Update analysis => FRF in the global frame
This commit is contained in:
@@ -3,7 +3,7 @@
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"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
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<html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en">
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<head>
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<!-- 2019-07-03 mer. 17:24 -->
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<!-- 2019-07-04 jeu. 17:49 -->
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<meta http-equiv="Content-Type" content="text/html;charset=utf-8" />
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<meta name="viewport" content="width=device-width, initial-scale=1" />
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<title>Modal Analysis - Measurement</title>
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@@ -280,30 +280,34 @@ for the JavaScript code in this tag.
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<h2>Table of Contents</h2>
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<div id="text-table-of-contents">
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<ul>
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<li><a href="#orgc2c9984">1. Instrumentation Used</a></li>
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<li><a href="#org8b7c836">2. Structure Preparation and Test Planning</a>
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<li><a href="#org85e5106">1. Goal</a></li>
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<li><a href="#org47f3940">2. Instrumentation Used</a></li>
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<li><a href="#org977bbae">3. Structure Preparation and Test Planning</a>
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<ul>
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<li><a href="#org4e2067f">2.1. Structure Preparation</a></li>
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<li><a href="#orge159695">2.2. Test Planing</a></li>
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<li><a href="#org93e879f">2.3. Location of the Accelerometers</a></li>
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<li><a href="#org155f119">2.4. Hammer Impacts</a></li>
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<li><a href="#orgde64688">3.1. Structure Preparation</a></li>
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<li><a href="#orgd83fa83">3.2. Test Planing</a></li>
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<li><a href="#org3e409fe">3.3. Location of the Accelerometers</a></li>
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<li><a href="#org7cb5221">3.4. Hammer Impacts</a></li>
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</ul>
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</li>
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<li><a href="#orgf4d3979">3. Signal Processing</a>
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<li><a href="#org2124857">4. Signal Processing</a>
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<ul>
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<li><a href="#org5818e34">3.1. Averaging</a></li>
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<li><a href="#orgc6aff30">3.2. Windowing</a></li>
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<li><a href="#org600eae0">4.1. Averaging</a></li>
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<li><a href="#org83d4969">4.2. Windowing</a></li>
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</ul>
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</li>
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<li><a href="#org58ab5bf">4. Force and Response signals</a>
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<li><a href="#org25e02aa">5. Force and Response signals</a>
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<ul>
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<li><a href="#org2216b49">4.1. Raw Force Data</a></li>
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<li><a href="#org91939ab">4.2. Raw Response Data</a></li>
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<li><a href="#orgc34cf19">4.3. Computation of one Frequency Response Function</a></li>
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<li><a href="#orgbf40925">5.1. Raw Force Data</a></li>
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<li><a href="#org2cf0a3e">5.2. Raw Response Data</a></li>
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<li><a href="#orgad3fa68">5.3. Computation of one Frequency Response Function</a></li>
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</ul>
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</li>
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<li><a href="#orgc91b48e">5. Frequency Response Functions and Coherence Results</a></li>
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<li><a href="#org6c0132c">6. Generation of a FRF matrix and a Coherence matrix from the measurements</a></li>
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<li><a href="#org4bffecd">6. Frequency Response Functions and Coherence Results</a></li>
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<li><a href="#org093edb6">7. <span class="todo TODO">TODO</span> Plot all coherences in one plot</a></li>
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<li><a href="#org23b7ad3">8. Generation of a FRF matrix and a Coherence matrix from the measurements</a></li>
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<li><a href="#org8a40769">9. Solid Bodies considered for further analysis</a></li>
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<li><a href="#org2405c81">10. Notes the solid body assumption</a></li>
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</ul>
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</div>
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</div>
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@@ -315,20 +319,29 @@ All the files (data and Matlab scripts) are accessible <a href="data/modal_frf_c
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</div>
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<div id="outline-container-orgc2c9984" class="outline-2">
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<h2 id="orgc2c9984"><span class="section-number-2">1</span> Instrumentation Used</h2>
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<div id="outline-container-org85e5106" class="outline-2">
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<h2 id="org85e5106"><span class="section-number-2">1</span> Goal</h2>
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<div class="outline-text-2" id="text-1">
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<p>
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The goal is to measure the dynamic of the Micro-Station and to extract Frequency Response Functions.
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</p>
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</div>
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</div>
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<div id="outline-container-org47f3940" class="outline-2">
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<h2 id="org47f3940"><span class="section-number-2">2</span> Instrumentation Used</h2>
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<div class="outline-text-2" id="text-2">
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<p>
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In order to perform to <b>Modal Analysis</b> and to obtain first a <b>Response Model</b>, the following devices are used:
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</p>
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<ul class="org-ul">
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<li>An <b>acquisition system</b> (OROS) with 24bits ADCs (figure <a href="#org8935b8f">1</a>)</li>
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<li>3 tri-axis <b>Accelerometers</b> (figure <a href="#org48dfe3a">2</a>) with parameters shown on table <a href="#org4d70939">1</a></li>
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<li>An <b>Instrumented Hammer</b> with various Tips (figure <a href="#org2a2c278">3</a>) (figure <a href="#org982b599">4</a>)</li>
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<li>An <b>acquisition system</b> (OROS) with 24bits ADCs (figure <a href="#org5f44420">1</a>)</li>
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<li>3 tri-axis <b>Accelerometers</b> (figure <a href="#orgc28a203">2</a>) with parameters shown on table <a href="#org8770ead">1</a></li>
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<li>An <b>Instrumented Hammer</b> with various Tips (figure <a href="#orge4928f3">3</a>) (figure <a href="#orge94631e">4</a>)</li>
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</ul>
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<div id="org8935b8f" class="figure">
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<div id="org5f44420" class="figure">
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<p><img src="img/instrumentation/oros.png" alt="oros.png" width="500px" />
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</p>
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<p><span class="figure-number">Figure 1: </span>Acquisition system: OROS</p>
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@@ -341,13 +354,13 @@ Anti-aliasing filters are also included in the system.
