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<h1 class="title">Flexible Joint - Test Bench</h1>
<div id="table-of-contents">
<h2>Table of Contents</h2>
<div id="text-table-of-contents">
<ul>
<li><a href="#org8a9965a">1. Finite Element Model</a>
<ul>
<li><a href="#org9997e43">1.1. Parameters</a></li>
</ul>
</li>
<li><a href="#orgb3ee02d">2. Measurement Setup</a></li>
<li><a href="#org2bac0e5">3. Shear</a>
<ul>
<li><a href="#orgd80536f">3.1. Setup</a></li>
<li><a href="#org05fbcb8">3.2. Effect of Shear</a></li>
<li><a href="#org50c7d50">3.3. Effect of Bending</a></li>
<li><a href="#org07a1333">3.4. Effect of Torsion</a></li>
</ul>
</li>
<li><a href="#org0219378">4. Axial</a>
<ul>
<li><a href="#org2ce5d55">4.1. Setup</a></li>
<li><a href="#org6f28ab5">4.2. Effect of Compression</a></li>
<li><a href="#org16b8bac">4.3. Effect of Bending</a></li>
<li><a href="#org8d4c003">4.4. Measured Displacement with realistic positioning errors</a></li>
</ul>
</li>
<li><a href="#orge1d596a">5. Bending Stiffness</a>
<ul>
<li><a href="#orgadb2b86">5.1. Setup</a></li>
<li><a href="#orgfa564b5">5.2. Effect of Bending</a></li>
<li><a href="#org27bf684">5.3. Effect of Shear</a></li>
<li><a href="#orgc75eebd">5.4. Effect of Torsion</a></li>
<li><a href="#org7b8ae81">5.5. Full stroke measured displacement and applied force as a function of \(H\)</a></li>
<li><a href="#org0466825">5.6. Conclusion</a></li>
</ul>
</li>
<li><a href="#orgebb4075">6. Torsion</a>
<ul>
<li><a href="#org1a1af6b">6.1. Distance to flex axis for torsion &gt;&gt; shear</a></li>
<li><a href="#org617a379">6.2. Force Location Precision for torsion &gt;&gt; bending</a></li>
</ul>
</li>
<li><a href="#orgcbfadc2">7. Force Sensor</a></li>
<li><a href="#orge58d454">8. Displacement Sensor</a>
<ul>
<li><a href="#org83e8392">8.1. Three displacement sensors to measure both rotations \(\theta_y,\theta_z\) and translation \(x\)</a></li>
</ul>
</li>
</ul>
</div>
</div>
<div id="outline-container-org8a9965a" class="outline-2">
<h2 id="org8a9965a"><span class="section-number-2">1</span> Finite Element Model</h2>
<div class="outline-text-2" id="text-1">
</div>
<div id="outline-container-org9997e43" class="outline-3">
<h3 id="org9997e43"><span class="section-number-3">1.1</span> Parameters</h3>
<div class="outline-text-3" id="text-1-1">
<div class="org-src-container">
<pre class="src src-matlab">ka = 94e6; <span class="org-comment">% Axial Stiffness [N/m]</span>
ks = 13e6; <span class="org-comment">% Shear Stiffness [N/m]</span>
kb = 5; <span class="org-comment">% Bending Stiffness [Nm/rad]</span>
kt = 260; <span class="org-comment">% Torsional Stiffness [Nm/rad]</span>
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab">Fa = 469; <span class="org-comment">% Axial Force before yield [N]</span>
Fs = 242; <span class="org-comment">% Shear Force before yield [N]</span>
Fb = 0.118; <span class="org-comment">% Bending Force before yield [Nm]</span>
Ft = 1.508; <span class="org-comment">% Torsional Force before yield [Nm]</span>
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab">Xa = Fa<span class="org-type">/</span>ka; <span class="org-comment">% Axial Stroke before yield [m]</span>
Xs = Fs<span class="org-type">/</span>ks; <span class="org-comment">% Shear Stroke before yield [m]</span>
Xb = Fb<span class="org-type">/</span>kb; <span class="org-comment">% Bending Stroke before yield [rad]</span>
Xt = Ft<span class="org-type">/</span>kt; <span class="org-comment">% Torsional Stroke before yield [rad]</span>
</pre>
</div>
<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
<colgroup>
<col class="org-left" />
<col class="org-right" />
<col class="org-right" />
<col class="org-right" />
</colgroup>
<thead>
<tr>
<th scope="col" class="org-left">&#xa0;</th>
