**4. Pendulum design and optimization**

The pendulums to be used in the experiment should be designed with the double target in mind of maximizing both accuracy and differential stability.

For accuracy, they should be "ideal," meaning that in their behavior they should not differ from the description that can be made with a mathematical model, supported by an adequate experimental characterization, in a way that can make *G* measurements uncertain by more than the desired accuracy. To this aim, all non-idealities affecting differential measurements between the two configurations of the field mass system should be considered. The main problem in this respect seems to be the uncertainty in the position of the center of mass of the pendulum given by the nonvanishing mass of the suspension relative to the bob. This shifts high the effective center of mass in a different way for the attraction of the Earth and that of the field mass system.

For differential frequency stability, which should exceed 10−12 for a full repetition period *T*R, three main characteristics should be optimized in design and realization. They are:


Stability against environmental changes may be particularly critical for temperature, if not addressed, because at least a few ppm per Kelvin must be assumed for the linear expansion coefficient of the suspension, unless some kind of compensation is made. This is a quite common practice in pendulum clocks, but the demand in this application is severe. Even in the case of a successful compensation by a factor of ten of a low-expansion suspension material (e.g., tungsten, with its 4.3 ppm/K), the requirement would still be for a temperature differential between the two pendulums of the order of a few K for the necessary 10−12 differential frequency stability. It is true that the two pendulums are contained in the same vacuum vessel, which can be temperature stabilized, but an excellent thermalization scheme must certainly be devised for the purpose. It is assumed here that gold plating of the inner surface of the vacuum vessel or if necessary cylindrical gold-plated mirrors focusing the two suspensions [14] onto each other are the best bet to this aim, but a detailed discussion of the problem is out of the scope of this paper. What instead can be done at the pendulum design phase is implementation of a temperature compensation scheme. To this aim, tungsten is used for the suspension of the bob, and aluminum is deployed in an expansion compensation structure as shown below.

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 minimization and consequent long-term stability.

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,

The pendulums to be used in the experiment should be designed with the double target in

For accuracy, they should be "ideal," meaning that in their behavior they should not differ from the description that can be made with a mathematical model, supported by an adequate experimental characterization, in a way that can make *G* measurements uncertain by more than the desired accuracy. To this aim, all non-idealities affecting differential measurements between the two configurations of the field mass system should be considered. The main problem in this respect seems to be the uncertainty in the position of the center of mass of the pendulum given by the nonvanishing mass of the suspension relative to the bob. This shifts high the effective center of mass in a different way for the attraction of the Earth and that of the field mass system.

For differential frequency stability, which should exceed 10−12 for a full repetition period *T*R,

**2.** Amplitude-to-frequency conversion, which induces frequency variations if the oscillation

**3.** Quality factor *Q* of the resonator, which is relevant in two ways: to obtain a long time constant in case of free decay operation [1] and to maximize stability with a given S/N ratio in

Stability against environmental changes may be particularly critical for temperature, if not addressed, because at least a few ppm per Kelvin must be assumed for the linear expansion coefficient of the suspension, unless some kind of compensation is made. This is a quite common practice in pendulum clocks, but the demand in this application is severe. Even in the case of a successful compensation by a factor of ten of a low-expansion suspension material (e.g., tungsten, with its 4.3 ppm/K), the requirement would still be for a temperature differential between the two pendulums of the order of a few K for the necessary 10−12 differential frequency stability. It is true that the two pendulums are contained in the same vacuum vessel, which can be temperature stabilized, but an excellent thermalization scheme must certainly be devised for the purpose. It is assumed here that gold plating of the inner surface of the vacuum vessel or if necessary cylindrical gold-plated mirrors focusing the two suspensions [14] onto each other are the best bet to this aim, but a detailed discussion of the problem is out of the scope of this paper. What instead can be done at the pendulum design phase is implementation of a temperature compensation scheme. To this aim, tungsten is used for the suspension of the bob, and aluminum is deployed in an expansion compensation structure as shown below.

three main characteristics should be optimized in design and realization. They are:

just like that of transverse horizontal positioning.

62 Metrology

**4. Pendulum design and optimization**

mind of maximizing both accuracy and differential stability.

