1114 lines
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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">Modal Analysis</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="#org356e5b0">1. Setup</a></li>
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<li><a href="#org8fdd762">2. Mode extraction and importation</a></li>
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<li><a href="#org689c124">3. Positions of the sensors</a></li>
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<li><a href="#org8972c45">4. Solids</a></li>
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<li><a href="#orgc79f768">5. From local coordinates to global coordinates</a></li>
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<li><a href="#orgaa2d870">6. Modal Matrices</a></li>
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<li><a href="#org16e1add">7. Modal Complexity</a></li>
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<li><a href="#orgc33fb24">8. Some notes about constraining the number of degrees of freedom</a></li>
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<li><a href="#orgd17ef15">9. <span class="todo TODO">TODO</span> Normalization of mode shapes?</a></li>
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<li><a href="#orgc7fe2ae">10. Compare Mode Shapes</a></li>
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<li><a href="#org1ec0de2">11. <span class="todo TODO">TODO</span> Synthesis of FRF curves</a></li>
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</ul>
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</div>
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<div id="outline-container-org356e5b0" class="outline-2">
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<h2 id="org356e5b0"><span class="section-number-2">1</span> Setup</h2>
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<div class="outline-text-2" id="text-1">
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<div id="org44c15fe" class="figure">
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<p><img src="figs/nass-modal-test.png" alt="nass-modal-test.png" />
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</p>
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<p><span class="figure-number">Figure 1: </span>Position and orientation of the accelerometer used</p>
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</div>
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</div>
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</div>
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<div id="outline-container-org8fdd762" class="outline-2">
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<h2 id="org8fdd762"><span class="section-number-2">2</span> Mode extraction and importation</h2>
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<div class="outline-text-2" id="text-2">
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<p>
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First, we split the big <code>modes.asc</code> files into sub text files using <code>bash</code>.
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</p>
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<div class="org-src-container">
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<pre class="src src-bash">sed <span class="org-string">'/^\s*[0-9]*[XYZ][+-]:/!d'</span> modal_analysis_updated/modes.asc > mat/mode_shapes.txt
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sed <span class="org-string">'/freq/!d'</span> modal_analysis_updated/modes.asc | sed <span class="org-string">'s/.* = \(.*\)Hz/\1/'</span> > mat/mode_freqs.txt
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sed <span class="org-string">'/damp/!d'</span> modal_analysis_updated/modes.asc | sed <span class="org-string">'s/.* = \(.*\)\%/\1/'</span> > mat/mode_damps.txt
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sed <span class="org-string">'/modal A/!d'</span> modal_analysis_updated/modes.asc | sed <span class="org-string">'s/.* =\s\+\([-0-9.e]\++[0-9]\+\)\([-+0-9.e]\+\)i/\1 \2/'</span> > mat/mode_modal_a.txt
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sed <span class="org-string">'/modal B/!d'</span> modal_analysis_updated/modes.asc | sed <span class="org-string">'s/.* =\s\+\([-0-9.e]\++[0-9]\+\)\([-+0-9.e]\+\)i/\1 \2/'</span> > mat/mode_modal_b.txt
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</pre>
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</div>
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<p>
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Then we import them on Matlab.
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</p>
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<div class="org-src-container">
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<pre class="src src-matlab">shapes = readtable<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'mat/mode_shapes.txt', 'ReadVariableNames'</span>, false<span class="org-rainbow-delimiters-depth-1">)</span>; <span class="org-comment">% [Sign / Real / Imag]</span>
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freqs = table2array<span class="org-rainbow-delimiters-depth-1">(</span>readtable<span class="org-rainbow-delimiters-depth-2">(</span><span class="org-string">'mat/mode_freqs.txt', 'ReadVariableNames'</span>, false<span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span>; <span class="org-comment">% in [Hz]</span>
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damps = table2array<span class="org-rainbow-delimiters-depth-1">(</span>readtable<span class="org-rainbow-delimiters-depth-2">(</span><span class="org-string">'mat/mode_damps.txt', 'ReadVariableNames'</span>, false<span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span>; <span class="org-comment">% in [%]</span>
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modal_a = table2array<span class="org-rainbow-delimiters-depth-1">(</span>readtable<span class="org-rainbow-delimiters-depth-2">(</span><span class="org-string">'mat/mode_modal_a.txt', 'ReadVariableNames'</span>, false<span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span>; <span class="org-comment">% [Real / Imag]</span>
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modal_a = complex<span class="org-rainbow-delimiters-depth-1">(</span>modal_a<span class="org-rainbow-delimiters-depth-2">(</span><span class="org-type">:</span>, <span class="org-highlight-numbers-number">1</span><span class="org-rainbow-delimiters-depth-2">)</span>, modal_a<span class="org-rainbow-delimiters-depth-2">(</span><span class="org-type">:</span>, <span class="org-highlight-numbers-number">2</span><span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span>;
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modal_b = table2array<span class="org-rainbow-delimiters-depth-1">(</span>readtable<span class="org-rainbow-delimiters-depth-2">(</span><span class="org-string">'mat/mode_modal_b.txt', 'ReadVariableNames'</span>, false<span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span>; <span class="org-comment">% [Real / Imag]</span>
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modal_b = complex<span class="org-rainbow-delimiters-depth-1">(</span>modal_b<span class="org-rainbow-delimiters-depth-2">(</span><span class="org-type">:</span>, <span class="org-highlight-numbers-number">1</span><span class="org-rainbow-delimiters-depth-2">)</span>, modal_b<span class="org-rainbow-delimiters-depth-2">(</span><span class="org-type">:</span>, <span class="org-highlight-numbers-number">2</span><span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span>;
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</pre>
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</div>
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<p>
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We guess the number of modes identified from the length of the imported data.