<th scope="col" class="org-right">Stiffness [N/um]</th>
<th scope="col" class="org-right">Max Force [N]</th>
<th scope="col" class="org-right">Stroke [um]</th>
</tr>
</thead>
<tbody>
<tr>
<td class="org-left">Axial</td>
<td class="org-right">94</td>
<td class="org-right">469</td>
<td class="org-right">5</td>
</tr>
<tr>
<td class="org-left">Shear</td>
<td class="org-right">13</td>
<td class="org-right">242</td>
<td class="org-right">19</td>
</tr>
</tbody>
</table>
<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
<colgroup>
<col class="org-left" />
<col class="org-right" />
<col class="org-right" />
<col class="org-right" />
</colgroup>
<thead>
<tr>
<th scope="col" class="org-left">&#xa0;</th>
<th scope="col" class="org-right">Stiffness [Nm/rad]</th>
<th scope="col" class="org-right">Max Torque [Nmm]</th>
<th scope="col" class="org-right">Stroke [mrad]</th>
</tr>
</thead>
<tbody>
<tr>
<td class="org-left">Bending</td>
<td class="org-right">5</td>
<td class="org-right">118</td>
<td class="org-right">24</td>
</tr>
<tr>
<td class="org-left">Torsional</td>
<td class="org-right">260</td>
<td class="org-right">1508</td>
<td class="org-right">6</td>
</tr>
</tbody>
</table>
</div>
</div>
</div>
<div id="outline-container-orgb3ee02d" class="outline-2">
<h2 id="orgb3ee02d"><span class="section-number-2">2</span> Measurement Setup</h2>
<div class="outline-text-2" id="text-2">
<p>
As shown in Figure <a href="#orged23794">1</a>, the x-axis is aligned with both the force axis and the measurement axis.
</p>
<div id="orged23794" class="figure">
<p><img src="figs/flexible_joint_test_bench_axis.png" alt="flexible_joint_test_bench_axis.png" />
</p>
<p><span class="figure-number">Figure 1: </span>Test bench geometry and axis</p>
</div>
</div>
</div>
<div id="outline-container-org2bac0e5" class="outline-2">
<h2 id="org2bac0e5"><span class="section-number-2">3</span> Shear</h2>
<div class="outline-text-2" id="text-3">
</div>
<div id="outline-container-orgd80536f" class="outline-3">
<h3 id="orgd80536f"><span class="section-number-3">3.1</span> Setup</h3>
<div class="outline-text-3" id="text-3-1">
<p>
The applied force as well as the measured displacement are aligned with the flexible joint&rsquo;s center (Figure <a href="#orged23794">1</a>).
</p>
<div id="org43c6a1e" class="figure">
<p><img src="figs/flex_joint_bench_shear.png" alt="flex_joint_bench_shear.png" />
</p>
<p><span class="figure-number">Figure 2: </span>Schematic of the test bench to measure the shear stiffness</p>
</div>
<p>
Ideally, only shear is induced by the force.
</p>
<p>
If the force is not exactly aligned with the rotation point of the flexible joint:
</p>
<ul class="org-ul">
<li>in the y-direction: some parasitic <b>torsional</b> motion will be induced</li>
<li>in the z-direction: some parasitic <b>bending</b> motion will be induced</li>
</ul>
<p>
We wish here to estimate how precise the force vector should be aligned with the flexible point in order for the motion measured is mainly due to shear effects and not torsional and bending effects.
</p>
</div>
</div>
<div id="outline-container-org05fbcb8" class="outline-3">
<h3 id="org05fbcb8"><span class="section-number-3">3.2</span> Effect of Shear</h3>
<div class="outline-text-3" id="text-3-2">
<p>
The shear effect is simply:
</p>
\begin{equation}
X_s = \frac{F}{k_s}
\end{equation}
</div>
</div>
<div id="outline-container-org50c7d50" class="outline-3">
<h3 id="org50c7d50"><span class="section-number-3">3.3</span> Effect of Bending</h3>
<div class="outline-text-3" id="text-3-3">
<p>
If the flexible joint is not perfectly positioned as shown in Figure <a href="#org2813737">3</a> (offset along the z-axis), the applied force will induce a bending motion in addition to the shear motion.
