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Estimation of thermal noise by a direct measurement of the mechanical conductance
28 February 2000
Physics Letters A 266 Ž2000. 228–233 www.elsevier.nlrlocaterphysleta
Estimation of thermal noise by a direct measurement of the mechanical conductance Naoko Ohishi ) , Shigemi Otsuka, Keita Kawabe 1, Kimio Tsubono Department of Physics, UniÕersity of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan Received 9 November 1999; accepted 7 December 1999 Communicated by P.R. Holland
Abstract We present a new method for estimating the thermal noise at off-resonant frequencies in a mechanical system. In this method, the mechanical conductance is directly measured at anti-resonant frequencies, and then the fluctuation-dissipation theorem is applied to the measured conductance in order to evaluate the thermal noise. To confirm the validity of this method, we demonstrated a direct measurement of the mechanical conductance of a 2-mode oscillator at an anti-resonant frequency. The measured conductance was in good agreement with the value calculated by the normal-mode expansion. q 2000 Elsevier Science B.V. All rights reserved. PACS: 05.40.Jc; 04.80.Nn Keywords: Thermal noise; Fluctuation–dissipation theorem; Anti-resonance; Gravitational waves
1. Introduction Mechanical thermal noise at off-resonant frequencies is a serious issue in precise experiments, such as laser interferometric gravitational wave detections w1–3x. In the TAMA300 gravitational wave detector w4x, for instance, thermal noise of the mirrors, whose resonant frequencies are higher than 20 kHz, is thought to be one of the dominant noise sources at the observation band from 150 to 450 Hz.
)
Corresponding author. E-mail address: [email protected] ŽN. Ohishi.. 1 Present address: Max-Planck-Institut fur ¨ Quantenoptik, Hans-Kopfermann Strasse 1, D-85748 Garching bei Munchen, ¨ Germany.
Experimentally, however, it is difficult to study thermal noise at off-resonant frequencies because the amplitude of the thermal noise is generally extremely small w5,6x. Therefore calculation methods based on the normal-mode expansion have been investigated in order to estimate the thermal noise w1,3x. In these methods, each resonance of the mechanical system is regarded as being a harmonic oscillator to which the thermal energy, k B Tr2, is divided. Though normalmode expansion is commonly used to estimate the thermal noise in gravitational wave detectors, there are some problems: Ž1. the normal-mode expansion is not correct unless the loss distribution is homogeneous, and Ž2. an assumption on the frequency dependence concerning the loss angle w7x, f Ž v ., is needed. In the TAMA300 detector, the quality factors of the mirrors are degraded by the magnets
0375-9601r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 Ž 9 9 . 0 0 8 6 7 - 1
N. Ohishi et al.r Physics Letters A 266 (2000) 228–233
attached to its surface, and the resultant quality factors vary with the modes w8x. This means that the loss distribution is not homogeneous. Moreover, the frequency dependence of the loss caused by the magnets is unpredictable. To make a correct estimation even if the loss distributes inhomogeneously in a mechanical system, new estimation methods which are based on the fluctuation–dissipation theorem, and in which the normal-mode expansion is not used, have been studied w9–11x. The fluctuation–dissipation theorem gives the power spectral density of the thermal motion, x 2 Ž v ., in terms of the mechanical conductance, s Ž v ., as x2 Ž v. s
4 k B Ts Ž v .
v2
,
Ž 1.
where k B is the Boltzmann constant and T is the temperature. The mechanical conductance, s Ž v ., is the real part of the admittance, Y Ž v .:
s Ž v . s Re Y Ž v . , Y Ž v . s Õ Ž v . rf Ž v . ,
Ž 2.
where Õ Ž v . and f Ž v . are the Fourier transform of the velocity and the applied force at a point of the generalized coordinates w12x. Though the thermal noise is calculated from the mechanical conductance, it is not easy to measure the conductance of a real mechanical system at off-resonant frequencies. When we measure the mechanical admittance, by applying force, f, to the system, and measuring its velocity, Õ, the imaginary part of the admittance, which is more than Q-times as large as the real part, is mixed with the real part by a small phase delay in the measuring system. Thus, it is difficult to separate the conductance by a direct measurement of the admittance. At resonances, however, the real part of the admittance becomes larger than the imaginary part, and can be precisely measured through measurements of the quality factors. We noticed that there are still other frequencies at which the imaginary part becomes smaller compared to the real part. At anti-resonances, the imaginary part of the admittance vanishes, while the real part, the conductance, remains. We expect that we can measure the conductance directly at these frequencies, and can evaluate the thermal noise by using Eq. Ž1..
