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Add analysis on soft granite suspension

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docs/optimal_stiffness_disturbances.html
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@ -4,7 +4,7 @@
"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>
<!-- 2020-04-07 mar. 15:57 --> <!-- 2020-04-07 mar. 17:10 -->
<meta http-equiv="Content-Type" content="text/html;charset=utf-8" /> <meta http-equiv="Content-Type" content="text/html;charset=utf-8" />
<meta name="viewport" content="width=device-width, initial-scale=1" /> <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> <title>Determination of the optimal nano-hexapod&rsquo;s stiffness for reducing the effect of disturbances</title>
@ -227,7 +227,9 @@
</script> </script>
<script> <script>
MathJax = { MathJax = {
tex: { macros: { tex: {
tags: 'ams',
macros: {
bm: ["\\boldsymbol{#1}",1], bm: ["\\boldsymbol{#1}",1],
} }
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@ -255,14 +257,20 @@
<li><a href="#org78dd34d">2.3. Sensitivity to Stages vibration (Filtering)</a></li> <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="#orgd4ea2f4">2.4. Effect of Ground motion (Transmissibility).</a></li>
<li><a href="#org0448746">2.5. Direct Forces (Compliance).</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> </ul>
</li> </li>
<li><a href="#org6527e58">3. Effect of granite stiffness</a> <li><a href="#org6527e58">3. Effect of granite stiffness</a>
<ul> <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="#org9215f81">3.2. Soft Granite</a></li>
<li><a href="#org8878556">3.3. Effect of the Granite transfer function</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> </ul>
</li> </li>
<li><a href="#org8a88fb0">4. Open Loop Budget Error</a> <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> <li><a href="#orgf2d36a1">5.1. Reduction thanks to feedback - Required bandwidth</a></li>
</ul> </ul>
</li> </li>
<li><a href="#org0953c03">6. Conclusion</a></li> <li><a href="#orga29f90b">6. Conclusion</a></li>
</ul> </ul>
</div> </div>
</div> </div>
@ -488,12 +496,16 @@ The effect of direct forces/torques applied on the sample (cable forces for inst
</div> </div>
</div> </div>
<div id="outline-container-orgea74617" class="outline-3"> <div id="outline-container-orge0160c0" class="outline-3">
<h3 id="orgea74617"><span class="section-number-3">2.6</span> Conclusion</h3> <h3 id="orge0160c0"><span class="section-number-3">2.6</span> Conclusion</h3>
<div class="outline-text-3" id="text-2-6"> <div class="outline-text-3" id="text-2-6">
<div class="important"> <div class="important">
<p> <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> </p>
</div> </div>
@ -507,15 +519,28 @@ The effect of direct forces/torques applied on the sample (cable forces for inst
<p> <p>
<a id="orgd4105b6"></a> <a id="orgd4105b6"></a>
</p> </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>
<div id="outline-container-orgd3e5fe1" class="outline-3"> <div id="outline-container-orgd3e5fe1" class="outline-3">
<h3 id="orgd3e5fe1"><span class="section-number-3">3.1</span> Analytical Analysis</h3> <h3 id="orgd3e5fe1"><span class="section-number-3">3.1</span> Analytical Analysis</h3>
<div class="outline-text-3" id="text-3-1"> <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"> <div id="org8fb9606" class="figure">
<p><img src="figs/2dof_system_granite_stiffness.png" alt="2dof_system_granite_stiffness.png" /> <p><img src="figs/2dof_system_granite_stiffness.png" alt="2dof_system_granite_stiffness.png" />
</p> </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> </div>
<p> <p>
@ -523,53 +548,148 @@ If we write the equation of motion of the system in Figure <a href="#org8fb9606"
</p> </p>
\begin{align} \begin{align}
m^\prime s^2 x^\prime &= (c^\prime s + k^\prime) (x - x^\prime) \\ 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} \end{align}
<p> <p>
