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Check the principle of reciprocity

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"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. 11:46 --> <!-- 2019-07-11 jeu. 16:25 -->
<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 +280,35 @@ 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="#org1379b74">1. Goal</a></li> <li><a href="#org7579c58">1. Goal</a></li>
<li><a href="#orgd38a775">2. Instrumentation Used</a></li> <li><a href="#org443821e">2. Instrumentation Used</a></li>
<li><a href="#orge0b6708">3. Structure Preparation and Test Planning</a> <li><a href="#org109ddf2">3. Structure Preparation and Test Planning</a>
<ul> <ul>
<li><a href="#org37f06d5">3.1. Structure Preparation</a></li> <li><a href="#org050f028">3.1. Structure Preparation</a></li>
<li><a href="#orgb7f0982">3.2. Test Planing</a></li> <li><a href="#org824ea1e">3.2. Test Planing</a></li>
<li><a href="#org164c06b">3.3. Location of the Accelerometers</a></li> <li><a href="#orgae947bb">3.3. Location of the Accelerometers</a></li>
<li><a href="#orge55aaed">3.4. Hammer Impacts</a></li> <li><a href="#org85336e7">3.4. Hammer Impacts</a></li>
</ul> </ul>
</li> </li>
<li><a href="#org94592f9">4. Signal Processing</a> <li><a href="#orgc7582b9">4. Signal Processing</a>
<ul> <ul>
<li><a href="#orgc9341a3">4.1. Averaging</a></li> <li><a href="#orga2b8b8e">4.1. Averaging</a></li>
<li><a href="#org62ef660">4.2. Windowing</a></li> <li><a href="#org0dbf1dd">4.2. Windowing</a></li>
</ul> </ul>
</li> </li>
<li><a href="#orgfcf409d">5. Force and Response signals</a> <li><a href="#org45d897a">5. Force and Response signals</a>
<ul> <ul>
<li><a href="#org7263d51">5.1. Raw Force Data</a></li> <li><a href="#org2a9b2e8">5.1. Raw Force Data</a></li>
<li><a href="#orgc9a27c3">5.2. Raw Response Data</a></li> <li><a href="#org85cf244">5.2. Raw Response Data</a></li>
<li><a href="#orgf1f892a">5.3. Computation of one Frequency Response Function</a></li> <li><a href="#orgacd507e">5.3. Computation of one Frequency Response Function</a></li>
</ul> </ul>
</li> </li>
<li><a href="#org0595bd4">6. Frequency Response Functions and Coherence Results</a></li> <li><a href="#org40c5c29">6. Frequency Response Functions and Coherence Results</a></li>
<li><a href="#org860e46e">7. Generation of a FRF matrix and a Coherence matrix from the measurements</a></li> <li><a href="#orge7a0dc1">7. 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="#org5ebd034">8. Plot showing the coherence of all the measurements</a></li>
<li><a href="#org6601f7d">9. Solid Bodies considered for further analysis</a></li> <li><a href="#org259382d">9. Solid Bodies considered for further analysis</a></li>
<li><a href="#org2b1c74a">10. Note about the solid body assumption</a></li> <li><a href="#orgc0c679a">10. Note about the solid body assumption</a></li>
<li><a href="#org95c7199">11. Verification of the principle of reciprocity</a></li>
</ul> </ul>
</div> </div>
</div> </div>
@ -319,8 +320,8 @@ All the files (data and Matlab scripts) are accessible <a href="data/modal_frf_c
</div> </div>
<div id="outline-container-org1379b74" class="outline-2"> <div id="outline-container-org7579c58" class="outline-2">
<h2 id="org1379b74"><span class="section-number-2">1</span> Goal</h2> <h2 id="org7579c58"><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 +329,20 @@ The goal is to measure the dynamic of the Micro-Station and to extract Frequency
</div> </div>
</div> </div>
<div id="outline-container-orgd38a775" class="outline-2"> <div id="outline-container-org443821e" class="outline-2">
<h2 id="orgd38a775"><span class="section-number-2">2</span> Instrumentation Used</h2> <h2 id="org443821e"><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="#org084bcd5">1</a>)</li> <li>An <b>acquisition system</b> (OROS) with 24bits ADCs (figure <a href="#orgd3ed416">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>3 tri-axis <b>Accelerometers</b> (figure <a href="#org5652fe1">2</a>) with parameters shown on table <a href="#orgc08239a">1</a></li>
<li>An <b>Instrumented Hammer</b> with various Tips (figure <a href="#orgb4bbf0d">3</a>) (figure <a href="#org9052254">4</a>)</li> <li>An <b>Instrumented Hammer</b> with various Tips (figure <a href="#orgba60092">3</a>) (figure <a href="#org1248b1f">4</a>)</li>
</ul> </ul>
<div id="org084bcd5" class="figure"> <div id="orgd3ed416" 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 +355,13 @@ Anti-aliasing filters are also included in the system.
