Update modal analysis, add .zip files (data and matlab files)
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modal-analysis/data/frf_processing.zip
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modal-analysis/data/frf_processing.zip
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modal-analysis/data/modal_frf_coh.zip
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modal-analysis/data/modal_frf_coh.zip
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@ -3,7 +3,7 @@
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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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<!-- 2019-07-05 ven. 10:16 -->
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<!-- 2019-07-05 ven. 11:06 -->
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<meta http-equiv="Content-Type" content="text/html;charset=utf-8" />
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<meta name="viewport" content="width=device-width, initial-scale=1" />
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<title>Modal Analysis - Processing of FRF</title>
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@ -280,15 +280,14 @@ for the JavaScript code in this tag.
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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="#orgd95e287">1. Importation of measured FRF curves</a></li>
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<li><a href="#org0649187">2. From accelerometer DOFs to solid body DOFs - Mathematics</a></li>
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<li><a href="#orgbfc68dd">3. What reference frame to choose?</a></li>
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<li><a href="#org5d9a38b">4. From accelerometer DOFs to solid body DOFs - Matlab Implementation</a></li>
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<li><a href="#orgbb81ded">5. Analysis of some FRF in the global coordinates</a></li>
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<li><a href="#orgb0c1df8">6. <span class="todo TODO">TODO</span> How to compare the relative motion of solid bodies</a></li>
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<li><a href="#org9c93ccf">7. Relative Motion in the global coordinates</a></li>
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<li><a href="#org7803dae">8. <span class="todo TODO">TODO</span> Compare original FRF measurements to transformed FRF in the global frame</a></li>
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<li><a href="#org45fdd6c">9. Verify that we find the original FRF from the FRF in the global coordinates</a></li>
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<li><a href="#org2cad1ec">1. Importation of measured FRF curves</a></li>
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<li><a href="#orga573b16">2. From accelerometer DOFs to solid body DOFs - Mathematics</a></li>
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<li><a href="#org93fc25e">3. What reference frame to choose?</a></li>
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<li><a href="#org15d9437">4. From accelerometer DOFs to solid body DOFs - Matlab Implementation</a></li>
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<li><a href="#org1d2db9a">5. Analysis of some FRF in the global coordinates</a></li>
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<li><a href="#org26b8f8f">6. Comparison of the relative motion of solid bodies</a></li>
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<li><a href="#org1f7f9fb">7. Verify that we find the original FRF from the FRF in the global coordinates</a></li>
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<li><a href="#org2cd1928">8. Saving of the FRF expressed in the global coordinates</a></li>
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</ul>
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</div>
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</div>
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@ -322,8 +321,15 @@ Thus, we are only interested in \(6 \times 6 = 36\) degrees of freedom.
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We here process the FRF matrix to go from the 69 measured DOFs to the wanted 36 DOFs.
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</p>
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<div id="outline-container-orgd95e287" class="outline-2">
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<h2 id="orgd95e287"><span class="section-number-2">1</span> Importation of measured FRF curves</h2>
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<div class="note">
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<p>
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All the files (data and Matlab scripts) are accessible <a href="data/frf_processing.zip">here</a>.
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</p>
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</div>
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<div id="outline-container-org2cad1ec" class="outline-2">
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<h2 id="org2cad1ec"><span class="section-number-2">1</span> Importation of measured FRF curves</h2>
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<div class="outline-text-2" id="text-1">
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<p>
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We load the measured FRF and Coherence matrices.
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@ -331,18 +337,18 @@ We also load the geometric parameters of the station: solid bodies considered an
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</p>
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<div class="org-src-container">
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<pre class="src src-matlab">load<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'./mat/frf_coh_matrices.mat', 'FRFs', 'COHs', 'freqs'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
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<pre class="src src-matlab">load<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'mat/frf_coh_matrices.mat', 'FRFs', 'COHs', 'freqs'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
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load<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'mat/geometry.mat', 'solids', 'solid_names', 'acc_pos'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
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</pre>
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</div>
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</div>
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</div>
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<div id="outline-container-org0649187" class="outline-2">
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<h2 id="org0649187"><span class="section-number-2">2</span> From accelerometer DOFs to solid body DOFs - Mathematics</h2>
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<div id="outline-container-orga573b16" class="outline-2">
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<h2 id="orga573b16"><span class="section-number-2">2</span> From accelerometer DOFs to solid body DOFs - Mathematics</h2>
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<div class="outline-text-2" id="text-2">
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<p>
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Let's consider the schematic shown on figure <a href="#org0345801">1</a> where we are measuring the motion of a (supposed) solid body at 4 distinct points in x-y-z.
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Let's consider the schematic shown on figure <a href="#org9f883a5">1</a> where we are measuring the motion of a (supposed) solid body at 4 distinct points in x-y-z.
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</p>
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<p>
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@ -350,14 +356,14 @@ The goal here is to link these \(4 \times 3 = 12\) measurements to the 6 DOFs of
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</p>
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<div id="org0345801" class="figure">
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<div id="org9f883a5" class="figure">
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<p><img src="figs/local_to_global_coordinates.png" alt="local_to_global_coordinates.png" />
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</p>
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<p><span class="figure-number">Figure 1: </span>Schematic of the measured motions of a solid body</p>
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</div>
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<p>
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From the figure <a href="#org0345801">1</a>, we can write:
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From the figure <a href="#org9f883a5">1</a>, we can write:
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</p>
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\begin{align*}
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\vec{v}_1 &= \vec{v} + \Omega \vec{p}_1\\
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@ -426,8 +432,8 @@ This inversion is equivalent to resolving a mean square problem.
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</div>
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</div>
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<div id="outline-container-orgbfc68dd" class="outline-2">
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<h2 id="orgbfc68dd"><span class="section-number-2">3</span> What reference frame to choose?</h2>
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<div id="outline-container-org93fc25e" class="outline-2">
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<h2 id="org93fc25e"><span class="section-number-2">3</span> What reference frame to choose?</h2>
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<div class="outline-text-2" id="text-3">
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<p>
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The question we wish here to answer is how to choose the reference frame \(\{O\}\) in which the DOFs of the solid bodies are defined.
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@ -447,7 +453,7 @@ The possibles choices are:
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<li><b>Base located at the joint position</b>: this is where we want to see the motion and estimate stiffness</li>
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</ul>
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<table id="org6509a46" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
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<table id="orgd8f2173" 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> Advantages and disadvantages for the choice of reference frame</caption>
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<colgroup>
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@ -491,8 +497,8 @@ As the easiest choice is to choose a common frame, we start with that solution.
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</div>
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</div>
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<div id="outline-container-org5d9a38b" class="outline-2">
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<h2 id="org5d9a38b"><span class="section-number-2">4</span> From accelerometer DOFs to solid body DOFs - Matlab Implementation</h2>
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<div id="outline-container-org15d9437" class="outline-2">
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<h2 id="org15d9437"><span class="section-number-2">4</span> From accelerometer DOFs to solid body DOFs - Matlab Implementation</h2>
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<div class="outline-text-2" id="text-4">
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<p>
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First, we initialize a new FRF matrix <code>FRFs_O</code> which is an \(n \times p \times q\) with:
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@ -503,6 +509,29 @@ First, we initialize a new FRF matrix <code>FRFs_O</code> which is an \(n \times
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<li>\(q\) is the number of frequency points \(\omega_i\)</li>
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</ul>
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<div class="important">
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<p>
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For each frequency point \(\omega_i\), the FRF matrix <code>FRFs_O</code> is a \(n\times p\) matrix:
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</p>
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\begin{equation}
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\text{FRF}_O(\omega_i) = \begin{bmatrix}
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\frac{D_{1,T_x}}{F_x}(\omega_i) & \frac{D_{1,T_x}}{F_y}(\omega_i) & \frac{D_{1,T_x}}{F_z}(\omega_i) \\
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\frac{D_{1,T_y}}{F_x}(\omega_i) & \frac{D_{1,T_y}}{F_y}(\omega_i) & \frac{D_{1,T_y}}{F_z}(\omega_i) \\
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\frac{D_{1,T_z}}{F_x}(\omega_i) & \frac{D_{1,T_z}}{F_y}(\omega_i) & \frac{D_{1,T_z}}{F_z}(\omega_i) \\
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\frac{D_{1,R_x}}{F_x}(\omega_i) & \frac{D_{1,R_x}}{F_y}(\omega_i) & \frac{D_{1,R_x}}{F_z}(\omega_i) \\
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\frac{D_{1,R_y}}{F_x}(\omega_i) & \frac{D_{1,R_y}}{F_y}(\omega_i) & \frac{D_{1,R_y}}{F_z}(\omega_i) \\
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\frac{D_{1,R_z}}{F_x}(\omega_i) & \frac{D_{1,R_z}}{F_y}(\omega_i) & \frac{D_{1,R_z}}{F_z}(\omega_i) \\
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\frac{D_{2,T_x}}{F_x}(\omega_i) & \frac{D_{2,T_x}}{F_y}(\omega_i) & \frac{D_{2,T_x}}{F_z}(\omega_i) \\
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\vdots & \vdots & \vdots \\
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\frac{D_{6,R_z}}{F_x}(\omega_i) & \frac{D_{6,R_z}}{F_y}(\omega_i) & \frac{D_{6,R_z}}{F_z}(\omega_i)
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\end{bmatrix}
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\end{equation}
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<p>
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where 1, 2, …, 6 corresponds to the 6 solid bodies.
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</p>
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</div>
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<div class="org-src-container">
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<pre class="src src-matlab">FRFs_O = zeros<span class="org-rainbow-delimiters-depth-1">(</span>length<span class="org-rainbow-delimiters-depth-2">(</span>solid_names<span class="org-rainbow-delimiters-depth-2">)</span><span class="org-type">*</span><span class="org-highlight-numbers-number">6</span>, <span class="org-highlight-numbers-number">3</span>, <span class="org-highlight-numbers-number">801</span><span class="org-rainbow-delimiters-depth-1">)</span>;
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</pre>
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@ -534,26 +563,26 @@ Then, as we know the positions of the accelerometers on each solid body, and we
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</div>
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</div>
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<div id="outline-container-orgbb81ded" class="outline-2">
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<h2 id="orgbb81ded"><span class="section-number-2">5</span> Analysis of some FRF in the global coordinates</h2>
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<div id="outline-container-org1d2db9a" class="outline-2">
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<h2 id="org1d2db9a"><span class="section-number-2">5</span> Analysis of some FRF in the global coordinates</h2>
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<div class="outline-text-2" id="text-5">
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<p>
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First, we can compare the motions of the 6 solid bodies in one direction (figure <a href="#orgc2878f8">2</a>)
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First, we can compare the motions of the 6 solid bodies in one direction (figure <a href="#org1dcf9e5">2</a>)
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</p>
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<p>
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We can also compare all the DOFs of one solid body (figure <a href="#orgfd0203f">3</a>).