**1.** Environmental sensitivity, especially versus temperature

amplitude is not constant

case of sustained oscillations

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

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 possible to achieve the necessary stability.

**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.

The period versus amplitude curve is compared to that of a mathematical pendulum in **Figure 10b**. The shape is still parabolic, but the vertex is moved from the vanishing amplitude point to an amplitude which can be chosen and adapted to the desired pendulum energy by suitably designing *D*, as shown in [15]. Period measurements, compared to the theory in **Figure 10b**, were taken on a 0.25-m-long pendulum with cylinders of 22 mm in diameter.

In fact, since at the apex of the parabola the two curves of **Figure 10b** show the same curvature, their local description is well known to be

$$\frac{\Delta\Gamma}{r\_{\text{max}}} - \frac{1}{\gamma\_6} \{\Delta\Theta\}^2 \tag{8}$$

avoids pulse timing and duration problems often encountered in the past by pendulum clock makers. The amplitude of such forcing term can be regulated in closed loop, by an automatic gain control (AGC) arrangement, to exactly compensate the dissipated power *P*d at the desired swing amplitude. This requires the average delivered mechanical power < *Fu* > to be

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

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

Since the energy **e** stored in the pendulum swing is proportional to the amplitude squared and *u* is linear with amplitude, it appears that the desired force is proportional to amplitude, as it might be intuitively expected. However, this is true only if *Q* is constant with amplitude, which turns out not to be the case for the adopted pendulum design. An analysis of what were felt to be the two main dissipation mechanisms for this structure was given in [1] and showed that in both cases the *Q* limitation is proportional to some power κ of the amplitude. In detail, periodic stretching of the wires under varying tension and their bending as they wrap and unwrap on the cylinders produce *Q* limitations which are inversely proportional to the square of the amplitude for the former (*Qs* ∝ *θ*−2, where *s* stands for stretching) and proportional to the amplitude's three-halves power for the latter (*Qb* ∝ *θ*3/2, where *b* stands for bending). Within that simplified theory, cyclic length variations of wires were overlooked, and only stretching under varying tension and bending on the cylinders were analyzed for small oscillations.

and features a maximum *Q* value at an angular swing amplitude *θ*max which can both be cal-

An example of such a *Q* dependence on amplitude is given in **Figure 11a** as calculated from Eq. (14) for a pendulum which could be suitable for the *G* experiment (L = 1 m and D = 4 mm), built with 4 μm Tungsten wires and a spherical 4.5 mm tungsten bob. The resulting peak force that is necessary to keep the bob swinging at the given amplitude according to Eq. (11) is shown in **Figure 11b**, where the strange effect appears that, in the branch before the mini-

A comprehensive campaign to confirm the theory in all conditions has not been completed yet as this book is going in press. In particular, *Q* values in excess of several millions were

mum, weaker forcing terms are needed to maintain greater amplitudes.

and *Qb*

is the static strain imposed on wires by the

is the intrinsic *Q* of the wire material. The

, respectively) were

Expressions obtained for the corresponding *Q* limitations (*Qs*

which shows that κ<sup>s</sup> = −2 and κ<sup>b</sup> = 3/2. Here ε0

total *Q* of the pendulum can then be obtained as

culated from Eqs. (12) through (14).

weight of the bob, *φ* is the wires' diameter, and *Qf*

(11)

65

(12)

(13)

(14)

which means that amplitude deviations Δ*θ* from the minimum period spot must not exceed 4 μ rad to keep the first term at the 10−12 level. On the other hand, a period-to-period amplitude reduction is unavoidable due to energy loss and is related to it by

$$\mathcal{L}\_{\frac{\Delta E}{\partial \mathbf{m} \text{str}}} = \mathcal{Z} \frac{\Delta \theta}{\theta\_{\text{mut}}} = \frac{2\pi}{Q} \tag{9}$$

which means that the minimum pendulum quality factor *Qm* necessary to keep the amplitude from decreasing more than the acceptable limit Δ*θ* in one period is

