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Remove subaxis, use subplot instead

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Thomas Dehaeze
2019-07-05 11:50:06 +02:00
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modal-analysis/frf_processing.html
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@@ -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>
<!-- 2019-07-05 ven. 11:06 -->
<!-- 2019-07-05 ven. 11:49 -->
<meta http-equiv="Content-Type" content="text/html;charset=utf-8" />
<meta name="viewport" content="width=device-width, initial-scale=1" />
<title>Modal Analysis - Processing of FRF</title>
@@ -280,14 +280,14 @@ for the JavaScript code in this tag.
<h2>Table of Contents</h2>
<div id="text-table-of-contents">
<ul>
<li><a href="#org2cad1ec">1. Importation of measured FRF curves</a></li>
<li><a href="#orga573b16">2. From accelerometer DOFs to solid body DOFs - Mathematics</a></li>
<li><a href="#org93fc25e">3. What reference frame to choose?</a></li>
<li><a href="#org15d9437">4. From accelerometer DOFs to solid body DOFs - Matlab Implementation</a></li>
<li><a href="#org1d2db9a">5. Analysis of some FRF in the global coordinates</a></li>
<li><a href="#org26b8f8f">6. Comparison of the relative motion of solid bodies</a></li>
<li><a href="#org1f7f9fb">7. Verify that we find the original FRF from the FRF in the global coordinates</a></li>
<li><a href="#org2cd1928">8. Saving of the FRF expressed in the global coordinates</a></li>
<li><a href="#org90f1199">1. Importation of measured FRF curves</a></li>
<li><a href="#orgad5a590">2. From accelerometer DOFs to solid body DOFs - Mathematics</a></li>
<li><a href="#org2cb14bf">3. What reference frame to choose?</a></li>
<li><a href="#org6dc37dd">4. From accelerometer DOFs to solid body DOFs - Matlab Implementation</a></li>
<li><a href="#org8ac4b88">5. Analysis of some FRF in the global coordinates</a></li>
<li><a href="#orgaeb8895">6. Comparison of the relative motion of solid bodies</a></li>
<li><a href="#orgfd3858f">7. Verify that we find the original FRF from the FRF in the global coordinates</a></li>
<li><a href="#org98351d2">8. Saving of the FRF expressed in the global coordinates</a></li>
</ul>
</div>
</div>
@@ -328,8 +328,8 @@ All the files (data and Matlab scripts) are accessible <a href="data/frf_process
</div>
<div id="outline-container-org2cad1ec" class="outline-2">
<h2 id="org2cad1ec"><span class="section-number-2">1</span> Importation of measured FRF curves</h2>
<div id="outline-container-org90f1199" class="outline-2">
<h2 id="org90f1199"><span class="section-number-2">1</span> Importation of measured FRF curves</h2>
<div class="outline-text-2" id="text-1">
<p>
We load the measured FRF and Coherence matrices.
@@ -344,11 +344,11 @@ load<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string
</div>
</div>
<div id="outline-container-orga573b16" class="outline-2">
<h2 id="orga573b16"><span class="section-number-2">2</span> From accelerometer DOFs to solid body DOFs - Mathematics</h2>
<div id="outline-container-orgad5a590" class="outline-2">
<h2 id="orgad5a590"><span class="section-number-2">2</span> From accelerometer DOFs to solid body DOFs - Mathematics</h2>
<div class="outline-text-2" id="text-2">
<p>
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.
Let's consider the schematic shown on figure <a href="#org2e4472b">1</a> where we are measuring the motion of a (supposed) solid body at 4 distinct points in x-y-z.
</p>
<p>
@@ -356,14 +356,14 @@ The goal here is to link these \(4 \times 3 = 12\) measurements to the 6 DOFs of
</p>
<div id="org9f883a5" class="figure">
<div id="org2e4472b" class="figure">
<p><img src="figs/local_to_global_coordinates.png" alt="local_to_global_coordinates.png" />
</p>
<p><span class="figure-number">Figure 1: </span>Schematic of the measured motions of a solid body</p>
</div>
<p>
From the figure <a href="#org9f883a5">1</a>, we can write:
From the figure <a href="#org2e4472b">1</a>, we can write:
</p>
\begin{align*}
\vec{v}_1 &= \vec{v} + \Omega \vec{p}_1\\
@@ -432,8 +432,8 @@ This inversion is equivalent to resolving a mean square problem.
</div>
</div>
<div id="outline-container-org93fc25e" class="outline-2">
<h2 id="org93fc25e"><span class="section-number-2">3</span> What reference frame to choose?</h2>
<div id="outline-container-org2cb14bf" class="outline-2">
<h2 id="org2cb14bf"><span class="section-number-2">3</span> What reference frame to choose?</h2>
<div class="outline-text-2" id="text-3">
<p>
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.