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We can also compare all the DOFs of one solid body (figure <a href="#org4750235">3</a>).
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</p>
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<div id="orgc2878f8" class="figure">
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<div id="org1dcf9e5" class="figure">
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<p><img src="figs/frf_all_bodies_one_direction.png" alt="frf_all_bodies_one_direction.png" />
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</p>
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<p><span class="figure-number">Figure 2: </span>FRFs of all the 6 solid bodies in one direction</p>
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</div>
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<div id="orgfd0203f" class="figure">
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<div id="org4750235" class="figure">
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<p><img src="figs/frf_one_body_all_directions.png" alt="frf_one_body_all_directions.png" />
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</p>
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<p><span class="figure-number">Figure 3: </span>FRFs of one solid body in all its DOFs</p>
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@ -561,136 +590,48 @@ We can also compare all the DOFs of one solid body (figure <a href="#orgfd0203f"
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</div>
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</div>
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<div id="outline-container-orgb0c1df8" class="outline-2">
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<h2 id="orgb0c1df8"><span class="section-number-2">6</span> <span class="todo TODO">TODO</span> How to compare the relative motion of solid bodies</h2>
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<div id="outline-container-org26b8f8f" class="outline-2">
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<h2 id="org26b8f8f"><span class="section-number-2">6</span> Comparison of the relative motion of solid bodies</h2>
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<div class="outline-text-2" id="text-6">
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<p>
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We have some of elements of the full FRF matrix:
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\[ \frac{D_{1x}}{F_x},\ \frac{D_{1y}}{F_x},\ \frac{D_{1z}}{F_x},\ \frac{D_{2x}}{F_x},\ \dots \]
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Now that the motion of all the solid bodies are expressed in the same frame, we should be able to <b>compare them</b>.
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This can be used to determine what joints direction between two solid bodies is stiff enough that we can fix this DoF.
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This could help reduce the order of the model and simplify the extraction of the model parameters from the measurements.
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</p>
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<p>
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\[ \frac{D_{1x}}{D_{2x}} = \frac{\frac{D_{1x}}{F_x}}{\frac{D_{2x}}{F_x}} \]
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Then, if \(\left| \frac{D_{1x}}{D_{2x}} \right| \approx 1\) in all the frequency band of interest, we can block the \(x\) motion between the solids 1 and 2.
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We decide to plot the "normalized relative motion" between solid bodies \(i\) and \(j\):
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\[ 0 < \Delta_{ij, x} = \frac{\left| D_{i,x} - D_{j,x} \right|}{|D_{i,x}| + |D_{j,x}|} < 1 \]
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</p>
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<p>
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\[ \frac{D_{2x} - D_{1x}}{D_{1x} + D_{2x}} = \frac{\frac{D_{2x}}{F_x} - \frac{D_{1x}}{F_x}}{\frac{D_{1x}}{F_x} + \frac{D_{2x}}{F_x}} \]
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Then, if \(\Delta_{ij,x} \ll 0\) in the frequency band of interest, we have that \(D_{ix} \approx D_{jx}\) and we can neglect that DOF between the two solid bodies \(i\) and \(j\).
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</p>
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<p>
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Then if \(\left| \frac{D_{2x} - D_{1x}}{D_{1x} + D_{2x}} \right| \ll 1\) in all the frequency band of interest, we can block the \(x\) motion between the solids 1 and 2.
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</p>
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</div>
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</div>
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<div id="outline-container-org9c93ccf" class="outline-2">
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<h2 id="org9c93ccf"><span class="section-number-2">7</span> Relative Motion in the global coordinates</h2>
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<div class="outline-text-2" id="text-7">
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<p>
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Below we plot the normalized relative motion between each stage:
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\[ 0 < \frac{\left| D_{ix} - D_{jx} \right|}{|D_{ix}| + |D_{jx}|} < 1 \]
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This normalized relative motion is shown on figure <a href="#org74b537d">4</a> for all the directions and for all the adjacent pair of solid bodies.
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</p>
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<div class="org-src-container">
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<pre class="src src-matlab">DOFs = <span class="org-rainbow-delimiters-depth-1">{</span><span class="org-string">'$T_x$', '$T_y$', '$T_z$', '</span>$<span class="org-type">\</span>theta_x$', '$<span class="org-type">\</span>theta_y$', '$<span class="org-type">\</span>theta_z$'<span class="org-rainbow-delimiters-depth-1">}</span>
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dirs_i = <span class="org-highlight-numbers-number">1</span><span class="org-type">:</span><span class="org-highlight-numbers-number">6</span>;
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exc_dir = <span class="org-highlight-numbers-number">1</span>;
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<span class="org-type">figure</span>;
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<span class="org-keyword">for</span> <span class="org-variable-name">i</span> = <span class="org-constant"><span class="org-highlight-numbers-number">2</span></span><span class="org-constant">:</span><span class="org-constant"><span class="org-highlight-numbers-number">6</span></span>
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subaxis<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">3</span>, <span class="org-highlight-numbers-number">2</span>, <span class="org-constant">i</span><span class="org-rainbow-delimiters-depth-1">)</span>;
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hold on;
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<span class="org-keyword">for</span> <span class="org-variable-name">dir_i</span> = <span class="org-constant">dirs_i</span>
|
||||
H = <span class="org-rainbow-delimiters-depth-1">(</span>squeeze<span class="org-rainbow-delimiters-depth-2">(</span>FRFs_O<span class="org-rainbow-delimiters-depth-3">(</span><span class="org-rainbow-delimiters-depth-4">(</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-4">)</span><span class="org-type">*</span><span class="org-highlight-numbers-number">6</span><span class="org-type">+</span>dir_i, exc_dir, <span class="org-type">:</span><span class="org-rainbow-delimiters-depth-3">)</span><span class="org-rainbow-delimiters-depth-2">)</span><span class="org-type">-</span>squeeze<span class="org-rainbow-delimiters-depth-2">(</span>FRFs_O<span class="org-rainbow-delimiters-depth-3">(</span><span class="org-rainbow-delimiters-depth-4">(</span><span class="org-constant">i</span><span class="org-type">-</span><span class="org-highlight-numbers-number">2</span><span class="org-rainbow-delimiters-depth-4">)</span><span class="org-type">*</span><span class="org-highlight-numbers-number">6</span><span class="org-type">+</span>dir_i, exc_dir, <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><span class="org-type">./</span><span class="org-rainbow-delimiters-depth-1">(</span>abs<span class="org-rainbow-delimiters-depth-2">(</span>squeeze<span class="org-rainbow-delimiters-depth-3">(</span>FRFs_O<span class="org-rainbow-delimiters-depth-4">(</span><span class="org-rainbow-delimiters-depth-5">(</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-5">)</span><span class="org-type">*</span><span class="org-highlight-numbers-number">6</span><span class="org-type">+</span>dir_i, exc_dir, <span class="org-type">:</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-type">+</span>abs<span class="org-rainbow-delimiters-depth-2">(</span>squeeze<span class="org-rainbow-delimiters-depth-3">(</span>FRFs_O<span class="org-rainbow-delimiters-depth-4">(</span><span class="org-rainbow-delimiters-depth-5">(</span><span class="org-constant">i</span><span class="org-type">-</span><span class="org-highlight-numbers-number">2</span><span class="org-rainbow-delimiters-depth-5">)</span><span class="org-type">*</span><span class="org-highlight-numbers-number">6</span><span class="org-type">+</span>dir_i, exc_dir, <span class="org-type">:</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-1">)</span>;
|
||||
plot<span class="org-rainbow-delimiters-depth-1">(</span>freqs, abs<span class="org-rainbow-delimiters-depth-2">(</span>H<span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
||||
<span class="org-keyword">end</span>
|
||||
hold off;
|
||||
<span class="org-type">set</span><span class="org-rainbow-delimiters-depth-1">(</span><span class="org-variable-name">gca</span>, <span class="org-string">'XScale', 'log'</span><span class="org-string"><span class="org-rainbow-delimiters-depth-1">)</span></span><span class="org-string">; set</span><span class="org-string"><span class="org-rainbow-delimiters-depth-1">(</span></span><span class="org-string">gca, 'YScale', 'lin'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
||||
xlim<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbow-delimiters-depth-2">[</span><span class="org-highlight-numbers-number">1</span>, <span class="org-highlight-numbers-number">200</span><span class="org-rainbow-delimiters-depth-2">]</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>;
|
||||
<span class="org-comment">% xlabel('Frequency [Hz]'); ylabel('Relative Motion');</span>
|
||||
title<span class="org-rainbow-delimiters-depth-1">(</span>sprintf<span class="org-rainbow-delimiters-depth-2">(</span><span class="org-string">'Normalized motion %s - %s'</span>, solid_names{<span class="org-constant">i</span><span class="org-type">-</span><span class="org-highlight-numbers-number">1</span>}, solid_names{<span class="org-constant">i</span>}));
|
||||
<span class="org-keyword">if</span> <span class="org-constant">i</span> <span class="org-type">></span> <span class="org-highlight-numbers-number">4</span>
|
||||
xlabel<span class="org-rainbow-delimiters-depth-3">(</span><span class="org-string">'Frequency </span><span class="org-string"><span class="org-rainbow-delimiters-depth-4">[</span></span><span class="org-string">Hz</span><span class="org-string"><span class="org-rainbow-delimiters-depth-4">]</span></span><span class="org-string">'</span><span class="org-rainbow-delimiters-depth-3">)</span>;
|
||||
<span class="org-keyword">else</span>
|
||||
<span class="org-type">set</span><span class="org-rainbow-delimiters-depth-3">(</span><span class="org-variable-name">gca</span>, <span class="org-string">'XTickLabel'</span>,<span class="org-rainbow-delimiters-depth-4">[]</span><span class="org-rainbow-delimiters-depth-3">)</span>;
|
||||
<span class="org-keyword">end</span>
|
||||
<span class="org-keyword">end</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-3">(</span></span><span class="org-constant">dirs_i</span><span class="org-constant"><span class="org-rainbow-delimiters-depth-3">)</span></span>
|
||||
legend_names<span class="org-rainbow-delimiters-depth-3">{</span><span class="org-constant">i</span><span class="org-rainbow-delimiters-depth-3">}</span> = DOFs<span class="org-rainbow-delimiters-depth-3">{</span>dirs_i<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-keyword">end</span>
|
||||
lgd = legend<span class="org-rainbow-delimiters-depth-3">(</span>legend_names<span class="org-rainbow-delimiters-depth-3">)</span>;
|
||||
|
||||
hL = subplot<span class="org-rainbow-delimiters-depth-3">(</span><span class="org-highlight-numbers-number">3</span>, <span class="org-highlight-numbers-number">2</span>, <span class="org-highlight-numbers-number">1</span><span class="org-rainbow-delimiters-depth-3">)</span>;
|
||||
poshL = <span class="org-type">get</span><span class="org-rainbow-delimiters-depth-3">(</span><span class="org-variable-name">hL</span>,<span class="org-string">'position'</span><span class="org-rainbow-delimiters-depth-3">)</span>;
|
||||
|
||||
<span class="org-type">set</span><span class="org-rainbow-delimiters-depth-3">(</span><span class="org-variable-name">lgd</span>,<span class="org-string">'position'</span>, poshL<span class="org-rainbow-delimiters-depth-3">)</span>;
|
||||
<span class="org-type">axis</span><span class="org-rainbow-delimiters-depth-3">(</span>hL, <span class="org-string">'off'</span><span class="org-rainbow-delimiters-depth-3">)</span>;
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
|
||||
<div id="org1cd90a2" class="figure">
|
||||
<div id="org74b537d" class="figure">
|
||||
<p><img src="figs/relative_motion_comparison.png" alt="relative_motion_comparison.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 4: </span>Relative motion between each stage</p>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org7803dae" class="outline-2">
|
||||
<h2 id="org7803dae"><span class="section-number-2">8</span> <span class="todo TODO">TODO</span> Compare original FRF measurements to transformed FRF in the global frame</h2>
|
||||
<div class="outline-text-2" id="text-8">
|
||||
<div class="warning">
|
||||
<p>
|
||||
We wish here to compare the FRF in order to verify if there is any mistake.
