q m 2 q 3m 3 . l 1
,
Bsy
3 Ž m1 q m 2 q 2 m 3 . g 2 Ž m1 q m 2 q 3m 3 . l 1
3 m1rl1 y m 2rl 2 q m 3 Ž 1rl1 y 1rl 2 . g q 12 k Ž 1rl12 q 1rl 22 . 2 Ž m1 q m 2 q 3m 3 . l 1
,
Cs
3 Ž m1 q m 2 q 2 m 3 . 2 Ž m1 q m 2 q 3m 3 . l 1
,
.
v 02 is the angular frequency of the FP, k is the elastic coefficient of the foils, b is the damping coefficient of the system, and w is the angle that the FP deviates from its static equilibrium position. The equivalent simple pendulum length of the FP is given by: l e s grv 02 s
2 Ž m1 q m 2 q 3m 3 . l 1 3 m1rl1 y m 2rl 2 q m 3 Ž 1rl1 y 1rl 2 . q 12 k Ž 1rl12 q 1rl 22 . rg
.
Ž 2.
S. Fan et al.r Physics Letters A 256 (1999) 132–140
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Fig. 3. Transfer function of the FP to the crustal tilt vibration.
From Eq. Ž1., we can obtain the transfer functions, Tt and Th , of the system to the crustal tilt vibrations and the horizontal vibrations as follows: Tt s
Th s
w 10
Av 2 yB
,
s
u0 l 1 w 20 Xg 0
(Ž v y v . q 4b v 2 2 0
2
2
l 1C v 2
s
(Ž v y v . q 4b v 2
2 2 0
Ž 3.
2
2
,
Ž 4.
2
where w 10 and w 20 represent the response amplitudes of the FP to the crustal tilt vibration and the horizontal vibration, respectively. Fig. 3 and Fig. 4 show the transfer functions Tt and Th , respectively. The typical parameters have been chosen in the figures as: l 1 s 0.18 m, m s 97.2 g, l s 0.22 m, m s 148.0 g, m s 2430.0 g, T s 12 s, and b s 0.1 radrs. It is clear in Fig. 3 that the FP is very sensitive to the low-frequency components Žbelow the natural frequency v 0 . of the crustal tilt vibrations and the amplification factor is about 200 while it is isolative to the high-frequency components Žabove the natural frequency v 0 .. As the frequencies of the tilt tides are much
Fig. 4. Transfer function of the FP to the crustal horizontal vibration.
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lower than the natural frequency of the FP, the FP could be used as a tiltmeter to detect the tilt tides. Besides, as shown in Fig.4, the effect of the horizontal vibrations, in contrast with the tilt vibrations, can be neglected even in the resonant frequency range of the FP.
4. Preliminary experimental results The sensitivity of the FP tiltmeter was calibrated. And the tilt tides were observed with two FP tiltmeters side-by-side. The second FP, which was constructed recently, is the same as the first one in dimensions. The preliminary experimental results show that the two observed tilt curves are similar and each of them is similar to the theoretical one which was calculated with the program ETGTAB Žcreated by H.-G. Wenzel, Geodaetisches Institut, Universitaet Karlsruhe, Germany. version 3.0 in the point of observation during the experiment. 4.1. Calibration The displacement sensitivity of the infrared transducer system used in the FP was calibrated as follows: the light shield was clipped on the movable part of an Abbe comparator and the frame with the ED was attached to the fixed part of it. Adjusting the horizontal screw of the Abbe comparator and then changing the relative position between the ED and the light shield, we can get the calibration curve as shown in Fig. 5. About every 5 minutes, the horizontal screw was adjusted, and each step represented a horizontal displacement D x s 20 mm, which corresponded to a voltage change D V f 400 mV of the output signal from the transducer system. Therefore the sensitivity of the displacement transducer used in our experiment was:
Dx
f 5.0 = 10y2 mmrmV, DV and the sensitivity of the FP tiltmeter was given as follows:
lx s
lu s
Dx l e DV
f 1.4 = 10y9 radrmV.