</p>
<div id="org2813737" class="figure">
<p><img src="figs/flex_joint_bench_pos_error.png" alt="flex_joint_bench_pos_error.png" />
</p>
<p><span class="figure-number">Figure 3: </span>Considering a position error \(\epsilon_z\) of the measurement and force application point with respect ot the flexible joint center (Side View)</p>
</div>
<p>
Let&rsquo;s estimate this effect.
</p>
<p>
The effect of the vertical position error \(\epsilon_z\) is the apparition of a &ldquo;bending&rdquo; torque:
</p>
\begin{equation}
M_b = \epsilon_z \cdot F
\end{equation}
<p>
This torque will induce a bending motion \(\theta_y\) of the flexible joint:
</p>
\begin{equation}
\theta_y = \frac{M_b}{k_b} = \frac{\epsilon_z \cdot F}{k_b}
\end{equation}
<p>
This rotation will induce an error \(\epsilon_{\theta_y}\) on the measured displacement:
</p>
\begin{equation}
\epsilon_{\theta_y} = \tan(\theta_y) \epsilon_z - L\left( 1 - \frac{1}{\cos(\theta_y)} \right)
\end{equation}
<p>
Supposing \(\theta_y\) small:
</p>
\begin{equation}
\epsilon_{\theta_y} \approx \frac{\epsilon_z^2 \cdot F}{k_b} \left( 1 + \frac{L}{2} \frac{F}{k_b} \right)
\end{equation}
<p>
And this error to be very small compared to the measured motion due to shear:
</p>
\begin{equation}
\epsilon_{\theta_y} \ll X_s
\end{equation}
<p>
Which is equivalent to:
</p>
\begin{equation}
\epsilon_z \ll \sqrt{\frac{k_b}{k_s} \frac{1}{1 + \frac{L}{2} \frac{F}{k_b}}}
\end{equation}
<p>
For realistic values, this is almost equivalent to:
</p>
\begin{equation}
\epsilon_z \ll \sqrt{\frac{k_b}{k_s}}
\end{equation}
<p>
Numerically:
</p>
<div class="org-src-container">
<pre class="src src-matlab">L = 5e<span class="org-type">-</span>2; <span class="org-comment">% [m]</span>
F = 20; <span class="org-comment">% [N]</span>
epsilon_z_max = sqrt(kb<span class="org-type">/</span>ks<span class="org-type">/</span>(1 <span class="org-type">+</span> L<span class="org-type">/</span>2<span class="org-type">*</span>F<span class="org-type">/</span>kb)); <span class="org-comment">% [m]</span>
</pre>
</div>
<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
<colgroup>
<col class="org-left" />
<col class="org-right" />
<col class="org-right" />
<col class="org-right" />
</colgroup>
<thead>
<tr>
<th scope="col" class="org-left">&#xa0;</th>
<th scope="col" class="org-right">10% error</th>
<th scope="col" class="org-right">1% error</th>
<th scope="col" class="org-right">0.1% error</th>
</tr>
</thead>
<tbody>
<tr>
<td class="org-left">\(\epsilon_z\,[\mu m]\)</td>
<td class="org-right">59.1</td>
<td class="org-right">5.9</td>
<td class="org-right">0.6</td>
</tr>
</tbody>
</table>
</div>
</div>
<div id="outline-container-org07a1333" class="outline-3">
<h3 id="org07a1333"><span class="section-number-3">3.4</span> Effect of Torsion</h3>
<div class="outline-text-3" id="text-3-4">
<p>
If the flexible joint is not perfectly positioned as shown in Figure <a href="#org36e2f7c">4</a> (offset along the y-axis), the applied force will induce a torsional motion in addition to the shear motion.
</p>
<div id="org36e2f7c" class="figure">
<p><img src="figs/flex_joint_bench_pos_error_bis.png" alt="flex_joint_bench_pos_error_bis.png" />
</p>
<p><span class="figure-number">Figure 4: </span>Considering a position error \(\epsilon_y\) of the measurement and force application point with respect ot the flexible joint center (top view)</p>
</div>
<p>
Let&rsquo;s estimate this effect.
