# Flexures for Kibble balances: Minimizing the effects of anelastic relaxation

Lorenz Keck1,2, Stephan Schlamminger1, René Theska2, Frank Seifert3, Darine
Haddad1 1 National Institute of Standards and Technology 100 Bureau Drive,
Gaithersburg, MD, USA  2 Precision Engineering Group, Technische Universität
Ilmenau, 98693 Ilmenau, Germany  3 University of Maryland, Joint Quantum
Institute, College Park, MD 20742, USA<EMAIL_ADDRESS><EMAIL_ADDRESS>
(September 2023)

###### Abstract

Modern weighing cells would not be possible without flexure technology.
Flexures are created by thinning metal in a specific shape so that it bends as
desired when an external torque is applied. In contrast to other pivots, such
as knife edges and different types of bearings, flexures provide rotary motion
without stick-slip friction. Flexures, however, suffer from anelastic effects,
most importantly a time-dependent torque that slowly decays after a large
excursion. This effect is known as anelastic relaxation. In traditional servo-
controlled mass comparators, the range of motion is minimal, and, hence,
anelastic relaxation is negligible. Employing the same technology in Kibble
balances challenges that assumption because the necessary velocity mode
requires a relatively large excursion of the balance.

Here, we present measurements of the anelastic relaxation that would occur
after excursions held for tens of minutes of time in a balance beam in closed-
loop operation. Procedures to mitigate the effect of spurious torque from
anelastic effects are discussed. We found a static shift of the flexure’s
internal equilibrium after a change in the stress and strain state. These
static shifts, although measurable, are small, and are uncritical in the
Kibble balance application at present.

Keywords: Maxwell model, generalized Maxwell model, relaxation, Kibble
balance, precision measurement, flexures

##  1 Introduction

Work led by T.J. Quinn, C.C. Speake, and colleagues at the International
Bureau of Weights and Measures (BIPM) in the late 1980s and early 1990s
popularized flexures for the use in weighing instruments [1, 2]. The team at
the BIPM showed that flexures [3] are superior over knife edges, the
traditional choice for the main and auxiliary pivots in beam balances and mass
comparators. Two planes come together in a single line to form a knife edge.
However, zooming in to a cross-sectional view, the meeting point is not a
sharp corner but rather a curve with a finite radius [4]. Hence, as the knife
edge moves, it effectively rolls. The rolling motion subtly alters the lever-
arm ratio of the beam, ultimately limiting the performance of the balance.
Replacing the central knife with a flexure prevents this lever arm change, as
the flexure can be designed to have a well-defined rotation axis. A drawback
of flexures, which Quinn, Speake, and coworkers were aware of and have
invested thoroughly, is the anelastic relaxation, which is also known as the
anelastic aftereffect.

The simplest model of a flexure can be thought of as a perfectly elastic
spring, in parallel with a lossy spring, usually described as a spring in
series with a viscous element, referred to as a dashpot, see Fig. 1. The
combination of a spring and a dashpot forms what is known as a Maxwell unit,
and when this Maxwell unit is combined in parallel with an ideal spring, it
constitutes the Maxwell model in material science. A material obeying the
systematic of the Maxwell model is called a Maxwell solid. When a sudden
strain is applied at $t=0$ to a Maxwell solid, a significant fraction of the
stress response is immediate, but the Maxwell unit contributes an
exponentially decaying stress with a time constant that is given by the ratio
of viscosity to the elastic modulus of the Maxwell unit. In mass comparators,
this anelastic after-effect can be kept small by avoiding large spring
excursions. In general, this can be achieved by limiting its mobility with
stops and by careful mass placement to avoid sudden excursions.

For Kibble balances, however, the story is different — large motions cannot be
avoided, because two measurement modes are required: force and velocity mode
[5]. The former is similar to weighing with a mass comparator, with the
difference that the weight of a test mass is compared directly to a magnetic
force produced by a coil. In the latter, the current-to-force coefficient of
the coil is determined by sweeping the coil through the magnetic field while
simultaneously measuring induced voltage and the coil’s velocity.

While it is not necessary to use the same mechanism for moving the coil and
weighing the mass, it is advantageous to do so. This observation was made by
Kibble and Robinson [6] and shall be abbreviated as Kibble-Robinson theory
(KRT).

Interestingly, the first two Kibble balances ever built [7, 8], one at the
National Physical Laboratory (NPL) in the United Kingdom, the other at the
National Institute of Standards and Technology (NIST) in the United States of
America, followed one of the fundamental principles of KRT, i.e., the same
mechanism for weighing and moving before the principle was known as such. The
balances at NPL and NIST used a beam and a wheel, respectively. Both were
supported by a central knife-edge since large motions were necessary. In both
experiments, the vertical travel of the coil was several centimeters,
requiring rotation about the central pivot of several degrees, and flexures
were not deemed suitable for this application. Although flexure-based weighing
cells have been used in Kibble balances, most of these designs ignore KRT,
because a different mechanism is used to move the coil than to perform the
precise weighing. Examples include the Kibble balances built in Switzerland,
Korea, and at the BIPM, [9, 10, 11]. Not taking advantage of KRT may come at a
performance cost — a balance that outperforms the original NPL balance
designed by Kibble and Robinson still needs to be built. Additionally, a
larger measurement uncertainty is not the only penalty for these designs. The
requirement for multiple mechanisms increases the mechanical complexity and
escalates the engineering, manufacturing, and operating costs.

A flexure-based Kibble balance that obeys KRT is the panacea for this field.
Hence, it’s not surprising that there is renewed interest in this topic.
Researchers at the Physikalische Technische Bundesanstalt [12], NPL [13], and
NIST [14] are actively working on such systems, and the BIPM Kibble balance
[15] is switching to one flexure mechanism for moving and weighing, as well.
Hence, it is time to revisit the anelastic relaxation of flexures and the
applicability and limitations in this use case.

After exercising the mechanism for the large deflections required for velocity
modes, anelastic drift from the flexures [2] limits the accuracy obtainable
from weighing in typical ABA weighing schemes (A= mass off, B= mass on).
Ideally, this effect needs to be characterized and compensated for to allow
for accuracy in weighing and systematic studies. An erasing or compensation
procedure shall be established to provide the means for the correct operation
of such mechanisms. Erasing procedures to reduce the influence of mechanical
hysteresis are well-known to researchers familiar with the Kibble balance. For
example, erasing procedures are needed in knife-edge balances, where plastic
deformation is the leading cause of hysteresis. The procedure is typically a
sinusoidal motion with decaying amplitude and is used to reduce errors caused
by disturbances during mass exchange [16, 17]. Such procedures, especially
when necessary periodically after each mass exchange, as in knife edge
balances, are very time-consuming and thus unfavorable.

In the following three sections, we present fundamental investigations into
the anelasticity of the flexure mechanism for the new Kibble balance at NIST
[14]. The theoretical work can be seen as the relevant extension of existing
models of anelasticity at low frequencies to describe the modes of operation
in the Kibble balance. These models are presented in connection with
experimental results from a working prototype of the new mechanism.
Measurements of the relaxation force from the flexures in servo-controlled
position feedback after defined disturbance are shown.

##  2 Review of anelasticity in the literature

Anelasticity and anelastic relaxation in polycrystalline metals appeared in
the literature in the late 19th century, notably by Boltzmann (1874) [18] and
Wiechert (1893) [19]. Wiechert seems to be the first to employ a generalized
form of relaxation model to metals to describe an anelastic solid. His model
consists of an ideal spring in parallel with multiple parallel Maxwell units.
This model, which shall be called generalized Maxwell model throughout this
manuscript, was later used in configurations of a finite number, $n$, of
parallel Maxwell units, e.g., in works by Nowick and Berry (1972) [20] or by
Beilby et al. (1998) [21] up to a continuum of relaxation stages and in work
by Quinn et al. (1992) to model low-frequency anelastic relaxation of
polycristalline metals [22].

One hundred years after Boltzmann and Wiechert, anelasticity and the search
for materials with low internal loss garnered new interest driven by precision
measurements, such as beam balances for mass comparison [4], torsion balances
for accurate determinations of the Newtonian gravitational constant $G$ [23],
and thermal noise limited test mass suspensions for gravitational wave
antennas [24]. Callen et al. [25] formulated the fluctuation-dissipation
theorem linking internal damping to thermal or Brownian noise in mechanical
suspensions and counts as one of the fundamental limits to the sensitivity of
high precision mechanical detectors [24]. In the context of $G$, Kuroda [26]
pointed out a possible measurement bias due to the anelastic effect in the
time-of-swing method.

Experiments to measure anelastic damping can be classified into dynamic and
static approaches. The former observes the ringdown of a free oscillation at
the resonance frequency of the mode under investigation using the logarithmic
decrement or the mechanical quality factor $Q$ to quantify the modulus defect
in a material directly [27, 28]. Using an inverted pendulum, for example,
anelastic effects in the material can be shown very clearly and investigated
for various frequencies [29, 30]. For these experiments, other sources of
damping, for example, gas pressure forces, [31] have to be reduced as much as
possible by performing these measurements in high or ultra-high vacuum
conditions.

Beilby et al. [21] presented an interesting method to characterize open-loop
static relaxation based on photoelastic measurements from optical
birefringence of glass ceramics under a change in stress. The anelastic
aftereffect was measured in a static experiment in optical borosilicate-crown
glass (BK7) and fused silica after a sharp increase in stress had been applied
and released. Beilby’s report also includes dynamic measurements by exciting
the eigenmode.

