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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>
<!-- 2021-04-16 ven. 18:28 --> <!-- 2021-04-19 lun. 15:02 -->
<meta http-equiv="Content-Type" content="text/html;charset=utf-8" /> <meta http-equiv="Content-Type" content="text/html;charset=utf-8" />
<title>Vibration Table</title> <title>Vibration Table</title>
<meta name="author" content="Dehaeze Thomas" /> <meta name="author" content="Dehaeze Thomas" />
@ -39,41 +39,41 @@
<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="#org2fecce4">1. Introduction</a></li> <li><a href="#orgf5ac764">1. Introduction</a></li>
<li><a href="#orga939dcb">2. Experimental Setup</a> <li><a href="#org5832600">2. Experimental Setup</a>
<ul> <ul>
<li><a href="#org7491bb1">2.1. CAD Model</a></li> <li><a href="#org088c0b5">2.1. CAD Model</a></li>
<li><a href="#orgf462a56">2.2. Instrumentation</a></li> <li><a href="#orge848908">2.2. Instrumentation</a></li>
<li><a href="#orgfb87c68">2.3. Suspended table</a></li> <li><a href="#orgae54ff5">2.3. Suspended table</a></li>
<li><a href="#org5f360e2">2.4. Inertial Sensors</a></li> <li><a href="#org435fee6">2.4. Inertial Sensors</a></li>
</ul> </ul>
</li> </li>
<li><a href="#orgcd95d84">3. Compute the 6DoF solid body motion from several inertial sensors</a> <li><a href="#orgf366ef9">3. Compute the 6DoF solid body motion from several inertial sensors</a>
<ul> <ul>
<li><a href="#org55b742b">3.1. Define accelerometers positions/orientations</a></li> <li><a href="#orgb03df82">3.1. Define accelerometers positions/orientations</a></li>
<li><a href="#org6a41e77">3.2. Transformation matrix from motion of the solid body to accelerometer measurements</a></li> <li><a href="#orgf872bb6">3.2. Transformation matrix from motion of the solid body to accelerometer measurements</a></li>
<li><a href="#org7548678">3.3. Compute the transformation matrix from accelerometer measurement to motion of the solid body</a></li> <li><a href="#orge54a621">3.3. Compute the transformation matrix from accelerometer measurement to motion of the solid body</a></li>
</ul> </ul>
</li> </li>
<li><a href="#org568ec42">4. Simscape Model</a> <li><a href="#orgae67fb9">4. Simscape Model</a>
<ul> <ul>
<li><a href="#org9dcca59">4.1. Simscape Sub-systems</a> <li><a href="#org83808e1">4.1. Simscape Sub-systems</a>
<ul> <ul>
<li><a href="#org5983163">4.1.1. Springs</a></li> <li><a href="#org430fcc4">4.1.1. Springs</a></li>
<li><a href="#orgda4ce0a">4.1.2. Inertial Shaker (IS20)</a></li> <li><a href="#org741008c">4.1.2. Inertial Shaker (IS20)</a></li>
<li><a href="#org3ec68ec">4.1.3. 3D accelerometer (356B18)</a></li> <li><a href="#org5b2f05e">4.1.3. 3D accelerometer (356B18)</a></li>
</ul> </ul>
</li> </li>
<li><a href="#org03afc98">4.2. Identification</a> <li><a href="#org37f2322">4.2. Identification</a>
<ul> <ul>
<li><a href="#org7eaac5b">4.2.1. Number of states</a></li> <li><a href="#org8ef7053">4.2.1. Number of states</a></li>
<li><a href="#org553cc8c">4.2.2. Resonance frequencies and mode shapes</a></li> <li><a href="#orgaedd6da">4.2.2. Resonance frequencies and mode shapes</a></li>
</ul> </ul>
</li> </li>
<li><a href="#org4c51a63">4.3. Verify transformation</a></li> <li><a href="#org0b133c2">4.3. Verify transformation</a></li>
</ul> </ul>
</li> </li>
<li><a href="#org630e0e9">5. Identification of the table&rsquo;s dynamics</a></li> <li><a href="#org71b8598">5. Identification of the table&rsquo;s dynamics</a></li>
</ul> </ul>
</div> </div>
</div> </div>
@ -81,33 +81,33 @@
<p>This report is also available as a <a href="./vibration-table.pdf">pdf</a>.</p> <p>This report is also available as a <a href="./vibration-table.pdf">pdf</a>.</p>
