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<h1 class="title">Modal Analysis - Derivation of Mathematical Models</h1>
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<div id="postamble" class="status">
<p class="author">Author: Dehaeze Thomas</p>
<p class="date">Created: 2019-07-03 mer. 13:53</p>
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#+TITLE: Modal Analysis - Derivation of Mathematical Models
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<!-- 2019-07-03 mer. 13:38 --> <!-- 2019-07-03 mer. 13:53 -->
<meta http-equiv="Content-Type" content="text/html;charset=utf-8" /> <meta http-equiv="Content-Type" content="text/html;charset=utf-8" />
<meta name="viewport" content="width=device-width, initial-scale=1" /> <meta name="viewport" content="width=device-width, initial-scale=1" />
<title>Modal Analysis</title> <title>Modal Analysis - Modal Parameter Extraction</title>
<meta name="generator" content="Org mode" /> <meta name="generator" content="Org mode" />
<meta name="author" content="Dehaeze Thomas" /> <meta name="author" content="Dehaeze Thomas" />
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@ -275,38 +275,38 @@ for the JavaScript code in this tag.
| |
<a accesskey="H" href="../index.html"> HOME </a> <a accesskey="H" href="../index.html"> HOME </a>
</div><div id="content"> </div><div id="content">
<h1 class="title">Modal Analysis</h1> <h1 class="title">Modal Analysis - Modal Parameter Extraction</h1>
<div id="table-of-contents"> <div id="table-of-contents">
<h2>Table of Contents</h2> <h2>Table of Contents</h2>
<div id="text-table-of-contents"> <div id="text-table-of-contents">
<ul> <ul>
<li><a href="#org222267a">1. Setup</a></li> <li><a href="#orgb1938e4">1. Setup</a></li>
<li><a href="#org6d2f9fb">2. Mode extraction and importation</a></li> <li><a href="#orgf3f308f">2. Mode extraction and importation</a></li>
<li><a href="#orgf667e97">3. Positions of the sensors</a></li> <li><a href="#org4189585">3. Positions of the sensors</a></li>
<li><a href="#org950ac1b">4. Solids</a></li> <li><a href="#org79f51e1">4. Solids</a></li>
<li><a href="#orgf857139">5. From local coordinates to global coordinates</a></li> <li><a href="#orge60b681">5. From local coordinates to global coordinates</a></li>
<li><a href="#org1f301af">6. Modal Matrices</a></li> <li><a href="#org40ea66a">6. Modal Matrices</a></li>
<li><a href="#orgc5754da">7. Modal Complexity</a></li> <li><a href="#org6f84eb7">7. Modal Complexity</a></li>
<li><a href="#org571cb5c">8. Some notes about constraining the number of degrees of freedom</a></li> <li><a href="#orgfc2cb08">8. Some notes about constraining the number of degrees of freedom</a></li>
<li><a href="#org63d4f29">9. <span class="todo TODO">TODO</span> Normalization of mode shapes?</a></li> <li><a href="#org39dc08b">9. <span class="todo TODO">TODO</span> Normalization of mode shapes?</a></li>
<li><a href="#org1cc7c6e">10. Compare Mode Shapes</a></li> <li><a href="#org865485d">10. Compare Mode Shapes</a></li>
<li><a href="#orgf13e17f">11. Importation of measured FRF curves</a></li> <li><a href="#orgf949d0b">11. Importation of measured FRF curves</a></li>
<li><a href="#org22af433">12. Importation of measured FRF curves to global FRF matrix</a></li> <li><a href="#orgc933603">12. Importation of measured FRF curves to global FRF matrix</a></li>
<li><a href="#org8686279">13. Analysis of some FRFs</a></li> <li><a href="#org887d0a8">13. Analysis of some FRFs</a></li>
<li><a href="#orged94f65">14. From local coordinates to global coordinates with the FRFs</a></li> <li><a href="#orgd5853aa">14. From local coordinates to global coordinates with the FRFs</a></li>
<li><a href="#orgc9405ef">15. Analysis of some FRF in the global coordinates</a></li> <li><a href="#orgac77284">15. Analysis of some FRF in the global coordinates</a></li>
<li><a href="#orgabaf3d0">16. Compare global coordinates to local coordinates</a></li> <li><a href="#orgb9dfde6">16. Compare global coordinates to local coordinates</a></li>
<li><a href="#org157bd81">17. Verify that we find the original FRF from the FRF in the global coordinates</a></li> <li><a href="#org2c2d94c">17. Verify that we find the original FRF from the FRF in the global coordinates</a></li>
<li><a href="#org6f98d62">18. <span class="todo TODO">TODO</span> Synthesis of FRF curves</a></li> <li><a href="#orgd0864b4">18. <span class="todo TODO">TODO</span> Synthesis of FRF curves</a></li>
</ul> </ul>
</div> </div>
</div> </div>
<div id="outline-container-org222267a" class="outline-2"> <div id="outline-container-orgb1938e4" class="outline-2">
<h2 id="org222267a"><span class="section-number-2">1</span> Setup</h2> <h2 id="orgb1938e4"><span class="section-number-2">1</span> Setup</h2>
<div class="outline-text-2" id="text-1"> <div class="outline-text-2" id="text-1">
<div id="org2e944fd" class="figure"> <div id="org4f6b563" class="figure">
<p><img src="figs/nass-modal-test.png" alt="nass-modal-test.png" /> <p><img src="figs/nass-modal-test.png" alt="nass-modal-test.png" />
</p> </p>
<p><span class="figure-number">Figure 1: </span>Position and orientation of the accelerometer used</p> <p><span class="figure-number">Figure 1: </span>Position and orientation of the accelerometer used</p>
@ -314,8 +314,8 @@ for the JavaScript code in this tag.