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</p>
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<div class="org-src-container">
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<pre class="src src-matlab">acc_n = <span class="org-highlight-numbers-number">23</span>; <span class="org-comment">% Number of accelerometers</span>
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dir_n = <span class="org-highlight-numbers-number">3</span>; <span class="org-comment">% Number of directions</span>
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dirs = <span class="org-string">'XYZ'</span>;
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|
|
mod_n = size<span class="org-rainbow-delimiters-depth-1">(</span>shapes,<span class="org-highlight-numbers-number">1</span><span class="org-rainbow-delimiters-depth-1">)</span><span class="org-type">/</span>acc_n<span class="org-type">/</span>dir_n; <span class="org-comment">% Number of modes</span>
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
As the mode shapes are split into 3 parts (direction plus sign, real part and imaginary part), we aggregate them into one array of complex numbers.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">T_sign = table2array<span class="org-rainbow-delimiters-depth-1">(</span>shapes<span class="org-rainbow-delimiters-depth-2">(</span><span class="org-type">:</span>, <span class="org-highlight-numbers-number">1</span><span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
T_real = table2array<span class="org-rainbow-delimiters-depth-1">(</span>shapes<span class="org-rainbow-delimiters-depth-2">(</span><span class="org-type">:</span>, <span class="org-highlight-numbers-number">2</span><span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
T_imag = table2array<span class="org-rainbow-delimiters-depth-1">(</span>shapes<span class="org-rainbow-delimiters-depth-2">(</span><span class="org-type">:</span>, <span class="org-highlight-numbers-number">3</span><span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
|
|
modes = zeros<span class="org-rainbow-delimiters-depth-1">(</span>mod_n, acc_n, dir_n<span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
|
|
<span class="org-keyword">for</span> <span class="org-variable-name">mod_i</span> = <span class="org-constant"><span class="org-highlight-numbers-number">1</span></span><span class="org-constant">:mod_n</span>
|
|
<span class="org-keyword">for</span> <span class="org-variable-name">acc_i</span> = <span class="org-constant"><span class="org-highlight-numbers-number">1</span></span><span class="org-constant">:acc_n</span>
|
|
<span class="org-comment">% Get the correct section of the signs</span>
|
|
T = T_sign<span class="org-rainbow-delimiters-depth-1">(</span>acc_n<span class="org-type">*</span>dir_n<span class="org-type">*</span><span class="org-rainbow-delimiters-depth-2">(</span>mod_i<span class="org-type">-</span><span class="org-highlight-numbers-number">1</span><span class="org-rainbow-delimiters-depth-2">)</span><span class="org-type">+</span><span class="org-highlight-numbers-number">1</span><span class="org-type">:</span>acc_n<span class="org-type">*</span>dir_n<span class="org-type">*</span>mod_i<span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
<span class="org-keyword">for</span> <span class="org-variable-name">dir_i</span> = <span class="org-constant"><span class="org-highlight-numbers-number">1</span></span><span class="org-constant">:dir_n</span>
|
|
<span class="org-comment">% Get the line corresponding to the sensor</span>
|
|
<span class="org-constant">i</span> = find<span class="org-rainbow-delimiters-depth-1">(</span>contains<span class="org-rainbow-delimiters-depth-2">(</span>T, sprintf<span class="org-rainbow-delimiters-depth-3">(</span>'<span class="org-comment">%i</span>%s',acc_i, dirs(dir_i))), <span class="org-highlight-numbers-number">1</span>, 'first')<span class="org-type">+</span>acc_n<span class="org-type">*</span>dir_n<span class="org-type">*</span>(mod_i<span class="org-type">-</span><span class="org-highlight-numbers-number">1</span>);
|
|
modes<span class="org-rainbow-delimiters-depth-4">(</span>mod_i, acc_i, dir_i<span class="org-rainbow-delimiters-depth-4">)</span> = str2num<span class="org-rainbow-delimiters-depth-4">(</span><span class="org-rainbow-delimiters-depth-5">[</span>T_sign<span class="org-rainbow-delimiters-depth-6">{</span><span class="org-constant">i</span><span class="org-rainbow-delimiters-depth-6">}(</span>end<span class="org-type">-</span><span class="org-highlight-numbers-number">1</span><span class="org-rainbow-delimiters-depth-6">)</span>, '<span class="org-highlight-numbers-number">1</span>'<span class="org-rainbow-delimiters-depth-5">]</span><span class="org-rainbow-delimiters-depth-4">)</span><span class="org-type">*</span>complex<span class="org-rainbow-delimiters-depth-4">(</span>T_real<span class="org-rainbow-delimiters-depth-5">(</span><span class="org-constant">i</span><span class="org-rainbow-delimiters-depth-5">)</span>,T_imag<span class="org-rainbow-delimiters-depth-5">(</span><span class="org-constant">i</span><span class="org-rainbow-delimiters-depth-5">)</span><span class="org-rainbow-delimiters-depth-4">)</span>;
|
|
<span class="org-keyword">end</span>
|
|
<span class="org-keyword">end</span>
|
|
<span class="org-keyword">end</span>
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
The obtained mode frequencies and damping are shown below.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">data2orgtable<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbow-delimiters-depth-2">[</span>freqs, damps<span class="org-rainbow-delimiters-depth-2">]</span>, <span class="org-rainbow-delimiters-depth-2">{}</span>, <span class="org-rainbow-delimiters-depth-2">{</span>'Frequency <span class="org-rainbow-delimiters-depth-3">[</span>Hz<span class="org-rainbow-delimiters-depth-3">]</span>', 'Damping <span class="org-rainbow-delimiters-depth-3">[</span><span class="org-comment">%]'}, ' </span>%.<span class="org-highlight-numbers-number">1f</span> ');
|
|
</pre>
|
|
</div>
|
|
|
|
<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
|
|
|
|
|
|
<colgroup>
|
|
<col class="org-right" />
|
|
|
|
<col class="org-right" />
|
|
</colgroup>
|
|
<thead>
|
|
<tr>
|
|
<th scope="col" class="org-right">Frequency [Hz]</th>
|
|
<th scope="col" class="org-right">Damping [%]</th>
|
|
</tr>
|
|
</thead>
|
|
<tbody>
|
|
<tr>
|
|
<td class="org-right">11.4</td>
|
|
<td class="org-right">8.7</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">18.5</td>
|
|
<td class="org-right">11.8</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">37.6</td>
|
|
<td class="org-right">6.4</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">39.4</td>
|
|
<td class="org-right">3.6</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">54.0</td>
|
|
<td class="org-right">0.2</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">56.1</td>
|
|
<td class="org-right">2.8</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">69.7</td>
|
|
<td class="org-right">4.6</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">71.6</td>
|
|
<td class="org-right">0.6</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">72.4</td>
|
|
<td class="org-right">1.6</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">84.9</td>
|
|
<td class="org-right">3.6</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">90.6</td>
|
|
<td class="org-right">0.3</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">91.0</td>
|
|
<td class="org-right">2.9</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">95.8</td>
|
|
<td class="org-right">3.3</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">105.4</td>
|
|