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</p>
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<div id="org48dfe3a" class="figure">
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<div id="orgc28a203" class="figure">
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<p><img src="img/instrumentation/accelero_M393B05.png" alt="accelero_M393B05.png" width="500px" />
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</p>
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<p><span class="figure-number">Figure 2: </span>Accelerometer used: M393B05</p>
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</div>
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<table id="org4d70939" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
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<table id="org8770ead" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
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<caption class="t-above"><span class="table-number">Table 1:</span> 393B05 Accelerometer Data Sheet</caption>
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<colgroup>
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@@ -390,14 +403,14 @@ It excites more the low frequency range where the coherence is low, the overall
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</p>
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<div id="org2a2c278" class="figure">
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<div id="orge4928f3" class="figure">
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<p><img src="img/instrumentation/instrumented_hammer.png" alt="instrumented_hammer.png" width="500px" />
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</p>
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<p><span class="figure-number">Figure 3: </span>Instrumented Hammer</p>
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</div>
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<div id="org982b599" class="figure">
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<div id="orge94631e" class="figure">
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<p><img src="img/instrumentation/hammer_tips.png" alt="hammer_tips.png" width="500px" />
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</p>
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<p><span class="figure-number">Figure 4: </span>Hammer tips</p>
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@@ -409,13 +422,13 @@ The accelerometers are glued on the structure.
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</div>
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</div>
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<div id="outline-container-org8b7c836" class="outline-2">
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<h2 id="org8b7c836"><span class="section-number-2">2</span> Structure Preparation and Test Planning</h2>
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<div class="outline-text-2" id="text-2">
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<div id="outline-container-org977bbae" class="outline-2">
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<h2 id="org977bbae"><span class="section-number-2">3</span> Structure Preparation and Test Planning</h2>
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<div class="outline-text-2" id="text-3">
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</div>
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<div id="outline-container-org4e2067f" class="outline-3">
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<h3 id="org4e2067f"><span class="section-number-3">2.1</span> Structure Preparation</h3>
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<div class="outline-text-3" id="text-2-1">
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<div id="outline-container-orgde64688" class="outline-3">
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<h3 id="orgde64688"><span class="section-number-3">3.1</span> Structure Preparation</h3>
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<div class="outline-text-3" id="text-3-1">
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<p>
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All the stages are turned ON.
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This is done for two reasons:
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@@ -457,9 +470,9 @@ All other elements have been remove from the granite such as another heavy posit
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</div>
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</div>
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<div id="outline-container-orge159695" class="outline-3">
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<h3 id="orge159695"><span class="section-number-3">2.2</span> Test Planing</h3>
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<div class="outline-text-3" id="text-2-2">
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<div id="outline-container-orgd83fa83" class="outline-3">
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<h3 id="orgd83fa83"><span class="section-number-3">3.2</span> Test Planing</h3>
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<div class="outline-text-3" id="text-3-2">
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<p>
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The goal is to identify the full \(N \times N\) FRF matrix (where \(N\) is the number of degree of freedom of the system).
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</p>
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@@ -491,9 +504,9 @@ The measurement thus consists of:
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</div>
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</div>
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<div id="outline-container-org93e879f" class="outline-3">
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<h3 id="org93e879f"><span class="section-number-3">2.3</span> Location of the Accelerometers</h3>
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<div class="outline-text-3" id="text-2-3">
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<div id="outline-container-org3e409fe" class="outline-3">
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||||
<h3 id="org3e409fe"><span class="section-number-3">3.3</span> Location of the Accelerometers</h3>
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<div class="outline-text-3" id="text-3-3">
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<p>
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4 tri-axis accelerometers are used for each solid body.
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</p>
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@@ -511,11 +524,11 @@ The position of the accelerometers are:
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</p>
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<ul class="org-ul">
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<li>4 on the first granite</li>
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<li>4 on the second granite (figure <a href="#org552acb7">5</a>)</li>
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<li>4 on top of the translation stage (figure <a href="#orgd191f19">6</a>)</li>
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<li>4 on the second granite (figure <a href="#org7b547b6">5</a>)</li>
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<li>4 on top of the translation stage (figure <a href="#orgf41551b">6</a>)</li>
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<li>4 on top of the tilt stage</li>
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<li>3 on top of the spindle</li>
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||||
<li>4 on top of the hexapod (figure <a href="#org0b791f2">7</a>)</li>
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<li>4 on top of the hexapod (figure <a href="#org168dfba">7</a>)</li>
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||||
</ul>
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<p>
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@@ -523,43 +536,43 @@ In total, 23 accelerometers are used: <b>69 DOFs are thus measured</b>.
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</p>
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<p>
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The position and orientation of all the accelerometers used are shown on figure <a href="#org29de56f">8</a>.
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The position and orientation of all the accelerometers used are shown on figure <a href="#org92f8aa8">8</a>.
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</p>
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<p>
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The precise determination of the position of each accelerometer is done using the SolidWorks model (shown on figure <a href="#org6cedca1">9</a>).
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||||
The precise determination of the position of each accelerometer is done using the SolidWorks model (shown on figure <a href="#org87fdd26">9</a>).
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</p>
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<div id="org552acb7" class="figure">
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<div id="org7b547b6" class="figure">
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<p><img src="img/accelerometers/accelerometers_granite2_overview.jpg" alt="accelerometers_granite2_overview.jpg" width="500px" />
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</p>
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<p><span class="figure-number">Figure 5: </span>Accelerometers located on the top granite</p>
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</div>
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<div id="orgd191f19" class="figure">
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<div id="orgf41551b" class="figure">
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<p><img src="img/accelerometers/accelerometers_ty_overview.jpg" alt="accelerometers_ty_overview.jpg" width="500px" />
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</p>
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<p><span class="figure-number">Figure 6: </span>Accelerometers located on top of the translation stage</p>
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</div>
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<div id="org0b791f2" class="figure">
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<div id="org168dfba" class="figure">
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<p><img src="img/accelerometers/accelerometers_hexa_overview.jpg" alt="accelerometers_hexa_overview.jpg" width="500px" />
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</p>
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<p><span class="figure-number">Figure 7: </span>Accelerometers located on the Hexapod</p>
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</div>
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<div id="org29de56f" class="figure">
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<div id="org92f8aa8" class="figure">
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<p><img src="figs/nass-modal-test.png" alt="nass-modal-test.png" width="800px" />
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</p>
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<p><span class="figure-number">Figure 8: </span>Position and orientation of the accelerometer used</p>
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</div>
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<div id="org6cedca1" class="figure">
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<div id="org87fdd26" class="figure">
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||||
<p><img src="img/location_accelerometers.png" alt="location_accelerometers.png" width="800px" />
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</p>
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||||
<p><span class="figure-number">Figure 9: </span>Position of the accelerometers using SolidWorks</p>
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@@ -569,9 +582,27 @@ The precise determination of the position of each accelerometer is done using th
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The precise position of all the 23 accelerometer with respect to a frame located at the point of interest (located 270mm above the top platform of the hexapod) is shown below. The values are in meter.
