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508 lines
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<a accesskey="h" href="./index.html"> UP </a>
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</div><div id="content">
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<h1 class="title">Modal Analysis - Derivation of Mathematical Models</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="#org93dde66">1. Type of Model</a></li>
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<li><a href="#orgfd7d140">2. <span class="todo TODO">TODO</span> Extract Physical Matrices</a></li>
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<li><a href="#orgdaaea25">3. Some notes about constraining the number of degrees of freedom</a></li>
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</ul>
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</div>
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</div>
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<div id="outline-container-org93dde66" class="outline-2">
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<h2 id="org93dde66"><span class="section-number-2">1</span> Type of Model</h2>
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<div class="outline-text-2" id="text-1">
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<p>
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The model that we want to obtain is a <b>multi-body model</b>.
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It is composed of several <b>solid bodies connected with springs and dampers</b>.
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The solid bodies are represented with different colors on figure <a href="#org4703119">1</a>.
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</p>
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<p>
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In the simscape model, the solid bodies are:
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</p>
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<ul class="org-ul">
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<li>the granite (1 or 2 solids)</li>
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<li>the translation stage</li>
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<li>the tilt stage</li>
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<li>the spindle and slip-ring</li>
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<li>the hexapod</li>
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</ul>
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<div id="org4703119" class="figure">
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<p><img src="img/nass_solidworks.png" alt="nass_solidworks.png" width="800px" />
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</p>
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<p><span class="figure-number">Figure 1: </span>CAD view of the ID31 Micro-Station</p>
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</div>
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<p>
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However, each of the DOF of the system may not be relevant for the modes present in the frequency band of interest.
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For instance, the translation stage may not vibrate in the Z direction for all the modes identified. Then, we can block this DOF and this simplifies the model.
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</p>
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<p>
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The modal identification done here will thus permit us to determine <b>which DOF can be neglected</b>.
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</p>
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</div>
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</div>
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<div id="outline-container-orgfd7d140" class="outline-2">
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<h2 id="orgfd7d140"><span class="section-number-2">2</span> <span class="todo TODO">TODO</span> Extract Physical Matrices</h2>
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<div class="outline-text-2" id="text-2">
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<p>
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<a class='org-ref-reference' href="#wang11_extrac_real_modes_physic_matric">wang11_extrac_real_modes_physic_matric</a>
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</p>
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<p>
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Let's recall that:
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\[ \Lambda = \begin{bmatrix}
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s_1 & & 0 \\
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& \ddots & \\
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0 & & s_N
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\end{bmatrix}_{N \times N}; \quad \Psi = \begin{bmatrix}
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& & \\
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\{\psi_1\} & \dots & \{\psi_N\} \\
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& &
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\end{bmatrix}_{M \times N} ; \quad A = \begin{bmatrix}
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a_1 & & 0 \\
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& \ddots & \\
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0 & & a_N
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\end{bmatrix}_{N \times N}; \]
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</p>
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\begin{align}
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M &= \frac{1}{2} \left[ \text{Re}(\Psi A^{-1} \Lambda \Psi^T ) \right]^{-1} \\
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C &= -2 M \text{Re}(\Psi A^{-1} \Lambda^2 A^{-1} \Psi^T ) M \\
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K &= -\frac{1}{2} \left[ \text{Re}(\Psi \Lambda^{-1} A^{-1} \Psi^T) \right]^{-1}
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\end{align}
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<div class="org-src-container">
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<pre class="src src-matlab">psi = eigen_vec_CoM;
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a = modal_a_M;
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lambda = eigen_val_M;
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M = <span class="org-highlight-numbers-number">0</span>.<span class="org-highlight-numbers-number">5</span><span class="org-type">*</span>inv<span class="org-rainbow-delimiters-depth-1">(</span>real<span class="org-rainbow-delimiters-depth-2">(</span>psi<span class="org-type">*</span>inv<span class="org-rainbow-delimiters-depth-3">(</span>a<span class="org-rainbow-delimiters-depth-3">)</span><span class="org-type">*</span>lambda<span class="org-type">*</span>psi'<span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span>;
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C = <span class="org-type">-</span><span class="org-highlight-numbers-number">2</span><span class="org-type">*</span>M<span class="org-type">*</span>real<span class="org-rainbow-delimiters-depth-1">(</span>psi<span class="org-type">*</span>inv<span class="org-rainbow-delimiters-depth-2">(</span>a<span class="org-rainbow-delimiters-depth-2">)</span><span class="org-type">*</span>lambda<span class="org-type">^</span><span class="org-highlight-numbers-number">2</span><span class="org-type">*</span>inv<span class="org-rainbow-delimiters-depth-2">(</span>a<span class="org-rainbow-delimiters-depth-2">)</span><span class="org-type">*</span>psi'<span class="org-rainbow-delimiters-depth-1">)</span><span class="org-type">*</span>M;
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K = <span class="org-type">-</span><span class="org-highlight-numbers-number">0</span>.<span class="org-highlight-numbers-number">5</span><span class="org-type">*</span>inv<span class="org-rainbow-delimiters-depth-1">(</span>real<span class="org-rainbow-delimiters-depth-2">(</span>psi<span class="org-type">*</span>inv<span class="org-rainbow-delimiters-depth-3">(</span>lambda<span class="org-rainbow-delimiters-depth-3">)</span><span class="org-type">*</span>inv<span class="org-rainbow-delimiters-depth-3">(</span>a<span class="org-rainbow-delimiters-depth-3">)</span><span class="org-type">*</span>psi'<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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From <a class='org-ref-reference' href="#ewins00_modal">ewins00_modal</a>
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</p>
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\begin{align}
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[M] &= [\Phi]^{-T} [I] [\Phi]^{-1} \\
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[K] &= [\Phi]^{-T} [\lambda_r^2] [\Phi]^{-1}
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\end{align}
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</div>
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</div>
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<div id="outline-container-orgdaaea25" class="outline-2">
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<h2 id="orgdaaea25"><span class="section-number-2">3</span> Some notes about constraining the number of degrees of freedom</h2>
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<div class="outline-text-2" id="text-3">
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<p>
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We want to have the two eigen matrices.
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</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>
|
|
<p>
|
|
|
|
<h1 class='org-ref-bib-h1'>Bibliography</h1>
|
|
<ul class='org-ref-bib'><li><a id="wang11_extrac_real_modes_physic_matric">[wang11_extrac_real_modes_physic_matric]</a> <a name="wang11_extrac_real_modes_physic_matric"></a>Tong Wang, Lingmi Zhang & Kong Fah Tee, Extraction of Real Modes and Physical Matrices From Modal Testing, <i>Earthquake Engineering and Engineering Vibration</i>, <b>10(2)</b>, 219-227 (2011). <a href="https://doi.org/10.1007/s11803-011-0060-6">link</a>. <a href="http://dx.doi.org/10.1007/s11803-011-0060-6">doi</a>.</li>
|
|
<li><a id="ewins00_modal">[ewins00_modal]</a> <a name="ewins00_modal"></a>Ewins, Modal testing: theory, practice and application, Wiley-Blackwell (2000).</li>
|
|
</ul>
|
|
</p>
|
|
</div>
|
|
<div id="postamble" class="status">
|
|
<p class="author">Author: Dehaeze Thomas</p>
|
|
<p class="date">Created: 2019-10-08 mar. 10:48</p>
|
|
<p class="validation"><a href="http://validator.w3.org/check?uri=referer">Validate</a></p>
|
|
</div>
|
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|
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