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TORSION BALANCE IN THE EXPERIMENT ON MEASUREMENT OF THE NEWTONIAN GRAVITATIONAL CONSTANT V. K. Milyukov 1 , Chen Tao 2 , A. P. Mironov 3 , Sternberg Astronomical Institute of Moscow University 13, Universitetskii prospect, 119992, Moscow, Russia. E-mail: milyukov@sai.msu.ru Abstract
Key words: gravitational constant, torsion balance, torsional mode couplings. The nonlinear behavior of the torsion balance has been studied. The numerical simulation of the motion of the torsion balance with five degrees of freedom has been done. It was shown, that swing modes are excited by the acting environmental noise. The coupling of the swing modes to the torsional mode has been revealed. The ways to suppress the effect of the mode couplings have been considered.

Introduction The traditional instrument for carrying-out high precision gravitational experiments, first of all, the measurement of the Newtonian gravitational constant, is a torsion balance. The gravitational constant is determined on measurement of the period of eigen oscillations of the torsion balance in a gravitational field of attractive masses, at two different positions of them. It is a so-called time-of-swing method. New technological approaches and optimization of the configuration of an experimental setups showed, that the gravitational constant can be measured at an accuracy level of 15-40 ppm. Such an accuracy is planned to reach in the experiment of Huazhong University of Science and Technology (China). Nevertheless, there are a number of problems which should be solved to achieve this goal. One of such problems is a precision of the measurement of the period of eigen oscillations of the torsion balance. The torsion balance is the complicated system with many degrees of freedom, and due to the nonlinear couplings between them, new oscillations on socalled coupled modes are appeared. This leads to perturbations of the basic torsional mode and as a consequence, to a big uncertainty of the determination of its period. The goal of this paper is to study the nonlinear behavior of the torsion balance and the excitation of swing modes by environmental random noise. Degrees of freedom of the torsion balance The detailed structure of the torsion balance, which is used in the HUST experiment on measurement of G, is shown in Fig. 1 [2]. The test mass, a rectangular bar, is suspended from point O by a tungsten fiber of length l. In order to describe the dynamics of the torsion balance, we define two coordinate systems. A stationary Cartesian coordinate system O-XYZ, with the origin at O, located in suspension point of torsion balance. The other coordinate system, O1-X1Y1Z1, is fixed rigidly with the test body ­ the torsion bar. Its origin is at O1 which is point of attachment of the fiber to the torsion bar. The center of mass of the torsion balance is located at point O2, at a distance l0 from point O1 in vertical direction. The suspension point is driven by the fluctuation forces (seismic noise), which cause random displacements of the suspension point (t), (t) and (t), accordingly along X, Y and Z directions. Other fluctuation force, acting directly to torsion bar (for example, random fluctuation of residual gas in vacuum chamber, variation of temperature etc), can provide the torsion rotation (t). In a such setting of problem the system has five degrees of freedom, which are marked as 1, 2, 3, 4, and 5, respectively. Here the parameters 4 and 5 represent the rotation angles around the axes X and Y, and describe the swing oscillations in the plane XZ. The parameters 1 and 2 represent the rotation angles around the axes X1 and Y1, and describe the swing oscillations in the plane YZ. The angle 3 represents rotation about the Zaxis and describes the principal torsion oscillations. Hence, the system, describing the motion of torsion balance, is five nonlinear equations, containing also fluctuating terms

1 2 3

Doctor of Sciences in Physics and Mathematics, Head of Laboratory. Student of Physical Department. Researcher.


-& 2& & & & ( J x + ml0 )& + 1 11 + mgl01 = -mll0&4 - ml0&0 (t ) + ( J y - J x )&2 3 ; 1

(1) (2)
2 3

& ( J y + ml )&2 +

2 0

-1 & 2 2

& & & + mgl 0 2 = mll 0&5 + ml 0&0 (t ) + ( J y - J x )& 3 ; 1

- &2 & J z &3 + 3 1 J z 3 + K e 3 2 &2 &2 - ( J x - J y )(1 - 2 ) 3 -& & ml 2&4 + 4 1 4 + mgl 4 & & ml 2& + -1 + mgl
5 5 5

&2 &2 && = ( J x - J y )(1 - 2 ) 3 + K e (t ) + 2( J x - J y )1 2
3 3

& && - 2 J z & 2 + ( J x - J y - 2 J z )1 2 ; 1

(3) (4) (5)

& & = - mll 0& - ml&0 (t ) ; 1 && && 5 = mll 0 2 - ml 0 (t ) .

