**2. The dynamic dual simple-pendulum approach**

The experiment illustrated here is based on a high-resolution technique, well known in frequency metrology [12], to measure very accurately small frequency differences between two almost synchronous sources. In fact, such small differences ∆*v* produce a variable time delay between the two waveforms, which add up to a full cycle in a time interval 1/∆*v*. The rate of change of the time delay yields directly the relative frequency difference.

Simple pendulums appear attractive for a *G* measurement based on this approach, because their small oscillation resonance frequency is directly proportional to the square root of the Earth's gravitational acceleration *g*, as is well known, which makes them particularly sensitive to a gravitational field variation induced in a controlled way by a displacement of field masses. We will call *y* the relative frequency change produced in this way. Resolution in this measurement is limited only by time delay resolution and differential frequency stability of the two sources. For example, if time resolution is 1 ns, a 1000 s run allows to determine the relative frequency difference to 10−12, provided its stability is adequate. This means that the two frequencies can wander around in parallel by more than that but their difference should not. This is important in considering the use of pendulum oscillators, because the gravity acceleration g is not constant in time due to a variety of causes, and so will be their frequency, which will then show instabilities not much below the 10−7 level [13]. Nevertheless, since such instabilities affect in a similar way all pendulum oscillators, particularly if they are in the same location, it can be expected to be quite possible that the differential instability of two equal pendulums oscillating not far from each other may be adequate for the projected resolution of the experiment under discussion.

Several different approaches have been followed in the realization of experiments aimed at its determination. A short summary can be found in the introduction of [1], where the experiment illustrated in this chapter was proposed, and for a deeper insight, a well-done recent comprehensive review [2] can be used for reference and comparison. It makes metrological sense to devise different experiments for the purpose, so that the set of possible systematic errors be not the same for all and the risk of undetected coherent biases among various *G* determinations be minimized. While refurbished and modernized versions of the original Cavendish torsion balance are still the most commonly adopted sensing device and at least one of them has demonstrated extremely high accuracy [3], experiments based on other configurations have also been developed, and a few of them have yielded some of the best results to date. The latter include a measurement based on a beam balance [4] and one based on a pair of simple pendulums used in the static mode [5]. Both achieved accuracy in the low 10−5 region. These three determinations of *G* agree within their stated uncertainty and are the most influential in the 2014 CODATA value, which, however, was attributed higher uncertainty due to the excessive disagreement of other results. A coordinate effort is being led by the recently established working group WG13 of UIAP, stimulated by a NIST initiative, aimed at improving the status of *G* metrology. The experiments coordinated in this effort are mainly based on the torsion balance approach because of its favorable S/N ratio, hoping to put to fruition the enormous amount of information on systematics affecting it, with the target of improving accuracy by possibly an order of magnitude. However, other approaches are also encouraged, and experiments based differently are monitored or even supported. The free-fall gravimeter [6–8] still appears very promising due to its unique absence of difficult-to-evaluate systematics, but results are still hard to come by, mainly due to the inherently low S/N ratio of these experiments. The experiment presented in this chapter is supported by NIST through its Precision Measurements Grant Program and is based on the adoption of a pair of simple pendulums as a detection device. The target is the determination of *G* with an accuracy of 10−5. The concept of the experiment has evolved from a pilot experiment carried on at Politecnico di Torino from 1998 to 2005, which used a single pendulum in vacuum and yielded preliminary results at 3% accuracy level [9–11].

54 Metrology

**2. The dynamic dual simple-pendulum approach**

change of the time delay yields directly the relative frequency difference.