$$Q\_m > \pi \frac{\theta\_{m\psi n}}{\Delta \theta} \tag{10}$$

For example, if *θmin* = 0.075 rad, as in the case of **Figure 10b**, the quality factor must be greater than about 104 to keep the period (and the frequency) within 10−12 for one period, and a *Q* of 107 will keep the desired frequency stability for less than an hour at most. Luckily, because it's only the differential frequency stability that must be very stable, this requirement applies only to the difference of the two pendulum quality factors, provided they are both oscillating at the sweet amplitude spot. If it can be assumed that both quality factors are the same within say 10%, a *Q* of 107 would be enough to guarantee that the desired differential stability is kept for a full working day. This would be a long enough time for two full cycles of the repetition rate of the experiment if the system of field masses is kept in one position for a couple of hours and then moved to the other position for another couple of hours. Such is the situation for an experiment based on a pair of pendulums operated in free decay mode, and it could possibly be improved more if the two quality factors are within 1% of each other, in which case the experiment could go on for almost a week. Modeling out the effect may also be possible to some extent, as silently assumed in [1], and might further increase the useful duration of the experiment between periodic operations of amplitude reset, but this gets more complicated.

Alternatively, at the light of the experimented difficulty in obtaining consistently the extremely high *Q* values which are needed for the discussed reasons if the free decay mode must be adopted, a sustained oscillation approach can be tried for the two pendulums. In this perspective, a synchronous forcing term must be applied to the pendulum, designed to exactly recover the energy lost by friction. The best for stability and most efficient way of doing this would be a sine-wave force *F* applied in phase with the velocity *u* of the bob. This approach avoids pulse timing and duration problems often encountered in the past by pendulum clock makers. The amplitude of such forcing term can be regulated in closed loop, by an automatic gain control (AGC) arrangement, to exactly compensate the dissipated power *P*d at the desired swing amplitude. This requires the average delivered mechanical power < *Fu* > to be

The period versus amplitude curve is compared to that of a mathematical pendulum in **Figure 10b**. The shape is still parabolic, but the vertex is moved from the vanishing amplitude point to an amplitude which can be chosen and adapted to the desired pendulum energy by suitably designing *D*, as shown in [15]. Period measurements, compared to the theory in **Figure 10b**, were taken on a 0.25-m-long pendulum with cylinders of 22 mm in

In fact, since at the apex of the parabola the two curves of **Figure 10b** show the same curva-

which means that amplitude deviations Δ*θ* from the minimum period spot must not exceed 4 μ rad to keep the first term at the 10−12 level. On the other hand, a period-to-period amplitude

which means that the minimum pendulum quality factor *Qm* necessary to keep the amplitude

For example, if *θmin* = 0.075 rad, as in the case of **Figure 10b**, the quality factor must be greater

for a full working day. This would be a long enough time for two full cycles of the repetition rate of the experiment if the system of field masses is kept in one position for a couple of hours and then moved to the other position for another couple of hours. Such is the situation for an experiment based on a pair of pendulums operated in free decay mode, and it could possibly be improved more if the two quality factors are within 1% of each other, in which case the experiment could go on for almost a week. Modeling out the effect may also be possible to some extent, as silently assumed in [1], and might further increase the useful duration of the experiment between periodic operations of amplitude reset, but this gets more complicated. Alternatively, at the light of the experimented difficulty in obtaining consistently the extremely high *Q* values which are needed for the discussed reasons if the free decay mode must be adopted, a sustained oscillation approach can be tried for the two pendulums. In this perspective, a synchronous forcing term must be applied to the pendulum, designed to exactly recover the energy lost by friction. The best for stability and most efficient way of doing this would be a sine-wave force *F* applied in phase with the velocity *u* of the bob. This approach

 will keep the desired frequency stability for less than an hour at most. Luckily, because it's only the differential frequency stability that must be very stable, this requirement applies only to the difference of the two pendulum quality factors, provided they are both oscillating at the sweet amplitude spot. If it can be assumed that both quality factors are the same within

to keep the period (and the frequency) within 10−12 for one period, and a *Q* of

would be enough to guarantee that the desired differential stability is kept

(8)

(9)

(10)

diameter.

