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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"> "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
<html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en"> <html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en">
<head> <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 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 - Processing of FRF</title> <title>Modal Analysis - Processing of FRF</title>
@ -280,14 +280,14 @@ for the JavaScript code in this tag.
<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="#org2cad1ec">1. Importation of measured FRF curves</a></li> <li><a href="#org90f1199">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="#orgad5a590">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="#org2cb14bf">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="#org6dc37dd">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="#org8ac4b88">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="#orgaeb8895">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="#orgfd3858f">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="#org98351d2">8. Saving of the FRF expressed in the global coordinates</a></li>
</ul> </ul>
</div> </div>
</div> </div>
@ -328,8 +328,8 @@ All the files (data and Matlab scripts) are accessible <a href="data/frf_process
</div> </div>
<div id="outline-container-org2cad1ec" class="outline-2"> <div id="outline-container-org90f1199" class="outline-2">
<h2 id="org2cad1ec"><span class="section-number-2">1</span> Importation of measured FRF curves</h2> <h2 id="org90f1199"><span class="section-number-2">1</span> Importation of measured FRF curves</h2>
<div class="outline-text-2" id="text-1"> <div class="outline-text-2" id="text-1">
<p> <p>
We load the measured FRF and Coherence matrices. 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> </div>
<div id="outline-container-orga573b16" class="outline-2"> <div id="outline-container-orgad5a590" class="outline-2">
<h2 id="orga573b16"><span class="section-number-2">2</span> From accelerometer DOFs to solid body DOFs - Mathematics</h2> <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"> <div class="outline-text-2" id="text-2">
<p> <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>
<p> <p>
@ -356,14 +356,14 @@ The goal here is to link these \(4 \times 3 = 12\) measurements to the 6 DOFs of
</p> </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><img src="figs/local_to_global_coordinates.png" alt="local_to_global_coordinates.png" />
</p> </p>
<p><span class="figure-number">Figure 1: </span>Schematic of the measured motions of a solid body</p> <p><span class="figure-number">Figure 1: </span>Schematic of the measured motions of a solid body</p>
</div> </div>
<p> <p>
From the figure <a href="#org9f883a5">1</a>, we can write: From the figure <a href="#org2e4472b">1</a>, we can write:
</p> </p>
\begin{align*} \begin{align*}
\vec{v}_1 &= \vec{v} + \Omega \vec{p}_1\\ \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> </div>
<div id="outline-container-org93fc25e" class="outline-2"> <div id="outline-container-org2cb14bf" class="outline-2">
<h2 id="org93fc25e"><span class="section-number-2">3</span> What reference frame to choose?</h2> <h2 id="org2cb14bf"><span class="section-number-2">3</span> What reference frame to choose?</h2>
<div class="outline-text-2" id="text-3"> <div class="outline-text-2" id="text-3">
<p> <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. 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> <li><b>Base located at the joint position</b>: this is where we want to see the motion and estimate stiffness</li>
</ul> </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> <caption class="t-above"><span class="table-number">Table 1:</span> Advantages and disadvantages for the choice of reference frame</caption>
<colgroup> <colgroup>
@ -497,8 +497,8 @@ As the easiest choice is to choose a common frame, we start with that solution.
</div> </div>
</div> </div>
<div id="outline-container-org15d9437" class="outline-2"> <div id="outline-container-org6dc37dd" class="outline-2">
<h2 id="org15d9437"><span class="section-number-2">4</span> From accelerometer DOFs to solid body DOFs - Matlab Implementation</h2> <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"> <div class="outline-text-2" id="text-4">
<p> <p>
First, we initialize a new FRF matrix <code>FRFs_O</code> which is an \(n \times p \times q\) with: 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> </div>
<div id="outline-container-org1d2db9a" class="outline-2"> <div id="outline-container-org8ac4b88" class="outline-2">
<h2 id="org1d2db9a"><span class="section-number-2">5</span> Analysis of some FRF in the global coordinates</h2> <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"> <div class="outline-text-2" id="text-5">
<p> <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>
<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> </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><img src="figs/frf_all_bodies_one_direction.png" alt="frf_all_bodies_one_direction.png" />
</p> </p>
<p><span class="figure-number">Figure 2: </span>FRFs of all the 6 solid bodies in one direction</p> <p><span class="figure-number">Figure 2: </span>FRFs of all the 6 solid bodies in one direction</p>
</div> </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><img src="figs/frf_one_body_all_directions.png" alt="frf_one_body_all_directions.png" />
</p> </p>
<p><span class="figure-number">Figure 3: </span>FRFs of one solid body in all its DOFs</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> </div>
<div id="outline-container-org26b8f8f" class="outline-2"> <div id="outline-container-orgaeb8895" class="outline-2">
<h2 id="org26b8f8f"><span class="section-number-2">6</span> Comparison of the relative motion of solid bodies</h2> <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"> <div class="outline-text-2" id="text-6">
<p> <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>. 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>
<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> </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><img src="figs/relative_motion_comparison.png" alt="relative_motion_comparison.png" />
</p> </p>
<p><span class="figure-number">Figure 4: </span>Relative motion between each stage</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> </div>
<div id="outline-container-org1f7f9fb" class="outline-2"> <div id="outline-container-orgfd3858f" 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> <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"> <div class="outline-text-2" id="text-7">
<p> <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. 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> <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>'; 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>; 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; 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> <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>
<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. All the FRF are matching very well in all the frequency range displayed.
</p> </p>
<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. The FRF are matching well until 100Hz.