Quinn et al. and Speake et al. measured anelastic properties in dynamic
ringdown experiments and static aftereffect relaxation in closed-loop null
measurement with particular interest in metal flexures, especially copper-
beryllium alloy C17200 with temper TF00 as mechanical suspension for beam and
torsion balances [22, 27, 28, 23]. Several practical aspects of their work
provide important foundations for the work. Also, we use a material of similar
metallurgic composition and treatment. Furthermore, Speake and Quinn noticed
that in such polycrystalline metal flexures of minimal notch thickness down to
at least $50\text{\,}\mathrm{\SIUnitSymbolMicro m}$, damping mainly results
from bulk effects and not dominance of surface effects [22], which contribute
more relevantly to high $Q$ materials such as sapphire or glass ceramics [32,
33] in fibers with large surface-to-volume ratio.

##  3 Physics of the Maxwell model

Figure 1: The Maxwell model. An ideal spring with an elastic modulus $E_{o}$
is parallel to a Maxwell unit. The Maxwell unit consists of an ideal spring
with elastic modulus $E_{1}$ in series with a viscous element. The viscous
element known as damper or dashpot produces a velocity-dependent stress. The
proportionality factor is given by the material coefficient of viscosity,
$\nu_{1}$. The $\nu_{1}$ and $E_{1}$ quotients yield a time constant
$\tau_{1}$. The stress $\sigma$ is applied in the vertical direction producing
a strain, $\varepsilon$, denoting the relative length extension from the
original spring length $L$.

In this section, we review the physics of the Maxwell model, see Fig. 1. The
stress of the damper, $\sigma=\nu_{1}\dot{\varepsilon}$, depends
proportionally on the derivative of the strain $\varepsilon$. The
proportionality constant is given by the material coefficient of viscosity,
$\nu_{1}$. The quotient, $\nu_{1}/E_{1}$ has dimension time and is abbreviated
as $\tau_{1}$. For the Maxwell unit, the strain is the sum of the strain
produced by the spring and the damper. This is also true for the derivative of
the strain. Hence, we obtain for the Maxwell unit,

$\dot{\varepsilon}=\frac{\dot{\sigma}_{\mathrm{MU}}}{E_{1}}+\frac{\sigma_{\mathrm{MU}}}{\nu_{1}}.$
(1)

Performing the Laplace transformation yields,

$s\breve{\varepsilon}=\frac{1}{E_{1}}s\breve{\sigma}_{\mathrm{MU}}+\frac{1}{\nu_{1}}\breve{\sigma}_{\mathrm{MU}}.$
(2)

We use the accent $\breve{}\;$ to denote the Laplace transform of the
corresponding variable.

Adding the stress contributed by the ideal spring, $\sigma=E_{o}\varepsilon$
to Eq. (2) yields the response function of the Maxwell model

$\breve{r}=\frac{\breve{\sigma}}{\breve{\varepsilon}}=\frac{(E_{o}+E_{1})s\tau_{1}+E_{o}}{s\tau_{1}+1}$
(3)

###  3.1 Response to two simple stimuli

The response function, Eq. 3, allows us to calculate the time domain response,
$\sigma(t)$ of any stimulus $\varepsilon(t)$, by calculating the inverse
Laplace transform of $\breve{r}\breve{\varepsilon}$. In the next few sections,
we investigate different stimuli. The total duration of all stimuli is
$\tau_{s}$. The first stimulus (subscript 1) that we investigate is, in the
time domain, of the form

$\varepsilon_{1}(t)=\left\\{\begin{array}[]{ll}0,&\mathrm{if}\;t<0\\\
\varepsilon_{a},&\mathrm{if}\;0\leq t<\tau_{s}\\\
0,&\mathrm{if}\;t\geq\tau_{s}\\\ \end{array}\right.$ (4)

In the Laplace domain, the above stimulus is

$\breve{\varepsilon}_{1}=\varepsilon_{a}\left(\frac{1}{s}-\frac{e^{-s\tau_{s}}}{s}\right).$
(5)

The inverse Laplace transform of the product $\breve{r}\breve{\varepsilon}$
for $t\geq\tau_{s}$ yields the stress response. It is

$\sigma_{1}(t)=-\varepsilon_{a}E_{1}e^{-t/\tau_{1}}\left(-1+e^{\tau_{s}/\tau_{1}}\right).$
(6)

The desired outcome of the calculation is the stress after a strain stimulus,
which can be obtained by shifting the time axis using $t=\tau_{s}+t^{\ast}$.
Then, the stress is given in the new time variable as

$\sigma_{1}(t^{\ast})=\sigma_{a1}e^{\displaystyle-t^{\ast}/\tau_{1}},\;\mbox{with}\;$
(7)
$\sigma_{a1}=-\varepsilon_{a}E_{1}\big{(}1-e^{\displaystyle-\tau_{s}/\tau_{1}}\big{)}$
(8)

consistent with Eq. (2) in [22]. The imprinted stress decays with time
constant $\tau_{1}$. The absolute initial amplitude, $|\sigma_{a1}|$, is
larger as $\tau_{s}$ is longer and converges to $-\varepsilon_{a}E_{1}$ for
$\tau_{s}\rightarrow\infty$. The top panel of Fig. 2 shows the development of
the stress as a function of time during and after the application of stimulus
one. After the stimulus, the stress decays from a negative value to zero with
a time constant $\tau_{1}$. The negative value appears as the strain is
suddenly released, see $t^{\ast}=0$ in Fig. 2.

Figure 2: Development of the stress (blue dashed line) for two strain stimuli.
The top panel is for a static deflection that was held for time $\tau_{S}$.
The bottom is a sinusoidal stimulus. At the same time, one period with
identical strain amplitude was applied to the flexure. The region of interest
is for $t^{\ast}>0$, which is shown in the insets. Note that the vertical axes
of the insets have the same extent, showing that
$|\sigma_{a2}|<|\sigma_{a1}|/2$.

We are not interested in a static deflection for the Kibble balance but
instead, a sinusoidal deflection as the coil moves up and down in the magnetic
field. To make the result comparable to the result above, we model a second
(subscript 2) strain impulse as

$\varepsilon_{2}(t)=\left\\{\begin{array}[]{ll}0,&\mathrm{if}\;t<0\\\
\varepsilon_{a}\sin{(2\pi nt/\tau_{s})},&\mathrm{if}\;0\leq t<\tau_{s}\\\
0,&\mathrm{if}\;t\geq\tau_{s}.\\\ \end{array}\right.$ (9)

Here $n$ is the number of integer periods the mechanism moves in a time
interval $\tau_{S}$. The motion given by Eq. 9 starts and ends at
$\varepsilon_{2}=0$, which is a reasonable assumption for the Kibble
experiment, where the velocity measurement is usually sandwiched in between
two force mode measurements for which the mechanism is controlled to the
equilibrium position, $\varepsilon=0$. In the $s$ domain, that impulse is

$\breve{\varepsilon}_{2}=\varepsilon_{a}\frac{2\pi
n/\tau_{s}}{s^{2}+4\pi^{2}n^{2}/\tau_{s}^{2}}\bigg{(}1-\exp{(-s\tau_{s})}\bigg{)}.$
(10)

The resulting stress for $t^{\ast}>0$ still relaxes as
$\sigma_{2}(t^{\ast})=\sigma_{a2}\exp{(-t^{\ast}/\tau_{1})}$, but the
amplitude is given by

$\sigma_{a2}=\varepsilon_{a}E_{1}\frac{2\pi
n\tau_{1}\tau_{s}}{\tau_{s}^{2}+4\pi
n^{2}\tau_{1}^{2}}\big{(}1-e^{\displaystyle-\tau_{s}/\tau_{1}}\big{)}=-\xi\sigma_{a1}.$
(11)

The lower panel of Fig. 2 shows the stress as a function of time for a
stimulus with $n=1$. It can be seen that, for the same signal duration, the
absolute value of the stress at $t^{\ast}$ is smaller than the one for a
static deflection. The sign of the stress is also different from the constant
excursion.

The attenuation of imprinted stress of a sinusoidal motion relative to a
constant excursion with the same amplitude is captured by the unitless ratio
$\xi$, given by the negative ratio of $\sigma_{a2}$ to $\sigma_{a1}$, which
computes to

$\xi=\frac{2\pi n\tau_{1}\tau_{s}}{\tau_{s}^{2}+4\pi^{2}n^{2}\tau_{1}^{2}}.$
(12)

We note that

$\xi\leq\frac{1}{2},$ (13)

which makes intuitive sense, because only half the time is spent straining the
flexure to either side. The maximum, $\xi=1/2$ is reached at $\tau_{s}=2\pi
n\tau_{1}$.

Figure 3 shows $\xi$ as a function of $\tau_{s}/\tau_{1}$ for different
integer numbers, $n$, of periods in the signal duration $\tau_{s}$. The maxima
are clearly visible. On either side of the maxima, $\xi$ drops like
$\tau_{s}/\tau_{1}$ or its inverse.

The consequence of this observation is that the sinusoidal motion attenuates
the relaxation effects of the Maxwell units with $\tau_{1}>>\tau_{s}/(2\pi n)$
and $\tau_{1}<<\tau_{s}/(2\pi n)$. Hence, in the Kibble balance with
sinusoidal coil motion, the Maxwell units of concern (largest $\xi$) have a
ratio $\tau_{s}/\tau_{1}$ of order $2\pi n$. Fortunately, as discussed in the
next section, the elastic energy stored in these units can be erased most
effectively.

Figure 3: The attenuation factor $\xi$ shows the ratio of the stress amplitude
after a sinusoidal motion versus a constant excursion of the same amplitude.
The value $\xi$ is plotted for different ratios of signal duration $\tau_{s}$
to relaxation time $\tau_{1}$. The integer $n$ shows how many complete periods
are in the signal duration.

