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Thomas Dehaeze
2020-04-01 16:17:26 +02:00
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@@ -4,7 +4,7 @@
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<!-- 2020-03-26 jeu. 17:24 -->
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<title>Effect of Uncertainty on the support&rsquo;s dynamics on the isolation platform dynamics</title>
@@ -202,50 +202,28 @@
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<ul>
<li><a href="#orgbe6e0b8">1. Simple Introductory Example</a>
<ul>
<li><a href="#org440a84d">1.1. Equations of motion</a></li>
<li><a href="#org283e8c3">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="#orgc2af076">1.5. Conclusion</a></li>
<li><a href="#orgd28ebd8">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="#org283e8c3">2.2. Equations of motion</a></li>
<li><a href="#org28c48e3">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>
<li><a href="#org67810a4">2.4. Equivalent Inverse Multiplicative Uncertainty</a></li>
<li><a href="#orge950395">2.5. Effect of the Isolation platform Stiffness</a></li>
<li><a href="#org6967854">2.6. Reduce the Uncertainty on the plant</a>
<ul>
<li><a href="#orgafebadd">2.6.1. Effect of the platform&rsquo;s stiffness \(k\)</a></li>
@@ -292,7 +270,7 @@ 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="#orgd28ebd8">2.7. Conclusion</a></li>
<li><a href="#org718f0f1">2.7. Conclusion</a></li>
</ul>
</li>
</ul>
@@ -346,8 +324,8 @@ The goal is to stabilize \(x\) using \(F\) in spite of uncertainty on the suppor
</div>
</div>
<div id="outline-container-org440a84d" class="outline-3">
<h3 id="org440a84d"><span class="section-number-3">1.1</span> Equations of motion</h3>
<div id="outline-container-org283e8c3" class="outline-3">
<h3 id="org283e8c3"><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:
@@ -451,8 +429,8 @@ The obtained dynamics from \(F\) to \(x\) for the three isolation platform are s
</div>
</div>
<div id="outline-container-orgc2af076" class="outline-3">
<h3 id="orgc2af076"><span class="section-number-3">1.5</span> Conclusion</h3>
<div id="outline-container-orgd28ebd8" class="outline-3">
<h3 id="orgd28ebd8"><span class="section-number-3">1.5</span> Conclusion</h3>
<div class="outline-text-3" id="text-1-5">
<div class="important">
<p>
@@ -497,8 +475,8 @@ Now let&rsquo;s consider the system consisting of a mass-spring-system (the isol
</div>
</div>
<div id="outline-container-org283e8c3" class="outline-3">
<h3 id="org283e8c3"><span class="section-number-3">2.2</span> Equations of motion</h3>
<div id="outline-container-org28c48e3" class="outline-3">
<h3 id="org28c48e3"><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:
@@ -581,7 +559,7 @@ The parameters are defined below.
</p>
<div class="org-src-container">
<pre class="src src-matlab">r0 = 0.5;
tau = 1<span class="org-type">/</span>(50<span class="org-type">*</span>2<span class="org-type">*</span><span class="org-constant">pi</span>);
tau = 1<span class="org-type">/</span>(100<span class="org-type">*</span>2<span class="org-type">*</span><span class="org-constant">pi</span>);
rinf = 10;
wI = (tau<span class="org-type">*</span>s <span class="org-type">+</span> r0)<span class="org-type">/</span>((tau<span class="org-type">/</span>rinf)<span class="org-type">*</span>s <span class="org-type">+</span> 1);
@@ -617,10 +595,48 @@ A set of uncertainty support&rsquo;s compliance transfer functions is shown in F
</div>
</div>
<div id="outline-container-orgf1c8c33" class="outline-3">
<h3 id="orgf1c8c33"><span class="section-number-3">2.4</span> Effect of the Isolation platform Stiffness.</h3>
<div id="outline-container-org67810a4" class="outline-3">
<h3 id="org67810a4"><span class="section-number-3">2.4</span> Equivalent Inverse Multiplicative Uncertainty</h3>
<div class="outline-text-3" id="text-2-4">
<p>
Let&rsquo;s express the uncertainty of the plant \(x/F\) as a function of the parameters as well as of the uncertainty on the platform&rsquo;s compliance:
</p>
\begin{align*}
\frac{x}{F} &= \frac{1}{ms^2 + cs + k + ms^2(cs + k)G_0^\prime(s)(1 + w_I(s)\Delta(s))}\\
&= \frac{1}{ms^2 + cs + k + ms^2(cs + k)G_0^\prime(s) + ms^2(cs + k)G_0^\prime(s) w_I(s)\Delta(s)}\\
&= \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*}
<div class="important">
<p>
We can the plant dynamics that as an inverse multiplicative uncertainty (Figure <a href="#orge738173">8</a>):
</p>
\begin{equation}
\frac{x}{F} = G_0(s) (1 + w_{iI}(s) \Delta(s))^{-1}
\end{equation}
<p>
with:
</p>
<ul class="org-ul">
<li>\(G_0(s) = \frac{1}{ms^2 + cs + k + ms^2(cs + k)G_0^\prime(s)}\)</li>
<li>\(w_{iI}(s) = \frac{ms^2(cs + k)G_0^\prime(s) w_I(s)}{ms^2 + cs + k + ms^2(cs + k)G_0^\prime(s)} = G_0(s) ms^2(cs + k)G_0^\prime(s) w_I(s)\)</li>
</ul>