64 Metrology

than about 104

say 10%, a *Q* of 107

107

ture, their local description is well known to be

reduction is unavoidable due to energy loss and is related to it by

from decreasing more than the acceptable limit Δ*θ* in one period is

$$
\epsilon < Fu > \quad \mathcal{P}\_{\mathfrak{g}} \quad \frac{\omega \Theta}{\varrho} \;. \tag{11}
$$

Since the energy **e** stored in the pendulum swing is proportional to the amplitude squared and *u* is linear with amplitude, it appears that the desired force is proportional to amplitude, as it might be intuitively expected. However, this is true only if *Q* is constant with amplitude, which turns out not to be the case for the adopted pendulum design. An analysis of what were felt to be the two main dissipation mechanisms for this structure was given in [1] and showed that in both cases the *Q* limitation is proportional to some power κ of the amplitude. In detail, periodic stretching of the wires under varying tension and their bending as they wrap and unwrap on the cylinders produce *Q* limitations which are inversely proportional to the square of the amplitude for the former (*Qs* ∝ *θ*−2, where *s* stands for stretching) and proportional to the amplitude's three-halves power for the latter (*Qb* ∝ *θ*3/2, where *b* stands for bending). Within that simplified theory, cyclic length variations of wires were overlooked, and only stretching under varying tension and bending on the cylinders were analyzed for small oscillations. Expressions obtained for the corresponding *Q* limitations (*Qs* and *Qb* , respectively) were

$$\frac{Q\_{\xi}}{Q\_{I}} = \frac{16}{9\sqrt{e\_{\Phi}}\theta^{\prime}}\tag{12}$$

$$\frac{Q\_h}{Q\_I} = \left(\frac{L\mathcal{D}}{\Phi^\times} \upsilon\_0 \mathcal{O}\right)^{s\_{f\_2}^\prime},\tag{13}$$

which shows that κ<sup>s</sup> = −2 and κ<sup>b</sup> = 3/2. Here ε0 is the static strain imposed on wires by the weight of the bob, *φ* is the wires' diameter, and *Qf* is the intrinsic *Q* of the wire material. The total *Q* of the pendulum can then be obtained as

$$\frac{1}{Q} = \frac{\rho\_{dz} + \Gamma\_{\text{d0}}}{\omega \cdot \mathfrak{E}} = \frac{1}{Q\_{\mathfrak{s}}} \parallel \frac{1}{Q\_{\mathfrak{n}}} \; \; \; \tag{14}$$

and features a maximum *Q* value at an angular swing amplitude *θ*max which can both be calculated from Eqs. (12) through (14).

An example of such a *Q* dependence on amplitude is given in **Figure 11a** as calculated from Eq. (14) for a pendulum which could be suitable for the *G* experiment (L = 1 m and D = 4 mm), built with 4 μm Tungsten wires and a spherical 4.5 mm tungsten bob. The resulting peak force that is necessary to keep the bob swinging at the given amplitude according to Eq. (11) is shown in **Figure 11b**, where the strange effect appears that, in the branch before the minimum, weaker forcing terms are needed to maintain greater amplitudes.

A comprehensive campaign to confirm the theory in all conditions has not been completed yet as this book is going in press. In particular, *Q* values in excess of several millions were

cylinders. These values are much smaller than the one adopted in the simulation of **Figure 11** to force the position of the *Q* maximum. Clearly, full confirmation of the *Q* theory must be

Another detail which is obviously relevant to this effect is the material of the suspension

diameter *φ* of the wires, in its own way. Unfortunately, mechanical characterization of fibers is less than complete in the open literature, particularly for what concerns mechanical losses

apparatus [17] on promising candidates, mainly aiming at characterization of mechanical losses, creep, and linearity [18]. Para-aramid, SiC, basaltic, and carbon fibers were analyzed [19], as well as steel and tungsten metal wires. Glass and fused quartz are still waiting in line. No doubt, a final decision on this important item must be integrated with the whole design

Three more very important items must be considered in the design of the pendulum because they have an impact on the operation of the device, if not on its effectiveness in detecting the gravitational field modulation. They are the mode map of the pendulum, the oscillation detection system, and the excitation mechanism in case of forced oscillations' operation mode.