@@ -453,7 +453,7 @@ The possibles choices are:
<li><b>Base located at the joint position</b>: this is where we want to see the motion and estimate stiffness</li>
</ul>
<table id="orgd8f2173" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
<table id="org8ace9f4" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
<caption class="t-above"><span class="table-number">Table 1:</span> Advantages and disadvantages for the choice of reference frame</caption>
<colgroup>
@@ -497,8 +497,8 @@ As the easiest choice is to choose a common frame, we start with that solution.
</div>
</div>
<div id="outline-container-org15d9437" class="outline-2">
<h2 id="org15d9437"><span class="section-number-2">4</span> From accelerometer DOFs to solid body DOFs - Matlab Implementation</h2>
<div id="outline-container-org6dc37dd" class="outline-2">
<h2 id="org6dc37dd"><span class="section-number-2">4</span> From accelerometer DOFs to solid body DOFs - Matlab Implementation</h2>
<div class="outline-text-2" id="text-4">
<p>
First, we initialize a new FRF matrix <code>FRFs_O</code> which is an \(n \times p \times q\) with:
@@ -563,26 +563,26 @@ Then, as we know the positions of the accelerometers on each solid body, and we
</div>
</div>
<div id="outline-container-org1d2db9a" class="outline-2">
<h2 id="org1d2db9a"><span class="section-number-2">5</span> Analysis of some FRF in the global coordinates</h2>
<div id="outline-container-org8ac4b88" class="outline-2">
<h2 id="org8ac4b88"><span class="section-number-2">5</span> Analysis of some FRF in the global coordinates</h2>
<div class="outline-text-2" id="text-5">
<p>
First, we can compare the motions of the 6 solid bodies in one direction (figure <a href="#org1dcf9e5">2</a>)
First, we can compare the motions of the 6 solid bodies in one direction (figure <a href="#orgb6c91b7">2</a>)
</p>
<p>
We can also compare all the DOFs of one solid body (figure <a href="#org4750235">3</a>).
We can also compare all the DOFs of one solid body (figure <a href="#orgcc6b483">3</a>).
</p>
<div id="org1dcf9e5" class="figure">
<div id="orgb6c91b7" class="figure">
<p><img src="figs/frf_all_bodies_one_direction.png" alt="frf_all_bodies_one_direction.png" />
</p>
<p><span class="figure-number">Figure 2: </span>FRFs of all the 6 solid bodies in one direction</p>
</div>
<div id="org4750235" class="figure">
<div id="orgcc6b483" class="figure">
<p><img src="figs/frf_one_body_all_directions.png" alt="frf_one_body_all_directions.png" />
</p>
<p><span class="figure-number">Figure 3: </span>FRFs of one solid body in all its DOFs</p>
@@ -590,8 +590,8 @@ We can also compare all the DOFs of one solid body (figure <a href="#org4750235"
</div>
</div>
<div id="outline-container-org26b8f8f" class="outline-2">
<h2 id="org26b8f8f"><span class="section-number-2">6</span> Comparison of the relative motion of solid bodies</h2>
<div id="outline-container-orgaeb8895" class="outline-2">
<h2 id="orgaeb8895"><span class="section-number-2">6</span> Comparison of the relative motion of solid bodies</h2>
<div class="outline-text-2" id="text-6">
<p>
Now that the motion of all the solid bodies are expressed in the same frame, we should be able to <b>compare them</b>.
@@ -609,11 +609,11 @@ Then, if \(\Delta_{ij,x} \ll 0\) in the frequency band of interest, we have that
</p>
<p>
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.
This normalized relative motion is shown on figure <a href="#orga181f24">4</a> for all the directions and for all the adjacent pair of solid bodies.
</p>
<div id="org74b537d" class="figure">
<div id="orga181f24" 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>
@@ -629,8 +629,8 @@ The relative motion of two solid bodies may be negligible when exciting the stru
</div>
</div>
<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 id="outline-container-orgfd3858f" class="outline-2">
<h2 id="orgfd3858f"><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.