229
In order to verify that the mechanical conductance can be measured at anti-resonant frequencies, we made a simple 2-mode oscillator comprising two blade springs and two masses, and measured the conductance at an anti-resonant frequency. The measured conductance was in good agreement with the value calculated by the normal-mode expansion while taking account of the measured quality factors.
2. Principle We explain how an anti-resonance occurs by using the mechanical transfer function, H Ž v ., instead of the admittance, Y Ž v .. Two quantities H Ž v . and Y Ž v . are related by the equation HŽ v. s
xŽ v.
s
YŽ v.
fŽ v.
iv
.
Ž 3.
Using H Ž v ., Eq. Ž1. is rewritten as 4 k BT x2 Ž v . sy Im H Ž v . . Ž 4. v Since the discussion here is qualitative, we use the method of the normal-mode expansion to describe the mechanical response. This description is quantitatively correct if the loss distributes homogeneously in a mechanical system. According to the normalmode expansion, the mechanical system is described as a superposition of harmonic oscillators. The mechanical transfer function, H Ž v ., is written as 1 HŽ v. s Ý , Ž 5. 2 2 2 i m i Ž v i y v . q i fi Ž v . v i where m i , v i , f i Ž v . are the reduced mass, the resonant angular frequency, and the loss angle of the i-th mode. The transfer function of the i-th mode, Hi Ž v ., below the resonant frequency Ž v < v i . is approximately Hi Ž v . ,
1 m i v i2
yi
fi Ž v . m i v i2
.
Ž 6.
At resonance Hi Ž v . s yi
Qi m i v i2
,
Ž 7.
where Q i is the inverse of the loss angle at the resonant frequency, Q i s fy1 Ž v i .. When the fre-
N. Ohishi et al.r Physics Letters A 266 (2000) 228–233
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quency becomes higher than the resonant frequency Ž v 4 v i ., Hi Ž v . , y
1 mi v 2
yi
f i Ž v . v i2 mi v 4
.
Ž 8.
As can be seen from Eq. Ž6. and Eq. Ž8., the real part of the mechanical transfer function is more than Q-times as large as the imaginary part at off-resonant frequencies, and the sign of the real part changes across the resonance. As a simple example, we consider a 2-mode oscillator of which the resonant angular frequencies are v 1 and v 2 . The mechanical transfer function is expressed as a summation of the two modes. Then, the real part of the transfer function between the two resonances Ž v 1 - v - v 2 . is approximated as Re H Ž v . , y
1 m1 v
1 2
q
m 2 v 22
.
Ž 9.
From Eq. Ž9., we notice that there is a frequency, vanti , where the real part of the transfer function vanishes,
vanti ,
(
m2 m1
v2 .
Ž 10 .
At this frequency, the transfer function turns out to be pure imaginary, H Ž vanti . , yi
f 1 Ž vanti . v 12 4 m1 vanti
q
f 2 Ž vanti . m 2 v 22
.
Ž 11 .
Thus, at the anti-resonant frequency, the imaginary part of the transfer function can be measured without contamination by the real part. As an example, a mechanical transfer function of a model 2-mode oscillator is shown in Fig. 1. Below the first resonance, the absolute value of the mechanical transfer function is almost constant. It increases Q1-times at the first resonance, and then decreases in inverse proportion to the square of the frequency. As the frequency increases, the effect of the second mode can not be negligible, and when the absolute value of the transfer function of the first mode, < y 1rm1 v 2 <, becomes equal to that of the second mode, <1rm 2 v 22 <, the real part of the mechanical transfer function of the whole system vanishes, while
Fig. 1. Example of the transfer function of a 2-mode oscillator; m1 is 10 g, m 2 is 1 g, resonant frequencies v 1 r2p s 10 Hz, v 2 r2p s 100 Hz, and f 1Ž v . s f 2 Ž v . s10y3 . The solid line shows the absolute value of the transfer function, and the dashed line shows the absolute value of the imaginary part of the transfer function. An anti-resonance appears at vanti r2p s 31.6 Hz.
the imaginary part remains. Thus, we can measure the imaginary part of the transfer function directly at anti-resonant frequencies and can estimate the thermal noise using Eq. Ž4..
