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Update modal analysis, add .zip files (data and matlab files)

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Thomas Dehaeze
2019-07-05 11:20:02 +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. 10:16 -->
<!-- 2019-07-05 ven. 11:06 -->
<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>
@@ -280,15 +280,14 @@ for the JavaScript code in this tag.
<h2>Table of Contents</h2>
<div id="text-table-of-contents">
<ul>
<li><a href="#orgd95e287">1. Importation of measured FRF curves</a></li>
<li><a href="#org0649187">2. From accelerometer DOFs to solid body DOFs - Mathematics</a></li>
<li><a href="#orgbfc68dd">3. What reference frame to choose?</a></li>
<li><a href="#org5d9a38b">4. From accelerometer DOFs to solid body DOFs - Matlab Implementation</a></li>
<li><a href="#orgbb81ded">5. Analysis of some FRF in the global coordinates</a></li>
<li><a href="#orgb0c1df8">6. <span class="todo TODO">TODO</span> How to compare the relative motion of solid bodies</a></li>
<li><a href="#org9c93ccf">7. Relative Motion in the global coordinates</a></li>
<li><a href="#org7803dae">8. <span class="todo TODO">TODO</span> Compare original FRF measurements to transformed FRF in the global frame</a></li>
<li><a href="#org45fdd6c">9. Verify that we find the original FRF from the FRF in the global coordinates</a></li>
<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>
</ul>
</div>
</div>
@@ -322,8 +321,15 @@ Thus, we are only interested in \(6 \times 6 = 36\) degrees of freedom.
We here process the FRF matrix to go from the 69 measured DOFs to the wanted 36 DOFs.
</p>
<div id="outline-container-orgd95e287" class="outline-2">
<h2 id="orgd95e287"><span class="section-number-2">1</span> Importation of measured FRF curves</h2>
<div class="note">
<p>
All the files (data and Matlab scripts) are accessible <a href="data/frf_processing.zip">here</a>.
</p>
</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 class="outline-text-2" id="text-1">
<p>
We load the measured FRF and Coherence matrices.
@@ -331,18 +337,18 @@ We also load the geometric parameters of the station: solid bodies considered an
</p>
<div class="org-src-container">
<pre class="src src-matlab">load<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'./mat/frf_coh_matrices.mat', 'FRFs', 'COHs', 'freqs'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
<pre class="src src-matlab">load<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'mat/frf_coh_matrices.mat', 'FRFs', 'COHs', 'freqs'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
load<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'mat/geometry.mat', 'solids', 'solid_names', 'acc_pos'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
</pre>
</div>
</div>
</div>
<div id="outline-container-org0649187" class="outline-2">
<h2 id="org0649187"><span class="section-number-2">2</span> From accelerometer DOFs to solid body DOFs - Mathematics</h2>
<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 class="outline-text-2" id="text-2">
<p>
Let's consider the schematic shown on figure <a href="#org0345801">1</a> where we are measuring the motion of a (supposed) solid body at 4 distinct points in x-y-z.
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.
</p>
<p>
@@ -350,14 +356,14 @@ The goal here is to link these \(4 \times 3 = 12\) measurements to the 6 DOFs of
</p>
<div id="org0345801" class="figure">
<div id="org9f883a5" 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="#org0345801">1</a>, we can write:
From the figure <a href="#org9f883a5">1</a>, we can write:
</p>
\begin{align*}
\vec{v}_1 &= \vec{v} + \Omega \vec{p}_1\\
@@ -426,8 +432,8 @@ This inversion is equivalent to resolving a mean square problem.
</div>
</div>
<div id="outline-container-orgbfc68dd" class="outline-2">
<h2 id="orgbfc68dd"><span class="section-number-2">3</span> What reference frame to choose?</h2>
<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 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.
@@ -447,7 +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="org6509a46" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
<table id="orgd8f2173" 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>
@@ -491,8 +497,8 @@ As the easiest choice is to choose a common frame, we start with that solution.