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They are contained in the <code>mat/id31_nanostation.cfg</code> file.
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</p>
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<div class="org-src-container">
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||||
<pre class="src src-bash">cat mat/id31_nanostation.cfg | grep NODES -A <span class="org-highlight-numbers-number">23</span> | sed <span class="org-string">'/\s\+[0-9]\+/!d'</span> | sed <span class="org-string">'s/\(.*\)\s\+0\s\+.\+/\1/'</span> > mat/acc_pos.txt
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</pre>
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</div>
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<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
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<p>
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We then import that on <code>matlab</code>, and sort them.
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</p>
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<div class="org-src-container">
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||||
<pre class="src src-matlab">acc_pos = readtable<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'mat/acc_pos.txt', 'ReadVariableNames'</span>, false<span class="org-rainbow-delimiters-depth-1">)</span>;
|
||||
acc_pos = table2array<span class="org-rainbow-delimiters-depth-1">(</span>acc_pos<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">4</span><span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
||||
<span class="org-rainbow-delimiters-depth-1">[</span><span class="org-type">~</span>, <span class="org-constant">i</span><span class="org-rainbow-delimiters-depth-1">]</span> = sort<span class="org-rainbow-delimiters-depth-1">(</span>acc_pos<span class="org-rainbow-delimiters-depth-2">(</span><span class="org-type">:</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>;
|
||||
acc_pos = acc_pos<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-constant">i</span>, <span class="org-highlight-numbers-number">2</span><span class="org-type">:</span><span class="org-highlight-numbers-number">4</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<p>
|
||||
The positions of the sensors relative to the point of interest are shown below (table <a href="#org65ae8c9">2</a>).
|
||||
</p>
|
||||
<table id="org65ae8c9" 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>
|
||||
|
||||
<colgroup>
|
||||
<col class="org-right" />
|
||||
@@ -585,186 +616,186 @@ They are contained in the <code>mat/id31_nanostation.cfg</code> file.
|
||||
<thead>
|
||||
<tr>
|
||||
<th scope="col" class="org-right">ID</th>
|
||||
<th scope="col" class="org-right">X[m]</th>
|
||||
<th scope="col" class="org-right">Y[m]</th>
|
||||
<th scope="col" class="org-right">Z[m]</th>
|
||||
<th scope="col" class="org-right">x [mm]</th>
|
||||
<th scope="col" class="org-right">y [mm]</th>
|
||||
<th scope="col" class="org-right">z [mm]</th>
|
||||
</tr>
|
||||
</thead>
|
||||
<tbody>
|
||||
<tr>
|
||||
<td class="org-right">1</td>
|
||||
<td class="org-right">-6.4000e-002</td>
|
||||
<td class="org-right">-6.4000e-002</td>
|
||||
<td class="org-right">-2.7000e-001</td>
|
||||
<td class="org-right">-64</td>
|
||||
<td class="org-right">-64</td>
|
||||
<td class="org-right">-270</td>
|
||||
</tr>
|
||||
|
||||
<tr>
|
||||
<td class="org-right">2</td>
|
||||
<td class="org-right">-6.4000e-002</td>
|
||||
<td class="org-right">6.4000e-002</td>
|
||||
<td class="org-right">-2.7000e-001</td>
|
||||
<td class="org-right">-64</td>
|
||||
<td class="org-right">64</td>
|
||||
<td class="org-right">-270</td>
|
||||
</tr>
|
||||
|
||||
<tr>
|
||||
<td class="org-right">3</td>
|
||||
<td class="org-right">6.4000e-002</td>
|
||||
<td class="org-right">6.4000e-002</td>
|
||||
<td class="org-right">-2.7000e-001</td>
|
||||
<td class="org-right">64</td>
|
||||
<td class="org-right">64</td>
|
||||
<td class="org-right">-270</td>
|
||||
</tr>
|
||||
|
||||
<tr>
|
||||
<td class="org-right">4</td>
|
||||
<td class="org-right">6.4000e-002</td>
|
||||
<td class="org-right">-6.4000e-002</td>
|
||||
<td class="org-right">-2.7000e-001</td>
|
||||
<td class="org-right">64</td>
|
||||
<td class="org-right">-64</td>
|
||||
<td class="org-right">-270</td>
|
||||
</tr>
|
||||
|
||||
<tr>
|
||||
<td class="org-right">5</td>
|
||||
<td class="org-right">-3.8500e-001</td>
|
||||
<td class="org-right">-3.0000e-001</td>
|
||||
<td class="org-right">-4.1680e-001</td>
|
||||
<td class="org-right">-385</td>
|
||||
<td class="org-right">-300</td>
|
||||
<td class="org-right">-417</td>
|
||||
</tr>
|
||||
|
||||
<tr>
|
||||
<td class="org-right">6</td>
|
||||
<td class="org-right">-4.2000e-001</td>
|
||||
<td class="org-right">2.8000e-001</td>
|
||||
<td class="org-right">-4.1680e-001</td>
|
||||
<td class="org-right">-420</td>
|
||||
<td class="org-right">280</td>
|
||||
<td class="org-right">-417</td>
|
||||
</tr>
|
||||
|
||||
<tr>
|
||||
<td class="org-right">7</td>
|
||||
<td class="org-right">4.2000e-001</td>
|
||||
<td class="org-right">2.8000e-001</td>
|
||||
<td class="org-right">-4.1680e-001</td>
|
||||
<td class="org-right">420</td>
|
||||
<td class="org-right">280</td>
|
||||
<td class="org-right">-417</td>
|
||||
</tr>
|
||||
|
||||
<tr>
|
||||
<td class="org-right">8</td>
|
||||
<td class="org-right">3.8000e-001</td>
|
||||
<td class="org-right">-3.0000e-001</td>
|
||||
<td class="org-right">-4.1680e-001</td>