|
||||
Can we really compare the motion of two solid bodies from Frequency Response Functions that clearly depends on the excitation point and direction?
|
||||
The relative motion of two solid bodies may be negligible when exciting the structure at on point and but at another point.
|
||||
</p>
|
||||
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">dir_names = <span class="org-rainbow-delimiters-depth-1">{</span><span class="org-string">'X', 'Y', 'Z', '</span>$<span class="org-type">\</span>theta_X$', '$<span class="org-type">\</span>theta_Y$', '$<span class="org-type">\</span>theta_Z$'<span class="org-rainbow-delimiters-depth-1">}</span>;
|
||||
|
||||
solid_i = <span class="org-highlight-numbers-number">6</span>;
|
||||
acc_dir_O = <span class="org-highlight-numbers-number">1</span>;
|
||||
acc_dir = <span class="org-highlight-numbers-number">1</span>;
|
||||
exc_dir = <span class="org-highlight-numbers-number">1</span>;
|
||||
|
||||
<span class="org-type">figure</span>;
|
||||
|
||||
ax1 = 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">solids.</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-2">{</span></span><span class="org-constant">solid_i</span><span class="org-constant"><span class="org-rainbow-delimiters-depth-2">}</span></span><span class="org-constant"><span class="org-rainbow-delimiters-depth-1">)</span></span>
|
||||
plot<span class="org-rainbow-delimiters-depth-1">(</span>freqs, abs<span class="org-rainbow-delimiters-depth-2">(</span>squeeze<span class="org-rainbow-delimiters-depth-3">(</span>FRFs<span class="org-rainbow-delimiters-depth-4">(</span>acc_dir<span class="org-type">+</span><span class="org-highlight-numbers-number">3</span><span class="org-type">*</span><span class="org-rainbow-delimiters-depth-5">(</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-5">)</span>, exc_dir, <span class="org-type">:</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-1">)</span>;
|
||||
<span class="org-keyword">end</span>
|
||||
plot<span class="org-rainbow-delimiters-depth-1">(</span>freqs, abs<span class="org-rainbow-delimiters-depth-2">(</span>squeeze<span class="org-rainbow-delimiters-depth-3">(</span>FRFs_O<span class="org-rainbow-delimiters-depth-4">(</span><span class="org-rainbow-delimiters-depth-5">(</span>solid_i<span class="org-type">-</span><span class="org-highlight-numbers-number">1</span><span class="org-rainbow-delimiters-depth-5">)</span><span class="org-type">*</span><span class="org-highlight-numbers-number">6</span><span class="org-type">+</span>acc_dir_O, exc_dir, <span class="org-type">:</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">'-k'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
||||
hold off;
|
||||
<span class="org-type">set</span><span class="org-rainbow-delimiters-depth-1">(</span><span class="org-variable-name">gca</span>, <span class="org-string">'XScale', 'log'</span><span class="org-string"><span class="org-rainbow-delimiters-depth-1">)</span></span><span class="org-string">; set</span><span class="org-string"><span class="org-rainbow-delimiters-depth-1">(</span></span><span class="org-string">gca, 'YScale', 'log'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
||||
<span class="org-type">set</span><span class="org-rainbow-delimiters-depth-1">(</span><span class="org-variable-name">gca</span>, <span class="org-string">'XTickLabel'</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">'Amplitude'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
||||
title<span class="org-rainbow-delimiters-depth-1">(</span>sprintf<span class="org-rainbow-delimiters-depth-2">(</span><span class="org-string">'%s motion measured by the Acc. vs %s motion computed in the common frame - %s'</span>, dir_names{acc_dir}, dir_names{acc_dir_O}, solid_names{solid_i}));
|
||||
|
||||
ax2 = subplot<span class="org-rainbow-delimiters-depth-3">(</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-3">)</span>;
|
||||
hold on;
|
||||
<span class="org-keyword">for</span> <span class="org-variable-name">i</span> = <span class="org-constant">solids.</span><span class="org-constant"><span class="org-rainbow-delimiters-depth-3">(</span></span><span class="org-constant">solid_names</span><span class="org-constant"><span class="org-rainbow-delimiters-depth-4">{</span></span><span class="org-constant">solid_i</span><span class="org-constant"><span class="org-rainbow-delimiters-depth-4">}</span></span><span class="org-constant"><span class="org-rainbow-delimiters-depth-3">)</span></span>
|
||||
plot<span class="org-rainbow-delimiters-depth-3">(</span>freqs, mod<span class="org-rainbow-delimiters-depth-4">(</span><span class="org-highlight-numbers-number">180</span><span class="org-type">+</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>phase<span class="org-rainbow-delimiters-depth-5">(</span>squeeze<span class="org-rainbow-delimiters-depth-6">(</span>FRFs<span class="org-rainbow-delimiters-depth-7">(</span>acc_dir<span class="org-type">+</span><span class="org-highlight-numbers-number">3</span><span class="org-type">*</span><span class="org-rainbow-delimiters-depth-8">(</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-8">)</span>, exc_dir, <span class="org-type">:</span><span class="org-rainbow-delimiters-depth-7">)</span><span class="org-rainbow-delimiters-depth-6">)</span><span class="org-rainbow-delimiters-depth-5">)</span>, <span class="org-highlight-numbers-number">360</span><span class="org-rainbow-delimiters-depth-4">)</span><span class="org-type">-</span><span class="org-highlight-numbers-number">180</span><span class="org-rainbow-delimiters-depth-3">)</span>;
|
||||
<span class="org-keyword">end</span>
|
||||
plot<span class="org-rainbow-delimiters-depth-3">(</span>freqs, mod<span class="org-rainbow-delimiters-depth-4">(</span><span class="org-highlight-numbers-number">180</span><span class="org-type">+</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>phase<span class="org-rainbow-delimiters-depth-5">(</span>squeeze<span class="org-rainbow-delimiters-depth-6">(</span>FRFs_O<span class="org-rainbow-delimiters-depth-7">(</span><span class="org-rainbow-delimiters-depth-8">(</span>solid_i<span class="org-type">-</span><span class="org-highlight-numbers-number">1</span><span class="org-rainbow-delimiters-depth-8">)</span><span class="org-type">*</span><span class="org-highlight-numbers-number">6</span><span class="org-type">+</span>acc_dir_O, exc_dir, <span class="org-type">:</span><span class="org-rainbow-delimiters-depth-7">)</span><span class="org-rainbow-delimiters-depth-6">)</span><span class="org-rainbow-delimiters-depth-5">)</span>, <span class="org-highlight-numbers-number">360</span><span class="org-rainbow-delimiters-depth-4">)</span><span class="org-type">-</span><span class="org-highlight-numbers-number">180</span>, <span class="org-string">'-k'</span><span class="org-rainbow-delimiters-depth-3">)</span>;
|
||||
hold off;
|
||||
ylim<span class="org-rainbow-delimiters-depth-3">(</span><span class="org-rainbow-delimiters-depth-4">[</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-4">]</span><span class="org-rainbow-delimiters-depth-3">)</span>; yticks<span class="org-rainbow-delimiters-depth-3">(</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-3">)</span>;
|
||||
xlabel<span class="org-rainbow-delimiters-depth-3">(</span><span class="org-string">'Frequency </span><span class="org-string"><span class="org-rainbow-delimiters-depth-4">[</span></span><span class="org-string">Hz</span><span class="org-string"><span class="org-rainbow-delimiters-depth-4">]</span></span><span class="org-string">'</span><span class="org-string"><span class="org-rainbow-delimiters-depth-3">)</span></span><span class="org-string">; ylabel</span><span class="org-string"><span class="org-rainbow-delimiters-depth-3">(</span></span><span class="org-string">'Phase </span><span class="org-string"><span class="org-rainbow-delimiters-depth-4">[</span></span><span class="org-string">deg</span><span class="org-string"><span class="org-rainbow-delimiters-depth-4">]</span></span><span class="org-string">'</span><span class="org-rainbow-delimiters-depth-3">)</span>;
|
||||
<span class="org-type">set</span><span class="org-rainbow-delimiters-depth-3">(</span><span class="org-variable-name">gca</span>, <span class="org-string">'xscale', 'log'</span><span class="org-rainbow-delimiters-depth-3">)</span>;
|
||||
|
||||
linkaxes<span class="org-rainbow-delimiters-depth-3">(</span><span class="org-rainbow-delimiters-depth-4">[</span>ax1,ax2<span class="org-rainbow-delimiters-depth-4">]</span>,<span class="org-string">'x'</span><span class="org-rainbow-delimiters-depth-3">)</span>;
|
||||
xlim<span class="org-rainbow-delimiters-depth-3">(</span><span class="org-rainbow-delimiters-depth-4">[</span><span class="org-highlight-numbers-number">1</span>, <span class="org-highlight-numbers-number">200</span><span class="org-rainbow-delimiters-depth-4">]</span><span class="org-rainbow-delimiters-depth-3">)</span>;
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org45fdd6c" class="outline-2">
|
||||
<h2 id="org45fdd6c"><span class="section-number-2">9</span> Verify that we find the original FRF from the FRF in the global coordinates</h2>
|
||||
<div class="outline-text-2" id="text-9">
|
||||
<div id="outline-container-org1f7f9fb" class="outline-2">
|
||||
<h2 id="org1f7f9fb"><span class="section-number-2">7</span> Verify that we find the original FRF from the FRF in the global coordinates</h2>
|
||||
<div class="outline-text-2" id="text-7">
|
||||
<p>
|
||||
We have computed the Frequency Response Functions Matrix <code>FRFs_O</code> representing the response of the 6 solid bodies in their 6 DOFs.