</div>
</div>
<div id="outline-container-org5d9a38b" class="outline-2">
<h2 id="org5d9a38b"><span class="section-number-2">4</span> From accelerometer DOFs to solid body DOFs - Matlab Implementation</h2>
<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 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:
@@ -503,6 +509,29 @@ First, we initialize a new FRF matrix <code>FRFs_O</code> which is an \(n \times
<li>\(q\) is the number of frequency points \(\omega_i\)</li>
</ul>
<div class="important">
<p>
For each frequency point \(\omega_i\), the FRF matrix <code>FRFs_O</code> is a \(n\times p\) matrix:
</p>
\begin{equation}
\text{FRF}_O(\omega_i) = \begin{bmatrix}
\frac{D_{1,T_x}}{F_x}(\omega_i) & \frac{D_{1,T_x}}{F_y}(\omega_i) & \frac{D_{1,T_x}}{F_z}(\omega_i) \\
\frac{D_{1,T_y}}{F_x}(\omega_i) & \frac{D_{1,T_y}}{F_y}(\omega_i) & \frac{D_{1,T_y}}{F_z}(\omega_i) \\
\frac{D_{1,T_z}}{F_x}(\omega_i) & \frac{D_{1,T_z}}{F_y}(\omega_i) & \frac{D_{1,T_z}}{F_z}(\omega_i) \\
\frac{D_{1,R_x}}{F_x}(\omega_i) & \frac{D_{1,R_x}}{F_y}(\omega_i) & \frac{D_{1,R_x}}{F_z}(\omega_i) \\
\frac{D_{1,R_y}}{F_x}(\omega_i) & \frac{D_{1,R_y}}{F_y}(\omega_i) & \frac{D_{1,R_y}}{F_z}(\omega_i) \\
\frac{D_{1,R_z}}{F_x}(\omega_i) & \frac{D_{1,R_z}}{F_y}(\omega_i) & \frac{D_{1,R_z}}{F_z}(\omega_i) \\
\frac{D_{2,T_x}}{F_x}(\omega_i) & \frac{D_{2,T_x}}{F_y}(\omega_i) & \frac{D_{2,T_x}}{F_z}(\omega_i) \\
\vdots & \vdots & \vdots \\
\frac{D_{6,R_z}}{F_x}(\omega_i) & \frac{D_{6,R_z}}{F_y}(\omega_i) & \frac{D_{6,R_z}}{F_z}(\omega_i)
\end{bmatrix}
\end{equation}
<p>
where 1, 2, &#x2026;, 6 corresponds to the 6 solid bodies.
</p>
</div>
<div class="org-src-container">
<pre class="src src-matlab">FRFs_O = zeros<span class="org-rainbow-delimiters-depth-1">(</span>length<span class="org-rainbow-delimiters-depth-2">(</span>solid_names<span class="org-rainbow-delimiters-depth-2">)</span><span class="org-type">*</span><span class="org-highlight-numbers-number">6</span>, <span class="org-highlight-numbers-number">3</span>, <span class="org-highlight-numbers-number">801</span><span class="org-rainbow-delimiters-depth-1">)</span>;
</pre>
@@ -534,26 +563,26 @@ Then, as we know the positions of the accelerometers on each solid body, and we
</div>
</div>
<div id="outline-container-orgbb81ded" class="outline-2">
<h2 id="orgbb81ded"><span class="section-number-2">5</span> Analysis of some FRF in the global coordinates</h2>
<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 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="#orgc2878f8">2</a>)
First, we can compare the motions of the 6 solid bodies in one direction (figure <a href="#org1dcf9e5">2</a>)
</p>
<p>
We can also compare all the DOFs of one solid body (figure <a href="#orgfd0203f">3</a>).
We can also compare all the DOFs of one solid body (figure <a href="#org4750235">3</a>).
</p>
<div id="orgc2878f8" class="figure">
<div id="org1dcf9e5" 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="orgfd0203f" class="figure">
<div id="org4750235" 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>
@@ -561,136 +590,48 @@ We can also compare all the DOFs of one solid body (figure <a href="#orgfd0203f"
</div>
</div>
<div id="outline-container-orgb0c1df8" class="outline-2">
<h2 id="orgb0c1df8"><span class="section-number-2">6</span> <span class="todo TODO">TODO</span> How to compare the relative motion of solid bodies</h2>
<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 class="outline-text-2" id="text-6">
<p>
We have some of elements of the full FRF matrix:
\[ \frac{D_{1x}}{F_x},\ \frac{D_{1y}}{F_x},\ \frac{D_{1z}}{F_x},\ \frac{D_{2x}}{F_x},\ \dots \]
Now that the motion of all the solid bodies are expressed in the same frame, we should be able to <b>compare them</b>.