</p> </p>
<div id="org2d07359" class="figure"> <div id="org5652fe1" 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="orgb383c7a" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides"> <table id="orgc08239a" 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 +404,14 @@ It excites more the low frequency range where the coherence is low, the overall
</p> </p>
<div id="orgb4bbf0d" class="figure"> <div id="orgba60092" 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="org9052254" class="figure"> <div id="org1248b1f" 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 +423,12 @@ The accelerometers are glued on the structure.
</div> </div>
</div> </div>
<div id="outline-container-orge0b6708" class="outline-2"> <div id="outline-container-org109ddf2" class="outline-2">
<h2 id="orge0b6708"><span class="section-number-2">3</span> Structure Preparation and Test Planning</h2> <h2 id="org109ddf2"><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-org37f06d5" class="outline-3"> <div id="outline-container-org050f028" class="outline-3">
<h3 id="org37f06d5"><span class="section-number-3">3.1</span> Structure Preparation</h3> <h3 id="org050f028"><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.
@ -470,17 +471,18 @@ All other elements have been remove from the granite such as another heavy posit
</div> </div>
</div> </div>
<div id="outline-container-orgb7f0982" class="outline-3"> <div id="outline-container-org824ea1e" class="outline-3">
<h3 id="orgb7f0982"><span class="section-number-3">3.2</span> Test Planing</h3> <h3 id="org824ea1e"><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 \(H\) (where \(N\) is the number of degree of freedom of the system):
</p> </p>
\begin{equation}
H_{jk} = \frac{X_j}{F_k}
\end{equation}
<p> <p>
However, the principle of reciprocity states that: However, from only one column or one line of the matrix, we can compute the other terms thanks to the principle of reciprocity.
\[ H_{jk} = \frac{X_j}{F_k} = H_{kj} = \frac{X_k}{F_j} \]
Thus, only one column or one line of the matrix has to be identified.
</p> </p>
<p> <p>
@ -491,7 +493,7 @@ Either we choose to identify \(\frac{X_k}{F_i}\) or \(\frac{X_i}{F_k}\) for any
We here choose to identify \(\frac{X_i}{F_k}\) for practical reasons: We here choose to identify \(\frac{X_i}{F_k}\) for practical reasons:
</p> </p>
<ul class="org-ul"> <ul class="org-ul">
<li>it is easier to glue the accelerometers on some stages than to excite this particular stage with the Hammer</li> <li>it is easier to glue the accelerometers on all the stages and excite only a one particular point than doing the opposite</li>
</ul> </ul>
<p> <p>
@ -501,11 +503,15 @@ The measurement thus consists of:
<li>always excite the structure at the same location with the Hammer</li> <li>always excite the structure at the same location with the Hammer</li>
<li>Move the accelerometers to measure all the DOF of the structure</li> <li>Move the accelerometers to measure all the DOF of the structure</li>
</ul> </ul>
<p>
We will measured 3 columns (3 impacts location) in order to have some redundancy.
</p>
</div> </div>
</div> </div>
<div id="outline-container-org164c06b" class="outline-3"> <div id="outline-container-orgae947bb" class="outline-3">
<h3 id="org164c06b"><span class="section-number-3">3.3</span> Location of the Accelerometers</h3> <h3 id="orgae947bb"><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 +530,11 @@ 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="#org3b8109d">5</a>)</li> <li>4 on the second granite (figure <a href="#org05b745d">5</a>)</li>
<li>4 on top of the translation stage (figure <a href="#org4b120a8">6</a>)</li> <li>4 on top of the translation stage (figure <a href="#org676fa5a">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="#org69dec1e">7</a>)</li> <li>4 on top of the hexapod (figure <a href="#orgd0b01da">7</a>)</li>
</ul> </ul>
<p> <p>
@ -536,43 +542,43 @@ 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="#orgbff47a7">8</a>. The position and orientation of all the accelerometers used are shown on figure <a href="#orgbc41472">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="#org5d34cd5">9</a>). The precise determination of the position of each accelerometer is done using the SolidWorks model (shown on figure <a href="#orgec3e0e6">9</a>).
</p> </p>
<div id="org3b8109d" class="figure"> <div id="org05b745d" 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="org4b120a8" class="figure"> <div id="org676fa5a" 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="org69dec1e" class="figure"> <div id="orgd0b01da" 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="orgbff47a7" class="figure"> <div id="orgbc41472" 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="org5d34cd5" class="figure"> <div id="orgec3e0e6" 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 +605,9 @@ 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="#org61987da">2</a>). The positions of the sensors relative to the point of interest are shown below (table <a href="#org9932674">2</a>).