<hr> <hr>
<div id="outline-container-org2fecce4" class="outline-2"> <div id="outline-container-orgf5ac764" class="outline-2">
<h2 id="org2fecce4"><span class="section-number-2">1</span> Introduction</h2> <h2 id="orgf5ac764"><span class="section-number-2">1</span> Introduction</h2>
<div class="outline-text-2" id="text-1"> <div class="outline-text-2" id="text-1">
<p> <p>
This document is divided as follows: This document is divided as follows:
</p> </p>
<ul class="org-ul"> <ul class="org-ul">
<li>Section <a href="#orga82ad00">2</a>: the experimental setup and all the instrumentation are described</li> <li>Section <a href="#org6a14e7a">2</a>: the experimental setup and all the instrumentation are described</li>
<li>Section <a href="#org37dfa0b">3</a>: the mathematics used to compute the 6DoF motion of a solid body from several inertial sensor is derived</li> <li>Section <a href="#orga0537a2">3</a>: the mathematics used to compute the 6DoF motion of a solid body from several inertial sensor is derived</li>
<li>Section <a href="#orgc494ecb">4</a>: a Simscape model of the vibration table is developed</li> <li>Section <a href="#org2b706f6">4</a>: a Simscape model of the vibration table is developed</li>
<li>Section <a href="#org275ad8b">5</a>: the table dynamics is identified and compared with the Simscape model</li> <li>Section <a href="#org5f4bcf1">5</a>: the table dynamics is identified and compared with the Simscape model</li>
</ul> </ul>
</div> </div>
</div> </div>
<div id="outline-container-orga939dcb" class="outline-2"> <div id="outline-container-org5832600" class="outline-2">
<h2 id="orga939dcb"><span class="section-number-2">2</span> Experimental Setup</h2> <h2 id="org5832600"><span class="section-number-2">2</span> Experimental Setup</h2>
<div class="outline-text-2" id="text-2"> <div class="outline-text-2" id="text-2">
<p> <p>
<a id="orga82ad00"></a> <a id="org6a14e7a"></a>
</p> </p>
</div> </div>
<div id="outline-container-org7491bb1" class="outline-3"> <div id="outline-container-org088c0b5" class="outline-3">
<h3 id="org7491bb1"><span class="section-number-3">2.1</span> CAD Model</h3> <h3 id="org088c0b5"><span class="section-number-3">2.1</span> CAD Model</h3>
<div class="outline-text-3" id="text-2-1"> <div class="outline-text-3" id="text-2-1">
<div id="org3881e63" class="figure"> <div id="org8048a79" class="figure">
<p><img src="figs/vibration-table-cad-view.png" alt="vibration-table-cad-view.png" /> <p><img src="figs/vibration-table-cad-view.png" alt="vibration-table-cad-view.png" />
</p> </p>
<p><span class="figure-number">Figure 1: </span>CAD View of the vibration table</p> <p><span class="figure-number">Figure 1: </span>CAD View of the vibration table</p>
@ -115,10 +115,10 @@ This document is divided as follows:
</div> </div>
</div> </div>
<div id="outline-container-orgf462a56" class="outline-3"> <div id="outline-container-orge848908" class="outline-3">
<h3 id="orgf462a56"><span class="section-number-3">2.2</span> Instrumentation</h3> <h3 id="orge848908"><span class="section-number-3">2.2</span> Instrumentation</h3>
<div class="outline-text-3" id="text-2-2"> <div class="outline-text-3" id="text-2-2">
<div class="note" id="org198ea81"> <div class="note" id="orgbb293bd">
<p> <p>
Here are the documentation of the equipment used for this vibration table: Here are the documentation of the equipment used for this vibration table:
</p> </p>
@ -136,8 +136,8 @@ Here are the documentation of the equipment used for this vibration table:
</div> </div>
</div> </div>
<div id="outline-container-orgfb87c68" class="outline-3"> <div id="outline-container-orgae54ff5" class="outline-3">