The experiment illustrated here is based on a high-resolution technique, well known in frequency metrology [12], to measure very accurately small frequency differences between two almost synchronous sources. In fact, such small differences ∆*v* produce a variable time delay between the two waveforms, which add up to a full cycle in a time interval 1/∆*v*. The rate of

Simple pendulums appear attractive for a *G* measurement based on this approach, because their small oscillation resonance frequency is directly proportional to the square root of the Earth's gravitational acceleration *g*, as is well known, which makes them particularly sensitive to a gravitational field variation induced in a controlled way by a displacement of field masses. We will call *y* the relative frequency change produced in this way. Resolution in this measurement is limited only by time delay resolution and differential frequency stability of In **Figure 1**, a sketch is shown of the expected evolution of time delay as the active field mass distribution is shifted back and forth between a geometrical configuration in which it increases the frequency of one pendulum and another antisymmetric one in which it increases the frequency of the other one. Suppose one pendulum is slightly slower than the other one (it always will be the case as two exactly equal lengths are very unlikely and even undesirable to avoid coupling). As time goes by, this slower oscillator will show increasing time delay with respect to the other, as indicated in **Figure 1** by the broken trend line. Now, when its frequency is increased by the field masses, it will get closer to that of the faster one, and its time delay rate of change (*DROC*) will decrease. The opposite will happen when the field masses increase the frequency of the other pendulum. The relevant information in this measurement is the difference between *DROC*s in the two configurations.

If *v*s0 and *vf*<sup>0</sup> are the undisturbed frequencies of slow and fast pendulums (*v*f0 − *v*s0 = ∆*v*<sup>0</sup> > 0), their difference will be modified by field masses as in Eq. (1) below, when the latter are next to the slow pendulum, and as in Eq. (2) when they are next to the fast one.

$$\Delta\boldsymbol{\nu}|\_{\rm s} = \boldsymbol{\nu}\_{f\mathbf{0}} \{ \mathbf{1} + \mathbf{y}\_{fwr} \} - \boldsymbol{\nu}\_{\rm s\mathbf{0}} \{ \mathbf{1} + \mathbf{y}\_{near} \} \tag{1}$$

$$|\Delta\boldsymbol{\nu}|\_{f} = \boldsymbol{\nu}\_{f0}(1 + \boldsymbol{\chi}\_{near}) - \boldsymbol{\nu}\_{s0}(1 + \boldsymbol{\chi}\_{far})\tag{2}$$

The DROC measurement is performed by measuring the time delay accumulated in *n* periods of the slow pendulum and dividing it by *nT*<sup>s</sup> , as illustrated in **Figure 2**.

Therefore, it turns out that the relationship between measured DROC and actual frequencies of the two oscillators is.

**Figure 1.** Time delay slope changes as field masses are moved back and forth between the two pendulums with repetition period *T*R.

**Figure 2.** Measurement scheme of the *DROC* (time delay rate of change).

$$DROC = \frac{\Delta t\_2 - \Delta t\_1}{nT\_\odot} = \frac{n(T\_\sf s - T\_{f})}{nT\_\odot} = \frac{\Delta \nu}{\nu\_f} \,, \tag{3}$$

oscillations of the bob along *x*, its angular frequency is given by the square root of (ag + aM)/*x*, with ag = *gx*/*L*. The relative frequency change *y* induced by field masses will then be (aM/a<sup>g</sup>

A peculiarity of the experiment discussed here, with field masses centered on both sides of

linear in *x* for small displacements. This fact gives this scheme a great advantage over other approaches, because it maximizes the effect exactly where field masses are closest to sensors. For example, this is not the case for free-fall experiments, which see the effect vanish along with aM where the sensing object spends most of its time, at the apogee of its parabolic flight. An analysis of the arrangement under discussion, with two equal masses symmetrically centered on either side of the bob rest position, yields for the effect on pendulum frequency.

where *L* is the pendulum length, *R* and a are radius and half distance of field masses between centers, *R*E is the Earth's radius, and shown densities are those of the Earth and field masses. Γ(0) is the value at *x* = 0 of a geometrical shape factor which is discussed below. It is interesting to point out in Eq. (6) that only the density of field masses is relevant for the size of the effect and not their total mass, other than in the fact that, for a given gap between them, the ratio *R/a* depends slightly on mass size. Also interesting is to notice that, other than hidden in the size of the gap that must host it, the test mass (which is the bob) does not appear in Eq. (6). This is because neither gravitational acceleration in play, from the Earth or from field masses,