Ž 5.
Ž 6.
From Fig. 5, we can see that the noise on each step was in the order of several tens of mV. This noise was just due to the calibration process because the person had to stand very nearly to operate the Abbe comparator for calibration. While the FP was installed on a foundation in our cavity laboratory, and the people was kept away from the laboratory during the experiment. Therefore the noise of the FP tiltmeter would be less than that
Fig. 5. Calibration curve of the infrared transducer system.
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Fig. 6. Output voltage curves of the infrared transducers of the FP tiltmeters in our cavity laboratory from 04:13 Nov.26, 1998 GMT to 04:08 Dec.09, 1998 GMT. Ža. Output voltage curve of the FP tiltmeter No. 1. Žb. Output voltage curve of the FP tiltmeter No. 2.
on the calibration curve. The resolving power of the system at present is mainly limited by the Analog-DigitalConvertor used in our recording system. Because the conversion voltage range of the Analog-Digital-Convertor
Fig. 7. Observed tilt curves of the two FP tiltmeters which were obtained by filtering the original output voltages through a high-pass filter with the cut-off frequency 5.6=10y6 Hz and then converting them into the tilt angles correspondingly according to the calibration result. Ža. Observed tilt curve of the FP tiltmeter No. 1. Žb. Observed tilt curve of the FP tiltmeter No. 2.
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Fig. 8. Theoretical tilt curve calculated with ETGTAB in the point of observation during the experiment from 04:13 Nov.26, 1998 GMT to 04:08 Dec.09, 1998 GMT.
of 12 bits was from -5 V to q5 V, the conversion error is about 2.5 mV. Therefore the resolving power of the FP tiltmeter is approximately given as follows:
Dumin s lu Verror f 3.5 = 10y9 rad.
Ž 7.
4.2. Measurement of tilt tides Two FP tiltmeters were installed on a foundation in our cavity laboratory. The foundation was built in tight connection with the crust and was isolated from the surroundings. So the outer interference was eliminated to the greatest possible extent. By the computer recording system, we obtained the output signals of the two FP tiltmeters as shown in Fig. 6. Through a highpass filter with the cut-off frequency 5.6 = 10y6 Hz, the extremely slow drifts in the raw data were removed and the observed tilt curves were obtained as shown in Fig. 7. Comparing Fig. 7 Ža. and Žb., we can see that the two observed tilt curves are similar. The program ETGTAB was used to calculate the theoretical tilt in the point of observation Ž30.54 o N, 114.35 o E, the azimuth is 260 o clockwise from north. during the experiment as shown in Fig. 8. Comparing Fig. 7 and Fig. 8, we can see that each observed tilt curve is similar to the theoretical one. We have access to local gravity tide curve as shown in Fig. 9, and we can see that it is correlated with the FP signal to some degree because both of them are caused by tidal force. The very low frequency variation in the FP signals in Fig. 6 might be mainly caused by the secular creep of the linkage foils and the slow changes of the temperature, and it is also easily imagined that it is just drift in the readout. In order to confirm that, some fixture was used to lock the position of the pendulum and the readout signal was observed as shown in Fig. 10. Comparing Fig. 10 and Fig. 6, we can see that during 200 hours the drift in the readout was only about 25 mV, which was much less than that in the FP output signals. Therefore
Fig. 9. Gravity tides curve observed from 04:13 Nov.26, 1998 GMT to 04:08 Dec.09, 1998 GMT with a superconducting gravimeter GWR TT-70 at local observation, which is about 5 km away from our cavity laboratory.
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Fig. 10. Output voltage curve of the readout system while the position of FP was locked with some fixture from 17:48 Mar.07, 1999 GMT to 01:48 Mar.16, 1999 GMT.
this drift might be due to the creep in the linkage foils and the variation of the temperature, which are being under investigation.