This can be used to determine what joints direction between two solid bodies is stiff enough that we can fix this DoF.
This could help reduce the order of the model and simplify the extraction of the model parameters from the measurements.
</p>
<p>
\[ \frac{D_{1x}}{D_{2x}} = \frac{\frac{D_{1x}}{F_x}}{\frac{D_{2x}}{F_x}} \]
Then, if \(\left| \frac{D_{1x}}{D_{2x}} \right| \approx 1\) in all the frequency band of interest, we can block the \(x\) motion between the solids 1 and 2.
We decide to plot the "normalized relative motion" between solid bodies \(i\) and \(j\):
\[ 0 < \Delta_{ij, x} = \frac{\left| D_{i,x} - D_{j,x} \right|}{|D_{i,x}| + |D_{j,x}|} < 1 \]
</p>
<p>
\[ \frac{D_{2x} - D_{1x}}{D_{1x} + D_{2x}} = \frac{\frac{D_{2x}}{F_x} - \frac{D_{1x}}{F_x}}{\frac{D_{1x}}{F_x} + \frac{D_{2x}}{F_x}} \]
Then, if \(\Delta_{ij,x} \ll 0\) in the frequency band of interest, we have that \(D_{ix} \approx D_{jx}\) and we can neglect that DOF between the two solid bodies \(i\) and \(j\).
</p>
<p>
Then if \(\left| \frac{D_{2x} - D_{1x}}{D_{1x} + D_{2x}} \right| \ll 1\) in all the frequency band of interest, we can block the \(x\) motion between the solids 1 and 2.
</p>
</div>
</div>
<div id="outline-container-org9c93ccf" class="outline-2">
<h2 id="org9c93ccf"><span class="section-number-2">7</span> Relative Motion in the global coordinates</h2>
<div class="outline-text-2" id="text-7">
<p>
Below we plot the normalized relative motion between each stage:
\[ 0 < \frac{\left| D_{ix} - D_{jx} \right|}{|D_{ix}| + |D_{jx}|} < 1 \]
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.
</p>
<div class="org-src-container">
<pre class="src src-matlab">DOFs = <span class="org-rainbow-delimiters-depth-1">{</span><span class="org-string">'$T_x$', '$T_y$', '$T_z$', '</span>$<span class="org-type">\</span>theta_x$', '$<span class="org-type">\</span>theta_y$', '$<span class="org-type">\</span>theta_z$'<span class="org-rainbow-delimiters-depth-1">}</span>
dirs_i = <span class="org-highlight-numbers-number">1</span><span class="org-type">:</span><span class="org-highlight-numbers-number">6</span>;
exc_dir = <span class="org-highlight-numbers-number">1</span>;
<span class="org-type">figure</span>;
<span class="org-keyword">for</span> <span class="org-variable-name">i</span> = <span class="org-constant"><span class="org-highlight-numbers-number">2</span></span><span class="org-constant">:</span><span class="org-constant"><span class="org-highlight-numbers-number">6</span></span>
subaxis<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">3</span>, <span class="org-highlight-numbers-number">2</span>, <span class="org-constant">i</span><span class="org-rainbow-delimiters-depth-1">)</span>;
hold on;
<span class="org-keyword">for</span> <span class="org-variable-name">dir_i</span> = <span class="org-constant">dirs_i</span>
H = <span class="org-rainbow-delimiters-depth-1">(</span>squeeze<span class="org-rainbow-delimiters-depth-2">(</span>FRFs_O<span class="org-rainbow-delimiters-depth-3">(</span><span class="org-rainbow-delimiters-depth-4">(</span><span class="org-constant">i</span><span class="org-type">-</span><span class="org-highlight-numbers-number">1</span><span class="org-rainbow-delimiters-depth-4">)</span><span class="org-type">*</span><span class="org-highlight-numbers-number">6</span><span class="org-type">+</span>dir_i, exc_dir, <span class="org-type">:</span><span class="org-rainbow-delimiters-depth-3">)</span><span class="org-rainbow-delimiters-depth-2">)</span><span class="org-type">-</span>squeeze<span class="org-rainbow-delimiters-depth-2">(</span>FRFs_O<span class="org-rainbow-delimiters-depth-3">(</span><span class="org-rainbow-delimiters-depth-4">(</span><span class="org-constant">i</span><span class="org-type">-</span><span