The shift of Eq. (6) is expected for small oscillations. However, neither acceleration is strictly linear, which yields the well-known non-isochronism of the simple pendulum plus, relevant for this experiment, a nontrivial tie with the extra gravitational pull. So, while it is easy to find frequency and shift for small oscillations, as the relative extra acceleration can then be considered constant over the swing, nontrivial calculations are

The field masses adopted for the experiment are cylinders of heavy metal positioned, for the "near" configuration, at either side of the bob as shown in **Figure 3a**. The metal could be platinum or, more cheaply, tungsten, but copper is chosen for budget reasons in the preliminary phases. The reason for adopting a cylindrical shape lies in the much higher uniformity of the additional recalling acceleration provided by this shape to a bob displaced from the rest point, with respect to the case of a spherical field mass shape. In **Figure 3b** a plot is given of such

for two tungsten field masses 85 mm in diameter and 117 mm long, spaced by an 8 mm gap

The resulting fractional frequency increase *y*near can be calculated with a suitable integration, which is quite straightforward for oscillation amplitudes not exceeding the uniformity region.

The expression of Γ(*x*) used in **Figure 3b**, written with *η* = *w*/*a* and *ξ* = *x*/*a*, is.

) versus bob displacement in a 1 m pendulum, calculated

and aM vanish at rest position, but their ratio does not, as both are

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

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the bob, is that both ag

depends in any way on the mass of the bob.

necessary for wider swings.

additional acceleration (relative to ag

to host a 5 mm spherical bob in between.

)/2.

57

(6)

(7)

and the difference between DROCs in the two configurations is then given by.

$$DROC\_f - DROC\_s = \frac{\nu\_{f0} + \nu\_{s0}}{\nu\_f} \left(\chi\_{near} - \chi\_{far}\right) \approx \left(2 - \frac{\Delta\nu\_0}{\nu\_f}\right) KG$$

In Eq. (4), the concept was introduced that relative frequency variations induced on pendulums by the field masses are proportional to the Newtonian constant *G* through a proportionality factor *K*. The analysis needed to identify the value of *K* is sketched in the next section. The value of G can be obtained by inverting Eq. (4) and is.

$$G = \frac{\text{DROC}\_f - \text{DROC}\_5}{\kappa \left(2 - \frac{\Delta \nu\_0}{\mathbf{v}\_f}\right)} \,. \tag{5}$$

Clearly, *K* must be known with relative uncertainty smaller than the 10−5 target *G* accuracy of this experiment. In fact, this may well be the ultimate accuracy limit of this approach.

As for measurement resolution, it is also shown in the next section that the relative effect on pendulum frequency obtainable by a geometrical change in mass distribution around the bob can be on the order of 10−7. It appears therefore clear that a target differential frequency stability of at least 10−12 should be looked for in designing the two oscillators. Other than that, the requirements for time interval measurement resolution are instead benign, both because the S/N ratio of pendulum signals is expected to be quite good (more on this in the following) and because the DROC type A uncertainty can be expected to improve as the averaging time to the power 3/2, much faster than the typical power ½ of averaging on white noise [1]. The intuitive explanation for this is in the fact that a linear regression on the scatterplot of delay data versus time will in fact yield a statistical uncertainty improving as the square root of the number of measurements (which is proportional to time), but then the result is divided by elapsed time to get the DROC, which produces the power 3/2 improvement law.

