729 lines
19 KiB
HTML
729 lines
19 KiB
HTML
<?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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<!-- 2020-06-14 dim. 12:22 -->
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<meta http-equiv="Content-Type" content="text/html;charset=utf-8" />
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<title>Finite Element Model with Simscape</title>
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<meta name="author" content="Dehaeze Thomas" />
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<script src="./js/readtheorg.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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<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">Finite Element Model with Simscape</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="#orge9e8230">1. Amplified Piezoelectric Actuator - 3D elements</a>
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<ul>
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<li><a href="#org710a905">1.1. Import Mass Matrix, Stiffness Matrix, and Interface Nodes Coordinates</a></li>
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<li><a href="#org25c7a89">1.2. Identification of the Dynamics</a></li>
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<li><a href="#org11dc96b">1.3. Comparison with Ansys</a></li>
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<li><a href="#orgfa4ae15">1.4. Force Sensor</a></li>
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</ul>
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</li>
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</ul>
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</div>
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</div>
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<div id="outline-container-orge9e8230" class="outline-2">
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<h2 id="orge9e8230"><span class="section-number-2">1</span> Amplified Piezoelectric Actuator - 3D elements</h2>
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<div class="outline-text-2" id="text-1">
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<p>
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The idea here is to:
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</p>
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<ul class="org-ul">
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<li>export a FEM of an amplified piezoelectric actuator from Ansys to Matlab</li>
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<li>import it into a Simscape model</li>
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<li>compare the obtained dynamics</li>
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<li>add 10kg mass on top of the actuator and identify the dynamics</li>
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<li>compare with results from Ansys where 10kg are directly added to the FEM</li>
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</ul>
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</div>
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<div id="outline-container-org710a905" class="outline-3">
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<h3 id="org710a905"><span class="section-number-3">1.1</span> Import Mass Matrix, Stiffness Matrix, and Interface Nodes Coordinates</h3>
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<div class="outline-text-3" id="text-1-1">
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<p>
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We first extract the stiffness and mass matrices.
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</p>
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<div class="org-src-container">
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<pre class="src src-matlab">K = extractMatrix('piezo_amplified_3d_K.txt');
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M = extractMatrix('piezo_amplified_3d_M.txt');
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</pre>
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</div>
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<p>
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Then, we extract the coordinates of the interface nodes.
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</p>
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<div class="org-src-container">
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<pre class="src src-matlab">[int_xyz, int_i, n_xyz, n_i, nodes] = extractNodes('piezo_amplified_3d.txt');
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</pre>
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</div>
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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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<tbody>
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<tr>
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<td class="org-left">Total number of Nodes</td>
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<td class="org-right">168959</td>
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</tr>
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<tr>
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<td class="org-left">Number of interface Nodes</td>
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<td class="org-right">13</td>
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</tr>
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<tr>
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<td class="org-left">Number of Modes</td>
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<td class="org-right">30</td>
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</tr>
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<tr>
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<td class="org-left">Size of M and K matrices</td>
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<td class="org-right">108</td>
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</tr>
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</tbody>
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</table>
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<div id="orgce84261" class="figure">
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<p><img src="figs/amplified_piezo_interface_nodes.png" alt="amplified_piezo_interface_nodes.png" />
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</p>
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<p><span class="figure-number">Figure 1: </span>Interface Nodes for the Amplified Piezo Actuator</p>
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</div>
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<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
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<caption class="t-above"><span class="table-number">Table 1:</span> Coordinates of the interface nodes</caption>
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<colgroup>
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<col class="org-right" />
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<col class="org-right" />
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<col class="org-right" />
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<col class="org-right" />
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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-right">Node i</th>
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<th scope="col" class="org-right">Node Number</th>
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<th scope="col" class="org-right">x [m]</th>
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<th scope="col" class="org-right">y [m]</th>
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<th scope="col" class="org-right">z [m]</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-right">1.0</td>
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<td class="org-right">168947.0</td>