3. Measurement of the mechanical conductance To confirm the validity of our estimation method, we demonstrated a measurement of the transfer function of a mechanical oscillator at an anti-resonant frequency, and compared the measured value with one calculated by the normal-mode expansion. A simple mechanical oscillator comprising two blade springs and masses ŽFig. 2. was used for the measurement. The springs and masses were made of brass. The upper blade spring was 6 mm in width, 200 mm in length, and 1.5 mm in thickness. The lower was 3 mm in width, 12 mm in length, and 0.3 mm in thickness. The upper mass was 20 g, and the lower mass was 4 g. The mechanical oscillator was fixed at the upper end of the upper blade spring. We measured the transfer function at the lower mass by applying force to the mass with a coil-magnet actuator and measuring its displacement with a Michelson interferometer. For this purpose, a small Nd magnet, which was 2 mm in diameter and 5 mm
N. Ohishi et al.r Physics Letters A 266 (2000) 228–233
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the quality factors, Q i . The values of v i and Q i are already known by measurements. For finding the reduced masses, m1 and m 2 , we then fit the absolute value of the measured transfer function with the following equation: < HŽ v.
1 m1
Ž
v 12 y v 2
. q i f 1Ž v . v 12 1
q m2
Ž
v 22 y v 2
. q i f 2 Ž v . v 22
.
Ž 12 .
Fig. 2. Experimental setup for measuring the mechanical conductance of the 2-mode oscillator. The oscillator comprises of two blade springs and two masses. To measure the transfer function at the lower mass, we apply force to it with a coil-magnet actuator, and measure its displacement with a Michelson laser interferometer. BS, beam splitter; PD, photo detector.
in length, was glued to the center of the lower mass and a small mirror was glued to the other side. The reference mirror of the Michelson interferometer was suspended as a double pendulum. After we locked the interferometer, we put small signals into the feedback loop and measured the open-loop transfer function using a spectrum analyzer. After removing the effects of the servo filters, which had been measured separately, we obtained the mechanical transfer function at the lower mass. Fig. 3 shows the measured mechanical transfer function. The open circles are the absolute value of the measured transfer function. There are two resonances at 7.5 Hz and 52.5 Hz; and in between, an anti-resonance is observed at 26.9 Hz. The measured quality factors of the first and second modes were 8.7 = 10 2 and 8.3 = 10 2 . These values support the structure damping model w7x of the loss. A higher resonance is observed at 96 Hz, and another anti-resonance is also observed at 95 Hz. In order to compare the measured transfer function with the calculation based on the normal-mode expansion, we need to know the values of the resonant frequencies, v i ; the reduced masses, m i ; and
The reduced masses of the first mode, m1 , was determined to be 26 g, and that of the second mode, m 2 , was 8 g. As shown in Fig. 3, the fit well reproduces the amplitude of the measured transfer function. Since all of the parameters are known, we calculate the imaginary part of the mechanical transfer function, and compare it with the measured value. Fig. 4 shows the real part and the imaginary part of the measured and calculated transfer functions around
Fig. 3. The transfer function measured at the lower mass of the 2-mode oscillator. The open circles are the absolute value of the measured mechanical transfer function. The solid line is the fit by Eq. Ž12.. The resonant frequencies are 7.5 Hz and 52.5 Hz, and an anti-resonance is seen at 26.9 Hz. A higher mode is observed at 96 Hz, and an anti-resonance is also observed at 95 Hz. The small figure is the expansion around the anti-resonant frequency. The minimum value of the absolute value of the transfer function is 2.4=10y6 mrN.
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N. Ohishi et al.r Physics Letters A 266 (2000) 228–233
cisely measured at an anti-resonance. In this case, the measured quality factors supported the structure damping model, and the calculation by the normalmode expansion agreed well with the measured value.
4. Discussion
Fig. 4. Measured transfer function around the anti-resonant frequency. The open circles are the absolute value of the real part of the measured transfer function, and the solid line is the fit by Eq. Ž12.. The closed circles are the absolute value of the imaginary part of the measured transfer function. At the anti-resonant frequency, the real part of the transfer function decreases and only the imaginary part remains. The dashed line is the imaginary part calculated by the normal-mode expansion taking account of the measured quality factors.
