**5. Attitude control**

One final problem must be addressed here to give a complete picture of the complexity of this apparently very simple experiment: the attitude of the whole apparatus with respect to the vertical direction, as defined by the Earth's local gravity vector, can affect the operation of pendulums and must therefore be guaranteed to be adequately accurate and stable in time, if necessary by active attitude control. Two different problems must be addressed in this respect as both the absolute tilt and its stability are relevant, in different ways.

was done to realize for the purpose a high-sensitivity tiltmeter [24] with sub-nanoradian resolution and special half-pole actuators based on thermoelectric devices [25]. The problem of substantial angular stability improvement by active control is not trivial because a tiltmeter with the necessary sensitivity was never seen with resonance-limited bandwidth above, say, 10 Hz, which provides not much more than a decade range in the Fourier frequency domain for open-loop gain increase from the zero-dB level of the loop bandwidth allowed by a typical 45° phase margin to the desired open-loop gain at the seismic peak sub-Hertz frequency. The halfpole actuator was developed for this reason, so that a loop-gain roll-off of 30 dB/dec could be achieved without resorting to a digital control, allowing in this way a guaranteed-stability servo

Measuring 'Big G', the Newtonian Constant, with a Frequency Metrology Approach

http://dx.doi.org/10.5772/intechopen.75635

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Nevertheless, the seismic angular noise problem is expected to be much more benign in case of acquisition of the whole sine-wave swing signal at a conveniently high sampling rate. For this reason, and because acquisition of the full sine wave is necessary anyway for oscillation support, unless the free ringdown approach is adopted, the choice was made to develop a suitable detection method for this purpose. The latter can be electromagnetic, based on the generation of an e.m.f. in a wire swinging through a magnetic field together with the suspenders of the bob, or optical, based on a linear position-sensitive detector deployed to translate

The first approach has the advantage of measuring the bob's velocity, with which the forcing term should be in phase, and is therefore to be preferred for phase accuracy of the oscillation loop but implies the risk of introducing in the experiment undesired electromagnetic forces

The second approach, on the contrary, is less risky of introducing undesired forces but requires the generation of a sine-wave signal in quadrature with the detected position for the implementation of the forcing term. This must be performed very accurately to obtain oscillations at exactly the resonance frequency because any error from quadrature would introduce a frequency shift, which in turn may build an error on *G* if it's not adequately common moded

Similar considerations hold for the forcing sine wave, which can be realized with just a current-carrying wire attached to the suspenders in a voice coil type of device in the first approach and if *Q* is high enough could be implemented by radiation pressure in the second.

A tentative accuracy budget for the experiment described here is given in [1]. Because of the highly efficient time and frequency metrology approach, only geometrical uncertainties are expected to be relevant at the level of 10−5, provided the necessary differential stability of 10−12 can be achieved. This is clearly a big "if," as discussed above, because it assumes that seismic and mode leakage problems are adequately solved. However, it can be in principle obtained if the limitation is electronic noise. It must be noted here for completeness that the

which could be greater than the weak gravitational force that must be measured.

operation with low-frequency open-loop gain in excess of 40 dB.

the bob's position in an electrical signal.

between the two pendulums.

**6. Accuracy budget**

The absolute verticality is important because the two cylinders must be guaranteed to be horizontal for the symmetry of the swing, which in turn guarantees the positioning of the minimum period in amplitude space (the curve of **Figure 10** was calculated for the case of two cylinders at the same level). Because the pendulums are two, this issue is complicated by the need to have both pairs of suspension cylinders aligned on the same horizontal plane.

The attitude stability is particularly critical in the case of low sampling-rate detection, like simple flyby time stamping at half periods, because of the heavy aliasing of seismic angular noise [1] that it produces. In **Figure 12**, a series of background seismic power spectrums is reported, as collected in different locations of the global seismographic network [21]. A peak at about 0.2 Hz appears in all of them, which is produced by low damping surface Rayleigh waves excited by ocean waves hammering the shores, extended with reduced intensity at higher frequencies. Because of their low frequency, it is very difficult to filter out such seismic angular noise contributions.

Work done to attack this problem includes passive and active attitude control [22, 23]. However, passive filtering was quickly understood to be inadequate for the purpose, not only because of its awkwardness at such low frequencies but also because of the need for stiffness of the structure holding pendulums to prevent detrimental effects of recoil from pendulums on attitude stability and damping itself. Active stabilization was then decided to be necessary, and work

**Figure 12.** Background seismic power spectrums, as collected in different locations of the global seismographic network. High noise at periods above 1s (i.e. Fourier frequencies below 1 Hz) is caused by far travelling Rayleigh surface-waves excited by ocean waves.

was done to realize for the purpose a high-sensitivity tiltmeter [24] with sub-nanoradian resolution and special half-pole actuators based on thermoelectric devices [25]. The problem of substantial angular stability improvement by active control is not trivial because a tiltmeter with the necessary sensitivity was never seen with resonance-limited bandwidth above, say, 10 Hz, which provides not much more than a decade range in the Fourier frequency domain for open-loop gain increase from the zero-dB level of the loop bandwidth allowed by a typical 45° phase margin to the desired open-loop gain at the seismic peak sub-Hertz frequency. The halfpole actuator was developed for this reason, so that a loop-gain roll-off of 30 dB/dec could be achieved without resorting to a digital control, allowing in this way a guaranteed-stability servo operation with low-frequency open-loop gain in excess of 40 dB.

Nevertheless, the seismic angular noise problem is expected to be much more benign in case of acquisition of the whole sine-wave swing signal at a conveniently high sampling rate. For this reason, and because acquisition of the full sine wave is necessary anyway for oscillation support, unless the free ringdown approach is adopted, the choice was made to develop a suitable detection method for this purpose. The latter can be electromagnetic, based on the generation of an e.m.f. in a wire swinging through a magnetic field together with the suspenders of the bob, or optical, based on a linear position-sensitive detector deployed to translate the bob's position in an electrical signal.

The first approach has the advantage of measuring the bob's velocity, with which the forcing term should be in phase, and is therefore to be preferred for phase accuracy of the oscillation loop but implies the risk of introducing in the experiment undesired electromagnetic forces which could be greater than the weak gravitational force that must be measured.

The second approach, on the contrary, is less risky of introducing undesired forces but requires the generation of a sine-wave signal in quadrature with the detected position for the implementation of the forcing term. This must be performed very accurately to obtain oscillations at exactly the resonance frequency because any error from quadrature would introduce a frequency shift, which in turn may build an error on *G* if it's not adequately common moded between the two pendulums.

Similar considerations hold for the forcing sine wave, which can be realized with just a current-carrying wire attached to the suspenders in a voice coil type of device in the first approach and if *Q* is high enough could be implemented by radiation pressure in the second.