<h3 id="orgfb87c68"><span class="section-number-3">2.3</span> Suspended table</h3> <h3 id="orgae54ff5"><span class="section-number-3">2.3</span> Suspended table</h3>
<div class="outline-text-3" id="text-2-3"> <div class="outline-text-3" id="text-2-3">
<dl class="org-dl"> <dl class="org-dl">
<dt>Dimensions</dt><dd>450 mm x 450 mm x 60 mm</dd> <dt>Dimensions</dt><dd>450 mm x 450 mm x 60 mm</dd>
@ -145,7 +145,7 @@ Here are the documentation of the equipment used for this vibration table:
</dl> </dl>
<div id="orga1448eb" class="figure"> <div id="org5bb879d" class="figure">
<p><img src="figs/B4545A_Compliance_inLb-780.png" alt="B4545A_Compliance_inLb-780.png" /> <p><img src="figs/B4545A_Compliance_inLb-780.png" alt="B4545A_Compliance_inLb-780.png" />
</p> </p>
<p><span class="figure-number">Figure 2: </span>Compliance of the B4545A optical table</p> <p><span class="figure-number">Figure 2: </span>Compliance of the B4545A optical table</p>
@ -153,8 +153,8 @@ Here are the documentation of the equipment used for this vibration table:
</div> </div>
</div> </div>
<div id="outline-container-org5f360e2" class="outline-3"> <div id="outline-container-org435fee6" class="outline-3">
<h3 id="org5f360e2"><span class="section-number-3">2.4</span> Inertial Sensors</h3> <h3 id="org435fee6"><span class="section-number-3">2.4</span> Inertial Sensors</h3>
<div class="outline-text-3" id="text-2-4"> <div class="outline-text-3" id="text-2-4">
<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides"> <table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
@ -181,18 +181,18 @@ Here are the documentation of the equipment used for this vibration table:
</div> </div>
</div> </div>
<div id="outline-container-orgcd95d84" class="outline-2"> <div id="outline-container-orgf366ef9" class="outline-2">
<h2 id="orgcd95d84"><span class="section-number-2">3</span> Compute the 6DoF solid body motion from several inertial sensors</h2> <h2 id="orgf366ef9"><span class="section-number-2">3</span> Compute the 6DoF solid body motion from several inertial sensors</h2>
<div class="outline-text-2" id="text-3"> <div class="outline-text-2" id="text-3">
<p> <p>
<a id="org37dfa0b"></a> <a id="orga0537a2"></a>
</p> </p>
<p> <p>
Let&rsquo;s consider a solid body with several accelerometers attached to it (Figure <a href="#orgdf3ed42">3</a>). Let&rsquo;s consider a solid body with several accelerometers attached to it (Figure <a href="#orgd7a7adf">3</a>).
</p> </p>
<div id="orgdf3ed42" class="figure"> <div id="orgd7a7adf" class="figure">
<p><img src="figs/local_to_global_coordinates.png" alt="local_to_global_coordinates.png" /> <p><img src="figs/local_to_global_coordinates.png" alt="local_to_global_coordinates.png" />
</p> </p>
<p><span class="figure-number">Figure 3: </span>Schematic of the measured motions of a solid body</p> <p><span class="figure-number">Figure 3: </span>Schematic of the measured motions of a solid body</p>
@ -222,7 +222,7 @@ The measurement of the individual vectors is defined as the vector \(\vec{a}\):
\end{equation} \end{equation}
<p> <p>
From the positions and orientations of the acceleremoters (defined in Section <a href="#org0c6bf9c">3.1</a>), it is quite straightforward to compute the accelerations measured by the sensors from the acceleration/angular acceleration of the solid body (Section <a href="#org2d30f57">3.2</a>). From the positions and orientations of the acceleremoters (defined in Section <a href="#org0125a45">3.1</a>), it is quite straightforward to compute the accelerations measured by the sensors from the acceleration/angular acceleration of the solid body (Section <a href="#orgb9fd993">3.2</a>).