</p> </p>
<table id="org61987da" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides"> <table id="org9932674" 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 +793,15 @@ The positions of the sensors relative to the point of interest are shown below (
</div> </div>
</div> </div>
<div id="outline-container-orge55aaed" class="outline-3"> <div id="outline-container-org85336e7" class="outline-3">
<h3 id="orge55aaed"><span class="section-number-3">3.4</span> Hammer Impacts</h3> <h3 id="org85336e7"><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="#orge3c8f2c">10</a>, <a href="#orga9f97bd">11</a> and <a href="#orgbf0886d">12</a>. The impact points are shown on figures <a href="#orgd9f9e9e">10</a>, <a href="#org4002508">11</a> and <a href="#org2dbf07c">12</a>.
</p> </p>
<p> <p>
@ -803,21 +809,21 @@ We chose this excitation point as it seems to excite all the modes in the freque
</p> </p>
<div id="orge3c8f2c" class="figure"> <div id="orgd9f9e9e" 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="orga9f97bd" class="figure"> <div id="org4002508" 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="orgbf0886d" class="figure"> <div id="org2dbf07c" 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 +832,19 @@ We chose this excitation point as it seems to excite all the modes in the freque
</div> </div>
</div> </div>
<div id="outline-container-org94592f9" class="outline-2"> <div id="outline-container-orgc7582b9" class="outline-2">
<h2 id="org94592f9"><span class="section-number-2">4</span> Signal Processing</h2> <h2 id="orgc7582b9"><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-orgc9341a3" class="outline-3"> <div id="outline-container-orga2b8b8e" class="outline-3">
<h3 id="orgc9341a3"><span class="section-number-3">4.1</span> Averaging</h3> <h3 id="orga2b8b8e"><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="#org328ce53">3</a>. The parameters for the impact test are shown on table <a href="#org548b6a0">3</a>.
</p> </p>
<table id="org328ce53" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides"> <table id="org548b6a0" 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 +882,15 @@ The parameters for the impact test are shown on table <a href="#org328ce53">3</a
</div> </div>
</div> </div>
<div id="outline-container-org62ef660" class="outline-3"> <div id="outline-container-org0dbf1dd" class="outline-3">
<h3 id="org62ef660"><span class="section-number-3">4.2</span> Windowing</h3> <h3 id="org0dbf1dd"><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="#orgac8e1ab">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="#org49a4fe0">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 +899,14 @@ The parameters are:
</ul> </ul>
<div id="orgac8e1ab" class="figure"> <div id="org49a4fe0" 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="#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. An exponential window (figure <a href="#orgd0a23d4">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 +923,7 @@ The parameters are:
</ul> </ul>
<div id="orga3cffc9" class="figure"> <div id="orgd0a23d4" 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 +932,8 @@ The parameters are:
</div> </div>
</div> </div>
<div id="outline-container-orgfcf409d" class="outline-2"> <div id="outline-container-org45d897a" class="outline-2">
<h2 id="orgfcf409d"><span class="section-number-2">5</span> Force and Response signals</h2> <h2 id="org45d897a"><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 +945,33 @@ Let's load some obtained data to look at the force and response signals.
</div> </div>
</div> </div>
<div id="outline-container-org7263d51" class="outline-3"> <div id="outline-container-org2a9b2e8" class="outline-3">
<h3 id="org7263d51"><span class="section-number-3">5.1</span> Raw Force Data</h3> <h3 id="org2a9b2e8"><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="#org2495eee">15</a>. We can see 10 impacts, one zoom on one impact is shown on figure <a href="#org5d907da">16</a>. The force input for the first measurement is shown on figure <a href="#org272eb14">15</a>. We can see 10 impacts, one zoom on one impact is shown on figure <a href="#org502f99a">16</a>.
</p> </p>
<p> <p>
The Fourier transform of the force is shown on figure <a href="#org904c868">17</a>. This has been obtained without any windowing. The Fourier transform of the force is shown on figure <a href="#org5df9406">17</a>. This has been obtained without any windowing.
</p> </p>
<div id="org2495eee" class="figure"> <div id="org272eb14" 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="org5d907da" class="figure"> <div id="org502f99a" 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="org904c868" class="figure"> <div id="org5df9406" 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 +979,33 @@ The Fourier transform of the force is shown on figure <a href="#org904c868">17</
</div> </div>
</div> </div>
<div id="outline-container-orgc9a27c3" class="outline-3"> <div id="outline-container-org85cf244" class="outline-3">
<h3 id="orgc9a27c3"><span class="section-number-3">5.2</span> Raw Response Data</h3> <h3 id="org85cf244"><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="#orge12758d">18</a>. One zoom on one response is shown on figure <a href="#org8eba24d">19</a>. The response signal for the first measurement is shown on figure <a href="#orge2002e9">18</a>. One zoom on one response is shown on figure <a href="#org677b009">19</a>.
