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Add analysis on soft granite suspension

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Thomas Dehaeze
2020-04-07 17:10:39 +02:00
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docs/optimal_stiffness_disturbances.html
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<!-- 2020-04-07 mar. 15:57 -->
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<meta http-equiv="Content-Type" content="text/html;charset=utf-8" />
<meta name="viewport" content="width=device-width, initial-scale=1" />
<title>Determination of the optimal nano-hexapod&rsquo;s stiffness for reducing the effect of disturbances</title>
@@ -227,7 +227,9 @@
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@@ -255,14 +257,20 @@
<li><a href="#org78dd34d">2.3. Sensitivity to Stages vibration (Filtering)</a></li>
<li><a href="#orgd4ea2f4">2.4. Effect of Ground motion (Transmissibility).</a></li>
<li><a href="#org0448746">2.5. Direct Forces (Compliance).</a></li>
<li><a href="#orgea74617">2.6. Conclusion</a></li>
<li><a href="#orge0160c0">2.6. Conclusion</a></li>
</ul>
</li>
<li><a href="#org6527e58">3. Effect of granite stiffness</a>
<ul>
<li><a href="#orgd3e5fe1">3.1. Analytical Analysis</a></li>
<li><a href="#orgd3e5fe1">3.1. Analytical Analysis</a>
<ul>
<li><a href="#orgbc34a65">3.1.1. Simple mass-spring-damper model</a></li>
<li><a href="#org4ddec32">3.1.2. General Case</a></li>
</ul>
</li>
<li><a href="#org9215f81">3.2. Soft Granite</a></li>
<li><a href="#org8878556">3.3. Effect of the Granite transfer function</a></li>
<li><a href="#orgb756362">3.4. Conclusion</a></li>
</ul>
</li>
<li><a href="#org8a88fb0">4. Open Loop Budget Error</a>
@@ -278,7 +286,7 @@
<li><a href="#orgf2d36a1">5.1. Reduction thanks to feedback - Required bandwidth</a></li>
</ul>
</li>
<li><a href="#org0953c03">6. Conclusion</a></li>
<li><a href="#orga29f90b">6. Conclusion</a></li>
</ul>
</div>
</div>
@@ -488,12 +496,16 @@ The effect of direct forces/torques applied on the sample (cable forces for inst
</div>
</div>
<div id="outline-container-orgea74617" class="outline-3">
<h3 id="orgea74617"><span class="section-number-3">2.6</span> Conclusion</h3>
<div id="outline-container-orge0160c0" class="outline-3">
<h3 id="orge0160c0"><span class="section-number-3">2.6</span> Conclusion</h3>
<div class="outline-text-3" id="text-2-6">
<div class="important">
<p>
Reducing the nano-hexapod stiffness generally lowers the sensitivity to stages vibration but increases the sensitivity to ground motion and direct forces.
</p>
<p>
In order to conclude on the optimal stiffness that will yield the smallest sample vibration, one has to include the level of disturbances. This is done in Section <a href="#org5d05990">4</a>.
</p>
</div>
@@ -507,15 +519,28 @@ The effect of direct forces/torques applied on the sample (cable forces for inst
<p>
<a id="orgd4105b6"></a>
</p>
<p>
In this section, we wish to see if a soft granite suspension could help in reducing the effect of disturbances on the position error of the sample.
</p>
</div>
<div id="outline-container-orgd3e5fe1" class="outline-3">
<h3 id="orgd3e5fe1"><span class="section-number-3">3.1</span> Analytical Analysis</h3>
<div class="outline-text-3" id="text-3-1">
</div>
<div id="outline-container-orgbc34a65" class="outline-4">
<h4 id="orgbc34a65"><span class="section-number-4">3.1.1</span> Simple mass-spring-damper model</h4>
<div class="outline-text-4" id="text-3-1-1">
<p>
Let&rsquo;s consider the system shown in Figure <a href="#org8fb9606">8</a> consisting of two stacked mass-spring-damper systems.
The bottom one represents the granite, and the top one all the positioning stages.
We want the smallest stage &ldquo;deformation&rdquo; \(d = x^\prime - x\) due to ground motion \(w\).