The first one may affect obtainable *Q* values and introduce fastidious coherent noise in the detection signal. In fact, if undesired oscillation modes get excited, albeit weakly, they can easily increase the effective total damping by sucking energy into dissipative mechanisms which do not belong to the main pendulum mode, lowering its *Q* as a consequence, and on the other hand, they force detection data processing to face spurious coherent signals which may reduce S/N ratio and ultimately affect resolution. Getting rid of spurious signals is impossible by the Nyquist theorem because of aliasing if the sampling frequency is not at least double the highest undesired mode frequency, which forces the handling of a massive amount of data in a full sine-wave detection system. The most difficult undesired modes to deal with, however, are the ones that are closest in frequency to the pendulum mode [20], because they are the ones that are most easily excited. In particular, the transverse mode, whose degeneracy with the pendulum mode is removed by the double-wire suspension structure, remains close to it

Other modes that should be focused on are the double pendulum mode and the similar balance wheel mode, which are more separated in frequency but are easily excited as soon as some imperfection appears in the suspension structure or in the excitation sys-

Given the boundary conditions emerging from the panorama spelled out here, care must be taken in designing and realizing excitation and detection systems for the two pendulums, to minimize the risk of getting undesired modes excited and affecting in this way damping and measurement resolution. Both optical and electromagnetic methods have been analyzed for both. All tested methods have their own advantages and problems, but all can serve the

and *ε*<sup>0</sup> in Eqs. (12) and (13) are material dependent, as well as the

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

, and hence on *φ*, while the

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

67

. Therefore, tests were carried out in the laboratory with a purposely built

carried out before the pendulum design can be finalized.

of the pendulum, as both analyzed loss mechanisms depend on ε0

bending loss, in particular, depends also explicitly on *φ*.

in frequency if the angle between wires is not too big.

wires, because both *Qf*

summarized by *Qf*

tem, if present.

purpose if well implemented.

**Figure 11.** (a) Q as a function of angular oscillation amplitude and (b) peak force value of the sinusoidal forcing term necessary to maintain the corresponding amplitude, calculated for a pendulum with 4-mm-diameter suspending cylinders and 1 m length, made with 4 μm Tungsten wires and a spherical 4.5 mm tungsten bob.

not observed yet in the limited range of configurations that were staged, which suggests that there may be other dissipation mechanisms worth studying, but it seems unlikely that a more complete analysis may not confirm the general shape of these curves. In fact, experimental results obtained by analyzing free decay ringdown amplitude data clearly show that such *Q* maximum exists. Further experiments are in progress, as well as the analysis of such additional dissipative phenomena as belt friction, squeeze film energy loss [16], and more trivially dissipation in the structure holding the experimental arrangement.

What must be underlined here is that, due to the *Q* behavior shown in **Figure 11**, Eq. (11) points to a criticality for AGC stability, because amplitude stabilization cannot be reached if increasing the amplitude requires a reduction of the forcing term. The derivative of the required force with respect to amplitude must be positive for AGC stability, which imposes the selection of a swing amplitude in a region where the dominant dissipation mechanism is such that κ < 1. If κ > 1 the AGC will be unstable, and if κ = 1 it will not be effective because the necessary force does not depend on amplitude. Conversely, given a desired swing amplitude, as dictated, for example, by the range of acceptable field uniformity of **Figure 7**, the design of pendulum suspensions should observe the specification of placing the desired amplitude in a range where AGC stability is guaranteed.

The best choice in this respect appears to be a design which positions the *θ*max at the desired oscillation amplitude, which is what was tried in the simulation of **Figure 11**, where the amplitude of maximum *Q* was made to correspond at 10 mm with a bob peak excursion which can be judged desirable from the calculated effect uniformity shown in **Figure 7**. However, it must be pointed out here that this design problem is still open because also the minimum of the period, as illustrated in **Figure 8**, must be placed by design at the same oscillation amplitude, which implies a tight restriction on the acceptable values of *D*, the diameter of suspension cylinders. These values are much smaller than the one adopted in the simulation of **Figure 11** to force the position of the *Q* maximum. Clearly, full confirmation of the *Q* theory must be carried out before the pendulum design can be finalized.

Another detail which is obviously relevant to this effect is the material of the suspension wires, because both *Qf* and *ε*<sup>0</sup> in Eqs. (12) and (13) are material dependent, as well as the diameter *φ* of the wires, in its own way. Unfortunately, mechanical characterization of fibers is less than complete in the open literature, particularly for what concerns mechanical losses summarized by *Qf* . Therefore, tests were carried out in the laboratory with a purposely built apparatus [17] on promising candidates, mainly aiming at characterization of mechanical losses, creep, and linearity [18]. Para-aramid, SiC, basaltic, and carbon fibers were analyzed [19], as well as steel and tungsten metal wires. Glass and fused quartz are still waiting in line. No doubt, a final decision on this important item must be integrated with the whole design of the pendulum, as both analyzed loss mechanisms depend on ε0 , and hence on *φ*, while the bending loss, in particular, depends also explicitly on *φ*.