#### **3. Field mass configuration**

For the calculation of the gravitational effect on frequency, the relative extra acceleration aM given to the bob by the system of field masses is the relevant parameter. In fact, for small oscillations of the bob along *x*, its angular frequency is given by the square root of (ag + aM)/*x*, with ag = *gx*/*L*. The relative frequency change *y* induced by field masses will then be (aM/a<sup>g</sup> )/2. A peculiarity of the experiment discussed here, with field masses centered on both sides of the bob, is that both ag and aM vanish at rest position, but their ratio does not, as both are linear in *x* for small displacements. This fact gives this scheme a great advantage over other approaches, because it maximizes the effect exactly where field masses are closest to sensors. For example, this is not the case for free-fall experiments, which see the effect vanish along with aM where the sensing object spends most of its time, at the apogee of its parabolic flight.

An analysis of the arrangement under discussion, with two equal masses symmetrically centered on either side of the bob rest position, yields for the effect on pendulum frequency.

(3)

(4)

(5)

and the difference between DROCs in the two configurations is then given by.

The value of G can be obtained by inverting Eq. (4) and is.

**Figure 2.** Measurement scheme of the *DROC* (time delay rate of change).

56 Metrology

to get the DROC, which produces the power 3/2 improvement law.

**3. Field mass configuration**

In Eq. (4), the concept was introduced that relative frequency variations induced on pendulums by the field masses are proportional to the Newtonian constant *G* through a proportionality factor *K*. The analysis needed to identify the value of *K* is sketched in the next section.

Clearly, *K* must be known with relative uncertainty smaller than the 10−5 target *G* accuracy of

As for measurement resolution, it is also shown in the next section that the relative effect on pendulum frequency obtainable by a geometrical change in mass distribution around the bob can be on the order of 10−7. It appears therefore clear that a target differential frequency stability of at least 10−12 should be looked for in designing the two oscillators. Other than that, the requirements for time interval measurement resolution are instead benign, both because the S/N ratio of pendulum signals is expected to be quite good (more on this in the following) and because the DROC type A uncertainty can be expected to improve as the averaging time to the power 3/2, much faster than the typical power ½ of averaging on white noise [1]. The intuitive explanation for this is in the fact that a linear regression on the scatterplot of delay data versus time will in fact yield a statistical uncertainty improving as the square root of the number of measurements (which is proportional to time), but then the result is divided by elapsed time

For the calculation of the gravitational effect on frequency, the relative extra acceleration aM given to the bob by the system of field masses is the relevant parameter. In fact, for small

this experiment. In fact, this may well be the ultimate accuracy limit of this approach.

$$
\hat{\rho} = \frac{\rho\_{\text{sf}}}{\rho\_{\text{E}}} \frac{L}{n\_{\text{E}}} \left(\frac{\kappa}{u}\right)^{2} \mathbb{I}'(0) \quad , \tag{6}
$$

where *L* is the pendulum length, *R* and a are radius and half distance of field masses between centers, *R*E is the Earth's radius, and shown densities are those of the Earth and field masses. Γ(0) is the value at *x* = 0 of a geometrical shape factor which is discussed below. It is interesting to point out in Eq. (6) that only the density of field masses is relevant for the size of the effect and not their total mass, other than in the fact that, for a given gap between them, the ratio *R/a* depends slightly on mass size. Also interesting is to notice that, other than hidden in the size of the gap that must host it, the test mass (which is the bob) does not appear in Eq. (6). This is because neither gravitational acceleration in play, from the Earth or from field masses, depends in any way on the mass of the bob.

The shift of Eq. (6) is expected for small oscillations. However, neither acceleration is strictly linear, which yields the well-known non-isochronism of the simple pendulum plus, relevant for this experiment, a nontrivial tie with the extra gravitational pull. So, while it is easy to find frequency and shift for small oscillations, as the relative extra acceleration can then be considered constant over the swing, nontrivial calculations are necessary for wider swings.

The field masses adopted for the experiment are cylinders of heavy metal positioned, for the "near" configuration, at either side of the bob as shown in **Figure 3a**. The metal could be platinum or, more cheaply, tungsten, but copper is chosen for budget reasons in the preliminary phases. The reason for adopting a cylindrical shape lies in the much higher uniformity of the additional recalling acceleration provided by this shape to a bob displaced from the rest point, with respect to the case of a spherical field mass shape. In **Figure 3b** a plot is given of such additional acceleration (relative to ag ) versus bob displacement in a 1 m pendulum, calculated for two tungsten field masses 85 mm in diameter and 117 mm long, spaced by an 8 mm gap to host a 5 mm spherical bob in between.