</p>
<p>
The effect of the horizontal position error \(\epsilon_y\) is the apparition of a &ldquo;torsional&rdquo; torque:
</p>
\begin{equation}
M_t = \epsilon_y \cdot F
\end{equation}
<p>
This torque will induce a torsional motion \(\theta_z\) of the flexible joint:
</p>
\begin{equation}
\theta_z = \frac{M_t}{k_t} = \frac{\epsilon_y \cdot F}{k_t}
\end{equation}
<p>
This rotation will induce an error \(\epsilon_{\theta_z}\) on the measured displacement:
</p>
\begin{equation}
\epsilon_{\theta_z} = \tan(\theta_z) \epsilon_y - L\left( 1 - \frac{1}{\cos(\theta_z)} \right)
\end{equation}
<p>
Supposing \(\theta_z\) small:
</p>
\begin{equation}
\epsilon_{\theta_z} \approx \frac{\epsilon_y^2 \cdot F}{k_t} \left( 1 + \frac{L}{2} \frac{F}{k_t} \right)
\end{equation}
<p>
We want this error to be very small compared to the measured motion due to shear:
</p>
\begin{equation}
\epsilon_{\theta_z} \ll X_s
\end{equation}
<p>
Which is equivalent to:
</p>
\begin{equation}
\epsilon_y \ll \sqrt{\frac{k_t}{k_s} \frac{1}{1 + \frac{L}{2} \frac{F}{k_t}}}
\end{equation}
<p>
For realistic values, this is almost equivalent to:
</p>
\begin{equation}
\epsilon_y \ll \sqrt{\frac{k_t}{k_s}}
\end{equation}
<p>
Numerically:
</p>
<div class="org-src-container">
<pre class="src src-matlab">L = 5e<span class="org-type">-</span>2; <span class="org-comment">% [m]</span>
F = 20; <span class="org-comment">% [N]</span>
epsilon_y_max = sqrt(kt<span class="org-type">/</span>ks<span class="org-type">/</span>(1 <span class="org-type">+</span> L<span class="org-type">/</span>2<span class="org-type">*</span>F<span class="org-type">/</span>kt)); <span class="org-comment">% [m]</span>
</pre>
</div>
<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
<colgroup>
<col class="org-left" />
<col class="org-right" />
<col class="org-right" />
<col class="org-right" />
</colgroup>
<thead>
<tr>
<th scope="col" class="org-left">&#xa0;</th>
<th scope="col" class="org-right">10% error</th>
<th scope="col" class="org-right">1% error</th>
<th scope="col" class="org-right">0.1% error</th>
</tr>
</thead>
<tbody>
<tr>
<td class="org-left">\(\epsilon_y\,[\mu m]\)</td>
<td class="org-right">446.8</td>
<td class="org-right">44.7</td>
<td class="org-right">4.5</td>
</tr>
</tbody>
</table>
</div>
</div>
</div>
<div id="outline-container-org0219378" class="outline-2">
<h2 id="org0219378"><span class="section-number-2">4</span> Axial</h2>
<div class="outline-text-2" id="text-4">
</div>
<div id="outline-container-org2ce5d55" class="outline-3">
<h3 id="org2ce5d55"><span class="section-number-3">4.1</span> Setup</h3>
<div class="outline-text-3" id="text-4-1">
<div id="org138c9ff" class="figure">
<p><img src="figs/flex_test_bench_axial.png" alt="flex_test_bench_axial.png" />
</p>
<p><span class="figure-number">Figure 5: </span>Schematic of the test bench to measure the Axial Stiffness</p>
</div>
</div>
</div>
<div id="outline-container-org6f28ab5" class="outline-3">
<h3 id="org6f28ab5"><span class="section-number-3">4.2</span> Effect of Compression</h3>
<div class="outline-text-3" id="text-4-2">
<p>
The axial (compression) effect is simply:
</p>
\begin{equation}
X_a = \frac{F}{k_a}
\end{equation}
</div>
</div>
<div id="outline-container-org16b8bac" class="outline-3">
<h3 id="org16b8bac"><span class="section-number-3">4.3</span> Effect of Bending</h3>
<div class="outline-text-3" id="text-4-3">
<p>
If the force is not exactly aligned with the rotation point of the flexible joint, some parasitic <b>bending</b> motion will be induced (Figure <a href="#orgbbb7a5d">6</a>).