###  3.2 The boxcar eraser

The erasing procedure is a prescribed motion that is performed after the
initial stimulus with the goal of minimizing the stress after both motions are
completed. We assume that the duration of the stimulus and erasing procedure
is $\tau_{s}$ and $\tau_{r}$, respectively. For the rectangular stimulus, an
erasing procedure can be found such that $\sigma(t)=0$ for
$t>\tau_{s}+\tau_{r}$. We define a new stimulus in the time domain, labeled
$\tilde{1}$ (tilde to show the case with erasing). In the time domain, it has
the following functional form:

$\varepsilon_{\tilde{1}}(t)=\left\\{\begin{array}[]{ll}0,&\mathrm{if}\;t<0\\\
\varepsilon_{a},&\mathrm{if}\;0\leq t<\tau_{s}\\\
\varepsilon_{r1},&\mathrm{if}\;\tau_{s}\geq t<\tau_{s}+\tau_{r}\\\
0,&\mathrm{if}\;t\geq\tau_{s}+\tau_{r}\\\ \end{array}\right.$ (14)

The strain amplitude $\varepsilon_{r1}$ is the erasing amplitude, which
depends on the erasing time $\tau_{r}$. The dependence can be found as
follows. The Laplace transformation of Eq. (14) is

$\breve{\varepsilon}_{\tilde{1}}=\frac{\varepsilon_{a}}{s}\left(1-e^{-s\tau_{s}}\right)+\frac{\varepsilon_{r1}}{s}\left(e^{-s\tau_{s}}-e^{-s(\tau_{s}+\tau_{r})}\right).$
(15)

From the inverse Laplace transform of the product of stimulus with response
function, we obtain, after shifting the time axis, this time by
$t^{\ast}=t-\tau_{s}-\tau_{r}$ again an exponential decay, identical to Eq.
(7), although with an amplitude

$\displaystyle\sigma_{a\tilde{1}}$ $\displaystyle=$
$\displaystyle-\varepsilon_{a}E_{1}e^{\displaystyle-(\tau_{s}+\tau_{r})/\tau_{1}}\big{(}-1+e^{\displaystyle-\tau_{s}/\tau_{1}}\big{)}$
(17)
$\displaystyle+\varepsilon_{r1}E_{1}\big{(}-1+e^{\displaystyle-\tau_{r}/\tau_{1}}\big{)}.$

This amplitude can be made zero by choosing

$\varepsilon_{r1}=\varepsilon_{a}\frac{1-e^{\displaystyle-\tau_{s}/\tau_{1}}}{1-e^{\displaystyle\tau_{r}/\tau_{1}}}.$
(18)

The absolute value of the ratio of $\varepsilon_{r1}$ to $\varepsilon_{a}$ is
shown in Fig. 4. In the limit $\tau_{1}\rightarrow\infty$, we obtain

$\lim_{\tau_{1}\rightarrow\infty}\varepsilon_{r1}=-\varepsilon_{a}\frac{\tau_{s}}{\tau_{r}}.$
(19)

The erasing amplitude is inversely proportional to the ratio of the erasing
time to stimulus time. In other words, in this case, the relaxation can be
erased with an opposite but otherwise identical signal to the stimulus. For
shorter erasing procedures the erasing amplitude needs to be scaled up
correspondingly.

Figure 4: The absolute value of the ratio of $\varepsilon_{r1}$ to
$\varepsilon_{a}$ as a function of erasing time to stimulus time for different
time constants $\tau_{1}$. For a sinusoidal stimulus the vertical axis of the
graph has to be multiplied by $\xi$ given by Eq. (12).

Another special case of Eq. (18) to consider is $\tau_{r}=\tau_{s}$. Then,

$\varepsilon_{r1}=-\varepsilon_{a}e^{\displaystyle-\tau_{s}/\tau_{1}}.$ (20)

This equation says that the erasing stimulus needs to be opposite to the
original stimulus, and the size is identical to how much the Maxwell unit has
been charged after $\tau_{s}$. For shorter times, $\tau_{r}$, the absolute
magnitude of the erasing amplitude becomes larger and vice versa.
Experimentally, both extremes are of limited utility. (1) There is no point in
using erasing times much longer than $3\tau_{1}$, because a similar effect can
be obtained by just waiting for the decay according to Eq. 7. (2) Short
erasing times require large motion amplitudes that may not be available
experimentally.

From the discussion above and Fig. 4 it becomes apparent that it is much
harder to erase Maxwell units with large time constants.

The same formalism can be applied to the sinusoidal stimulus, and not
surprisingly, the results are similar, and the erasing amplitude is scaled by
$-\xi$ i.e.,

$\varepsilon_{r2}=-\xi\varepsilon_{r1}.$ (21)

Figure 5: Development of the stress (blue dashed line) for the same two
stimuli shown in Fig. 2. This time, an erasing pulse of duration $\tau_{r}$
follows immediately after the stimulus. Here, the erasing pulse lasts a
quarter of the original stimulus, $\tau_{r}=\tau_{s}/4$. A zoomed plot of the
strain for $t^{\ast}>0$ is not necessary because then the stress produced by
the flexure is zero.

##  4 The generalized Maxwell model

Figure 6: The generalized Maxwell model. The main element is perfectly elastic
with a modulus $E_{o}$. In parallel to it is an infinite number of Maxwell
units, each with a spring of elastic modulus $E_{1}$, but different viscosity
values, $\nu_{j}$. Hence, each unit has a different time constant
$\tau_{j}=\nu_{j}/E_{1}$ for $j=1,2,\cdots,\infty$.

Previously, we discussed the stress relaxation behavior for the Maxwell model.
While this model is conducive to building intuition and deriving simple
equations, it falls short of reproducing the behavior of a realistic material.
A better model is an ideal spring in series with many Maxwell units having a
distribution of time constants as is shown in Fig. 6. This model is known as
the generalized Maxwell model. We follow in the footsteps of Speake and Quinn,
who have discussed such models in the literature [2, 23, 34].

Figure 7:  Bode plot of the anelastic part of the response function, given by
$\rho:=(\frac{\breve{\sigma}}{\breve{\varepsilon}}-E_{0})/E_{1}$. The solid
line gives the response function for a generalized Maxwell model with time
constants distributed between $\tau_{l}=$10\text{\,}\mathrm{s}$$ and
$\tau_{u}=$5000\text{\,}\mathrm{s}$$, see Eq. (26). The dashed-dotted line
shows the response function for the Maxwell model with a time constant that
corresponds to the geometric mean of the limits of the distribution, i.e.,
$\tau_{1}=\sqrt{\tau_{l}\tau_{u}}$. The inset in the top panel gives the
Nyquist plot of the two responses. The response of the generalized Maxwell
model is vertically compressed compared to that of the Maxwell model. The
lower plot shows the imaginary part of the response normalized by $E_{1}$.

First, let’s assume we have $n$ discrete sets of Maxwell units, each with an
elastic modulus $E_{1}$ and a viscosity $\nu_{j}$, yielding a time constant
$\tau_{j}=\nu_{j}/E_{1}$. Hence, the response function is given by

$\frac{\breve{\sigma}}{\breve{\varepsilon}}=E_{o}+E_{1}\sum_{j=1}^{n}\frac{s\tau_{j}}{s\tau_{j}+1}.$
(22)

By replacing the sum with an integral with lower limit $\tau_{l}$ and upper
limit $\tau_{u}$ and using a distribution function $f(\tau)$, we obtain

$\frac{\breve{\sigma}}{\breve{\varepsilon}}=E_{o}+E_{1}\int_{\tau_{l}}^{\tau_{u}}f(\tau)\frac{s\tau}{s\tau+1}\mathrm{d}\tau$
(23)

Note that the distribution function is normalized such that

$\int_{\tau_{l}}^{\tau_{u}}f(\tau)\mathrm{d}\tau=1.$ (24)

Following [2], a good candidate is

$f(\tau)=\frac{1}{\ln{\left(\tau_{u}/\tau_{l}\right)}}\frac{1}{\tau}.$ (25)

With this distribution function, the density of Maxwell units decreases for
longer time constants – a reasonable assumption that is found in many
mechanisms that produce $1/f$ noise. The response function simplifies to

$\frac{\breve{\sigma}}{\breve{\varepsilon}}=E_{o}+E_{1}\frac{\ln{\left(1+\tau_{u}s\right)}-\ln{\left(1+\tau_{l}s\right)}}{\ln{\left(\tau_{u}/\tau_{l}\right)}}.$
(26)

The second summand of the equation above is shown as a Bode Plot in Fig. 7.
For comparison, the equivalent term for the Maxwell model with an effective
time constant of $\sqrt{\tau_{u}\tau_{l}}$ is shown also. Qualitatively, both
Bode plots look similar and show high pass behavior. However, in the region of
interest, for angular frequencies between $1/\tau_{u}$ and $1/\tau_{l}$ the
curves differ. More importantly, the generalized Maxwell model has about
$10\text{\,}\mathrm{dB}$ more gain for low frequencies.