|
||||
</p>
|
||||
@ -737,24 +678,24 @@ We then compare the original FRF measured for each accelerometer with the recove
|
||||
</p>
|
||||
|
||||
<p>
|
||||
The FRF for the 4 accelerometers on the Hexapod are compared on figure <a href="#orgdf7e1b1">5</a>.
|
||||
The FRF for the 4 accelerometers on the Hexapod are compared on figure <a href="#orgacee11c">5</a>.
|
||||
All the FRF are matching very well in all the frequency range displayed.
|
||||
</p>
|
||||
|
||||
<p>
|
||||
The FRF for accelerometers located on the translation stage are compared on figure <a href="#orgc07a0c4">6</a>.
|
||||
The FRF for accelerometers located on the translation stage are compared on figure <a href="#orgeff91f2">6</a>.
|
||||
The FRF are matching well until 100Hz.
|
||||
</p>
|
||||
|
||||
|
||||
<div id="orgdf7e1b1" class="figure">
|
||||
<div id="orgacee11c" class="figure">
|
||||
<p><img src="figs/recovered_frf_comparison_hexa.png" alt="recovered_frf_comparison_hexa.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 5: </span>Comparison of the original FRF with the recovered ones - Hexapod</p>
|
||||
</div>
|
||||
|
||||
|
||||
<div id="orgc07a0c4" class="figure">
|
||||
<div id="orgeff91f2" class="figure">
|
||||
<p><img src="figs/recovered_frf_comparison_ty.png" alt="recovered_frf_comparison_ty.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 6: </span>Comparison of the original FRF with the recovered ones - Ty</p>
|
||||
@ -771,13 +712,23 @@ This confirms the fact that the stages are indeed behaving as a solid body in th
|
||||
This valid the fact that a multi-body model can be used to represent the dynamics of the micro-station.
|
||||
</p>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org2cd1928" class="outline-2">
|
||||
<h2 id="org2cd1928"><span class="section-number-2">8</span> Saving of the FRF expressed in the global coordinates</h2>
|
||||
<div class="outline-text-2" id="text-8">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">save<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'mat/frf_o.mat', 'FRFs_O'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<div id="postamble" class="status">
|
||||
<p class="author">Author: Dehaeze Thomas</p>
|
||||
<p class="date">Created: 2019-07-05 ven. 10:16</p>
|
||||
<p class="date">Created: 2019-07-05 ven. 11:06</p>
|
||||
<p class="validation"><a href="http://validator.w3.org/check?uri=referer">Validate</a></p>
|
||||
</div>
|
||||
</body>
|
||||
|
||||
@ -25,6 +25,8 @@
|
||||
#+PROPERTY: header-args:matlab+ :exports both
|
||||
#+PROPERTY: header-args:matlab+ :eval no-export
|
||||
#+PROPERTY: header-args:matlab+ :output-dir figs
|
||||
#+PROPERTY: header-args:matlab+ :tangle matlab/frf_processing.m
|
||||
#+PROPERTY: header-args:matlab+ :mkdirp yes
|
||||
|
||||
#+PROPERTY: header-args:shell :eval no-export
|
||||
|
||||
@ -56,6 +58,22 @@ Thus, we are only interested in $6 \times 6 = 36$ degrees of freedom.
|
||||
|
||||
We here process the FRF matrix to go from the 69 measured DOFs to the wanted 36 DOFs.
|
||||
|
||||
* ZIP file containing the data and matlab files :ignore:
|
||||
#+begin_src bash :exports none :results none
|
||||
if [ matlab/frf_processing.m -nt data/frf_processing.zip ]; then
|
||||
cp matlab/frf_processing.m frf_processing.m;
|
||||
zip data/frf_processing \
|
||||
mat/frf_coh_matrices.mat \
|
||||
mat/geometry.mat \
|
||||
frf_processing.m
|
||||
rm frf_processing.m;
|
||||
fi
|
||||
#+end_src
|
||||
|
||||
#+begin_note
|
||||
All the files (data and Matlab scripts) are accessible [[file:data/frf_processing.zip][here]].
|
||||
#+end_note
|
||||
|
||||
* Matlab Init :noexport:ignore:
|
||||
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
|
||||
<<matlab-dir>>
|
||||
@ -70,7 +88,7 @@ We load the measured FRF and Coherence matrices.
|
||||
We also load the geometric parameters of the station: solid bodies considered and the position of the accelerometers.
|
||||
|
||||
#+begin_src matlab
|
||||
load('./mat/frf_coh_matrices.mat', 'FRFs', 'COHs', 'freqs');
|
||||
load('mat/frf_coh_matrices.mat', 'FRFs', 'COHs', 'freqs');
|
||||
load('mat/geometry.mat', 'solids', 'solid_names', 'acc_pos');
|
||||
#+end_src
|
||||
|
||||
@ -201,6 +219,24 @@ First, we initialize a new FRF matrix =FRFs_O= which is an $n \times p \times q$
|
||||
- $p$ is the number of excitation inputs: $3$
|
||||
- $q$ is the number of frequency points $\omega_i$
|
||||
|
||||
#+begin_important
|
||||
For each frequency point $\omega_i$, the FRF matrix =FRFs_O= is a $n\times p$ matrix:
|
||||
\begin{equation}
|
||||
\text{FRF}_O(\omega_i) = \begin{bmatrix}
|
||||
\frac{D_{1,T_x}}{F_x}(\omega_i) & \frac{D_{1,T_x}}{F_y}(\omega_i) & \frac{D_{1,T_x}}{F_z}(\omega_i) \\
|
||||
\frac{D_{1,T_y}}{F_x}(\omega_i) & \frac{D_{1,T_y}}{F_y}(\omega_i) & \frac{D_{1,T_y}}{F_z}(\omega_i) \\
|
||||
\frac{D_{1,T_z}}{F_x}(\omega_i) & \frac{D_{1,T_z}}{F_y}(\omega_i) & \frac{D_{1,T_z}}{F_z}(\omega_i) \\
|
||||
\frac{D_{1,R_x}}{F_x}(\omega_i) & \frac{D_{1,R_x}}{F_y}(\omega_i) & \frac{D_{1,R_x}}{F_z}(\omega_i) \\
|
||||
\frac{D_{1,R_y}}{F_x}(\omega_i) & \frac{D_{1,R_y}}{F_y}(\omega_i) & \frac{D_{1,R_y}}{F_z}(\omega_i) \\
|
||||
\frac{D_{1,R_z}}{F_x}(\omega_i) & \frac{D_{1,R_z}}{F_y}(\omega_i) & \frac{D_{1,R_z}}{F_z}(\omega_i) \\
|
||||
\frac{D_{2,T_x}}{F_x}(\omega_i) & \frac{D_{2,T_x}}{F_y}(\omega_i) & \frac{D_{2,T_x}}{F_z}(\omega_i) \\
|
||||
\vdots & \vdots & \vdots \\
|
||||
\frac{D_{6,R_z}}{F_x}(\omega_i) & \frac{D_{6,R_z}}{F_y}(\omega_i) & \frac{D_{6,R_z}}{F_z}(\omega_i)
|
||||
\end{bmatrix}
|
||||
\end{equation}
|
||||
where 1, 2, ..., 6 corresponds to the 6 solid bodies.
|
||||
#+end_important
|
||||
|
||||
#+begin_src matlab
|
||||
FRFs_O = zeros(length(solid_names)*6, 3, 801);
|
||||
#+end_src
|
||||
@ -233,7 +269,7 @@ We can also compare all the DOFs of one solid body (figure [[fig:frf_one_body_al
|
||||
|
||||
#+begin_src matlab :exports none
|
||||
exc_names = {'$F_x$', '$F_y$', '$F_z$'};
|
||||
DOFs = {'$T_x$', '$T_y$', '$T_z$', '$\theta_x$', '$\theta_y$', '$\theta_z$'}
|
||||
DOFs = {'$T_x$', '$T_y$', '$T_z$', '$\theta_x$', '$\theta_y$', '$\theta_z$'};
|
||||
solids_i = 1:6;
|
||||
dir_i = 1;
|
||||
exc_dir = 1;
|
||||
@ -276,7 +312,7 @@ We can also compare all the DOFs of one solid body (figure [[fig:frf_one_body_al
|
||||
[[file:figs/frf_all_bodies_one_direction.png]]
|
||||
|
||||
#+begin_src matlab :exports none
|
||||
DOFs = {'$T_x$', '$T_y$', '$T_z$', '$\theta_x$', '$\theta_y$', '$\theta_z$'}
|
||||
DOFs = {'$T_x$', '$T_y$', '$T_z$', '$\theta_x$', '$\theta_y$', '$\theta_z$'};
|
||||
solid_i = 3;
|
||||
dirs_i = 1:6;
|
||||
exc_dir = 1;
|
||||
@ -318,23 +354,20 @@ We can also compare all the DOFs of one solid body (figure [[fig:frf_one_body_al
|
||||
#+CAPTION: FRFs of one solid body in all its DOFs
|
||||
[[file:figs/frf_one_body_all_directions.png]]
|
||||
|
||||
* TODO How to compare the relative motion of solid bodies
|
||||
We have some of elements of the full FRF matrix:
|
||||
\[ \frac{D_{1x}}{F_x},\ \frac{D_{1y}}{F_x},\ \frac{D_{1z}}{F_x},\ \frac{D_{2x}}{F_x},\ \dots \]
|
||||
* Comparison of the relative motion of solid bodies
|
||||
Now that the motion of all the solid bodies are expressed in the same frame, we should be able to *compare them*.
|
||||
This can be used to determine what joints direction between two solid bodies is stiff enough that we can fix this DoF.
|
||||
This could help reduce the order of the model and simplify the extraction of the model parameters from the measurements.
|
||||
|
||||
\[ \frac{D_{1x}}{D_{2x}} = \frac{\frac{D_{1x}}{F_x}}{\frac{D_{2x}}{F_x}} \]
|
||||
Then, if $\left| \frac{D_{1x}}{D_{2x}} \right| \approx 1$ in all the frequency band of interest, we can block the $x$ motion between the solids 1 and 2.