</div>
<div id="orge738173" class="figure">
<p><img src="figs/inverse_uncertainty_set.png" alt="inverse_uncertainty_set.png" />
</p>
<p><span class="figure-number">Figure 8: </span>Inverse Multiplicative Uncertainty</p>
</div>
</div>
</div>
<div id="outline-container-orge950395" class="outline-3">
<h3 id="orge950395"><span class="section-number-3">2.5</span> Effect of the Isolation platform Stiffness</h3>
<div class="outline-text-3" id="text-2-5">
<p>
Let&rsquo;s first fix the mass of the payload to be isolated:
</p>
<div class="org-src-container">
@@ -645,6 +661,8 @@ Soft Isolation Platform:
c_soft = 0.1<span class="org-type">*</span>sqrt(m<span class="org-type">*</span>k_soft);
G_soft = 1<span class="org-type">/</span>(m<span class="org-type">*</span>s<span class="org-type">^</span>2 <span class="org-type">+</span> c_soft<span class="org-type">*</span>s <span class="org-type">+</span> k_soft <span class="org-type">+</span> m<span class="org-type">*</span>s<span class="org-type">^</span>2<span class="org-type">*</span>(c_soft<span class="org-type">*</span>s <span class="org-type">+</span> k_soft)<span class="org-type">*</span>Gp);
G0_soft = 1<span class="org-type">/</span>(m<span class="org-type">*</span>s<span class="org-type">^</span>2 <span class="org-type">+</span> c_soft<span class="org-type">*</span>s <span class="org-type">+</span> k_soft <span class="org-type">+</span> m<span class="org-type">*</span>s<span class="org-type">^</span>2<span class="org-type">*</span>(c_soft<span class="org-type">*</span>s <span class="org-type">+</span> k_soft)<span class="org-type">*</span>Gp0);
wiI_soft = Gp0<span class="org-type">*</span>m<span class="org-type">*</span>s<span class="org-type">^</span>2<span class="org-type">*</span>(c_soft<span class="org-type">*</span>s <span class="org-type">+</span> k_soft)<span class="org-type">*</span>G0_soft<span class="org-type">*</span>wI;
</pre>
</div>
@@ -656,6 +674,8 @@ Mid Isolation Platform
c_mid = 0.1<span class="org-type">*</span>sqrt(m<span class="org-type">*</span>k_mid);
G_mid = 1<span class="org-type">/</span>(m<span class="org-type">*</span>s<span class="org-type">^</span>2 <span class="org-type">+</span> c_mid<span class="org-type">*</span>s <span class="org-type">+</span> k_mid <span class="org-type">+</span> m<span class="org-type">*</span>s<span class="org-type">^</span>2<span class="org-type">*</span>(c_mid<span class="org-type">*</span>s <span class="org-type">+</span> k_mid)<span class="org-type">*</span>Gp);
G0_mid = 1<span class="org-type">/</span>(m<span class="org-type">*</span>s<span class="org-type">^</span>2 <span class="org-type">+</span> c_mid<span class="org-type">*</span>s <span class="org-type">+</span> k_mid <span class="org-type">+</span> m<span class="org-type">*</span>s<span class="org-type">^</span>2<span class="org-type">*</span>(c_mid<span class="org-type">*</span>s <span class="org-type">+</span> k_mid)<span class="org-type">*</span>Gp0);
wiI_mid = Gp0<span class="org-type">*</span>m<span class="org-type">*</span>s<span class="org-type">^</span>2<span class="org-type">*</span>(c_mid<span class="org-type">*</span>s <span class="org-type">+</span> k_mid)<span class="org-type">*</span>G0_mid<span class="org-type">*</span>wI;
</pre>
</div>
@@ -667,18 +687,20 @@ Stiff Isolation Platform
c_stiff = 0.1<span class="org-type">*</span>sqrt(m<span class="org-type">*</span>k_stiff);
G_stiff = 1<span class="org-type">/</span>(m<span class="org-type">*</span>s<span class="org-type">^</span>2 <span class="org-type">+</span> c_stiff<span class="org-type">*</span>s <span class="org-type">+</span> k_stiff <span class="org-type">+</span> m<span class="org-type">*</span>s<span class="org-type">^</span>2<span class="org-type">*</span>(c_stiff<span class="org-type">*</span>s <span class="org-type">+</span> k_stiff)<span class="org-type">*</span>Gp);
G0_stiff = 1<span class="org-type">/</span>(m<span class="org-type">*</span>s<span class="org-type">^</span>2 <span class="org-type">+</span> c_stiff<span class="org-type">*</span>s <span class="org-type">+</span> k_stiff <span class="org-type">+</span> m<span class="org-type">*</span>s<span class="org-type">^</span>2<span class="org-type">*</span>(c_stiff<span class="org-type">*</span>s <span class="org-type">+</span> k_stiff)<span class="org-type">*</span>Gp0);
wiI_stiff = Gp0<span class="org-type">*</span>m<span class="org-type">*</span>s<span class="org-type">^</span>2<span class="org-type">*</span>(c_stiff<span class="org-type">*</span>s <span class="org-type">+</span> k_stiff)<span class="org-type">*</span>G0_stiff<span class="org-type">*</span>wI;
</pre>
</div>
<p>
The obtained transfer functions \(x/F\) for each of the three platforms are shown in Figure <a href="#org89aa89f">8</a>.
