Add analysis on soft granite suspension
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docs/figs/general_system_granite_stiffness.pdf
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@ -4,7 +4,7 @@
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"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
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<html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en">
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<head>
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<!-- 2020-04-07 mar. 15:57 -->
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<!-- 2020-04-07 mar. 17:10 -->
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<meta http-equiv="Content-Type" content="text/html;charset=utf-8" />
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<meta name="viewport" content="width=device-width, initial-scale=1" />
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<title>Determination of the optimal nano-hexapod’s stiffness for reducing the effect of disturbances</title>
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@ -227,7 +227,9 @@
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</script>
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<script>
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MathJax = {
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tex: { macros: {
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tex: {
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tags: 'ams',
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macros: {
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bm: ["\\boldsymbol{#1}",1],
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}
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}
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@ -255,14 +257,20 @@
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<li><a href="#org78dd34d">2.3. Sensitivity to Stages vibration (Filtering)</a></li>
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<li><a href="#orgd4ea2f4">2.4. Effect of Ground motion (Transmissibility).</a></li>
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<li><a href="#org0448746">2.5. Direct Forces (Compliance).</a></li>
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<li><a href="#orgea74617">2.6. Conclusion</a></li>
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<li><a href="#orge0160c0">2.6. Conclusion</a></li>
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</ul>
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</li>
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<li><a href="#org6527e58">3. Effect of granite stiffness</a>
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<ul>
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<li><a href="#orgd3e5fe1">3.1. Analytical Analysis</a></li>
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<li><a href="#orgd3e5fe1">3.1. Analytical Analysis</a>
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<ul>
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<li><a href="#orgbc34a65">3.1.1. Simple mass-spring-damper model</a></li>
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<li><a href="#org4ddec32">3.1.2. General Case</a></li>
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</ul>
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</li>
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<li><a href="#org9215f81">3.2. Soft Granite</a></li>
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<li><a href="#org8878556">3.3. Effect of the Granite transfer function</a></li>
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<li><a href="#orgb756362">3.4. Conclusion</a></li>
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</ul>
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</li>
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<li><a href="#org8a88fb0">4. Open Loop Budget Error</a>
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@ -278,7 +286,7 @@
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<li><a href="#orgf2d36a1">5.1. Reduction thanks to feedback - Required bandwidth</a></li>
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</ul>
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</li>
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<li><a href="#org0953c03">6. Conclusion</a></li>
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<li><a href="#orga29f90b">6. Conclusion</a></li>
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</ul>
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</div>
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</div>
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@ -488,12 +496,16 @@ The effect of direct forces/torques applied on the sample (cable forces for inst
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</div>
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</div>
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<div id="outline-container-orgea74617" class="outline-3">
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<h3 id="orgea74617"><span class="section-number-3">2.6</span> Conclusion</h3>
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<div id="outline-container-orge0160c0" class="outline-3">
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<h3 id="orge0160c0"><span class="section-number-3">2.6</span> Conclusion</h3>
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<div class="outline-text-3" id="text-2-6">
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<div class="important">
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<p>
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Reducing the nano-hexapod stiffness generally lowers the sensitivity to stages vibration but increases the sensitivity to ground motion and direct forces.
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</p>
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<p>
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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>.
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</p>
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</div>
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@ -507,15 +519,28 @@ The effect of direct forces/torques applied on the sample (cable forces for inst
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<p>
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<a id="orgd4105b6"></a>
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</p>
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<p>
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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.
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</p>
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</div>
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<div id="outline-container-orgd3e5fe1" class="outline-3">
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<h3 id="orgd3e5fe1"><span class="section-number-3">3.1</span> Analytical Analysis</h3>
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<div class="outline-text-3" id="text-3-1">
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</div>
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<div id="outline-container-orgbc34a65" class="outline-4">
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<h4 id="orgbc34a65"><span class="section-number-4">3.1.1</span> Simple mass-spring-damper model</h4>
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<div class="outline-text-4" id="text-3-1-1">
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<p>
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Let’s consider the system shown in Figure <a href="#org8fb9606">8</a> consisting of two stacked mass-spring-damper systems.