|
||||
We decide to plot the "normalized relative motion" between solid bodies $i$ and $j$:
|
||||
\[ 0 < \Delta_{ij, x} = \frac{\left| D_{i,x} - D_{j,x} \right|}{|D_{i,x}| + |D_{j,x}|} < 1 \]
|
||||
|
||||
\[ \frac{D_{2x} - D_{1x}}{D_{1x} + D_{2x}} = \frac{\frac{D_{2x}}{F_x} - \frac{D_{1x}}{F_x}}{\frac{D_{1x}}{F_x} + \frac{D_{2x}}{F_x}} \]
|
||||
Then, if $\Delta_{ij,x} \ll 0$ in the frequency band of interest, we have that $D_{ix} \approx D_{jx}$ and we can neglect that DOF between the two solid bodies $i$ and $j$.
|
||||
|
||||
Then if $\left| \frac{D_{2x} - D_{1x}}{D_{1x} + D_{2x}} \right| \ll 1$ in all the frequency band of interest, we can block the $x$ motion between the solids 1 and 2.
|
||||
This normalized relative motion is shown on figure [[fig:relative_motion_comparison]] for all the directions and for all the adjacent pair of solid bodies.
|
||||
|
||||
* Relative Motion in the global coordinates
|
||||
Below we plot the normalized relative motion between each stage:
|
||||
\[ 0 < \frac{\left| D_{ix} - D_{jx} \right|}{|D_{ix}| + |D_{jx}|} < 1 \]
|
||||
|
||||
#+begin_src matlab
|
||||
DOFs = {'$T_x$', '$T_y$', '$T_z$', '$\theta_x$', '$\theta_y$', '$\theta_z$'}
|
||||
#+begin_src matlab :exports none
|
||||
DOFs = {'$T_x$', '$T_y$', '$T_z$', '$\theta_x$', '$\theta_y$', '$\theta_z$'};
|
||||
|
||||
dirs_i = 1:6;
|
||||
exc_dir = 1;
|
||||
@ -381,45 +414,10 @@ Below we plot the normalized relative motion between each stage:
|
||||
#+CAPTION: Relative motion between each stage
|
||||
[[file:figs/relative_motion_comparison.png]]
|
||||
|
||||
* TODO Compare original FRF measurements to transformed FRF in the global frame
|
||||
We wish here to compare the FRF in order to verify if there is any mistake.
|
||||
|
||||
#+begin_src matlab
|
||||
dir_names = {'X', 'Y', 'Z', '$\theta_X$', '$\theta_Y$', '$\theta_Z$'};
|
||||
|
||||
solid_i = 6;
|
||||
acc_dir_O = 1;
|
||||
acc_dir = 1;
|
||||
exc_dir = 1;
|
||||
|
||||
figure;
|
||||
|
||||
ax1 = subplot(2, 1, 1);
|
||||
hold on;
|
||||
for i = solids.(solid_names{solid_i})
|
||||
plot(freqs, abs(squeeze(FRFs(acc_dir+3*(i-1), exc_dir, :))));
|
||||
end
|
||||
plot(freqs, abs(squeeze(FRFs_O((solid_i-1)*6+acc_dir_O, exc_dir, :))), '-k');
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
set(gca, 'XTickLabel',[]);
|
||||
ylabel('Amplitude');
|
||||
title(sprintf('%s motion measured by the Acc. vs %s motion computed in the common frame - %s', dir_names{acc_dir}, dir_names{acc_dir_O}, solid_names{solid_i}));
|
||||
|
||||
ax2 = subplot(2, 1, 2);
|
||||
hold on;
|
||||
for i = solids.(solid_names{solid_i})
|
||||
plot(freqs, mod(180+180/pi*phase(squeeze(FRFs(acc_dir+3*(i-1), exc_dir, :))), 360)-180);
|
||||
end
|
||||
plot(freqs, mod(180+180/pi*phase(squeeze(FRFs_O((solid_i-1)*6+acc_dir_O, exc_dir, :))), 360)-180, '-k');
|
||||
hold off;
|
||||
ylim([-180, 180]); yticks(-180:90:180);
|
||||
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
|
||||
set(gca, 'xscale', 'log');
|
||||
|
||||
linkaxes([ax1,ax2],'x');
|
||||
xlim([1, 200]);
|
||||
#+end_src
|
||||
#+begin_warning
|
||||
Can we really compare the motion of two solid bodies from Frequency Response Functions that clearly depends on the excitation point and direction?
|
||||
The relative motion of two solid bodies may be negligible when exciting the structure at on point and but at another point.
|
||||
#+end_warning
|
||||
|
||||
* Verify that we find the original FRF from the FRF in the global coordinates
|
||||
We have computed the Frequency Response Functions Matrix =FRFs_O= representing the response of the 6 solid bodies in their 6 DOFs.
|
||||
@ -446,7 +444,6 @@ This will help us to determine if:
|
||||
% We get the position of the accelerometer expressed in frame O
|
||||
pos = acc_pos(acc_i, :)';
|
||||
posX = [0 pos(3) -pos(2); -pos(3) 0 pos(1) ; pos(2) -pos(1) 0];
|
||||
[0 acc_pos(i, 3) -acc_pos(i, 2) ; -acc_pos(i, 3) 0 acc_pos(i, 1) ; acc_pos(i, 2) -acc_pos(i, 1) 0]
|
||||
|
||||
FRF_recovered(3*(acc_i-1)+1:3*(acc_i-1)+3, exc_dir, :) = v0 + posX*W0;
|
||||
end
|
||||
@ -464,7 +461,7 @@ The FRF are matching well until 100Hz.
|
||||
|
||||
#+begin_src matlab :exports none
|
||||
exc_names = {'$F_x$', '$F_y$', '$F_z$'};
|
||||
DOFs = {'$T_x$', '$T_y$', '$T_z$', '$\theta_x$', '$\theta_y$', '$\theta_z$'}
|
||||
DOFs = {'$T_x$', '$T_y$', '$T_z$', '$\theta_x$', '$\theta_y$', '$\theta_z$'};
|
||||
|
||||
solid_i = 6;
|
||||
exc_dir = 1;
|
||||
@ -516,7 +513,7 @@ The FRF are matching well until 100Hz.
|
||||
|
||||
#+begin_src matlab :exports none
|
||||
exc_names = {'$F_x$', '$F_y$', '$F_z$'};
|
||||
DOFs = {'$T_x$', '$T_y$', '$T_z$', '$\theta_x$', '$\theta_y$', '$\theta_z$'}
|
||||
DOFs = {'$T_x$', '$T_y$', '$T_z$', '$\theta_x$', '$\theta_y$', '$\theta_z$'};
|
||||
|
||||
solid_i = 3;
|
||||
exc_dir = 1;
|
||||
@ -574,107 +571,7 @@ The FRF are matching well until 100Hz.
|
||||
This valid the fact that a multi-body model can be used to represent the dynamics of the micro-station.
|
||||
#+end_important
|
||||
|
||||
* Importation of measured FRF curves :noexport:ignore:
|
||||
There are 24 measurements files corresponding to 24 series of impacts:
|
||||
- 3 directions, 8 sets of 3 accelerometers
|
||||
For each measurement file, the FRF and coherence between the impact and the 9 accelerations measured.
|
||||
|
||||
In reality: 4 sets of 10 things
|
||||
|
||||
* Saving of the FRF expressed in the global coordinates
|
||||
#+begin_src matlab
|
||||
a = load('mat/meas_frf_coh_1.mat');
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab
|
||||
figure;
|
||||
|
||||
ax1 = subplot(2, 1, 1);
|
||||
hold on;
|
||||
plot(a.FFT1_AvXSpc_2_1_RMS_X_Val, a.FFT1_AvXSpc_2_1_RMS_Y_Mod)
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
set(gca, 'XTickLabel',[]);
|
||||
ylabel('Amplitude');
|
||||
title(sprintf('From %s, to %s', FFT1_AvXSpc_2_1_RfName, FFT1_AvXSpc_2_1_RpName))
|
||||
|
||||
ax2 = subplot(2, 1, 2);
|
||||
hold on;
|
||||
plot(a.FFT1_AvXSpc_2_1_RMS_X_Val, a.FFT1_AvXSpc_2_1_RMS_Y_Phas)
|
||||
hold off;
|
||||
ylim([-180, 180]); yticks(-180:90:180);
|
||||
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
|
||||
set(gca, 'xscale', 'log');
|
||||
|
||||
linkaxes([ax1,ax2],'x');
|
||||
xlim([1, 200]);
|
||||
#+end_src
|
||||
|
||||
* Analysis of some FRFs :noexport:ignore:
|
||||
#+begin_src matlab
|
||||
acc_i = 3;
|
||||
acc_dir = 1;
|
||||
exc_dir = 1;
|
||||
|
||||
figure;
|
||||
|
||||
ax1 = subplot(2, 1, 1);
|
||||
hold on;
|
||||
plot(freqs, abs(squeeze(FRFs(acc_dir+3*(acc_i-1), exc_dir, :))));
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
set(gca, 'XTickLabel',[]);
|
||||
ylabel('Amplitude');
|
||||
|
||||
ax2 = subplot(2, 1, 2);
|
||||
hold on;
|
||||
plot(freqs, mod(180+180/pi*phase(squeeze(FRFs(acc_dir+3*(acc_i-1), exc_dir, :))), 360)-180);
|
||||
hold off;
|
||||
ylim([-180, 180]); yticks(-180:90:180);
|
||||
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
|
||||
set(gca, 'xscale', 'log');
|
||||
|
||||
linkaxes([ax1,ax2],'x');
|
||||
xlim([1, 200]);
|
||||
#+end_src
|
||||
|
||||
|
||||
#+begin_src matlab
|
||||
figure;
|
||||
hold on;
|
||||
for i = 1:3*n_acc
|
||||
plot(freqs, squeeze(COHs(i, 1, :)), 'color', [0, 0, 0, 0.2]);
|
||||
end
|
||||
hold off;
|
||||
xlabel('Frequency [Hz]');
|
||||
ylabel('Coherence [\%]');
|
||||
#+end_src
|
||||
|
||||
|
||||
Composite Response Function.
|
||||
|
||||
We here sum the norm instead of the complex numbers.