</p> </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><img src="figs/recovered_frf_comparison_hexa.png" alt="recovered_frf_comparison_hexa.png" />
</p> </p>
<p><span class="figure-number">Figure 5: </span>Comparison of the original FRF with the recovered ones - Hexapod</p> <p><span class="figure-number">Figure 5: </span>Comparison of the original FRF with the recovered ones - Hexapod</p>
</div> </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><img src="figs/recovered_frf_comparison_ty.png" alt="recovered_frf_comparison_ty.png" />
</p> </p>
<p><span class="figure-number">Figure 6: </span>Comparison of the original FRF with the recovered ones - Ty</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> </div>
<div id="outline-container-org2cd1928" class="outline-2"> <div id="outline-container-org98351d2" class="outline-2">
<h2 id="org2cd1928"><span class="section-number-2">8</span> Saving of the FRF expressed in the global coordinates</h2> <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="outline-text-2" id="text-8">
<div class="org-src-container"> <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 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>
<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: 2019-07-05 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> <p class="validation"><a href="http://validator.w3.org/check?uri=referer">Validate</a></p>
</div> </div>
</body> </body>

10
modal-analysis/frf_processing.org
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@ -276,7 +276,7 @@ We can also compare all the DOFs of one solid body (figure [[fig:frf_one_body_al
figure; figure;
ax1 = subaxis(2, 1, 1); ax1 = subplot(2, 1, 1);
hold on; hold on;
for solid_i = solids_i for solid_i = solids_i
plot(freqs, abs(squeeze(FRFs_O((solid_i-1)*6+dir_i, exc_dir, :))), 'DisplayName', solid_names{solid_i}); plot(freqs, abs(squeeze(FRFs_O((solid_i-1)*6+dir_i, exc_dir, :))), 'DisplayName', solid_names{solid_i});
@ -288,7 +288,7 @@ We can also compare all the DOFs of one solid body (figure [[fig:frf_one_body_al
legend('Location', 'northwest'); legend('Location', 'northwest');
title(sprintf('FRF between %s and %s', exc_names{exc_dir}, DOFs{dir_i})); title(sprintf('FRF between %s and %s', exc_names{exc_dir}, DOFs{dir_i}));
ax2 = subaxis(2, 1, 2); ax2 = subplot(2, 1, 2);
hold on; hold on;
for solid_i = solids_i 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); plot(freqs, mod(180+180/pi*phase(squeeze(FRFs_O((solid_i-1)*6+dir_i, exc_dir, :))), 360)-180);
@ -375,7 +375,7 @@ This normalized relative motion is shown on figure [[fig:relative_motion_compari
figure; figure;
for i = 2:6 for i = 2:6
subaxis(3, 2, i); subplot(3, 2, i);
hold on; hold on;
for dir_i = dirs_i 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, :)))); 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, :))));
@ -473,7 +473,7 @@ The FRF are matching well until 100Hz.
for i = 1:length(accs_i) for i = 1:length(accs_i)
acc_i = accs_i(i); acc_i = accs_i(i);
subaxis(2, 2, i); subplot(2, 2, i);
hold on; hold on;
for dir_i = 1:3 for dir_i = 1:3
@ -525,7 +525,7 @@ The FRF are matching well until 100Hz.
for i = 1:length(accs_i) for i = 1:length(accs_i)
acc_i = accs_i(i); acc_i = accs_i(i);
subaxis(2, 2, i); subplot(2, 2, i);
hold on; hold on;
for dir_i = 1:3 for dir_i = 1:3

BIN
modal-analysis/mat/frf_coh_matrices.mat
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Binary file not shown.

10
modal-analysis/matlab/frf_processing.m
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@ -75,7 +75,7 @@ exc_dir = 1;
figure; figure;
ax1 = subaxis(2, 1, 1); ax1 = subplot(2, 1, 1);
hold on; hold on;
for solid_i = solids_i for solid_i = solids_i
plot(freqs, abs(squeeze(FRFs_O((solid_i-1)*6+dir_i, exc_dir, :))), 'DisplayName', solid_names{solid_i}); plot(freqs, abs(squeeze(FRFs_O((solid_i-1)*6+dir_i, exc_dir, :))), 'DisplayName', solid_names{solid_i});
@ -87,7 +87,7 @@ ylabel('Amplitude');
legend('Location', 'northwest'); legend('Location', 'northwest');
title(sprintf('FRF between %s and %s', exc_names{exc_dir}, DOFs{dir_i})); title(sprintf('FRF between %s and %s', exc_names{exc_dir}, DOFs{dir_i}));
ax2 = subaxis(2, 1, 2); ax2 = subplot(2, 1, 2);
hold on; hold on;
for solid_i = solids_i 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); plot(freqs, mod(180+180/pi*phase(squeeze(FRFs_O((solid_i-1)*6+dir_i, exc_dir, :))), 360)-180);
@ -160,7 +160,7 @@ exc_dir = 1;
figure; figure;
for i = 2:6 for i = 2:6
subaxis(3, 2, i); subplot(3, 2, i);
hold on; hold on;
for dir_i = dirs_i 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, :)))); 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, :))));
@ -244,7 +244,7 @@ figure;
for i = 1:length(accs_i) for i = 1:length(accs_i)
acc_i = accs_i(i); acc_i = accs_i(i);
subaxis(2, 2, i); subplot(2, 2, i);
hold on; hold on;
for dir_i = 1:3 for dir_i = 1:3
@ -292,7 +292,7 @@ figure;
for i = 1:length(accs_i) for i = 1:length(accs_i)
acc_i = accs_i(i); acc_i = accs_i(i);
subaxis(2, 2, i); subplot(2, 2, i);
hold on; hold on;
for dir_i = 1:3 for dir_i = 1:3

4
modal-analysis/matlab/modal_frf_coh.m
View File

@ -417,7 +417,7 @@ acc_i = [1 , 4 ;
figure; figure;
for i = 1:size(acc_i, 1) for i = 1:size(acc_i, 1)
subaxis(3, 3, i); subplot(3, 3, i);
hold on; 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, 1)-1), exc_dir, :))))
plot(freqs, abs(squeeze(FRFs(meas_dir+3*(acc_i(i, 2)-1), exc_dir, :)))) plot(freqs, abs(squeeze(FRFs(meas_dir+3*(acc_i(i, 2)-1), exc_dir, :))))
@ -459,7 +459,7 @@ acc_i = [1, 2;
figure; figure;
for i = 1:size(acc_i, 1) for i = 1:size(acc_i, 1)
subaxis(3, 3, i); subplot(3, 3, i);
hold on; 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, 1)-1), exc_dir, :))))
plot(freqs, abs(squeeze(FRFs(meas_dir+3*(acc_i(i, 2)-1), exc_dir, :)))) plot(freqs, abs(squeeze(FRFs(meas_dir+3*(acc_i(i, 2)-1), exc_dir, :))))

255
modal-analysis/measurement.html
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@ -3,7 +3,7 @@
"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd"> "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
<html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en"> <html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en">
<head> <head>
<!-- 2019-07-04 jeu. 17:49 --> <!-- 2019-07-05 ven. 11:46 -->
<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 - Measurement</title> <title>Modal Analysis - Measurement</title>
@ -280,34 +280,34 @@ for the JavaScript code in this tag.