</div> </div>
</div> </div>
<div id="outline-container-org6d2f9fb" class="outline-2"> <div id="outline-container-orgf3f308f" class="outline-2">
<h2 id="org6d2f9fb"><span class="section-number-2">2</span> Mode extraction and importation</h2> <h2 id="orgf3f308f"><span class="section-number-2">2</span> Mode extraction and importation</h2>
<div class="outline-text-2" id="text-2"> <div class="outline-text-2" id="text-2">
<p> <p>
First, we split the big <code>modes.asc</code> files into sub text files using <code>bash</code>. First, we split the big <code>modes.asc</code> files into sub text files using <code>bash</code>.
@ -511,8 +511,8 @@ The obtained mode frequencies and damping are shown below.
</div> </div>
</div> </div>
<div id="outline-container-orgf667e97" class="outline-2"> <div id="outline-container-org4189585" class="outline-2">
<h2 id="orgf667e97"><span class="section-number-2">3</span> Positions of the sensors</h2> <h2 id="org4189585"><span class="section-number-2">3</span> Positions of the sensors</h2>
<div class="outline-text-2" id="text-3"> <div class="outline-text-2" id="text-3">
<p> <p>
We process the file exported from the <code>modal</code> software containing the positions of the sensors using <code>bash</code>. We process the file exported from the <code>modal</code> software containing the positions of the sensors using <code>bash</code>.
@ -701,8 +701,8 @@ The positions of the sensors relative to the point of interest are shown below.
</div> </div>
</div> </div>
<div id="outline-container-org950ac1b" class="outline-2"> <div id="outline-container-org79f51e1" class="outline-2">
<h2 id="org950ac1b"><span class="section-number-2">4</span> Solids</h2> <h2 id="org79f51e1"><span class="section-number-2">4</span> Solids</h2>
<div class="outline-text-2" id="text-4"> <div class="outline-text-2" id="text-4">
<p> <p>
We consider the following solid bodies: We consider the following solid bodies:
@ -717,7 +717,7 @@ We consider the following solid bodies:
</ul> </ul>
<p> <p>
We create a structure <code>solids</code> that contains the accelerometer number of each solid bodies (as shown on figure <a href="#org2e944fd">1</a>). We create a structure <code>solids</code> that contains the accelerometer number of each solid bodies (as shown on figure <a href="#org4f6b563">1</a>).
</p> </p>
<div class="org-src-container"> <div class="org-src-container">
<pre class="src src-matlab">solids = <span class="org-rainbow-delimiters-depth-1">{}</span>; <pre class="src src-matlab">solids = <span class="org-rainbow-delimiters-depth-1">{}</span>;
@ -734,8 +734,8 @@ solid_names = fields<span class="org-rainbow-delimiters-depth-1">(</span>solids<
</div> </div>
</div> </div>
<div id="outline-container-orgf857139" class="outline-2"> <div id="outline-container-orge60b681" class="outline-2">
<h2 id="orgf857139"><span class="section-number-2">5</span> From local coordinates to global coordinates</h2> <h2 id="orge60b681"><span class="section-number-2">5</span> From local coordinates to global coordinates</h2>
<div class="outline-text-2" id="text-5"> <div class="outline-text-2" id="text-5">
<div class="figure"> <div class="figure">
@ -835,8 +835,8 @@ This inversion is equivalent to a mean square problem.