<td class="org-right">3.3</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">106.8</td>
|
|
<td class="org-right">1.9</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">112.6</td>
|
|
<td class="org-right">3.0</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">116.8</td>
|
|
<td class="org-right">2.7</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">124.1</td>
|
|
<td class="org-right">0.6</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">145.4</td>
|
|
<td class="org-right">1.6</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">150.1</td>
|
|
<td class="org-right">2.2</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">164.7</td>
|
|
<td class="org-right">1.4</td>
|
|
</tr>
|
|
</tbody>
|
|
</table>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org689c124" class="outline-2">
|
|
<h2 id="org689c124"><span class="section-number-2">3</span> Positions of the sensors</h2>
|
|
<div class="outline-text-2" id="text-3">
|
|
<p>
|
|
We process the file exported from the <code>modal</code> software containing the positions of the sensors using <code>bash</code>.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-bash">cat modal_analysis_updated/id31_nanostation_modified.cfg | grep NODES -A <span class="org-highlight-numbers-number">23</span> | sed <span class="org-string">'/\s\+[0-9]\+/!d'</span> | sed <span class="org-string">'s/\(.*\)\s\+0\s\+.\+/\1/'</span> > mat/acc_pos.txt
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
We then import that on <code>matlab</code>, and sort them.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">acc_pos = readtable<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'mat/acc_pos.txt', 'ReadVariableNames'</span>, false<span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
acc_pos = table2array<span class="org-rainbow-delimiters-depth-1">(</span>acc_pos<span class="org-rainbow-delimiters-depth-2">(</span><span class="org-type">:</span>, <span class="org-highlight-numbers-number">1</span><span class="org-type">:</span><span class="org-highlight-numbers-number">4</span><span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
<span class="org-rainbow-delimiters-depth-1">[</span><span class="org-type">~</span>, <span class="org-constant">i</span><span class="org-rainbow-delimiters-depth-1">]</span> = sort<span class="org-rainbow-delimiters-depth-1">(</span>acc_pos<span class="org-rainbow-delimiters-depth-2">(</span><span class="org-type">:</span>, <span class="org-highlight-numbers-number">1</span><span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
acc_pos = acc_pos<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-constant">i</span>, <span class="org-highlight-numbers-number">2</span><span class="org-type">:</span><span class="org-highlight-numbers-number">4</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
The positions of the sensors relative to the point of interest are shown below.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">data2orgtable<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">1000</span><span class="org-type">*</span>acc_pos, <span class="org-rainbow-delimiters-depth-2">{}</span>, <span class="org-rainbow-delimiters-depth-2">{</span>'x <span class="org-rainbow-delimiters-depth-3">[</span>mm<span class="org-rainbow-delimiters-depth-3">]</span>', 'y <span class="org-rainbow-delimiters-depth-3">[</span>mm<span class="org-rainbow-delimiters-depth-3">]</span>', 'z <span class="org-rainbow-delimiters-depth-3">[</span>mm<span class="org-rainbow-delimiters-depth-3">]</span>'<span class="org-rainbow-delimiters-depth-2">}</span>, ' %.<span class="org-highlight-numbers-number">0f</span> ');
|
|
</pre>
|
|
</div>
|
|
|
|
<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
|
|
|
|
|
|
<colgroup>
|
|
<col class="org-right" />
|
|
|
|
<col class="org-right" />
|
|
|
|
<col class="org-right" />
|
|
</colgroup>
|
|
<thead>
|
|
<tr>
|
|
<th scope="col" class="org-right">x [mm]</th>
|
|
<th scope="col" class="org-right">y [mm]</th>
|
|
<th scope="col" class="org-right">z [mm]</th>
|
|
</tr>
|
|
</thead>
|
|
<tbody>
|
|
<tr>
|
|
<td class="org-right">-64</td>
|
|
<td class="org-right">-64</td>
|
|
<td class="org-right">-296</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">-64</td>
|
|
<td class="org-right">64</td>
|
|
<td class="org-right">-296</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">64</td>
|
|
<td class="org-right">64</td>
|
|
<td class="org-right">-296</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">64</td>
|
|
<td class="org-right">-64</td>
|
|
<td class="org-right">-296</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">-385</td>
|
|
<td class="org-right">-300</td>
|
|
<td class="org-right">-417</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">-420</td>
|
|
<td class="org-right">280</td>
|
|
<td class="org-right">-417</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">420</td>
|
|
<td class="org-right">280</td>
|
|
<td class="org-right">-417</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">380</td>
|
|
<td class="org-right">-300</td>
|
|
<td class="org-right">-417</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">-475</td>
|
|
<td class="org-right">-414</td>
|
|
<td class="org-right">-427</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">-465</td>
|
|
<td class="org-right">407</td>
|
|
<td class="org-right">-427</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">475</td>
|
|
<td class="org-right">424</td>
|
|
<td class="org-right">-427</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">475</td>
|
|
<td class="org-right">-419</td>
|
|
<td class="org-right">-427</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">-320</td>
|
|
<td class="org-right">-446</td>
|
|
<td class="org-right">-786</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">-480</td>
|
|
<td class="org-right">534</td>
|
|
<td class="org-right">-786</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">450</td>
|
|
<td class="org-right">534</td>
|
|
<td class="org-right">-786</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">295</td>
|
|
<td class="org-right">-481</td>
|
|
<td class="org-right">-786</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">-730</td>
|
|
<td class="org-right">-526</td>
|
|
<td class="org-right">-951</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">-735</td>
|
|
<td class="org-right">814</td>
|
|
<td class="org-right">-951</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">875</td>
|
|
<td class="org-right">799</td>
|
|
<td class="org-right">-951</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">865</td>
|
|
<td class="org-right">-506</td>
|
|
<td class="org-right">-951</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">-155</td>
|
|
<td class="org-right">-90</td>
|
|
<td class="org-right">-594</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">0</td>
|
|
<td class="org-right">180</td>
|
|
<td class="org-right">-594</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">155</td>
|
|
<td class="org-right">-90</td>
|
|
<td class="org-right">-594</td>
|
|
</tr>
|
|
</tbody>
|
|
</table>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org8972c45" class="outline-2">
|
|
<h2 id="org8972c45"><span class="section-number-2">4</span> Solids</h2>
|
|
<div class="outline-text-2" id="text-4">
|
|
<p>
|
|
We consider the following solid bodies:
|
|
</p>
|
|
<ul class="org-ul">
|
|
<li>Bottom Granite</li>
|
|
<li>Top Granite</li>
|
|
<li>Translation Stage</li>
|
|
<li>Tilt Stage</li>
|
|
<li>Spindle</li>
|
|
<li>Hexapod</li>
|
|
</ul>
|
|
|
|
<p>
|
|
We create a structure <code>solids</code> that contains the accelerometer number of each solid bodies (as shown on figure <a href="#org44c15fe">1</a>).