</p> </p>
<p> <p>
The Fourier transform of the response signals is shown on figure <a href="#orgf445c3f">20</a>. This has been obtained without any windowing. The Fourier transform of the response signals is shown on figure <a href="#orgade9101">20</a>. This has been obtained without any windowing.
</p> </p>
<div id="orge12758d" class="figure"> <div id="orge2002e9" 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="org8eba24d" class="figure"> <div id="org677b009" 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="orgf445c3f" class="figure"> <div id="orgade9101" 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 +1013,15 @@ The Fourier transform of the response signals is shown on figure <a href="#orgf4
</div> </div>
</div> </div>
<div id="outline-container-orgf1f892a" class="outline-3"> <div id="outline-container-orgacd507e" class="outline-3">
<h3 id="orgf1f892a"><span class="section-number-3">5.3</span> Computation of one Frequency Response Function</h3> <h3 id="orgacd507e"><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="#orgd26bd12">21</a>). We then compare the result obtained with the FRF computed by the modal software (figure <a href="#orga66271c">21</a>).
</p> </p>
<p> <p>
@ -1032,7 +1038,7 @@ In the following analysis, FRF computed from the software will be used.
</div> </div>
<div id="orgd26bd12" class="figure"> <div id="orga66271c" 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 +1047,8 @@ In the following analysis, FRF computed from the software will be used.
</div> </div>
</div> </div>
<div id="outline-container-org0595bd4" class="outline-2"> <div id="outline-container-org40c5c29" class="outline-2">
<h2 id="org0595bd4"><span class="section-number-2">6</span> Frequency Response Functions and Coherence Results</h2> <h2 id="org40c5c29"><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 +1063,22 @@ First, we load the data.
</div> </div>
<p> <p>
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. And we plot on figure <a href="#org27a4dde">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="#orgf92314e">22</a>. The coherence associated is shown on figure <a href="#org27a4dde">22</a>.
</p> </p>
<div id="orgf92314e" class="figure"> <div id="org27a4dde" 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="org19a9a76" class="figure"> <div id="orgd5c11c0" 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>
@ -1080,8 +1086,8 @@ The coherence associated is shown on figure <a href="#orgf92314e">22</a>.
</div> </div>
</div> </div>
<div id="outline-container-org860e46e" class="outline-2"> <div id="outline-container-orge7a0dc1" class="outline-2">
<h2 id="org860e46e"><span class="section-number-2">7</span> Generation of a FRF matrix and a Coherence matrix from the measurements</h2> <h2 id="orge7a0dc1"><span class="section-number-2">7</span> Generation of a FRF matrix and a Coherence matrix from the measurements</h2>
<div class="outline-text-2" id="text-7"> <div class="outline-text-2" id="text-7">
<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>.
@ -1098,9 +1104,9 @@ The frequency response matrix is an \(n \times p \times q\):
<p> <p>
Thus, the FRF matrix is an \(69 \times 3 \times 801\) matrix. Thus, the FRF matrix is an \(69 \times 3 \times 801\) matrix.
We do the same thing for the coherence matrix.
</p> </p>
<div class="important"> <div class="important">
<p> <p>
For each frequency point \(\omega_i\), we obtain a 2D matrix: For each frequency point \(\omega_i\), we obtain a 2D matrix:
@ -1176,16 +1182,17 @@ 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-orgc8738ff" class="outline-2">
<h2 id="orgc8738ff"><span class="section-number-2">8</span> Plot showing the coherence of all the measurements</h2> <div id="outline-container-org5ebd034" class="outline-2">
<h2 id="org5ebd034"><span class="section-number-2">8</span> Plot showing the coherence of all the measurements</h2>
<div class="outline-text-2" id="text-8"> <div class="outline-text-2" id="text-8">
<p> <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. 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>. The result is shown on figure <a href="#org820d336">24</a>.
</p> </p>
<div id="org433aa7b" class="figure"> <div id="org820d336" class="figure">
<p><img src="figs/all_coherence.png" alt="all_coherence.png" /> <p><img src="figs/all_coherence.png" alt="all_coherence.png" />
</p> </p>
<p><span class="figure-number">Figure 24: </span>Plot of the coherence of all the measurements</p> <p><span class="figure-number">Figure 24: </span>Plot of the coherence of all the measurements</p>
@ -1193,8 +1200,8 @@ The result is shown on figure <a href="#org433aa7b">24</a>.