</p>
<div id="orgbbb7a5d" class="figure">
<p><img src="figs/flex_joint_bench_pos_error_axial.png" alt="flex_joint_bench_pos_error_axial.png" />
</p>
<p><span class="figure-number">Figure 6: </span>Error of position of the force vector and measurement point with respect to the flexible joint&rsquo;s center</p>
</div>
<p>
We wish here to estimate how close to the rotation point the force vector should be aligned for the motion measured to be mainly due to shear effects and not bending effects.
</p>
<p>
Suppose there is a distance \(\epsilon_z\) form the center of rotation, it will induce a torque equal to:
</p>
\begin{equation}
M_b = \epsilon_z \cdot F
\end{equation}
<p>
Therefore, the rotation of the top part will be:
</p>
\begin{equation}
X_b = \frac{M_b}{k_b} = \frac{\epsilon_z \cdot F}{k_b}
\end{equation}
<p>
And the induced measured displacement is:
</p>
\begin{equation}
D_b = \tan(X_b) \epsilon_z - L\left( 1 - \frac{1}{\cos(X_b)} \right) \\
\end{equation}
<p>
Supposing \(X_b\) small:
</p>
\begin{equation}
D_b \approx \frac{\epsilon_z^2 \cdot F}{k_b} \left( 1 + \frac{L}{2} \frac{F}{k_b} \right)
\end{equation}
<p>
We want the induced displacement due to the bending much smaller than the induced displacement due to the compression:
</p>
\begin{equation}
D_b \ll Xs
\end{equation}
<p>
Which is equivalent to (taking \(\tan x \approx x\)):
</p>
\begin{equation}
\epsilon_z \ll \sqrt{\frac{k_b}{k_a} \frac{1}{1 + \frac{L}{2} \frac{F}{k_b}}}
\end{equation}
<p>
For realistic values, this is almost equivalent to:
</p>
\begin{equation}
\epsilon_z \ll \sqrt{\frac{k_b}{k_a}}
\end{equation}
<p>
Note that an error in \(z\) is chosen here, but the same results are obtained for errors in the \(y\) direction.
What matters is the distance from the axis \(F\) to the flexible joint&rsquo;s center.
</p>
<p>
Numerically:
</p>
<div class="org-src-container">
<pre class="src src-matlab">L = 10e<span class="org-type">-</span>2; <span class="org-comment">% [m]</span>
F = 50; <span class="org-comment">% [N]</span>
d_off_eq = sqrt(kb<span class="org-type">/</span>ka<span class="org-type">/</span>(1 <span class="org-type">+</span> L<span class="org-type">/</span>2<span class="org-type">*</span>F<span class="org-type">/</span>kb))
</pre>
</div>
<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
<colgroup>
<col class="org-left" />
<col class="org-right" />
<col class="org-right" />
<col class="org-right" />
</colgroup>
<thead>
<tr>
<th scope="col" class="org-left">&#xa0;</th>
<th scope="col" class="org-right">10% error</th>
<th scope="col" class="org-right">1% error</th>
<th scope="col" class="org-right">0.1% error</th>
</tr>
</thead>
<tbody>
<tr>
<td class="org-left">\(\epsilon_y\,[\mu m]\)</td>
<td class="org-right">18.8</td>
<td class="org-right">1.9</td>
<td class="org-right">0.2</td>
</tr>
<tr>
<td class="org-left">\(\epsilon_z\,[\mu m]\)</td>
<td class="org-right">18.8</td>
<td class="org-right">1.9</td>
<td class="org-right">0.2</td>
</tr>
</tbody>
</table>
<p>
Same as for the shear measurement, it is very difficult to be that precise, and therefore rotations should also be measured and compensated for.
</p>
</div>
</div>
<div id="outline-container-org8d4c003" class="outline-3">
<h3 id="org8d4c003"><span class="section-number-3">4.4</span> Measured Displacement with realistic positioning errors</h3>
<div class="outline-text-3" id="text-4-4">
<p>
Suppose there is a positioning error of \(\epsilon_z = 0.1\,mm\).
</p>
\begin{equation}
X_b = \frac{\epsilon_z \cdot F}{k_b}
\end{equation}
<div class="org-src-container">
<pre class="src src-matlab">Fmax = Fa;
Ez = 0.1e<span class="org-type">-</span>3;
Xb_max = Ez<span class="org-type">*</span>Fmax<span class="org-type">/</span>kb;
</pre>
</div>
<pre class="example">
Xbmax = 9.4 [mrad]
</pre>
<p>
Which is about half of the maximum stroke in bending.
