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<title>Flexible Joint - Measurement of the Bending Stiffness</title>
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<h1 class="title">Flexible Joint - Measurement of the Bending Stiffness</h1>
<div id="table-of-contents">
<h2>Table of Contents</h2>
<div id="text-table-of-contents">
<ul>
<li><a href="#orge8827dd">1. Finite Element Model</a></li>
<li><a href="#orgc947253">2. Setup</a></li>
<li><a href="#orgf13af64">3. Effect of Bending</a></li>
<li><a href="#orgca610d4">4. Computation of the bending stiffness</a></li>
<li><a href="#orga6a9fed">5. Effect of Shear</a></li>
<li><a href="#orgb361e5e">6. Effect of Torsion</a></li>
<li><a href="#org939c423">7. Full stroke measured displacement and applied force as a function of \(H\)</a></li>
<li><a href="#org333fa3f">8. Negligible bending of the supporting bar</a></li>
<li><a href="#orgf9b93dc">9. Conclusion</a></li>
</ul>
</div>
</div>
<p>
The goal is to design a test bench to measure the bending stiffness of a flexible joint with 1% accuracy.
</p>
<div id="outline-container-orge8827dd" class="outline-2">
<h2 id="orge8827dd"><span class="section-number-2">1</span> Finite Element Model</h2>
<div class="outline-text-2" id="text-1">
<p>
<a id="org6f8fcb1"></a>
</p>
<p>
From the Finite Element Model, the stiffnesses and strokes of the flexible joint have been computed.
</p>
<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 id="outline-container-orgc947253" class="outline-2">
<h2 id="orgc947253"><span class="section-number-2">2</span> Setup</h2>
<div class="outline-text-2" id="text-2">
<p>
<a id="org86d0b98"></a>
</p>
<p>
Let&rsquo;s say a force is applied on top of the flexible joint with a distance \(H_F\) with the joint&rsquo;s center.
</p>
<p>
The displacement of the flexible joint is also measure at an height \(H_D\) from the joint&rsquo;s center.
</p>
<div id="org1d32223" class="figure">
<p><img src="figs/flexible_joint_test_bench_bending_setup.png" alt="flexible_joint_test_bench_bending_setup.png" />
</p>
<p><span class="figure-number">Figure 1: </span>Figure caption</p>
</div>
</div>
</div>
<div id="outline-container-orgf13af64" class="outline-2">
<h2 id="orgf13af64"><span class="section-number-2">3</span> Effect of Bending</h2>
<div class="outline-text-2" id="text-3">
<p>
<a id="org2781d1d"></a>
</p>
<p>
The torque applied is:
</p>
\begin{equation}
M_b = F \cdot H_F
\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_F}{k_b}
\end{equation}
<p>
This rotation is then measured by the displacement sensor.
The measured displacement is:
</p>
\begin{equation}
D_b = H_D \tan(R_b) = H_D \tan\left( \frac{F \cdot H_F}{k_b} \right) \label{eq:bending_meaured_disp}
\end{equation}
</div>
</div>
<div id="outline-container-orgca610d4" class="outline-2">
<h2 id="orgca610d4"><span class="section-number-2">4</span> Computation of the bending stiffness</h2>
<div class="outline-text-2" id="text-4">
<p>
From equation \eqref{eq:bending_meaured_disp}, we can compute the bending stiffness:
</p>
\begin{equation}
k_b = \frac{\tan^{-1}\left( \frac{D_b}{H_D} \right)}{F \cdot H_F}
\end{equation}
<p>
And therefore, to precisely measure \(k_b\), we need to:
</p>
<ul class="org-ul">
<li>precisely measure the motion \(D_b\)</li>
<li>precisely measure the applied force \(F\)</li>
<li>precisely know the height from the flexible joint&rsquo;s center to the measurement point \(H_D\)</li>
<li>precisely know the height from the flexible joint&rsquo;s center to the force application point \(H_F\)</li>
</ul>
<p>
If there are estimation errors for \(H_D\) or \(H_F\) as shown in Figure <a href="#orgd840605">2</a>, this will induce an error for the estimation of the stiffness.
</p>
<p>
For 1% accuracy estimation of \(k_b\), we can write the following approximate requirements:
</p>
<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
<colgroup>
<col class="org-left" />
<col class="org-right" />
</colgroup>
<thead>
<tr>
<th scope="col" class="org-left">&#xa0;</th>
<th scope="col" class="org-right">Accuracy</th>
</tr>
</thead>
<tbody>
<tr>
<td class="org-left">Force Measurement</td>
<td class="org-right">1%</td>
</tr>
<tr>
<td class="org-left">Displacement Measurement</td>
<td class="org-right">1%</td>
</tr>
<tr>
<td class="org-left">\(H_D\)</td>
<td class="org-right">1%</td>
</tr>
<tr>
<td class="org-left">\(H_F\)</td>
<td class="org-right">1%</td>
</tr>
</tbody>
</table>
<div id="orgd840605" class="figure">
<p><img src="figs/bending_effect_error_vertical.png" alt="bending_effect_error_vertical.png" />
</p>
<p><span class="figure-number">Figure 2: </span>Error in the estimation of the height of the force sensor and displacement sensor</p>
</div>
</div>
</div>
<div id="outline-container-orga6a9fed" class="outline-2">
<h2 id="orga6a9fed"><span class="section-number-2">5</span> Effect of Shear</h2>
<div class="outline-text-2" id="text-5">
<p>
<a id="orgd2827a2"></a>
</p>
<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_D \tan\left( \frac{F \cdot H_F}{k_b} \right) \gg \frac{F}{k_s}
\end{equation}
<p>
Here to simplify, we suppose \(FH_F/k_b \ll 1\) (which is the case in practice), and we suppose \(H_D = H_F = H\).