The quotient $\frac{\breve{\sigma}}{\breve{\varepsilon}}$ describes the
transfer function of strain to stress (the modulus). The imaginary part is the
relevant portion responsible for the anelastic effect. The literature [2, 34]
introduces the relative modulus defect as the imaginary part of the modulus of
the generalized Maxwell model, which we will abbreviate with $\eta$, as

$\eta=\frac{\displaystyle\mathrm{Im}\left(E_{1}\int_{\tau_{l}}^{\tau_{u}}f(\tau)\frac{s\tau}{s\tau+1}\mathrm{d}\tau\right)}{E_{o}}.$
(27)

Using $s=i\omega$ with $i$ being the imaginary unit, the value of the modulus
defect at the geometric middle of the angular frequency range
$\omega=(\tau_{l}\tau_{u})^{-1/2}$ can be found to be

$\eta\approx\frac{\Delta E}{E_{o}}\;\;\mbox{with}\;\;\Delta
E:=\frac{\pi}{2\ln\left(\frac{\displaystyle\tau_{u}}{\displaystyle\tau_{l}}\right)}E_{1}.$
(28)

As is indicated in Fig. 7 the module defect stays nearly constant at that
value over a wide frequency range. A frequency-independent modulus defect at
the level given by Eq. 28 is assumed in the literature [22, 23, 34] and we
continue this tradition.

Reasonable assumptions for $\tau_{l}$ and $\tau_{u}$ are
$10\text{\,}\mathrm{s}$ and $5000\text{\,}\mathrm{s}$, respectively. With
these time constants, $\pi/(2\ln{\left(\tau_{u}/\tau_{l}\right)})$ evaluates
to 0.253, which is given by the green dotted line in the lower panel of Fig.
7.

With the response function at hand, the anelastic relaxation after a
rectangular strain given by Eq. (4) is applied can be found. Substituting the
time scale, $t=\tau_{s}+t^{\ast}$ such that $t^{\ast}=0$ after the applied
displacement, the stress is

$\sigma_{1}(t^{\ast})=\frac{\displaystyle\varepsilon_{a}E_{1}}{\ln\left(\frac{\displaystyle\tau_{l}}{\displaystyle\tau_{u}}\right)}\Bigg{(}\text{Ei}\left(-\frac{t^{\ast}}{\tau_{l}}\right)-\text{Ei}\left(-\frac{t^{\ast}}{\tau_{u}}\right)\Bigg{.}\\\
\Bigg{.}-\text{Ei}\left(-\frac{t^{\ast}+\tau_{s}}{\tau_{l}}\right)+\text{Ei}\left(-\frac{t^{\ast}+\tau_{s}}{\tau_{u}}\right)\Bigg{)},$
(29)

where Ei denotes the exponential integral given by

$\text{Ei}(x)=\int_{-\infty}^{x}\frac{e^{t}}{t}\,\mathrm{d}t.$ (30)

Equation 29 has previously been given in a slightly different form in [22], as
Eq. (14a).

###  4.1 The boxcar eraser

In this section, we study the effect of the boxcar eraser given by Eq. (14),
replacing $\varepsilon_{r1}$ with $\varepsilon_{r3}$ for the generalized
Maxwell model. In this case, the response for time $t^{\ast}>0$ is

$\sigma_{3}(t^{\ast})=\frac{\displaystyle
E_{1}}{\ln\left(\frac{\displaystyle\tau_{l}}{\displaystyle\tau_{u}}\right)}\Bigg{(}\varepsilon_{a}\text{Ei}\left(-\frac{t^{\ast}+\tau_{r}}{\tau_{l}}\right)\Bigg{.}\\\
-\varepsilon_{a}\text{Ei}\left(-\frac{t^{\ast}+\tau_{r}}{\tau_{u}}\right)-\varepsilon_{a}\text{Ei}\left(-\frac{t^{\ast}+\tau_{r}+\tau_{s}}{\tau_{l}}\right)\\\
+\varepsilon_{a}\text{Ei}\left(-\frac{t^{\ast}+\tau_{r}+\tau_{s}}{\tau_{u}}\right)+\varepsilon_{r3}\text{Ei}\left(-\frac{t^{\ast}}{\tau_{l}}\right)-\varepsilon_{r3}\text{Ei}\left(-\frac{t^{\ast}}{\tau_{u}}\right)\\\
\Bigg{.}-\varepsilon_{r3}\text{Ei}\left(-\frac{t^{\ast}+\tau_{r}}{\tau_{l}}\right)+\varepsilon_{r3}\text{Ei}\left(-\frac{t^{\ast}+\tau_{r}}{\tau_{u}}\right)\Bigg{)}.$
(31)

Equation (31) is plotted for different ratios
$\varepsilon_{r3}/\varepsilon_{a}$ and for the specific values of
$\tau_{r}=\tau_{s}/4$ and $\tau_{s}=$1920\text{\,}\mathrm{s}$$ in Fig. 8. The
boxcar eraser does not completely remove the anelastic relaxation. It is also
interesting to note that for long $t^{\ast}$, i.e., $t^{\ast}>\tau_{s}$, the
remaining stress is independent of the erasing amplitude and is dominated by
the initial excursion.

Figure 8: Stress given by Eq. (31) for the generalized Maxwell model after
applying the boxcar eraser. Here $\tau_{r}=\tau_{s}/4$ and
$\tau_{s}=$1920\text{\,}\mathrm{s}$$. The lines of different colors show
different erasing amplitudes ranging from 0 to $-\varepsilon_{a}/2$. The main
plot shows the data on a logarithmic time scale. The inset shows the same data
in a linear time scale.

Since perfect erasing is not attainable, we aim to find the best possible
erasing procedure. To measure the effectiveness of the boxcar eraser we define
a figure of merit,

$f_{\mathrm{om}}=\int_{\tau_{l}}^{\tau_{u}}\big{(}\sigma(t^{\ast})\big{)}^{2}\,\mathrm{d}t^{\ast}.$
(32)

Figure 9: The figure of merit defined by Eq. 32 as a color coded plot versus
$\varepsilon_{r3}/\varepsilon_{a}$ on the horizontal axis and
$\tau_{r}/\tau_{s}$ on the vertical axis. The plot on the right gives the
figure of merit as a function of $\tau_{r}/\tau_{s}$ for
$\varepsilon_{r3}/\varepsilon_{a}-0.25$. The plot on the top shows the figure
of merit as a function of $\varepsilon_{r3}/\varepsilon_{a}$ for
$\tau_{r}=\tau_{s}/4$. The square and circular markers denote the minimum
along these sectional lines. The global minimum occurs at
$\tau_{r}=0.95/\tau_{s}$ and $\varepsilon_{r3}=0$. It is marked with a black
diamond.

Figure 9 shows a heatmap of $f_{\mathrm{om}}$ as a function of
$\tau_{r}/\tau_{s}$ and $\varepsilon_{r3}/\varepsilon_{a}$. One can see a
curved valley. Two points are marked in the valley. The plots to the top and
the right of the heat map show the figure of merit along the horizontal and
vertical dashed lines. Both marks are at the minimum along the respective
dimensions. Note that the minimum marked with the blue circle is much lower
than that marked with the green square. The diamond on the top right marks the
point where $\varepsilon_{r3}=0$. This point is equivalent to no erasing and
only entails waiting.

The stress as a function of $t^{\ast}$ for the three locations in the heatmap
is shown in Fig. 10.

Figure 10: The stress after the erasing procedure using the boxcar eraser. The
curves correspond to the three marked parameters in Fig. 9.

##  5 Effect on the measurement

Previously, we calculated the drift in the stress produced by a linear flexure
after it has been exercised sinusoidally. Here, we would like to apply this
knowledge to a Kibble balance, and in order to achieve this goal, three layers
of mechanics need to be revisited. (1) The basic equilibrium mechanics of the
flexure-based balance. (2) The calculation of the stiffness of a flexural
element. (3) The effect of one or more Maxwell units on the stiffness. Most of
the details discussed below can be found in the literature [27, 34]. For
convenience, we reiterate the main finding using coherent notation with the
rest of the article.

###  5.1 The static equilibrium of the flexure-based balance

Figure 11: A diagram of an equal arm balance beam suspended by a main flexure
with the gravitational acceleration pointing in the negative $z$ direction.
All flexures are shown in red. To simplify the model, the end flexures are
assumed to deflect by $\varphi$, which is a good assumption for thin flexures.

Figure 11 shows a schematic diagram of a beam balance. A balance beam with a
total length of $L$ ($250\text{\,}\mathrm{mm}$ in our case) and a mass $m_{b}$
is suspended from a main flexure with torsional stiffness $\kappa_{m}$. The
beam is terminated with end flexures, each with a torsional stiffness of
$\kappa_{e}$ supporting a mass $m_{p}$. The parameter $r_{f}$ denotes the
distance between the line defined by the rotation axes of both end flexures
and the main flexure’s rotation axis. The force (blue arrow) on the left side
of the beam balance can be controlled to maintain the right tip of the beam to
an operator-defined position $z=L\sin\varphi\approx L\varphi$.

The potential energy of the system in the small-angle approximation is

$V\approx
V_{o}+\frac{1}{2}m_{b}gh\varphi^{2}+m_{p}gr_{f}\varphi^{2}+\frac{1}{2}\kappa_{m}\varphi^{2}+\kappa_{e}\varphi^{2}.$
(33)

The restoring force imposed at an end flexure as a function of $z$ is the
negative derivative of the energy with respect to $z$. It is $F=-\partial
V/\partial z=-\partial F/\partial\varphi\cdot\partial\varphi/\partial z$, with
$\partial\varphi/\partial z=1/L$. Hence,

$F\approx-\frac{m_{b}gh}{L}\varphi-2\frac{m_{p}gr_{f}}{L}\varphi-\frac{\kappa_{m}}{L}\varphi-2\frac{\kappa_{e}}{L}\varphi.$
(34)

By taking a second derivative with respect to $z$, the linear spring constant
can be calculated via $K=-\partial F/\partial z=-\partial
F/\partial\varphi\cdot\partial\varphi/\partial z$, the spring constant can be
calculated. We find

$K\approx\underbrace{\frac{m_{b}gh}{L^{2}}+2\frac{m_{p}gr_{f}}{L^{2}}}_{\mathrm{gravitational}}+\underbrace{\frac{\kappa_{m}}{L^{2}}+2\frac{\kappa_{e}}{L^{2}}}_{\mathrm{flexural}}.$
(35)

Note that $K$ has two principal contributions: a gravitational component,
given by the first two terms, and a flexural component, given by the last two
terms. Only the latter suffers from loss and anelasticity.