the anti-resonant frequency. The open circles are the absolute value of the measured real part of the transfer function and the solid line is that calculated by the normal-mode expansion. The closed circles are the absolute value of the measured imaginary part and the dashed line is that calculated by the normal-mode expansion. The real part of the mechanical transfer function decreases as the frequency becomes close to the anti-resonance, and only the imaginary part remains at the anti-resonant frequency. The imaginary part is measured precisely around the anti-resonance, in the frequency range of 0.2 Hz. The measured imaginary part agrees well with the calculated value by the normal-mode expansion. We think this agreement occurred because the quality factors measured at resonances were almost constant, and the loss distributed homogeneously in the system. The additional loss caused by the magnet and the mirror was negligible in this case. From this measurement, we confirmed that the imaginary part of the mechanical transfer function is well separated from the real part and can be pre-
We have proposed a method for estimating the thermal noise at off-resonant frequencies. This method is valid even if the loss distribution is inhomogeneous and needs no assumption concerning the frequency dependence of the loss. In this method, contributions from all of the modes are taken into account, while a limited number of quality factors are measured and included in the normal-mode expansion method. We, then, examine the required sensitivity of the measuring system for obtaining the conductance at anti-resonant frequencies. When we apply force to a mechanical system, the amplitude of the induced displacement is expressed as a product of the absolute value of the transfer function of the mechanical system and the applied force, f. If we measure the displacement at an anti-resonant frequency with a bandwidth of Dn , the required sensitivity, x sensŽ vanti ., is given as follows: x sens Ž vanti . s
Im H Ž vanti . = f
'Dn
.
Ž 13 .
In the demonstration, the absolute value of the measured transfer function at the anti-resonant frequency was 2.4 = 10y6 mrN, and the magnitude of the applied force was 3 = 10y3 N. With the bandwidth of 10 mHz, the required sensitivity of the measuring system is 7 = 10y8 mr 'Hz . On the other hand, the estimated amplitude of the thermal motion, which is calculated by Eq. Ž4., is 1.5 = 10y1 4 mr 'Hz . Though it is not easy to measure this level of thermal noise directly, it was rather easy to measure the above value of the mechanical conductance. Next, we study whether we can make an anti-resonance at issued frequency. We consider here only the mirrors used in the laser interferometric gravitational wave detectors. The mirrors are suspended by wires, and the system has a pendulum mode, violin modes, and mirror internal modes; the combination
N. Ohishi et al.r Physics Letters A 266 (2000) 228–233
of these modes produces anti-resonances. Though the fundamental violin mode is several hundreds Hz w13x and it is close to the observation band, the reduced mass of the violin modes are too heavy to separate the anti-resonant frequency from the resonant frequency. Therefore, we examine the frequency of an anti-resonance which is made by pendulum mode and mirror internal modes. Experimentally, we observed an anti-resonance at 2 kHz with a mirror which has the same size as the mirrors used in the TAMA300 laser interferometer. Although this frequency is higher than the observation band, the estimation of the thermal noise at this frequency will be useful for studying the effect of inhomogeneous loss caused by magnets attached to the mirrors and the wires by which the mirrors are suspended.
5. Conclusion We have reported on a new estimation method of the thermal noise at off-resonant frequencies. In this method, we apply the fluctuation–dissipation theorem to the mechanical conductance measured at anti-resonant frequencies. To confirm the validity of our method, we demonstrated a measurement of the mechanical conductance of a 2-mode oscillator at an anti-resonant frequency. The mechanical conductance was precisely measured at the anti-resonant frequency, and it was in good agreement with the value calculated by the normal-mode expansion in this case. Though it is not easy to measure the
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estimated level of thermal noise directly, it was rather easy to measure the mechanical conductance. This method will be useful when we study the off-resonant thermal noise of mechanical systems with inhomogeneous loss. Acknowledgements A part of this research is supported by a Grant-inAid for Creative Basic Research of the Ministry of Education Ž10NP0801.. References w1x w2x w3x w4x
w5x w6x w7x w8x
w9x w10x w11x w12x w13x
A. Gillespie, F. Raab, Phys. Rev. D 52 Ž1995. 577. A. Gillespie, Ph.D. thesis, 1995. F. Bondu, J.-Y. Vinet, Phys. Lett. A 198 Ž1995. 74. K. Tsubono, the TAMA Collaboration, Proc. TAMA Workshop on Gravitational Wave Detection, Saitama, November 1996, Universal Academy Press, 1997, p. 183. G.I. Gonzalez, P.R. Saulson, Phys. Lett. A 201 Ž1995. 12. ´ M. Kajima, N. Kusumi, S. Moriwaki, N. Mio, Phys. Lett. A 264 Ž1999. 251. P.R. Saulson, Phys. Rev. D 42 Ž1990. 2437. N. Ohishi, S. Otsuka, K. Kawabe, K. Tsubono, in: Proc. TAMA Workshop on Gravitational Wave Detection, Saitama, November 1996, Universal Academy Press, 1997, p. 369. N. Nakagawa, B.A. Auld, E. Gustafson, M.M. Fejer, Rev. Sci. Instrum. 68 Ž1997. 3553. Y. Levin, Phys. Rev. D 57 Ž1998. 659. F. Bondu, P. Hello, J.-Y. Vinet, Phys. Lett. A 246 Ž1998. 227. H.B. Callen, R.F. Greene, Phys. Rev. 86 Ž1952. 702. A. Gillespie, F. Raab, Phys. Lett. A 178 Ž1993. 357.