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<td class="org-right">0.0</td>
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<td class="org-right">0.03</td>
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<td class="org-right">0.0</td>
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</tr>
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<tr>
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<td class="org-right">2.0</td>
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<td class="org-right">168949.0</td>
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<td class="org-right">0.0</td>
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<td class="org-right">-0.03</td>
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<td class="org-right">0.0</td>
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</tr>
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<tr>
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<td class="org-right">3.0</td>
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<td class="org-right">168950.0</td>
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<td class="org-right">-0.035</td>
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<td class="org-right">0.0</td>
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<td class="org-right">0.0</td>
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</tr>
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<tr>
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<td class="org-right">4.0</td>
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<td class="org-right">168951.0</td>
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<td class="org-right">-0.028</td>
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<td class="org-right">0.0</td>
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<td class="org-right">0.0</td>
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</tr>
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<tr>
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<td class="org-right">5.0</td>
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<td class="org-right">168952.0</td>
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<td class="org-right">-0.021</td>
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<td class="org-right">0.0</td>
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<td class="org-right">0.0</td>
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</tr>
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<tr>
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<td class="org-right">6.0</td>
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<td class="org-right">168953.0</td>
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<td class="org-right">-0.014</td>
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<td class="org-right">0.0</td>
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<td class="org-right">0.0</td>
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</tr>
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<tr>
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<td class="org-right">7.0</td>
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<td class="org-right">168954.0</td>
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<td class="org-right">-0.007</td>
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<td class="org-right">0.0</td>
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<td class="org-right">0.0</td>
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</tr>
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<tr>
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<td class="org-right">8.0</td>
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<td class="org-right">168955.0</td>
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<td class="org-right">0.0</td>
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<td class="org-right">0.0</td>
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<td class="org-right">0.0</td>
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</tr>
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<tr>
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<td class="org-right">9.0</td>
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<td class="org-right">168956.0</td>
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<td class="org-right">0.007</td>
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<td class="org-right">0.0</td>
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<td class="org-right">0.0</td>
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</tr>
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<tr>
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<td class="org-right">10.0</td>
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<td class="org-right">168957.0</td>
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<td class="org-right">0.014</td>
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<td class="org-right">0.0</td>
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<td class="org-right">0.0</td>
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</tr>
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<tr>
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<td class="org-right">11.0</td>
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<td class="org-right">168958.0</td>
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<td class="org-right">0.021</td>
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<td class="org-right">0.0</td>
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<td class="org-right">0.0</td>
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</tr>
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<tr>
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<td class="org-right">12.0</td>
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<td class="org-right">168959.0</td>
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<td class="org-right">0.035</td>
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<td class="org-right">0.0</td>
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<td class="org-right">0.0</td>
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</tr>
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<tr>
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<td class="org-right">13.0</td>
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<td class="org-right">168960.0</td>
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<td class="org-right">0.028</td>
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<td class="org-right">0.0</td>
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<td class="org-right">0.0</td>
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</tr>
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</tbody>
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</table>
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<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
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<caption class="t-above"><span class="table-number">Table 2:</span> First 10x10 elements of the Stiffness matrix</caption>
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<colgroup>
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<col class="org-right" />
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<col class="org-right" />
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<col class="org-right" />
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<col class="org-right" />
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<col class="org-right" />
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<col class="org-right" />
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<col class="org-right" />
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<col class="org-right" />
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<col class="org-right" />