The resulting fractional frequency increase *y*near can be calculated with a suitable integration, which is quite straightforward for oscillation amplitudes not exceeding the uniformity region. The expression of Γ(*x*) used in **Figure 3b**, written with *η* = *w*/*a* and *ξ* = *x*/*a*, is.

$$
\Gamma(\mathbf{x}) = \frac{3}{4} \frac{a}{\hbar \xi} \begin{pmatrix} 1 \\ \frac{1}{\sqrt{1 + (\eta - \xi)^2}} & \frac{1}{\sqrt{1 + (\eta + \zeta)^2}} \end{pmatrix} \tag{7}
$$

gives raise to the relative frequency change *y*far of the "far" configuration which appears in Eq. (4). As a matter of fact, this effect is not so small, given the fact that the two pendulums are

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

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59

In order to facilitate this calculation, while increasing the signal by a factor of two, the idea was conceived to design the field mass system as a periodic structure. In fact, it can easily be shown that increasing *w*, the length of the field mass cylinders, would cause a signal reduction which would take the signal to vanish if the length is taken to infinity. This happens because such a structure would produce no field gradient in the longitudinal direction. Only a modulation along *x* of the mass density can produce a field gradient. The periodic structure which is planned, with a density switch between *ρ*M and zero (or the lower density of another material) for every length of 2 *w*, will produce a periodic field gradient along *x* which vanishes at the center of all regions of uniform density. A pendulum centered at such vanishing gradient points will experience an increased frequency when positioned in correspondence with the higher density material and a symmetrically decreased frequency when positioned

By placing the two pendulums inside the vacuum vessel at a distance 2 *w* from each other, within a periodic field mass system so conceived, as shown in **Figure 5**, the measured DROC will be doubled because while one pendulum is pulled up, the other one is pulled down. The opposite will then happen after the whole field mass system is displaced by 2 *w* to invert the

In practice, an infinite length of the field mass system cannot obviously be deployed, and the structure must be truncated at some point. In **Figure 6**, a calculation is shown of the expected relative gravitational extra acceleration in the case of a nine-mass-long truncated periodic structure. The material of field masses was assumed to be tungsten, dimensions were the

Details of acceleration uniformity around the rest point are given in **Figure 7** for both positions of the two pendulums, at the center of the middle field masses (upper curve) and at the center of the first air gap at their right (lower curve). It can be noticed that the latter is asymmetric. This is because the truncated periodic structure is asymmetric with respect to that point, with five masses on one side and only four on the other one. However, the effect on frequency of such asymmetry is expected to vanish to first order, as long as the rest point of the pendulum is correctly centered. In any case, centering of the pendulums will be important for accuracy as much as uniformity of the extra acceleration. Nevertheless, it can be noticed that a subtraction of the slanting baseline in the lower curve will make it appear very similar

**Figure 5.** Scheme of the periodic field mass principle. Rest positions of the two bobs are shown (black dots). The circle in the middle represents the outline of the lower vacuum chamber through whose tunnels, shown in **Figure 4**, the field

same of **Figure 3**, and the density of air in between masses was neglected.

contained in the same vacuum vessel of **Figure 4**.

in correspondence with the lower density one.

centering of the two pendulums.

mass systems go.

**Figure 3.** (a) Sketch of the near arrangement of two cylindrical field masses and (b) variations along x, elongation of the bob from rest position, of the recall acceleration aM produced by the two field masses divided by the relevant *g* component ag .