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The bottom one represents the granite, and the top one all the positioning stages.
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We want the smallest stage “deformation” \(d = x^\prime - x\) due to ground motion \(w\).
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</p>
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<div id="org8fb9606" class="figure">
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<p><img src="figs/2dof_system_granite_stiffness.png" alt="2dof_system_granite_stiffness.png" />
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</p>
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<p><span class="figure-number">Figure 8: </span>Figure caption</p>
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<p><span class="figure-number">Figure 8: </span>Mass Spring Damper system consisting of a granite and a positioning stage</p>
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</div>
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<p>
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@ -523,53 +548,148 @@ If we write the equation of motion of the system in Figure <a href="#org8fb9606"
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</p>
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\begin{align}
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m^\prime s^2 x^\prime &= (c^\prime s + k^\prime) (x - x^\prime) \\
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ms^2 x &= (c^\prime s + k^\prime) (x^\prime - x) + (cs + k) (x_w - x)
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ms^2 x &= (c^\prime s + k^\prime) (x^\prime - x) + (cs + k) (w - x)
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\end{align}
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<p>
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If we note \(d = x^\prime - x\), we obtain:
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</p>
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\begin{equation}
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\label{org4396920}
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\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)}
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\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)}
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\end{equation}
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</div>
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</div>
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<div id="outline-container-org4ddec32" class="outline-4">
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<h4 id="org4ddec32"><span class="section-number-4">3.1.2</span> General Case</h4>
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<div class="outline-text-4" id="text-3-1-2">
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<p>
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Let’s now considering a general positioning stage defined by:
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</p>
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<ul class="org-ul">
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<li>\(G^\prime(s) = \frac{F}{x}\): its mechanical “impedance”</li>
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<li>\(D^\prime(s) = \frac{d}{x}\): its “deformation” transfer function</li>
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</ul>
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<div id="org9702e0f" class="figure">
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<p><img src="figs/general_system_granite_stiffness.png" alt="general_system_granite_stiffness.png" />
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</p>
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<p><span class="figure-number">Figure 9: </span>Mass Spring Damper representing the granite and a general representation of positioning stages</p>
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</div>
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<p>
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The equation of motion are:
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</p>
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\begin{align}
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ms^2 x &= (cs + k) (x - w) - F \\
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F &= G^\prime(s) x \\
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d &= D^\prime(s) x
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\end{align}
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<p>
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where:
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</p>
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<ul class="org-ul">
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<li>\(F\) is the force applied by the position stages on the granite mass</li>
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</ul>
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<div class="important">
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<p>
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We can express \(d\) as a function of \(w\):
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</p>
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\begin{equation}
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\frac{d}{w} = \frac{D^\prime(s) (cs + k)}{ms^2 + cs + k + G^\prime(s)}
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\end{equation}
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<p>
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This is the transfer function that we would like to minimize.
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</p>
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</div>
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<p>
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Let’s verify this formula for a simple mass/spring/damper positioning stage.
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In that case, we have:
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</p>
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\begin{align*}
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D^\prime(s) &= \frac{d}{x} = \frac{- m^\prime s^2}{m^\prime s^2 + c^\prime s + k^\prime} \\
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G^\prime(s) &= \frac{F}{x} = \frac{m^\prime s^2(c^\prime s + k)}{m^\prime s^2 + c^\prime s + k^\prime}
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\end{align*}
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<p>
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And finally:
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</p>
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\begin{equation}
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\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)}
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\end{equation}
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<p>
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which is the same as the previously derived equation.
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</p>
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</div>
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</div>
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</div>
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<div id="outline-container-org9215f81" class="outline-3">
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<h3 id="org9215f81"><span class="section-number-3">3.2</span> Soft Granite</h3>
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<div class="outline-text-3" id="text-3-2">
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<p>
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Let’s initialize a soft granite that will act as an isolation stage from ground motion.