###  5.2 Flexural stiffness

Figure 12: Parameters of a flexure with load suspended in tension at the free
end. The local acceleration $g$ points in the negative $z$ direction. The
symbols with 0 are values at the flexure end ($s=L$).

Here, the angular stiffness $\kappa$ of a loaded flexure as a function of
load, geometry, and elastic modulus, which is assumed to be constant, is
calculated. Following the approach using nonlinear equations of large
deflections in bending used in [35] for flexure of arbitrary shape under load
in tension the following two relationships hold

$\displaystyle\frac{\mbox{d}M(s)}{\mbox{d}s}=F_{\mathrm{w,0}}\sin{\big{(}\theta(s)\big{)}}$
$\displaystyle\;\mathrm{and}\;$
$\displaystyle\frac{\mbox{d}\theta(s)}{\mbox{d}s}=r(s),$ (36)
$\displaystyle\mbox{with}\;\;r(s)=\frac{M(s)}{EI_{\mathrm{z}}(s)},$ and
$\displaystyle\;I_{\mathrm{z}}(s)=\frac{b(s)h(s)^{3}}{12},$ (37)

where the second moment of area $I_{z}(s)$ can vary along the flexure length
parameterized by $s$. The variables used in the calculation are shown in
Figure 12.

The differential equations above can be solved for $\theta(s)$. The bending of
the flexure in the $x,z$ coordinate system is obtained using

$\frac{\mbox{d}z(s)}{\mbox{d}s}=-\cos{\big{(}\theta(s)\big{)}}\;\;\mbox{and}\;\;\frac{\mbox{d}x(s)}{\mbox{d}s}=\sin{\big{(}\theta(s)\big{)}}.$
(38)

This system of ordinary partial differential equations can be solved using
numerical tools, such as the shooting method [36]. We have a Python We have
plans to make the program publicly available at the end of 2024 code that
allows us to calculate arbitrarily shaped flexures. The end flexures can be
approximated by the same calculation by reversing the boundary conditions to
the ones shown in Fig. 12, which is a valid assumption at a long pendulum
length of tens of centimeters of the suspended masses $m_{p}$ [1].

Now, $\kappa$ can be extracted with the energy method, e.g., [27, 34] from the
second derivative of the elastic energy with respect to the deflection angle
$\theta_{0}$ at the free end, i.e.,

$\kappa=\frac{\mbox{d}^{2}V_{\mathrm{el}}}{\mbox{d}\theta_{0}^{2}}\;\;\mathrm{with}\;\;V_{\mathrm{el}}=\int_{0}^{L}\frac{M(s)^{2}}{2EI_{z}(s)}\mbox{d}s.$
(39)

In general, $\kappa$ can only be obtained numerically. The data describing the
geometries for the two types of flexure used in the experiment are given in
Table 1. The numerically obtained values of $\kappa$ are shown as a function
of mass load in Fig. 13. Additionally, $\kappa$ for the actual load are
printed in Table 2.

Table 1: The characteristic dimensions for the elliptical notch contour of the main flexure and the circular contour for the end flexures the beam balance. Dimension | Main flexure | End flexure
---|---|---
| nominal | meas. | nominal | meas.
Min. notch |  $50$ $\mathrm{\SIUnitSymbolMicro m}$ |  $40$ $\mathrm{\SIUnitSymbolMicro m}$ |  $100$ $\mathrm{\SIUnitSymbolMicro m}$ |  $94$$\mathrm{\SIUnitSymbolMicro m}$
Major axis |  $5$ $\mathrm{mm}$ | n.a. |  $2.8$ $\mathrm{mm}$ | n.a.
Minor axis |  $0.7$ $\mathrm{mm}$ | n.a. |  $2.8$ $\mathrm{mm}$ | n.a.
Width |  $37.2$ $\mathrm{mm}$ | n.a. |  $7.6$ $\mathrm{mm}$ | n.a.

###  5.3 Time dependent elastic modulus

The Maxwell model shows anelastic relaxation. Here, we investigate how such a
change in $E$ affects $\kappa$. Therefore, we separate our elastic modulus
into time-independent and time-dependent parts,

$E=E_{0}+\Delta E,$ (40)

where $\Delta E$ depends on time and is much smaller than $E_{0}$. Since
$\Delta E\ll E_{0}$, we can Taylor expand $\kappa_{\mathrm{el}}$. It is

$\kappa_{\mathrm{el}}(E_{0}+\Delta
E)\approx\underbrace{\kappa(E_{0})}_{:=\kappa_{0}}+\underbrace{\left.\Delta
E\frac{\mathrm{d}\kappa}{\mathrm{d}E}\right|_{E=E_{0}}}_{:=\Delta\kappa}.$
(41)

Equation 41 also defines the static and time-dependent parts of
$\kappa_{\mathrm{el}}$. The derivative of $\kappa_{\mathrm{el}}$ with respect
to $E$ can be obtained by numerically evaluating $\kappa_{\mathrm{el}}$ for
discrete values of $E$ around $E_{0}$. Since it is an important quantity, we
name it the modulus sensitivity, because it describes the sensitivity of
$\kappa$ to elastic modulus. We abbreviate it with

$S_{\mathrm{mod}}:=\left.\frac{\mathrm{d}\kappa}{\mathrm{d}E}\right|_{E=E_{0}}$
(42)

The elastic modulus for calculation of the elastic stiffness $\kappa$ was
corrected from plane stress to a plane strain assumption [38] by
multiplication with $1/(1-\nu^{2})$, where $\nu$ is Poisson’s ratio which is
0.3 for Copper Beryllium. The results of these calculations for the two types
of flexures used in the experiment below are shown in Fig. 13.

For an unloaded flexure, $S_{\mathrm{mod}}=\kappa_{0}/E_{0}$, indicated by the
dotted horizontal lines in Fig. 13. The higher the load, the smaller the
derivative. The relative fraction of the anelastic effect can be diluted by
loading the flexure. For Kibble balance design, we advocate for the largest
possible anelastic dilution, i.e., very heavily loaded flexures.

Figure 13: Top plot: Results of Eq. 39 using the solutions to numerical
solutions of the coupled differential equations in Eq. 36 as input. The points
are calculated as a function of $m$, where $F=mg$. Bottom plot: By slightly
varying the elastic modulus in the calculation shown in the top panel, the
modulus sensitivity $S_{\mathrm{mod}}$ can be obtained. It is plotted as a
function of load. The calculation was performed for the main flexure (circles)
and end flexure (squares).

###  5.4 The modulus defect

Starting from the groundwork laid out in the three preceding sub-sections, the
modulus defect can be calculated from the experimental observation. The
experiments discussed below measure the force on the balance $F$ as a function
of $z$. From that, we can calculate a spring constant $K=F/z$. It is given by
the gravitational and perfectly elastic terms in Eq. 35. The remaining time-
dependent part, $\Delta K=\Delta F/z$, can easily be converted to
$\Delta\kappa=L^{2}\Delta K$; see Eq. 35. According to Eq. 41,
$\Delta\kappa=\Delta ES_{\mathrm{mod}}$ and a combination of the last two
equations yields,

$\eta=\frac{\Delta E}{E_{o}}=\frac{1}{E_{o}}\frac{L^{2}\Delta
K}{S_{\mathrm{mod}}},$ (43)

where the numerically calculated values for the derivative are in the
denominator. The numerical values used here are shown in Tab. 2.

In the experiment, we can only test the combination of flexures. And just as
the spring constants add in Eq. 35 so do the modulus sensitivities. As is
shown in Table 2, the modulus sensitivity of the whole balance (tot) is
dominated by the end flexures. The two end flexures contribute
$93.8\text{\,}\mathrm{\char 37\relax}$ to the total $S_{\mathrm{mod}}$. So,
the experimental results given below are mostly a statement about the end
flexures. Here, the end flexures are relatively short, leading to negligible
anelastic dilution. Small anelastic dilution values are typically not desired
for Kibble balance experiments. For the investigation presented here, however,
a small dilution is favorable because it emphasizes the anelastic behavior,
which is the point of this article.

Table 2: The characteristic stiffness and sensitivity on the elastic modulus of main flexure m and end flexures e for the loads given in the last column. Flexure | $\kappa$ | $S_{\mathrm{mod}}$ | $F/g$
---|---|---|---
| ($\mathrm{mN}\text{\,}\mathrm{m}\text{/}\mathrm{rad}$) | ($\mathrm{m}\mathrm{m}^{3}\mathrm{/}\mathrm{rad}$) | ($\mathrm{kg}$)
Main (m) | $78\text{\,}$ | $1.2\text{\times}{10}^{-4}\text{\,}$ | $15\text{\,}$
End (e) | $130\text{\,}$ | $1.0\text{\times}{10}^{-3}\text{\,}$ | $6\text{\,}$
total $(_{\mathrm{tot}})$ | $338\text{\,}$ | $2.13\text{\times}{10}^{-3}\text{\,}$ | n.a.

###  5.5 Previous measurements of the modulus defect

Several measurements of the modulus defect conducted with often hardened
tempers of Copper Beryllium alloy C17200 can be found in the literature. Table
3 presents an overview of these results. Measurements were performed in
bending and torsion, yielding comparable results. The earliest measurement
produces a high result, which was subsequently explained by clamping losses.
One study found a dependence on load.