It should be pointed out here that shape factor of the cylindrical field masses was chosen in this calculation to optimize the uniformity of the effect, as shown in **Figure 3b**. The absolute dimension of the masses, instead, was designed to best fit the chosen geometry of the vacuum chamber, whose relevant part of the realization is shown in **Figure 4**. The chamber is realized with commercial 10 inch ConFlat flanges for the vertical body assembly, which will host both pendulums. Two thin steel tubes were welded across it to provide tunnels for the passage of the movable field mass system. These tubes are 100 mm in diameter and cannot therefore host cylinders greater than say 90 mm in diameter, together with their cradle which will be necessary for their management.

While the expression of Eq. (7) is valid for one pendulum in the "near" configuration, the effect on the other pendulum of field masses in that position must also be studied because it

**Figure 4.** Detail of the lower chamber of the UHV vacuum system, showing the two thin steel tubes that allow management of field masses without feedthroughs by keeping them outside the vacuum vessel.

gives raise to the relative frequency change *y*far of the "far" configuration which appears in Eq. (4). As a matter of fact, this effect is not so small, given the fact that the two pendulums are contained in the same vacuum vessel of **Figure 4**.

In order to facilitate this calculation, while increasing the signal by a factor of two, the idea was conceived to design the field mass system as a periodic structure. In fact, it can easily be shown that increasing *w*, the length of the field mass cylinders, would cause a signal reduction which would take the signal to vanish if the length is taken to infinity. This happens because such a structure would produce no field gradient in the longitudinal direction. Only a modulation along *x* of the mass density can produce a field gradient. The periodic structure which is planned, with a density switch between *ρ*M and zero (or the lower density of another material) for every length of 2 *w*, will produce a periodic field gradient along *x* which vanishes at the center of all regions of uniform density. A pendulum centered at such vanishing gradient points will experience an increased frequency when positioned in correspondence with the higher density material and a symmetrically decreased frequency when positioned in correspondence with the lower density one.

By placing the two pendulums inside the vacuum vessel at a distance 2 *w* from each other, within a periodic field mass system so conceived, as shown in **Figure 5**, the measured DROC will be doubled because while one pendulum is pulled up, the other one is pulled down. The opposite will then happen after the whole field mass system is displaced by 2 *w* to invert the centering of the two pendulums.

It should be pointed out here that shape factor of the cylindrical field masses was chosen in this calculation to optimize the uniformity of the effect, as shown in **Figure 3b**. The absolute dimension of the masses, instead, was designed to best fit the chosen geometry of the vacuum chamber, whose relevant part of the realization is shown in **Figure 4**. The chamber is realized with commercial 10 inch ConFlat flanges for the vertical body assembly, which will host both pendulums. Two thin steel tubes were welded across it to provide tunnels for the passage of the movable field mass system. These tubes are 100 mm in diameter and cannot therefore host cylinders greater than say 90 mm in diameter, together with their cradle which will be necessary for their management.

**Figure 3.** (a) Sketch of the near arrangement of two cylindrical field masses and (b) variations along x, elongation of the bob from rest position, of the recall acceleration aM produced by the two field masses divided by the relevant *g*

component ag

58 Metrology

.

While the expression of Eq. (7) is valid for one pendulum in the "near" configuration, the effect on the other pendulum of field masses in that position must also be studied because it

**Figure 4.** Detail of the lower chamber of the UHV vacuum system, showing the two thin steel tubes that allow

management of field masses without feedthroughs by keeping them outside the vacuum vessel.

In practice, an infinite length of the field mass system cannot obviously be deployed, and the structure must be truncated at some point. In **Figure 6**, a calculation is shown of the expected relative gravitational extra acceleration in the case of a nine-mass-long truncated periodic structure. The material of field masses was assumed to be tungsten, dimensions were the same of **Figure 3**, and the density of air in between masses was neglected.