The obtained transfer functions \(x/F\) for each of the three platforms are shown in Figure <a href="#org89aa89f">9</a>.
</p>
<div id="org89aa89f" class="figure">
<p><img src="figs/plant_uncertainty_stiffness_isolator.png" alt="plant_uncertainty_stiffness_isolator.png" />
</p>
<p><span class="figure-number">Figure 8: </span>Obtained plant for the three isolators (<a href="./figs/plant_uncertainty_stiffness_isolator.png">png</a>, <a href="./figs/plant_uncertainty_stiffness_isolator.pdf">pdf</a>)</p>
<p><span class="figure-number">Figure 9: </span>Obtained plant for the three isolators (<a href="./figs/plant_uncertainty_stiffness_isolator.png">png</a>, <a href="./figs/plant_uncertainty_stiffness_isolator.pdf">pdf</a>)</p>
</div>
<p>
@@ -688,44 +710,6 @@ This is due to the fact that with the current model, at high frequency, the supp
</div>
</div>
<div id="outline-container-org67810a4" class="outline-3">
<h3 id="org67810a4"><span class="section-number-3">2.5</span> Equivalent Inverse Multiplicative Uncertainty</h3>
<div class="outline-text-3" id="text-2-5">
<p>
Let&rsquo;s express the uncertainty of the plant \(x/F\) as a function of the parameters as well as of the uncertainty on the platform&rsquo;s compliance:
</p>
\begin{align*}
\frac{x}{F} &= \frac{1}{ms^2 + cs + k + ms^2(cs + k)G_0^\prime(s)(1 + w_I(s)\Delta(s))}\\
&= \frac{1}{ms^2 + cs + k + ms^2(cs + k)G_0^\prime(s) + ms^2(cs + k)G_0^\prime(s) w_I(s)\Delta(s)}\\
&= \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*}
<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}
<p>
with:
</p>
<ul class="org-ul">
<li>\(G_0(s) = \frac{1}{ms^2 + cs + k + ms^2(cs + k)G_0^\prime(s)}\)</li>
<li>\(w_{iI}(s) = \frac{ms^2(cs + k)G_0^\prime(s) w_I(s)}{ms^2 + cs + k + ms^2(cs + k)G_0^\prime(s)} = G_0(s) ms^2(cs + k)G_0^\prime(s) w_I(s)\)</li>
</ul>
</div>
<div id="orge738173" class="figure">
<p><img src="figs/inverse_uncertainty_set.png" alt="inverse_uncertainty_set.png" />
</p>
<p><span class="figure-number">Figure 9: </span>Inverse Multiplicative Uncertainty</p>
</div>
</div>
</div>
<div id="outline-container-org6967854" class="outline-3">
<h3 id="org6967854"><span class="section-number-3">2.6</span> Reduce the Uncertainty on the plant</h3>
<div class="outline-text-3" id="text-2-6">
@@ -828,8 +812,8 @@ 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-orgd28ebd8" class="outline-3">
<h3 id="orgd28ebd8"><span class="section-number-3">2.7</span> Conclusion</h3>
<div id="outline-container-org718f0f1" class="outline-3">
<h3 id="org718f0f1"><span class="section-number-3">2.7</span> Conclusion</h3>
<div class="outline-text-3" id="text-2-7">
<div class="important">
<p>
@@ -857,7 +841,7 @@ Thus, if a stiff isolation platform is used, the recommendation is to have the l
</div>
<div id="postamble" class="status">
<p class="author">Author: Dehaeze Thomas</p>
<p class="date">Created: 2020-03-26 jeu. 17:24</p>
<p class="date">Created: 2020-04-01 mer. 16:14</p>
</div>
</body>
</html>