Table 3:  Published values of the modulus defect, $\Delta E/E_{\circ}$, for CuBe flexures sorted by year of publication.  Ref. |  Comment | $\Delta E/E_{\circ}$
---|---|---
[2] |  bending, beam balance | $1.1\text{\times}{10}^{-3}\text{\,}$
[27] |  bending and torsion. No dependence on load was found. | $4.3\text{\times}{10}^{-5}\text{\,}$
[28] |  torsion, measured as a function of load. An update to [27] where no load dependence was reported |  $4.3\text{\times}{10}^{-5}\text{\,}$ \- $6.0\text{\times}{10}^{-5}\text{\,}$
[23] |  torsion, attributes the high value in[2] to losses in the flexure mount and affirms this value also for bending | $5\text{\times}{10}^{-5}\text{\,}$
[30] |  bending, inverted pendulum | $5\text{\times}{10}^{-5}\text{\,}$
this work |  see experimental section | $1.2\text{\times}{10}^{-4}\text{\,}$

##  6 Magic flexures

For the flexures investigated here, the modulus sensitivity
$S_{\mathrm{mod}}=\mathrm{d}\kappa/\mathrm{d}E$ decreases with increasing load
as is shown in the lower panel of Fig. 13. The question that remains is
whether it is possible to build a flexure whose modulus sensitivity is zero.
We call such flexures magic because they should have very little anelastic
relaxation. Flexures that do not reach $S_{\mathrm{mod}}=0$ but have a minimum
as a function of the suspended load $F_{W,0}$, we call almost magic.

While Fig. 13 is the result of a numerical solution of the differential
equation Eq. 36, we strive to intuitively understand the mechanism that makes
certain flexures magic. To this end, we study the bending of a flat flexure of
length $L$ with constant cross section $b\times h$ under applied torque. The
$x$ coordinate of the flexure normalized by $x(L)$ as a function of $s$ is
shown in Fig. 14 for two load cases and for two different values of the
elastic modulus. A larger elastic modulus moves the bending further away from
the end of the flexure.

Imagine now that the cross-section of the flexure is engineered such that the
area moment of inertia, $I_{z}$, is smaller as the bending moves up, the
stiffening due to an increase of $E$ can be counteracted by a decrease in
$I_{z}$. Clearly, this is a delicate balance. But even if the effects do not
cancel exactly, the dependence of $\kappa$ on $E$ can be made small.

The parameter that helps to tune a flexure to its magic condition is the load
$F_{W,0}$ on the flexure. As shown in Fig. 14 for a flat flexure, changing the
load has the opposite effect of changing the modulus. A larger load moves the
bending down on the flexure. Most importantly, we note that the bending point
in a flexure can be moved by changing the load.

Figure 14: Bending of a flat flexure under a torque of
$1\text{\,}\mathrm{mN}\text{\,}\mathrm{m}$. The dimensions of the flexure were
arbitrarily chosen to be $L=$6\text{\,}\mathrm{mm}$$,
$t=$50\text{\,}\mathrm{\SIUnitSymbolMicro m}$$, and
$b=$20\text{\,}\mathrm{mm}$$ . The solution to the differential equation can
be found, e.g. in [34]. To emphasize the region where the flexure bends, the
curves are normalized to the values $x(L)$, which are different for the four
cases. They are in order of the legend
$0.17\text{\,}\mathrm{\SIUnitSymbolMicro m}$,
$0.09\text{\,}\mathrm{\SIUnitSymbolMicro m}$,
$0.19\text{\,}\mathrm{\SIUnitSymbolMicro m}$, and
$0.10\text{\,}\mathrm{\SIUnitSymbolMicro m}$.

While it is instructive to explain the emergence of magic on a flat flexure,
magic flexure can only be achieved by engineering the moment area of inertia
$I_{z}(s)$ to change as a function of $s$. For these cases, however,
analytical solutions to the differential equations governing the bending are
either elusive or mathematically so complex that most insight is lost.

The only way forward is a numerical calculation. Here, we use the code and
mathematics described in the previous section to calculate the main flexure.

Figure 15 shows the deflection as a function of the load on the flexure given
by the weight $mg$ of a point mass at the end of the flexure with mass values
$m$ ranging from $0\text{\,}\mathrm{kg}$ to $50\text{\,}\mathrm{kg}$.
Different from the flat flexure (see Fig. 14), and as a consequence of the
non-constant $I_{z}$ the bending point moves down the flexure as the mass load
increases. The unloaded flexure bends approximately in the middle of the
flexure, $s=$5\text{\,}\mathrm{mm}$$, where the flexure is thinnest. As the
mass load increases, the bending point moves down (increase in $s$) where the
flexure becomes thicker and hence stiffer. The bending is, of course,
distributed throughout the flexure. To indicate where the bending occurs the
normal stress is shown in the lower panel of Fig. 15. The maximum of the
(bending) stress is a good indication of where the bending occurs.

The complement to Fig. 15 is Fig. 16 where the deflection and the stress of
the same flexure are plotted for different values of $E_{o}$ under a constant
load of $15\text{\,}\mathrm{kg}$. We observe the opposite. As $E$ increases
the bending moves towards the middle of the flexure where it is thinnest and
bending happens at a lower torque.

With this combination, it is possible to find a load where the stiffness of
the flexure does not change as $E$ increases slightly. Usually, an increase in
$E$ would increase $\kappa$, but as the location of bending moves towards a
thinner part of the flexure, it simultaneously decreases. By choosing the
right mass, these two effects can offset each other yielding a constant
$\kappa$, i.e., a vanishing $S_{\mathrm{mod}}$.

Figure 15: Normalized deflection (upper graph) and normal stress (lower graph)
of the main flexure numerically calculated from the differential equation
system in Equation 38 under variation of the suspended load $F_{w}=mg$. The
angle $\theta_{0}$ was imposed as a boundary condition at the free end and the
calculation was performed with $E_{o}=$128\text{\,}\mathrm{GPa}$$. The lower
graph shows the normalized bending stress along the flexure.  Figure 16:
Normalized deflection (upper graph) and bending stress normalized to the
maximum (lower graph) of the main flexure obtained from our Python code by
changing the elastic modulus $E_{o}$. As before, the angle $\theta_{0}$ was
imposed at the free end as a boundary condition. A load with
$m=$15\text{\,}\mathrm{kg}$$ is suspended from the flexure.

To put this to practice, we calculate $\kappa$ and $S_{\mathrm{mod}}$ for the
main flexure for loads up to $100\text{\,}\mathrm{kg}$. The results are shown
in Fig. 17. To verify the Python code, several points were also calculated
using commercial finite element analysis (FEA) software. While there are small
differences between the two calculations that we attribute to negligible
numerical inaccuracies in both methods, the overall trends agree. Both
calculations predict the existence of a magic flexure. The main flexure
becomes magic at a load of $56\text{\,}\mathrm{kg}$.

Much more work needs to be done to understand magic flexures fully. We believe
they have low loss, i.e., high $Q$, and very little anelastic relaxation, our
focus in this article. Hence, Kibble balances might be a perfect application
for magic flexures. The investigation presented in this section is theoretical
and numerical in nature, and a careful experimental investigation is
unfortunately outside the scope of this work. We hope to dive into this
exciting research in years to come after the Quantum Electro-Mechanical
Metrology Suite (QEMMS) [14] has been built and qualified.

Figure 17: The spring constant $\kappa$ and the modulus sensitivity as a
function of load for the main flexures for a larger range of loads than in
Fig. 13. The orange circles are obtained by numerically solving the
differential equation for the given flexure shape. The green triangles were
obtained with finite element analysis (FEA) software. Besides small numerical
inaccuracies, the general trend agrees, and more importantly, both methods
find
$S_{\mathrm{mod}}=$0\text{\,}\mathrm{m}\mathrm{m}^{3}\mathrm{/}\mathrm{rad}$$
for $m\approx$56\text{\,}\mathrm{kg}$$. At this load in tension, the flexure
has a safety factor of $3.5$ to yield, and even with a bending angle of
$\theta_{0}=$7\text{\,}{}^{\circ}$$, a safety factor of $\approx 2$ is
present.

##  7 Experimental evidence

The goal of the measurements is to understand anelastic relaxation and verify
the effectiveness of the boxcar eraser. Here, we seek to arrive at a
measurement sequence that reduces the effect of the anelastic relaxation on a
classical ABA measurement, i.e., mass-off, mass-on, and mass-off measurement
to below $2\text{\,}\mathrm{nN}$. Furthermore, we would like to compare our
obtained value of the modulus defect to published values. An agreement would
indicate that our mechanism is free of clamping loss.

###  7.1 Description of the apparatus

For the experiments discussed below, the QEMMS Kibble balance was modified to
only consist of a single beam, i.e., the guiding mechanism and the wheel were
disconnected. With this disconnect, the system becomes much easier, and only
the central flexure and the two end flexures are under investigation with this
setup.

The beam balance mechanism, a position detector for feedback operation, and an
electromagnetic actuator are housed in a vacuum chamber kept at
$0.15\text{\,}\mathrm{mPa}$. A photo of the apparatus is shown in Figure 18.

Figure 18: Photograph of one end flexure through a window of the evacuated
vacuum chamber The tapered piece of aluminum to the left is the structural
part of the beam (mass $m_{b}$) connected to the main flexure, and the
threaded rod connects to a dummy mass (mass $m_{p}$). The total suspended load
is $m_{b}+2m_{p}\approx$15\text{\,}\mathrm{kg}$$.