<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="#org85e5106">1. Goal</a></li> <li><a href="#org1379b74">1. Goal</a></li>
<li><a href="#org47f3940">2. Instrumentation Used</a></li> <li><a href="#orgd38a775">2. Instrumentation Used</a></li>
<li><a href="#org977bbae">3. Structure Preparation and Test Planning</a> <li><a href="#orge0b6708">3. Structure Preparation and Test Planning</a>
<ul> <ul>
<li><a href="#orgde64688">3.1. Structure Preparation</a></li> <li><a href="#org37f06d5">3.1. Structure Preparation</a></li>
<li><a href="#orgd83fa83">3.2. Test Planing</a></li> <li><a href="#orgb7f0982">3.2. Test Planing</a></li>
<li><a href="#org3e409fe">3.3. Location of the Accelerometers</a></li> <li><a href="#org164c06b">3.3. Location of the Accelerometers</a></li>
<li><a href="#org7cb5221">3.4. Hammer Impacts</a></li> <li><a href="#orge55aaed">3.4. Hammer Impacts</a></li>
</ul> </ul>
</li> </li>
<li><a href="#org2124857">4. Signal Processing</a> <li><a href="#org94592f9">4. Signal Processing</a>
<ul> <ul>
<li><a href="#org600eae0">4.1. Averaging</a></li> <li><a href="#orgc9341a3">4.1. Averaging</a></li>
<li><a href="#org83d4969">4.2. Windowing</a></li> <li><a href="#org62ef660">4.2. Windowing</a></li>
</ul> </ul>
</li> </li>
<li><a href="#org25e02aa">5. Force and Response signals</a> <li><a href="#orgfcf409d">5. Force and Response signals</a>
<ul> <ul>
<li><a href="#orgbf40925">5.1. Raw Force Data</a></li> <li><a href="#org7263d51">5.1. Raw Force Data</a></li>
<li><a href="#org2cf0a3e">5.2. Raw Response Data</a></li> <li><a href="#orgc9a27c3">5.2. Raw Response Data</a></li>
<li><a href="#orgad3fa68">5.3. Computation of one Frequency Response Function</a></li> <li><a href="#orgf1f892a">5.3. Computation of one Frequency Response Function</a></li>
</ul> </ul>
</li> </li>
<li><a href="#org4bffecd">6. Frequency Response Functions and Coherence Results</a></li> <li><a href="#org0595bd4">6. Frequency Response Functions and Coherence Results</a></li>
<li><a href="#org093edb6">7. <span class="todo TODO">TODO</span> Plot all coherences in one plot</a></li> <li><a href="#org860e46e">7. Generation of a FRF matrix and a Coherence matrix from the measurements</a></li>
<li><a href="#org23b7ad3">8. Generation of a FRF matrix and a Coherence matrix from the measurements</a></li> <li><a href="#orgc8738ff">8. Plot showing the coherence of all the measurements</a></li>
<li><a href="#org8a40769">9. Solid Bodies considered for further analysis</a></li> <li><a href="#org6601f7d">9. Solid Bodies considered for further analysis</a></li>
<li><a href="#org2405c81">10. Notes the solid body assumption</a></li> <li><a href="#org2b1c74a">10. Note about the solid body assumption</a></li>
</ul> </ul>
</div> </div>
</div> </div>
@ -319,8 +319,8 @@ All the files (data and Matlab scripts) are accessible <a href="data/modal_frf_c
</div> </div>
<div id="outline-container-org85e5106" class="outline-2"> <div id="outline-container-org1379b74" class="outline-2">
<h2 id="org85e5106"><span class="section-number-2">1</span> Goal</h2> <h2 id="org1379b74"><span class="section-number-2">1</span> Goal</h2>
<div class="outline-text-2" id="text-1"> <div class="outline-text-2" id="text-1">
<p> <p>
The goal is to measure the dynamic of the Micro-Station and to extract Frequency Response Functions. The goal is to measure the dynamic of the Micro-Station and to extract Frequency Response Functions.
@ -328,20 +328,20 @@ The goal is to measure the dynamic of the Micro-Station and to extract Frequency
</div> </div>
</div> </div>
<div id="outline-container-org47f3940" class="outline-2"> <div id="outline-container-orgd38a775" class="outline-2">
<h2 id="org47f3940"><span class="section-number-2">2</span> Instrumentation Used</h2> <h2 id="orgd38a775"><span class="section-number-2">2</span> Instrumentation Used</h2>
<div class="outline-text-2" id="text-2"> <div class="outline-text-2" id="text-2">
<p> <p>
In order to perform to <b>Modal Analysis</b> and to obtain first a <b>Response Model</b>, the following devices are used: In order to perform to <b>Modal Analysis</b> and to obtain first a <b>Response Model</b>, the following devices are used:
</p> </p>
<ul class="org-ul"> <ul class="org-ul">
<li>An <b>acquisition system</b> (OROS) with 24bits ADCs (figure <a href="#org5f44420">1</a>)</li> <li>An <b>acquisition system</b> (OROS) with 24bits ADCs (figure <a href="#org084bcd5">1</a>)</li>
<li>3 tri-axis <b>Accelerometers</b> (figure <a href="#orgc28a203">2</a>) with parameters shown on table <a href="#org8770ead">1</a></li> <li>3 tri-axis <b>Accelerometers</b> (figure <a href="#org2d07359">2</a>) with parameters shown on table <a href="#orgb383c7a">1</a></li>
<li>An <b>Instrumented Hammer</b> with various Tips (figure <a href="#orge4928f3">3</a>) (figure <a href="#orge94631e">4</a>)</li> <li>An <b>Instrumented Hammer</b> with various Tips (figure <a href="#orgb4bbf0d">3</a>) (figure <a href="#org9052254">4</a>)</li>
</ul> </ul>
<div id="org5f44420" class="figure"> <div id="org084bcd5" class="figure">
<p><img src="img/instrumentation/oros.png" alt="oros.png" width="500px" /> <p><img src="img/instrumentation/oros.png" alt="oros.png" width="500px" />
</p> </p>
<p><span class="figure-number">Figure 1: </span>Acquisition system: OROS</p> <p><span class="figure-number">Figure 1: </span>Acquisition system: OROS</p>
@ -354,13 +354,13 @@ Anti-aliasing filters are also included in the system.
</p> </p>
<div id="orgc28a203" class="figure"> <div id="org2d07359" class="figure">
<p><img src="img/instrumentation/accelero_M393B05.png" alt="accelero_M393B05.png" width="500px" /> <p><img src="img/instrumentation/accelero_M393B05.png" alt="accelero_M393B05.png" width="500px" />
</p> </p>
<p><span class="figure-number">Figure 2: </span>Accelerometer used: M393B05</p> <p><span class="figure-number">Figure 2: </span>Accelerometer used: M393B05</p>
</div> </div>
<table id="org8770ead" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides"> <table id="orgb383c7a" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
<caption class="t-above"><span class="table-number">Table 1:</span> 393B05 Accelerometer Data Sheet</caption> <caption class="t-above"><span class="table-number">Table 1:</span> 393B05 Accelerometer Data Sheet</caption>
<colgroup> <colgroup>
@ -403,14 +403,14 @@ It excites more the low frequency range where the coherence is low, the overall
</p> </p>
<div id="orge4928f3" class="figure"> <div id="orgb4bbf0d" class="figure">
<p><img src="img/instrumentation/instrumented_hammer.png" alt="instrumented_hammer.png" width="500px" /> <p><img src="img/instrumentation/instrumented_hammer.png" alt="instrumented_hammer.png" width="500px" />
</p> </p>
<p><span class="figure-number">Figure 3: </span>Instrumented Hammer</p> <p><span class="figure-number">Figure 3: </span>Instrumented Hammer</p>
</div> </div>
<div id="orge94631e" class="figure"> <div id="org9052254" class="figure">
<p><img src="img/instrumentation/hammer_tips.png" alt="hammer_tips.png" width="500px" /> <p><img src="img/instrumentation/hammer_tips.png" alt="hammer_tips.png" width="500px" />
</p> </p>
<p><span class="figure-number">Figure 4: </span>Hammer tips</p> <p><span class="figure-number">Figure 4: </span>Hammer tips</p>
@ -422,12 +422,12 @@ The accelerometers are glued on the structure.