Physics Letters A 266 Ž2000. 228–233 www.elsevier.nlrlocaterphysleta
Estimation of thermal noise by a direct measurement of the mechanical conductance Naoko Ohishi ) , Shigemi Otsuka, Keita Kawabe 1, Kimio Tsubono Department of Physics, UniÕersity of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan Received 9 November 1999; accepted 7 December 1999 Communicated by P.R. Holland
Abstract We present a new method for estimating the thermal noise at off-resonant frequencies in a mechanical system. In this method, the mechanical conductance is directly measured at anti-resonant frequencies, and then the fluctuation-dissipation theorem is applied to the measured conductance in order to evaluate the thermal noise. To confirm the validity of this method, we demonstrated a direct measurement of the mechanical conductance of a 2-mode oscillator at an anti-resonant frequency. The measured conductance was in good agreement with the value calculated by the normal-mode expansion. q 2000 Elsevier Science B.V. All rights reserved. PACS: 05.40.Jc; 04.80.Nn Keywords: Thermal noise; Fluctuation–dissipation theorem; Anti-resonance; Gravitational waves
1. Introduction Mechanical thermal noise at off-resonant frequencies is a serious issue in precise experiments, such as laser interferometric gravitational wave detections w1–3x. In the TAMA300 gravitational wave detector w4x, for instance, thermal noise of the mirrors, whose resonant frequencies are higher than 20 kHz, is thought to be one of the dominant noise sources at the observation band from 150 to 450 Hz.
)
Corresponding author. E-mail address: [email protected] ŽN. Ohishi.. 1 Present address: Max-Planck-Institut fur ¨ Quantenoptik, Hans-Kopfermann Strasse 1, D-85748 Garching bei Munchen, ¨ Germany.
Experimentally, however, it is difficult to study thermal noise at off-resonant frequencies because the amplitude of the thermal noise is generally extremely small w5,6x. Therefore calculation methods based on the normal-mode expansion have been investigated in order to estimate the thermal noise w1,3x. In these methods, each resonance of the mechanical system is regarded as being a harmonic oscillator to which the thermal energy, k B Tr2, is divided. Though normalmode expansion is commonly used to estimate the thermal noise in gravitational wave detectors, there are some problems: Ž1. the normal-mode expansion is not correct unless the loss distribution is homogeneous, and Ž2. an assumption on the frequency dependence concerning the loss angle w7x, f Ž v ., is needed. In the TAMA300 detector, the quality factors of the mirrors are degraded by the magnets
0375-9601r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 Ž 9 9 . 0 0 8 6 7 - 1
N. Ohishi et al.r Physics Letters A 266 (2000) 228–233
attached to its surface, and the resultant quality factors vary with the modes w8x. This means that the loss distribution is not homogeneous. Moreover, the frequency dependence of the loss caused by the magnets is unpredictable. To make a correct estimation even if the loss distributes inhomogeneously in a mechanical system, new estimation methods which are based on the fluctuation–dissipation theorem, and in which the normal-mode expansion is not used, have been studied w9–11x. The fluctuation–dissipation theorem gives the power spectral density of the thermal motion, x 2 Ž v ., in terms of the mechanical conductance, s Ž v ., as x2 Ž v. s
4 k B Ts Ž v .
v2
,
Ž 1.
where k B is the Boltzmann constant and T is the temperature. The mechanical conductance, s Ž v ., is the real part of the admittance, Y Ž v .:
s Ž v . s Re Y Ž v . , Y Ž v . s Õ Ž v . rf Ž v . ,
Ž 2.