If we note \(d = x^\prime - x\), we obtain: If we note \(d = x^\prime - x\), we obtain:
</p> </p>
\begin{equation} \begin{equation}
\label{org4396920} \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)}
\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)}
\end{equation} \end{equation}
</div> </div>
</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"> <div id="outline-container-org9215f81" class="outline-3">
<h3 id="org9215f81"><span class="section-number-3">3.2</span> Soft Granite</h3> <h3 id="org9215f81"><span class="section-number-3">3.2</span> Soft Granite</h3>
<div class="outline-text-3" id="text-3-2"> <div class="outline-text-3" id="text-3-2">
<p> <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> </p>
<div class="org-src-container"> <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 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> </pre>
</div> </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> </div>
<div id="outline-container-org8878556" class="outline-3"> <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> <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>
</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"> <div id="outline-container-org8a88fb0" class="outline-2">
<h2 id="org8a88fb0"><span class="section-number-2">4</span> Open Loop Budget Error</h2> <h2 id="org8a88fb0"><span class="section-number-2">4</span> Open Loop Budget Error</h2>
<div class="outline-text-2" id="text-4"> <div class="outline-text-2" id="text-4">
@ -846,13 +966,13 @@ xlim([1, 500]);
</div> </div>
</div> </div>
</div> </div>
<div id="outline-container-org0953c03" class="outline-2"> <div id="outline-container-orga29f90b" class="outline-2">
<h2 id="org0953c03"><span class="section-number-2">6</span> Conclusion</h2> <h2 id="orga29f90b"><span class="section-number-2">6</span> Conclusion</h2>
</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: 2020-04-07 mar. 15:57</p> <p class="date">Created: 2020-04-07 mar. 17:10</p>
</div> </div>
</body> </body>
</html> </html>

189
org/optimal_stiffness_disturbances.org
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@ -389,12 +389,22 @@ The effect of direct forces/torques applied on the sample (cable forces for inst
** Conclusion ** Conclusion
#+begin_important #+begin_important
Reducing the nano-hexapod stiffness generally lowers the sensitivity to stages vibration but increases the sensitivity to ground motion and direct forces.
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 [[sec:open_loop_budget_error]].
#+end_important #+end_important
* Effect of granite stiffness * Effect of granite stiffness
<<sec:granite_stiffness>> <<sec:granite_stiffness>>
** Introduction :ignore:
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.
** Analytical Analysis ** Analytical Analysis
*** Simple mass-spring-damper model
Let's consider the system shown in Figure [[fig:2dof_system_granite_stiffness]] 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 "deformation" $d = x^\prime - x$ due to ground motion $w$.
#+begin_src latex :file 2dof_system_granite_stiffness.pdf #+begin_src latex :file 2dof_system_granite_stiffness.pdf
\begin{tikzpicture} \begin{tikzpicture}
% ==================== % ====================
@ -415,7 +425,7 @@ The effect of direct forces/torques applied on the sample (cable forces for inst
% ==================== % ====================
\draw (-0.5*\massw, 0) -- (0.5*\massw, 0); \draw (-0.5*\massw, 0) -- (0.5*\massw, 0);
\draw[dashed] (0.5*\massw, 0) -- ++(\dispw, 0) coordinate(dlow); \draw[dashed] (0.5*\massw, 0) -- ++(\dispw, 0) coordinate(dlow);
\draw[->] (0.5*\massw+0.5*\dispw, 0) -- ++(0, \disph) node[right]{$x_{w}$}; \draw[->] (0.5*\massw+0.5*\dispw, 0) -- ++(0, \disph) node[right]{$w$};
% ==================== % ====================
% Micro Station % Micro Station
@ -425,8 +435,8 @@ The effect of direct forces/torques applied on the sample (cable forces for inst