5. Discussion The FP used as a tiltmeter has been studied in theory and demonstrated in prototype form. The sensitivity of the FP to the tilt tides is stable due to its symmetric structure and flat linkage foils. Preliminary experimental result shows that the resolving power of the FP to the tilt tides has reached in the order of about 3.5 = 10y9 rad. However, some problems should be solved, which include the drift of the readout due to the creep of foils, the calibration of the sensitivity with a standard dilatometer or a PZT, and the influences due to the air current and the temperature variation. Above works are still under investigations in our laboratory.
Acknowledgements The authors gratefully thank G.Q. Hu, W.X. Cai and D.G. Blair for their useful discussions. This work was arose from the collaboration program between HUST and UWA Žthe University of Western Australia., and supported by the National Natural Science Foundation of China under Grants 49674246 and 19425008.
References w1x w2x w3x w4x w5x w6x w7x w8x w9x w10x w11x w12x w13x w14x
W.A. Prottero Jr., J.M. Goodkind, J. Geophys. Res. 77 Ž1972. 926. H. Benioff, Geol. Soc. Am. Bull. 70 Ž1959. 1019. R.C. Shopland, R.H. KirKlin, Bull. Seismol. Soc. Am. 60 Ž1970. 105. J.E. Fix, J.R. Sherwin, Bull. Seismol. Soc. Am. 60 Ž1970. 1803. P.H. Sydenham, J. Phys. E. 2 Ž1969. 1095. G.C.P. King, R. Bilham, Bull. Seismol. Soc. Am. 66 Ž1976. 2039. T.C. Moore et al., Final Rep. AFOSR-TR-75-0021 ŽNTIS ADrA-003710., Develco. Inc., Palo Alto, CA, 1974. Chi Shunliang, Acta Seismologic Sinica Žin Chinese. 4 Ž1982. 98. M.T. Gladwin, Rev. Sci. Instrum. 55 Ž1984. 2011. S. Sakata, H. Sato, J. Phys. Earth. 34 Ž1986. Suppl. S129. R.H.G. Hart et al., J. Geophys. Res. 101 Ž1996. 25, 553. N.R. Goulty et al., Geophys. J.R. Astron. Soc. 39 Ž1974. 269. S. Takemoto, Bull. Disaster. Prev. Res. Inst. Kyoto Univ. 29 Ž1979. 65. V. Rautenberg et al., Geophys. Res. Lett. 24 Ž1997. 893.
140 w15x w16x w17x w18x w19x w20x w21x w22x w23x w24x w25x w26x w27x w28x w29x
S. Fan et al.r Physics Letters A 256 (1999) 132–140 D.C. Agnew, Rev. Geophys. 24 Ž1986. 579. J. Verbaandert, P. Melchior, Royal de Belgique, Monogr. Obs. R. Belg. 7 Ž1961. 1. J.C. Harrison, J. Lovine, Rep. AFGL-TR-81-0304 ŽNTIS AD108-865-17., Air. Force Geophysics Laboratory, Bedford, MA, 1981. W.X. Cai et al., Crustal Deformation and Earthquake Žin Chinese. 2 Ž1982. 69. J. Yang et al., Acta Seismologic Sinica Žin Chinese. 5 Ž1983. 446. R.V. Allen, Bull. Seismol. Soc. Am. 62 Ž1972. 815. R.V. Jones, J.C.S. Richards, J. Phys. E. b. Ž1973. 589. F. Wyatt et al., Geophys. Rev. Lett. 9 Ž1982. 743. G.L. Cooper, Am. Inst of Aeronaut. and Astronautic, Anoheim, CA, 1970. J.A. Westphal et al., Rev. Sci. Instrum. 54 Ž1983. 415. C.E. Liu, J.L. Zhong, Acta Seismologic Sinica Žin Chinese. 6 Ž1984. 111. C.E. Liu, S.B. Deng, Acta Geophysical Sinica Žin Chinese. 28 Ž1985. 502. M. Kato, Bull. Disaster. Prev. Res. Inst. Kyoto Univ. 27 Ž1977. 155. D.G. Blair et al., Phys. Lett. A 193 Ž1994. 223. J.F. Liu et al., Phys. Lett. A 228 Ž1997. 243.