|
||||
<td class="org-right">380</td>
|
||||
<td class="org-right">-300</td>
|
||||
<td class="org-right">-417</td>
|
||||
</tr>
|
||||
|
||||
<tr>
|
||||
<td class="org-right">9</td>
|
||||
<td class="org-right">-4.7500e-001</td>
|
||||
<td class="org-right">-4.1400e-001</td>
|
||||
<td class="org-right">-4.2730e-001</td>
|
||||
<td class="org-right">-475</td>
|
||||
<td class="org-right">-414</td>
|
||||
<td class="org-right">-427</td>
|
||||
</tr>
|
||||
|
||||
<tr>
|
||||
<td class="org-right">10</td>
|
||||
<td class="org-right">-4.6500e-001</td>
|
||||
<td class="org-right">4.0700e-001</td>
|
||||
<td class="org-right">-4.2730e-001</td>
|
||||
<td class="org-right">-465</td>
|
||||
<td class="org-right">407</td>
|
||||
<td class="org-right">-427</td>
|
||||
</tr>
|
||||
|
||||
<tr>
|
||||
<td class="org-right">11</td>
|
||||
<td class="org-right">4.7500e-001</td>
|
||||
<td class="org-right">4.2400e-001</td>
|
||||
<td class="org-right">-4.2730e-001</td>
|
||||
<td class="org-right">475</td>
|
||||
<td class="org-right">424</td>
|
||||
<td class="org-right">-427</td>
|
||||
</tr>
|
||||
|
||||
<tr>
|
||||
<td class="org-right">12</td>
|
||||
<td class="org-right">4.7500e-001</td>
|
||||
<td class="org-right">-4.1900e-001</td>
|
||||
<td class="org-right">-4.2730e-001</td>
|
||||
<td class="org-right">475</td>
|
||||
<td class="org-right">-419</td>
|
||||
<td class="org-right">-427</td>
|
||||
</tr>
|
||||
|
||||
<tr>
|
||||
<td class="org-right">13</td>
|
||||
<td class="org-right">-3.2000e-001</td>
|
||||
<td class="org-right">-4.4600e-001</td>
|
||||
<td class="org-right">-7.8560e-001</td>
|
||||
<td class="org-right">-320</td>
|
||||
<td class="org-right">-446</td>
|
||||
<td class="org-right">-786</td>
|
||||
</tr>
|
||||
|
||||
<tr>
|
||||
<td class="org-right">14</td>
|
||||
<td class="org-right">-4.8000e-001</td>
|
||||
<td class="org-right">5.3400e-001</td>
|
||||
<td class="org-right">-7.8560e-001</td>
|
||||
<td class="org-right">-480</td>
|
||||
<td class="org-right">534</td>
|
||||
<td class="org-right">-786</td>
|
||||
</tr>
|
||||
|
||||
<tr>
|
||||
<td class="org-right">15</td>
|
||||
<td class="org-right">4.5000e-001</td>
|
||||
<td class="org-right">5.3400e-001</td>
|
||||
<td class="org-right">-7.8560e-001</td>
|
||||
<td class="org-right">450</td>
|
||||
<td class="org-right">534</td>
|
||||
<td class="org-right">-786</td>
|
||||
</tr>
|
||||
|
||||
<tr>
|
||||
<td class="org-right">16</td>
|
||||
<td class="org-right">2.9500e-001</td>
|
||||
<td class="org-right">-4.8100e-001</td>
|
||||
<td class="org-right">-7.8560e-001</td>
|
||||
<td class="org-right">295</td>
|
||||
<td class="org-right">-481</td>
|
||||
<td class="org-right">-786</td>
|
||||
</tr>
|
||||
|
||||
<tr>
|
||||
<td class="org-right">17</td>
|
||||
<td class="org-right">-7.3000e-001</td>
|
||||
<td class="org-right">-5.2600e-001</td>
|
||||
<td class="org-right">-9.5060e-001</td>
|
||||
<td class="org-right">-730</td>
|
||||
<td class="org-right">-526</td>
|
||||
<td class="org-right">-951</td>
|
||||
</tr>
|
||||
|
||||
<tr>
|
||||
<td class="org-right">18</td>
|
||||
<td class="org-right">-7.3500e-001</td>
|
||||
<td class="org-right">8.1400e-001</td>
|
||||
<td class="org-right">-9.5060e-001</td>
|
||||
<td class="org-right">-735</td>
|
||||
<td class="org-right">814</td>
|
||||
<td class="org-right">-951</td>
|
||||
</tr>
|
||||
|
||||
<tr>
|
||||
<td class="org-right">19</td>
|
||||
<td class="org-right">8.7500e-001</td>
|
||||
<td class="org-right">7.9900e-001</td>
|
||||
<td class="org-right">-9.5060e-001</td>
|
||||
<td class="org-right">875</td>
|
||||
<td class="org-right">799</td>
|
||||
<td class="org-right">-951</td>
|
||||
</tr>
|
||||
|
||||
<tr>
|
||||
<td class="org-right">20</td>
|
||||
<td class="org-right">8.6490e-001</td>
|
||||
<td class="org-right">-5.0600e-001</td>
|
||||
<td class="org-right">-9.5060e-001</td>
|
||||
<td class="org-right">865</td>
|
||||
<td class="org-right">-506</td>
|
||||
<td class="org-right">-951</td>
|
||||
</tr>
|
||||
|
||||
<tr>
|
||||
<td class="org-right">21</td>
|
||||
<td class="org-right">-1.5500e-001</td>
|
||||
<td class="org-right">-9.0000e-002</td>
|
||||
<td class="org-right">-5.9400e-001</td>
|
||||
<td class="org-right">-155</td>
|
||||
<td class="org-right">-90</td>
|
||||
<td class="org-right">-594</td>
|
||||
</tr>
|
||||
|
||||
<tr>
|
||||
<td class="org-right">22</td>
|
||||
<td class="org-right">0.0000e+000</td>
|
||||
<td class="org-right">1.8000e-001</td>
|
||||
<td class="org-right">-5.9400e-001</td>
|
||||
<td class="org-right">0</td>
|
||||
<td class="org-right">180</td>
|
||||
<td class="org-right">-594</td>
|
||||
</tr>
|
||||
|
||||
<tr>
|
||||
<td class="org-right">23</td>
|
||||
<td class="org-right">1.5500e-001</td>
|
||||
<td class="org-right">-9.0000e-002</td>
|
||||
<td class="org-right">-5.9400e-001</td>
|
||||
<td class="org-right">155</td>
|
||||
<td class="org-right">-90</td>
|
||||
<td class="org-right">-594</td>
|
||||
</tr>
|
||||
</tbody>
|
||||
</table>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org155f119" class="outline-3">
|
||||
<h3 id="org155f119"><span class="section-number-3">2.4</span> Hammer Impacts</h3>
|
||||
<div class="outline-text-3" id="text-2-4">
|
||||
<div id="outline-container-org7cb5221" class="outline-3">
|
||||
<h3 id="org7cb5221"><span class="section-number-3">3.4</span> Hammer Impacts</h3>
|
||||
<div class="outline-text-3" id="text-3-4">
|
||||
<p>
|
||||
Only 3 impact points are used.