Details of acceleration uniformity around the rest point are given in **Figure 7** for both positions of the two pendulums, at the center of the middle field masses (upper curve) and at the center of the first air gap at their right (lower curve). It can be noticed that the latter is asymmetric. This is because the truncated periodic structure is asymmetric with respect to that point, with five masses on one side and only four on the other one. However, the effect on frequency of such asymmetry is expected to vanish to first order, as long as the rest point of the pendulum is correctly centered. In any case, centering of the pendulums will be important for accuracy as much as uniformity of the extra acceleration. Nevertheless, it can be noticed that a subtraction of the slanting baseline in the lower curve will make it appear very similar

**Figure 5.** Scheme of the periodic field mass principle. Rest positions of the two bobs are shown (black dots). The circle in the middle represents the outline of the lower vacuum chamber through whose tunnels, shown in **Figure 4**, the field mass systems go.

is the real target of the present work. Such an operation will most likely belong to a national metrology institute and not to a university. What this effort wants to prove is that no real obstacle exists in this approach on the way to an accuracy of 10−5, other than problems that

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

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More benign is the requirement on positioning of the bob's trajectory with respect to active masses. In fact, it turns out that both in the horizontal and vertical direction, the extra acceleration features an extreme versus trajectory positioning, as shown in **Figures 8** and **9**, respectively: a minimum in the center for the lateral direction and a maximum a little above masses'

The vertical displacement of the maximum is due to the extra vertical pull down that field masses exert if they are moved lower than the bob, which adds to Earth's gravity and hence

the masses' gravity centers, which turns out to be almost 3 mm for the assumed masses. The

**Figure 8.** Relative variation of extra pull for lateral displacement of the bob's trajectory from the symmetry plane

**Figure 9.** Relative variation of the relevant effect for vertical displacements of the bob's trajectory from the plane of mass

)/3 *L* above

61

to recall force, until they get too far down to be relevant. Such maximum is (*a*<sup>2</sup> + *w*<sup>2</sup>

may come from accuracy and stability of the field mass system.

gravity centers for the vertical.

between field masses.

centers.

**Figure 6.** Calculated relative extra acceleration for a pendulum positioned at *x* from the center of a periodic field mass structure truncated to nine masses.

**Figure 7.** Relative uniformity of the extra acceleration for a pendulum positioned at the center of the middle masses (upper display) and one positioned at the center of the first gap (lower display), as a function of *x*, distance from the center of a nine-mass-long periodic structure. Uniformity was optimized here by trimming *η* = *w*/*a*.

to the upper curve for what concerns uniformity. Since both remain within +/− 10−5 up to 7 mm either side of the center, integration of the extra gravitational effect will be straightforward for peak pendulum oscillation amplitudes up to 7 mm if the target accuracy is at the 10−5 level.

In any case, all geometrical characteristics of the field mass system affect the proportionality constant *K* of Eq. (4), including the uniformity of their mass density and their stability in operational environmental conditions (like temperature expansion or deformation under mechanical stress). Adequate care must then be taken in design, realization, and handling of the field mass system.

To be truthful, in this respect, the experiment presented here is no different from any other experiment that was or will be tried to measure *G*. Revisiting the geometry of the field mass system for accuracy optimization will then be necessary after the concept is proven, which is the real target of the present work. Such an operation will most likely belong to a national metrology institute and not to a university. What this effort wants to prove is that no real obstacle exists in this approach on the way to an accuracy of 10−5, other than problems that may come from accuracy and stability of the field mass system.

More benign is the requirement on positioning of the bob's trajectory with respect to active masses. In fact, it turns out that both in the horizontal and vertical direction, the extra acceleration features an extreme versus trajectory positioning, as shown in **Figures 8** and **9**, respectively: a minimum in the center for the lateral direction and a maximum a little above masses' gravity centers for the vertical.

The vertical displacement of the maximum is due to the extra vertical pull down that field masses exert if they are moved lower than the bob, which adds to Earth's gravity and hence to recall force, until they get too far down to be relevant. Such maximum is (*a*<sup>2</sup> + *w*<sup>2</sup> )/3 *L* above the masses' gravity centers, which turns out to be almost 3 mm for the assumed masses. The

**Figure 8.** Relative variation of extra pull for lateral displacement of the bob's trajectory from the symmetry plane between field masses.

to the upper curve for what concerns uniformity. Since both remain within +/− 10−5 up to 7 mm either side of the center, integration of the extra gravitational effect will be straightforward for peak pendulum oscillation amplitudes up to 7 mm if the target accuracy is at the 10−5 level.