</div> </div>
</div> </div>
<div id="outline-container-org977bbae" class="outline-2"> <div id="outline-container-orge0b6708" class="outline-2">
<h2 id="org977bbae"><span class="section-number-2">3</span> Structure Preparation and Test Planning</h2> <h2 id="orge0b6708"><span class="section-number-2">3</span> Structure Preparation and Test Planning</h2>
<div class="outline-text-2" id="text-3"> <div class="outline-text-2" id="text-3">
</div> </div>
<div id="outline-container-orgde64688" class="outline-3"> <div id="outline-container-org37f06d5" class="outline-3">
<h3 id="orgde64688"><span class="section-number-3">3.1</span> Structure Preparation</h3> <h3 id="org37f06d5"><span class="section-number-3">3.1</span> Structure Preparation</h3>
<div class="outline-text-3" id="text-3-1"> <div class="outline-text-3" id="text-3-1">
<p> <p>
All the stages are turned ON. All the stages are turned ON.
@ -470,8 +470,8 @@ All other elements have been remove from the granite such as another heavy posit
</div> </div>
</div> </div>
<div id="outline-container-orgd83fa83" class="outline-3"> <div id="outline-container-orgb7f0982" class="outline-3">
<h3 id="orgd83fa83"><span class="section-number-3">3.2</span> Test Planing</h3> <h3 id="orgb7f0982"><span class="section-number-3">3.2</span> Test Planing</h3>
<div class="outline-text-3" id="text-3-2"> <div class="outline-text-3" id="text-3-2">
<p> <p>
The goal is to identify the full \(N \times N\) FRF matrix (where \(N\) is the number of degree of freedom of the system). The goal is to identify the full \(N \times N\) FRF matrix (where \(N\) is the number of degree of freedom of the system).
@ -504,8 +504,8 @@ The measurement thus consists of:
</div> </div>
</div> </div>
<div id="outline-container-org3e409fe" class="outline-3"> <div id="outline-container-org164c06b" class="outline-3">
<h3 id="org3e409fe"><span class="section-number-3">3.3</span> Location of the Accelerometers</h3> <h3 id="org164c06b"><span class="section-number-3">3.3</span> Location of the Accelerometers</h3>
<div class="outline-text-3" id="text-3-3"> <div class="outline-text-3" id="text-3-3">
<p> <p>
4 tri-axis accelerometers are used for each solid body. 4 tri-axis accelerometers are used for each solid body.
@ -524,11 +524,11 @@ The position of the accelerometers are:
</p> </p>
<ul class="org-ul"> <ul class="org-ul">
<li>4 on the first granite</li> <li>4 on the first granite</li>
<li>4 on the second granite (figure <a href="#org7b547b6">5</a>)</li> <li>4 on the second granite (figure <a href="#org3b8109d">5</a>)</li>
<li>4 on top of the translation stage (figure <a href="#orgf41551b">6</a>)</li> <li>4 on top of the translation stage (figure <a href="#org4b120a8">6</a>)</li>
<li>4 on top of the tilt stage</li> <li>4 on top of the tilt stage</li>
<li>3 on top of the spindle</li> <li>3 on top of the spindle</li>
<li>4 on top of the hexapod (figure <a href="#org168dfba">7</a>)</li> <li>4 on top of the hexapod (figure <a href="#org69dec1e">7</a>)</li>
</ul> </ul>
<p> <p>
@ -536,43 +536,43 @@ In total, 23 accelerometers are used: <b>69 DOFs are thus measured</b>.
</p> </p>
<p> <p>
The position and orientation of all the accelerometers used are shown on figure <a href="#org92f8aa8">8</a>. The position and orientation of all the accelerometers used are shown on figure <a href="#orgbff47a7">8</a>.
</p> </p>
<p> <p>
The precise determination of the position of each accelerometer is done using the SolidWorks model (shown on figure <a href="#org87fdd26">9</a>). The precise determination of the position of each accelerometer is done using the SolidWorks model (shown on figure <a href="#org5d34cd5">9</a>).
</p> </p>
<div id="org7b547b6" class="figure"> <div id="org3b8109d" class="figure">
<p><img src="img/accelerometers/accelerometers_granite2_overview.jpg" alt="accelerometers_granite2_overview.jpg" width="500px" /> <p><img src="img/accelerometers/accelerometers_granite2_overview.jpg" alt="accelerometers_granite2_overview.jpg" width="500px" />
</p> </p>
<p><span class="figure-number">Figure 5: </span>Accelerometers located on the top granite</p> <p><span class="figure-number">Figure 5: </span>Accelerometers located on the top granite</p>
</div> </div>
<div id="orgf41551b" class="figure"> <div id="org4b120a8" class="figure">
<p><img src="img/accelerometers/accelerometers_ty_overview.jpg" alt="accelerometers_ty_overview.jpg" width="500px" /> <p><img src="img/accelerometers/accelerometers_ty_overview.jpg" alt="accelerometers_ty_overview.jpg" width="500px" />
</p> </p>
<p><span class="figure-number">Figure 6: </span>Accelerometers located on top of the translation stage</p> <p><span class="figure-number">Figure 6: </span>Accelerometers located on top of the translation stage</p>
</div> </div>
<div id="org168dfba" class="figure"> <div id="org69dec1e" class="figure">
<p><img src="img/accelerometers/accelerometers_hexa_overview.jpg" alt="accelerometers_hexa_overview.jpg" width="500px" /> <p><img src="img/accelerometers/accelerometers_hexa_overview.jpg" alt="accelerometers_hexa_overview.jpg" width="500px" />
</p> </p>
<p><span class="figure-number">Figure 7: </span>Accelerometers located on the Hexapod</p> <p><span class="figure-number">Figure 7: </span>Accelerometers located on the Hexapod</p>
</div> </div>
<div id="org92f8aa8" class="figure"> <div id="orgbff47a7" class="figure">
<p><img src="figs/nass-modal-test.png" alt="nass-modal-test.png" width="800px" /> <p><img src="figs/nass-modal-test.png" alt="nass-modal-test.png" width="800px" />
</p> </p>
<p><span class="figure-number">Figure 8: </span>Position and orientation of the accelerometer used</p> <p><span class="figure-number">Figure 8: </span>Position and orientation of the accelerometer used</p>
</div> </div>
<div id="org87fdd26" class="figure"> <div id="org5d34cd5" class="figure">
<p><img src="img/location_accelerometers.png" alt="location_accelerometers.png" width="800px" /> <p><img src="img/location_accelerometers.png" alt="location_accelerometers.png" width="800px" />
</p> </p>
<p><span class="figure-number">Figure 9: </span>Position of the accelerometers using SolidWorks</p> <p><span class="figure-number">Figure 9: </span>Position of the accelerometers using SolidWorks</p>
@ -599,9 +599,9 @@ acc_pos = acc_pos<span class="org-rainbow-delimiters-depth-1">(</span><span clas
</div> </div>
<p> <p>
The positions of the sensors relative to the point of interest are shown below (table <a href="#org65ae8c9">2</a>). The positions of the sensors relative to the point of interest are shown below (table <a href="#org61987da">2</a>).