|
||||
</p>
|
||||
|
||||
<p>
|
||||
The impact points are shown on figures <a href="#org6cb6a5e">10</a>, <a href="#org9228a0f">11</a> and <a href="#orgf3797df">12</a>.
|
||||
The impact points are shown on figures <a href="#orge9d3fcb">10</a>, <a href="#org689764c">11</a> and <a href="#orgb4d4ec1">12</a>.
|
||||
</p>
|
||||
|
||||
<p>
|
||||
@@ -772,21 +803,21 @@ We chose this excitation point as it seems to excite all the modes in the freque
|
||||
</p>
|
||||
|
||||
|
||||
<div id="org6cb6a5e" class="figure">
|
||||
<div id="orge9d3fcb" class="figure">
|
||||
<p><img src="img/impacts/hammer_x.gif" alt="hammer_x.gif" width="300px" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 10: </span>Hammer Blow in the X direction</p>
|
||||
</div>
|
||||
|
||||
|
||||
<div id="org9228a0f" class="figure">
|
||||
<div id="org689764c" class="figure">
|
||||
<p><img src="img/impacts/hammer_y.gif" alt="hammer_y.gif" width="300px" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 11: </span>Hammer Blow in the Y direction</p>
|
||||
</div>
|
||||
|
||||
|
||||
<div id="orgf3797df" class="figure">
|
||||
<div id="orgb4d4ec1" class="figure">
|
||||
<p><img src="img/impacts/hammer_z.gif" alt="hammer_z.gif" width="300px" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 12: </span>Hammer Blow in the Z direction</p>
|
||||
@@ -795,20 +826,20 @@ We chose this excitation point as it seems to excite all the modes in the freque
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgf4d3979" class="outline-2">
|
||||
<h2 id="orgf4d3979"><span class="section-number-2">3</span> Signal Processing</h2>
|
||||
<div class="outline-text-2" id="text-3">
|
||||
<div id="outline-container-org2124857" class="outline-2">
|
||||
<h2 id="org2124857"><span class="section-number-2">4</span> Signal Processing</h2>
|
||||
<div class="outline-text-2" id="text-4">
|
||||
</div>
|
||||
<div id="outline-container-org5818e34" class="outline-3">
|
||||
<h3 id="org5818e34"><span class="section-number-3">3.1</span> Averaging</h3>
|
||||
<div class="outline-text-3" id="text-3-1">
|
||||
<div id="outline-container-org600eae0" class="outline-3">
|
||||
<h3 id="org600eae0"><span class="section-number-3">4.1</span> Averaging</h3>
|
||||
<div class="outline-text-3" id="text-4-1">
|
||||
<p>
|
||||
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="#orga755bcd">2</a>.
|
||||
The parameters for the impact test are shown on table <a href="#orgc7dedc2">3</a>.
|
||||
</p>
|
||||
|
||||
<table id="orga755bcd" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
|
||||
<caption class="t-above"><span class="table-number">Table 2:</span> Impact test parameters</caption>
|
||||
<table id="orgc7dedc2" 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>
|
||||
|
||||
<colgroup>
|
||||
<col class="org-left" />
|
||||
@@ -845,15 +876,15 @@ The parameters for the impact test are shown on table <a href="#orga755bcd">2</a
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgc6aff30" class="outline-3">
|
||||
<h3 id="orgc6aff30"><span class="section-number-3">3.2</span> Windowing</h3>
|
||||
<div class="outline-text-3" id="text-3-2">
|
||||
<div id="outline-container-org83d4969" class="outline-3">
|
||||
<h3 id="org83d4969"><span class="section-number-3">4.2</span> Windowing</h3>
|
||||
<div class="outline-text-3" id="text-4-2">
|
||||
<p>
|
||||
Windowing is used on the force and response signals.
|
||||
</p>
|
||||
|
||||
<p>
|
||||
A boxcar window (figure <a href="#orgd10fc79">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="#org36f5bec">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:
|
||||
</p>
|
||||
<ul class="org-ul">
|
||||
@@ -862,14 +893,14 @@ The parameters are:
|
||||
</ul>
|
||||
|
||||
|
||||
<div id="orgd10fc79" class="figure">
|
||||
<div id="org36f5bec" class="figure">
|
||||
<p><img src="figs/windowing_force_signal.png" alt="windowing_force_signal.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 13: </span>Window used for the force signal</p>
|
||||
</div>
|
||||
|
||||
<p>
|
||||
An exponential window (figure <a href="#orgdd629ee">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="#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.
|
||||
The parameters are:
|
||||
</p>
|
||||
<ul class="org-ul">
|
||||
@@ -886,7 +917,7 @@ The parameters are:
|
||||
</ul>
|
||||
|
||||
|
||||
<div id="orgdd629ee" class="figure">
|
||||
<div id="orgf99890f" class="figure">
|
||||
<p><img src="figs/windowing_response_signal.png" alt="windowing_response_signal.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 14: </span>Window used for the response signals</p>
|
||||
@@ -895,9 +926,9 @@ The parameters are:
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org58ab5bf" class="outline-2">
|
||||
<h2 id="org58ab5bf"><span class="section-number-2">4</span> Force and Response signals</h2>
|
||||
<div class="outline-text-2" id="text-4">
|
||||
<div id="outline-container-org25e02aa" class="outline-2">
|
||||
<h2 id="org25e02aa"><span class="section-number-2">5</span> Force and Response signals</h2>
|
||||
<div class="outline-text-2" id="text-5">
|
||||
<p>
|
||||
Let's load some obtained data to look at the force and response signals.
|
||||
</p>
|
||||
@@ -908,33 +939,33 @@ Let's load some obtained data to look at the force and response signals.