|
||||
|
||||
#+begin_src matlab
|
||||
HHx = squeeze(sum(abs(FRFs(:, 1, :))));
|
||||
HHy = squeeze(sum(abs(FRFs(:, 2, :))));
|
||||
HHz = squeeze(sum(abs(FRFs(:, 3, :))));
|
||||
HH = squeeze(sum([HHx, HHy, HHz], 2));
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab
|
||||
exc_dir = 3;
|
||||
|
||||
figure;
|
||||
hold on;
|
||||
for i = 1:3*n_acc
|
||||
plot(freqs, abs(squeeze(FRFs(i, exc_dir, :))), 'color', [0, 0, 0, 0.2]);
|
||||
end
|
||||
plot(freqs, abs(HHx));
|
||||
plot(freqs, abs(HHy));
|
||||
plot(freqs, abs(HHz));
|
||||
plot(freqs, abs(HH), 'k');
|
||||
hold off;
|
||||
set(gca, 'XScale', 'lin'); set(gca, 'YScale', 'lin');
|
||||
xlabel('Frequency [Hz]'); ylabel('Amplitude');
|
||||
xlim([1, 200]);
|
||||
save('mat/frf_o.mat', 'FRFs_O');
|
||||
#+end_src
|
||||
|
||||
@ -3,7 +3,7 @@
|
||||
"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
|
||||
<html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en">
|
||||
<head>
|
||||
<!-- 2019-07-03 mer. 13:58 -->
|
||||
<!-- 2019-07-05 ven. 10:17 -->
|
||||
<meta http-equiv="Content-Type" content="text/html;charset=utf-8" />
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1" />
|
||||
<title>Modal Analysis of the ID31 Micro-Station</title>
|
||||
@ -256,22 +256,22 @@ for the JavaScript code in this tag.
|
||||
<h1 class="title">Modal Analysis of the ID31 Micro-Station</h1>
|
||||
|
||||
<p>
|
||||
The goal is to experimentally extract a <b>Spatial Model</b> (mass, damping, stiffness) of the structure (shown on figure <a href="#orgf65e30f">1</a>) in order to tune the Multi-Body model.
|
||||
The goal is to experimentally extract a <b>Spatial Model</b> (mass, damping, stiffness) of the structure (shown on figure <a href="#org811aaee">1</a>) in order to tune the Multi-Body model.
|
||||
</p>
|
||||
|
||||
|
||||
<div id="orgf65e30f" class="figure">
|
||||
<div id="org811aaee" class="figure">
|
||||
<p><img src="img/nass_picture.png" alt="nass_picture.png" width="500px" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 1: </span>Picture of the ID31 Micro-Station. (1) Granite (2) Translation Stage (3) Tilt Stage (4) Hexapod (5) Dummy Mass</p>
|
||||
</div>
|
||||
|
||||
<p>
|
||||
The procedure is represented on figure <a href="#org28538ba">2</a> where we go from left to right.
|
||||
The procedure is represented on figure <a href="#org60c2278">2</a> where we go from left to right.
|
||||
</p>
|
||||
|
||||
|
||||
<div id="org28538ba" class="figure">
|
||||
<div id="org60c2278" class="figure">
|
||||
<p><img src="img/vibration_analysis_procedure.png" alt="vibration_analysis_procedure.png" width="400px" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 2: </span>Vibration Analysis Procedure</p>
|
||||
@ -300,13 +300,14 @@ The modal analysis of the ID31 Micro-station thus consists of several parts:
|
||||
</p>
|
||||
<ul class="org-ul">
|
||||
<li><a href="measurement.html">Frequency Response Measurements</a></li>
|
||||
<li><a href="frf_processing.html">Frequency Response Analysis and Processing</a></li>
|
||||
<li><a href="modal_extraction.html">Modal Parameter Extraction</a></li>
|
||||
<li><a href="mathematical_model.html">Derivation of Mathematical Model</a></li>
|
||||
</ul>
|
||||
</div>
|
||||
<div id="postamble" class="status">
|
||||
<p class="author">Author: Dehaeze Thomas</p>
|
||||
<p class="date">Created: 2019-07-03 mer. 13:58</p>
|
||||
<p class="date">Created: 2019-07-05 ven. 10:17</p>
|
||||
<p class="validation"><a href="http://validator.w3.org/check?uri=referer">Validate</a></p>
|
||||
</div>
|
||||
</body>
|
||||
|
||||
@ -46,5 +46,6 @@ The modes we want to identify are those in the frequency range between 0Hz and 1
|
||||
|
||||
The modal analysis of the ID31 Micro-station thus consists of several parts:
|
||||
- [[file:measurement.org][Frequency Response Measurements]]
|
||||
- [[file:frf_processing.org][Frequency Response Analysis and Processing]]
|
||||
- [[file:modal_extraction.org][Modal Parameter Extraction]]
|
||||
- [[file:mathematical_model.org][Derivation of Mathematical Model]]
|
||||
|
||||
Binary file not shown.
BIN
modal-analysis/mat/frf_o.mat
Normal file
BIN
modal-analysis/mat/frf_o.mat
Normal file
Binary file not shown.
BIN
modal-analysis/mat/geometry.mat
Normal file
BIN
modal-analysis/mat/geometry.mat
Normal file
Binary file not shown.
@ -1,2 +0,0 @@
|
||||
|
||||
(termite:6464): GLib-WARNING **: 10:26:13.129: GChildWatchSource: Exit status of a child process was requested but ECHILD was received by waitpid(). See the documentation of g_child_watch_source_new() for possible causes.
|
||||
325
modal-analysis/matlab/frf_processing.m
Normal file
325
modal-analysis/matlab/frf_processing.m
Normal file
@ -0,0 +1,325 @@
|
||||
%% Clear Workspace and Close figures
|
||||
clear; close all; clc;
|
||||
|
||||
%% Intialize Laplace variable
|
||||
s = zpk('s');
|
||||
|
||||
% Importation of measured FRF curves
|
||||
% We load the measured FRF and Coherence matrices.
|
||||
% We also load the geometric parameters of the station: solid bodies considered and the position of the accelerometers.
|
||||
|
||||
|
||||
load('mat/frf_coh_matrices.mat', 'FRFs', 'COHs', 'freqs');
|
||||
load('mat/geometry.mat', 'solids', 'solid_names', 'acc_pos');
|
||||
|
||||
% From accelerometer DOFs to solid body DOFs - Matlab Implementation
|
||||
% First, we initialize a new FRF matrix =FRFs_O= which is an $n \times p \times q$ with:
|
||||
% - $n$ is the number of DOFs of the considered 6 solid-bodies: $6 \times 6 = 36$
|
||||
% - $p$ is the number of excitation inputs: $3$
|
||||
% - $q$ is the number of frequency points $\omega_i$
|
||||
|
||||
% #+begin_important
|
||||
% For each frequency point $\omega_i$, the FRF matrix =FRFs_O= is a $n\times p$ matrix:
|
||||
% \begin{equation}
|
||||
% \text{FRF}_O(\omega_i) = \begin{bmatrix}
|
||||
% \frac{D_{1,T_x}}{F_x}(\omega_i) & \frac{D_{1,T_x}}{F_y}(\omega_i) & \frac{D_{1,T_x}}{F_z}(\omega_i) \\
|
||||
% \frac{D_{1,T_y}}{F_x}(\omega_i) & \frac{D_{1,T_y}}{F_y}(\omega_i) & \frac{D_{1,T_y}}{F_z}(\omega_i) \\
|
||||
% \frac{D_{1,T_z}}{F_x}(\omega_i) & \frac{D_{1,T_z}}{F_y}(\omega_i) & \frac{D_{1,T_z}}{F_z}(\omega_i) \\
|
||||
% \frac{D_{1,R_x}}{F_x}(\omega_i) & \frac{D_{1,R_x}}{F_y}(\omega_i) & \frac{D_{1,R_x}}{F_z}(\omega_i) \\
|
||||
% \frac{D_{1,R_y}}{F_x}(\omega_i) & \frac{D_{1,R_y}}{F_y}(\omega_i) & \frac{D_{1,R_y}}{F_z}(\omega_i) \\
|
||||
% \frac{D_{1,R_z}}{F_x}(\omega_i) & \frac{D_{1,R_z}}{F_y}(\omega_i) & \frac{D_{1,R_z}}{F_z}(\omega_i) \\
|
||||
% \frac{D_{2,T_x}}{F_x}(\omega_i) & \frac{D_{2,T_x}}{F_y}(\omega_i) & \frac{D_{2,T_x}}{F_z}(\omega_i) \\
|
||||
% \vdots & \vdots & \vdots \\
|
||||
% \frac{D_{6,R_z}}{F_x}(\omega_i) & \frac{D_{6,R_z}}{F_y}(\omega_i) & \frac{D_{6,R_z}}{F_z}(\omega_i)
|
||||
% \end{bmatrix}
|
||||
% \end{equation}
|
||||
% where 1, 2, ..., 6 corresponds to the 6 solid bodies.
|
||||
% #+end_important
|
||||
|
||||
|
||||
FRFs_O = zeros(length(solid_names)*6, 3, 801);
|
||||
|
||||
|
||||
|
||||
% Then, as we know the positions of the accelerometers on each solid body, and we have the response of those accelerometers, we can use the equations derived in the previous section to determine the response of each solid body expressed in the frame $\{O\}$.
|
||||
|
||||
for solid_i = 1:length(solid_names)
|
||||
solids_i = solids.(solid_names{solid_i});
|
||||
|
||||
A = zeros(3*length(solids_i), 6);
|
||||
for i = 1:length(solids_i)
|
||||
acc_i = solids_i(i);
|
||||
|
||||
A(3*(i-1)+1:3*i, 1:3) = eye(3);
|
||||
A(3*(i-1)+1:3*i, 4:6) = [ 0 acc_pos(acc_i, 3) -acc_pos(acc_i, 2) ;
|
||||
-acc_pos(acc_i, 3) 0 acc_pos(acc_i, 1) ;
|
||||
acc_pos(acc_i, 2) -acc_pos(acc_i, 1) 0];
|
||||
end
|
||||
|
||||
for exc_dir = 1:3
|
||||
FRFs_O((solid_i-1)*6+1:solid_i*6, exc_dir, :) = A\squeeze(FRFs((solids_i(1)-1)*3+1:solids_i(end)*3, exc_dir, :));
|
||||
end
|
||||
end
|
||||
|
||||
% Analysis of some FRF in the global coordinates
|
||||
% First, we can compare the motions of the 6 solid bodies in one direction (figure [[fig:frf_all_bodies_one_direction]])
|
||||
|
||||
% We can also compare all the DOFs of one solid body (figure [[fig:frf_one_body_all_directions]]).