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">solids = <span class="org-rainbow-delimiters-depth-1">{}</span>;
|
|
solids.granite_bot = <span class="org-rainbow-delimiters-depth-1">[</span><span class="org-highlight-numbers-number">17</span>, <span class="org-highlight-numbers-number">18</span>, <span class="org-highlight-numbers-number">19</span>, <span class="org-highlight-numbers-number">20</span><span class="org-rainbow-delimiters-depth-1">]</span>;
|
|
solids.granite_top = <span class="org-rainbow-delimiters-depth-1">[</span><span class="org-highlight-numbers-number">13</span>, <span class="org-highlight-numbers-number">14</span>, <span class="org-highlight-numbers-number">15</span>, <span class="org-highlight-numbers-number">16</span><span class="org-rainbow-delimiters-depth-1">]</span>;
|
|
solids.ty = <span class="org-rainbow-delimiters-depth-1">[</span><span class="org-highlight-numbers-number">9</span>, <span class="org-highlight-numbers-number">10</span>, <span class="org-highlight-numbers-number">11</span>, <span class="org-highlight-numbers-number">12</span><span class="org-rainbow-delimiters-depth-1">]</span>;
|
|
solids.ry = <span class="org-rainbow-delimiters-depth-1">[</span><span class="org-highlight-numbers-number">5</span>, <span class="org-highlight-numbers-number">6</span>, <span class="org-highlight-numbers-number">7</span>, <span class="org-highlight-numbers-number">8</span><span class="org-rainbow-delimiters-depth-1">]</span>;
|
|
solids.rz = <span class="org-rainbow-delimiters-depth-1">[</span><span class="org-highlight-numbers-number">21</span>, <span class="org-highlight-numbers-number">22</span>, <span class="org-highlight-numbers-number">23</span><span class="org-rainbow-delimiters-depth-1">]</span>;
|
|
solids.hexa = <span class="org-rainbow-delimiters-depth-1">[</span><span class="org-highlight-numbers-number">1</span>, <span class="org-highlight-numbers-number">2</span>, <span class="org-highlight-numbers-number">3</span>, <span class="org-highlight-numbers-number">4</span><span class="org-rainbow-delimiters-depth-1">]</span>;
|
|
|
|
solid_names = fields<span class="org-rainbow-delimiters-depth-1">(</span>solids<span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orgc79f768" class="outline-2">
|
|
<h2 id="orgc79f768"><span class="section-number-2">5</span> From local coordinates to global coordinates</h2>
|
|
<div class="outline-text-2" id="text-5">
|
|
|
|
<div class="figure">
|
|
<p><img src="Figures/local_to_global_coordinates.png" alt="local_to_global_coordinates.png" />
|
|
</p>
|
|
</div>
|
|
|
|
<p>
|
|
From the figure above, we can write:
|
|
</p>
|
|
\begin{align*}
|
|
\vec{v}_1 &= \vec{v} + \Omega \vec{p}_1\\
|
|
\vec{v}_2 &= \vec{v} + \Omega \vec{p}_2\\
|
|
\vec{v}_3 &= \vec{v} + \Omega \vec{p}_3\\
|
|
\vec{v}_4 &= \vec{v} + \Omega \vec{p}_4
|
|
\end{align*}
|
|
|
|
<p>
|
|
With
|
|
</p>
|
|
\begin{equation}
|
|
\Omega = \begin{bmatrix}
|
|
0 & -\Omega_z & \Omega_y \\
|
|
\Omega_z & 0 & -\Omega_x \\
|
|
-\Omega_y & \Omega_x & 0
|
|
\end{bmatrix}
|
|
\end{equation}
|
|
<p>
|
|
\(\vec{v}\) and \(\Omega\) represent to velocity and rotation of the solid expressed in the frame \(\{O\}\).
|
|
</p>
|
|
|
|
<p>
|
|
We can rearrange the equations in a matrix form:
|
|
</p>
|
|
|
|
\begin{equation}
|
|
\left[\begin{array}{ccc|ccc}
|
|
1 & 0 & 0 & 0 & p_{1z} & -p_{1y} \\
|
|
0 & 1 & 0 & -p_{1z} & 0 & p_{1x} \\
|
|
0 & 0 & 1 & p_{1y} & -p_{1x} & 0 \\ \hline
|
|
& \vdots & & & \vdots & \\ \hline
|
|
1 & 0 & 0 & 0 & p_{4z} & -p_{4y} \\
|
|
0 & 1 & 0 & -p_{4z} & 0 & p_{4x} \\
|
|
0 & 0 & 1 & p_{4y} & -p_{4x} & 0
|
|
\end{array}\right] \begin{bmatrix}
|
|
v_x \\ v_y \\ v_z \\ \hline \Omega_x \\ \Omega_y \\ \Omega_z
|
|
\end{bmatrix} = \begin{bmatrix}
|
|
v_{1x} \\ v_{1y} \\ v_{1z} \\\hline \vdots \\\hline v_{4x} \\ v_{4y} \\ v_{4z}
|
|
\end{bmatrix}
|
|
\end{equation}
|
|
|
|
<p>
|
|
and then we obtain the velocity and rotation of the solid in the wanted frame \(\{O\}\):
|
|
</p>
|
|
\begin{equation}
|
|
\begin{bmatrix}
|
|
v_x \\ v_y \\ v_z \\ \hline \Omega_x \\ \Omega_y \\ \Omega_z
|
|
\end{bmatrix} =
|
|
\left[\begin{array}{ccc|ccc}
|
|
1 & 0 & 0 & 0 & p_{1z} & -p_{1y} \\
|
|
0 & 1 & 0 & -p_{1z} & 0 & p_{1x} \\
|
|
0 & 0 & 1 & p_{1y} & -p_{1x} & 0 \\ \hline
|
|
& \vdots & & & \vdots & \\ \hline
|
|
1 & 0 & 0 & 0 & p_{4z} & -p_{4y} \\
|
|
0 & 1 & 0 & -p_{4z} & 0 & p_{4x} \\
|
|
0 & 0 & 1 & p_{4y} & -p_{4x} & 0
|
|
\end{array}\right]^{-1} \begin{bmatrix}
|
|
v_{1x} \\ v_{1y} \\ v_{1z} \\\hline \vdots \\\hline v_{4x} \\ v_{4y} \\ v_{4z}
|
|
\end{bmatrix}
|
|
\end{equation}
|
|
|
|
<p>
|
|
This inversion is equivalent to a mean square problem.
