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figs/flexible_joint_geometry.pdf
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figs/flexible_joint_geometry.pdf
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<!-- 2020-12-15 mar. 22:32 -->
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<meta http-equiv="Content-Type" content="text/html;charset=utf-8" />
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<title>Flexible Joint - Test Bench</title>
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||||
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<meta name="author" content="Dehaeze Thomas" />
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<div id="org-div-home-and-up">
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<a accesskey="h" href="../index.html"> UP </a>
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|
||||
<a accesskey="H" href="../index.html"> HOME </a>
|
||||
</div><div id="content">
|
||||
<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="#org3ec0fb6">1. Test Bench Description</a>
|
||||
<ul>
|
||||
<li><a href="#org1c35c31">1.1. Flexible joint Geometry</a></li>
|
||||
<li><a href="#org943ed6d">1.2. Required external applied force</a></li>
|
||||
<li><a href="#org866642a">1.3. Required actuator stroke and sensors range</a></li>
|
||||
<li><a href="#org4789077">1.4. First try with the APA95ML</a></li>
|
||||
</ul>
|
||||
</li>
|
||||
<li><a href="#org75cb5e5">2. Experimental measurement</a></li>
|
||||
</ul>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org3ec0fb6" class="outline-2">
|
||||
<h2 id="org3ec0fb6"><span class="section-number-2">1</span> Test Bench Description</h2>
|
||||
<div class="outline-text-2" id="text-1">
|
||||
<p>
|
||||
The main characteristic of the flexible joint that we want to measure is its bending stiffness \(k_{R_x} \approx k_{R_y}\).
|
||||
</p>
|
||||
|
||||
<p>
|
||||
To do so, a test bench is used.
|
||||
Specifications of the test bench to precisely measure the bending stiffness are described in this section.
|
||||
</p>
|
||||
|
||||
<p>
|
||||
The basic idea is to measured the angular deflection of the flexible joint as a function of the applied torque.
|
||||
</p>
|
||||
|
||||
|
||||
<div id="org12e0ba4" class="figure">
|
||||
<p><img src="figs/test-bench-schematic.png" alt="test-bench-schematic.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 1: </span>Schematic of the test bench to measure the bending stiffness of the flexible joints</p>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org1c35c31" class="outline-3">
|
||||
<h3 id="org1c35c31"><span class="section-number-3">1.1</span> Flexible joint Geometry</h3>
|
||||
<div class="outline-text-3" id="text-1-1">
|
||||
<p>
|
||||
The flexible joint used for the Nano-Hexapod is shown in Figure <a href="#org907b319">2</a>.
|
||||
Its bending stiffness is foreseen to be \(k_{R_y}\approx 20\,\frac{Nm}{rad}\) and its stroke \(\theta_{y,\text{max}}\approx 20\,mrad\).
|
||||
</p>
|
||||
|
||||
|
||||
<div id="org907b319" class="figure">
|
||||
<p><img src="figs/flexible_joint_geometry.png" alt="flexible_joint_geometry.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 2: </span>Geometry of the flexible joint</p>
|
||||
</div>
|
||||
|
||||
<p>
|
||||
The height between the flexible point (center of the joint) and the point where external forces are applied is \(h = 20\,mm\).
|
||||
</p>
|
||||
|
||||
<p>
|
||||
Let’s define the parameters on Matlab.
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">kRx = 20; <span class="org-comment">% Bending Stiffness [Nm/rad]</span>
|
||||
Rxmax = 20e<span class="org-type">-</span>3; <span class="org-comment">% Bending Stroke [rad]</span>
|
||||
h = 20e<span class="org-type">-</span>3; <span class="org-comment">% Height [m]</span>
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org943ed6d" class="outline-3">
|
||||
<h3 id="org943ed6d"><span class="section-number-3">1.2</span> Required external applied force</h3>
|
||||
<div class="outline-text-3" id="text-1-2">
|
||||
<p>
|
||||
The bending \(\theta_y\) of the flexible joint due to the force \(F_x\) is:
|
||||
</p>
|
||||
\begin{equation}
|
||||
\theta_y = \frac{M_y}{k_{R_y}} = \frac{F_x h}{k_{R_y}}
|
||||
\end{equation}
|
||||
|
||||
<p>
|
||||
Therefore, the applied force to test the full range of the flexible joint is:
|
||||
</p>
|
||||
\begin{equation}
|
||||
F_{x,\text{max}} = \frac{k_{R_y} \theta_{y,\text{max}}}{h}
|
||||
\end{equation}
|
||||
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">Fxmax = kRx<span class="org-type">*</span>Rxmax<span class="org-type">/</span>h; <span class="org-comment">% Force to induce maximum stroke [N]</span>
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<p>
|
||||
And we obtain:
|
||||
</p>
|
||||
\begin{equation} F_{max} = 20.0\, [N] \end{equation}
|
||||
|
||||
<p>
|
||||
The measurement range of the force sensor should then be higher than \(20\,N\).
