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8.2 KiB
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259 lines
8.2 KiB
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<!-- 2020-11-12 jeu. 10:29 -->
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<title>Modal Analysis - Derivation of Mathematical Models</title>
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<meta name="author" content="Dehaeze Thomas" />
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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 - 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="#org2a87d8f">1. Type of Model</a></li>
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<li><a href="#orgdb0ed87">2. Extract Physical Matrices</a></li>
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<li><a href="#orgcc897f4">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-org2a87d8f" class="outline-2">
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<h2 id="org2a87d8f"><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="#orga016307">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="orga016307" 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-orgdb0ed87" class="outline-2">
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<h2 id="orgdb0ed87"><span class="section-number-2">2</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 href="#citeproc_bib_item_2">Wang, Zhang, and Tee 2011</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 = 0.5<span class="org-type">*</span>inv(real(psi<span class="org-type">*</span>inv(a)<span class="org-type">*</span>lambda<span class="org-type">*</span>psi<span class="org-type">'</span>));
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C = <span class="org-type">-</span>2<span class="org-type">*</span>M<span class="org-type">*</span>real(psi<span class="org-type">*</span>inv(a)<span class="org-type">*</span>lambda<span class="org-type">^</span>2<span class="org-type">*</span>inv(a)<span class="org-type">*</span>psi<span class="org-type">'</span>)<span class="org-type">*</span>M;
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K = <span class="org-type">-</span>0.5<span class="org-type">*</span>inv(real(psi<span class="org-type">*</span>inv(lambda)<span class="org-type">*</span>inv(a)<span class="org-type">*</span>psi<span class="org-type">'</span>));
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</pre>
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</div>
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<p>
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From (<a href="#citeproc_bib_item_1">Ewins 2000</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-orgcc897f4" class="outline-2">
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<h2 id="orgcc897f4"><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>
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<p>
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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.
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Thus, if we consider 21 modes, we should restrict our system to have only 21 DOFs.
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</p>
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<p>
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Actually, we are measured 6 DOFs of 6 solids, thus we have 36 DOFs.
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</p>
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<p>
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From the mode shapes animations, it seems that in the frequency range of interest, the two marbles can be considered as one solid.
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We thus have 5 solids and 30 DOFs.
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</p>
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<p>
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In order to determine which DOF can be neglected, two solutions seems possible:
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</p>
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<ul class="org-ul">
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<li>compare the mode shapes</li>
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<li>compare the FRFs</li>
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</ul>
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<p>
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The question is: in which base (frame) should be express the modes shapes and FRFs?
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Is it meaningful to compare mode shapes as they give no information about the amplitudes of vibration?
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</p>
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<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
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<colgroup>
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<col class="org-left" />
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<col class="org-right" />
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<col class="org-right" />
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<col class="org-right" />
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<col class="org-left" />
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</colgroup>
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<thead>
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<tr>
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<th scope="col" class="org-left">Stage</th>
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<th scope="col" class="org-right">Motion DOFs</th>
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<th scope="col" class="org-right">Parasitic DOF</th>
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<th scope="col" class="org-right">Total DOF</th>
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<th scope="col" class="org-left">Description of DOF</th>
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</tr>
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</thead>
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<tbody>
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<tr>
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<td class="org-left">Granite</td>
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<td class="org-right">0</td>
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<td class="org-right">3</td>
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<td class="org-right">3</td>
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<td class="org-left"> </td>
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</tr>
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<tr>
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<td class="org-left">Ty</td>
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<td class="org-right">1</td>
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<td class="org-right">2</td>
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<td class="org-right">3</td>
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<td class="org-left">Ty, Rz</td>
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</tr>
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<tr>
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<td class="org-left">Ry</td>
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<td class="org-right">1</td>
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<td class="org-right">2</td>
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<td class="org-right">3</td>
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<td class="org-left">Ry,</td>
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</tr>
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<tr>
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<td class="org-left">Rz</td>
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<td class="org-right">1</td>
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<td class="org-right">2</td>
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<td class="org-right">3</td>
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<td class="org-left">Rz, Rx, Ry</td>
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</tr>
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<tr>
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<td class="org-left">Hexapod</td>
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<td class="org-right">6</td>
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<td class="org-right">0</td>
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<td class="org-right">6</td>
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<td class="org-left">Txyz, Rxyz</td>
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</tr>
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</tbody>
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<tbody>
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<tr>
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<td class="org-left"> </td>
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<td class="org-right">9</td>
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<td class="org-right">9</td>
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<td class="org-right">18</td>
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<td class="org-left"> </td>
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</tr>
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</tbody>
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</table>
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</div>
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</div>
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<p>
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</p>
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<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><h2 class='citeproc-org-bib-h2'>Bibliography</h2>
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<div class="csl-bib-body">
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<div class="csl-entry"><a name="citeproc_bib_item_1"></a>Ewins, DJ. 2000. <i>Modal Testing: Theory, Practice and Application</i>. <i>Research Studies Pre, 2nd Ed., ISBN-13</i>. Baldock, Hertfordshire, England Philadelphia, PA: Wiley-Blackwell.</div>
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<div class="csl-entry"><a name="citeproc_bib_item_2"></a>Wang, Tong, Lingmi Zhang, and Kong Fah Tee. 2011. “Extraction of Real Modes and Physical Matrices from Modal Testing.” <i>Earthquake Engineering and Engineering Vibration</i> 10 (2):219–27. <a href="https://doi.org/10.1007/s11803-011-0060-6">https://doi.org/10.1007/s11803-011-0060-6</a>.</div>
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</div>
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</div>
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<div id="postamble" class="status">
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<p class="author">Author: Dehaeze Thomas</p>
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<p class="date">Created: 2020-11-12 jeu. 10:29</p>
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