class="org-highlight-numbers-number">2</span><span class="org-rainbow-delimiters-depth-4">)</span><span class="org-type">*</span><span class="org-highlight-numbers-number">6</span><span class="org-type">+</span>dir_i, exc_dir, <span class="org-type">:</span><span class="org-rainbow-delimiters-depth-3">)</span><span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span><span class="org-type">./</span><span class="org-rainbow-delimiters-depth-1">(</span>abs<span class="org-rainbow-delimiters-depth-2">(</span>squeeze<span class="org-rainbow-delimiters-depth-3">(</span>FRFs_O<span class="org-rainbow-delimiters-depth-4">(</span><span class="org-rainbow-delimiters-depth-5">(</span><span class="org-constant">i</span><span class="org-type">-</span><span class="org-highlight-numbers-number">1</span><span class="org-rainbow-delimiters-depth-5">)</span><span class="org-type">*</span><span class="org-highlight-numbers-number">6</span><span class="org-type">+</span>dir_i, exc_dir, <span class="org-type">:</span><span class="org-rainbow-delimiters-depth-4">)</span><span class="org-rainbow-delimiters-depth-3">)</span><span class="org-rainbow-delimiters-depth-2">)</span><span class="org-type">+</span>abs<span class="org-rainbow-delimiters-depth-2">(</span>squeeze<span class="org-rainbow-delimiters-depth-3">(</span>FRFs_O<span class="org-rainbow-delimiters-depth-4">(</span><span class="org-rainbow-delimiters-depth-5">(</span><span class="org-constant">i</span><span class="org-type">-</span><span class="org-highlight-numbers-number">2</span><span class="org-rainbow-delimiters-depth-5">)</span><span class="org-type">*</span><span class="org-highlight-numbers-number">6</span><span class="org-type">+</span>dir_i, exc_dir, <span class="org-type">:</span><span class="org-rainbow-delimiters-depth-4">)</span><span class="org-rainbow-delimiters-depth-3">)</span><span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span>;
plot<span class="org-rainbow-delimiters-depth-1">(</span>freqs, abs<span class="org-rainbow-delimiters-depth-2">(</span>H<span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span>;
<span class="org-keyword">end</span>
hold off;
<span class="org-type">set</span><span class="org-rainbow-delimiters-depth-1">(</span><span class="org-variable-name">gca</span>, <span class="org-string">'XScale', 'log'</span><span class="org-string"><span class="org-rainbow-delimiters-depth-1">)</span></span><span class="org-string">; set</span><span class="org-string"><span class="org-rainbow-delimiters-depth-1">(</span></span><span class="org-string">gca, 'YScale', 'lin'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
xlim<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbow-delimiters-depth-2">[</span><span class="org-highlight-numbers-number">1</span>, <span class="org-highlight-numbers-number">200</span><span class="org-rainbow-delimiters-depth-2">]</span><span class="org-rainbow-delimiters-depth-1">)</span>; ylim<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbow-delimiters-depth-2">[</span><span class="org-highlight-numbers-number">0</span>, <span class="org-highlight-numbers-number">1</span><span class="org-rainbow-delimiters-depth-2">]</span><span class="org-rainbow-delimiters-depth-1">)</span>;
<span class="org-comment">% xlabel('Frequency [Hz]'); ylabel('Relative Motion');</span>
title<span class="org-rainbow-delimiters-depth-1">(</span>sprintf<span class="org-rainbow-delimiters-depth-2">(</span><span class="org-string">'Normalized motion %s - %s'</span>, solid_names{<span class="org-constant">i</span><span class="org-type">-</span><span class="org-highlight-numbers-number">1</span>}, solid_names{<span class="org-constant">i</span>}));