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org2216b49" class="outline-3">
|
||||
<h3 id="org2216b49"><span class="section-number-3">4.1</span> Raw Force Data</h3>
|
||||
<div class="outline-text-3" id="text-4-1">
|
||||
<div id="outline-container-orgbf40925" class="outline-3">
|
||||
<h3 id="orgbf40925"><span class="section-number-3">5.1</span> Raw Force Data</h3>
|
||||
<div class="outline-text-3" id="text-5-1">
|
||||
<p>
|
||||
The force input for the first measurement is shown on figure <a href="#org22bc8af">15</a>. We can see 10 impacts, one zoom on one impact is shown on figure <a href="#org6947c62">16</a>.
|
||||
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>.
|
||||
</p>
|
||||
|
||||
<p>
|
||||
The Fourier transform of the force is shown on figure <a href="#org3431ecd">17</a>. This has been obtained without any windowing.
|
||||
The Fourier transform of the force is shown on figure <a href="#orga594684">17</a>. This has been obtained without any windowing.
|
||||
</p>
|
||||
|
||||
|
||||
<div id="org22bc8af" class="figure">
|
||||
<div id="org01af530" class="figure">
|
||||
<p><img src="figs/raw_data_force.png" alt="raw_data_force.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 15: </span>Raw Force Data from Hammer Blow</p>
|
||||
</div>
|
||||
|
||||
|
||||
<div id="org6947c62" class="figure">
|
||||
<div id="org90778c7" class="figure">
|
||||
<p><img src="figs/raw_data_force_zoom.png" alt="raw_data_force_zoom.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 16: </span>Raw Force Data from Hammer Blow - Zoom</p>
|
||||
</div>
|
||||
|
||||
|
||||
<div id="org3431ecd" class="figure">
|
||||
<div id="orga594684" class="figure">
|
||||
<p><img src="figs/fourier_transfor_force_impact.png" alt="fourier_transfor_force_impact.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 17: </span>Fourier Transform of the 10 force impacts for the first measurement</p>
|
||||
@@ -942,33 +973,33 @@ The Fourier transform of the force is shown on figure <a href="#org3431ecd">17</
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org91939ab" class="outline-3">
|
||||
<h3 id="org91939ab"><span class="section-number-3">4.2</span> Raw Response Data</h3>
|
||||
<div class="outline-text-3" id="text-4-2">
|
||||
<div id="outline-container-org2cf0a3e" class="outline-3">
|
||||
<h3 id="org2cf0a3e"><span class="section-number-3">5.2</span> Raw Response Data</h3>
|
||||
<div class="outline-text-3" id="text-5-2">
|
||||
<p>
|
||||
The response signal for the first measurement is shown on figure <a href="#orgf99383e">18</a>. One zoom on one response is shown on figure <a href="#orgb1fed61">19</a>.
|
||||
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>.
|
||||
</p>
|
||||
|
||||
<p>
|
||||
The Fourier transform of the response signals is shown on figure <a href="#org9e5f2eb">20</a>. This has been obtained without any windowing.
|
||||
The Fourier transform of the response signals is shown on figure <a href="#org716c433">20</a>. This has been obtained without any windowing.
|
||||
</p>
|
||||
|
||||
|
||||
<div id="orgf99383e" class="figure">
|
||||
<div id="org2612da5" class="figure">
|
||||
<p><img src="figs/raw_data_acceleration.png" alt="raw_data_acceleration.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 18: </span>Raw Acceleration Data from Accelerometer</p>
|
||||
</div>
|
||||
|
||||
|
||||
<div id="orgb1fed61" class="figure">
|
||||
<div id="org4bfecd1" class="figure">
|
||||
<p><img src="figs/raw_data_acceleration_zoom.png" alt="raw_data_acceleration_zoom.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 19: </span>Raw Acceleration Data from Accelerometer - Zoom</p>
|
||||
</div>
|
||||
|
||||
|
||||
<div id="org9e5f2eb" class="figure">
|
||||
<div id="org716c433" class="figure">
|
||||
<p><img src="figs/fourier_transform_response_signals.png" alt="fourier_transform_response_signals.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 20: </span>Fourier transform of the measured response signals</p>
|
||||
@@ -976,15 +1007,15 @@ The Fourier transform of the response signals is shown on figure <a href="#org9e
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgc34cf19" class="outline-3">
|
||||
<h3 id="orgc34cf19"><span class="section-number-3">4.3</span> Computation of one Frequency Response Function</h3>
|
||||
<div class="outline-text-3" id="text-4-3">
|
||||
<div id="outline-container-orgad3fa68" class="outline-3">
|
||||
<h3 id="orgad3fa68"><span class="section-number-3">5.3</span> Computation of one Frequency Response Function</h3>
|
||||
<div class="outline-text-3" id="text-5-3">
|
||||
<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.
|
||||
</p>
|
||||
|
||||
<p>
|
||||
We then compare the result obtained with the FRF computed by the modal software (figure <a href="#org01c1f1c">21</a>).
|
||||
We then compare the result obtained with the FRF computed by the modal software (figure <a href="#orgc57fff5">21</a>).
|
||||
</p>
|
||||
|
||||
<p>
|
||||
@@ -1001,7 +1032,7 @@ In the following analysis, FRF computed from the software will be used.
|
||||
</div>
|
||||
|
||||
|
||||
<div id="org01c1f1c" class="figure">
|
||||
<div id="orgc57fff5" class="figure">
|
||||
<p><img src="figs/frf_comparison_software.png" alt="frf_comparison_software.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 21: </span>Comparison of the computed FRF from the Fourier transform and using the modal software</p>
|
||||
@@ -1010,9 +1041,9 @@ In the following analysis, FRF computed from the software will be used.