where Õ Ž v . and f Ž v . are the Fourier transform of the velocity and the applied force at a point of the generalized coordinates w12x. Though the thermal noise is calculated from the mechanical conductance, it is not easy to measure the conductance of a real mechanical system at off-resonant frequencies. When we measure the mechanical admittance, by applying force, f, to the system, and measuring its velocity, Õ, the imaginary part of the admittance, which is more than Q-times as large as the real part, is mixed with the real part by a small phase delay in the measuring system. Thus, it is difficult to separate the conductance by a direct measurement of the admittance. At resonances, however, the real part of the admittance becomes larger than the imaginary part, and can be precisely measured through measurements of the quality factors. We noticed that there are still other frequencies at which the imaginary part becomes smaller compared to the real part. At anti-resonances, the imaginary part of the admittance vanishes, while the real part, the conductance, remains. We expect that we can measure the conductance directly at these frequencies, and can evaluate the thermal noise by using Eq. Ž1..
229
In order to verify that the mechanical conductance can be measured at anti-resonant frequencies, we made a simple 2-mode oscillator comprising two blade springs and two masses, and measured the conductance at an anti-resonant frequency. The measured conductance was in good agreement with the value calculated by the normal-mode expansion while taking account of the measured quality factors.
2. Principle We explain how an anti-resonance occurs by using the mechanical transfer function, H Ž v ., instead of the admittance, Y Ž v .. Two quantities H Ž v . and Y Ž v . are related by the equation HŽ v. s
xŽ v.
s
YŽ v.
fŽ v.
iv
.
Ž 3.
Using H Ž v ., Eq. Ž1. is rewritten as 4 k BT x2 Ž v . sy Im H Ž v . . Ž 4. v Since the discussion here is qualitative, we use the method of the normal-mode expansion to describe the mechanical response. This description is quantitatively correct if the loss distributes homogeneously in a mechanical system. According to the normalmode expansion, the mechanical system is described as a superposition of harmonic oscillators. The mechanical transfer function, H Ž v ., is written as 1 HŽ v. s Ý , Ž 5. 2 2 2 i m i Ž v i y v . q i fi Ž v . v i where m i , v i , f i Ž v . are the reduced mass, the resonant angular frequency, and the loss angle of the i-th mode. The transfer function of the i-th mode, Hi Ž v ., below the resonant frequency Ž v < v i . is approximately Hi Ž v . ,
1 m i v i2
yi
fi Ž v . m i v i2
.
Ž 6.
At resonance Hi Ž v . s yi
Qi m i v i2
,
Ž 7.
where Q i is the inverse of the loss angle at the resonant frequency, Q i s fy1 Ž v i .. When the fre-
N. Ohishi et al.r Physics Letters A 266 (2000) 228–233
230
quency becomes higher than the resonant frequency Ž v 4 v i ., Hi Ž v . , y
1 mi v 2
yi
f i Ž v . v i2 mi v 4
.
Ž 8.
As can be seen from Eq. Ž6. and Eq. Ž8., the real part of the mechanical transfer function is more than Q-times as large as the imaginary part at off-resonant frequencies, and the sign of the real part changes across the resonance. As a simple example, we consider a 2-mode oscillator of which the resonant angular frequencies are v 1 and v 2 . The mechanical transfer function is expressed as a summation of the two modes. Then, the real part of the transfer function between the two resonances Ž v 1 - v - v 2 . is approximated as Re H Ž v . , y
1 m1 v
1 2
q
m 2 v 22
.
Ž 9.
From Eq. Ž9., we notice that there is a frequency, vanti , where the real part of the transfer function vanishes,
vanti ,
(
m2 m1
v2 .
Ž 10 .
At this frequency, the transfer function turns out to be pure imaginary, H Ž vanti . , yi
f 1 Ž vanti . v 12 4 m1 vanti
q
f 2 Ž vanti . m 2 v 22
.
Ž 11 .