From this, we can easily build a transformation matrix \(M\), such that: From this, we can easily build a transformation matrix \(M\), such that:
</p> </p>
\begin{equation} \begin{equation}
@ -230,7 +230,7 @@ From this, we can easily build a transformation matrix \(M\), such that:
\end{equation} \end{equation}
<p> <p>
If the matrix is invertible, we can just take the inverse in order to obtain the transformation matrix giving the 6dof acceleration of the solid body from the accelerometer measurements (Section <a href="#orgf420152">3.3</a>): If the matrix is invertible, we can just take the inverse in order to obtain the transformation matrix giving the 6dof acceleration of the solid body from the accelerometer measurements (Section <a href="#orgdaf30e9">3.3</a>):
</p> </p>
\begin{equation} \begin{equation}
{}^O\vec{x} = M^{-1} \cdot \vec{a} {}^O\vec{x} = M^{-1} \cdot \vec{a}
@ -241,11 +241,11 @@ If it is not invertible, then it means that it is not possible to compute all 6d
The solution is then to change the location/orientation of some of the accelerometers. The solution is then to change the location/orientation of some of the accelerometers.
</p> </p>
</div> </div>
<div id="outline-container-org55b742b" class="outline-3"> <div id="outline-container-orgb03df82" class="outline-3">
<h3 id="org55b742b"><span class="section-number-3">3.1</span> Define accelerometers positions/orientations</h3> <h3 id="orgb03df82"><span class="section-number-3">3.1</span> Define accelerometers positions/orientations</h3>
<div class="outline-text-3" id="text-3-1"> <div class="outline-text-3" id="text-3-1">
<p> <p>
<a id="org0c6bf9c"></a> <a id="org0125a45"></a>
Let&rsquo;s first define the position and orientation of all measured accelerations with respect to a defined frame \(\{O\}\). Let&rsquo;s first define the position and orientation of all measured accelerations with respect to a defined frame \(\{O\}\).
</p> </p>
@ -260,10 +260,10 @@ Let&rsquo;s first define the position and orientation of all measured accelerati
</div> </div>
<p> <p>
There are summarized in Table <a href="#org36e16bb">1</a>. There are summarized in Table <a href="#orgdf64746">1</a>.
</p> </p>
<table id="org36e16bb" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides"> <table id="orgdf64746" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
<caption class="t-above"><span class="table-number">Table 1:</span> Positions of the accelerometers fixed to the vibration table with respect to \(\{O\}\)</caption> <caption class="t-above"><span class="table-number">Table 1:</span> Positions of the accelerometers fixed to the vibration table with respect to \(\{O\}\)</caption>
<colgroup> <colgroup>
@ -339,10 +339,10 @@ We then define the direction of the measured accelerations (unit vectors):
</div> </div>
<p> <p>
They are summarized in Table <a href="#org33ec6f3">2</a>. They are summarized in Table <a href="#org13c9bf3">2</a>.
</p> </p>
<table id="org33ec6f3" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides"> <table id="org13c9bf3" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
<caption class="t-above"><span class="table-number">Table 2:</span> Orientations of the accelerometers fixed to the vibration table expressed in \(\{O\}\)</caption> <caption class="t-above"><span class="table-number">Table 2:</span> Orientations of the accelerometers fixed to the vibration table expressed in \(\{O\}\)</caption>
<colgroup> <colgroup>
@ -406,11 +406,11 @@ They are summarized in Table <a href="#org33ec6f3">2</a>.