</p> </p>
<table id="org65ae8c9" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides"> <table id="org61987da" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
<caption class="t-above"><span class="table-number">Table 2:</span> position of the accelerometers</caption> <caption class="t-above"><span class="table-number">Table 2:</span> position of the accelerometers</caption>
<colgroup> <colgroup>
@ -787,15 +787,15 @@ The positions of the sensors relative to the point of interest are shown below (
</div> </div>
</div> </div>
<div id="outline-container-org7cb5221" class="outline-3"> <div id="outline-container-orge55aaed" class="outline-3">
<h3 id="org7cb5221"><span class="section-number-3">3.4</span> Hammer Impacts</h3> <h3 id="orge55aaed"><span class="section-number-3">3.4</span> Hammer Impacts</h3>
<div class="outline-text-3" id="text-3-4"> <div class="outline-text-3" id="text-3-4">
<p> <p>
Only 3 impact points are used. Only 3 impact points are used.
</p> </p>
<p> <p>
The impact points are shown on figures <a href="#orge9d3fcb">10</a>, <a href="#org689764c">11</a> and <a href="#orgb4d4ec1">12</a>. The impact points are shown on figures <a href="#orge3c8f2c">10</a>, <a href="#orga9f97bd">11</a> and <a href="#orgbf0886d">12</a>.
</p> </p>
<p> <p>
@ -803,21 +803,21 @@ We chose this excitation point as it seems to excite all the modes in the freque
</p> </p>
<div id="orge9d3fcb" class="figure"> <div id="orge3c8f2c" class="figure">
<p><img src="img/impacts/hammer_x.gif" alt="hammer_x.gif" width="300px" /> <p><img src="img/impacts/hammer_x.gif" alt="hammer_x.gif" width="300px" />
</p> </p>
<p><span class="figure-number">Figure 10: </span>Hammer Blow in the X direction</p> <p><span class="figure-number">Figure 10: </span>Hammer Blow in the X direction</p>
</div> </div>
<div id="org689764c" class="figure"> <div id="orga9f97bd" class="figure">
<p><img src="img/impacts/hammer_y.gif" alt="hammer_y.gif" width="300px" /> <p><img src="img/impacts/hammer_y.gif" alt="hammer_y.gif" width="300px" />
</p> </p>
<p><span class="figure-number">Figure 11: </span>Hammer Blow in the Y direction</p> <p><span class="figure-number">Figure 11: </span>Hammer Blow in the Y direction</p>
</div> </div>
<div id="orgb4d4ec1" class="figure"> <div id="orgbf0886d" class="figure">
<p><img src="img/impacts/hammer_z.gif" alt="hammer_z.gif" width="300px" /> <p><img src="img/impacts/hammer_z.gif" alt="hammer_z.gif" width="300px" />
</p> </p>
<p><span class="figure-number">Figure 12: </span>Hammer Blow in the Z direction</p> <p><span class="figure-number">Figure 12: </span>Hammer Blow in the Z direction</p>
@ -826,19 +826,19 @@ We chose this excitation point as it seems to excite all the modes in the freque
</div> </div>
</div> </div>
<div id="outline-container-org2124857" class="outline-2"> <div id="outline-container-org94592f9" class="outline-2">
<h2 id="org2124857"><span class="section-number-2">4</span> Signal Processing</h2> <h2 id="org94592f9"><span class="section-number-2">4</span> Signal Processing</h2>
<div class="outline-text-2" id="text-4"> <div class="outline-text-2" id="text-4">
</div> </div>
<div id="outline-container-org600eae0" class="outline-3"> <div id="outline-container-orgc9341a3" class="outline-3">
<h3 id="org600eae0"><span class="section-number-3">4.1</span> Averaging</h3> <h3 id="orgc9341a3"><span class="section-number-3">4.1</span> Averaging</h3>
<div class="outline-text-3" id="text-4-1"> <div class="outline-text-3" id="text-4-1">
<p> <p>
The measurements are averaged 10 times corresponding to 10 hammer impacts in order to reduce the effect of random noise. The measurements are averaged 10 times corresponding to 10 hammer impacts in order to reduce the effect of random noise.
The parameters for the impact test are shown on table <a href="#orgc7dedc2">3</a>. The parameters for the impact test are shown on table <a href="#org328ce53">3</a>.
</p> </p>
<table id="orgc7dedc2" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides"> <table id="org328ce53" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
<caption class="t-above"><span class="table-number">Table 3:</span> Impact test parameters</caption> <caption class="t-above"><span class="table-number">Table 3:</span> Impact test parameters</caption>
<colgroup> <colgroup>
@ -876,15 +876,15 @@ The parameters for the impact test are shown on table <a href="#orgc7dedc2">3</a
</div> </div>
</div> </div>
<div id="outline-container-org83d4969" class="outline-3"> <div id="outline-container-org62ef660" class="outline-3">
<h3 id="org83d4969"><span class="section-number-3">4.2</span> Windowing</h3> <h3 id="org62ef660"><span class="section-number-3">4.2</span> Windowing</h3>
<div class="outline-text-3" id="text-4-2"> <div class="outline-text-3" id="text-4-2">
<p> <p>
Windowing is used on the force and response signals. Windowing is used on the force and response signals.
</p> </p>
<p> <p>
A boxcar window (figure <a href="#org36f5bec">13</a>) is used for the force signal as once the impact on the structure is done, the measured signal is meaningless. A boxcar window (figure <a href="#orgac8e1ab">13</a>) is used for the force signal as once the impact on the structure is done, the measured signal is meaningless.