</p>
<div id="org8fb9606" class="figure">
<p><img src="figs/2dof_system_granite_stiffness.png" alt="2dof_system_granite_stiffness.png" />
</p>
<p><span class="figure-number">Figure 8: </span>Figure caption</p>
<p><span class="figure-number">Figure 8: </span>Mass Spring Damper system consisting of a granite and a positioning stage</p>
</div>
<p>
@@ -523,53 +548,148 @@ If we write the equation of motion of the system in Figure <a href="#org8fb9606"
</p>
\begin{align}
m^\prime s^2 x^\prime &= (c^\prime s + k^\prime) (x - x^\prime) \\
ms^2 x &= (c^\prime s + k^\prime) (x^\prime - x) + (cs + k) (x_w - x)
ms^2 x &= (c^\prime s + k^\prime) (x^\prime - x) + (cs + k) (w - x)
\end{align}
<p>
If we note \(d = x^\prime - x\), we obtain:
</p>
\begin{equation}
\label{org4396920}
\frac{d}{x_w} = \frac{-m^\prime s^2 (cs + k)}{ (m^\prime s^2 + c^\prime s + k^\prime) (ms^2 + cs + k) + m^\prime s^2(c^\prime s + k^\prime)}
\frac{d}{w} = \frac{-m^\prime s^2 (cs + k)}{ (m^\prime s^2 + c^\prime s + k^\prime) (ms^2 + cs + k) + m^\prime s^2(c^\prime s + k^\prime)}
\end{equation}
</div>
</div>
<div id="outline-container-org4ddec32" class="outline-4">
<h4 id="org4ddec32"><span class="section-number-4">3.1.2</span> General Case</h4>
<div class="outline-text-4" id="text-3-1-2">
<p>
Let&rsquo;s now considering a general positioning stage defined by:
</p>
<ul class="org-ul">
<li>\(G^\prime(s) = \frac{F}{x}\): its mechanical &ldquo;impedance&rdquo;</li>
<li>\(D^\prime(s) = \frac{d}{x}\): its &ldquo;deformation&rdquo; transfer function</li>
</ul>
<div id="org9702e0f" class="figure">
<p><img src="figs/general_system_granite_stiffness.png" alt="general_system_granite_stiffness.png" />
</p>
<p><span class="figure-number">Figure 9: </span>Mass Spring Damper representing the granite and a general representation of positioning stages</p>
</div>
<p>
The equation of motion are:
</p>
\begin{align}
ms^2 x &= (cs + k) (x - w) - F \\
F &= G^\prime(s) x \\
d &= D^\prime(s) x
\end{align}
<p>
where:
</p>
<ul class="org-ul">
<li>\(F\) is the force applied by the position stages on the granite mass</li>
</ul>
<div class="important">
<p>
We can express \(d\) as a function of \(w\):
</p>
\begin{equation}
\frac{d}{w} = \frac{D^\prime(s) (cs + k)}{ms^2 + cs + k + G^\prime(s)}
\end{equation}
<p>
This is the transfer function that we would like to minimize.
</p>
</div>
<p>
Let&rsquo;s verify this formula for a simple mass/spring/damper positioning stage.
In that case, we have:
</p>
\begin{align*}
D^\prime(s) &= \frac{d}{x} = \frac{- m^\prime s^2}{m^\prime s^2 + c^\prime s + k^\prime} \\
G^\prime(s) &= \frac{F}{x} = \frac{m^\prime s^2(c^\prime s + k)}{m^\prime s^2 + c^\prime s + k^\prime}
\end{align*}
<p>
And finally:
</p>
\begin{equation}
\frac{d}{w} = \frac{-m^\prime s^2 (cs + k)}{ (m^\prime s^2 + c^\prime s + k^\prime) (ms^2 + cs + k) + m^\prime s^2(c^\prime s + k^\prime)}
\end{equation}
<p>
which is the same as the previously derived equation.
</p>
</div>
</div>
</div>
<div id="outline-container-org9215f81" class="outline-3">
<h3 id="org9215f81"><span class="section-number-3">3.2</span> Soft Granite</h3>
<div class="outline-text-3" id="text-3-2">
<p>
Let&rsquo;s initialize a soft granite that will act as an isolation stage from ground motion.
Let&rsquo;s initialize a soft granite and see how the sensitivity to disturbances will change.