</p>
<p>
The obtained condition is then:
</p>
\begin{equation}
H \gg \frac{k_b}{k_s}
\end{equation}
<table id="org0b1af59" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
<caption class="t-above"><span class="table-number">Table 1:</span> Height \(H\) to have less than certain amount of error due to shear effects</caption>
<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">\(H\,[mm]\)</td>
<td class="org-right">6</td>
<td class="org-right">62</td>
<td class="org-right">620</td>
</tr>
</tbody>
</table>
<p>
In order to limit the effect of shear of less than 1%, the height from the joint&rsquo;s center to the force application point and to the measurement point should be larger than 62mm.
</p>
<div class="org-src-container">
<pre class="src src-matlab">H = 62e<span class="org-type">-</span>3; <span class="org-comment">% [m]</span>
</pre>
</div>
</div>
</div>
<div id="outline-container-orgb361e5e" class="outline-2">
<h2 id="orgb361e5e"><span class="section-number-2">6</span> Effect of Torsion</h2>
<div class="outline-text-2" id="text-6">
<p>
<a id="orge2b2893"></a>
</p>
<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 (Figure <a href="#orgdc4ca5c">3</a>).
</p>
<div id="orgdc4ca5c" class="figure">
<p><img src="figs/bending_effect_torsion.png" alt="bending_effect_torsion.png" />
</p>
<p><span class="figure-number">Figure 3: </span>Horizontal position error of the force sensor and displacement sensor</p>
</div>
<p>
Let&rsquo;s note the offset of the force sensor \(\epsilon_{F,y}\) and the offset of the measurement point \(\epsilon_{D,y}\).
The vertical torque (torsion) will be equal to:
</p>
\begin{equation}
M_t = F \cdot \epsilon_{F,y}
\end{equation}
<p>
And the induced torsion:
</p>
\begin{equation}
R_t = \frac{M_t}{k_t} = \frac{F \cdot \epsilon_{F,y}}{k_t}
\end{equation}
<p>
The effect on the measured displacement is:
</p>
\begin{equation}
D_t = \epsilon_{D,y} \tan \left( R_t \right) = \epsilon_{D,y} \tan\left( \frac{F \cdot \epsilon_{F,y}}{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}
H_D \tan\left( \frac{F \cdot H_F}{k_b} \right) \gg \epsilon_{D,y} \tan\left( \frac{F \cdot \epsilon_{F,y}}{k_t} \right)
\end{equation}
<p>
Supposing \(H_F = H_D = H\) and \(\epsilon_{F,y} = \epsilon_{D,y} = \epsilon_{y}\), the condition becomes:
</p>
\begin{equation}
\epsilon_{y} \ll H \sqrt{\frac{k_t}{k_b}}
\end{equation}
<table id="orga97cd00" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
<caption class="t-above"><span class="table-number">Table 2:</span> Maximum lateral position error \(\epsilon_y\) to have less than certain amount of error due to torsion effects</caption>
<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">\(\epsilon_y\,[mm]\)</td>
<td class="org-right">44.7</td>
<td class="org-right">4.5</td>
<td class="org-right">0.4</td>
</tr>
</tbody>
</table>
<p>
For 1% error, the lateral positioning errors \(\epsilon_y\) for both the force sensor and the displacement sensor should be less than 4.5mm.
</p>
</div>
</div>
<div id="outline-container-org939c423" class="outline-2">
<h2 id="org939c423"><span class="section-number-2">7</span> Full stroke measured displacement and applied force as a function of \(H\)</h2>
<div class="outline-text-2" id="text-7">
<p>
Applying a force with a large height \(H\) means the induced rotation (for constant force) will be larger.
This also means that the measured displacement \(D_b\) will also be larger.
</p>
<p>
Note that we here suppose the force axis is co-linear with the measurement axis (\(H_F = H_D = H\)).
</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 as a function of \(H\)
\[ F_{\text{max}} \approx \frac{X_b \cdot k_b}{H} \]</li>
</ul>
<div id="orgd2df8f9" class="figure">
<p><img src="figs/force_motion_function_H.png" alt="force_motion_function_H.png" />
</p>
<p><span class="figure-number">Figure 4: </span>Applied force \(F_{\text{max}}\) and measured displacement \(D_b\) as a function of \(H\)</p>
</div>
<p>
With an offset of 62mm, we obtained values shown in Table <a href="#orgfcc8e9b">3</a>.