\draw[fill=white] (-0.5*\massw, \spaceh) rectangle (0.5*\massw, \spaceh+\massh) node[pos=0.5]{$m$}; \draw[fill=white] (-0.5*\massw, \spaceh) rectangle (0.5*\massw, \spaceh+\massh) node[pos=0.5]{$m$};
% Spring, Damper, and Actuator % Spring, Damper, and Actuator
\draw[spring] (-0.4*\massw, 0) -- (-0.4*\massw, \spaceh) node[midway, left=0.1]{$k$}; \draw[spring] (-0.3*\massw, 0) -- (-0.3*\massw, \spaceh) node[midway, left=0.1]{$k$};
\draw[damper] (0, 0) -- ( 0, \spaceh) node[midway, left=0.2]{$c$}; \draw[damper] ( 0.3*\massw, 0) -- ( 0.3*\massw, \spaceh) node[midway, left=0.2]{$c$};
% Displacements % Displacements
\draw[dashed] (0.5*\massw, \spaceh) -- ++(\dispw, 0); \draw[dashed] (0.5*\massw, \spaceh) -- ++(\dispw, 0);
@ -434,8 +444,8 @@ The effect of direct forces/torques applied on the sample (cable forces for inst
% Legend % Legend
\draw[decorate, decoration={brace, amplitude=8pt}, xshift=\brach] % \draw[decorate, decoration={brace, amplitude=8pt}, xshift=\brach] %
(-0.5*\massw, \bracs) -- (-0.5*\massw, \spaceh+\massh-\bracs) % (-0.5*\massw, \bracs) -- (-0.5*\massw, \spaceh+\massh-\bracs) %
node[midway,rotate=90,anchor=south,yshift=10pt,align=center]{Granite}; node[midway,rotate=90,anchor=south,yshift=10pt,align=center]{Granite};
\end{scope} \end{scope}
% ==================== % ====================
@ -446,8 +456,8 @@ The effect of direct forces/torques applied on the sample (cable forces for inst
\draw[fill=white] (-0.5*\massw, \spaceh) rectangle (0.5*\massw, \spaceh+\massh) node[pos=0.5]{$m^\prime$}; \draw[fill=white] (-0.5*\massw, \spaceh) rectangle (0.5*\massw, \spaceh+\massh) node[pos=0.5]{$m^\prime$};
% Spring, Damper, and Actuator % Spring, Damper, and Actuator
\draw[spring] (-0.4*\massw, 0) -- (-0.4*\massw, \spaceh) node[midway, left=0.1]{$k^\prime$}; \draw[spring] (-0.3*\massw, 0) -- (-0.3*\massw, \spaceh) node[midway, left=0.1]{$k^\prime$};
\draw[damper] (0, 0) -- ( 0, \spaceh) node[midway, left=0.2]{$c^\prime$}; \draw[damper] ( 0.3*\massw, 0) -- ( 0.3*\massw, \spaceh) node[midway, left=0.2]{$c^\prime$};
% Displacements % Displacements
\draw[dashed] (0.5*\massw, \spaceh) -- ++(\dispw, 0) coordinate(dhigh); \draw[dashed] (0.5*\massw, \spaceh) -- ++(\dispw, 0) coordinate(dhigh);
@ -455,44 +465,123 @@ The effect of direct forces/torques applied on the sample (cable forces for inst
% Legend % Legend
\draw[decorate, decoration={brace, amplitude=8pt}, xshift=\brach] % \draw[decorate, decoration={brace, amplitude=8pt}, xshift=\brach] %
(-0.5*\massw, \bracs) -- (-0.5*\massw, \spaceh+\massh-\bracs) % (-0.5*\massw, \bracs) -- (-0.5*\massw, \spaceh+\massh-\bracs) %
node[midway,rotate=90,anchor=south,yshift=10pt,align=center]{Positioning\\Stages}; node[midway,rotate=90,anchor=south,yshift=10pt,align=center]{Positioning\\Stages};
\end{scope} \end{scope}
\end{tikzpicture} \end{tikzpicture}
#+end_src #+end_src
#+name: fig:2dof_system_granite_stiffness #+name: fig:2dof_system_granite_stiffness
#+caption: Figure caption #+caption: Mass Spring Damper system consisting of a granite and a positioning stage
#+RESULTS: #+RESULTS:
[[file:figs/2dof_system_granite_stiffness.png]] [[file:figs/2dof_system_granite_stiffness.png]]
If we write the equation of motion of the system in Figure [[fig:2dof_system_granite_stiffness]], we obtain: If we write the equation of motion of the system in Figure [[fig:2dof_system_granite_stiffness]], we obtain:
\begin{align} \begin{align}
m^\prime s^2 x^\prime &= (c^\prime s + k^\prime) (x - x^\prime) \\ 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} \end{align}
If we note $d = x^\prime - x$, we obtain: If we note $d = x^\prime - x$, we obtain:
#+name: eq:plant_ground_transmissibility
\begin{equation} \begin{equation}
\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} \end{equation}
*** General Case
Let's now considering a general positioning stage defined by:
- $G^\prime(s) = \frac{F}{x}$: its mechanical "impedance"