</div> </div>
</div> </div>
<div id="outline-container-org6601f7d" class="outline-2"> <div id="outline-container-org259382d" class="outline-2">
<h2 id="org6601f7d"><span class="section-number-2">9</span> Solid Bodies considered for further analysis</h2> <h2 id="org259382d"><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:
@ -1209,7 +1216,7 @@ 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="#orgbff47a7">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="#orgbc41472">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>;
@ -1234,49 +1241,49 @@ Finally, we save that into a <code>.mat</code> file.
</div> </div>
</div> </div>
<div id="outline-container-org2b1c74a" class="outline-2"> <div id="outline-container-orgc0c679a" class="outline-2">
<h2 id="org2b1c74a"><span class="section-number-2">10</span> Note about the solid body assumption</h2> <h2 id="orgc0c679a"><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="#org8f78f19">25</a>. This is illustrated on figure <a href="#org700c0ff">25</a>.
</p> </p>
<div id="org8f78f19" class="figure"> <div id="org700c0ff" 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 25: </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="#org8f78f19">25</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="#org700c0ff">25</a> is defined by its displacement \(\delta p\) and rotation \(\vec{\Omega}\) with respect to the reference frame \(\{O\}\).
</p> </p>
<p> <p>
The motions at points \(1\) and \(2\) are: The motions at points \(1\) and \(2\) are:
</p> </p>
\begin{align*} \begin{align*}
v_1 &= v + \Omega \times p_1 \\ \delta p_1 &= \delta p + \Omega \times p_1 \\
v_2 &= v + \Omega \times p_2 \delta p_2 &= \delta p + \Omega \times p_2
\end{align*} \end{align*}
<p> <p>
Taking only the \(x\) direction: Taking only the \(x\) direction:
</p> </p>
\begin{align*} \begin{align*}
v_{x1} &= v + \Omega_y p_{z1} - \Omega_z p_{y1} \\ \delta p_{x1} &= \delta p_x + \Omega_y p_{z1} - \Omega_z p_{y1} \\
v_{x2} &= v + \Omega_y p_{z2} - \Omega_z p_{y2} \delta p_{x2} &= \delta p_x + \Omega_y p_{z2} - \Omega_z p_{y2}
\end{align*} \end{align*}
<p> <p>
However, we have \(p_{1y} = p_{2y}\) and \(p_{1z} = p_{2z}\) because of the co-linearity of the two sensors in the \(x\) direction, and thus we obtain However, we have \(p_{1y} = p_{2y}\) and \(p_{1z} = p_{2z}\) because of the co-linearity of the two sensors in the \(x\) direction, and thus we obtain
</p> </p>
\begin{equation} \begin{equation}
v_{x1} = v_{x2} \delta p_{x1} = \delta p_{x2}
\end{equation} \end{equation}
<div class="important"> <div class="important">
@ -1291,15 +1298,15 @@ We can verify that the rigid body assumption is correct by comparing the measure
</p> </p>
<p> <p>
From the table <a href="#org61987da">2</a>, we can guess which sensors will give the same results in the X and Y directions. From the table <a href="#org9932674">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="#org3f45f65">26</a> and in the Y direction on figure <a href="#org01521f0">27</a>. Comparison of such measurements in the X direction is shown on figure <a href="#org8826aed">26</a> and in the Y direction on figure <a href="#org2a938cd">27</a>.
</p> </p>
<div id="org3f45f65" class="figure"> <div id="org8826aed" 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 26: </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>
@ -1307,7 +1314,7 @@ Comparison of such measurements in the X direction is shown on figure <a href="#
<div id="org01521f0" class="figure"> <div id="org2a938cd" 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 27: </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>
@ -1318,13 +1325,72 @@ Comparison of such measurements in the X direction is shown on figure <a href="#
From the two figures above, we are more confident about the rigid body assumption in the frequency band of interest. From the two figures above, we are more confident about the rigid body assumption in the frequency band of interest.
</p> </p>
</div>
</div>
</div>
<div id="outline-container-org95c7199" class="outline-2">
<h2 id="org95c7199"><span class="section-number-2">11</span> Verification of the principle of reciprocity</h2>
<div class="outline-text-2" id="text-11">
<p>
Because we expect our system to follow the principle of reciprocity.
That is to say the response \(X_j\) at some degree of freedom \(j\) due to a force \(F_k\) applied on DOF \(k\) should be the same as the response \(X_k\) due to a force \(F_j\):
\[ H_{jk} = \frac{X_j}{F_k} = \frac{X_k}{F_j} = H_{kj} \]
</p>
<p>
This comes from the fact that we expect to have symmetric mass, stiffness and damping matrices.