@@ -664,7 +664,6 @@ This will help us to determine if:
<span class="org-comment">% We get the position of the accelerometer expressed in frame O</span>
pos = acc_pos<span class="org-rainbow-delimiters-depth-1">(</span>acc_i, <span class="org-type">:</span><span class="org-rainbow-delimiters-depth-1">)</span>';
posX = <span class="org-rainbow-delimiters-depth-1">[</span><span class="org-highlight-numbers-number">0</span> pos<span class="org-rainbow-delimiters-depth-2">(</span><span class="org-highlight-numbers-number">3</span><span class="org-rainbow-delimiters-depth-2">)</span> <span class="org-type">-</span>pos<span class="org-rainbow-delimiters-depth-2">(</span><span class="org-highlight-numbers-number">2</span><span class="org-rainbow-delimiters-depth-2">)</span>; <span class="org-type">-</span>pos<span class="org-rainbow-delimiters-depth-2">(</span><span class="org-highlight-numbers-number">3</span><span class="org-rainbow-delimiters-depth-2">)</span> <span class="org-highlight-numbers-number">0</span> pos<span class="org-rainbow-delimiters-depth-2">(</span><span class="org-highlight-numbers-number">1</span><span class="org-rainbow-delimiters-depth-2">)</span> ; pos<span class="org-rainbow-delimiters-depth-2">(</span><span class="org-highlight-numbers-number">2</span><span class="org-rainbow-delimiters-depth-2">)</span> <span class="org-type">-</span>pos<span class="org-rainbow-delimiters-depth-2">(</span><span class="org-highlight-numbers-number">1</span><span class="org-rainbow-delimiters-depth-2">)</span> <span class="org-highlight-numbers-number">0</span><span class="org-rainbow-delimiters-depth-1">]</span>;
<span class="org-rainbow-delimiters-depth-1">[</span><span class="org-highlight-numbers-number">0</span> acc_pos<span class="org-rainbow-delimiters-depth-2">(</span><span class="org-constant">i</span>, <span class="org-highlight-numbers-number">3</span><span class="org-rainbow-delimiters-depth-2">)</span> <span class="org-type">-</span>acc_pos<span class="org-rainbow-delimiters-depth-2">(</span><span class="org-constant">i</span>, <span class="org-highlight-numbers-number">2</span><span class="org-rainbow-delimiters-depth-2">)</span> ; <span class="org-type">-</span>acc_pos<span class="org-rainbow-delimiters-depth-2">(</span><span class="org-constant">i</span>, <span class="org-highlight-numbers-number">3</span><span class="org-rainbow-delimiters-depth-2">)</span> <span class="org-highlight-numbers-number">0</span> acc_pos<span class="org-rainbow-delimiters-depth-2">(</span><span class="org-constant">i</span>, <span class="org-highlight-numbers-number">1</span><span class="org-rainbow-delimiters-depth-2">)</span> ; acc_pos<span class="org-rainbow-delimiters-depth-2">(</span><span class="org-constant">i</span>, <span class="org-highlight-numbers-number">2</span><span class="org-rainbow-delimiters-depth-2">)</span> <span class="org-type">-</span>acc_pos<span class="org-rainbow-delimiters-depth-2">(</span><span class="org-constant">i</span>, <span class="org-highlight-numbers-number">1</span><span class="org-rainbow-delimiters-depth-2">)</span> <span class="org-highlight-numbers-number">0</span><span class="org-rainbow-delimiters-depth-1">]</span>
FRF_recovered<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">3</span><span class="org-type">*</span><span class="org-rainbow-delimiters-depth-2">(</span>acc_i<span class="org-type">-</span><span class="org-highlight-numbers-number">1</span><span class="org-rainbow-delimiters-depth-2">)</span><span class="org-type">+</span><span class="org-highlight-numbers-number">1</span><span class="org-type">:</span><span class="org-highlight-numbers-number">3</span><span class="org-type">*</span><span class="org-rainbow-delimiters-depth-2">(</span>acc_i<span class="org-type">-</span><span class="org-highlight-numbers-number">1</span><span class="org-rainbow-delimiters-depth-2">)</span><span class="org-type">+</span><span class="org-highlight-numbers-number">3</span>, exc_dir, <span class="org-type">:</span><span class="org-rainbow-delimiters-depth-1">)</span> = v0 <span class="org-type">+</span> posX<span class="org-type">*</span>W0;
<span class="org-keyword">end</span>
@@ -678,24 +677,24 @@ 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="#orgacee11c">5</a>.
The FRF for the 4 accelerometers on the Hexapod are compared on figure <a href="#orgf852bb5">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="#orgeff91f2">6</a>.
The FRF for accelerometers located on the translation stage are compared on figure <a href="#org9d0e673">6</a>.
The FRF are matching well until 100Hz.
</p>
<div id="orgacee11c" class="figure">
<div id="orgf852bb5" 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="orgeff91f2" class="figure">
<div id="org9d0e673" 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>
@@ -716,8 +715,8 @@ This valid the fact that a multi-body model can be used to represent the dynamic
</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 id="outline-container-org98351d2" class="outline-2">
<h2 id="org98351d2"><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>;
@@ -728,7 +727,7 @@ This valid the fact that a multi-body model can be used to represent the dynamic
</div>
<div id="postamble" class="status">
<p class="author">Author: Dehaeze Thomas</p>
<p class="date">Created: 2019-07-05 ven. 11:06</p>
<p class="date">Created: 2019-07-05 ven. 11:49</p>
<p class="validation"><a href="http://validator.w3.org/check?uri=referer">Validate</a></p>
</div>
</body>