</div> </div>
</div> </div>
<div id="outline-container-org6a41e77" class="outline-3"> <div id="outline-container-orgf872bb6" class="outline-3">
<h3 id="org6a41e77"><span class="section-number-3">3.2</span> Transformation matrix from motion of the solid body to accelerometer measurements</h3> <h3 id="orgf872bb6"><span class="section-number-3">3.2</span> Transformation matrix from motion of the solid body to accelerometer measurements</h3>
<div class="outline-text-3" id="text-3-2"> <div class="outline-text-3" id="text-3-2">
<p> <p>
<a id="org2d30f57"></a> <a id="orgb9fd993"></a>
</p> </p>
<p> <p>
@ -475,7 +475,7 @@ a_i = \begin{bmatrix}
And finally we can combine the 6 (line) vectors for the 6 accelerometers to write that in a matrix form. And finally we can combine the 6 (line) vectors for the 6 accelerometers to write that in a matrix form.
We obtain Eq. \eqref{eq:M_matrix}. We obtain Eq. \eqref{eq:M_matrix}.
</p> </p>
<div class="important" id="org1060d47"> <div class="important" id="org1f576e5">
<p> <p>
The transformation from solid body acceleration \({}^O\vec{x}\) from sensor measured acceleration \(\vec{a}\) is: The transformation from solid body acceleration \({}^O\vec{x}\) from sensor measured acceleration \(\vec{a}\) is:
</p> </p>
@ -510,10 +510,10 @@ Let&rsquo;s define such matrix using matlab:
</div> </div>
<p> <p>
The obtained matrix is shown in Table <a href="#org5d00f9c">3</a>. The obtained matrix is shown in Table <a href="#orgb7789db">3</a>.
</p> </p>
<table id="org5d00f9c" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides"> <table id="orgb7789db" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
<caption class="t-above"><span class="table-number">Table 3:</span> Effect of a displacement/rotation on the 6 measurements</caption> <caption class="t-above"><span class="table-number">Table 3:</span> Effect of a displacement/rotation on the 6 measurements</caption>
<colgroup> <colgroup>
@ -607,11 +607,11 @@ The obtained matrix is shown in Table <a href="#org5d00f9c">3</a>.
</div> </div>
</div> </div>
<div id="outline-container-org7548678" class="outline-3"> <div id="outline-container-orge54a621" class="outline-3">
<h3 id="org7548678"><span class="section-number-3">3.3</span> Compute the transformation matrix from accelerometer measurement to motion of the solid body</h3> <h3 id="orge54a621"><span class="section-number-3">3.3</span> Compute the transformation matrix from accelerometer measurement to motion of the solid body</h3>
<div class="outline-text-3" id="text-3-3"> <div class="outline-text-3" id="text-3-3">
<p> <p>
<a id="orgf420152"></a> <a id="orgdaf30e9"></a>
</p> </p>
<p> <p>
@ -630,10 +630,10 @@ We therefore need the determinant of \(M\) to be non zero:
</div> </div>
<p> <p>
The obtained inverse of the matrix is shown in Table <a href="#org322aa13">4</a>. The obtained inverse of the matrix is shown in Table <a href="#orgb3868cf">4</a>.
</p> </p>
<table id="org322aa13" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides"> <table id="orgb3868cf" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
<caption class="t-above"><span class="table-number">Table 4:</span> Compute the displacement/rotation from the 6 measurements</caption> <caption class="t-above"><span class="table-number">Table 4:</span> Compute the displacement/rotation from the 6 measurements</caption>
<colgroup> <colgroup>
@ -728,28 +728,28 @@ The obtained inverse of the matrix is shown in Table <a href="#org322aa13">4</a>
</div> </div>
</div> </div>
<div id="outline-container-org568ec42" class="outline-2"> <div id="outline-container-orgae67fb9" class="outline-2">
<h2 id="org568ec42"><span class="section-number-2">4</span> Simscape Model</h2> <h2 id="orgae67fb9"><span class="section-number-2">4</span> Simscape Model</h2>
<div class="outline-text-2" id="text-4"> <div class="outline-text-2" id="text-4">
<p> <p>
<a id="orgc494ecb"></a> <a id="org2b706f6"></a>
</p> </p>
<p> <p>
In this section, the Simscape model of the vibration table is described. In this section, the Simscape model of the vibration table is described.