|
||||
</p>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org866642a" class="outline-3">
|
||||
<h3 id="org866642a"><span class="section-number-3">1.3</span> Required actuator stroke and sensors range</h3>
|
||||
<div class="outline-text-3" id="text-1-3">
|
||||
<p>
|
||||
The flexible joint is designed to allow a bending motion of \(\pm 20\,mrad\).
|
||||
The corresponding actuator stroke to impose such motion is:
|
||||
</p>
|
||||
|
||||
<p>
|
||||
\[ d_{x,\text{max}} = h \tan(R_{x,\text{max}}) \]
|
||||
</p>
|
||||
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">dxmax = h<span class="org-type">*</span>tan(Rxmax);
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
\begin{equation} d_{max} = 0.4\, [mm] \end{equation}
|
||||
|
||||
<p>
|
||||
In order to test the full range of the flexible joint, the stroke of the actuator should be higher than \(0.4\,mm\).
|
||||
The measurement range of the displacement sensor should also be higher than \(0.4\,mm\).
|
||||
</p>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org4789077" class="outline-3">
|
||||
<h3 id="org4789077"><span class="section-number-3">1.4</span> First try with the APA95ML</h3>
|
||||
<div class="outline-text-3" id="text-1-4">
|
||||
<p>
|
||||
The APA95ML as a stroke of \(100\,\mu m\) and the encoder in parallel can easily measure the required stroke.
|
||||
</p>
|
||||
|
||||
<p>
|
||||
Suppose the full stroke of the APA can be used to bend the flexible joint (ideal case), the measured force will be:
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">Fxmax = kRx<span class="org-type">*</span>100e<span class="org-type">-</span>6<span class="org-type">/</span>h<span class="org-type">^</span>2; <span class="org-comment">% Force at maximum stroke [N]</span>
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
\begin{equation} F_{max} = 5.0\, [N] \end{equation}
|
||||
|
||||
<p>
|
||||
And the tested angular range is:
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">Rmax = tan(100e<span class="org-type">-</span>6<span class="org-type">/</span>h);
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
\begin{equation} \theta_{max} = 5.0\, [mrad] \end{equation}
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org75cb5e5" class="outline-2">
|
||||
<h2 id="org75cb5e5"><span class="section-number-2">2</span> Experimental measurement</h2>
|
||||
</div>
|
||||
</div>
|
||||
<div id="postamble" class="status">
|
||||
<p class="author">Author: Dehaeze Thomas</p>
|
||||
<p class="date">Created: 2020-12-15 mar. 22:32</p>
|
||||
</div>
|
||||
</body>
|
||||
</html>
|
||||
158
index.org
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index.org
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||||
#+TITLE: Flexible Joint - Test Bench
|
||||
:DRAWER:
|
||||
#+LANGUAGE: en
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||||
#+EMAIL: dehaeze.thomas@gmail.com
|
||||
#+AUTHOR: Dehaeze Thomas
|
||||
|
||||
#+HTML_LINK_HOME: ../index.html
|
||||
#+HTML_LINK_UP: ../index.html
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||||
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||||
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|
||||
#+PROPERTY: header-args:latex+ :post pdf2svg(file=*this*, ext="png")
|
||||
:END:
|
||||
|
||||
* Test Bench Description
|
||||
** Introduction :ignore:
|
||||
|
||||
The main characteristic of the flexible joint that we want to measure is its bending stiffness $k_{R_x} \approx k_{R_y}$.
|
||||
|
||||
To do so, a test bench is used.
|
||||
Specifications of the test bench to precisely measure the bending stiffness are described in this section.
|
||||
|
||||
The basic idea is to measured the angular deflection of the flexible joint as a function of the applied torque.
|
||||
|
||||
#+name: fig:test-bench-schematic
|
||||
#+caption: Schematic of the test bench to measure the bending stiffness of the flexible joints
|
||||
[[file:figs/test-bench-schematic.png]]
|
||||
|
||||
** Matlab Init :noexport:ignore:
|
||||
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
|
||||
<<matlab-dir>>
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab :exports none :results silent :noweb yes
|
||||
<<matlab-init>>
|
||||
#+end_src
|
||||
|
||||
** Flexible joint Geometry
|
||||
The flexible joint used for the Nano-Hexapod is shown in Figure [[fig:flexible_joint_geometry]].