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<col class="org-right" />
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</colgroup>
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<tbody>
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<tr>
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<td class="org-right">300000000.0</td>
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<td class="org-right">-30000.0</td>
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<td class="org-right">8000.0</td>
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<td class="org-right">-200.0</td>
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<td class="org-right">-30.0</td>
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<td class="org-right">-60000.0</td>
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<td class="org-right">20000000.0</td>
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<td class="org-right">-4000.0</td>
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<td class="org-right">500.0</td>
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<td class="org-right">8</td>
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</tr>
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<tr>
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<td class="org-right">-30000.0</td>
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<td class="org-right">100000000.0</td>
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<td class="org-right">400.0</td>
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<td class="org-right">30.0</td>
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<td class="org-right">200.0</td>
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<td class="org-right">-1</td>
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<td class="org-right">4000.0</td>
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<td class="org-right">-8000000.0</td>
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<td class="org-right">800.0</td>
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<td class="org-right">7</td>
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</tr>
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<tr>
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<td class="org-right">8000.0</td>
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<td class="org-right">400.0</td>
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<td class="org-right">50000000.0</td>
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<td class="org-right">-800000.0</td>
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<td class="org-right">-300.0</td>
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<td class="org-right">-40.0</td>
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<td class="org-right">300.0</td>
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<td class="org-right">100.0</td>
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<td class="org-right">5000000.0</td>
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<td class="org-right">40000.0</td>
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</tr>
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<tr>
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<td class="org-right">-200.0</td>
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<td class="org-right">30.0</td>
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<td class="org-right">-800000.0</td>
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<td class="org-right">20000.0</td>
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<td class="org-right">5</td>
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<td class="org-right">1</td>
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<td class="org-right">-10.0</td>
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<td class="org-right">-2</td>
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<td class="org-right">-40000.0</td>
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<td class="org-right">-300.0</td>
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</tr>
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<tr>
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<td class="org-right">-30.0</td>
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<td class="org-right">200.0</td>
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<td class="org-right">-300.0</td>
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<td class="org-right">5</td>
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<td class="org-right">40000.0</td>
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<td class="org-right">0.3</td>
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<td class="org-right">-4</td>
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<td class="org-right">-10.0</td>
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<td class="org-right">40.0</td>
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<td class="org-right">0.4</td>
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</tr>
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<tr>
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<td class="org-right">-60000.0</td>
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|
<td class="org-right">-1</td>
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<td class="org-right">-40.0</td>
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<td class="org-right">1</td>
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<td class="org-right">0.3</td>
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<td class="org-right">3000.0</td>
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<td class="org-right">7000.0</td>
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<td class="org-right">0.8</td>
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<td class="org-right">-1</td>
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<td class="org-right">0.0003</td>
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</tr>
|
|
|
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<tr>
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<td class="org-right">20000000.0</td>
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|
<td class="org-right">4000.0</td>
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|
<td class="org-right">300.0</td>
|
|
<td class="org-right">-10.0</td>
|
|
<td class="org-right">-4</td>
|
|
<td class="org-right">7000.0</td>
|
|
<td class="org-right">300000000.0</td>
|
|
<td class="org-right">20000.0</td>
|
|
<td class="org-right">3000.0</td>
|
|
<td class="org-right">80.0</td>
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</tr>
|
|
|
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<tr>
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<td class="org-right">-4000.0</td>
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<td class="org-right">-8000000.0</td>
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<td class="org-right">100.0</td>
|
|
<td class="org-right">-2</td>
|
|
<td class="org-right">-10.0</td>
|
|
<td class="org-right">0.8</td>
|
|
<td class="org-right">20000.0</td>
|
|
<td class="org-right">100000000.0</td>
|
|
<td class="org-right">-4000.0</td>
|
|
<td class="org-right">-100.0</td>
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|
</tr>
|
|
|
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<tr>
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<td class="org-right">500.0</td>
|
|
<td class="org-right">800.0</td>
|
|
<td class="org-right">5000000.0</td>
|
|
<td class="org-right">-40000.0</td>
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<td class="org-right">40.0</td>
|
|
<td class="org-right">-1</td>
|
|
<td class="org-right">3000.0</td>
|
|
<td class="org-right">-4000.0</td>
|
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<td class="org-right">50000000.0</td>
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<td class="org-right">800000.0</td>
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</tr>