</p>
<div class="org-src-container">
<pre class="src src-matlab">initializeGranite(<span class="org-string">'K'</span>, 5e5<span class="org-type">*</span>ones(3,1), <span class="org-string">'C'</span>, 5e3<span class="org-type">*</span>ones(3,1));
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab">Ks = logspace(3,9,7); <span class="org-comment">% [N/m]</span>
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-keyword">for</span> <span class="org-variable-name"><span class="org-constant">i</span></span> = <span class="org-constant">1:length(Ks)</span>
initializeNanoHexapod(<span class="org-string">'k'</span>, Ks(<span class="org-constant">i</span>));
G = linearize(mdl, io);
G.InputName = {<span class="org-string">'Dwx'</span>, <span class="org-string">'Dwy'</span>, <span class="org-string">'Dwz'</span>, <span class="org-string">'Fty_x'</span>, <span class="org-string">'Fty_z'</span>, <span class="org-string">'Frz_z'</span>, <span class="org-string">'Fdx'</span>, <span class="org-string">'Fdy'</span>, <span class="org-string">'Fdz'</span>, <span class="org-string">'Mdx'</span>, <span class="org-string">'Mdy'</span>, <span class="org-string">'Mdz'</span>};
G.OutputName = {<span class="org-string">'Ex'</span>, <span class="org-string">'Ey'</span>, <span class="org-string">'Ez'</span>, <span class="org-string">'Erx'</span>, <span class="org-string">'Ery'</span>, <span class="org-string">'Erz'</span>};
Gdr(<span class="org-constant">i</span>) = {minreal(G)};
<span class="org-keyword">end</span>
</pre>
</div>
</div>
</div>
<div id="outline-container-org8878556" class="outline-3">
<h3 id="org8878556"><span class="section-number-3">3.3</span> Effect of the Granite transfer function</h3>
<div class="outline-text-3" id="text-3-3">
<p>
From Figure <a href="#org38146da">10</a>, we can see that having a &ldquo;soft&rdquo; granite suspension greatly lowers the sensitivity to ground motion.
The sensitivity is indeed lowered starting from the resonance of the granite on its soft suspension (few Hz here).
</p>
<p>
From Figures <a href="#orgc4c14fb">11</a> and <a href="#org533cc4b">12</a>, we see that the change of granite suspension does not change a lot the sensitivity to both direct forces and stage vibrations.
</p>
<div id="org38146da" class="figure">
<p><img src="figs/opt_stiff_soft_granite_Dw.png" alt="opt_stiff_soft_granite_Dw.png" />
</p>
<p><span class="figure-number">Figure 10: </span>Change of sensibility to Ground motion when using a stiff Granite (solid curves) and a soft Granite (dashed curves) (<a href="./figs/opt_stiff_soft_granite_Dw.png">png</a>, <a href="./figs/opt_stiff_soft_granite_Dw.pdf">pdf</a>)</p>
</div>
<div id="orgc4c14fb" class="figure">
<p><img src="figs/opt_stiff_soft_granite_Frz.png" alt="opt_stiff_soft_granite_Frz.png" />
</p>
<p><span class="figure-number">Figure 11: </span>Change of sensibility to Spindle vibrations when using a stiff Granite (solid curves) and a soft Granite (dashed curves) (<a href="./figs/opt_stiff_soft_granite_Frz.png">png</a>, <a href="./figs/opt_stiff_soft_granite_Frz.pdf">pdf</a>)</p>
</div>
<div id="org533cc4b" class="figure">
<p><img src="figs/opt_stiff_soft_granite_Fd.png" alt="opt_stiff_soft_granite_Fd.png" />
</p>
<p><span class="figure-number">Figure 12: </span>Change of sensibility to direct forces when using a stiff Granite (solid curves) and a soft Granite (dashed curves) (<a href="./figs/opt_stiff_soft_granite_Fd.png">png</a>, <a href="./figs/opt_stiff_soft_granite_Fd.pdf">pdf</a>)</p>
</div>
</div>
</div>
<div id="outline-container-orgb756362" class="outline-3">
<h3 id="orgb756362"><span class="section-number-3">3.4</span> Conclusion</h3>
<div class="outline-text-3" id="text-3-4">
<div class="important">
<p>
Having a soft granite suspension could greatly improve the sensitivity the ground motion and thus the level of sample vibration if it is found that ground motion is the limiting factor.
</p>
</div>
</div>
</div>
</div>
<div id="outline-container-org8a88fb0" class="outline-2">
<h2 id="org8a88fb0"><span class="section-number-2">4</span> Open Loop Budget Error</h2>
<div class="outline-text-2" id="text-4">
@@ -846,13 +966,13 @@ xlim([1, 500]);
</div>
</div>
</div>
<div id="outline-container-org0953c03" class="outline-2">
<h2 id="org0953c03"><span class="section-number-2">6</span> Conclusion</h2>
<div id="outline-container-orga29f90b" class="outline-2">
<h2 id="orga29f90b"><span class="section-number-2">6</span> Conclusion</h2>
</div>
</div>
<div id="postamble" class="status">
<p class="author">Author: Dehaeze Thomas</p>
<p class="date">Created: 2020-04-07 mar. 15:57</p>
<p class="date">Created: 2020-04-07 mar. 17:10</p>
</div>
</body>
</html>