Here m is the mass of the torsion balance, JX, JY, JZ are the inertial moments of the torsion bar relatively the axes of X1, Y1 and Z1, respectively; i are the time constants for each degree of freedom, K e is the torsion elastic constant, (t) -- fluctuation function, describing random action on torsion degree of freedom. Equations (1) and (4), and (2) and (5), describe swing oscillations in two perpendicular planes, ZX and ZY, and equation (3) describes torsion oscillations. Analytic solution of this system of equations was done in [1,2]. Here the system of equations has been solved by numerical method and simulation of the motion of the torsion balance has been done.

Fig. 1. The torsion balance and the coordinate system are chosen to describe its motion.

Fig. 2. Swing oscillations on plane ZY, which excited by seismic noise with amplitude about 1mGal.

Swing oscillations It was studied the character of excitation of the swing oscillations of the torsion balance. The numerical experiment has shown, that oscillations on swing degrees of freedom are exited by the random noise of a seismic origin and occur with the amplitude, varying in time. (Fig. 2). The spectral analysis of swing oscillations has shown, that the oscillations on the each swing degree of freedom are beating of all quasi-harmonic swing modes with random amplitude changing in time. The swing frequencies are defined by geometrical parameters of the torsion balance. The random character of the swing oscillations is defined by the seismic noise affected on the suspension point. It was shown also, that even with damping, due to the action of seismic noise, the swing oscillations are the steady process. Torsion oscillations and mode couplings Oscillations on torsion degree of freedom are described by Eq. (3). The terms in the right part of this equation are nonlinear combinations of random quasi-harmonic swing modes. So, the right part of this equation determines the forced oscillations of torsion balance, which are torsional mode couplings, or coupled modes. The frequencies of coupled modes are simply linear combinations of swing-mode frequencies. The torsion balance, represented in Fig. 1, has an asymmetrical configuration, which is characterized by inequality of all inertial moments, i.e. JX JY JZ. The swing-mode frequencies of the asymmetrical torsion balance are also different.


Some of them can be close to each other. Their linear combination will determine the low-frequency coupled mode, which can disturb the torsion mode. The numerical simulation of the motion of the torsion balance with five degrees of freedom, described by the system of equations (1)-(5), has been done. The spectral analysis of the torsional degree of freedom revealed a number of harmonics (Fig. 3). The most intensive mode with the frequency 1.755 в 10 of the torsion oscillations. Other modes are torsional mode couplings. Coupled modes with the frequencies of 2 в 10 -4 - 5 в 10 -3 Hz are closest to the eigen mode and can disturb the last one. Therefore the problem of the high precision measurement of the torsional-mode frequency has to be solved on the condition of maximal suppression of coupled modes. The traditional method to remove the effects of low-frequency torsional mode couplings is employing a magnetic damper in the torsion system to overcome seismic noise and consequently to suppress the intensity of swing oscillations of asymmetrical modes. This method is called the amplitude Fig. 3. The spectrum of torsion -3 torsion balance. Peak at 1.755в10 Hz is eigen mode of torsion suppression and is used in the experimental setup, oscillations, peaks marked with index 1-9 are coupled modes. mentioned above. The other way, which we have proposed for suppression of combined modes is called frequency suppression. The main idea of this method is to choose the geometry of torsion balance in such a way, so to "move" the frequencies of coupled modes in a high frequency range. In this case the influence of the coupled modes can be reduced to minimum. In particular, symmetrical configuration of the torsion balance (JX = JY JZ) leads to degeneration of swing modes, i.e. swing frequencies in plane ZX coincide with frequencies in plane ZY. Conclusions In high precision gravitational experiments with torsion balance, as a measurement of the Newtonian gravitation constant, is necessary to take in attention the complicated motion of the torsion balance, in particular, the torsion mode couplings. Our investigation shown, that this effect in the torsion balance behavior arises directly from driven combination of the swing modes of the balance, and the swing modes are excited by the acting environmental noise. To successfully suppress these types of mode coupling in the torsion balance, the amplitude or frequency suppression has to be used in experimental setup. This work is supported by the Russian Foundation for Basic Research (No 05-02-39014), the National Basic Research Program of China (No 2003CB716300) and the National Science Foundation of China (No 10121503). References
1. Milyukov V. K. The theory of motion of the torsion balance in inhomogeneous gravitational field under the action of random noise. In: Problems of gravitation and elementary particles. Moscow: Energoizdat, 1981, V. 12, P. 128. 2. X.-D. Fan et al. Coupled modes of the torsion pendulum, Phys. Lett. A (2007), doi: 10.1016/j.physleta.2007.08.020.
-3

Hz is the eigen mode