|
||||
|
||||
|
||||
exc_names = {'$F_x$', '$F_y$', '$F_z$'};
|
||||
DOFs = {'$T_x$', '$T_y$', '$T_z$', '$\theta_x$', '$\theta_y$', '$\theta_z$'};
|
||||
solids_i = 1:6;
|
||||
dir_i = 1;
|
||||
exc_dir = 1;
|
||||
|
||||
figure;
|
||||
|
||||
ax1 = subaxis(2, 1, 1);
|
||||
hold on;
|
||||
for solid_i = solids_i
|
||||
plot(freqs, abs(squeeze(FRFs_O((solid_i-1)*6+dir_i, exc_dir, :))), 'DisplayName', solid_names{solid_i});
|
||||
end
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
set(gca, 'XTickLabel',[]);
|
||||
ylabel('Amplitude');
|
||||
legend('Location', 'northwest');
|
||||
title(sprintf('FRF between %s and %s', exc_names{exc_dir}, DOFs{dir_i}));
|
||||
|
||||
ax2 = subaxis(2, 1, 2);
|
||||
hold on;
|
||||
for solid_i = solids_i
|
||||
plot(freqs, mod(180+180/pi*phase(squeeze(FRFs_O((solid_i-1)*6+dir_i, exc_dir, :))), 360)-180);
|
||||
end
|
||||
hold off;
|
||||
ylim([-180, 180]); yticks(-180:90:180);
|
||||
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
|
||||
set(gca, 'xscale', 'log');
|
||||
|
||||
linkaxes([ax1,ax2],'x');
|
||||
xlim([1, 200]);
|
||||
|
||||
|
||||
|
||||
% #+NAME: fig:frf_all_bodies_one_direction
|
||||
% #+CAPTION: FRFs of all the 6 solid bodies in one direction
|
||||
% [[file:figs/frf_all_bodies_one_direction.png]]
|
||||
|
||||
|
||||
DOFs = {'$T_x$', '$T_y$', '$T_z$', '$\theta_x$', '$\theta_y$', '$\theta_z$'};
|
||||
solid_i = 3;
|
||||
dirs_i = 1:6;
|
||||
exc_dir = 1;
|
||||
|
||||
figure;
|
||||
|
||||
ax1 = subplot(2, 1, 1);
|
||||
hold on;
|
||||
for dir_i = dirs_i
|
||||
plot(freqs, abs(squeeze(FRFs_O((solid_i-1)*6+dir_i, exc_dir, :))), 'DisplayName', DOFs{dir_i});
|
||||
end
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
set(gca, 'XTickLabel',[]);
|
||||
ylabel('Amplitude');
|
||||
legend('Location', 'northwest');
|
||||
title(sprintf('Motion of %s due to %s', solid_names{solid_i}, exc_names{exc_dir}));
|
||||
|
||||
ax2 = subplot(2, 1, 2);
|
||||
hold on;
|
||||
for dir_i = dirs_i
|
||||
plot(freqs, mod(180+180/pi*phase(squeeze(FRFs_O((solid_i-1)*6+dir_i, exc_dir, :))), 360)-180);
|
||||
end
|
||||
hold off;
|
||||
ylim([-180, 180]); yticks(-180:90:180);
|
||||
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
|
||||
set(gca, 'xscale', 'log');
|
||||
|
||||
linkaxes([ax1,ax2],'x');
|
||||
xlim([1, 200]);
|
||||
|
||||
% Comparison of the relative motion of solid bodies
|
||||
% Now that the motion of all the solid bodies are expressed in the same frame, we should be able to *compare them*.
|
||||
% This can be used to determine what joints direction between two solid bodies is stiff enough that we can fix this DoF.
|
||||
% This could help reduce the order of the model and simplify the extraction of the model parameters from the measurements.
|
||||
|
||||
% We decide to plot the "normalized relative motion" between solid bodies $i$ and $j$:
|
||||
% \[ 0 < \Delta_{ij, x} = \frac{\left| D_{i,x} - D_{j,x} \right|}{|D_{i,x}| + |D_{j,x}|} < 1 \]
|
||||
|
||||
% Then, if $\Delta_{ij,x} \ll 0$ in the frequency band of interest, we have that $D_{ix} \approx D_{jx}$ and we can neglect that DOF between the two solid bodies $i$ and $j$.
|
||||
|
||||
% This normalized relative motion is shown on figure [[fig:relative_motion_comparison]] for all the directions and for all the adjacent pair of solid bodies.
|
||||
|
||||
|
||||
DOFs = {'$T_x$', '$T_y$', '$T_z$', '$\theta_x$', '$\theta_y$', '$\theta_z$'};
|
||||
|
||||
dirs_i = 1:6;
|
||||
exc_dir = 1;
|
||||
|
||||
figure;
|
||||
|
||||
for i = 2:6
|
||||
subaxis(3, 2, i);
|
||||
hold on;
|
||||
for dir_i = dirs_i
|
||||
H = (squeeze(FRFs_O((i-1)*6+dir_i, exc_dir, :))-squeeze(FRFs_O((i-2)*6+dir_i, exc_dir, :)))./(abs(squeeze(FRFs_O((i-1)*6+dir_i, exc_dir, :)))+abs(squeeze(FRFs_O((i-2)*6+dir_i, exc_dir, :))));
|
||||
plot(freqs, abs(H));
|
||||
end
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
||||
xlim([1, 200]); ylim([0, 1]);
|
||||
% xlabel('Frequency [Hz]'); ylabel('Relative Motion');
|
||||
title(sprintf('Normalized motion %s - %s', solid_names{i-1}, solid_names{i}));
|
||||
if i > 4
|
||||
xlabel('Frequency [Hz]');
|
||||
else
|
||||
set(gca, 'XTickLabel',[]);
|
||||
end
|
||||
end
|
||||
|
||||
for i = 1:length(dirs_i)
|
||||
legend_names{i} = DOFs{dirs_i(i)};
|
||||
end
|
||||
lgd = legend(legend_names);
|
||||
|
||||
hL = subplot(3, 2, 1);
|
||||
poshL = get(hL,'position');
|
||||
|
||||
set(lgd,'position', poshL);
|
||||
axis(hL, 'off');
|
||||
|
||||
% Verify that we find the original FRF from the FRF in the global coordinates
|
||||
% We have computed the Frequency Response Functions Matrix =FRFs_O= representing the response of the 6 solid bodies in their 6 DOFs.
|
||||
|
||||
% From the response of one body in its 6 DOFs, we should be able to compute the FRF of each of its accelerometer fixed to it during the measurement.
|
||||
|
||||
% We can then compare the result with the original measurements.
|
||||
% This will help us to determine if:
|
||||
% - the previous inversion used is correct
|
||||
% - the solid body assumption is correct in the frequency band of interest
|
||||
|
||||
|
||||
FRF_recovered = zeros(size(FRFs));
|
||||
|
||||
% For each excitation direction
|
||||
for exc_dir = 1:3
|
||||
% For each solid
|
||||
for solid_i = 1:length(solid_names)
|
||||
v0 = squeeze(FRFs_O((solid_i-1)*6+1:(solid_i-1)*6+3, exc_dir, :));
|
||||
W0 = squeeze(FRFs_O((solid_i-1)*6+4:(solid_i-1)*6+6, exc_dir, :));
|
||||
|
||||
% For each accelerometer attached to the current solid
|
||||
for acc_i = solids.(solid_names{solid_i})
|
||||
% We get the position of the accelerometer expressed in frame O
|
||||
pos = acc_pos(acc_i, :)';
|
||||
posX = [0 pos(3) -pos(2); -pos(3) 0 pos(1) ; pos(2) -pos(1) 0];
|
||||
|
||||
FRF_recovered(3*(acc_i-1)+1:3*(acc_i-1)+3, exc_dir, :) = v0 + posX*W0;
|
||||
end
|
||||
end
|
||||
end
|
||||
|
||||
|
||||
|
||||
% We then compare the original FRF measured for each accelerometer with the recovered FRF from the global FRF matrix in the common frame.
|
||||
|
||||
% The FRF for the 4 accelerometers on the Hexapod are compared on figure [[fig:recovered_frf_comparison_hexa]].
|
||||
% All the FRF are matching very well in all the frequency range displayed.
|
||||
|
||||
% The FRF for accelerometers located on the translation stage are compared on figure [[fig:recovered_frf_comparison_ty]].
|
||||
% The FRF are matching well until 100Hz.
|
||||
|
||||
|
||||
exc_names = {'$F_x$', '$F_y$', '$F_z$'};
|
||||
DOFs = {'$T_x$', '$T_y$', '$T_z$', '$\theta_x$', '$\theta_y$', '$\theta_z$'};
|
||||
|
||||
solid_i = 6;
|
||||
exc_dir = 1;
|
||||
|
||||
accs_i = solids.(solid_names{solid_i});
|
||||
|
||||
figure;
|
||||
|
||||
for i = 1:length(accs_i)
|
||||
acc_i = accs_i(i);
|
||||
|
||||
subaxis(2, 2, i);
|
||||
|
||||
hold on;
|
||||
for dir_i = 1:3
|
||||
plot(freqs, abs(squeeze(FRFs(3*(acc_i-1)+dir_i, exc_dir, :))), '-', 'DisplayName', DOFs{dir_i});
|
||||
end
|
||||
set(gca,'ColorOrderIndex',1)
|
||||
for dir_i = 1:3
|
||||
plot(freqs, abs(squeeze(FRF_recovered(3*(acc_i-1)+dir_i, exc_dir, :))), '--', 'HandleVisibility', 'off');
|
||||
end
|
||||
hold off;
|
||||
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
if i > 2
|
||||
xlabel('Frequency [Hz]');
|
||||
else
|
||||
set(gca, 'XTickLabel',[]);
|
||||
end
|
||||
|
||||
if rem(i, 2) == 1
|
||||
ylabel('Amplitude');
|
||||
end
|
||||
|
||||
xlim([1, 200]);
|
||||
title(sprintf('Accelerometer %i', accs_i(i)));
|
||||
legend('location', 'northwest');
|
||||
end
|
||||
|
||||
|
||||
|
||||
% #+NAME: fig:recovered_frf_comparison_hexa
|
||||
% #+CAPTION: Comparison of the original FRF with the recovered ones - Hexapod
|
||||
% [[file:figs/recovered_frf_comparison_hexa.png]]
|
||||
|
||||
|
||||
exc_names = {'$F_x$', '$F_y$', '$F_z$'};
|
||||
DOFs = {'$T_x$', '$T_y$', '$T_z$', '$\theta_x$', '$\theta_y$', '$\theta_z$'};
|
||||
|
||||
solid_i = 3;
|
||||
exc_dir = 1;
|
||||
|
||||
accs_i = solids.(solid_names{solid_i});
|
||||
|
||||
figure;
|
||||
|
||||
for i = 1:length(accs_i)
|
||||
acc_i = accs_i(i);
|
||||
|
||||
subaxis(2, 2, i);
|
||||
|
||||
hold on;
|
||||
for dir_i = 1:3
|
||||
plot(freqs, abs(squeeze(FRFs(3*(acc_i-1)+dir_i, exc_dir, :))), '-', 'DisplayName', DOFs{dir_i});
|
||||
end
|
||||
set(gca,'ColorOrderIndex',1)
|
||||
for dir_i = 1:3
|
||||
plot(freqs, abs(squeeze(FRF_recovered(3*(acc_i-1)+dir_i, exc_dir, :))), '--', 'HandleVisibility', 'off');
|
||||
end
|
||||
hold off;
|
||||
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
if i > 2
|
||||
xlabel('Frequency [Hz]');
|
||||
else
|
||||
set(gca, 'XTickLabel',[]);
|
||||
end
|
||||
|
||||
if rem(i, 2) == 1
|
||||
ylabel('Amplitude');
|
||||
end
|
||||
|
||||
xlim([1, 200]);
|
||||
title(sprintf('Accelerometer %i', accs_i(i)));
|
||||
legend('location', 'northwest');
|
||||
end
|
||||
|
||||
% Saving of the FRF expressed in the global coordinates
|
||||
|
||||
save('mat/frf_o.mat', 'FRFs_O');
|
||||
@ -13,12 +13,6 @@ acc_pos = table2array(acc_pos(:, 1:4));
|
||||
[~, i] = sort(acc_pos(:, 1));
|
||||
acc_pos = acc_pos(i, 2:4);
|
||||
|
||||
|
||||
|
||||
% The positions of the sensors relative to the point of interest are shown below.