Thus, at the anti-resonant frequency, the imaginary part of the transfer function can be measured without contamination by the real part. As an example, a mechanical transfer function of a model 2-mode oscillator is shown in Fig. 1. Below the first resonance, the absolute value of the mechanical transfer function is almost constant. It increases Q1-times at the first resonance, and then decreases in inverse proportion to the square of the frequency. As the frequency increases, the effect of the second mode can not be negligible, and when the absolute value of the transfer function of the first mode, < y 1rm1 v 2 <, becomes equal to that of the second mode, <1rm 2 v 22 <, the real part of the mechanical transfer function of the whole system vanishes, while
Fig. 1. Example of the transfer function of a 2-mode oscillator; m1 is 10 g, m 2 is 1 g, resonant frequencies v 1 r2p s 10 Hz, v 2 r2p s 100 Hz, and f 1Ž v . s f 2 Ž v . s10y3 . The solid line shows the absolute value of the transfer function, and the dashed line shows the absolute value of the imaginary part of the transfer function. An anti-resonance appears at vanti r2p s 31.6 Hz.
the imaginary part remains. Thus, we can measure the imaginary part of the transfer function directly at anti-resonant frequencies and can estimate the thermal noise using Eq. Ž4..
3. Measurement of the mechanical conductance To confirm the validity of our estimation method, we demonstrated a measurement of the transfer function of a mechanical oscillator at an anti-resonant frequency, and compared the measured value with one calculated by the normal-mode expansion. A simple mechanical oscillator comprising two blade springs and masses ŽFig. 2. was used for the measurement. The springs and masses were made of brass. The upper blade spring was 6 mm in width, 200 mm in length, and 1.5 mm in thickness. The lower was 3 mm in width, 12 mm in length, and 0.3 mm in thickness. The upper mass was 20 g, and the lower mass was 4 g. The mechanical oscillator was fixed at the upper end of the upper blade spring. We measured the transfer function at the lower mass by applying force to the mass with a coil-magnet actuator and measuring its displacement with a Michelson interferometer. For this purpose, a small Nd magnet, which was 2 mm in diameter and 5 mm
N. Ohishi et al.r Physics Letters A 266 (2000) 228–233
231
the quality factors, Q i . The values of v i and Q i are already known by measurements. For finding the reduced masses, m1 and m 2 , we then fit the absolute value of the measured transfer function with the following equation: < HŽ v.
1 m1
Ž
v 12 y v 2
. q i f 1Ž v . v 12 1
q m2
Ž
v 22 y v 2
. q i f 2 Ž v . v 22
.
Ž 12 .
Fig. 2. Experimental setup for measuring the mechanical conductance of the 2-mode oscillator. The oscillator comprises of two blade springs and two masses. To measure the transfer function at the lower mass, we apply force to it with a coil-magnet actuator, and measure its displacement with a Michelson laser interferometer. BS, beam splitter; PD, photo detector.
in length, was glued to the center of the lower mass and a small mirror was glued to the other side. The reference mirror of the Michelson interferometer was suspended as a double pendulum. After we locked the interferometer, we put small signals into the feedback loop and measured the open-loop transfer function using a spectrum analyzer. After removing the effects of the servo filters, which had been measured separately, we obtained the mechanical transfer function at the lower mass. Fig. 3 shows the measured mechanical transfer function. The open circles are the absolute value of the measured transfer function. There are two resonances at 7.5 Hz and 52.5 Hz; and in between, an anti-resonance is observed at 26.9 Hz. The measured quality factors of the first and second modes were 8.7 = 10 2 and 8.3 = 10 2 . These values support the structure damping model w7x of the loss. A higher resonance is observed at 96 Hz, and another anti-resonance is also observed at 95 Hz. In order to compare the measured transfer function with the calculation based on the normal-mode expansion, we need to know the values of the resonant frequencies, v i ; the reduced masses, m i ; and
The reduced masses of the first mode, m1 , was determined to be 26 g, and that of the second mode, m 2 , was 8 g. As shown in Fig. 3, the fit well reproduces the amplitude of the measured transfer function. Since all of the parameters are known, we calculate the imaginary part of the mechanical transfer function, and compare it with the measured value. Fig. 4 shows the real part and the imaginary part of the measured and calculated transfer functions around
Fig. 3. The transfer function measured at the lower mass of the 2-mode oscillator. The open circles are the absolute value of the measured mechanical transfer function. The solid line is the fit by Eq. Ž12.. The resonant frequencies are 7.5 Hz and 52.5 Hz, and an anti-resonance is seen at 26.9 Hz. A higher mode is observed at 96 Hz, and an anti-resonance is also observed at 95 Hz. The small figure is the expansion around the anti-resonant frequency. The minimum value of the absolute value of the transfer function is 2.4=10y6 mrN.
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cisely measured at an anti-resonance. In this case, the measured quality factors supported the structure damping model, and the calculation by the normalmode expansion agreed well with the measured value.