The parameters are: The parameters are:
</p> </p>
<ul class="org-ul"> <ul class="org-ul">
@ -893,14 +893,14 @@ The parameters are:
</ul> </ul>
<div id="org36f5bec" class="figure"> <div id="orgac8e1ab" class="figure">
<p><img src="figs/windowing_force_signal.png" alt="windowing_force_signal.png" /> <p><img src="figs/windowing_force_signal.png" alt="windowing_force_signal.png" />
</p> </p>
<p><span class="figure-number">Figure 13: </span>Window used for the force signal</p> <p><span class="figure-number">Figure 13: </span>Window used for the force signal</p>
</div> </div>
<p> <p>
An exponential window (figure <a href="#orgf99890f">14</a>) is used for the response signal as we are measuring transient signals and most of the information is located at the beginning of the signal. An exponential window (figure <a href="#orga3cffc9">14</a>) is used for the response signal as we are measuring transient signals and most of the information is located at the beginning of the signal.
The parameters are: The parameters are:
</p> </p>
<ul class="org-ul"> <ul class="org-ul">
@ -917,7 +917,7 @@ The parameters are:
</ul> </ul>
<div id="orgf99890f" class="figure"> <div id="orga3cffc9" class="figure">
<p><img src="figs/windowing_response_signal.png" alt="windowing_response_signal.png" /> <p><img src="figs/windowing_response_signal.png" alt="windowing_response_signal.png" />
</p> </p>
<p><span class="figure-number">Figure 14: </span>Window used for the response signals</p> <p><span class="figure-number">Figure 14: </span>Window used for the response signals</p>
@ -926,8 +926,8 @@ The parameters are:
</div> </div>
</div> </div>
<div id="outline-container-org25e02aa" class="outline-2"> <div id="outline-container-orgfcf409d" class="outline-2">
<h2 id="org25e02aa"><span class="section-number-2">5</span> Force and Response signals</h2> <h2 id="orgfcf409d"><span class="section-number-2">5</span> Force and Response signals</h2>
<div class="outline-text-2" id="text-5"> <div class="outline-text-2" id="text-5">
<p> <p>
Let's load some obtained data to look at the force and response signals. Let's load some obtained data to look at the force and response signals.
@ -939,33 +939,33 @@ Let's load some obtained data to look at the force and response signals.
</div> </div>
</div> </div>
<div id="outline-container-orgbf40925" class="outline-3"> <div id="outline-container-org7263d51" class="outline-3">
<h3 id="orgbf40925"><span class="section-number-3">5.1</span> Raw Force Data</h3> <h3 id="org7263d51"><span class="section-number-3">5.1</span> Raw Force Data</h3>
<div class="outline-text-3" id="text-5-1"> <div class="outline-text-3" id="text-5-1">
<p> <p>
The force input for the first measurement is shown on figure <a href="#org01af530">15</a>. We can see 10 impacts, one zoom on one impact is shown on figure <a href="#org90778c7">16</a>. The force input for the first measurement is shown on figure <a href="#org2495eee">15</a>. We can see 10 impacts, one zoom on one impact is shown on figure <a href="#org5d907da">16</a>.
</p> </p>
<p> <p>
The Fourier transform of the force is shown on figure <a href="#orga594684">17</a>. This has been obtained without any windowing. The Fourier transform of the force is shown on figure <a href="#org904c868">17</a>. This has been obtained without any windowing.
</p> </p>
<div id="org01af530" class="figure"> <div id="org2495eee" class="figure">
<p><img src="figs/raw_data_force.png" alt="raw_data_force.png" /> <p><img src="figs/raw_data_force.png" alt="raw_data_force.png" />
</p> </p>
<p><span class="figure-number">Figure 15: </span>Raw Force Data from Hammer Blow</p> <p><span class="figure-number">Figure 15: </span>Raw Force Data from Hammer Blow</p>
</div> </div>
<div id="org90778c7" class="figure"> <div id="org5d907da" class="figure">
<p><img src="figs/raw_data_force_zoom.png" alt="raw_data_force_zoom.png" /> <p><img src="figs/raw_data_force_zoom.png" alt="raw_data_force_zoom.png" />
</p> </p>
<p><span class="figure-number">Figure 16: </span>Raw Force Data from Hammer Blow - Zoom</p> <p><span class="figure-number">Figure 16: </span>Raw Force Data from Hammer Blow - Zoom</p>
</div> </div>
<div id="orga594684" class="figure"> <div id="org904c868" class="figure">
<p><img src="figs/fourier_transfor_force_impact.png" alt="fourier_transfor_force_impact.png" /> <p><img src="figs/fourier_transfor_force_impact.png" alt="fourier_transfor_force_impact.png" />
</p> </p>
<p><span class="figure-number">Figure 17: </span>Fourier Transform of the 10 force impacts for the first measurement</p> <p><span class="figure-number">Figure 17: </span>Fourier Transform of the 10 force impacts for the first measurement</p>
@ -973,33 +973,33 @@ The Fourier transform of the force is shown on figure <a href="#orga594684">17</
</div> </div>
</div> </div>
<div id="outline-container-org2cf0a3e" class="outline-3"> <div id="outline-container-orgc9a27c3" class="outline-3">
<h3 id="org2cf0a3e"><span class="section-number-3">5.2</span> Raw Response Data</h3> <h3 id="orgc9a27c3"><span class="section-number-3">5.2</span> Raw Response Data</h3>
<div class="outline-text-3" id="text-5-2"> <div class="outline-text-3" id="text-5-2">
<p> <p>
The response signal for the first measurement is shown on figure <a href="#org2612da5">18</a>. One zoom on one response is shown on figure <a href="#org4bfecd1">19</a>. The response signal for the first measurement is shown on figure <a href="#orge12758d">18</a>. One zoom on one response is shown on figure <a href="#org8eba24d">19</a>.
</p> </p>
<p> <p>
The Fourier transform of the response signals is shown on figure <a href="#org716c433">20</a>. This has been obtained without any windowing. The Fourier transform of the response signals is shown on figure <a href="#orgf445c3f">20</a>. This has been obtained without any windowing.