</p>
<p>
The
</p>
\begin{equation}
D_b = \tan(X_b) \epsilon_z - L\left( 1 - \frac{1}{\cos(X_b)} \right) \\
\end{equation}
</div>
</div>
</div>
<div id="outline-container-orge1d596a" class="outline-2">
<h2 id="orge1d596a"><span class="section-number-2">5</span> Bending Stiffness</h2>
<div class="outline-text-2" id="text-5">
</div>
<div id="outline-container-orgadb2b86" class="outline-3">
<h3 id="orgadb2b86"><span class="section-number-3">5.1</span> Setup</h3>
<div class="outline-text-3" id="text-5-1">
<p>
Let&rsquo;s say a force is applied on top of the flexible joint with a distance \(H\) with the joint&rsquo;s center.
</p>
<p>
The displacement of the flexible joint is also measure at the same height \(H\).
</p>
</div>
</div>
<div id="outline-container-orgfa564b5" class="outline-3">
<h3 id="orgfa564b5"><span class="section-number-3">5.2</span> Effect of Bending</h3>
<div class="outline-text-3" id="text-5-2">
<p>
The torque applied is:
</p>
\begin{equation}
M_b = F \cdot H
\end{equation}
<p>
The flexible joint is experiencing a rotation \(R_b\) due to the torque \(M_b\):
</p>
\begin{equation}
R_b = \frac{M_b}{k_b} = \frac{F \cdot H}{k_b}
\end{equation}
<p>
This rotation is then measured by the displacement sensor coaxial with the force actuator.
The measured displacement is:
</p>
\begin{equation}
D_b \approx H \tan (R_b) \approx \frac{F \cdot H^2}{k_b}
\end{equation}
</div>
</div>
<div id="outline-container-org27bf684" class="outline-3">
<h3 id="org27bf684"><span class="section-number-3">5.3</span> Effect of Shear</h3>
<div class="outline-text-3" id="text-5-3">
<p>
The effect of Shear on the measured displacement is simply:
</p>
\begin{equation}
D_s = \frac{F}{k_s}
\end{equation}
<p>
We would like to have this displacement much smaller than the displacement induced by the bending effects:
</p>
\begin{equation}
D_b \gg D_s
\end{equation}
<p>
Which is equivalent as to have:
</p>
\begin{equation}
H \gg \sqrt{\frac{k_b}{k_s}}
\end{equation}
<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
<colgroup>
<col class="org-left" />
<col class="org-right" />
<col class="org-right" />
<col class="org-right" />
</colgroup>
<thead>
<tr>
<th scope="col" class="org-left">&#xa0;</th>
<th scope="col" class="org-right">10% error</th>
<th scope="col" class="org-right">1% error</th>
<th scope="col" class="org-right">0.1% error</th>
</tr>
</thead>
<tbody>
<tr>
<td class="org-left">D offset [mm]</td>
<td class="org-right">6</td>
<td class="org-right">62</td>
<td class="org-right">620</td>
</tr>
</tbody>
</table>
<p>
We should therefore apply a force at least 50mm higher than the joint&rsquo;s center for the shear effects to be negligible.
</p>
</div>
</div>
<div id="outline-container-orgc75eebd" class="outline-3">
<h3 id="orgc75eebd"><span class="section-number-3">5.4</span> Effect of Torsion</h3>
<div class="outline-text-3" id="text-5-4">
<p>
If the application force is not aligned with the vertical axis of the flexible joint, this will induce a torsion motion that will induce a measurement error.
</p>
<p>
Let&rsquo;s note the offset \(x_{off}\):
</p>
\begin{equation}
D_t \approx \frac{x_{off}^2 \cdot F}{k_t} \left( 1 + \frac{L}{2} \frac{F}{k_t} \right)
\end{equation}
<p>
And we would like to have:
</p>
\begin{equation}
D_b \gg D_t
\end{equation}
<p>
Which is equivalent as to have:
</p>
\begin{equation}
x_{off} \ll H \sqrt{\frac{k_t}{k_b}}
\end{equation}
<p>
Let&rsquo;s take \(H = 50/,mm\).
