195 lines
6.0 KiB
HTML
195 lines
6.0 KiB
HTML
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<?xml version="1.0" encoding="utf-8"?>
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<?xml version="1.0" encoding="utf-8"?>
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<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
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"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
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<html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en">
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<head>
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<!-- 2020-05-06 mer. 16:02 -->
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<meta http-equiv="Content-Type" content="text/html;charset=utf-8" />
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<title>Centrifugal Forces</title>
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<meta name="generator" content="Org mode" />
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<meta name="author" content="Dehaeze Thomas" />
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<script src="./js/bootstrap.min.js"></script>
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<script src="./js/jquery.stickytableheaders.min.js"></script>
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<script>MathJax = {
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tex: {
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tags: 'ams',
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macros: {bm: ["\\boldsymbol{#1}",1],}
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<script type="text/javascript" src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script>
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</head>
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<body>
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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>
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</div><div id="content">
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<h1 class="title">Centrifugal Forces</h1>
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<div id="table-of-contents">
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<h2>Table of Contents</h2>
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<div id="text-table-of-contents">
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<ul>
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<li><a href="#orga1e0a81">1. Parameters</a></li>
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<li><a href="#orge469792">2. Centrifugal forces for light and heavy sample</a></li>
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<li><a href="#orgb479647">3. Centrifugal forces as a function of the rotation speed</a></li>
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<li><a href="#orgd6375c3">4. Maximum rotation speed as a function of the mass</a></li>
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</ul>
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</div>
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</div>
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<p>
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In this document, we wish to estimate the centrifugal forces due to the spindle’s rotation when the sample’s center of mass is off-centered with respect to the rotation axis.
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</p>
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<p>
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This is the case then the sample is moved by the micro-hexapod.
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</p>
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<p>
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The centrifugal forces are defined as represented Figure <a href="#org1cf510f">1</a> where:
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</p>
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<ul class="org-ul">
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<li>\(M\) is the total mass of the rotating elements in \([kg]\)</li>
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<li>\(\omega\) is the rotation speed in \([rad/s]\)</li>
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<li>\(r\) is the distance to the rotation axis in \([m]\)</li>
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</ul>
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<div id="org1cf510f" class="figure">
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<p><img src="./figs/centrifugal.png" alt="centrifugal.png" />
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</p>
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<p><span class="figure-number">Figure 1: </span>Centrifugal forces</p>
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</div>
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<div id="outline-container-orga1e0a81" class="outline-2">
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<h2 id="orga1e0a81"><span class="section-number-2">1</span> Parameters</h2>
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<div class="outline-text-2" id="text-1">
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<p>
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We define some parameters for the computation.
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</p>
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<p>
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The mass of the sample can vary from \(1\,kg\) to \(50\,kg\) to which is added to mass of the metrology reflector and the nano-hexapod’s top platform (here set to \(15\,kg\)).
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</p>
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<div class="org-src-container">
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<pre class="src src-matlab">M_light = 16; % mass of excentred parts mooving [kg]
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M_heavy = 65; % [kg]
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</pre>
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</div>
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<p>
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For the light mass, the rotation speed is 60rpm whereas for the heavy mass, it is equal to 1rpm.
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</p>
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<div class="org-src-container">
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<pre class="src src-matlab">w_light = 2*pi; % rotational speed [rad/s]
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w_heavy = 2*pi/60; % rotational speed [rad/s]
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</pre>
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</div>
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<p>
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Finally, we consider a mass eccentricity of \(10\,mm\).
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</p>
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<div class="org-src-container">
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<pre class="src src-matlab">R = 0.01; % Excentricity [m]
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</pre>
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</div>
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</div>
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</div>
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<div id="outline-container-orge469792" class="outline-2">
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<h2 id="orge469792"><span class="section-number-2">2</span> Centrifugal forces for light and heavy sample</h2>
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<div class="outline-text-2" id="text-2">
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<p>
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From the formula \(F_c = m \omega^2 r\), we obtain the values shown below.
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</p>
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<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
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<colgroup>
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<col class="org-left" />
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<col class="org-right" />
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</colgroup>
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<thead>
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<tr>
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<th scope="col" class="org-left"> </th>
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<th scope="col" class="org-right">Force [N]</th>
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</tr>
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</thead>
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<tbody>
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<tr>
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<td class="org-left">light</td>
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<td class="org-right">6.32</td>
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</tr>
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<tr>
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<td class="org-left">heavy</td>
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<td class="org-right">0.01</td>
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</tr>
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</tbody>
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</table>
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</div>
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</div>
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<div id="outline-container-orgb479647" class="outline-2">
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<h2 id="orgb479647"><span class="section-number-2">3</span> Centrifugal forces as a function of the rotation speed</h2>
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<div class="outline-text-2" id="text-3">
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<p>
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The centrifugal forces as a function of the rotation speed for light and heavy sample is shown on Figure <a href="#orgdb75476">2</a>.
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</p>
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<div id="orgdb75476" class="figure">
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<p><img src="figs/centrifugal_forces_rpm.png" alt="centrifugal_forces_rpm.png" />
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</p>
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<p><span class="figure-number">Figure 2: </span>Centrifugal forces function of the rotation speed</p>
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</div>
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</div>
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</div>
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<div id="outline-container-orgd6375c3" class="outline-2">
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<h2 id="orgd6375c3"><span class="section-number-2">4</span> Maximum rotation speed as a function of the mass</h2>
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<div class="outline-text-2" id="text-4">
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<p>
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We plot the maximum rotation speed as a function of the mass for different maximum force that we can use to counteract the centrifugal forces (Figure <a href="#org522f2af">3</a>).
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</p>
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<p>
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From a specified maximum allowed centrifugal force (here set to \(10\,[N]\)), the maximum rotation speed as a function of the sample’s mass is shown in Figure <a href="#org522f2af">3</a>.
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</p>
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<div class="org-src-container">
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<pre class="src src-matlab">F_max = 10; % Maximum accepted centrifugal forces [N]
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R = 0.01;
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M_sample = 0:1:100;
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M_reflector = 15;
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</pre>
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</div>
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<div id="org522f2af" class="figure">
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<p><img src="figs/max_force_rpm.png" alt="max_force_rpm.png" />
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</p>
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<p><span class="figure-number">Figure 3: </span>Maximum rotation speed as a function of the sample mass for an allowed centrifugal force of \(100\,[N]\)</p>
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</div>
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</div>
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</div>
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</div>
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<div id="postamble" class="status">
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<p class="author">Author: Dehaeze Thomas</p>
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<p class="date">Created: 2020-05-06 mer. 16:02</p>
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</div>
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</body>
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</html>
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