|
||||
|
||||
data2orgtable([[1:23]', 1000*acc_pos], {}, {'ID', 'x [mm]', 'y [mm]', 'z [mm]'}, ' %.0f ');
|
||||
|
||||
% Windowing
|
||||
% Windowing is used on the force and response signals.
|
||||
|
||||
@ -328,6 +322,22 @@ freqs = meas.FFT1_Coh_10_1_RMS_X_Val;
|
||||
|
||||
save('./mat/frf_coh_matrices.mat', 'FRFs', 'COHs', 'freqs');
|
||||
|
||||
% Plot showing the coherence of all the measurements
|
||||
% Now that we have defined a Coherence matrix, we can plot each of its elements to have an idea of the overall coherence and thus, quality of the measurement.
|
||||
% The result is shown on figure [[fig:all_coherence]].
|
||||
|
||||
|
||||
n_acc = 23;
|
||||
|
||||
figure;
|
||||
hold on;
|
||||
for i = 1:3*n_acc
|
||||
plot(freqs, squeeze(COHs(i, 1, :)), 'color', [0, 0, 0, 0.2]);
|
||||
end
|
||||
hold off;
|
||||
xlabel('Frequency [Hz]');
|
||||
ylabel('Coherence [\%]');
|
||||
|
||||
% Solid Bodies considered for further analysis
|
||||
% We consider the following solid bodies for further analysis:
|
||||
% - Bottom Granite
|
||||
@ -340,8 +350,8 @@ save('./mat/frf_coh_matrices.mat', 'FRFs', 'COHs', 'freqs');
|
||||
% We create a =matlab= structure =solids= that contains the accelerometers ID connected to each solid bodies (as shown on figure [[fig:nass-modal-test]]).
|
||||
|
||||
solids = {};
|
||||
solids.granite_bot = [17, 18, 19, 20];
|
||||
solids.granite_top = [13, 14, 15, 16];
|
||||
solids.gbot = [17, 18, 19, 20];
|
||||
solids.gtop = [13, 14, 15, 16];
|
||||
solids.ty = [9, 10, 11, 12];
|
||||
solids.ry = [5, 6, 7, 8];
|
||||
solids.rz = [21, 22, 23];
|
||||
@ -354,3 +364,116 @@ solid_names = fields(solids);
|
||||
% Finally, we save that into a =.mat= file.
|
||||
|
||||
save('mat/geometry.mat', 'solids', 'solid_names', 'acc_pos');
|
||||
|
||||
|
||||
|
||||
% #+name: fig:aligned_accelerometers
|
||||
% #+caption: Aligned measurement of the motion of a solid body
|
||||
% #+RESULTS:
|
||||
% [[file:figs/aligned_accelerometers.png]]
|
||||
|
||||
% The motion of the rigid body of figure [[fig:aligned_accelerometers]] is defined by the velocity $\vec{v}$ and rotation $\vec{\Omega}$ with respect to the reference frame $\{O\}$.
|
||||
|
||||
% The motions at points $1$ and $2$ are:
|
||||
% \begin{align*}
|
||||
% v_1 &= v + \Omega \times p_1 \\
|
||||
% v_2 &= v + \Omega \times p_2
|
||||
% \end{align*}
|
||||
|
||||
% Taking only the $x$ direction:
|
||||
% \begin{align*}
|
||||
% v_{x1} &= v + \Omega_y p_{z1} - \Omega_z p_{y1} \\
|
||||
% v_{x2} &= v + \Omega_y p_{z2} - \Omega_z p_{y2}
|
||||
% \end{align*}
|
||||
|
||||
% However, we have $p_{1y} = p_{2y}$ and $p_{1z} = p_{2z}$ because of the co-linearity of the two sensors in the $x$ direction, and thus we obtain
|
||||
% \begin{equation}
|
||||
% v_{x1} = v_{x2}
|
||||
% \end{equation}
|
||||
|
||||
% #+begin_important
|
||||
% Two sensors that are measuring the motion of a rigid body in the direction of the line linking the two sensors should measure the same quantity.
|
||||
% #+end_important
|
||||
|
||||
% We can verify that the rigid body assumption is correct by comparing the measurement of the sensors.
|
||||
|
||||
% From the table [[tab:position_accelerometers]], we can guess which sensors will give the same results in the X and Y directions.
|
||||
|
||||
% Comparison of such measurements in the X direction is shown on figure [[fig:compare_acc_x_dir]] and in the Y direction on figure [[fig:compare_acc_y_dir]].
|
||||
|
||||
|
||||
meas_dir = 1;
|
||||
exc_dir = 1;
|
||||
|
||||
acc_i = [1 , 4 ;
|
||||
2 , 3 ;
|
||||
5 , 8 ;
|
||||
6 , 7 ;
|
||||
9 , 12;
|
||||
10, 11;
|
||||
14, 15;
|
||||
18, 19;
|
||||
21, 23];
|
||||
|
||||
figure;
|
||||
for i = 1:size(acc_i, 1)
|
||||
subaxis(3, 3, i);
|
||||
hold on;
|
||||
plot(freqs, abs(squeeze(FRFs(meas_dir+3*(acc_i(i, 1)-1), exc_dir, :))))
|
||||
plot(freqs, abs(squeeze(FRFs(meas_dir+3*(acc_i(i, 2)-1), exc_dir, :))))
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
if i > 6
|
||||
xlabel('Frequency [Hz]');
|
||||
else
|
||||
set(gca, 'XTickLabel',[]);
|
||||
end
|
||||
|
||||
if rem(i, 3) == 1
|
||||
ylabel('Amplitude');
|
||||
end
|
||||
xlim([1, 200]);
|
||||
title(sprintf('Acc %i and %i - X', acc_i(i, 1), acc_i(i, 2)));
|
||||
end
|
||||
|
||||
|
||||
|
||||
% #+NAME: fig:compare_acc_x_dir
|
||||
% #+CAPTION: Compare accelerometers align in the X direction
|
||||
% [[file:figs/compare_acc_x_dir.png]]
|
||||
|
||||
|
||||
|
||||
meas_dir = 2;
|
||||
exc_dir = 1;
|
||||
|
||||
acc_i = [1, 2;
|
||||
5, 6;
|
||||
7, 8;
|
||||
9, 10;
|
||||
11, 12;
|
||||
13, 14;
|
||||
15, 16;
|
||||
17, 18;
|
||||
19, 20];
|
||||
|
||||
figure;
|
||||
for i = 1:size(acc_i, 1)
|
||||
subaxis(3, 3, i);
|
||||
hold on;
|
||||
plot(freqs, abs(squeeze(FRFs(meas_dir+3*(acc_i(i, 1)-1), exc_dir, :))))
|
||||
plot(freqs, abs(squeeze(FRFs(meas_dir+3*(acc_i(i, 2)-1), exc_dir, :))))
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
if i > 6
|
||||
xlabel('Frequency [Hz]');
|
||||
else
|
||||
set(gca, 'XTickLabel',[]);
|
||||
end
|
||||
|
||||
if rem(i, 3) == 1
|
||||
ylabel('Amplitude');
|
||||
end
|
||||
xlim([1, 200]);
|
||||
title(sprintf('Acc %i and %i - Y', acc_i(i, 1), acc_i(i, 2)));
|
||||
end
|
||||
|
||||
@ -214,7 +214,7 @@ We then import that on =matlab=, and sort them.
|
||||
#+end_src
|
||||
|
||||
The positions of the sensors relative to the point of interest are shown below (table [[tab:position_accelerometers]]).
|
||||
#+begin_src matlab :exports results :results value table replace :post addhdr(*this*)
|
||||
#+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*)
|
||||
data2orgtable([[1:23]', 1000*acc_pos], {}, {'ID', 'x [mm]', 'y [mm]', 'z [mm]'}, ' %.0f ');
|
||||
#+end_src
|
||||
|
||||
|
||||
@ -25,6 +25,8 @@
|
||||
#+PROPERTY: header-args:matlab+ :exports both
|
||||
#+PROPERTY: header-args:matlab+ :eval no-export
|
||||
#+PROPERTY: header-args:matlab+ :output-dir figs
|
||||
#+PROPERTY: header-args:matlab+ :tangle matlab/modal_extraction.m
|
||||
#+PROPERTY: header-args:matlab+ :mkdirp yes
|
||||
|
||||
#+PROPERTY: header-args:shell :eval no-export
|
||||
|
||||
@ -39,6 +41,23 @@
|
||||
#+PROPERTY: header-args:latex+ :output-dir figs
|
||||
:END:
|
||||
|
||||
The goal here is to extract the modal parameters describing the modes of station being studied.
|
||||
|
||||
* ZIP file containing the data and matlab files :ignore:
|
||||
#+begin_src bash :exports none :results none
|
||||
if [ matlab/modal_extraction.m -nt data/modal_extraction.zip ]; then
|
||||
cp matlab/modal_extraction.m modal_extraction.m;
|
||||
zip data/modal_extraction \
|
||||
mat/data.mat \
|
||||
modal_extraction.m
|
||||
rm modal_extraction.m;
|
||||
fi
|
||||
#+end_src
|
||||
|
||||
#+begin_note
|
||||
All the files (data and Matlab scripts) are accessible [[file:data/modal_extraction.zip][here]].
|
||||
#+end_note
|
||||
|
||||
* Matlab Init :noexport:ignore:
|
||||
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
|
||||
<<matlab-dir>>
|
||||
@ -1828,7 +1847,6 @@ We here sum the norm instead of the complex numbers.
|
||||
#+CAPTION: Composite Response Function
|
||||
[[file:figs/composite_response_function.png]]
|
||||
|
||||
|
||||
* TODO Singular Value Decomposition - Modal Indication Function
|
||||
Show the same plot as in the modal software.
|
||||
This helps to identify double modes.
|
||||
|
||||
Loading…
x
Reference in New Issue
Block a user