</p> </p>
<div id="org1d7efeb" class="figure"> <div id="org79ee770" class="figure">
<p><img src="figs/simscape_vibration_table.png" alt="simscape_vibration_table.png" /> <p><img src="figs/simscape_vibration_table.png" alt="simscape_vibration_table.png" />
</p> </p>
<p><span class="figure-number">Figure 4: </span>3D representation of the simscape model</p> <p><span class="figure-number">Figure 4: </span>3D representation of the simscape model</p>
</div> </div>
</div> </div>
<div id="outline-container-org9dcca59" class="outline-3"> <div id="outline-container-org83808e1" class="outline-3">
<h3 id="org9dcca59"><span class="section-number-3">4.1</span> Simscape Sub-systems</h3> <h3 id="org83808e1"><span class="section-number-3">4.1</span> Simscape Sub-systems</h3>
<div class="outline-text-3" id="text-4-1"> <div class="outline-text-3" id="text-4-1">
<p> <p>
<a id="org69420ab"></a> <a id="orgc93c68c"></a>
</p> </p>
<p> <p>
@ -757,11 +757,11 @@ Parameters for sub-components of the simscape model are defined below.
</p> </p>
</div> </div>
<div id="outline-container-org5983163" class="outline-4"> <div id="outline-container-org430fcc4" class="outline-4">
<h4 id="org5983163"><span class="section-number-4">4.1.1</span> Springs</h4> <h4 id="org430fcc4"><span class="section-number-4">4.1.1</span> Springs</h4>
<div class="outline-text-4" id="text-4-1-1"> <div class="outline-text-4" id="text-4-1-1">
<p> <p>
<a id="org7d51a21"></a> <a id="org4ce6c6c"></a>
</p> </p>
<p> <p>
@ -781,14 +781,22 @@ spring.cz = 1e1; <span class="org-comment">% Z- Damping [N/(m/s)]</span>
spring.z0 = 32e<span class="org-type">-</span>3; <span class="org-comment">% Equilibrium z-length [m]</span> spring.z0 = 32e<span class="org-type">-</span>3; <span class="org-comment">% Equilibrium z-length [m]</span>
</pre> </pre>
</div> </div>
<p>
And we can increase the &ldquo;equilibrium position&rdquo; of the vertical springs to take into account the gravity forces and start closer to equilibrium:
</p>
<div class="org-src-container">
<pre class="src src-matlab">spring.dl = (30.5918<span class="org-type">/</span>4)<span class="org-type">*</span>9.80665<span class="org-type">/</span>spring.kz;
</pre>
</div>
</div> </div>
</div> </div>
<div id="outline-container-orgda4ce0a" class="outline-4"> <div id="outline-container-org741008c" class="outline-4">
<h4 id="orgda4ce0a"><span class="section-number-4">4.1.2</span> Inertial Shaker (IS20)</h4> <h4 id="org741008c"><span class="section-number-4">4.1.2</span> Inertial Shaker (IS20)</h4>
<div class="outline-text-4" id="text-4-1-2"> <div class="outline-text-4" id="text-4-1-2">
<p> <p>
<a id="org64d06b0"></a> <a id="org65750ec"></a>
</p> </p>
<p> <p>
@ -804,7 +812,7 @@ The inertial mass is guided inside the housing and an actuator (coil and magnet)
The &ldquo;reacting&rdquo; force on the support is then used as an excitation. The &ldquo;reacting&rdquo; force on the support is then used as an excitation.
</p> </p>
<table id="org13c1e94" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides"> <table id="orgbcb82ac" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
<caption class="t-above"><span class="table-number">Table 5:</span> Summary of the IS20 datasheet</caption> <caption class="t-above"><span class="table-number">Table 5:</span> Summary of the IS20 datasheet</caption>
<colgroup> <colgroup>
@ -842,7 +850,7 @@ The &ldquo;reacting&rdquo; force on the support is then used as an excitation.
</table> </table>
<p> <p>
From the datasheet in Table <a href="#org13c1e94">5</a>, we can estimate the parameters of the physical shaker. From the datasheet in Table <a href="#orgbcb82ac">5</a>, we can estimate the parameters of the physical shaker.