<span class="org-keyword">if</span> <span class="org-constant">i</span> <span class="org-type">&gt;</span> <span class="org-highlight-numbers-number">4</span>
xlabel<span class="org-rainbow-delimiters-depth-3">(</span><span class="org-string">'Frequency </span><span class="org-string"><span class="org-rainbow-delimiters-depth-4">[</span></span><span class="org-string">Hz</span><span class="org-string"><span class="org-rainbow-delimiters-depth-4">]</span></span><span class="org-string">'</span><span class="org-rainbow-delimiters-depth-3">)</span>;
<span class="org-keyword">else</span>
<span class="org-type">set</span><span class="org-rainbow-delimiters-depth-3">(</span><span class="org-variable-name">gca</span>, <span class="org-string">'XTickLabel'</span>,<span class="org-rainbow-delimiters-depth-4">[]</span><span class="org-rainbow-delimiters-depth-3">)</span>;
<span class="org-keyword">end</span>
<span class="org-keyword">end</span>
<span class="org-keyword">for</span> <span class="org-variable-name">i</span> = <span class="org-constant"><span class="org-highlight-numbers-number">1</span></span><span class="org-constant">:length</span><span class="org-constant"><span class="org-rainbow-delimiters-depth-3">(</span></span><span class="org-constant">dirs_i</span><span class="org-constant"><span class="org-rainbow-delimiters-depth-3">)</span></span>
legend_names<span class="org-rainbow-delimiters-depth-3">{</span><span class="org-constant">i</span><span class="org-rainbow-delimiters-depth-3">}</span> = DOFs<span class="org-rainbow-delimiters-depth-3">{</span>dirs_i<span class="org-rainbow-delimiters-depth-4">(</span><span class="org-constant">i</span><span class="org-rainbow-delimiters-depth-4">)</span><span class="org-rainbow-delimiters-depth-3">}</span>;
<span class="org-keyword">end</span>
lgd = legend<span class="org-rainbow-delimiters-depth-3">(</span>legend_names<span class="org-rainbow-delimiters-depth-3">)</span>;
hL = subplot<span class="org-rainbow-delimiters-depth-3">(</span><span class="org-highlight-numbers-number">3</span>, <span class="org-highlight-numbers-number">2</span>, <span class="org-highlight-numbers-number">1</span><span class="org-rainbow-delimiters-depth-3">)</span>;
poshL = <span class="org-type">get</span><span class="org-rainbow-delimiters-depth-3">(</span><span class="org-variable-name">hL</span>,<span class="org-string">'position'</span><span class="org-rainbow-delimiters-depth-3">)</span>;
<span class="org-type">set</span><span class="org-rainbow-delimiters-depth-3">(</span><span class="org-variable-name">lgd</span>,<span class="org-string">'position'</span>, poshL<span class="org-rainbow-delimiters-depth-3">)</span>;
<span class="org-type">axis</span><span class="org-rainbow-delimiters-depth-3">(</span>hL, <span class="org-string">'off'</span><span class="org-rainbow-delimiters-depth-3">)</span>;
</pre>
</div>
<div id="org1cd90a2" class="figure">
<div id="org74b537d" class="figure">
<p><img src="figs/relative_motion_comparison.png" alt="relative_motion_comparison.png" />
</p>
<p><span class="figure-number">Figure 4: </span>Relative motion between each stage</p>
</div>
</div>
</div>
<div id="outline-container-org7803dae" class="outline-2">
<h2 id="org7803dae"><span class="section-number-2">8</span> <span class="todo TODO">TODO</span> Compare original FRF measurements to transformed FRF in the global frame</h2>
<div class="outline-text-2" id="text-8">
<div class="warning">
<p>
We wish here to compare the FRF in order to verify if there is any mistake.
Can we really compare the motion of two solid bodies from Frequency Response Functions that clearly depends on the excitation point and direction?
The relative motion of two solid bodies may be negligible when exciting the structure at on point and but at another point.