</p> </p>
<div id="org2612da5" class="figure"> <div id="orge12758d" class="figure">
<p><img src="figs/raw_data_acceleration.png" alt="raw_data_acceleration.png" /> <p><img src="figs/raw_data_acceleration.png" alt="raw_data_acceleration.png" />
</p> </p>
<p><span class="figure-number">Figure 18: </span>Raw Acceleration Data from Accelerometer</p> <p><span class="figure-number">Figure 18: </span>Raw Acceleration Data from Accelerometer</p>
</div> </div>
<div id="org4bfecd1" class="figure"> <div id="org8eba24d" class="figure">
<p><img src="figs/raw_data_acceleration_zoom.png" alt="raw_data_acceleration_zoom.png" /> <p><img src="figs/raw_data_acceleration_zoom.png" alt="raw_data_acceleration_zoom.png" />
</p> </p>
<p><span class="figure-number">Figure 19: </span>Raw Acceleration Data from Accelerometer - Zoom</p> <p><span class="figure-number">Figure 19: </span>Raw Acceleration Data from Accelerometer - Zoom</p>
</div> </div>
<div id="org716c433" class="figure"> <div id="orgf445c3f" class="figure">
<p><img src="figs/fourier_transform_response_signals.png" alt="fourier_transform_response_signals.png" /> <p><img src="figs/fourier_transform_response_signals.png" alt="fourier_transform_response_signals.png" />
</p> </p>
<p><span class="figure-number">Figure 20: </span>Fourier transform of the measured response signals</p> <p><span class="figure-number">Figure 20: </span>Fourier transform of the measured response signals</p>
@ -1007,15 +1007,15 @@ The Fourier transform of the response signals is shown on figure <a href="#org71
</div> </div>
</div> </div>
<div id="outline-container-orgad3fa68" class="outline-3"> <div id="outline-container-orgf1f892a" class="outline-3">
<h3 id="orgad3fa68"><span class="section-number-3">5.3</span> Computation of one Frequency Response Function</h3> <h3 id="orgf1f892a"><span class="section-number-3">5.3</span> Computation of one Frequency Response Function</h3>
<div class="outline-text-3" id="text-5-3"> <div class="outline-text-3" id="text-5-3">
<p> <p>
Now that we have obtained the Fourier transform of both the force input and the response signal, we can compute the Frequency Response Function from the force to the acceleration. Now that we have obtained the Fourier transform of both the force input and the response signal, we can compute the Frequency Response Function from the force to the acceleration.
</p> </p>
<p> <p>
We then compare the result obtained with the FRF computed by the modal software (figure <a href="#orgc57fff5">21</a>). We then compare the result obtained with the FRF computed by the modal software (figure <a href="#orgd26bd12">21</a>).
</p> </p>
<p> <p>
@ -1032,7 +1032,7 @@ In the following analysis, FRF computed from the software will be used.
</div> </div>
<div id="orgc57fff5" class="figure"> <div id="orgd26bd12" class="figure">
<p><img src="figs/frf_comparison_software.png" alt="frf_comparison_software.png" /> <p><img src="figs/frf_comparison_software.png" alt="frf_comparison_software.png" />
</p> </p>
<p><span class="figure-number">Figure 21: </span>Comparison of the computed FRF from the Fourier transform and using the modal software</p> <p><span class="figure-number">Figure 21: </span>Comparison of the computed FRF from the Fourier transform and using the modal software</p>
@ -1041,8 +1041,8 @@ In the following analysis, FRF computed from the software will be used.
</div> </div>
</div> </div>
<div id="outline-container-org4bffecd" class="outline-2"> <div id="outline-container-org0595bd4" class="outline-2">
<h2 id="org4bffecd"><span class="section-number-2">6</span> Frequency Response Functions and Coherence Results</h2> <h2 id="org0595bd4"><span class="section-number-2">6</span> Frequency Response Functions and Coherence Results</h2>
<div class="outline-text-2" id="text-6"> <div class="outline-text-2" id="text-6">
<p> <p>
Let's see one computed Frequency Response Function and one coherence in order to attest the quality of the measurement. Let's see one computed Frequency Response Function and one coherence in order to attest the quality of the measurement.
@ -1057,22 +1057,22 @@ First, we load the data.
</div> </div>
<p> <p>
And we plot on figure <a href="#org0618b41">22</a> the frequency response function from the force applied in the \(X\) direction at the location of the accelerometer number 11 to the acceleration in the \(X\) direction as measured by the first accelerometer located on the top platform of the hexapod. And we plot on figure <a href="#orgf92314e">22</a> the frequency response function from the force applied in the \(X\) direction at the location of the accelerometer number 11 to the acceleration in the \(X\) direction as measured by the first accelerometer located on the top platform of the hexapod.
</p> </p>
<p> <p>
The coherence associated is shown on figure <a href="#org0618b41">22</a>. The coherence associated is shown on figure <a href="#orgf92314e">22</a>.
</p> </p>
<div id="org0618b41" class="figure"> <div id="orgf92314e" class="figure">
<p><img src="figs/frf_result_example.png" alt="frf_result_example.png" /> <p><img src="figs/frf_result_example.png" alt="frf_result_example.png" />
</p> </p>
<p><span class="figure-number">Figure 22: </span>Example of one measured FRF</p> <p><span class="figure-number">Figure 22: </span>Example of one measured FRF</p>
</div> </div>
<div id="org8a24269" class="figure"> <div id="org19a9a76" class="figure">
<p><img src="figs/coh_result_example.png" alt="coh_result_example.png" /> <p><img src="figs/coh_result_example.png" alt="coh_result_example.png" />
</p> </p>
<p><span class="figure-number">Figure 23: </span>Example of one measured Coherence</p> <p><span class="figure-number">Figure 23: </span>Example of one measured Coherence</p>
@ -1080,12 +1080,9 @@ The coherence associated is shown on figure <a href="#org0618b41">22</a>.
</div> </div>
</div> </div>
<div id="outline-container-org093edb6" class="outline-2"> <div id="outline-container-org860e46e" class="outline-2">
<h2 id="org093edb6"><span class="section-number-2">7</span> <span class="todo TODO">TODO</span> Plot all coherences in one plot</h2> <h2 id="org860e46e"><span class="section-number-2">7</span> Generation of a FRF matrix and a Coherence matrix from the measurements</h2>
</div> <div class="outline-text-2" id="text-7">
<div id="outline-container-org23b7ad3" class="outline-2">
<h2 id="org23b7ad3"><span class="section-number-2">8</span> Generation of a FRF matrix and a Coherence matrix from the measurements</h2>
<div class="outline-text-2" id="text-8">
<p> <p>
We want here to combine all the Frequency Response Functions measured into one big array called the <b>Frequency Response Matrix</b>. We want here to combine all the Frequency Response Functions measured into one big array called the <b>Frequency Response Matrix</b>.
</p> </p>
@ -1179,8 +1176,25 @@ And we save the obtained FRF matrix and Coherence matrix in a <code>.mat</code>
</div> </div>
</div> </div>
</div> </div>
<div id="outline-container-org8a40769" class="outline-2"> <div id="outline-container-orgc8738ff" class="outline-2">
<h2 id="org8a40769"><span class="section-number-2">9</span> Solid Bodies considered for further analysis</h2> <h2 id="orgc8738ff"><span class="section-number-2">8</span> Plot showing the coherence of all the measurements</h2>
<div class="outline-text-2" id="text-8">
<p>
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 <a href="#org433aa7b">24</a>.