</p> </p>
<p> <p>
@ -859,11 +867,11 @@ shaker.c = 0.2<span class="org-type">*</span>sqrt(shaker.k<span class="org-type"
</div> </div>
</div> </div>
<div id="outline-container-org3ec68ec" class="outline-4"> <div id="outline-container-org5b2f05e" class="outline-4">
<h4 id="org3ec68ec"><span class="section-number-4">4.1.3</span> 3D accelerometer (356B18)</h4> <h4 id="org5b2f05e"><span class="section-number-4">4.1.3</span> 3D accelerometer (356B18)</h4>
<div class="outline-text-4" id="text-4-1-3"> <div class="outline-text-4" id="text-4-1-3">
<p> <p>
<a id="org55e06a7"></a> <a id="org23e3d96"></a>
</p> </p>
<p> <p>
@ -878,7 +886,7 @@ An accelerometer consists of 2 solids:
The relative motion between the housing and the inertial mass gives a measurement of the acceleration of the measured body (up to the suspension mode of the inertial mass). The relative motion between the housing and the inertial mass gives a measurement of the acceleration of the measured body (up to the suspension mode of the inertial mass).
</p> </p>
<table id="orgc0a070c" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides"> <table id="org60e803f" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
<caption class="t-above"><span class="table-number">Table 6:</span> Summary of the 356B18 datasheet</caption> <caption class="t-above"><span class="table-number">Table 6:</span> Summary of the 356B18 datasheet</caption>
<colgroup> <colgroup>
@ -982,16 +990,16 @@ The accelerometer model can be chosen by setting the <code>type</code> property:
</div> </div>
</div> </div>
<div id="outline-container-org03afc98" class="outline-3"> <div id="outline-container-org37f2322" class="outline-3">
<h3 id="org03afc98"><span class="section-number-3">4.2</span> Identification</h3> <h3 id="org37f2322"><span class="section-number-3">4.2</span> Identification</h3>
<div class="outline-text-3" id="text-4-2"> <div class="outline-text-3" id="text-4-2">
<p> <p>
<a id="org089a338"></a> <a id="org5252ce8"></a>
</p> </p>
</div> </div>
<div id="outline-container-org7eaac5b" class="outline-4"> <div id="outline-container-org8ef7053" class="outline-4">
<h4 id="org7eaac5b"><span class="section-number-4">4.2.1</span> Number of states</h4> <h4 id="org8ef7053"><span class="section-number-4">4.2.1</span> Number of states</h4>
<div class="outline-text-4" id="text-4-2-1"> <div class="outline-text-4" id="text-4-2-1">
<p> <p>
Let&rsquo;s first use perfect 3d accelerometers: Let&rsquo;s first use perfect 3d accelerometers:
@ -1067,8 +1075,8 @@ This corresponds to 6 states for each triaxial accelerometers.
</div> </div>
</div> </div>
<div id="outline-container-org553cc8c" class="outline-4"> <div id="outline-container-orgaedd6da" class="outline-4">
<h4 id="org553cc8c"><span class="section-number-4">4.2.2</span> Resonance frequencies and mode shapes</h4> <h4 id="orgaedd6da"><span class="section-number-4">4.2.2</span> Resonance frequencies and mode shapes</h4>
<div class="outline-text-4" id="text-4-2-2"> <div class="outline-text-4" id="text-4-2-2">
<p> <p>
Let&rsquo;s now identify the resonance frequency and mode shapes associated with the suspension modes of the optical table. Let&rsquo;s now identify the resonance frequency and mode shapes associated with the suspension modes of the optical table.
@ -1115,11 +1123,11 @@ And the associated response of the optical table
</div> </div>
<p> <p>
The results are shown in Table <a href="#orgfc9b1dd">7</a>. The results are shown in Table <a href="#org7e240b1">7</a>.
The motion associated to the mode shapes are just indicative. The motion associated to the mode shapes are just indicative.
</p> </p>
<table id="orgfc9b1dd" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides"> <table id="org7e240b1" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
<caption class="t-above"><span class="table-number">Table 7:</span> Resonance frequency and approximation of the mode shapes</caption> <caption class="t-above"><span class="table-number">Table 7:</span> Resonance frequency and approximation of the mode shapes</caption>
<colgroup> <colgroup>
@ -1214,8 +1222,8 @@ The motion associated to the mode shapes are just indicative.