</p>
<p>
In order to access the quality of the data and the validity of the measured FRF, we then check that the reciprocity between \(H_{jk}\) and \(H_{kj}\) is of an acceptable level.
</p>
<p>
We can verify this reciprocity using 3 different pairs of response/force.
</p>
<div id="orga8de4fa" class="figure">
<p><img src="figs/principle_reciprocity.png" alt="principle_reciprocity.png" />
</p>
<p><span class="figure-number">Figure 28: </span>Verification of the principle of reciprocity</p>
</div>
<p>
From figure <a href="#orga8de4fa">28</a>, it seems that the principle of reciprocity is valid from 50Hz to 100Hz.
Above that, the difference between the two could be due to:
</p>
<ul class="org-ul">
<li>rotational DOFs? Local flexibility?</li>
</ul>
<p>
Below, it could be due to:
</p>
<ul class="org-ul">
<li>not exciting and measuring the same DOF</li>
</ul>
<div class="important">
<p>
One should note here that the excitation is not applied on the same DOF as the measured response.
This could be seen on figures <a href="#orgd9f9e9e">10</a>, <a href="#org4002508">11</a> and <a href="#org2dbf07c">12</a> that show the excitation points, and on figure <a href="#org676fa5a">6</a> where the accelerometer on the bottom left shows the one used for the principle of reciprocity validation above.
Thus, technically we cannot verify the principle of reciprocity here.
</p>
</div>
<div class="warning">
<p>
As it is usually very important to measure point response \(\frac{X_j}{F_j}\), that is to say exciting and measuring the <b>same</b> DOF, should the measurements be redone?
</p>
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<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. 11:46</p> <p class="date">Created: 2019-07-11 jeu. 16:25</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>
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modal-analysis/measurement.org
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@ -138,21 +138,24 @@ All the stages are moved to their zero position (Ty, Ry, Rz, Slip-Ring, Hexapod)
All other elements have been remove from the granite such as another heavy positioning system. All other elements have been remove from the granite such as another heavy positioning system.
** Test Planing ** Test Planing
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 $H$ (where $N$ is the number of degree of freedom of the system):
\begin{equation}
H_{jk} = \frac{X_j}{F_k}
\end{equation}
However, the principle of reciprocity states that: However, from only one column or one line of the matrix, we can compute the other terms thanks to the principle of reciprocity.
\[ H_{jk} = \frac{X_j}{F_k} = H_{kj} = \frac{X_k}{F_j} \]
Thus, only one column or one line of the matrix has to be identified.
Either we choose to identify $\frac{X_k}{F_i}$ or $\frac{X_i}{F_k}$ for any chosen $k$ and for $i = 1,\ ...,\ N$. Either we choose to identify $\frac{X_k}{F_i}$ or $\frac{X_i}{F_k}$ for any chosen $k$ and for $i = 1,\ ...,\ N$.
We here choose to identify $\frac{X_i}{F_k}$ for practical reasons: We here choose to identify $\frac{X_i}{F_k}$ for practical reasons:
- it is easier to glue the accelerometers on some stages than to excite this particular stage with the Hammer - it is easier to glue the accelerometers on all the stages and excite only a one particular point than doing the opposite
The measurement thus consists of: The measurement thus consists of:
- always excite the structure at the same location with the Hammer - always excite the structure at the same location with the Hammer
- Move the accelerometers to measure all the DOF of the structure - Move the accelerometers to measure all the DOF of the structure
We will measured 3 columns (3 impacts location) in order to have some redundancy.
** Location of the Accelerometers ** Location of the Accelerometers
4 tri-axis accelerometers are used for each solid body. 4 tri-axis accelerometers are used for each solid body.
@ -561,7 +564,7 @@ The coherence associated is shown on figure [[fig:frf_result_example]].
plot(meas1.FFT1_AvSpc_2_RMS_X_Val, meas1.FFT1_AvXSpc_2_1_RMS_Y_Mod); plot(meas1.FFT1_AvSpc_2_RMS_X_Val, meas1.FFT1_AvXSpc_2_1_RMS_Y_Mod);
set(gca, 'Yscale', 'log'); set(gca, 'Yscale', 'log');
set(gca, 'XTickLabel',[]); set(gca, 'XTickLabel',[]);
ylabel('Magnitude'); ylabel('Magnitude [$\frac{m/s^2}{N}$]');
ax2 = subplot(2, 1, 2); ax2 = subplot(2, 1, 2);
plot(meas1.FFT1_AvSpc_2_RMS_X_Val, meas1.FFT1_AvXSpc_2_1_RMS_Y_Phas); plot(meas1.FFT1_AvSpc_2_RMS_X_Val, meas1.FFT1_AvXSpc_2_1_RMS_Y_Phas);
@ -606,7 +609,7 @@ The frequency response matrix is an $n \times p \times q$:
- $q$ is the number of frequency points $\omega_i$ - $q$ is the number of frequency points $\omega_i$
Thus, the FRF matrix is an $69 \times 3 \times 801$ matrix. Thus, the FRF matrix is an $69 \times 3 \times 801$ matrix.