4. Discussion
Fig. 4. Measured transfer function around the anti-resonant frequency. The open circles are the absolute value of the real part of the measured transfer function, and the solid line is the fit by Eq. Ž12.. The closed circles are the absolute value of the imaginary part of the measured transfer function. At the anti-resonant frequency, the real part of the transfer function decreases and only the imaginary part remains. The dashed line is the imaginary part calculated by the normal-mode expansion taking account of the measured quality factors.
the anti-resonant frequency. The open circles are the absolute value of the measured real part of the transfer function and the solid line is that calculated by the normal-mode expansion. The closed circles are the absolute value of the measured imaginary part and the dashed line is that calculated by the normal-mode expansion. The real part of the mechanical transfer function decreases as the frequency becomes close to the anti-resonance, and only the imaginary part remains at the anti-resonant frequency. The imaginary part is measured precisely around the anti-resonance, in the frequency range of 0.2 Hz. The measured imaginary part agrees well with the calculated value by the normal-mode expansion. We think this agreement occurred because the quality factors measured at resonances were almost constant, and the loss distributed homogeneously in the system. The additional loss caused by the magnet and the mirror was negligible in this case. From this measurement, we confirmed that the imaginary part of the mechanical transfer function is well separated from the real part and can be pre-
We have proposed a method for estimating the thermal noise at off-resonant frequencies. This method is valid even if the loss distribution is inhomogeneous and needs no assumption concerning the frequency dependence of the loss. In this method, contributions from all of the modes are taken into account, while a limited number of quality factors are measured and included in the normal-mode expansion method. We, then, examine the required sensitivity of the measuring system for obtaining the conductance at anti-resonant frequencies. When we apply force to a mechanical system, the amplitude of the induced displacement is expressed as a product of the absolute value of the transfer function of the mechanical system and the applied force, f. If we measure the displacement at an anti-resonant frequency with a bandwidth of Dn , the required sensitivity, x sensŽ vanti ., is given as follows: x sens Ž vanti . s
Im H Ž vanti . = f
'Dn
.
Ž 13 .
In the demonstration, the absolute value of the measured transfer function at the anti-resonant frequency was 2.4 = 10y6 mrN, and the magnitude of the applied force was 3 = 10y3 N. With the bandwidth of 10 mHz, the required sensitivity of the measuring system is 7 = 10y8 mr 'Hz . On the other hand, the estimated amplitude of the thermal motion, which is calculated by Eq. Ž4., is 1.5 = 10y1 4 mr 'Hz . Though it is not easy to measure this level of thermal noise directly, it was rather easy to measure the above value of the mechanical conductance. Next, we study whether we can make an anti-resonance at issued frequency. We consider here only the mirrors used in the laser interferometric gravitational wave detectors. The mirrors are suspended by wires, and the system has a pendulum mode, violin modes, and mirror internal modes; the combination
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of these modes produces anti-resonances. Though the fundamental violin mode is several hundreds Hz w13x and it is close to the observation band, the reduced mass of the violin modes are too heavy to separate the anti-resonant frequency from the resonant frequency. Therefore, we examine the frequency of an anti-resonance which is made by pendulum mode and mirror internal modes. Experimentally, we observed an anti-resonance at 2 kHz with a mirror which has the same size as the mirrors used in the TAMA300 laser interferometer. Although this frequency is higher than the observation band, the estimation of the thermal noise at this frequency will be useful for studying the effect of inhomogeneous loss caused by magnets attached to the mirrors and the wires by which the mirrors are suspended.
5. Conclusion We have reported on a new estimation method of the thermal noise at off-resonant frequencies. In this method, we apply the fluctuation–dissipation theorem to the mechanical conductance measured at anti-resonant frequencies. To confirm the validity of our method, we demonstrated a measurement of the mechanical conductance of a 2-mode oscillator at an anti-resonant frequency. The mechanical conductance was precisely measured at the anti-resonant frequency, and it was in good agreement with the value calculated by the normal-mode expansion in this case. Though it is not easy to measure the
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estimated level of thermal noise directly, it was rather easy to measure the mechanical conductance. This method will be useful when we study the off-resonant thermal noise of mechanical systems with inhomogeneous loss. Acknowledgements A part of this research is supported by a Grant-inAid for Creative Basic Research of the Ministry of Education Ž10NP0801.. References w1x w2x w3x w4x
w5x w6x w7x w8x
w9x w10x w11x w12x w13x
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