</div> </div>
</div> </div>
<div id="outline-container-org4c51a63" class="outline-3"> <div id="outline-container-org0b133c2" class="outline-3">
<h3 id="org4c51a63"><span class="section-number-3">4.3</span> Verify transformation</h3> <h3 id="org0b133c2"><span class="section-number-3">4.3</span> Verify transformation</h3>
<div class="outline-text-3" id="text-4-3"> <div class="outline-text-3" id="text-4-3">
<div class="org-src-container"> <div class="org-src-container">
<pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Options for Linearized</span></span> <pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Options for Linearized</span></span>
@ -1229,7 +1237,7 @@ mdl = <span class="org-string">'vibration_table'</span>;
clear io; io_i = 1; clear io; io_i = 1;
io(io_i) = linio([mdl, <span class="org-string">'/F'</span>], 1, <span class="org-string">'openinput'</span>); io_i = io_i <span class="org-type">+</span> 1; io(io_i) = linio([mdl, <span class="org-string">'/F'</span>], 1, <span class="org-string">'openinput'</span>); io_i = io_i <span class="org-type">+</span> 1;
io(io_i) = linio([mdl, <span class="org-string">'/acc'</span>], 1, <span class="org-string">'openoutput'</span>); io_i = io_i <span class="org-type">+</span> 1; io(io_i) = linio([mdl, <span class="org-string">'/acc'</span>], 1, <span class="org-string">'openoutput'</span>); io_i = io_i <span class="org-type">+</span> 1;
io(io_i) = linio([mdl, <span class="org-string">'/Absolute_Accelerometer'</span>], 1, <span class="org-string">'openoutput'</span>); io_i = io_i <span class="org-type">+</span> 1; io(io_i) = linio([mdl, <span class="org-string">'/acc_O'</span>], 1, <span class="org-string">'openoutput'</span>); io_i = io_i <span class="org-type">+</span> 1;
<span class="org-matlab-cellbreak"><span class="org-comment">%% Run the linearization</span></span> <span class="org-matlab-cellbreak"><span class="org-comment">%% Run the linearization</span></span>
G = linearize(mdl, io, 0.0, options); G = linearize(mdl, io, 0.0, options);
@ -1258,18 +1266,18 @@ bodeFig({G_acc(6), G_id(6)})
</div> </div>
</div> </div>
<div id="outline-container-org630e0e9" class="outline-2"> <div id="outline-container-org71b8598" class="outline-2">
<h2 id="org630e0e9"><span class="section-number-2">5</span> Identification of the table&rsquo;s dynamics</h2> <h2 id="org71b8598"><span class="section-number-2">5</span> Identification of the table&rsquo;s dynamics</h2>
<div class="outline-text-2" id="text-5"> <div class="outline-text-2" id="text-5">
<p> <p>
<a id="org275ad8b"></a> <a id="org5f4bcf1"></a>
</p> </p>
</div> </div>
</div> </div>
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Dehaeze Thomas</p> <p class="author">Author: Dehaeze Thomas</p>
<p class="date">Created: 2021-04-16 ven. 18:28</p> <p class="date">Created: 2021-04-19 lun. 15:02</p>
</div> </div>
</body> </body>
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@ -456,6 +456,11 @@ spring.cz = 1e1; % Z- Damping [N/(m/s)]
spring.z0 = 32e-3; % Equilibrium z-length [m] spring.z0 = 32e-3; % Equilibrium z-length [m]
#+end_src #+end_src
And we can increase the "equilibrium position" of the vertical springs to take into account the gravity forces and start closer to equilibrium:
#+begin_src matlab
spring.dl = (30.5918/4)*9.80665/spring.kz;
#+end_src
*** Inertial Shaker (IS20) *** Inertial Shaker (IS20)
<<sec:simscape_inertial_shaker>> <<sec:simscape_inertial_shaker>>

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