- $D^\prime(s) = \frac{d}{x}$: its "deformation" transfer function
#+begin_src latex :file general_system_granite_stiffness.pdf
\begin{tikzpicture}
\def\massw{2.2} % Width of the masses
\def\massh{0.8} % Height of the masses
\def\spaceh{1.2} % Height of the springs/dampers
\def\dispw{0.3} % Width of the dashed line for the displacement
\def\disph{0.5} % Height of the arrow for the displacements
\def\bracs{0.05} % Brace spacing vertically
\def\brach{-10pt} % Brace shift horizontaly
% Mass
\draw[fill=white] (-0.5*\massw, \spaceh) rectangle node[left=6pt]{$m$} (0.5*\massw, \spaceh+\massh);
% Spring, Damper, and Actuator
\draw[spring] (-0.3*\massw, 0) -- (-0.3*\massw, \spaceh) node[midway, left=0.1]{$k$};
\draw[damper] ( 0.3*\massw, 0) -- ( 0.3*\massw, \spaceh) node[midway, left=0.2]{$c$};
% Ground
\draw (-0.5*\massw, 0) -- (0.5*\massw, 0);
% Groud Motion
\draw[dashed] (0.5*\massw, 0) -- ++(\dispw, 0);
\draw[->] (0.5*\massw+0.5*\dispw, 0) -- ++(0, \disph) node[right]{$w$};
% Displacements
\draw[dashed] (0.5*\massw, \spaceh+\massh) -- ++(2*\dispw, 0) coordinate(dhigh);
\draw[->] (0.5*\massw+1.5*\dispw, \spaceh+\massh) -- ++(0, \disph) node[right]{$x$};
% Legend
\draw[decorate, decoration={brace, amplitude=8pt}, xshift=\brach] %
(-0.5*\massw, \bracs) -- (-0.5*\massw, \spaceh+\massh-\bracs) %
node[midway,rotate=90,anchor=south,yshift=10pt,align=center]{Granite};
\begin{scope}[shift={(0, \spaceh+\massh)}]
\node[piezo={2.2}{1.5}{6}, anchor=south] (piezo) at (0, 0){};
\draw[->] (0,0)node[branch]{} -- ++(0, -0.6)node[above right]{$F$}
\draw[<->] (1.1+0.5*\dispw,0) -- node[midway, right]{$d$} ++(0,1.5);
\draw[decorate, decoration={brace, amplitude=8pt}, xshift=\brach] %
($(piezo.south west) + (-10pt, 0)$) -- ($(piezo.north west) + (-10pt, 0)$) %
node[midway,rotate=90,anchor=south,yshift=10pt,align=center]{Positioning\\Stages};
\end{scope}
\end{tikzpicture}
#+end_src
#+name: fig:general_system_granite_stiffness
#+caption: Mass Spring Damper representing the granite and a general representation of positioning stages
#+RESULTS:
[[file:figs/general_system_granite_stiffness.png]]
The equation of motion are:
\begin{align}
ms^2 x &= (cs + k) (x - w) - F \\
F &= G^\prime(s) x \\
d &= D^\prime(s) x
\end{align}
where:
- $F$ is the force applied by the position stages on the granite mass
#+begin_important
We can express $d$ as a function of $w$:
\begin{equation}
\frac{d}{w} = \frac{D^\prime(s) (cs + k)}{ms^2 + cs + k + G^\prime(s)}
\end{equation}
This is the transfer function that we would like to minimize.
#+end_important
Let's verify this formula for a simple mass/spring/damper positioning stage.
In that case, we have:
\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*}
And finally:
\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}
which is the same as the previously derived equation.
** Soft Granite ** Soft Granite
Let's initialize a soft granite that will act as an isolation stage from ground motion. Let's initialize a soft granite and see how the sensitivity to disturbances will change.
#+begin_src matlab #+begin_src matlab
initializeGranite('K', 5e5*ones(3,1), 'C', 5e3*ones(3,1)); initializeGranite('K', 5e5*ones(3,1), 'C', 5e3*ones(3,1));
#+end_src #+end_src
#+begin_src matlab
Ks = logspace(3,9,7); % [N/m]
#+end_src
#+begin_src matlab :exports none #+begin_src matlab :exports none
Gdr = {zeros(length(Ks), 1)}; Gdr = {zeros(length(Ks), 1)};
#+end_src
#+begin_src matlab
for i = 1:length(Ks) for i = 1:length(Ks)
initializeNanoHexapod('k', Ks(i)); initializeNanoHexapod('k', Ks(i));
@ -504,6 +593,11 @@ Let's initialize a soft granite that will act as an isolation stage from ground
#+end_src #+end_src
** Effect of the Granite transfer function ** Effect of the Granite transfer function
From Figure [[fig:opt_stiff_soft_granite_Dw]], we can see that having a "soft" 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).