</div> </div>
</div> </div>
<div id="outline-container-org1f301af" class="outline-2"> <div id="outline-container-org40ea66a" class="outline-2">
<h2 id="org1f301af"><span class="section-number-2">6</span> Modal Matrices</h2> <h2 id="org40ea66a"><span class="section-number-2">6</span> Modal Matrices</h2>
<div class="outline-text-2" id="text-6"> <div class="outline-text-2" id="text-6">
<p> <p>
We want to obtain the two following matrices: We want to obtain the two following matrices:
@ -867,11 +867,11 @@ eigen_vector_M = reshape<span class="org-rainbow-delimiters-depth-1">(</span>mod
</div> </div>
</div> </div>
<div id="outline-container-orgc5754da" class="outline-2"> <div id="outline-container-org6f84eb7" class="outline-2">
<h2 id="orgc5754da"><span class="section-number-2">7</span> Modal Complexity</h2> <h2 id="org6f84eb7"><span class="section-number-2">7</span> Modal Complexity</h2>
<div class="outline-text-2" id="text-7"> <div class="outline-text-2" id="text-7">
<p> <p>
A method of displaying <b>modal complexity</b> is by plotting the elements of the eigenvector on an <b>Argand diagram</b>, such as the ones shown in figure <a href="#org04ca94d">3</a>. A method of displaying <b>modal complexity</b> is by plotting the elements of the eigenvector on an <b>Argand diagram</b>, such as the ones shown in figure <a href="#orgcde45b7">3</a>.
</p> </p>
<p> <p>
@ -880,26 +880,26 @@ The obtained area of this polygon is then compared with the area of the circle w
</p> </p>
<p> <p>
A little complex mode is shown on figure <a href="#org04ca94d">3</a> whereas an highly complex mode is shown on figure <a href="#orgb5f9171">4</a>. A little complex mode is shown on figure <a href="#orgcde45b7">3</a> whereas an highly complex mode is shown on figure <a href="#org9e99786">4</a>.
The complexity of all the modes are compared on figure <a href="#orgae9d053">5</a>. The complexity of all the modes are compared on figure <a href="#orgada7038">5</a>.
</p> </p>
<div id="org04ca94d" class="figure"> <div id="orgcde45b7" class="figure">
<p><img src="figs/modal_complexity_small.png" alt="modal_complexity_small.png" /> <p><img src="figs/modal_complexity_small.png" alt="modal_complexity_small.png" />
</p> </p>
<p><span class="figure-number">Figure 3: </span>Modal Complexity of one mode with small complexity</p> <p><span class="figure-number">Figure 3: </span>Modal Complexity of one mode with small complexity</p>
</div> </div>
<div id="orgb5f9171" class="figure"> <div id="org9e99786" class="figure">
<p><img src="figs/modal_complexity_high.png" alt="modal_complexity_high.png" /> <p><img src="figs/modal_complexity_high.png" alt="modal_complexity_high.png" />
</p> </p>
<p><span class="figure-number">Figure 4: </span>Modal Complexity of one higly complex mode</p> <p><span class="figure-number">Figure 4: </span>Modal Complexity of one higly complex mode</p>
</div> </div>
<div id="orgae9d053" class="figure"> <div id="orgada7038" class="figure">
<p><img src="figs/modal_complexities.png" alt="modal_complexities.png" /> <p><img src="figs/modal_complexities.png" alt="modal_complexities.png" />
</p> </p>
<p><span class="figure-number">Figure 5: </span>Modal complexity for each mode</p> <p><span class="figure-number">Figure 5: </span>Modal complexity for each mode</p>
@ -907,8 +907,8 @@ The complexity of all the modes are compared on figure <a href="#orgae9d053">5</
</div> </div>
</div> </div>
<div id="outline-container-org571cb5c" class="outline-2"> <div id="outline-container-orgfc2cb08" class="outline-2">
<h2 id="org571cb5c"><span class="section-number-2">8</span> Some notes about constraining the number of degrees of freedom</h2> <h2 id="orgfc2cb08"><span class="section-number-2">8</span> Some notes about constraining the number of degrees of freedom</h2>
<div class="outline-text-2" id="text-8"> <div class="outline-text-2" id="text-8">
<p> <p>
We want to have the two eigen matrices. We want to have the two eigen matrices.