This mechanism is used to suspend and guide a coil and can further be used as
a high-precision force comparator. The salient aspects regarding the design
are outlined in [14]. Most important for this work is that the flexure
elements (main and end) are made from hardened (TF00 temper) Copper Beryllium
alloy C17200 (containing about $1.8\text{\,}\mathrm{\char 37\relax}$ to
$2\text{\,}\mathrm{\char 37\relax}$ beryllium). In such a modular design
(Aluminum frame and beam, but CuBe flexures), it is crucial to prevent stress
gradients in the interface of flexure and connecting members, which, if poorly
designed, give rise to additional damping, anelastic aftereffect, and possibly
also static hysteresis [27]. To avoid clamping directly on flexible elements
the flexures are monolithic, i.e., the thin part and the interface piece
bolted onto the trusses are machined from a single coupon of material.

The central flexure was wire cut from bulk material, which was solution heat
treated and then precipitation hardened to provide a TF00 temper. It has an
elliptical notch contour with the dimensions in Table 1.

For the end flexures, two-dimensional flexure cubes with side length
$19.05\text{\,}\mathrm{mm}$ were used as the ones comprising the coil
suspension in the NIST-4 Kibble balance [39]. The cube is visible in the
photograph shown in Fig. 18.

The schematic of the block diagram of the control is shown in Fig. 19. For the
data reduction and analysis, two quantities are important. The position of the
balance, $z_{m}$ and the force $F$ needed to control the balance to that
position. The position $z_{m}$ is the vertical position of a mass suspended
from the left end flexure. The force is obtained by multiplying the coil
current $i_{c}$ with the geometric factor $Bl$. The coil current is measured
as a voltage drop over a calibrated resistor. The geometric factor has been
calibrated by measuring a mass.

Figure 19: Block diagram of the signals in the experiment. The host PC
provides the control set point $z_{\mathrm{SP}}$ to a proportional, integral,
derivative (PID) controller on a field programmable gate arrays (FPGA), which
compares it with the actual position reading from a quadrature encoded
(AquadB) interferometer signal. The output of the controller is a voltage
$u_{\mathrm{PID}}$ connected to a current source board, which directs a
current $i_{\mathrm{C}}$ through the coil and a calibrated resistor. A digital
volt meter (V) measures current as a voltage drop over the known resistor and
the coil generates a known force $F_{\mathrm{C}}$ in an electromagnetic setup
with a fixed permanent magnet. This force acts upon the mechanism and can keep
the mechanism in feedback controlled null state. The mechanism displacement
$z_{\mathrm{M}}$ from the ideal position $z_{\mathrm{SP}}$ is monitored.

###  7.2 Sources of measurement uncertainties

The main sources of measurement and analysis uncertainties are summarized
below:

  1. 1.

Fit uncertainty, mostly due to the defined start of $t^{\ast}$. We fit a
perfect mathematical description to an imperfect experiment.

  2. 2.

Since velocity mode was not available yet, we calibrated the $Bl$ with a
standard mass, whose value was known with a relative uncertainty of
$1\text{\times}{10}^{-3}\text{\,}$. We assign this number to the relative
uncertainty of the force measurements.

  3. 3.

The measurements discussed below were made with very little current in the
coil as the instrument was balanced to better than
$300\text{\,}\mathrm{\SIUnitSymbolMicro N}$ leading to two benefits:

     * •

Thermal effects due to ohmic heating in the coil were negligible.

     * •

Magnetic hysteresis is not a concern at these small currents, up to
$4\text{\,}\mathrm{\SIUnitSymbolMicro A}$ at static holding.

  4. 4.

Thermal effects due to evacuation and room temperature fluctuations affect the
measurements at long-time scales. To account for these, a linear drift is
removed from the measurements during data processing.

  5. 5.

Similarly, ground tilt can shift the center of mass and can cause long-term
drift in the required holding current in the balance. Similarly to thermal
effects, tilt produces a $1/f$ effect and is subtracted by linear fitting to
the data.

  6. 6.

The flexure clamps are designed to act on rigid pieces of metal, never on
flexing elements.

  7. 7.

When moving the balance from one position to another, there is a trade-off
between the duration of movement and the overshoot from the commanded position
due to the large inertia of the structure. The overshoot is of order
$<$2\text{\,}\mathrm{\char 37\relax}$$ and lasts less than
$10\text{\,}\mathrm{s}$. Hence, such overshoots only trigger the very short
time constants of a generalized Maxwell solid and are negligible for the long
time constants that we are interested in; See Fig.3.

  8. 8.

The coil current is supplied by wires attached to the balance mechanism. To
minimize parasitic elastic and hysteretic effects produced by these wires,
they are very thin (AWG 50 about $25\text{\,}\mathrm{\SIUnitSymbolMicro m}$ in
diameter, including insulation). They are routed parallel to the flexing axis
of the main pivot. The wire stiffness is further reduced by using extra wire,
about $50\text{\,}\mathrm{mm}$, which is coiled into a spring shape. The wires
running from the wheel to the coil are wound to springs as well, long, and do
not move horizontally more than $1.5\text{\,}\mathrm{mm}$ for the largest
imposed displacements of the mechanism.

Despite the limitations discussed above, the measurement results provide
valueable information for the anelastic relaxation.

###  7.3  Verifying the generalized Maxwell Model

The first series of measurements is summarized in Fig. 20. For the day-long
experiment, the balance position, $z$, was controlled to $\pm
2\text{\,}\mathrm{mm}$, $\pm 5\text{\,}\mathrm{mm}$, and $\pm
10\text{\,}\mathrm{mm}$. In each position, the beam was held for 32 minutes.
After that, the force required to maintain the balance at $z=0$ was measured
for three hours.

Two steps of post-processing were applied to the data. From all measurements,
a constant balance drift of $-1.3\text{\,}\mathrm{\SIUnitSymbolMicro
N}\text{\,}{\mathrm{h}}^{-1}$ was subtracted. The experimental run was started
within 24 hours of pumping the system to vacuum. The temperature change caused
by the evacuation led to the aforementioned drift, which was numerically taken
out. Usually, after a few days in the vacuum, the drift has vanished.

Second, each trace was shifted vertically such that the fit result converges
to a $F=0$ for $t\rightarrow\infty$. Subtracting these offsets makes the plot
less discombobulated and shows clearly that the relaxation amplitude is
proportional to the initial deflection. This processing step hides one
observation, though. For the large amplitudes, i.e., $\pm
10\text{\,}\mathrm{mm}$, the equilibrium position of the balance changes as
much as $3.5\text{\,}\mathrm{\SIUnitSymbolMicro N}$. These irreversible
changes were investigated separately and are discussed in detail in Sec. 8.

Figure 20: The anelastic relaxation for six different excursion amplitudes measured with the simple beam balance. The measurement procedure is shown in the inset. The top graph of the inset shows the balance excursions in mm. Each excursion is held for 32 minutes. After the balance has been served to zero, the force required to hold the balance at zero is recorded for 3 hours, indicated by the shaded regions. The corresponding data points are plotted as the colored data points in the main plot. The black lines are fits to Eq. (44) using $F_{A}$ as prefactor.  Table 4: Amplitude of the anelastic relaxation. The original excursion is given in the first column. The amplitude of the anelastic relaxation is labeled $F_{A}$. The column with the header $\eta$ gives the modulus defect. It is obtained using Eq. (45). Applying the boxcar eraser reduces the amplitude to $F_{B}$. The ratio of the reduction is given in the last column. $z_{A}$ | $F_{A}$ | $\eta$ | $F_{B}$ | $F_{B}/F_{A}$
---|---|---|---|---
($\mathrm{mm}$) | ($\mathrm{\SIUnitSymbolMicro N}$) |  | ($\mathrm{\SIUnitSymbolMicro N}$) | ($\mathrm{\char 37\relax}$)
$-10$ | $7.1$ | $1.6\text{\times}{10}^{-4}\text{\,}$ | $-2.5$ | $-36$
$-5$ | $2.3$ | $1.1\text{\times}{10}^{-4}\text{\,}$ | $-0.9$ | $-39$
$-2$ | $1.0$ | $1.1\text{\times}{10}^{-4}\text{\,}$ | $-0.2$ | $-19$
$2$ | $-0.8$ | $9.2\text{\times}{10}^{-5}\text{\,}$ | $0.2$ | $-25$
$5$ | $-2.5$ | $1.1\text{\times}{10}^{-4}\text{\,}$ | $0.8$ | $-32$
$10$ | $-6.4$ | $1.5\text{\times}{10}^{-4}\text{\,}$ | $2.6$ | $-40$
Average: | $1.2\text{\times}{10}^{-4}\text{\,}$ |  |  $-27$
Std. Dev.: | $2.7\text{\times}{10}^{-5}\text{\,}$ |  | $8$

A least squares fit was used to fit the function

$F(t^{\ast})=F_{A}\Bigg{(}\text{Ei}\left(-\frac{t^{\ast}}{\tau_{l}}\right)-\text{Ei}\left(-\frac{t^{\ast}}{\tau_{u}}\right)\Bigg{.}\\\
\Bigg{.}-\text{Ei}\left(-\frac{t^{\ast}+\tau_{s}}{\tau_{l}}\right)+\text{Ei}\left(-\frac{t^{\ast}+\tau_{s}}{\tau_{u}}\right)\Bigg{)}$
(44)

to the data. The functional form of this equation, with the exception of the
prefactor, and hence the unit, is identical to Eq. (29). The only free
parameter is the amplitude $F_{A}$, as the time constants were fixed to be
$\tau_{l}=$10\text{\,}\mathrm{s}$$ and $\tau_{u}=$5000\text{\,}\mathrm{s}$$.