</p>
<div id="org433aa7b" class="figure">
<p><img src="figs/all_coherence.png" alt="all_coherence.png" />
</p>
<p><span class="figure-number">Figure 24: </span>Plot of the coherence of all the measurements</p>
</div>
</div>
</div>
<div id="outline-container-org6601f7d" class="outline-2">
<h2 id="org6601f7d"><span class="section-number-2">9</span> Solid Bodies considered for further analysis</h2>
<div class="outline-text-2" id="text-9"> <div class="outline-text-2" id="text-9">
<p> <p>
We consider the following solid bodies for further analysis: We consider the following solid bodies for further analysis:
@ -1195,7 +1209,7 @@ We consider the following solid bodies for further analysis:
</ul> </ul>
<p> <p>
We create a <code>matlab</code> structure <code>solids</code> that contains the accelerometers ID connected to each solid bodies (as shown on figure <a href="#org92f8aa8">8</a>). We create a <code>matlab</code> structure <code>solids</code> that contains the accelerometers ID connected to each solid bodies (as shown on figure <a href="#orgbff47a7">8</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>;
@ -1219,26 +1233,27 @@ Finally, we save that into a <code>.mat</code> file.
</div> </div>
</div> </div>
</div> </div>
<div id="outline-container-org2405c81" class="outline-2">
<h2 id="org2405c81"><span class="section-number-2">10</span> Notes the solid body assumption</h2> <div id="outline-container-org2b1c74a" class="outline-2">
<h2 id="org2b1c74a"><span class="section-number-2">10</span> Note about the solid body assumption</h2>
<div class="outline-text-2" id="text-10"> <div class="outline-text-2" id="text-10">
<p> <p>
If we measure the motion of a rigid body along a direction \(\vec{x}\) using 2 sensors that are co-linear with the same direction \(\vec{x}\) (\(\vec{p}_2 = \vec{p}_1 + \alpha \vec{x}\)), they will measured the same quantity. If we measure the motion of a rigid body along a direction \(\vec{x}\) using 2 sensors that are co-linear with the same direction \(\vec{x}\) (\(\vec{p}_2 = \vec{p}_1 + \alpha \vec{x}\)), they will measured the same quantity.
</p> </p>
<p> <p>
This is illustrated on figure <a href="#org4627880">24</a>. This is illustrated on figure <a href="#org8f78f19">25</a>.
</p> </p>
<div id="org4627880" class="figure"> <div id="org8f78f19" class="figure">
<p><img src="figs/aligned_accelerometers.png" alt="aligned_accelerometers.png" /> <p><img src="figs/aligned_accelerometers.png" alt="aligned_accelerometers.png" />
</p> </p>
<p><span class="figure-number">Figure 24: </span>Aligned measurement of the motion of a solid body</p> <p><span class="figure-number">Figure 25: </span>Aligned measurement of the motion of a solid body</p>
</div> </div>
<p> <p>
The motion of the rigid body of figure <a href="#org4627880">24</a> is defined by the velocity \(\vec{v}\) and rotation \(\vec{\Omega}\) with respect to the reference frame \(\{O\}\). The motion of the rigid body of figure <a href="#org8f78f19">25</a> is defined by the velocity \(\vec{v}\) and rotation \(\vec{\Omega}\) with respect to the reference frame \(\{O\}\).
</p> </p>
<p> <p>
@ -1266,7 +1281,7 @@ However, we have \(p_{1y} = p_{2y}\) and \(p_{1z} = p_{2z}\) because of the co-l
<div class="important"> <div class="important">
<p> <p>
Two sensors that are measuring the co-linear motion of a rigid body should measure the same quantity. 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.
</p> </p>
</div> </div>
@ -1276,31 +1291,31 @@ We can verify that the rigid body assumption is correct by comparing the measure
</p> </p>
<p> <p>
From the table <a href="#org65ae8c9">2</a>, we can guess which sensors will give the same results in the X and Y directions. From the table <a href="#org61987da">2</a>, we can guess which sensors will give the same results in the X and Y directions.
</p> </p>
<p> <p>
Comparison of such measurements in the X direction is shown on figure <a href="#orgac307f7">25</a> and in the Y direction on figure <a href="#org3f01b7c">26</a>. Comparison of such measurements in the X direction is shown on figure <a href="#org3f45f65">26</a> and in the Y direction on figure <a href="#org01521f0">27</a>.
</p> </p>
<div id="orgac307f7" class="figure"> <div id="org3f45f65" class="figure">
<p><img src="figs/compare_acc_x_dir.png" alt="compare_acc_x_dir.png" /> <p><img src="figs/compare_acc_x_dir.png" alt="compare_acc_x_dir.png" />
</p> </p>
<p><span class="figure-number">Figure 25: </span>Compare accelerometers align in the X direction</p> <p><span class="figure-number">Figure 26: </span>Compare accelerometers align in the X direction</p>
</div> </div>
<div id="org3f01b7c" class="figure"> <div id="org01521f0" class="figure">
<p><img src="figs/compare_acc_y_dir.png" alt="compare_acc_y_dir.png" /> <p><img src="figs/compare_acc_y_dir.png" alt="compare_acc_y_dir.png" />
</p> </p>
<p><span class="figure-number">Figure 26: </span>Compare accelerometers align in the Y direction</p> <p><span class="figure-number">Figure 27: </span>Compare accelerometers align in the Y direction</p>
</div> </div>
<div class="important"> <div class="important">
<p> <p>
From the two figures above, we are more confident about the rigid body assumption in the frequency band of interest.
</p> </p>
</div> </div>
@ -1309,7 +1324,7 @@ Comparison of such measurements in the X direction is shown on figure <a href="#
</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: 2019-07-04 jeu. 17:49</p> <p class="date">Created: 2019-07-05 ven. 11:46</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>

4
modal-analysis/measurement.org
View File

@ -812,7 +812,7 @@ Comparison of such measurements in the X direction is shown on figure [[fig:comp
figure; figure;
for i = 1:size(acc_i, 1) for i = 1:size(acc_i, 1)
subaxis(3, 3, i); subplot(3, 3, i);
hold on; 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, 1)-1), exc_dir, :))))
plot(freqs, abs(squeeze(FRFs(meas_dir+3*(acc_i(i, 2)-1), exc_dir, :)))) plot(freqs, abs(squeeze(FRFs(meas_dir+3*(acc_i(i, 2)-1), exc_dir, :))))
@ -858,7 +858,7 @@ Comparison of such measurements in the X direction is shown on figure [[fig:comp
figure; figure;
for i = 1:size(acc_i, 1) for i = 1:size(acc_i, 1)
subaxis(3, 3, i); subplot(3, 3, i);
hold on; 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, 1)-1), exc_dir, :))))
plot(freqs, abs(squeeze(FRFs(meas_dir+3*(acc_i(i, 2)-1), exc_dir, :)))) plot(freqs, abs(squeeze(FRFs(meas_dir+3*(acc_i(i, 2)-1), exc_dir, :))))