We do the same thing for the coherence matrix.
#+begin_important #+begin_important
For each frequency point $\omega_i$, we obtain a 2D matrix: For each frequency point $\omega_i$, we obtain a 2D matrix:
@ -672,6 +675,7 @@ And we save the obtained FRF matrix and Coherence matrix in a =.mat= file.
#+begin_src matlab #+begin_src matlab
save('./mat/frf_coh_matrices.mat', 'FRFs', 'COHs', 'freqs'); save('./mat/frf_coh_matrices.mat', 'FRFs', 'COHs', 'freqs');
#+end_src #+end_src
* Plot showing the coherence of all the measurements * Plot showing the coherence of all the measurements
Now that we have defined a Coherence matrix, we can plot each of its elements to have an idea of the overall coherence and thus, quality of the measurement. Now that we have defined a Coherence matrix, we can plot each of its elements to have an idea of the overall coherence and thus, quality of the measurement.
The result is shown on figure [[fig:all_coherence]]. The result is shown on figure [[fig:all_coherence]].
@ -755,8 +759,8 @@ This is illustrated on figure [[fig:aligned_accelerometers]].
\coordinate[] (p1) at (-1.5, 1.5); \coordinate[] (p1) at (-1.5, 1.5);
\coordinate[] (p2) at ( 1.5, 1.5); \coordinate[] (p2) at ( 1.5, 1.5);
\draw[->] (p1)node[]{$\bullet$}node[above]{$p_1$} -- ++(1, 0)node[above]{$v_{x1}$}; \draw[->] (p1)node[]{$\bullet$}node[above]{$p_1$} -- ++(1, 0)node[above]{$\delta p_{x1}$};
\draw[->] (p2)node[]{$\bullet$}node[above]{$p_2$} -- ++(1, 0)node[above]{$v_{x2}$}; \draw[->] (p2)node[]{$\bullet$}node[above]{$p_2$} -- ++(1, 0)node[above]{$\delta p_{x2}$};
\draw[dashed] ($(p1)+(-1, 0)$) -- ($(p2)+(2, 0)$); \draw[dashed] ($(p1)+(-1, 0)$) -- ($(p2)+(2, 0)$);
\end{tikzpicture} \end{tikzpicture}
@ -767,23 +771,23 @@ This is illustrated on figure [[fig:aligned_accelerometers]].
#+RESULTS: #+RESULTS:
[[file:figs/aligned_accelerometers.png]] [[file:figs/aligned_accelerometers.png]]
The motion of the rigid body of figure [[fig:aligned_accelerometers]] is defined by the velocity $\vec{v}$ and rotation $\vec{\Omega}$ with respect to the reference frame $\{O\}$. The motion of the rigid body of figure [[fig:aligned_accelerometers]] is defined by its displacement $\delta p$ and rotation $\vec{\Omega}$ with respect to the reference frame $\{O\}$.