@ -1020,8 +1020,8 @@ Is it meaningful to compare mode shapes as they give no information about the am
</div> </div>
</div> </div>
<div id="outline-container-org63d4f29" class="outline-2"> <div id="outline-container-org39dc08b" class="outline-2">
<h2 id="org63d4f29"><span class="section-number-2">9</span> <span class="todo TODO">TODO</span> Normalization of mode shapes?</h2> <h2 id="org39dc08b"><span class="section-number-2">9</span> <span class="todo TODO">TODO</span> Normalization of mode shapes?</h2>
<div class="outline-text-2" id="text-9"> <div class="outline-text-2" id="text-9">
<p> <p>
We normalize each column of the eigen vector matrix. We normalize each column of the eigen vector matrix.
@ -1034,8 +1034,8 @@ Then, each eigenvector as a norm of 1.
</div> </div>
</div> </div>
<div id="outline-container-org1cc7c6e" class="outline-2"> <div id="outline-container-org865485d" class="outline-2">
<h2 id="org1cc7c6e"><span class="section-number-2">10</span> Compare Mode Shapes</h2> <h2 id="org865485d"><span class="section-number-2">10</span> Compare Mode Shapes</h2>
<div class="outline-text-2" id="text-10"> <div class="outline-text-2" id="text-10">
<p> <p>
Let's say we want to see for the first mode which DOFs can be neglected. Let's say we want to see for the first mode which DOFs can be neglected.
@ -1089,8 +1089,8 @@ test = mode_shapes_O<span class="org-rainbow-delimiters-depth-1">(</span><span c
</div> </div>
</div> </div>
<div id="outline-container-orgf13e17f" class="outline-2"> <div id="outline-container-orgf949d0b" class="outline-2">
<h2 id="orgf13e17f"><span class="section-number-2">11</span> Importation of measured FRF curves</h2> <h2 id="orgf949d0b"><span class="section-number-2">11</span> Importation of measured FRF curves</h2>
<div class="outline-text-2" id="text-11"> <div class="outline-text-2" id="text-11">
<p> <p>
There are 24 measurements files corresponding to 24 series of impacts: There are 24 measurements files corresponding to 24 series of impacts:
@ -1138,8 +1138,8 @@ xlim<span class="org-rainbow-delimiters-depth-3">(</span><span class="org-rainbo
</div> </div>
</div> </div>
<div id="outline-container-org22af433" class="outline-2"> <div id="outline-container-orgc933603" class="outline-2">
<h2 id="org22af433"><span class="section-number-2">12</span> Importation of measured FRF curves to global FRF matrix</h2> <h2 id="orgc933603"><span class="section-number-2">12</span> Importation of measured FRF curves to global FRF matrix</h2>
<div class="outline-text-2" id="text-12"> <div class="outline-text-2" id="text-12">
<p> <p>
FRF matrix \(n \times p\): FRF matrix \(n \times p\):
@ -1215,8 +1215,8 @@ freqs = meas.FFT1_Coh_10_1_RMS_X_Val;
</div> </div>
</div> </div>
<div id="outline-container-org8686279" class="outline-2"> <div id="outline-container-org887d0a8" class="outline-2">
<h2 id="org8686279"><span class="section-number-2">13</span> Analysis of some FRFs</h2> <h2 id="org887d0a8"><span class="section-number-2">13</span> Analysis of some FRFs</h2>
<div class="outline-text-2" id="text-13"> <div class="outline-text-2" id="text-13">
<div class="org-src-container"> <div class="org-src-container">
<pre class="src src-matlab">acc_i = <span class="org-highlight-numbers-number">3</span>; <pre class="src src-matlab">acc_i = <span class="org-highlight-numbers-number">3</span>;
@ -1297,8 +1297,8 @@ xlim<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbo
</div> </div>
</div> </div>
<div id="outline-container-orged94f65" class="outline-2"> <div id="outline-container-orgd5853aa" class="outline-2">
<h2 id="orged94f65"><span class="section-number-2">14</span> From local coordinates to global coordinates with the FRFs</h2> <h2 id="orgd5853aa"><span class="section-number-2">14</span> From local coordinates to global coordinates with the FRFs</h2>
<div class="outline-text-2" id="text-14"> <div class="outline-text-2" id="text-14">
<div class="org-src-container"> <div class="org-src-container">
<pre class="src src-matlab"><span class="org-comment">% Number of Solids * DOF for each solid / Number of excitation / frequency points</span> <pre class="src src-matlab"><span class="org-comment">% Number of Solids * DOF for each solid / Number of excitation / frequency points</span>
@ -1325,8 +1325,8 @@ FRFs_O = zeros<span class="org-rainbow-delimiters-depth-1">(</span>length<span c
</div> </div>
</div> </div>
