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Thomas Dehaeze
2020-03-26 17:25:43 +01:00
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"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
<html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en">
<head>
<!-- 2020-03-25 mer. 19:20 -->
<!-- 2020-03-26 jeu. 17:24 -->
<meta http-equiv="Content-Type" content="text/html;charset=utf-8" />
<meta name="viewport" content="width=device-width, initial-scale=1" />
<title>Effect of Uncertainty on the support&rsquo;s dynamics on the isolation platform dynamics</title>
@@ -271,17 +271,17 @@ for the JavaScript code in this tag.
<ul>
<li><a href="#orgbe6e0b8">1. Simple Introductory Example</a>
<ul>
<li><a href="#orgf5c9fe3">1.1. Equations of motion</a></li>
<li><a href="#org440a84d">1.1. Equations of motion</a></li>
<li><a href="#org8bd2a4a">1.2. Initialization of the support dynamics</a></li>
<li><a href="#orgefb9b71">1.3. Initialization of the isolation platform</a></li>
<li><a href="#org3bc4ad1">1.4. Comparison</a></li>
<li><a href="#org2200b2c">1.5. Conclusion</a></li>
<li><a href="#orgc2af076">1.5. Conclusion</a></li>
</ul>
</li>
<li><a href="#orge1d3484">2. Generalization to arbitrary dynamics</a>
<ul>
<li><a href="#org3948d1f">2.1. Introduction</a></li>
<li><a href="#org96daf87">2.2. Equations of motion</a></li>
<li><a href="#org283e8c3">2.2. Equations of motion</a></li>
<li><a href="#orgc20cabb">2.3. Compliance of the Support</a></li>
<li><a href="#orgf1c8c33">2.4. Effect of the Isolation platform Stiffness.</a></li>
<li><a href="#org67810a4">2.5. Equivalent Inverse Multiplicative Uncertainty</a></li>
@@ -292,7 +292,7 @@ for the JavaScript code in this tag.
<li><a href="#orgd2fc303">2.6.3. Effect of the platform&rsquo;s mass \(m\)</a></li>
</ul>
</li>
<li><a href="#org9c46c2a">2.7. Conclusion</a></li>
<li><a href="#orgd28ebd8">2.7. Conclusion</a></li>
</ul>
</li>
</ul>
@@ -346,8 +346,8 @@ The goal is to stabilize \(x\) using \(F\) in spite of uncertainty on the suppor
</div>
</div>
<div id="outline-container-orgf5c9fe3" class="outline-3">
<h3 id="orgf5c9fe3"><span class="section-number-3">1.1</span> Equations of motion</h3>
<div id="outline-container-org440a84d" class="outline-3">
<h3 id="org440a84d"><span class="section-number-3">1.1</span> Equations of motion</h3>
<div class="outline-text-3" id="text-1-1">
<p>
If we write the equation of motion of the system in Figure <a href="#org41bc770">1</a>, we obtain:
@@ -451,12 +451,12 @@ The obtained dynamics from \(F\) to \(x\) for the three isolation platform are s
</div>
</div>
<div id="outline-container-org2200b2c" class="outline-3">
<h3 id="org2200b2c"><span class="section-number-3">1.5</span> Conclusion</h3>
<div id="outline-container-orgc2af076" class="outline-3">
<h3 id="orgc2af076"><span class="section-number-3">1.5</span> Conclusion</h3>
<div class="outline-text-3" id="text-1-5">
<div class="important">
<p>
The soft platform dynamics does not seems to depend on the dynamics of the support.
The soft platform dynamics does not seems to depend on the dynamics of the support nor to be affect by the dynamic uncertainty of the support.
</p>
</div>
@@ -497,8 +497,8 @@ Now let&rsquo;s consider the system consisting of a mass-spring-system (the isol
</div>
</div>
<div id="outline-container-org96daf87" class="outline-3">
<h3 id="org96daf87"><span class="section-number-3">2.2</span> Equations of motion</h3>
<div id="outline-container-org283e8c3" class="outline-3">
<h3 id="org283e8c3"><span class="section-number-3">2.2</span> Equations of motion</h3>
<div class="outline-text-3" id="text-2-2">
<p>
We have to following equations of motion:
@@ -674,6 +674,7 @@ G_stiff = 1<span class="org-type">/</span>(m<span class="org-type">*</span>s<spa
The obtained transfer functions \(x/F\) for each of the three platforms are shown in Figure <a href="#org89aa89f">8</a>.
</p>
<div id="org89aa89f" class="figure">
<p><img src="figs/plant_uncertainty_stiffness_isolator.png" alt="plant_uncertainty_stiffness_isolator.png" />
</p>
@@ -699,10 +700,10 @@ Let&rsquo;s express the uncertainty of the plant \(x/F\) as a function of the pa
&= \frac{1}{ms^2 + cs + k + ms^2(cs + k)G_0^\prime(s)} \cdot \frac{1}{1 + \frac{ms^2(cs + k)G_0^\prime(s) w_I(s)}{ms^2 + cs + k + ms^2(cs + k)G_0^\prime(s)} \Delta(s)}\\
\end{align*}
<p>
We can rewrite that as an inverse multiplicative uncertainty (Figure <a href="#orge738173">9</a>):
</p>
<div class="important">
<p>
We can the plant dynamics that as an inverse multiplicative uncertainty (Figure <a href="#orge738173">9</a>):
</p>
\begin{equation}
\frac{x}{F} = G_0(s) (1 + w_{iI}(s) \Delta(s))^{-1}
\end{equation}
@@ -827,9 +828,10 @@ Let&rsquo;s fix \(k = 10^7\ [N/m]\), \(\xi = \frac{c}{2\sqrt{km}} = 0.1\) and se
</div>
</div>
<div id="outline-container-org9c46c2a" class="outline-3">
<h3 id="org9c46c2a"><span class="section-number-3">2.7</span> Conclusion</h3>
<div id="outline-container-orgd28ebd8" class="outline-3">
<h3 id="orgd28ebd8"><span class="section-number-3">2.7</span> Conclusion</h3>
<div class="outline-text-3" id="text-2-7">
<div class="important">
<p>
If the goal is to have an acceptable (\(<10\%\)) uncertainty on the plant until the highest frequency, two design choice for the isolation platform are possible:
</p>
@@ -847,13 +849,15 @@ If a very stiff isolation platform is used, the uncertainty will be high around
It will then be high around \(\omega_0\) and probably be higher than one.
Thus, if a stiff isolation platform is used, the recommendation is to have the largest possible resonance frequency, as the control bandwidth will be limited by the first resonance of the isolation platform (if not already limited by the resonance of the support).
</p>
</div>
</div>
</div>
</div>
</div>
<div id="postamble" class="status">
<p class="author">Author: Dehaeze Thomas</p>
<p class="date">Created: 2020-03-25 mer. 19:20</p>
<p class="date">Created: 2020-03-26 jeu. 17:24</p>
</div>
</body>
</html>