Physics Letters A 256 Ž1999. 132–140
Response of a folded pendulum to tilt tides Shuhua Fan a , Yunxia Cai a , Shuchao Wu a , Jun Luo a
a,b
, Houtse Hsu
b
Department of Physics, Huazhong UniÕersity of Science & Technology, Wuhan 430074, People’s Republic of China b Institute of Geodesy and Geophysics, Academia Sinica, Wuhan 430077, People’s Republic of China Received 9 June 1998; received in revised form 30 March 1999; accepted 5 April 1999 Communicated by P.R. Holland
Abstract A new application of the folded pendulum as a tiltmeter is proposed and its response to the crustal vibrations is studied. Theoretical analysis shows that the folded pendulum is very sensitive to the low-frequency components of the crustal tilt vibrations. Preliminary experimental results show that the resolving power of the folded pendulum to the tilt tides is in the order of about 3.5 = 10y9 rad. q 1999 Elsevier Science B.V. All rights reserved. PACS: 93.85.tq; 91.10.tq Keywords: Folded pendulum; Tiltmeter; Solid earth tide
1. Introduction Because of the tidal force, the earth crustal elastic deformation will occur and the solid earth tides will be produced. It is significant to measure the earth tides because it not only is an important way to understand a number of geophysical phenomena and the internal structure of the earth, but is also very valuable in the research on great natural disasters, such as the earthquakes and volcanic eruptions. The solid earth tides include gravity tides, strain tides and tilt tides. A superconducting gravimeter with the precision of several nGal is the most sensitive gravimeter for measuring the gravity tides w1x. The strain tides can be measured by the rod strainmeters w2–4x, wire strainmeters w5,6x, borehole strainmeters w7–11x and laser strainmeters w12–14x. The tilt tides can be measured by both short-base and long-base tiltmeters w15x. The short-base tiltmeters include the horizontal pendulums w16–19x, simple pendulums w20,21x, diamagnetic tiltmeters w22x and bubble tiltmeters w23,24x. The long-base tiltmeters are the liquid tiltmeters w25–27x. For the measurement of the tilt tides, many long-base tiltmeters such as water-tube tiltmeters have been used at stations. They all use the surface of a liquid to define the horizontal plane and then measure tilt variations by monitoring the motion of this surface. The sensitivity of a liquid tiltmeter, in general speaking, is dependent on the horizontal distance between its two containers. The longer the distance is, the higher its sensitivity is. So the liquid tiltmeter works well at detecting the average value of the tilts in a large scale. Unfortunately, some influences other than tilts, for example, the density variations caused by temperature differences, the surface tension acting on the walls of the liquid containers, the symmetry of the containers and transducers and so on, could make the liquid surface move as 0375-9601r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 Ž 9 9 . 0 0 2 2 3 - 6
S. Fan et al.r Physics Letters A 256 (1999) 132–140
133
well. Therefore extra careful consideration should be given to these influences. A typical short-base tiltmeter is the horizontal pendulum, whose design is that the bob swings at the end of an arm attached to the frame in such a way that the bob moves along a nearly horizontal circle. This is an astatic system: the restoring force is kept so small that slight external forces can cause large motions. If the angle of inclination of the arm is small, the horizontal pendulum will magnify the tilts considerably. But, a small tilt perpendicular to the direction of sensitivity can alter the sensitivity of the horizontal pendulum significantly, so the frequent calibration is needed. At present, the folded pendulum ŽFP. is studied in laboratory mainly for vibration isolation for the laser interferometric gravitational waves detector and much progress has been achieved recently w28,29x. In this article, we proposed to use a FP as a sensitive tiltmeter to measure the earth tilt tides due to its following advantages. First, it is very sensitive to the low-frequency components of the crustal tilt vibrations, namely tilt tides. Second, it is insensitive, in contrast with a horizontal pendulum, to the tilts perpendicular to the direction of its sensitivity due to the flat linkage foils. Finally, its dimension is about half a meter or less so that it can be installed in a limited space as a usual short-base tiltmeter. The preliminary experimental result shows that the resolving power of a FP tiltmeter can reach about 3.5 = 10y9 rad.