</p>
<div class="org-src-container">
<pre class="src src-matlab">dir_names = <span class="org-rainbow-delimiters-depth-1">{</span><span class="org-string">'X', 'Y', 'Z', '</span>$<span class="org-type">\</span>theta_X$', '$<span class="org-type">\</span>theta_Y$', '$<span class="org-type">\</span>theta_Z$'<span class="org-rainbow-delimiters-depth-1">}</span>;
solid_i = <span class="org-highlight-numbers-number">6</span>;
acc_dir_O = <span class="org-highlight-numbers-number">1</span>;
acc_dir = <span class="org-highlight-numbers-number">1</span>;
exc_dir = <span class="org-highlight-numbers-number">1</span>;
<span class="org-type">figure</span>;
ax1 = subplot<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">2</span>, <span class="org-highlight-numbers-number">1</span>, <span class="org-highlight-numbers-number">1</span><span class="org-rainbow-delimiters-depth-1">)</span>;
hold on;
<span class="org-keyword">for</span> <span class="org-variable-name">i</span> = <span class="org-constant">solids.</span><span class="org-constant"><span class="org-rainbow-delimiters-depth-1">(</span></span><span class="org-constant">solid_names</span><span class="org-constant"><span class="org-rainbow-delimiters-depth-2">{</span></span><span class="org-constant">solid_i</span><span class="org-constant"><span class="org-rainbow-delimiters-depth-2">}</span></span><span class="org-constant"><span class="org-rainbow-delimiters-depth-1">)</span></span>
plot<span class="org-rainbow-delimiters-depth-1">(</span>freqs, abs<span class="org-rainbow-delimiters-depth-2">(</span>squeeze<span class="org-rainbow-delimiters-depth-3">(</span>FRFs<span class="org-rainbow-delimiters-depth-4">(</span>acc_dir<span class="org-type">+</span><span class="org-highlight-numbers-number">3</span><span class="org-type">*</span><span class="org-rainbow-delimiters-depth-5">(</span><span class="org-constant">i</span><span class="org-type">-</span><span class="org-highlight-numbers-number">1</span><span class="org-rainbow-delimiters-depth-5">)</span>, exc_dir, <span class="org-type">:</span><span class="org-rainbow-delimiters-depth-4">)</span><span class="org-rainbow-delimiters-depth-3">)</span><span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span>;
<span class="org-keyword">end</span>
plot<span class="org-rainbow-delimiters-depth-1">(</span>freqs, abs<span class="org-rainbow-delimiters-depth-2">(</span>squeeze<span class="org-rainbow-delimiters-depth-3">(</span>FRFs_O<span class="org-rainbow-delimiters-depth-4">(</span><span class="org-rainbow-delimiters-depth-5">(</span>solid_i<span class="org-type">-</span><span class="org-highlight-numbers-number">1</span><span class="org-rainbow-delimiters-depth-5">)</span><span class="org-type">*</span><span class="org-highlight-numbers-number">6</span><span class="org-type">+</span>acc_dir_O, exc_dir, <span class="org-type">:</span><span class="org-rainbow-delimiters-depth-4">)</span><span class="org-rainbow-delimiters-depth-3">)</span><span class="org-rainbow-delimiters-depth-2">)</span>, <span class="org-string">'-k'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
hold off;
<span class="org-type">set</span><span class="org-rainbow-delimiters-depth-1">(</span><span class="org-variable-name">gca</span>, <span class="org-string">'XScale', 'log'</span><span class="org-string"><span class="org-rainbow-delimiters-depth-1">)</span></span><span class="org-string">; set</span><span class="org-string"><span class="org-rainbow-delimiters-depth-1">(</span></span><span class="org-string">gca, 'YScale', 'log'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
<span class="org-type">set</span><span class="org-rainbow-delimiters-depth-1">(</span><span class="org-variable-name">gca</span>, <span class="org-string">'XTickLabel'</span>,<span class="org-rainbow-delimiters-depth-2">[]</span><span class="org-rainbow-delimiters-depth-1">)</span>;
ylabel<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'Amplitude'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
title<span class="org-rainbow-delimiters-depth-1">(</span>sprintf<span class="org-rainbow-delimiters-depth-2">(</span><span class="org-string">'%s motion measured by the Acc. vs %s motion computed in the common frame - %s'</span>, dir_names{acc_dir}, dir_names{acc_dir_O}, solid_names{solid_i}));