|
|
</p>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">mode_shapes_O = zeros<span class="org-rainbow-delimiters-depth-1">(</span>mod_n, length<span class="org-rainbow-delimiters-depth-2">(</span>solid_names<span class="org-rainbow-delimiters-depth-2">)</span>, <span class="org-highlight-numbers-number">6</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
|
|
<span class="org-keyword">for</span> <span class="org-variable-name">mod_i</span> = <span class="org-constant"><span class="org-highlight-numbers-number">1</span></span><span class="org-constant">:mod_n</span>
|
|
<span class="org-keyword">for</span> <span class="org-variable-name">solid_i</span> = <span class="org-constant"><span class="org-highlight-numbers-number">1</span></span><span class="org-constant">:length</span><span class="org-constant"><span class="org-rainbow-delimiters-depth-1">(</span></span><span class="org-constant">solid_names</span><span class="org-constant"><span class="org-rainbow-delimiters-depth-1">)</span></span>
|
|
solids_i = solids.<span class="org-rainbow-delimiters-depth-1">(</span>solid_names<span class="org-rainbow-delimiters-depth-2">{</span>solid_i<span class="org-rainbow-delimiters-depth-2">}</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
|
|
Y = reshape<span class="org-rainbow-delimiters-depth-1">(</span>squeeze<span class="org-rainbow-delimiters-depth-2">(</span>modes<span class="org-rainbow-delimiters-depth-3">(</span>mod_i, solids_i, <span class="org-type">:</span><span class="org-rainbow-delimiters-depth-3">)</span><span class="org-rainbow-delimiters-depth-2">)</span>', <span class="org-rainbow-delimiters-depth-2">[]</span>, <span class="org-highlight-numbers-number">1</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
|
|
A = zeros<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">3</span><span class="org-type">*</span>length<span class="org-rainbow-delimiters-depth-2">(</span>solids_i<span class="org-rainbow-delimiters-depth-2">)</span>, <span class="org-highlight-numbers-number">6</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
<span class="org-keyword">for</span> <span class="org-variable-name">i</span> = <span class="org-constant"><span class="org-highlight-numbers-number">1</span></span><span class="org-constant">:length</span><span class="org-constant"><span class="org-rainbow-delimiters-depth-1">(</span></span><span class="org-constant">solids_i</span><span class="org-constant"><span class="org-rainbow-delimiters-depth-1">)</span></span>
|
|
A<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">3</span><span class="org-type">*</span><span class="org-rainbow-delimiters-depth-2">(</span><span class="org-constant">i</span><span class="org-type">-</span><span class="org-highlight-numbers-number">1</span><span class="org-rainbow-delimiters-depth-2">)</span><span class="org-type">+</span><span class="org-highlight-numbers-number">1</span><span class="org-type">:</span><span class="org-highlight-numbers-number">3</span><span class="org-type">*</span><span class="org-constant">i</span>, <span class="org-highlight-numbers-number">1</span><span class="org-type">:</span><span class="org-highlight-numbers-number">3</span><span class="org-rainbow-delimiters-depth-1">)</span> = eye<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">3</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
|
|
A<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">3</span><span class="org-type">*</span><span class="org-rainbow-delimiters-depth-2">(</span><span class="org-constant">i</span><span class="org-type">-</span><span class="org-highlight-numbers-number">1</span><span class="org-rainbow-delimiters-depth-2">)</span><span class="org-type">+</span><span class="org-highlight-numbers-number">1</span><span class="org-type">:</span><span class="org-highlight-numbers-number">3</span><span class="org-type">*</span><span class="org-constant">i</span>, <span class="org-highlight-numbers-number">4</span><span class="org-type">:</span><span class="org-highlight-numbers-number">6</span><span class="org-rainbow-delimiters-depth-1">)</span> = <span class="org-rainbow-delimiters-depth-1">[</span><span class="org-highlight-numbers-number">0</span> acc_pos<span class="org-rainbow-delimiters-depth-2">(</span><span class="org-constant">i</span>, <span class="org-highlight-numbers-number">3</span><span class="org-rainbow-delimiters-depth-2">)</span> <span class="org-type">-</span>acc_pos<span class="org-rainbow-delimiters-depth-2">(</span><span class="org-constant">i</span>, <span class="org-highlight-numbers-number">2</span><span class="org-rainbow-delimiters-depth-2">)</span> ; <span class="org-type">-</span>acc_pos<span class="org-rainbow-delimiters-depth-2">(</span><span class="org-constant">i</span>, <span class="org-highlight-numbers-number">3</span><span class="org-rainbow-delimiters-depth-2">)</span> <span class="org-highlight-numbers-number">0</span> acc_pos<span class="org-rainbow-delimiters-depth-2">(</span><span class="org-constant">i</span>, <span class="org-highlight-numbers-number">1</span><span class="org-rainbow-delimiters-depth-2">)</span> ; acc_pos<span class="org-rainbow-delimiters-depth-2">(</span><span class="org-constant">i</span>, <span class="org-highlight-numbers-number">2</span><span class="org-rainbow-delimiters-depth-2">)</span> <span class="org-type">-</span>acc_pos<span class="org-rainbow-delimiters-depth-2">(</span><span class="org-constant">i</span>, <span class="org-highlight-numbers-number">1</span><span class="org-rainbow-delimiters-depth-2">)</span> <span class="org-highlight-numbers-number">0</span><span class="org-rainbow-delimiters-depth-1">]</span>;
|
|
<span class="org-keyword">end</span>
|
|
|
|
mode_shapes_O<span class="org-rainbow-delimiters-depth-1">(</span>mod_i, solid_i, <span class="org-type">:</span><span class="org-rainbow-delimiters-depth-1">)</span> = A<span class="org-type">\</span>Y;
|
|
<span class="org-keyword">end</span>
|
|
<span class="org-keyword">end</span>
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orgaa2d870" class="outline-2">
|
|
<h2 id="orgaa2d870"><span class="section-number-2">6</span> Modal Matrices</h2>
|
|
<div class="outline-text-2" id="text-6">
|
|
<p>
|
|
We want to obtain the two following matrices:
|
|
\[ \Omega = \begin{bmatrix}
|
|
\omega_1^2 & & 0 \\
|
|
& \ddots & \\
|
|
0 & & \omega_n^2
|
|
\end{bmatrix}; \quad \Psi = \begin{bmatrix}
|
|
& & \\
|
|
\{\psi_1\} & \dots & \{\psi_n\} \\
|
|
& &
|
|
\end{bmatrix} \]
|
|
</p>
|
|
|
|
<ul class="org-ul">
|
|
<li class="off"><code>[ ]</code> How to add damping to the eigen value matrix?</li>
|
|
</ul>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">eigen_value_M = diag<span class="org-rainbow-delimiters-depth-1">(</span>freqs<span class="org-type">*</span><span class="org-highlight-numbers-number">2</span><span class="org-type">*</span><span class="org-constant">pi</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
eigen_vector_M = reshape<span class="org-rainbow-delimiters-depth-1">(</span>mode_shapes_O, <span class="org-rainbow-delimiters-depth-2">[</span>mod_n, <span class="org-highlight-numbers-number">6</span><span class="org-type">*</span>length<span class="org-rainbow-delimiters-depth-3">(</span>solid_names<span class="org-rainbow-delimiters-depth-3">)</span><span class="org-rainbow-delimiters-depth-2">]</span><span class="org-rainbow-delimiters-depth-1">)</span>';
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
\[ \{\psi_1\} = \begin{Bmatrix} \psi_{1_x} & \psi_{2_x} & \dots & \psi_{6_x} & \psi_{1_x} & \dots & \psi_{1\Omega_x} & \dots & \psi_{6\Omega_z} \end{Bmatrix}^T \]
|
|
</p>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org16e1add" class="outline-2">
|
|
<h2 id="org16e1add"><span class="section-number-2">7</span> Modal Complexity</h2>
|
|
<div class="outline-text-2" id="text-7">
|
|