2. Principle and design of the FP A FP, as proposed by Blair et al. w28x, consists of a positive pendulum and an inverse pendulum as shown in Fig. 1. The positive pendulum works as an ordinary simple pendulum. That is, whenever it deviates from the equilibrium position, the gravity force acting on it will tend to pull it back to its equilibrium position. On the contrary, the inverse pendulum will move further away from the equilibrium position once it deviates from there. The positive pendulum and the inverse one are connected by a horizontal platform to form a FP. Thus the negative elastic coefficient of the inverse pendulum partially counterbalances the positive one of the positive pendulum. The total elastic coefficient of the system can then be greatly reduced and the FP could have a quite low natural frequency. Of course, the longer the period is, the worse the stability of the system is and the more difficult to adjust it. The practical FP used in our experiment consists of an 18 cm long positive pendulum and a 22 cm long inverse one, which are connected by a horizontal platform whose dimensions are 30 cm = 20 cm = 1.5 cm. These movable components and the fixed frame and base are all made of aluminum. They are linked to each other with flat beryllium bronze foils, which are 2.5 mm wide and 50 mm thick. The separation between the frame and the platform is measured with a precise infrared displacement transducer. The infrared emitter and the photodiode ŽED. are fixed on the frame and the light shield is fixed on the platform. So the separation between the frame and the platform changes with the movement of the crust, which results in the changes of the output
Fig. 1. The schematic diagram of the FP.
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Fig. 2. The layout of the effect of the crustal vibrations on the FP and the coordinate system.
voltage of the photodiode. The output signal is amplified and transmitted both to an X–Y chart recorder in analog record and to a computer in digital record.
3. Response to the crustal vibration The effect of the crustal vibrations on the FP and the coordinate system can be graphically shown in Fig. 2: AC, BD and CD represent respectively the positive pendulum, the inverse pendulum and the horizontal platform, whose lengths and masses are also shown in the figure. AO’ and BO’ represent the frame and the base of the FP, respectively. X g , Yg and u indicate the horizontal and vertical components of the crustal vibrations and the angle of the crustal tilt vibrations, respectively. According to the coordinate system shown in Fig. 2, we can obtain the generalized Lagrangian equation of the FP system in linear approximation if we consider X g , Yg and u as the first order quantities as follows:
w¨ q 2 bw q v 02 w q Au¨q Bu q CX¨g s 0, .
Ž 1.
where As
m1 Ž l 1 q 3l 2 . q 2 m 2 l 2 q 6 m 3 l 2
v 02 s
2 Ž m1 q m 2 q 3m 3 . l 1
,
Bsy
3 Ž m1 q m 2 q 2 m 3 . g 2 Ž m1 q m 2 q 3m 3 . l 1
3 m1rl1 y m 2rl 2 q m 3 Ž 1rl1 y 1rl 2 . g q 12 k Ž 1rl12 q 1rl 22 . 2 Ž m1 q m 2 q 3m 3 . l 1
,
Cs
3 Ž m1 q m 2 q 2 m 3 . 2 Ž m1 q m 2 q 3m 3 . l 1
,
.
v 02 is the angular frequency of the FP, k is the elastic coefficient of the foils, b is the damping coefficient of the system, and w is the angle that the FP deviates from its static equilibrium position. The equivalent simple pendulum length of the FP is given by: l e s grv 02 s
2 Ž m1 q m 2 q 3m 3 . l 1 3 m1rl1 y m 2rl 2 q m 3 Ž 1rl1 y 1rl 2 . q 12 k Ž 1rl12 q 1rl 22 . rg
.