</p>
<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
<colgroup>
<col class="org-left" />
<col class="org-right" />
<col class="org-right" />
<col class="org-right" />
</colgroup>
<thead>
<tr>
<th scope="col" class="org-left">&#xa0;</th>
<th scope="col" class="org-right">10% error</th>
<th scope="col" class="org-right">1% error</th>
<th scope="col" class="org-right">0.1% error</th>
</tr>
</thead>
<tbody>
<tr>
<td class="org-left">X offset [mm]</td>
<td class="org-right">36.1</td>
<td class="org-right">3.6</td>
<td class="org-right">0.4</td>
</tr>
</tbody>
</table>
<p>
The effect of torsion should be negligible for a positioning accuracy of ~1mm.
</p>
</div>
</div>
<div id="outline-container-org7b8ae81" class="outline-3">
<h3 id="org7b8ae81"><span class="section-number-3">5.5</span> Full stroke measured displacement and applied force as a function of \(H\)</h3>
<div class="outline-text-3" id="text-5-5">
<p>
Applying a force with a large offset \(H\) means the induced rotation (for constant force) will be larger.
This also means that the measured displacement \(D_b\) will also be larger.
</p>
\begin{equation}
D_b \approx H \tan (R_b)
\end{equation}
<p>
Note that we here suppose the force axis is co-linear with the measurement axis.
</p>
<p>
Let&rsquo;s compute:
</p>
<ul class="org-ul">
<li>\(D_b\) as a function of \(H\)
\[ D_b \approx H \tan (R_b) \]</li>
<li>the applied force \(F_{\text{max}}\) to induce the maximum rotation
\[ F_{\text{max}} \approx \frac{X_b \cdot k_b}{H} \]</li>
</ul>
<div class="org-src-container">
<pre class="src src-matlab">H = linspace(0, 100e<span class="org-type">-</span>3, 1000);
Db = H<span class="org-type">*</span>tan(Xb);
Fmax = Xb<span class="org-type">*</span>kb<span class="org-type">./</span>H;
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-type">figure</span>;
yyaxis left
plot(1e3<span class="org-type">*</span>H, Fmax);
ylabel(<span class="org-string">'Maximum Force [$N$]'</span>);
ylim([0, 100]);
yyaxis right
plot(1e3<span class="org-type">*</span>H, 1e6<span class="org-type">*</span>Db);
ylabel(<span class="org-string">'Measurement Range [$\mu m$]'</span>);
xlabel(<span class="org-string">'Offset $H$ [$mm$]'</span>);
</pre>
</div>
</div>
</div>
<div id="outline-container-org0466825" class="outline-3">
<h3 id="org0466825"><span class="section-number-3">5.6</span> Conclusion</h3>
<div class="outline-text-3" id="text-5-6">
<ul class="org-ul">
<li>Applied Force</li>
<li>Measured Displacement</li>
<li>Requirements in terms of positioning</li>
</ul>
</div>
</div>
</div>
<div id="outline-container-orgebb4075" class="outline-2">
<h2 id="orgebb4075"><span class="section-number-2">6</span> Torsion</h2>
<div class="outline-text-2" id="text-6">
</div>
<div id="outline-container-org1a1af6b" class="outline-3">
<h3 id="org1a1af6b"><span class="section-number-3">6.1</span> Distance to flex axis for torsion &gt;&gt; shear</h3>
</div>
<div id="outline-container-org617a379" class="outline-3">
<h3 id="org617a379"><span class="section-number-3">6.2</span> Force Location Precision for torsion &gt;&gt; bending</h3>
</div>
</div>
<div id="outline-container-orgcbfadc2" class="outline-2">
<h2 id="orgcbfadc2"><span class="section-number-2">7</span> Force Sensor</h2>
<div class="outline-text-2" id="text-7">
<p>
Requirements:
</p>
<ul class="org-ul">
<li>Large enough measurement range</li>
<li>Small enough measurement noise and errors</li>
</ul>
<p>
The largest applied force is foreseen to be around 500N.
The smallest (full range) force is foreseen to be around 2N.
</p>
<p>
If we want an accuracy of 1% (even for the small measurement range), the force sensor should be accurate down to ~0.02N and with a full range of 500N.
This corresponds to an accuracy of 0.004%.