The motions at points $1$ and $2$ are: The motions at points $1$ and $2$ are:
\begin{align*} \begin{align*}
v_1 &= v + \Omega \times p_1 \\ \delta p_1 &= \delta p + \Omega \times p_1 \\
v_2 &= v + \Omega \times p_2 \delta p_2 &= \delta p + \Omega \times p_2
\end{align*} \end{align*}
Taking only the $x$ direction: Taking only the $x$ direction:
\begin{align*} \begin{align*}
v_{x1} &= v + \Omega_y p_{z1} - \Omega_z p_{y1} \\ \delta p_{x1} &= \delta p_x + \Omega_y p_{z1} - \Omega_z p_{y1} \\
v_{x2} &= v + \Omega_y p_{z2} - \Omega_z p_{y2} \delta p_{x2} &= \delta p_x + \Omega_y p_{z2} - \Omega_z p_{y2}
\end{align*} \end{align*}
However, we have $p_{1y} = p_{2y}$ and $p_{1z} = p_{2z}$ because of the co-linearity of the two sensors in the $x$ direction, and thus we obtain However, we have $p_{1y} = p_{2y}$ and $p_{1z} = p_{2z}$ because of the co-linearity of the two sensors in the $x$ direction, and thus we obtain
\begin{equation} \begin{equation}
v_{x1} = v_{x2} \delta p_{x1} = \delta p_{x2}
\end{equation} \end{equation}
#+begin_important #+begin_important
@ -825,7 +829,7 @@ Comparison of such measurements in the X direction is shown on figure [[fig:comp
end end
if rem(i, 3) == 1 if rem(i, 3) == 1
ylabel('Amplitude'); ylabel('Amplitude [$\frac{m/s^2}{N}$]');
end end
xlim([1, 200]); xlim([1, 200]);
title(sprintf('Acc %i and %i - X', acc_i(i, 1), acc_i(i, 2))); title(sprintf('Acc %i and %i - X', acc_i(i, 1), acc_i(i, 2)));
@ -871,7 +875,7 @@ Comparison of such measurements in the X direction is shown on figure [[fig:comp
end end
if rem(i, 3) == 1 if rem(i, 3) == 1
ylabel('Amplitude'); ylabel('Amplitude [$\frac{m/s^2}{N}$]');
end end
xlim([1, 200]); xlim([1, 200]);
title(sprintf('Acc %i and %i - Y', acc_i(i, 1), acc_i(i, 2))); title(sprintf('Acc %i and %i - Y', acc_i(i, 1), acc_i(i, 2)));
@ -890,3 +894,65 @@ Comparison of such measurements in the X direction is shown on figure [[fig:comp
#+begin_important #+begin_important
From the two figures above, we are more confident about the rigid body assumption in the frequency band of interest. From the two figures above, we are more confident about the rigid body assumption in the frequency band of interest.
#+end_important #+end_important
* Verification of the principle of reciprocity
Because we expect our system to follow the principle of reciprocity.
That is to say the response $X_j$ at some degree of freedom $j$ due to a force $F_k$ applied on DOF $k$ should be the same as the response $X_k$ due to a force $F_j$:
\[ H_{jk} = \frac{X_j}{F_k} = \frac{X_k}{F_j} = H_{kj} \]
This comes from the fact that we expect to have symmetric mass, stiffness and damping matrices.
In order to access the quality of the data and the validity of the measured FRF, we then check that the reciprocity between $H_{jk}$ and $H_{kj}$ is of an acceptable level.
We can verify this reciprocity using 3 different pairs of response/force.
#+begin_src matlab :exports none
dir_names = {'X', 'Y', 'Z'};
figure;
for i = 1:3
subplot(3, 1, i)
a = mod(i, 3)+1;
b = mod(i-2, 3)+1;
hold on;
plot(freqs, abs(squeeze(FRFs(3*(11-1)+a, b, :))), 'DisplayName', sprintf('$\\frac{F_%s}{D_%s}$', dir_names{a}, dir_names{b}));
plot(freqs, abs(squeeze(FRFs(3*(11-1)+b, a, :))), 'DisplayName', sprintf('$\\frac{F_%s}{D_%s}$', dir_names{b}, dir_names{a}));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
if i == 3
xlabel('Frequency [Hz]');
else
set(gca, 'XTickLabel',[]);
end
if i == 2
ylabel('Amplitude [$\frac{m/s^2}{N}$]');
end
xlim([1, 200]);
legend('location', 'northwest');
end
#+end_src
#+HEADER: :tangle no :exports results :results none :noweb yes
#+begin_src matlab :var filepath="figs/principle_reciprocity.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
<<plt-matlab>>
#+end_src
#+NAME: fig:principle_reciprocity
#+CAPTION: Verification of the principle of reciprocity
[[file:figs/principle_reciprocity.png]]
From figure [[fig:principle_reciprocity]], it seems that the principle of reciprocity is valid from 50Hz to 100Hz.
Above that, the difference between the two could be due to:
- rotational DOFs? Local flexibility?
Below, it could be due to:
- not exciting and measuring the same DOF
#+begin_important
One should note here that the excitation is not applied on the same DOF as the measured response.
This could be seen on figures [[fig:hammer_x]], [[fig:hammer_y]] and [[fig:hammer_z]] that show the excitation points, and on figure [[fig:accelerometers_ty_overview]] where the accelerometer on the bottom left shows the one used for the principle of reciprocity validation above.
Thus, technically we cannot verify the principle of reciprocity here.
#+end_important
#+begin_warning
As it is usually very important to measure point response $\frac{X_j}{F_j}$, that is to say exciting and measuring the *same* DOF, should the measurements be redone?
On the modal software that is used to extract modal parameters, it is suppose that we are exciting the *exact* same DOF as the one measured by the accelerometer 11. As is it not the case in practice, could this induce large errors in the modal model?
#+end_warning