<p>
|
|
Complexity of one mode
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">mod_i = <span class="org-highlight-numbers-number">1</span>;
|
|
i_max = convhull<span class="org-rainbow-delimiters-depth-1">(</span>real<span class="org-rainbow-delimiters-depth-2">(</span>eigen_vector_M<span class="org-rainbow-delimiters-depth-3">(</span><span class="org-type">:</span>, mod_i<span class="org-rainbow-delimiters-depth-3">)</span><span class="org-rainbow-delimiters-depth-2">)</span>, imag<span class="org-rainbow-delimiters-depth-2">(</span>eigen_vector_M<span class="org-rainbow-delimiters-depth-3">(</span><span class="org-type">:</span>, mod_i<span class="org-rainbow-delimiters-depth-3">)</span><span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
radius = max<span class="org-rainbow-delimiters-depth-1">(</span>abs<span class="org-rainbow-delimiters-depth-2">(</span>eigen_vector_M<span class="org-rainbow-delimiters-depth-3">(</span><span class="org-type">:</span>, mod_i<span class="org-rainbow-delimiters-depth-3">)</span><span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
theta = linspace<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">0</span>, <span class="org-highlight-numbers-number">2</span><span class="org-type">*</span><span class="org-constant">pi</span>, <span class="org-highlight-numbers-number">100</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
|
|
<span class="org-type">figure</span>;
|
|
hold on;
|
|
plot<span class="org-rainbow-delimiters-depth-1">(</span>radius<span class="org-type">*</span>cos<span class="org-rainbow-delimiters-depth-2">(</span>theta<span class="org-rainbow-delimiters-depth-2">)</span>, radius<span class="org-type">*</span>sin<span class="org-rainbow-delimiters-depth-2">(</span>theta<span class="org-rainbow-delimiters-depth-2">)</span>, <span class="org-string">'-'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
plot<span class="org-rainbow-delimiters-depth-1">(</span>real<span class="org-rainbow-delimiters-depth-2">(</span>eigen_vector_M<span class="org-rainbow-delimiters-depth-3">(</span>i_max, mod_i<span class="org-rainbow-delimiters-depth-3">)</span><span class="org-rainbow-delimiters-depth-2">)</span>, imag<span class="org-rainbow-delimiters-depth-2">(</span>eigen_vector_M<span class="org-rainbow-delimiters-depth-3">(</span>i_max, mod_i<span class="org-rainbow-delimiters-depth-3">)</span><span class="org-rainbow-delimiters-depth-2">)</span>, <span class="org-string">'-'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
plot<span class="org-rainbow-delimiters-depth-1">(</span>real<span class="org-rainbow-delimiters-depth-2">(</span>eigen_vector_M<span class="org-rainbow-delimiters-depth-3">(</span><span class="org-type">:</span>, mod_i<span class="org-rainbow-delimiters-depth-3">)</span><span class="org-rainbow-delimiters-depth-2">)</span>, imag<span class="org-rainbow-delimiters-depth-2">(</span>eigen_vector_M<span class="org-rainbow-delimiters-depth-3">(</span><span class="org-type">:</span>, mod_i<span class="org-rainbow-delimiters-depth-3">)</span><span class="org-rainbow-delimiters-depth-2">)</span>, <span class="org-string">'ko'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
hold off;
|
|
xlabel<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'Real Part'</span><span class="org-string"><span class="org-rainbow-delimiters-depth-1">)</span></span><span class="org-string">; ylabel</span><span class="org-string"><span class="org-rainbow-delimiters-depth-1">(</span></span><span class="org-string">'Imaginary Part'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
<span class="org-type">axis</span> manual equal
|
|
</pre>
|
|
</div>
|
|
|
|
|
|
<div id="org51e10de" class="figure">
|
|
<p><img src="figs/modal_complexity.png" alt="modal_complexity.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 3: </span>Modal Complexity of one mode</p>
|
|
</div>
|
|
|
|
<p>
|
|
Complexity function of the mode order.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">modes_complexity = zeros<span class="org-rainbow-delimiters-depth-1">(</span>mod_n, <span class="org-highlight-numbers-number">1</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
<span class="org-keyword">for</span> <span class="org-variable-name">mod_i</span> = <span class="org-constant"><span class="org-highlight-numbers-number">1</span></span><span class="org-constant">:mod_n</span>
|
|
<span class="org-constant">i</span> = convhull<span class="org-rainbow-delimiters-depth-1">(</span>real<span class="org-rainbow-delimiters-depth-2">(</span>eigen_vector_M<span class="org-rainbow-delimiters-depth-3">(</span><span class="org-type">:</span>, mod_i<span class="org-rainbow-delimiters-depth-3">)</span><span class="org-rainbow-delimiters-depth-2">)</span>, imag<span class="org-rainbow-delimiters-depth-2">(</span>eigen_vector_M<span class="org-rainbow-delimiters-depth-3">(</span><span class="org-type">:</span>, mod_i<span class="org-rainbow-delimiters-depth-3">)</span><span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
area_complex = polyarea<span class="org-rainbow-delimiters-depth-1">(</span>real<span class="org-rainbow-delimiters-depth-2">(</span>eigen_vector_M<span class="org-rainbow-delimiters-depth-3">(</span><span class="org-constant">i</span>, mod_i<span class="org-rainbow-delimiters-depth-3">)</span><span class="org-rainbow-delimiters-depth-2">)</span>, imag<span class="org-rainbow-delimiters-depth-2">(</span>eigen_vector_M<span class="org-rainbow-delimiters-depth-3">(</span><span class="org-constant">i</span>, mod_i<span class="org-rainbow-delimiters-depth-3">)</span><span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
area_circle = <span class="org-constant">pi</span><span class="org-type">*</span>max<span class="org-rainbow-delimiters-depth-1">(</span>abs<span class="org-rainbow-delimiters-depth-2">(</span>eigen_vector_M<span class="org-rainbow-delimiters-depth-3">(</span><span class="org-type">:</span>, mod_i<span class="org-rainbow-delimiters-depth-3">)</span><span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span><span class="org-type">^</span><span class="org-highlight-numbers-number">2</span>;
|
|
modes_complexity<span class="org-rainbow-delimiters-depth-1">(</span>mod_i<span class="org-rainbow-delimiters-depth-1">)</span> = area_complex<span class="org-type">/</span>area_circle;
|
|
<span class="org-keyword">end</span>
|
|
|
|
<span class="org-type">figure</span>;
|
|
plot<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">1</span><span class="org-type">:</span>mod_n, modes_complexity, <span class="org-string">'ok'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
ylim<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbow-delimiters-depth-2">[</span><span class="org-highlight-numbers-number">0</span>, <span class="org-highlight-numbers-number">1</span><span class="org-rainbow-delimiters-depth-2">]</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
xlabel<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'Mode Number'</span><span class="org-string"><span class="org-rainbow-delimiters-depth-1">)</span></span><span class="org-string">; ylabel</span><span class="org-string"><span class="org-rainbow-delimiters-depth-1">(</span></span><span class="org-string">'Modal Complexity'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
</pre>
|
|
</div>
|
|
|
|
|
|
<div id="org7388c3e" class="figure">
|
|
<p><img src="figs/modal_complexities.png" alt="modal_complexities.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 4: </span>Modal complexity for each mode</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orgc33fb24" class="outline-2">
|
|
<h2 id="orgc33fb24"><span class="section-number-2">8</span> Some notes about constraining the number of degrees of freedom</h2>
|
|
<div class="outline-text-2" id="text-8">
|
|
<p>
|
|
We want to have the two eigen matrices.