ax2 = subplot<span class="org-rainbow-delimiters-depth-3">(</span><span class="org-highlight-numbers-number">2</span>, <span class="org-highlight-numbers-number">1</span>, <span class="org-highlight-numbers-number">2</span><span class="org-rainbow-delimiters-depth-3">)</span>;
hold on;
<span class="org-keyword">for</span> <span class="org-variable-name">i</span> = <span class="org-constant">solids.</span><span class="org-constant"><span class="org-rainbow-delimiters-depth-3">(</span></span><span class="org-constant">solid_names</span><span class="org-constant"><span class="org-rainbow-delimiters-depth-4">{</span></span><span class="org-constant">solid_i</span><span class="org-constant"><span class="org-rainbow-delimiters-depth-4">}</span></span><span class="org-constant"><span class="org-rainbow-delimiters-depth-3">)</span></span>
plot<span class="org-rainbow-delimiters-depth-3">(</span>freqs, mod<span class="org-rainbow-delimiters-depth-4">(</span><span class="org-highlight-numbers-number">180</span><span class="org-type">+</span><span class="org-highlight-numbers-number">180</span><span class="org-type">/</span><span class="org-constant">pi</span><span class="org-type">*</span>phase<span class="org-rainbow-delimiters-depth-5">(</span>squeeze<span class="org-rainbow-delimiters-depth-6">(</span>FRFs<span class="org-rainbow-delimiters-depth-7">(</span>acc_dir<span class="org-type">+</span><span class="org-highlight-numbers-number">3</span><span class="org-type">*</span><span class="org-rainbow-delimiters-depth-8">(</span><span class="org-constant">i</span><span class="org-type">-</span><span class="org-highlight-numbers-number">1</span><span class="org-rainbow-delimiters-depth-8">)</span>, exc_dir, <span class="org-type">:</span><span class="org-rainbow-delimiters-depth-7">)</span><span class="org-rainbow-delimiters-depth-6">)</span><span class="org-rainbow-delimiters-depth-5">)</span>, <span class="org-highlight-numbers-number">360</span><span class="org-rainbow-delimiters-depth-4">)</span><span class="org-type">-</span><span class="org-highlight-numbers-number">180</span><span class="org-rainbow-delimiters-depth-3">)</span>;
<span class="org-keyword">end</span>
plot<span class="org-rainbow-delimiters-depth-3">(</span>freqs, mod<span class="org-rainbow-delimiters-depth-4">(</span><span class="org-highlight-numbers-number">180</span><span class="org-type">+</span><span class="org-highlight-numbers-number">180</span><span class="org-type">/</span><span class="org-constant">pi</span><span class="org-type">*</span>phase<span class="org-rainbow-delimiters-depth-5">(</span>squeeze<span class="org-rainbow-delimiters-depth-6">(</span>FRFs_O<span class="org-rainbow-delimiters-depth-7">(</span><span class="org-rainbow-delimiters-depth-8">(</span>solid_i<span class="org-type">-</span><span class="org-highlight-numbers-number">1</span><span class="org-rainbow-delimiters-depth-8">)</span><span class="org-type">*</span><span class="org-highlight-numbers-number">6</span><span class="org-type">+</span>acc_dir_O, exc_dir, <span class="org-type">:</span><span class="org-rainbow-delimiters-depth-7">)</span><span class="org-rainbow-delimiters-depth-6">)</span><span class="org-rainbow-delimiters-depth-5">)</span>, <span class="org-highlight-numbers-number">360</span><span class="org-rainbow-delimiters-depth-4">)</span><span class="org-type">-</span><span class="org-highlight-numbers-number">180</span>, <span class="org-string">'-k'</span><span class="org-rainbow-delimiters-depth-3">)</span>;
hold off;
ylim<span class="org-rainbow-delimiters-depth-3">(</span><span class="org-rainbow-delimiters-depth-4">[</span><span class="org-type">-</span><span class="org-highlight-numbers-number">180</span>, <span class="org-highlight-numbers-number">180</span><span class="org-rainbow-delimiters-depth-4">]</span><span class="org-rainbow-delimiters-depth-3">)</span>; yticks<span class="org-rainbow-delimiters-depth-3">(</span><span class="org-type">-</span><span class="org-highlight-numbers-number">180</span><span class="org-type">:</span><span class="org-highlight-numbers-number">90</span><span class="org-type">:</span><span class="org-highlight-numbers-number">180</span><span class="org-rainbow-delimiters-depth-3">)</span>;