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Let’s initialize a soft granite and see how the sensitivity to disturbances will change.
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</p>
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<div class="org-src-container">
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<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));
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</pre>
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</div>
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<div class="org-src-container">
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<pre class="src src-matlab">Ks = logspace(3,9,7); <span class="org-comment">% [N/m]</span>
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</pre>
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</div>
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<div class="org-src-container">
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<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>
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initializeNanoHexapod(<span class="org-string">'k'</span>, Ks(<span class="org-constant">i</span>));
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G = linearize(mdl, io);
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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>};
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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>};
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Gdr(<span class="org-constant">i</span>) = {minreal(G)};
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<span class="org-keyword">end</span>
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</pre>
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</div>
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</div>
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</div>
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<div id="outline-container-org8878556" class="outline-3">
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<h3 id="org8878556"><span class="section-number-3">3.3</span> Effect of the Granite transfer function</h3>
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<div class="outline-text-3" id="text-3-3">
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<p>
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From Figure <a href="#org38146da">10</a>, we can see that having a “soft” granite suspension greatly lowers the sensitivity to ground motion.
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The sensitivity is indeed lowered starting from the resonance of the granite on its soft suspension (few Hz here).
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</p>
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<p>
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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.
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</p>
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<div id="org38146da" class="figure">
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<p><img src="figs/opt_stiff_soft_granite_Dw.png" alt="opt_stiff_soft_granite_Dw.png" />
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</p>
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<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>
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</div>
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<div id="orgc4c14fb" class="figure">
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<p><img src="figs/opt_stiff_soft_granite_Frz.png" alt="opt_stiff_soft_granite_Frz.png" />
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</p>
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<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>
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</div>
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<div id="org533cc4b" class="figure">
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<p><img src="figs/opt_stiff_soft_granite_Fd.png" alt="opt_stiff_soft_granite_Fd.png" />
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</p>
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<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>
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</div>
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</div>
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</div>
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<div id="outline-container-orgb756362" class="outline-3">
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<h3 id="orgb756362"><span class="section-number-3">3.4</span> Conclusion</h3>
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<div class="outline-text-3" id="text-3-4">
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<div class="important">
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<p>
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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.
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</p>
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</div>
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</div>
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</div>
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</div>
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<div id="outline-container-org8a88fb0" class="outline-2">
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<h2 id="org8a88fb0"><span class="section-number-2">4</span> Open Loop Budget Error</h2>
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<div class="outline-text-2" id="text-4">
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@ -846,13 +966,13 @@ xlim([1, 500]);
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</div>
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</div>
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</div>
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<div id="outline-container-org0953c03" class="outline-2">
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<h2 id="org0953c03"><span class="section-number-2">6</span> Conclusion</h2>
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<div id="outline-container-orga29f90b" class="outline-2">
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<h2 id="orga29f90b"><span class="section-number-2">6</span> Conclusion</h2>
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</div>
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</div>
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<div id="postamble" class="status">
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<p class="author">Author: Dehaeze Thomas</p>
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<p class="date">Created: 2020-04-07 mar. 15:57</p>
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<p class="date">Created: 2020-04-07 mar. 17:10</p>
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</div>
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</body>
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</html>
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@ -389,12 +389,22 @@ The effect of direct forces/torques applied on the sample (cable forces for inst
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** Conclusion
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#+begin_important
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Reducing the nano-hexapod stiffness generally lowers the sensitivity to stages vibration but increases the sensitivity to ground motion and direct forces.
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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]].
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#+end_important
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* Effect of granite stiffness
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<<sec:granite_stiffness>>
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** Introduction :ignore:
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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.
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||||
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||||
** 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.
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The bottom one represents the granite, and the top one all the positioning stages.
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We want the smallest stage "deformation" $d = x^\prime - x$ due to ground motion $w$.