|
|
</p>
|
|
|
|
<p>
|
|
They should have the same size \(n \times n\) where \(n\) is the number of modes as well as the number of degrees of freedom.
|
|
Thus, if we consider 21 modes, we should restrict our system to have only 21 DOFs.
|
|
</p>
|
|
|
|
<p>
|
|
Actually, we are measured 6 DOFs of 6 solids, thus we have 36 DOFs.
|
|
</p>
|
|
|
|
<p>
|
|
From the mode shapes animations, it seems that in the frequency range of interest, the two marbles can be considered as one solid.
|
|
We thus have 5 solids and 30 DOFs.
|
|
</p>
|
|
|
|
|
|
<p>
|
|
In order to determine which DOF can be neglected, two solutions seems possible:
|
|
</p>
|
|
<ul class="org-ul">
|
|
<li>compare the mode shapes</li>
|
|
<li>compare the FRFs</li>
|
|
</ul>
|
|
|
|
<p>
|
|
The question is: in which base (frame) should be express the modes shapes and FRFs?
|
|
Is it meaningful to compare mode shapes as they give no information about the amplitudes of vibration?
|
|
</p>
|
|
|
|
|
|
<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" />
|
|
|
|
<col class="org-left" />
|
|
</colgroup>
|
|
<thead>
|
|
<tr>
|
|
<th scope="col" class="org-left">Stage</th>
|
|
<th scope="col" class="org-right">Motion DOFs</th>
|
|
<th scope="col" class="org-right">Parasitic DOF</th>
|
|
<th scope="col" class="org-right">Total DOF</th>
|
|
<th scope="col" class="org-left">Description of DOF</th>
|
|
</tr>
|
|
</thead>
|
|
<tbody>
|
|
<tr>
|
|
<td class="org-left">Granite</td>
|
|
<td class="org-right">0</td>
|
|
<td class="org-right">3</td>
|
|
<td class="org-right">3</td>
|
|
<td class="org-left"> </td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-left">Ty</td>
|
|
<td class="org-right">1</td>
|
|
<td class="org-right">2</td>
|
|
<td class="org-right">3</td>
|
|
<td class="org-left">Ty, Rz</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-left">Ry</td>
|
|
<td class="org-right">1</td>
|
|
<td class="org-right">2</td>
|
|
<td class="org-right">3</td>
|
|
<td class="org-left">Ry,</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-left">Rz</td>
|
|
<td class="org-right">1</td>
|
|
<td class="org-right">2</td>
|
|
<td class="org-right">3</td>
|
|
<td class="org-left">Rz, Rx, Ry</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-left">Hexapod</td>
|
|
<td class="org-right">6</td>
|
|
<td class="org-right">0</td>
|
|
<td class="org-right">6</td>
|
|
<td class="org-left">Txyz, Rxyz</td>
|
|
</tr>
|
|
</tbody>
|
|
<tbody>
|
|
<tr>
|
|
<td class="org-left"> </td>
|
|
<td class="org-right">9</td>
|
|
<td class="org-right">9</td>
|
|
<td class="org-right">18</td>
|
|
<td class="org-left"> </td>
|
|
</tr>
|
|
</tbody>
|
|
</table>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orgd17ef15" class="outline-2">
|
|
<h2 id="orgd17ef15"><span class="section-number-2">9</span> <span class="todo TODO">TODO</span> Normalization of mode shapes?</h2>
|
|
<div class="outline-text-2" id="text-9">
|
|
<p>
|
|
We normalize each column of the eigen vector matrix.
|
|
Then, each eigenvector as a norm of 1.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">eigen_vector_M = eigen_vector_M<span class="org-type">./</span>vecnorm<span class="org-rainbow-delimiters-depth-1">(</span>eigen_vector_M<span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orgc7fe2ae" class="outline-2">
|
|
<h2 id="orgc7fe2ae"><span class="section-number-2">10</span> Compare Mode Shapes</h2>
|
|
<div class="outline-text-2" id="text-10">
|
|
<p>
|
|
Let's say we want to see for the first mode which DOFs can be neglected.
|
|
In order to do so, we should estimate the motion of each stage in particular directions.
|
|
If we look at the z motion for instance, we will find that we cannot neglect that motion (because of the tilt causing z motion).