|
||||
Its bending stiffness is foreseen to be $k_{R_y}\approx 20\,\frac{Nm}{rad}$ and its stroke $\theta_{y,\text{max}}\approx 20\,mrad$.
|
||||
|
||||
#+name: fig:flexible_joint_geometry
|
||||
#+caption: Geometry of the flexible joint
|
||||
[[file:figs/flexible_joint_geometry.png]]
|
||||
|
||||
The height between the flexible point (center of the joint) and the point where external forces are applied is $h = 20\,mm$.
|
||||
|
||||
Let's define the parameters on Matlab.
|
||||
#+begin_src matlab
|
||||
kRx = 20; % Bending Stiffness [Nm/rad]
|
||||
Rxmax = 20e-3; % Bending Stroke [rad]
|
||||
h = 20e-3; % Height [m]
|
||||
#+end_src
|
||||
|
||||
** Required external applied force
|
||||
|
||||
The bending $\theta_y$ of the flexible joint due to the force $F_x$ is:
|
||||
\begin{equation}
|
||||
\theta_y = \frac{M_y}{k_{R_y}} = \frac{F_x h}{k_{R_y}}
|
||||
\end{equation}
|
||||
|
||||
Therefore, the applied force to test the full range of the flexible joint is:
|
||||
\begin{equation}
|
||||
F_{x,\text{max}} = \frac{k_{R_y} \theta_{y,\text{max}}}{h}
|
||||
\end{equation}
|
||||
|
||||
#+begin_src matlab
|
||||
Fxmax = kRx*Rxmax/h; % Force to induce maximum stroke [N]
|
||||
#+end_src
|
||||
|
||||
And we obtain:
|
||||
#+begin_src matlab :results value raw replace :exports results
|
||||
sprintf('\\begin{equation} F_{max} = %.1f\\, [N] \\end{equation}', Fxmax)
|
||||
#+end_src
|
||||
|
||||
#+RESULTS:
|
||||
\begin{equation} F_{max} = 20.0\, [N] \end{equation}
|
||||
|
||||
The measurement range of the force sensor should then be higher than $20\,N$.
|
||||
|
||||
** Required actuator stroke and sensors range
|
||||
|
||||
The flexible joint is designed to allow a bending motion of $\pm 20\,mrad$.
|
||||
The corresponding actuator stroke to impose such motion is:
|
||||
|
||||
\[ d_{x,\text{max}} = h \tan(R_{x,\text{max}}) \]
|
||||
|
||||
#+begin_src matlab
|
||||
dxmax = h*tan(Rxmax);
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab :results value raw replace :exports results
|
||||
sprintf('\\begin{equation} d_{max} = %.1f\\, [mm] \\end{equation}', 1e3*dxmax)
|
||||
#+end_src
|
||||
|
||||
#+RESULTS:
|
||||
\begin{equation} d_{max} = 0.4\, [mm] \end{equation}
|
||||
|
||||
In order to test the full range of the flexible joint, the stroke of the actuator should be higher than $0.4\,mm$.
|
||||
The measurement range of the displacement sensor should also be higher than $0.4\,mm$.
|
||||
|
||||
** First try with the APA95ML
|
||||
|
||||
The APA95ML as a stroke of $100\,\mu m$ and the encoder in parallel can easily measure the required stroke.
|
||||
|
||||
Suppose the full stroke of the APA can be used to bend the flexible joint (ideal case), the measured force will be:
|
||||
#+begin_src matlab
|
||||
Fxmax = kRx*100e-6/h^2; % Force at maximum stroke [N]
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab :results value raw replace :exports results
|
||||
sprintf('\\begin{equation} F_{max} = %.1f\\, [N] \\end{equation}', Fxmax)
|
||||
#+end_src
|
||||
|
||||
#+RESULTS:
|
||||
\begin{equation} F_{max} = 5.0\, [N] \end{equation}
|
||||
|
||||
And the tested angular range is:
|
||||
#+begin_src matlab
|
||||
Rmax = tan(100e-6/h);
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab :results value raw replace :exports results
|
||||
sprintf('\\begin{equation} \\theta_{max} = %.1f\\, [mrad] \\end{equation}', 1e3*Rmax)
|
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#+end_src
|
||||
|
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#+RESULTS:
|
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\begin{equation} \theta_{max} = 5.0\, [mrad] \end{equation}
|
||||
|
||||
* Experimental measurement
|
||||
Loading…
Reference in New Issue
Block a user