</p>
<p>
If we limit ourselves to 100N of applied force, the required accuracy of 0.02N corresponds to 0.02% of accuracy which is more realistic.
</p>
<p>
Another solution would be to use 2 force sensors.
</p>
<div class="org-src-container">
<pre class="src src-matlab">Fmax = 500; <span class="org-comment">% [N]</span>
Fmin = 2; <span class="org-comment">% [N]</span>
Fres = Fmin<span class="org-type">/</span>100; <span class="org-comment">% [N]</span>
<span class="org-comment">% Dynamic range</span>
Fmax<span class="org-type">/</span>Fres
</pre>
</div>
<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
<colgroup>
<col class="org-left" />
<col class="org-right" />
<col class="org-right" />
<col class="org-right" />
<col class="org-left" />
</colgroup>
<thead>
<tr>
<th scope="col" class="org-left">Model</th>
<th scope="col" class="org-right">Range [N]</th>
<th scope="col" class="org-right">Accuracy</th>
<th scope="col" class="org-right">Error [N]</th>
<th scope="col" class="org-left">Price [eur]</th>
</tr>
</thead>
<tbody>
<tr>
<td class="org-left"><a href="https://www.hbm.com/en/3364/s2m-reliable-high-precision-s-type-force-load-cell/">S2M</a></td>
<td class="org-right">100</td>
<td class="org-right">0.02%</td>
<td class="org-right">0.02</td>
<td class="org-left">&#xa0;</td>
</tr>
<tr>
<td class="org-left"><a href="https://www.hbm.com/en/3364/s2m-reliable-high-precision-s-type-force-load-cell/">S2M</a></td>
<td class="org-right">500</td>
<td class="org-right">0.02%</td>
<td class="org-right">0.1</td>
<td class="org-left">&#xa0;</td>
</tr>
<tr>
<td class="org-left"><a href="https://www.althensensors.com/sensors/force-sensors/cylindrical-force-sensors/4510/alf241-axial-compensated-force-sensor/">ALF241</a></td>
<td class="org-right">100</td>
<td class="org-right">0.05%</td>
<td class="org-right">0.05</td>
<td class="org-left">&#xa0;</td>
</tr>
<tr>
<td class="org-left"><a href="https://en.pm-instrumentation.com/1500-cellule-de-force-plate-125-a-1500-n-faible-sensibilite-aux-efforts-transverses#documents">PM1500</a></td>
<td class="org-right">500</td>
<td class="org-right">0.05%</td>
<td class="org-right">0.5</td>
<td class="org-left">&#xa0;</td>
</tr>
</tbody>
</table>
<p>
<a href="https://www.hbm.com/en/2925/top-transfer-reference-force-transducer-maximum-precision/">https://www.hbm.com/en/2925/top-transfer-reference-force-transducer-maximum-precision/</a>
</p>
<ul class="org-ul">
<li>State of the art but big</li>
</ul>
<p>
<a href="https://www.burster.com//en/sensors/load-cells/high-precision-load-cells/p/detail/8527">https://www.burster.com//en/sensors/load-cells/high-precision-load-cells/p/detail/8527</a>
</p>
<ul class="org-ul">
<li>500N range =&gt; 0.25N error</li>
</ul>
</div>
</div>
<div id="outline-container-orge58d454" class="outline-2">
<h2 id="orge58d454"><span class="section-number-2">8</span> Displacement Sensor</h2>
<div class="outline-text-2" id="text-8">
</div>
<div id="outline-container-org83e8392" class="outline-3">
<h3 id="org83e8392"><span class="section-number-3">8.1</span> Three displacement sensors to measure both rotations \(\theta_y,\theta_z\) and translation \(x\)</h3>
<div class="outline-text-3" id="text-8-1">
<div id="orgd133a31" class="figure">
<p><img src="figs/3_point_measurement.png" alt="3_point_measurement.png" />
</p>
<p><span class="figure-number">Figure 7: </span>Figure caption</p>
</div>
<ul class="org-ul">
<li>Capacitive sensors or interferometers</li>
<li>Choice of the position of the three sensors =&gt; will change rotation sensitivity and range</li>
</ul>
</div>
</div>
</div>
</div>
<div id="postamble" class="status">
<p class="author">Author: Dehaeze Thomas</p>
<p class="date">Created: 2021-01-15 ven. 15:20</p>
</div>
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