Ž 2.
S. Fan et al.r Physics Letters A 256 (1999) 132–140
135
Fig. 3. Transfer function of the FP to the crustal tilt vibration.
From Eq. Ž1., we can obtain the transfer functions, Tt and Th , of the system to the crustal tilt vibrations and the horizontal vibrations as follows: Tt s
Th s
w 10
Av 2 yB
,
s
u0 l 1 w 20 Xg 0
(Ž v y v . q 4b v 2 2 0
2
2
l 1C v 2
s
(Ž v y v . q 4b v 2
2 2 0
Ž 3.
2
2
,
Ž 4.
2
where w 10 and w 20 represent the response amplitudes of the FP to the crustal tilt vibration and the horizontal vibration, respectively. Fig. 3 and Fig. 4 show the transfer functions Tt and Th , respectively. The typical parameters have been chosen in the figures as: l 1 s 0.18 m, m s 97.2 g, l s 0.22 m, m s 148.0 g, m s 2430.0 g, T s 12 s, and b s 0.1 radrs. It is clear in Fig. 3 that the FP is very sensitive to the low-frequency components Žbelow the natural frequency v 0 . of the crustal tilt vibrations and the amplification factor is about 200 while it is isolative to the high-frequency components Žabove the natural frequency v 0 .. As the frequencies of the tilt tides are much
Fig. 4. Transfer function of the FP to the crustal horizontal vibration.
S. Fan et al.r Physics Letters A 256 (1999) 132–140
136
lower than the natural frequency of the FP, the FP could be used as a tiltmeter to detect the tilt tides. Besides, as shown in Fig.4, the effect of the horizontal vibrations, in contrast with the tilt vibrations, can be neglected even in the resonant frequency range of the FP.
4. Preliminary experimental results The sensitivity of the FP tiltmeter was calibrated. And the tilt tides were observed with two FP tiltmeters side-by-side. The second FP, which was constructed recently, is the same as the first one in dimensions. The preliminary experimental results show that the two observed tilt curves are similar and each of them is similar to the theoretical one which was calculated with the program ETGTAB Žcreated by H.-G. Wenzel, Geodaetisches Institut, Universitaet Karlsruhe, Germany. version 3.0 in the point of observation during the experiment. 4.1. Calibration The displacement sensitivity of the infrared transducer system used in the FP was calibrated as follows: the light shield was clipped on the movable part of an Abbe comparator and the frame with the ED was attached to the fixed part of it. Adjusting the horizontal screw of the Abbe comparator and then changing the relative position between the ED and the light shield, we can get the calibration curve as shown in Fig. 5. About every 5 minutes, the horizontal screw was adjusted, and each step represented a horizontal displacement D x s 20 mm, which corresponded to a voltage change D V f 400 mV of the output signal from the transducer system. Therefore the sensitivity of the displacement transducer used in our experiment was:
Dx
f 5.0 = 10y2 mmrmV, DV and the sensitivity of the FP tiltmeter was given as follows:
lx s
lu s
Dx l e DV
f 1.4 = 10y9 radrmV.
Ž 5.
Ž 6.
From Fig. 5, we can see that the noise on each step was in the order of several tens of mV. This noise was just due to the calibration process because the person had to stand very nearly to operate the Abbe comparator for calibration. While the FP was installed on a foundation in our cavity laboratory, and the people was kept away from the laboratory during the experiment. Therefore the noise of the FP tiltmeter would be less than that
Fig. 5. Calibration curve of the infrared transducer system.