xlabel<span class="org-rainbow-delimiters-depth-3">(</span><span class="org-string">'Frequency </span><span class="org-string"><span class="org-rainbow-delimiters-depth-4">[</span></span><span class="org-string">Hz</span><span class="org-string"><span class="org-rainbow-delimiters-depth-4">]</span></span><span class="org-string">'</span><span class="org-string"><span class="org-rainbow-delimiters-depth-3">)</span></span><span class="org-string">; ylabel</span><span class="org-string"><span class="org-rainbow-delimiters-depth-3">(</span></span><span class="org-string">'Phase </span><span class="org-string"><span class="org-rainbow-delimiters-depth-4">[</span></span><span class="org-string">deg</span><span class="org-string"><span class="org-rainbow-delimiters-depth-4">]</span></span><span class="org-string">'</span><span class="org-rainbow-delimiters-depth-3">)</span>;
<span class="org-type">set</span><span class="org-rainbow-delimiters-depth-3">(</span><span class="org-variable-name">gca</span>, <span class="org-string">'xscale', 'log'</span><span class="org-rainbow-delimiters-depth-3">)</span>;
linkaxes<span class="org-rainbow-delimiters-depth-3">(</span><span class="org-rainbow-delimiters-depth-4">[</span>ax1,ax2<span class="org-rainbow-delimiters-depth-4">]</span>,<span class="org-string">'x'</span><span class="org-rainbow-delimiters-depth-3">)</span>;
xlim<span class="org-rainbow-delimiters-depth-3">(</span><span class="org-rainbow-delimiters-depth-4">[</span><span class="org-highlight-numbers-number">1</span>, <span class="org-highlight-numbers-number">200</span><span class="org-rainbow-delimiters-depth-4">]</span><span class="org-rainbow-delimiters-depth-3">)</span>;
</pre>
</div>
</div>
</div>
<div id="outline-container-org45fdd6c" class="outline-2">
<h2 id="org45fdd6c"><span class="section-number-2">9</span> Verify that we find the original FRF from the FRF in the global coordinates</h2>
<div class="outline-text-2" id="text-9">
<div id="outline-container-org1f7f9fb" class="outline-2">
<h2 id="org1f7f9fb"><span class="section-number-2">7</span> Verify that we find the original FRF from the FRF in the global coordinates</h2>
<div class="outline-text-2" id="text-7">
<p>
We have computed the Frequency Response Functions Matrix <code>FRFs_O</code> representing the response of the 6 solid bodies in their 6 DOFs.
</p>
@@ -737,24 +678,24 @@ We then compare the original FRF measured for each accelerometer with the recove
</p>
<p>
The FRF for the 4 accelerometers on the Hexapod are compared on figure <a href="#orgdf7e1b1">5</a>.
The FRF for the 4 accelerometers on the Hexapod are compared on figure <a href="#orgacee11c">5</a>.
All the FRF are matching very well in all the frequency range displayed.
</p>
<p>
The FRF for accelerometers located on the translation stage are compared on figure <a href="#orgc07a0c4">6</a>.
The FRF for accelerometers located on the translation stage are compared on figure <a href="#orgeff91f2">6</a>.
The FRF are matching well until 100Hz.
</p>
<div id="orgdf7e1b1" class="figure">
<div id="orgacee11c" class="figure">
<p><img src="figs/recovered_frf_comparison_hexa.png" alt="recovered_frf_comparison_hexa.png" />
</p>
<p><span class="figure-number">Figure 5: </span>Comparison of the original FRF with the recovered ones - Hexapod</p>
</div>
<div id="orgc07a0c4" class="figure">
<div id="orgeff91f2" class="figure">
<p><img src="figs/recovered_frf_comparison_ty.png" alt="recovered_frf_comparison_ty.png" />
</p>
<p><span class="figure-number">Figure 6: </span>Comparison of the original FRF with the recovered ones - Ty</p>
@@ -771,13 +712,23 @@ This confirms the fact that the stages are indeed behaving as a solid body in th
This valid the fact that a multi-body model can be used to represent the dynamics of the micro-station.
</p>
</div>
</div>
</div>
<div id="outline-container-org2cd1928" class="outline-2">
<h2 id="org2cd1928"><span class="section-number-2">8</span> Saving of the FRF expressed in the global coordinates</h2>
<div class="outline-text-2" id="text-8">
<div class="org-src-container">
<pre class="src src-matlab">save<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'mat/frf_o.mat', 'FRFs_O'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
</pre>
</div>
</div>
</div>
</div>
<div id="postamble" class="status">
<p class="author">Author: Dehaeze Thomas</p>
<p class="date">Created: 2019-07-05 ven. 10:16</p>
<p class="date">Created: 2019-07-05 ven. 11:06</p>
<p class="validation"><a href="http://validator.w3.org/check?uri=referer">Validate</a></p>
</div>
</body>