|
|
|
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<tr>
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<td class="org-right">8</td>
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<td class="org-right">7</td>
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<td class="org-right">40000.0</td>
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<td class="org-right">-300.0</td>
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<td class="org-right">0.4</td>
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<td class="org-right">0.0003</td>
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<td class="org-right">80.0</td>
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<td class="org-right">-100.0</td>
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<td class="org-right">800000.0</td>
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<td class="org-right">20000.0</td>
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</tr>
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</tbody>
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</table>
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<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
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<caption class="t-above"><span class="table-number">Table 3:</span> First 10x10 elements of the Mass matrix</caption>
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<colgroup>
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<col class="org-right" />
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<col class="org-right" />
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<col class="org-right" />
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<col class="org-right" />
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<col class="org-right" />
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<col class="org-right" />
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<col class="org-right" />
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<col class="org-right" />
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<col class="org-right" />
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<col class="org-right" />
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</colgroup>
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<tbody>
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<tr>
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<td class="org-right">0.03</td>
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<td class="org-right">2e-06</td>
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<td class="org-right">-2e-07</td>
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<td class="org-right">1e-08</td>
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<td class="org-right">2e-08</td>
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<td class="org-right">0.0002</td>
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<td class="org-right">-0.001</td>
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<td class="org-right">2e-07</td>
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<td class="org-right">-8e-08</td>
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<td class="org-right">-9e-10</td>
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</tr>
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<tr>
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<td class="org-right">2e-06</td>
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<td class="org-right">0.02</td>
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<td class="org-right">-5e-07</td>
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<td class="org-right">7e-09</td>
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<td class="org-right">3e-08</td>
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<td class="org-right">2e-08</td>
|
|
<td class="org-right">-3e-07</td>
|
|
<td class="org-right">0.0003</td>
|
|
<td class="org-right">-1e-08</td>
|
|
<td class="org-right">1e-10</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">-2e-07</td>
|
|
<td class="org-right">-5e-07</td>
|
|
<td class="org-right">0.02</td>
|
|
<td class="org-right">-9e-05</td>
|
|
<td class="org-right">4e-09</td>
|
|
<td class="org-right">-1e-08</td>
|
|
<td class="org-right">2e-07</td>
|
|
<td class="org-right">-2e-08</td>
|
|
<td class="org-right">-0.0006</td>
|
|
<td class="org-right">-5e-06</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">1e-08</td>
|
|
<td class="org-right">7e-09</td>
|
|
<td class="org-right">-9e-05</td>
|
|
<td class="org-right">1e-06</td>
|
|
<td class="org-right">6e-11</td>
|
|
<td class="org-right">4e-10</td>
|
|
<td class="org-right">-1e-09</td>
|
|
<td class="org-right">3e-11</td>
|
|
<td class="org-right">5e-06</td>
|
|
<td class="org-right">3e-08</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">2e-08</td>
|
|
<td class="org-right">3e-08</td>
|
|
<td class="org-right">4e-09</td>
|
|
<td class="org-right">6e-11</td>
|
|
<td class="org-right">1e-06</td>
|
|
<td class="org-right">2e-10</td>
|
|
<td class="org-right">-2e-09</td>
|
|
<td class="org-right">2e-10</td>
|
|
<td class="org-right">-7e-09</td>
|
|
<td class="org-right">-4e-11</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">0.0002</td>
|
|
<td class="org-right">2e-08</td>
|
|
<td class="org-right">-1e-08</td>
|
|
<td class="org-right">4e-10</td>
|
|
<td class="org-right">2e-10</td>
|
|
<td class="org-right">2e-06</td>
|
|
<td class="org-right">-2e-06</td>
|
|
<td class="org-right">-1e-09</td>
|
|
<td class="org-right">-7e-10</td>
|
|
<td class="org-right">-9e-12</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">-0.001</td>
|
|
<td class="org-right">-3e-07</td>
|
|
<td class="org-right">2e-07</td>
|
|
<td class="org-right">-1e-09</td>
|
|
<td class="org-right">-2e-09</td>
|
|
<td class="org-right">-2e-06</td>
|
|
<td class="org-right">0.03</td>
|
|
<td class="org-right">-2e-06</td>
|
|
<td class="org-right">-1e-07</td>
|
|
<td class="org-right">-5e-09</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">2e-07</td>
|
|
<td class="org-right">0.0003</td>
|
|
<td class="org-right">-2e-08</td>
|
|
<td class="org-right">3e-11</td>
|
|
<td class="org-right">2e-10</td>
|
|
<td class="org-right">-1e-09</td>
|
|
<td class="org-right">-2e-06</td>
|
|
<td class="org-right">0.02</td>
|
|
<td class="org-right">-8e-07</td>
|
|
<td class="org-right">-1e-08</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">-8e-08</td>
|
|
<td class="org-right">-1e-08</td>
|
|
<td class="org-right">-0.0006</td>
|
|
<td class="org-right">5e-06</td>
|
|
<td class="org-right">-7e-09</td>
|
|
<td class="org-right">-7e-10</td>
|
|
<td class="org-right">-1e-07</td>
|
|
<td class="org-right">-8e-07</td>
|
|
<td class="org-right">0.02</td>
|
|
<td class="org-right">9e-05</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">-9e-10</td>
|
|
<td class="org-right">1e-10</td>
|
|
<td class="org-right">-5e-06</td>
|
|
<td class="org-right">3e-08</td>
|
|
<td class="org-right">-4e-11</td>
|
|
<td class="org-right">-9e-12</td>
|
|
<td class="org-right">-5e-09</td>
|
|
<td class="org-right">-1e-08</td>
|
|
<td class="org-right">9e-05</td>
|
|
<td class="org-right">1e-06</td>
|
|
</tr>
|
|
</tbody>
|
|
</table>
|
|
|
|
<p>
|
|
Using <code>K</code>, <code>M</code> and <code>int_xyz</code>, we can use the <code>Reduced Order Flexible Solid</code> simscape block.
|
|
</p>
|
|
</div>
|
|
</div>
|
|
|
|
|
|
<div id="outline-container-org25c7a89" class="outline-3">
|
|
<h3 id="org25c7a89"><span class="section-number-3">1.2</span> Identification of the Dynamics</h3>
|
|
<div class="outline-text-3" id="text-1-2">
|
|
<p>
|
|
The flexible element is imported using the <code>Reduced Order Flexible Solid</code> simscape block.
|
|
</p>
|
|
|
|
<p>
|
|
To model the actuator, an <code>Internal Force</code> block is added between the first and second nodes.
|
|
To model the sensors, a <code>Relative Motion Sensor</code> block is added between the second and the third nodes.