|
|
</p>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">mode_i = <span class="org-highlight-numbers-number">3</span>;
|
|
dof_i = <span class="org-highlight-numbers-number">6</span>;
|
|
|
|
mode = eigen_vector_M<span class="org-rainbow-delimiters-depth-1">(</span>dof_i<span class="org-type">:</span><span class="org-highlight-numbers-number">6</span><span class="org-type">:</span>end, mode_i<span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
|
|
<span class="org-type">figure</span>;
|
|
hold on;
|
|
<span class="org-keyword">for</span> <span class="org-variable-name">i</span>=<span class="org-constant"><span class="org-highlight-numbers-number">1</span></span><span class="org-constant">:length</span><span class="org-constant"><span class="org-rainbow-delimiters-depth-1">(</span></span><span class="org-constant">mode</span><span class="org-constant"><span class="org-rainbow-delimiters-depth-1">)</span></span>
|
|
plot<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbow-delimiters-depth-2">[</span><span class="org-highlight-numbers-number">0</span>, real<span class="org-rainbow-delimiters-depth-3">(</span>mode<span class="org-rainbow-delimiters-depth-4">(</span><span class="org-constant">i</span><span class="org-rainbow-delimiters-depth-4">)</span><span class="org-rainbow-delimiters-depth-3">)</span><span class="org-rainbow-delimiters-depth-2">]</span>, <span class="org-rainbow-delimiters-depth-2">[</span><span class="org-highlight-numbers-number">0</span>, imag<span class="org-rainbow-delimiters-depth-3">(</span>mode<span class="org-rainbow-delimiters-depth-4">(</span><span class="org-constant">i</span><span class="org-rainbow-delimiters-depth-4">)</span><span class="org-rainbow-delimiters-depth-3">)</span><span class="org-rainbow-delimiters-depth-2">]</span>, <span class="org-string">'-', 'DisplayName'</span>, solid_names<span class="org-rainbow-delimiters-depth-2">{</span><span class="org-constant">i</span><span class="org-rainbow-delimiters-depth-2">}</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
<span class="org-keyword">end</span>
|
|
hold off;
|
|
legend<span class="org-rainbow-delimiters-depth-1">()</span>;
|
|
</pre>
|
|
</div>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab"><span class="org-type">figure</span>;
|
|
subplot<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">2</span>, <span class="org-highlight-numbers-number">1</span>, <span class="org-highlight-numbers-number">1</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
hold on;
|
|
<span class="org-keyword">for</span> <span class="org-variable-name">i</span>=<span class="org-constant"><span class="org-highlight-numbers-number">1</span></span><span class="org-constant">:length</span><span class="org-constant"><span class="org-rainbow-delimiters-depth-1">(</span></span><span class="org-constant">mode</span><span class="org-constant"><span class="org-rainbow-delimiters-depth-1">)</span></span>
|
|
plot<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">1</span>, norm<span class="org-rainbow-delimiters-depth-2">(</span>mode<span class="org-rainbow-delimiters-depth-3">(</span><span class="org-constant">i</span><span class="org-rainbow-delimiters-depth-3">)</span><span class="org-rainbow-delimiters-depth-2">)</span>, <span class="org-string">'o'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
<span class="org-keyword">end</span>
|
|
hold off;
|
|
ylabel<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'Amplitude'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
|
|
subplot<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">2</span>, <span class="org-highlight-numbers-number">1</span>, <span class="org-highlight-numbers-number">2</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
hold on;
|
|
<span class="org-keyword">for</span> <span class="org-variable-name">i</span>=<span class="org-constant"><span class="org-highlight-numbers-number">1</span></span><span class="org-constant">:length</span><span class="org-constant"><span class="org-rainbow-delimiters-depth-1">(</span></span><span class="org-constant">mode</span><span class="org-constant"><span class="org-rainbow-delimiters-depth-1">)</span></span>
|
|
plot<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">1</span>, <span class="org-highlight-numbers-number">180</span><span class="org-type">/</span><span class="org-constant">pi</span><span class="org-type">*</span>angle<span class="org-rainbow-delimiters-depth-2">(</span>mode<span class="org-rainbow-delimiters-depth-3">(</span><span class="org-constant">i</span><span class="org-rainbow-delimiters-depth-3">)</span><span class="org-rainbow-delimiters-depth-2">)</span>, <span class="org-string">'o', 'DisplayName'</span>, solid_names<span class="org-rainbow-delimiters-depth-2">{</span><span class="org-constant">i</span><span class="org-rainbow-delimiters-depth-2">}</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
<span class="org-keyword">end</span>
|
|
hold off;
|
|
ylim<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbow-delimiters-depth-2">[</span><span class="org-type">-</span><span class="org-highlight-numbers-number">180</span>, <span class="org-highlight-numbers-number">180</span><span class="org-rainbow-delimiters-depth-2">]</span><span class="org-rainbow-delimiters-depth-1">)</span>; yticks<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbow-delimiters-depth-2">[</span><span class="org-type">-</span><span class="org-highlight-numbers-number">180</span><span class="org-type">:</span><span class="org-highlight-numbers-number">90</span><span class="org-type">:</span><span class="org-highlight-numbers-number">180</span><span class="org-rainbow-delimiters-depth-2">]</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
ylabel<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'Phase </span><span class="org-string"><span class="org-rainbow-delimiters-depth-2">[</span></span><span class="org-string">deg</span><span class="org-string"><span class="org-rainbow-delimiters-depth-2">]</span></span><span class="org-string">'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
legend<span class="org-rainbow-delimiters-depth-1">()</span>;
|
|
</pre>
|
|
</div>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">test = mode_shapes_O<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">10</span>, <span class="org-highlight-numbers-number">1</span>, <span class="org-type">:</span><span class="org-rainbow-delimiters-depth-1">)</span><span class="org-type">/</span>norm<span class="org-rainbow-delimiters-depth-1">(</span>squeeze<span class="org-rainbow-delimiters-depth-2">(</span>mode_shapes_O<span class="org-rainbow-delimiters-depth-3">(</span><span class="org-highlight-numbers-number">10</span>, <span class="org-highlight-numbers-number">1</span>, <span class="org-type">:</span><span class="org-rainbow-delimiters-depth-3">)</span><span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
test = mode_shapes_O<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">10</span>, <span class="org-highlight-numbers-number">2</span>, <span class="org-type">:</span><span class="org-rainbow-delimiters-depth-1">)</span><span class="org-type">/</span>norm<span class="org-rainbow-delimiters-depth-1">(</span>squeeze<span class="org-rainbow-delimiters-depth-2">(</span>mode_shapes_O<span class="org-rainbow-delimiters-depth-3">(</span><span class="org-highlight-numbers-number">10</span>, <span class="org-highlight-numbers-number">2</span>, <span class="org-type">:</span><span class="org-rainbow-delimiters-depth-3">)</span><span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org1ec0de2" class="outline-2">
|
|
<h2 id="org1ec0de2"><span class="section-number-2">11</span> <span class="todo TODO">TODO</span> Synthesis of FRF curves</h2>
|
|
</div>
|
|
</div>
|
|
<div id="postamble" class="status">
|
|
<p class="author">Author: Dehaeze Thomas</p>
|
|
<p class="date">Created: 2019-07-02 mar. 17:45</p>
|
|
<p class="validation"><a href="http://validator.w3.org/check?uri=referer">Validate</a></p>
|
|
</div>
|
|
</body>
|
|
</html>
|