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgc91b48e" class="outline-2">
|
||||
<h2 id="orgc91b48e"><span class="section-number-2">5</span> Frequency Response Functions and Coherence Results</h2>
|
||||
<div class="outline-text-2" id="text-5">
|
||||
<div id="outline-container-org4bffecd" class="outline-2">
|
||||
<h2 id="org4bffecd"><span class="section-number-2">6</span> Frequency Response Functions and Coherence Results</h2>
|
||||
<div class="outline-text-2" id="text-6">
|
||||
<p>
|
||||
Let's see one computed Frequency Response Function and one coherence in order to attest the quality of the measurement.
|
||||
</p>
|
||||
@@ -1026,22 +1057,22 @@ First, we load the data.
|
||||
</div>
|
||||
|
||||
<p>
|
||||
And we plot on figure <a href="#org2962423">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="#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.
|
||||
</p>
|
||||
|
||||
<p>
|
||||
The coherence associated is shown on figure <a href="#org2962423">22</a>.
|
||||
The coherence associated is shown on figure <a href="#org0618b41">22</a>.
|
||||
</p>
|
||||
|
||||
|
||||
<div id="org2962423" class="figure">
|
||||
<div id="org0618b41" class="figure">
|
||||
<p><img src="figs/frf_result_example.png" alt="frf_result_example.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 22: </span>Example of one measured FRF</p>
|
||||
</div>
|
||||
|
||||
|
||||
<div id="org46c579a" class="figure">
|
||||
<div id="org8a24269" class="figure">
|
||||
<p><img src="figs/coh_result_example.png" alt="coh_result_example.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 23: </span>Example of one measured Coherence</p>
|
||||
@@ -1049,9 +1080,12 @@ The coherence associated is shown on figure <a href="#org2962423">22</a>.
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org6c0132c" class="outline-2">
|
||||
<h2 id="org6c0132c"><span class="section-number-2">6</span> Generation of a FRF matrix and a Coherence matrix from the measurements</h2>
|
||||
<div class="outline-text-2" id="text-6">
|
||||
<div id="outline-container-org093edb6" 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>
|
||||
</div>
|
||||
<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>
|
||||
We want here to combine all the Frequency Response Functions measured into one big array called the <b>Frequency Response Matrix</b>.
|
||||
</p>
|
||||
@@ -1128,9 +1162,6 @@ COHs = zeros<span class="org-rainbow-delimiters-depth-1">(</span><span class="or
|
||||
meas_factor = meas_factor<span class="org-type">*</span><span class="org-rainbow-delimiters-depth-4">(</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-keyword">end</span>
|
||||
|
||||
% FRFs(acc_i<span class="org-type">+</span>n_acc<span class="org-type">*</span>(find(dirs<span class="org-type">==</span>meas_dir)<span class="org-type">-</span><span class="org-highlight-numbers-number">1</span>), find(dirs<span class="org-type">==</span>exc_dir), <span class="org-type">:</span>) = exc_factor<span class="org-type">*</span>meas_factor<span class="org-type">*</span>meas.(sprintf('FFT1_H1_<span class="org-comment">%i_1_Y_ReIm', j));</span>
|
||||
% COHs(acc_i<span class="org-type">+</span>n_acc<span class="org-type">*</span>(find(dirs<span class="org-type">==</span>meas_dir)<span class="org-type">-</span><span class="org-highlight-numbers-number">1</span>), find(dirs<span class="org-type">==</span>exc_dir), <span class="org-type">:</span>) = meas.(sprintf('FFT1_Coh_<span class="org-comment">%i_1_RMS_Y_Val', j));</span>
|
||||
|
||||
FRFs<span class="org-rainbow-delimiters-depth-4">(</span>find<span class="org-rainbow-delimiters-depth-5">(</span>dirs<span class="org-type">==</span>meas_dir<span class="org-rainbow-delimiters-depth-5">)</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-5">(</span>acc_i<span class="org-type">-</span><span class="org-highlight-numbers-number">1</span><span class="org-rainbow-delimiters-depth-5">)</span>, find<span class="org-rainbow-delimiters-depth-5">(</span>dirs<span class="org-type">==</span>exc_dir<span class="org-rainbow-delimiters-depth-5">)</span>, <span class="org-type">:</span><span class="org-rainbow-delimiters-depth-4">)</span> = exc_factor<span class="org-type">*</span>meas_factor<span class="org-type">*</span>meas.<span class="org-rainbow-delimiters-depth-4">(</span>sprintf<span class="org-rainbow-delimiters-depth-5">(</span><span class="org-string">'FFT1_H1_%i_1_Y_ReIm'</span>, <span class="org-constant">j</span>));
|
||||
COHs<span class="org-rainbow-delimiters-depth-6">(</span>find<span class="org-rainbow-delimiters-depth-7">(</span>dirs<span class="org-type">==</span>meas_dir<span class="org-rainbow-delimiters-depth-7">)</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-7">(</span>acc_i<span class="org-type">-</span><span class="org-highlight-numbers-number">1</span><span class="org-rainbow-delimiters-depth-7">)</span>, find<span class="org-rainbow-delimiters-depth-7">(</span>dirs<span class="org-type">==</span>exc_dir<span class="org-rainbow-delimiters-depth-7">)</span>, <span class="org-type">:</span><span class="org-rainbow-delimiters-depth-6">)</span> = meas.<span class="org-rainbow-delimiters-depth-6">(</span>sprintf<span class="org-rainbow-delimiters-depth-7">(</span><span class="org-string">'FFT1_Coh_%i_1_RMS_Y_Val'</span>, <span class="org-constant">j</span>));
|
||||
<span class="org-keyword">end</span>
|
||||
@@ -1145,13 +1176,140 @@ And we save the obtained FRF matrix and Coherence matrix in a <code>.mat</code>
|
||||
<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_coh_matrices.mat', 'FRFs', 'COHs', 'freqs'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<div id="outline-container-org8a40769" class="outline-2">
|
||||
<h2 id="org8a40769"><span class="section-number-2">9</span> Solid Bodies considered for further analysis</h2>
|
||||
<div class="outline-text-2" id="text-9">
|
||||
<p>
|
||||
We consider the following solid bodies for further analysis:
|
||||
</p>
|
||||
<ul class="org-ul">
|
||||
<li>Bottom Granite</li>
|
||||
<li>Top Granite</li>
|
||||
<li>Translation Stage</li>
|
||||
<li>Tilt Stage</li>
|
||||
<li>Spindle</li>
|
||||
<li>Hexapod</li>
|
||||
</ul>
|
||||
|
||||
<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>).