|
|
</p>
|
|
|
|
<p>
|
|
Two masses are fixed at the ends of the piezo-electric stack actuator.
|
|
</p>
|
|
|
|
<p>
|
|
We first set the mass to be zero.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">m = 0;
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
The dynamics is identified from the applied force to the measured relative displacement.
|
|
</p>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">%% Name of the Simulink File
|
|
mdl = 'piezo_amplified_3d';
|
|
|
|
%% Input/Output definition
|
|
clear io; io_i = 1;
|
|
io(io_i) = linio([mdl, '/F'], 1, 'openinput'); io_i = io_i + 1;
|
|
io(io_i) = linio([mdl, '/y'], 1, 'openoutput'); io_i = io_i + 1;
|
|
|
|
Gh = linearize(mdl, io);
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
Then, we add 10Kg of mass:
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">m = 10;
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
And the dynamics is identified.
|
|
</p>
|
|
|
|
<p>
|
|
The two identified dynamics are compared in Figure <a href="#orga2eee2b">2</a>.
|
|
</p>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">%% Name of the Simulink File
|
|
mdl = 'piezo_amplified_3d';
|
|
|
|
%% Input/Output definition
|
|
clear io; io_i = 1;
|
|
io(io_i) = linio([mdl, '/F'], 1, 'openinput'); io_i = io_i + 1;
|
|
io(io_i) = linio([mdl, '/y'], 1, 'openoutput'); io_i = io_i + 1;
|
|
|
|
Ghm = linearize(mdl, io);
|
|
</pre>
|
|
</div>
|
|
|
|
|
|
<div id="orga2eee2b" class="figure">
|
|
<p><img src="figs/dynamics_act_disp_comp_mass.png" alt="dynamics_act_disp_comp_mass.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 2: </span>Dynamics from \(F\) to \(d\) without a payload and with a 10kg payload</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org11dc96b" class="outline-3">
|
|
<h3 id="org11dc96b"><span class="section-number-3">1.3</span> Comparison with Ansys</h3>
|
|
<div class="outline-text-3" id="text-1-3">
|
|
<p>
|
|
Let’s import the results from an Harmonic response analysis in Ansys.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">Gresp0 = readtable('FEA_HarmResponse_00kg.txt');
|
|
Gresp10 = readtable('FEA_HarmResponse_10kg.txt');
|
|
</pre>
|
|
</div>
|
|
|
|
|
|
<div id="orgee19caf" class="figure">
|
|
<p><img src="figs/dynamics_force_disp_comp_anasys.png" alt="dynamics_force_disp_comp_anasys.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 3: </span>Comparison of the obtained dynamics using Simscape with the harmonic response analysis using Ansys</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orgfa4ae15" class="outline-3">
|
|
<h3 id="orgfa4ae15"><span class="section-number-3">1.4</span> Force Sensor</h3>
|
|
<div class="outline-text-3" id="text-1-4">
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">m = 0;
|
|
</pre>
|
|
</div>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">%% Name of the Simulink File
|
|
mdl = 'piezo_amplified_3d';
|
|
|
|
%% Input/Output definition
|
|
clear io; io_i = 1;
|
|
io(io_i) = linio([mdl, '/Fa'], 1, 'openinput'); io_i = io_i + 1;
|
|
io(io_i) = linio([mdl, '/Fs'], 1, 'openoutput'); io_i = io_i + 1;
|
|
|
|
Gf = linearize(mdl, io);
|
|
</pre>
|
|
</div>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">m = 10;
|
|
</pre>
|
|
</div>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">%% Name of the Simulink File
|
|
mdl = 'piezo_amplified_3d';
|
|
|
|
%% Input/Output definition
|
|
clear io; io_i = 1;
|
|
io(io_i) = linio([mdl, '/Fa'], 1, 'openinput'); io_i = io_i + 1;
|
|
io(io_i) = linio([mdl, '/Fs'], 1, 'openoutput'); io_i = io_i + 1;
|
|
|
|
Gfm = linearize(mdl, io);
|
|
</pre>
|
|
</div>
|
|
|
|
|
|
<div id="orgcfb5aa6" class="figure">
|
|
<p><img src="figs/dynamics_force_force_sensor_comp_mass.png" alt="dynamics_force_force_sensor_comp_mass.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 4: </span>Dynamics from \(F\) to \(F_m\) for \(m=0\) and \(m = 10kg\)</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
<div id="postamble" class="status">
|
|
<p class="author">Author: Dehaeze Thomas</p>
|
|
<p class="date">Created: 2020-06-14 dim. 12:22</p>
|
|
</div>
|
|
</body>
|
|
</html>
|