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#+begin_src latex :file 2dof_system_granite_stiffness.pdf
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\begin{tikzpicture}
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% ====================
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@ -415,7 +425,7 @@ The effect of direct forces/torques applied on the sample (cable forces for inst
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% ====================
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\draw (-0.5*\massw, 0) -- (0.5*\massw, 0);
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\draw[dashed] (0.5*\massw, 0) -- ++(\dispw, 0) coordinate(dlow);
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\draw[->] (0.5*\massw+0.5*\dispw, 0) -- ++(0, \disph) node[right]{$x_{w}$};
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\draw[->] (0.5*\massw+0.5*\dispw, 0) -- ++(0, \disph) node[right]{$w$};
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% ====================
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% Micro Station
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@ -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]{$m$};
|
||||
|
||||
% Spring, Damper, and Actuator
|
||||
\draw[spring] (-0.4*\massw, 0) -- (-0.4*\massw, \spaceh) node[midway, left=0.1]{$k$};
|
||||
\draw[damper] (0, 0) -- ( 0, \spaceh) node[midway, left=0.2]{$c$};
|
||||
\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]{$c$};
|
||||
|
||||
% Displacements
|
||||
\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
|
||||
\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};
|
||||
(-0.5*\massw, \bracs) -- (-0.5*\massw, \spaceh+\massh-\bracs) %
|
||||
node[midway,rotate=90,anchor=south,yshift=10pt,align=center]{Granite};
|
||||
\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]{$m^\prime$};
|
||||
|
||||
% Spring, Damper, and Actuator
|
||||
\draw[spring] (-0.4*\massw, 0) -- (-0.4*\massw, \spaceh) node[midway, left=0.1]{$k^\prime$};
|
||||
\draw[damper] (0, 0) -- ( 0, \spaceh) node[midway, left=0.2]{$c^\prime$};
|
||||
\draw[spring] (-0.3*\massw, 0) -- (-0.3*\massw, \spaceh) node[midway, left=0.1]{$k^\prime$};
|
||||
\draw[damper] ( 0.3*\massw, 0) -- ( 0.3*\massw, \spaceh) node[midway, left=0.2]{$c^\prime$};
|
||||
|
||||
% Displacements
|
||||
\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
|
||||
\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]{Positioning\\Stages};
|
||||
(-0.5*\massw, \bracs) -- (-0.5*\massw, \spaceh+\massh-\bracs) %
|
||||
node[midway,rotate=90,anchor=south,yshift=10pt,align=center]{Positioning\\Stages};
|
||||
\end{scope}
|
||||
\end{tikzpicture}
|
||||
#+end_src
|
||||
|
||||
#+name: fig:2dof_system_granite_stiffness
|
||||
#+caption: Figure caption
|
||||
#+caption: Mass Spring Damper system consisting of a granite and a positioning stage
|
||||
#+RESULTS:
|
||||
[[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:
|
||||
\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}
|
||||
|
||||
If we note $d = x^\prime - x$, we obtain:
|
||||
#+name: eq:plant_ground_transmissibility
|
||||
\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}
|
||||
|
||||
*** 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
|
||||
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
|
||||
initializeGranite('K', 5e5*ones(3,1), 'C', 5e3*ones(3,1));
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab
|
||||
Ks = logspace(3,9,7); % [N/m]
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab :exports none
|
||||
Gdr = {zeros(length(Ks), 1)};
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab
|
||||
for i = 1:length(Ks)
|
||||
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
|
||||
|
||||
** 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
|
||||
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;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
ylabel('Amplitude [m/m]'); xlabel('Frequency [Hz]');
|
||||
legend('location', 'southeast');
|
||||
legend('location', 'southwest');
|
||||
#+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
|
||||
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;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
ylabel('Amplitude [m/N]'); xlabel('Frequency [Hz]');
|
||||
legend('location', 'southeast');
|
||||
legend('location', 'southwest');
|
||||
#+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
|
||||
<<sec:open_loop_budget_error>>
|
||||
|
||||
Loading…
Reference in New Issue
Block a user