</p>
<table id="orgfcc8e9b" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
<caption class="t-above"><span class="table-number">Table 3:</span> Maximum displacement and maximum applied force for \(H = 62\,[mm]\)</caption>
<colgroup>
<col class="org-right" />
<col class="org-right" />
</colgroup>
<thead>
<tr>
<th scope="col" class="org-right">\(D_b\,[mm]\)</th>
<th scope="col" class="org-right">\(F_m\,[N]\)</th>
</tr>
</thead>
<tbody>
<tr>
<td class="org-right">1.5</td>
<td class="org-right">1.9</td>
</tr>
</tbody>
</table>
</div>
</div>
<div id="outline-container-org333fa3f" class="outline-2">
<h2 id="org333fa3f"><span class="section-number-2">8</span> Negligible bending of the supporting bar</h2>
<div class="outline-text-2" id="text-8">
<p>
This should be confirmed with FEM.
</p>
</div>
</div>
<div id="outline-container-orgf9b93dc" class="outline-2">
<h2 id="orgf9b93dc"><span class="section-number-2">9</span> Conclusion</h2>
<div class="outline-text-2" id="text-9">
<table id="org9bcb31c" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
<caption class="t-above"><span class="table-number">Table 4:</span> Conclusions in terms of forces and displacement measurements</caption>
<colgroup>
<col class="org-left" />
<col class="org-left" />
<col class="org-left" />
</colgroup>
<thead>
<tr>
<th scope="col" class="org-left">&#xa0;</th>
<th scope="col" class="org-left">Range</th>
<th scope="col" class="org-left">Accuracy</th>
</tr>
</thead>
<tbody>
<tr>
<td class="org-left">Force Measurement</td>
<td class="org-left">2 N</td>
<td class="org-left">1% = 0.02 N</td>
</tr>
<tr>
<td class="org-left">Displacement Measurement</td>
<td class="org-left">1.5 mm</td>
<td class="org-left">1% = 15 um</td>
</tr>
</tbody>
</table>
<table id="org59ecc8a" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
<caption class="t-above"><span class="table-number">Table 5:</span> Conclusions in terms of required positioning accuracy</caption>
<colgroup>
<col class="org-left" />
<col class="org-left" />
<col class="org-left" />
<col class="org-left" />
</colgroup>
<thead>
<tr>
<th scope="col" class="org-left">&#xa0;</th>
<th scope="col" class="org-left">Value</th>
<th scope="col" class="org-left">Precision</th>
<th scope="col" class="org-left">Comment</th>
</tr>
</thead>
<tbody>
<tr>
<td class="org-left">\(H_D\)</td>
<td class="org-left">62mm</td>
<td class="org-left">&#xa0;</td>
<td class="org-left">For negligible Shear</td>
</tr>
<tr>
<td class="org-left">\(H_F\)</td>
<td class="org-left">62mm</td>
<td class="org-left">&#xa0;</td>
<td class="org-left">Same</td>
</tr>
<tr>
<td class="org-left">\(\epsilon_y\)</td>
<td class="org-left">0</td>
<td class="org-left">4.5mm</td>
<td class="org-left">For negligible Torsion</td>
</tr>
<tr>
<td class="org-left">\(\epsilon_z\)</td>
<td class="org-left">0</td>
<td class="org-left">1% = 0.6mm</td>
<td class="org-left">For torque estimation precision</td>
</tr>
</tbody>
</table>
<p>
Load cells:
</p>
<ul class="org-ul">
<li><a href="https://www.te.com/usa-en/product-CAT-FLS0020.html?q=&amp;n=510872&amp;d=681051%20564247&amp;type=products&amp;samples=N&amp;inStoreWithoutPL=false&amp;instock=N#mdp-tabs-content">https://www.te.com/usa-en/product-CAT-FLS0020.html?q=&amp;n=510872&amp;d=681051%20564247&amp;type=products&amp;samples=N&amp;inStoreWithoutPL=false&amp;instock=N#mdp-tabs-content</a></li>
<li><a href="https://www.digikey.com/en/products/detail/honeywell-sensing-and-productivity-solutions/FSS005WNSB/6056404">https://www.digikey.com/en/products/detail/honeywell-sensing-and-productivity-solutions/FSS005WNSB/6056404</a></li>
</ul>
<p>
Displacement sensors:
</p>
<ul class="org-ul">
<li><a href="http://www.vaco-france.com/outillage-de-metrologie/comparateur-a-cadran-de-precision.htm">http://www.vaco-france.com/outillage-de-metrologie/comparateur-a-cadran-de-precision.htm</a></li>
<li><a href="https://fr.rs-online.com/web/p/comparateurs/1940101/">https://fr.rs-online.com/web/p/comparateurs/1940101/</a></li>
</ul>
</div>
</div>
</div>
<div id="postamble" class="status">
<p class="author">Author: Dehaeze Thomas</p>
<p class="date">Created: 2021-01-15 ven. 23:32</p>
</div>
</body>
</html>
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