198 lines
6.5 KiB
HTML
198 lines
6.5 KiB
HTML
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<!-- 2021-02-20 sam. 23:09 -->
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<title>Centrifugal Forces</title>
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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>
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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="#org5199302">1. Parameters</a></li>
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<li><a href="#orga2ea10d">2. Centrifugal forces for light and heavy sample</a></li>
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<li><a href="#orgf375b50">3. Centrifugal forces as a function of the rotation speed</a></li>
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<li><a href="#orge7fb13d">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="#org91ed599">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="org91ed599" 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-org5199302" class="outline-2">
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<h2 id="org5199302"><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; <span class="org-comment">% mass of excentred parts mooving [kg]</span>
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M_heavy = 65; <span class="org-comment">% [kg]</span>
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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<span class="org-type">*</span><span class="org-constant">pi</span>; <span class="org-comment">% rotational speed [rad/s]</span>
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w_heavy = 2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">/</span>60; <span class="org-comment">% rotational speed [rad/s]</span>
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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; <span class="org-comment">% Excentricity [m]</span>
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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-orga2ea10d" class="outline-2">
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<h2 id="orga2ea10d"><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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<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-orgf375b50" class="outline-2">
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<h2 id="orgf375b50"><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="#org87b7644">2</a>.
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</p>
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<div id="org87b7644" 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-orge7fb13d" class="outline-2">
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<h2 id="orge7fb13d"><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="#org8fe6a07">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="#org8fe6a07">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; <span class="org-comment">% Maximum accepted centrifugal forces [N]</span>
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R = 0.01;
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M_sample = 0<span class="org-type">:</span>1<span class="org-type">:</span>100;
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M_reflector = 15;
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</pre>
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</div>
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<div id="org8fe6a07" 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: 2021-02-20 sam. 23:09</p>
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</div>
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</body>
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</html>
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