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">solids = <span class="org-rainbow-delimiters-depth-1">{}</span>;
|
||||
solids.gbot = <span class="org-rainbow-delimiters-depth-1">[</span><span class="org-highlight-numbers-number">17</span>, <span class="org-highlight-numbers-number">18</span>, <span class="org-highlight-numbers-number">19</span>, <span class="org-highlight-numbers-number">20</span><span class="org-rainbow-delimiters-depth-1">]</span>;
|
||||
solids.gtop = <span class="org-rainbow-delimiters-depth-1">[</span><span class="org-highlight-numbers-number">13</span>, <span class="org-highlight-numbers-number">14</span>, <span class="org-highlight-numbers-number">15</span>, <span class="org-highlight-numbers-number">16</span><span class="org-rainbow-delimiters-depth-1">]</span>;
|
||||
solids.ty = <span class="org-rainbow-delimiters-depth-1">[</span><span class="org-highlight-numbers-number">9</span>, <span class="org-highlight-numbers-number">10</span>, <span class="org-highlight-numbers-number">11</span>, <span class="org-highlight-numbers-number">12</span><span class="org-rainbow-delimiters-depth-1">]</span>;
|
||||
solids.ry = <span class="org-rainbow-delimiters-depth-1">[</span><span class="org-highlight-numbers-number">5</span>, <span class="org-highlight-numbers-number">6</span>, <span class="org-highlight-numbers-number">7</span>, <span class="org-highlight-numbers-number">8</span><span class="org-rainbow-delimiters-depth-1">]</span>;
|
||||
solids.rz = <span class="org-rainbow-delimiters-depth-1">[</span><span class="org-highlight-numbers-number">21</span>, <span class="org-highlight-numbers-number">22</span>, <span class="org-highlight-numbers-number">23</span><span class="org-rainbow-delimiters-depth-1">]</span>;
|
||||
solids.hexa = <span class="org-rainbow-delimiters-depth-1">[</span><span class="org-highlight-numbers-number">1</span>, <span class="org-highlight-numbers-number">2</span>, <span class="org-highlight-numbers-number">3</span>, <span class="org-highlight-numbers-number">4</span><span class="org-rainbow-delimiters-depth-1">]</span>;
|
||||
|
||||
solid_names = fields<span class="org-rainbow-delimiters-depth-1">(</span>solids<span class="org-rainbow-delimiters-depth-1">)</span>;
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<p>
|
||||
Finally, we save that into a <code>.mat</code> file.
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">save<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-org2405c81" class="outline-2">
|
||||
<h2 id="org2405c81"><span class="section-number-2">10</span> Notes the solid body assumption</h2>
|
||||
<div class="outline-text-2" id="text-10">
|
||||
<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.
|
||||
</p>
|
||||
|
||||
<p>
|
||||
This is illustrated on figure <a href="#org4627880">24</a>.
|
||||
</p>
|
||||
|
||||
|
||||
<div id="org4627880" class="figure">
|
||||
<p><img src="figs/aligned_accelerometers.png" alt="aligned_accelerometers.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 24: </span>Aligned measurement of the motion of a solid body</p>
|
||||
</div>
|
||||
|
||||
<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 \(\{O\}\).
|
||||
</p>
|
||||
|
||||
<p>
|
||||
The motions at points \(1\) and \(2\) are:
|
||||
</p>
|
||||
\begin{align*}
|
||||
v_1 &= v + \Omega \times p_1 \\
|
||||
v_2 &= v + \Omega \times p_2
|
||||
\end{align*}
|
||||
|
||||
<p>
|
||||
Taking only the \(x\) direction:
|
||||
</p>
|
||||
\begin{align*}
|
||||
v_{x1} &= v + \Omega_y p_{z1} - \Omega_z p_{y1} \\
|
||||
v_{x2} &= v + \Omega_y p_{z2} - \Omega_z p_{y2}
|
||||
\end{align*}
|
||||
|
||||
<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
|
||||
</p>
|
||||
\begin{equation}
|
||||
v_{x1} = v_{x2}
|
||||
\end{equation}
|
||||
|
||||
<div class="important">
|
||||
<p>
|
||||
Two sensors that are measuring the co-linear motion of a rigid body should measure the same quantity.
|
||||
</p>
|
||||
|
||||
</div>
|
||||
|
||||
<p>
|
||||
We can verify that the rigid body assumption is correct by comparing the measurement of the sensors.
|
||||
</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.
|
||||
</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>.
|
||||
</p>
|
||||
|
||||
|
||||
<div id="orgac307f7" class="figure">
|
||||
<p><img src="figs/compare_acc_x_dir.png" alt="compare_acc_x_dir.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 25: </span>Compare accelerometers align in the X direction</p>
|
||||
</div>
|
||||
|
||||
|
||||
|
||||
<div id="org3f01b7c" class="figure">
|
||||
<p><img src="figs/compare_acc_y_dir.png" alt="compare_acc_y_dir.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 26: </span>Compare accelerometers align in the Y direction</p>
|
||||
</div>
|
||||
|
||||
<div class="important">
|
||||
<p>
|
||||
|
||||
</p>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<div id="postamble" class="status">
|
||||
<p class="author">Author: Dehaeze Thomas</p>
|
||||
<p class="date">Created: 2019-07-03 mer. 17:24</p>
|
||||
<p class="date">Created: 2019-07-04 jeu. 17:49</p>
|
||||
<p class="validation"><a href="http://validator.w3.org/check?uri=referer">Validate</a></p>
|
||||
</div>
|
||||
</body>
|
||||
|
||||
Reference in New Issue
Block a user