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Fig. 6. Output voltage curves of the infrared transducers of the FP tiltmeters in our cavity laboratory from 04:13 Nov.26, 1998 GMT to 04:08 Dec.09, 1998 GMT. Ža. Output voltage curve of the FP tiltmeter No. 1. Žb. Output voltage curve of the FP tiltmeter No. 2.
on the calibration curve. The resolving power of the system at present is mainly limited by the Analog-DigitalConvertor used in our recording system. Because the conversion voltage range of the Analog-Digital-Convertor
Fig. 7. Observed tilt curves of the two FP tiltmeters which were obtained by filtering the original output voltages through a high-pass filter with the cut-off frequency 5.6=10y6 Hz and then converting them into the tilt angles correspondingly according to the calibration result. Ža. Observed tilt curve of the FP tiltmeter No. 1. Žb. Observed tilt curve of the FP tiltmeter No. 2.
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Fig. 8. Theoretical tilt curve calculated with ETGTAB in the point of observation during the experiment from 04:13 Nov.26, 1998 GMT to 04:08 Dec.09, 1998 GMT.
of 12 bits was from -5 V to q5 V, the conversion error is about 2.5 mV. Therefore the resolving power of the FP tiltmeter is approximately given as follows:
Dumin s lu Verror f 3.5 = 10y9 rad.
Ž 7.
4.2. Measurement of tilt tides Two FP tiltmeters were installed on a foundation in our cavity laboratory. The foundation was built in tight connection with the crust and was isolated from the surroundings. So the outer interference was eliminated to the greatest possible extent. By the computer recording system, we obtained the output signals of the two FP tiltmeters as shown in Fig. 6. Through a highpass filter with the cut-off frequency 5.6 = 10y6 Hz, the extremely slow drifts in the raw data were removed and the observed tilt curves were obtained as shown in Fig. 7. Comparing Fig. 7 Ža. and Žb., we can see that the two observed tilt curves are similar. The program ETGTAB was used to calculate the theoretical tilt in the point of observation Ž30.54 o N, 114.35 o E, the azimuth is 260 o clockwise from north. during the experiment as shown in Fig. 8. Comparing Fig. 7 and Fig. 8, we can see that each observed tilt curve is similar to the theoretical one. We have access to local gravity tide curve as shown in Fig. 9, and we can see that it is correlated with the FP signal to some degree because both of them are caused by tidal force. The very low frequency variation in the FP signals in Fig. 6 might be mainly caused by the secular creep of the linkage foils and the slow changes of the temperature, and it is also easily imagined that it is just drift in the readout. In order to confirm that, some fixture was used to lock the position of the pendulum and the readout signal was observed as shown in Fig. 10. Comparing Fig. 10 and Fig. 6, we can see that during 200 hours the drift in the readout was only about 25 mV, which was much less than that in the FP output signals. Therefore
Fig. 9. Gravity tides curve observed from 04:13 Nov.26, 1998 GMT to 04:08 Dec.09, 1998 GMT with a superconducting gravimeter GWR TT-70 at local observation, which is about 5 km away from our cavity laboratory.
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Fig. 10. Output voltage curve of the readout system while the position of FP was locked with some fixture from 17:48 Mar.07, 1999 GMT to 01:48 Mar.16, 1999 GMT.
this drift might be due to the creep in the linkage foils and the variation of the temperature, which are being under investigation.
5. Discussion The FP used as a tiltmeter has been studied in theory and demonstrated in prototype form. The sensitivity of the FP to the tilt tides is stable due to its symmetric structure and flat linkage foils. Preliminary experimental result shows that the resolving power of the FP to the tilt tides has reached in the order of about 3.5 = 10y9 rad. However, some problems should be solved, which include the drift of the readout due to the creep of foils, the calibration of the sensitivity with a standard dilatometer or a PZT, and the influences due to the air current and the temperature variation. Above works are still under investigations in our laboratory.
Acknowledgements The authors gratefully thank G.Q. Hu, W.X. Cai and D.G. Blair for their useful discussions. This work was arose from the collaboration program between HUST and UWA Žthe University of Western Australia., and supported by the National Natural Science Foundation of China under Grants 49674246 and 19425008.
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