Three more very important items must be considered in the design of the pendulum because they have an impact on the operation of the device, if not on its effectiveness in detecting the gravitational field modulation. They are the mode map of the pendulum, the oscillation detection system, and the excitation mechanism in case of forced oscillations' operation mode.

not observed yet in the limited range of configurations that were staged, which suggests that there may be other dissipation mechanisms worth studying, but it seems unlikely that a more complete analysis may not confirm the general shape of these curves. In fact, experimental results obtained by analyzing free decay ringdown amplitude data clearly show that such *Q* maximum exists. Further experiments are in progress, as well as the analysis of such additional dissipative phenomena as belt friction, squeeze film energy loss [16], and more trivially

**Figure 11.** (a) Q as a function of angular oscillation amplitude and (b) peak force value of the sinusoidal forcing term necessary to maintain the corresponding amplitude, calculated for a pendulum with 4-mm-diameter suspending

What must be underlined here is that, due to the *Q* behavior shown in **Figure 11**, Eq. (11) points to a criticality for AGC stability, because amplitude stabilization cannot be reached if increasing the amplitude requires a reduction of the forcing term. The derivative of the required force with respect to amplitude must be positive for AGC stability, which imposes the selection of a swing amplitude in a region where the dominant dissipation mechanism is such that κ < 1. If κ > 1 the AGC will be unstable, and if κ = 1 it will not be effective because the necessary force does not depend on amplitude. Conversely, given a desired swing amplitude, as dictated, for example, by the range of acceptable field uniformity of **Figure 7**, the design of pendulum suspensions should observe the specification of placing the desired amplitude in a

The best choice in this respect appears to be a design which positions the *θ*max at the desired oscillation amplitude, which is what was tried in the simulation of **Figure 11**, where the amplitude of maximum *Q* was made to correspond at 10 mm with a bob peak excursion which can be judged desirable from the calculated effect uniformity shown in **Figure 7**. However, it must be pointed out here that this design problem is still open because also the minimum of the period, as illustrated in **Figure 8**, must be placed by design at the same oscillation amplitude, which implies a tight restriction on the acceptable values of *D*, the diameter of suspension

dissipation in the structure holding the experimental arrangement.

cylinders and 1 m length, made with 4 μm Tungsten wires and a spherical 4.5 mm tungsten bob.

range where AGC stability is guaranteed.

66 Metrology

The first one may affect obtainable *Q* values and introduce fastidious coherent noise in the detection signal. In fact, if undesired oscillation modes get excited, albeit weakly, they can easily increase the effective total damping by sucking energy into dissipative mechanisms which do not belong to the main pendulum mode, lowering its *Q* as a consequence, and on the other hand, they force detection data processing to face spurious coherent signals which may reduce S/N ratio and ultimately affect resolution. Getting rid of spurious signals is impossible by the Nyquist theorem because of aliasing if the sampling frequency is not at least double the highest undesired mode frequency, which forces the handling of a massive amount of data in a full sine-wave detection system. The most difficult undesired modes to deal with, however, are the ones that are closest in frequency to the pendulum mode [20], because they are the ones that are most easily excited. In particular, the transverse mode, whose degeneracy with the pendulum mode is removed by the double-wire suspension structure, remains close to it in frequency if the angle between wires is not too big.

Other modes that should be focused on are the double pendulum mode and the similar balance wheel mode, which are more separated in frequency but are easily excited as soon as some imperfection appears in the suspension structure or in the excitation system, if present.

Given the boundary conditions emerging from the panorama spelled out here, care must be taken in designing and realizing excitation and detection systems for the two pendulums, to minimize the risk of getting undesired modes excited and affecting in this way damping and measurement resolution. Both optical and electromagnetic methods have been analyzed for both. All tested methods have their own advantages and problems, but all can serve the purpose if well implemented.