From Figures [[fig:opt_stiff_soft_granite_Frz]] and [[fig:opt_stiff_soft_granite_Fd]], we see that the change of granite suspension does not change a lot the sensitivity to both direct forces and stage vibrations.
#+begin_src matlab :exports none #+begin_src matlab :exports none
freqs = logspace(0, 3, 1000); freqs = logspace(0, 3, 1000);
@ -520,9 +614,18 @@ Let's initialize a soft granite that will act as an isolation stage from ground
hold off; hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [m/m]'); xlabel('Frequency [Hz]'); ylabel('Amplitude [m/m]'); xlabel('Frequency [Hz]');
legend('location', 'southeast'); legend('location', 'southwest');
#+end_src #+end_src
#+header: :tangle no :exports results :results none :noweb yes
#+begin_src matlab :var filepath="figs/opt_stiff_soft_granite_Dw.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
<<plt-matlab>>
#+end_src
#+name: fig:opt_stiff_soft_granite_Dw
#+caption: Change of sensibility to Ground motion when using a stiff Granite (solid curves) and a soft Granite (dashed curves) ([[./figs/opt_stiff_soft_granite_Dw.png][png]], [[./figs/opt_stiff_soft_granite_Dw.pdf][pdf]])
[[file:figs/opt_stiff_soft_granite_Dw.png]]
#+begin_src matlab :exports none #+begin_src matlab :exports none
freqs = logspace(0, 3, 1000); freqs = logspace(0, 3, 1000);
@ -539,9 +642,51 @@ Let's initialize a soft granite that will act as an isolation stage from ground
hold off; hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [m/N]'); xlabel('Frequency [Hz]'); ylabel('Amplitude [m/N]'); xlabel('Frequency [Hz]');
legend('location', 'southeast'); legend('location', 'southwest');
#+end_src #+end_src
#+header: :tangle no :exports results :results none :noweb yes
#+begin_src matlab :var filepath="figs/opt_stiff_soft_granite_Frz.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
<<plt-matlab>>
#+end_src
#+name: fig:opt_stiff_soft_granite_Frz
#+caption: Change of sensibility to Spindle vibrations when using a stiff Granite (solid curves) and a soft Granite (dashed curves) ([[./figs/opt_stiff_soft_granite_Frz.png][png]], [[./figs/opt_stiff_soft_granite_Frz.pdf][pdf]])
[[file:figs/opt_stiff_soft_granite_Frz.png]]
#+begin_src matlab :exports none
freqs = logspace(0, 3, 1000);
figure;
hold on;
for i = 1:length(Ks)
set(gca,'ColorOrderIndex',i);
plot(freqs, abs(squeeze(freqresp(Gd{i}( 'Ez', 'Fdz'), freqs, 'Hz'))), '-', ...
'DisplayName', sprintf('$k = %.0g$ [N/m]', Ks(i)));
set(gca,'ColorOrderIndex',i);
plot(freqs, abs(squeeze(freqresp(Gdr{i}('Ez', 'Fdz'), freqs, 'Hz'))), '--', ...
'HandleVisibility', 'off');
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('$E_{z}/F_{d,z}$ [m/N]'); xlabel('Frequency [Hz]');
legend('location', 'northeast');
#+end_src
#+header: :tangle no :exports results :results none :noweb yes
#+begin_src matlab :var filepath="figs/opt_stiff_soft_granite_Fd.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
<<plt-matlab>>
#+end_src
#+name: fig:opt_stiff_soft_granite_Fd
#+caption: Change of sensibility to direct forces when using a stiff Granite (solid curves) and a soft Granite (dashed curves) ([[./figs/opt_stiff_soft_granite_Fd.png][png]], [[./figs/opt_stiff_soft_granite_Fd.pdf][pdf]])
[[file:figs/opt_stiff_soft_granite_Fd.png]]
** Conclusion
#+begin_important
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.
#+end_important
* Open Loop Budget Error * Open Loop Budget Error
<<sec:open_loop_budget_error>> <<sec:open_loop_budget_error>>