From $F_{A}$ the modulus defect, $\eta$ can be obtained by combining Eq. (43)
with $\Delta K=F_{A}/z_{a}$ and is found to be

$\eta=-\frac{1}{E_{o}}\frac{L^{2}}{S_{\mathrm{mod}}}\frac{F_{A}}{z_{a}},$ (45)

where $z_{a}$ is the original excursion.

A close inspection of Fig. 20 leaves us with several observations. (1) The
anelastic after-effect can take a long time. The plot shows three hours after
the excursion, and the anelastic effect has not completely decayed. (2) Eq.
(44) is a reasonable fit to the measured traces. For the fits used in Fig. 20,
$\tau_{l}$ and $\tau_{u}$ have been fixed to $10\text{\,}\mathrm{s}$ and
$5000\text{\,}\mathrm{s}$, respectively. (3) The size of the anelastic after-
effect is roughly proportional to the original excursion, consistent with Eq.
(44), which is proportional to $z_{A}$. Table 4 gives the fitted amplitude for
the four measurements.

The calculated modulus defects are shown in Table 4. While there is
considerable scatter in the data, the average is
$\eta=$1.2\text{\times}{10}^{-4}\text{\,}$\pm$2.7\text{\times}{10}^{-5}\text{\,}$$,
where we used the sample standard deviation as an uncertainty. This number is
a factor of two to three larger than the results previously reported in the
literature; see Table 3. Given the difficult nature of precisely measuring the
modulus defect, this agreement is satisfactory. Furthermore, It can be
concluded that the experiment is not dominated by clamping loss, because this
would increase the modulus defect by a decade.

###  7.4  Verifying the boxcar eraser

Here, the effectiveness of the box car eraser is tested. The experiment was
conducted in the same way as previously. The balance was held for 32 minutes
each at the $z$-positions, $\pm 2\text{\,}\mathrm{mm}$, $\pm
5\text{\,}\mathrm{mm}$, and $\pm 10\text{\,}\mathrm{mm}$. After this time, the
balance is controlled to the negative value of the original setpoint and left
there for 8 minutes. After that, it is controlled to 0, and the force is
measured.

The results are shown in Fig. 21. To allow for easy comparison to the traces
without erasing, the original fits (black lines in Fig. 20) are shown here as
dashed lines with corresponding colors. After the boxcar eraser, the
amplitudes are much smaller, and the integral between the curve and the $F=0$
line is much smaller. The new amplitudes, $F_{B}$ are given in Tab. 4. One can
see that, on average, the relaxation amplitude is a bit more than a quarter of
the original amplitude. Note the minus sign in the table. After erasing the
flexure, the sign of the anelastic relaxation.

Figure 21: The balance is stressed the same was as in Fig. 20. This time,
after holding the original excursion for 32, the balance is controlled to the
negative excursion and held there for 8 minutes, see the top graph in the
inset. The colored data points show the measured force on the balance for
three hours after erasing. The black lines are fits to Eq. (31). The dashed
colored lines are the fits to the data without erasing procedure, i.e., the
black lines in Fig. 20.

##  8 Irreversible effects

Analyzing the results of the measurements discussed in the preceding section,
we noticed that, especially after large displacement, the flexure did not come
back to the same zero position – even after a long time. This effect is known
to scientists and technicians working with flexure balances and is another
reason, why large excursions of flexures are generally avoided. It is as if
the spring had been reset after a large excursion. In the Kibble balance, such
a reset of the spring after velocity mode is generally not a problem as long
as the spring stays consistent in the subsequent mass-on and mass-off
measurements. However, the mass manipulation for the mass-on and mass-off
measurements also induces disturbances to the balance, and it would be
important to know if the disturbances are large enough to cause a systematic
bias.

At this point, we do not know the precise mechanism that causes the
irreversible effects. Many causes are discussed in the literature, such as
dislocation pinning, work hardening, clamp effects, stick-slip effects, and
more.

Despite not knowing the physical mechanism, we can measure the irreversible
effect and discuss how much it affects the Kibble experiment. Fig. 22 shows
the results of three such measurements. For each measurement, the balance was
held at a given excursion for $30\text{\,}\mathrm{s}$ and then controlled back
to zero, where the force required to hold the balance at zero was recorded.
The drift corrected force as a function of excursion is shown in Fig. 22.
Three data sets were taken, two with larger excursions from
$1\text{\,}\mathrm{mm}$ to more than $10\text{\,}\mathrm{mm}$ and one set with
excursions from $0.1\text{\,}\mathrm{mm}$ to $1\text{\,}\mathrm{mm}$. Besides
a small discontinuity, between the set with smaller excursions and the two
with larger excursions, caused probably by the drift subtractions, the data
follow an exponential law with an exponent of 1.73. For a
$50\text{\,}\mathrm{\SIUnitSymbolMicro m}$ excursion, the irreversible effect
contributes a force of $$2\text{\,}\mathrm{nN}$\pm$1.5\text{\,}\mathrm{nN}$$.
The figure also shows, for comparison, the effect caused by the modulus
defect, which is proportional to excursion; see the dotted red line. For small
displacement, the anelastic effect immediately after bending, i.e.,
$t^{\ast}=0$ is much larger than the irreversible effect. A
$50\text{\,}\mathrm{\SIUnitSymbolMicro m}$ excursion causes an anelastic
aftereffect of $0.2\text{\,}\mathrm{\SIUnitSymbolMicro N}$ or about 100 times
larger than the irreversible effect. Fortunately, as has been discussed in
this article, the anelastic effect decays away and can be mitigated with an
erasing procedure.

Figure 22: Points with error bars denote the irreversible effect measured
after an excursion of the balance given by the horizontal axis in
$\mathrm{\SIUnitSymbolMicro m}$. The different colors and symbols of the
points denote three different data sets. Each excursion was held for
$30\text{\,}\mathrm{s}$. The solid black line is a power law fit to the data,
and the shaded area gives the 1-$\sigma$ uncertainty band of the fit line. For
comparison, the red dotted line gives the magnitude of the anelastic-after
effect at $t^{\ast}=0$, which is for all excursions shown here much larger
than the irreversible effect.

##  9 Conclusion

The purpose of this article is to explore the usage and limitations of
flexure-based Kibble balances with the hopes of breaking ground for the
widespread usage of flexures in Kibble balance, similar to what the seminal
paper [2] has done for conventional balances. To achieve this goal, we have
reviewed the basic considerations necessary to understand the effect of
anelastic relaxation on the Kibble balance measurement. With the Laplace
transformation, it is possible to calculate the after-effect of sinusoidal
excitation on the stress produced by the flexure. To mitigate the anelastic
after-effect on the Kibble balance measurement, an erasing procedure can be
employed after velocity mode. The simplest erasing procedure is the boxcar
eraser, holding the flexure at a specific deflection for a given time. A
numerical calculation of the boxcar eraser was performed, and pairs of optimal
deflection amplitude and hold time were found.

The theoretical framework has been supported by measurements on a simple beam
balance with a total of three flexures, one main and two end flexures. After
straining the balance, the stress followed the relaxation predicted by theory.
The application of a simple boxcar eraser significantly and consistent with
the theory reduces the relaxation stress. After erasing, the largest excursion
left an initial stress amplitude in the balance of approximately
$2.5\text{\,}\mathrm{\SIUnitSymbolMicro N}$. This amplitude can be suppressed
by a filter in the data analysis such as the one discussed in [40]. It can
reduce the effect of drift using realistic measurement times to
$2.4\text{\times}{10}^{-5}\text{\,}$ times the initial amplitude. Combining
these two numbers leaves a systematic bias of $0.36\text{\,}\mathrm{nN}$ for
the Kibble balance experiment. This number is much smaller than the
$2\text{\,}\mathrm{nN}$ that were initially budgeted for the anelastic
uncertainty for the Quantum Electro-Mechanical Metrology Suite, the new Kibble
balance under construction at NIST.

From the measurements, a modulus defect of the hardened Copper Beryllium alloy
C17200 with TF00 temper can be calculated. We find it to be
$\eta=$1.2\text{\times}{10}^{-4}\text{\,}$\pm$3\text{\times}{10}^{-5}\text{\,}$$
slightly larger than the values reported in the literature previously.

While investigating the anelastic relaxation irreversible effects most
prominent after large excursions were found. After a large excursion, the
force required to servo the balance to the null position changed. Such effects
are not unusual and were also seen previously in a Kibble balance with a knife
edge, NIST-3 [41] in agreement with a statement made in[22]. At this point, we
have not found a convincing explanation for the irreversible effect. A
systematic investigation showed that the force change has a power law
dependence on the excursion with an exponent of 1.73.

The irreversible effect after excursions in velocity mode will be common to
all weighings and should not cause a systematic bias. A possible systematic
bias can occur by the irreversible effect caused by balance excursions that
occur during mass exchanges. Since the excursions for the mass-on and mass-off
measurements have different signs, a systematic bias will remain.

Lastly, while thinking about these flexures and the physics that describes
them, we had the idea of a magic flexure — a flexure geometry that promises to
have very little anelastic relaxation. Unfortunately, at this point, we do not
have the resources to investigate the idea experimentally. A preliminary
numerical investigation, however, implies that it is possible to build
flexures with very low loss. The key idea is to shape the cross-section of the
flexure such that the region of maximal bending moves to a thinner part as the
elastic modulus increases.

Flexures are amazing elements for precision mechanics, and it is time for
widespread adaptation of them in the main and auxiliary pivots for Kibble
balance. We firmly believe flexures are the best choice for pivots in high-
precision mechanical experiments.

##  10 Acknowledgement

We would like to thank our colleagues Bill Luecke and Jon Pratt at the
National Institute of Standards and Technology for their helpful comments on
the manuscript.

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