<div id="outline-container-orgc9405ef" class="outline-2"> <div id="outline-container-orgac77284" class="outline-2">
<h2 id="orgc9405ef"><span class="section-number-2">15</span> Analysis of some FRF in the global coordinates</h2> <h2 id="orgac77284"><span class="section-number-2">15</span> Analysis of some FRF in the global coordinates</h2>
<div class="outline-text-2" id="text-15"> <div class="outline-text-2" id="text-15">
<div class="org-src-container"> <div class="org-src-container">
<pre class="src src-matlab">solid_i = <span class="org-highlight-numbers-number">6</span>; <pre class="src src-matlab">solid_i = <span class="org-highlight-numbers-number">6</span>;
@ -1358,8 +1358,8 @@ xlim<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbo
</div> </div>
</div> </div>
<div id="outline-container-orgabaf3d0" class="outline-2"> <div id="outline-container-orgb9dfde6" class="outline-2">
<h2 id="orgabaf3d0"><span class="section-number-2">16</span> Compare global coordinates to local coordinates</h2> <h2 id="orgb9dfde6"><span class="section-number-2">16</span> Compare global coordinates to local coordinates</h2>
<div class="outline-text-2" id="text-16"> <div class="outline-text-2" id="text-16">
<div class="org-src-container"> <div class="org-src-container">
<pre class="src src-matlab">solid_i = <span class="org-highlight-numbers-number">1</span>; <pre class="src src-matlab">solid_i = <span class="org-highlight-numbers-number">1</span>;
@ -1398,8 +1398,8 @@ xlim<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbo
</div> </div>
</div> </div>
<div id="outline-container-org157bd81" class="outline-2"> <div id="outline-container-org2c2d94c" class="outline-2">
<h2 id="org157bd81"><span class="section-number-2">17</span> Verify that we find the original FRF from the FRF in the global coordinates</h2> <h2 id="org2c2d94c"><span class="section-number-2">17</span> Verify that we find the original FRF from the FRF in the global coordinates</h2>
<div class="outline-text-2" id="text-17"> <div class="outline-text-2" id="text-17">
<p> <p>
From the computed FRF of the Hexapod in its 6 DOFs, compute the FRF of the accelerometer 1 fixed to the Hexapod during the measurement. From the computed FRF of the Hexapod in its 6 DOFs, compute the FRF of the accelerometer 1 fixed to the Hexapod during the measurement.
@ -1414,21 +1414,28 @@ From the computed FRF of the Hexapod in its 6 DOFs, compute the FRF of the accel
</div> </div>
<div id="orgb8684e6" class="figure"> <div id="org2303cea" class="figure">
<p><img src="figs/compare_original_meas_with_recovered.png" alt="compare_original_meas_with_recovered.png" /> <p><img src="figs/compare_original_meas_with_recovered.png" alt="compare_original_meas_with_recovered.png" />
</p> </p>
<p><span class="figure-number">Figure 6: </span>Comparison of the original measured FRFs with the recovered FRF from the FRF in the common cartesian frame</p> <p><span class="figure-number">Figure 6: </span>Comparison of the original measured FRFs with the recovered FRF from the FRF in the common cartesian frame</p>
</div>
<div class="important">
<p>
The reduction of the number of degrees of freedom from 69 (23 accelerometers with each 3DOF) to 36 (6 solid bodies with 6 DOF) seems to work well.
</p>
</div> </div>
</div> </div>
</div> </div>
<div id="outline-container-org6f98d62" class="outline-2"> <div id="outline-container-orgd0864b4" class="outline-2">
<h2 id="org6f98d62"><span class="section-number-2">18</span> <span class="todo TODO">TODO</span> Synthesis of FRF curves</h2> <h2 id="orgd0864b4"><span class="section-number-2">18</span> <span class="todo TODO">TODO</span> Synthesis of FRF curves</h2>
</div> </div>
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Dehaeze Thomas</p> <p class="author">Author: Dehaeze Thomas</p>
<p class="date">Created: 2019-07-03 mer. 13:38</p> <p class="date">Created: 2019-07-03 mer. 13:53</p>
<p class="validation"><a href="http://validator.w3.org/check?uri=referer">Validate</a></p> <p class="validation"><a href="http://validator.w3.org/check?uri=referer">Validate</a></p>
</div> </div>
</body> </body>

2
modal-analysis/modes_analysis.org → modal-analysis/modal_extraction.org
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@ -1,4 +1,4 @@
#+TITLE: Modal Analysis #+TITLE: Modal Analysis - Modal Parameter Extraction
:DRAWER: :DRAWER:
#+STARTUP: overview #+STARTUP: overview