**Figure 7.** Relative uniformity of the extra acceleration for a pendulum positioned at the center of the middle masses (upper display) and one positioned at the center of the first gap (lower display), as a function of *x*, distance from the

center of a nine-mass-long periodic structure. Uniformity was optimized here by trimming *η* = *w*/*a*.

**Figure 6.** Calculated relative extra acceleration for a pendulum positioned at *x* from the center of a periodic field mass

structure truncated to nine masses.

60 Metrology

In any case, all geometrical characteristics of the field mass system affect the proportionality constant *K* of Eq. (4), including the uniformity of their mass density and their stability in operational environmental conditions (like temperature expansion or deformation under mechanical stress). Adequate care must then be taken in design, realization, and handling of the field mass system. To be truthful, in this respect, the experiment presented here is no different from any other experiment that was or will be tried to measure *G*. Revisiting the geometry of the field mass system for accuracy optimization will then be necessary after the concept is proven, which

**Figure 9.** Relative variation of the relevant effect for vertical displacements of the bob's trajectory from the plane of mass centers.

relative shift is below 7∙10−4, which must be evaluated only to 1% for an accuracy contribution well below 10−5, and the vertical positioning tolerance is 0.2 mm either side of the maximum, just like that of transverse horizontal positioning.

Amplitude-to-frequency (or period) conversion is a well-known problem of pendulum clocks, because period-to-period instability of the kick turns into frequency instability through such connection, and famous in this perspective is the solution proposed by Christiaan Huygens in his 1657 patent of making an initial ribbon section of the suspension wrap on cycloidal profiles each side as it swings back and forth. However, neither Salomon Coster (who built the first such device, still shown in Boerhaave Museum in Leiden) nor anyone later appeared to be able to take full advantage of Huygens' idea, presumably because realizing a faithfully cycloidal profile is very difficult, as its curvature diverges in the cusp, where the shape is most important for small oscillations, which is where pendulum clocks are operated for wear

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In the model chosen for this experiment, pictured in **Figure 10a** with the bob in between field masses, the pendulum suspensions are made of tungsten wires hanging between two cylinders on which they wrap and unwrap. The wires are two for each pendulum, converging on the bob, for removal of the degeneracy of the two orthogonal modes, and the wire section above the cylinders is dimensioned for temperature compensation in a scheme that includes

Cylinders are technologically very easy to fabricate, contrary to the cycloidal case, and very good ones are common in modern machines, which makes them easy to obtain and cheap. In this work, dowel pins and specifically wrist pins are employed. The latter are very well rectified and have a hard surface because they must bear high forces with little friction in connecting pistons to rods in ICE power trains. As for amplitude-to-frequency conversion, deploying circular profiles does not realize a completely isochronous pendulum like Huygens showed true for a cycloidal profile; nevertheless, they produce a period vs. amplitude curve which shows a minimum at a certain amplitude value which is related to the diameter *D* of the cylinders. For that magic amplitude, the pendulum is then locally isochronous, and operation exactly at that amplitude shows no amplitude-to-frequency conversion. This means that the effect on frequency of amplitude variations vanishes if the amplitude is set correctly and that it depends quadratically on the amplitude error from that magic value in a way that makes it

**Figure 10.** (a) Picture of the pendulum configuration chosen for this work, with the bob hanging between field masses, and (b) period versus amplitude curve of such pendulum compared to the one of a mathematical pendulum. Length is about 250 mm and D is 22 mm. The experimental points are superimposed on the measured section of the curve.

minimization and consequent long-term stability.

possible to achieve the necessary stability.

an aluminum structure to fix the length of the upper part of the wires.
