805 lines
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805 lines
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<a accesskey="h" href="index.html"> UP </a>
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<a accesskey="H" href="index.html"> HOME </a>
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</div><div id="content">
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<h1 class="title">Determination of the optimal nano-hexapod’s stiffness for reducing the effect of disturbances</h1>
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<div id="table-of-contents">
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<h2>Table of Contents</h2>
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<div id="text-table-of-contents">
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<ul>
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<li><a href="#org72be3da">1. Spectral Density of Disturbances label:sec:psd_disturbances</a>
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<ul>
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<li><a href="#orge84ae10">1.1. Load of the identified disturbances</a></li>
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<li><a href="#org0bec7fc">1.2. Plots</a></li>
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</ul>
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</li>
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<li><a href="#orgc44cf7e">2. Effect of disturbances on the position error</a>
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<ul>
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<li><a href="#org524df41">2.1. Initialization</a></li>
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<li><a href="#orgaf88c9f">2.2. Identification</a></li>
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<li><a href="#org2b8e5e0">2.3. Plots</a></li>
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<li><a href="#org71f73c5">2.4. Save</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="#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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</ul>
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</li>
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<li><a href="#org8a88fb0">4. Open Loop Budget Error</a>
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<ul>
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<li><a href="#org2ee8120">4.1. Load of the identified disturbances and transfer functions</a></li>
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<li><a href="#orgb1a3177">4.2. Equations</a></li>
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<li><a href="#org63eade9">4.3. Results</a></li>
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<li><a href="#orgef96b89">4.4. Cumulative Amplitude Spectrum</a></li>
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</ul>
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</li>
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<li><a href="#org34c0f38">5. Closed Loop Budget Error</a>
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<ul>
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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="#orgea74617">6. Conclusion</a></li>
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</ul>
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</div>
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</div>
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<ul class="org-ul">
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<li><a href="#org17d3d6a">1</a></li>
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<li><a href="#orgf9e4300">2</a></li>
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<li><a href="#orgd4105b6">3</a></li>
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<li><a href="#org5d05990">4</a></li>
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<li><a href="#orgd3503fb">5</a></li>
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</ul>
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<ul class="org-ul">
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<li><a href="#sec:psd_disturbances">sec:psd_disturbances</a></li>
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<li><a href="#sec:effect_disturbances">sec:effect_disturbances</a></li>
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<li><a href="#sec:granite_stiffness">sec:granite_stiffness</a></li>
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<li><a href="#sec:open_loop_budget_error">sec:open_loop_budget_error</a></li>
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<li><a href="#sec:closed_loop_budget_error">sec:closed_loop_budget_error</a></li>
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</ul>
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<div id="outline-container-org72be3da" class="outline-2">
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<h2 id="org72be3da"><span class="section-number-2">1</span> Spectral Density of Disturbances <div id="sec:psd_disturbances"></div></h2>
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<div class="outline-text-2" id="text-1">
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<p>
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<a id="org17d3d6a"></a>
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</p>
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<p>
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The level of disturbances has been identified form experiments.
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This is detailed in <a href="disturbances.html">this</a> document.
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</p>
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</div>
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<div id="outline-container-orge84ae10" class="outline-3">
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<h3 id="orge84ae10"><span class="section-number-3">1.1</span> Load of the identified disturbances</h3>
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<div class="outline-text-3" id="text-1-1">
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<div class="org-src-container">
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<pre class="src src-matlab">load(<span class="org-string">'./mat/dist_psd.mat'</span>, <span class="org-string">'dist_f'</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-org0bec7fc" class="outline-3">
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<h3 id="org0bec7fc"><span class="section-number-3">1.2</span> Plots</h3>
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</div>
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</div>
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<div id="outline-container-orgc44cf7e" class="outline-2">
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<h2 id="orgc44cf7e"><span class="section-number-2">2</span> Effect of disturbances on the position error</h2>
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<div class="outline-text-2" id="text-2">
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<p>
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<a id="orgf9e4300"></a>
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</p>
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</div>
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<div id="outline-container-org524df41" class="outline-3">
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<h3 id="org524df41"><span class="section-number-3">2.1</span> Initialization</h3>
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<div class="outline-text-3" id="text-2-1">
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<p>
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We initialize all the stages with the default parameters.
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</p>
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<div class="org-src-container">
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<pre class="src src-matlab">initializeGround();
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initializeGranite();
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initializeTy();
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initializeRy();
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initializeRz();
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initializeMicroHexapod();
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initializeAxisc();
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initializeMirror();
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</pre>
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</div>
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<p>
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We use a sample mass of 10kg.
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</p>
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<div class="org-src-container">
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<pre class="src src-matlab">initializeSample(<span class="org-string">'mass'</span>, 10);
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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">initializeSimscapeConfiguration(<span class="org-string">'gravity'</span>, <span class="org-constant">true</span>);
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initializeDisturbances(<span class="org-string">'enable'</span>, <span class="org-constant">false</span>);
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initializeLoggingConfiguration(<span class="org-string">'log'</span>, <span class="org-string">'none'</span>);
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initializeController();
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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-orgaf88c9f" class="outline-3">
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<h3 id="orgaf88c9f"><span class="section-number-3">2.2</span> Identification</h3>
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<div class="outline-text-3" id="text-2-2">
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<p>
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Inputs:
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</p>
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<ul class="org-ul">
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<li><code>Dwx</code>: Ground displacement in the \(x\) direction</li>
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<li><code>Dwy</code>: Ground displacement in the \(y\) direction</li>
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<li><code>Dwz</code>: Ground displacement in the \(z\) direction</li>
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<li><code>Fty_x</code>: Forces applied by the Translation stage in the \(x\) direction</li>
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<li><code>Fty_z</code>: Forces applied by the Translation stage in the \(z\) direction</li>
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<li><code>Frz_z</code>: Forces applied by the Spindle in the \(z\) direction</li>
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<li><code>Fd</code>: Direct forces applied at the center of mass of the Payload</li>
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</ul>
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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-matlab-cellbreak"><span class="org-comment">%% Name of the Simulink File</span></span>
|
|
mdl = <span class="org-string">'nass_model'</span>;
|
|
|
|
<span class="org-matlab-cellbreak"><span class="org-comment">%% Micro-Hexapod</span></span>
|
|
clear io; io_i = 1;
|
|
io(io_i) = linio([mdl, <span class="org-string">'/Disturbances'</span>], 1, <span class="org-string">'openinput'</span>, [], <span class="org-string">'Dwx'</span>); io_i = io_i <span class="org-type">+</span> 1; <span class="org-comment">% X Ground motion</span>
|
|
io(io_i) = linio([mdl, <span class="org-string">'/Disturbances'</span>], 1, <span class="org-string">'openinput'</span>, [], <span class="org-string">'Dwy'</span>); io_i = io_i <span class="org-type">+</span> 1; <span class="org-comment">% Y Ground motion</span>
|
|
io(io_i) = linio([mdl, <span class="org-string">'/Disturbances'</span>], 1, <span class="org-string">'openinput'</span>, [], <span class="org-string">'Dwz'</span>); io_i = io_i <span class="org-type">+</span> 1; <span class="org-comment">% Z Ground motion</span>
|
|
io(io_i) = linio([mdl, <span class="org-string">'/Disturbances'</span>], 1, <span class="org-string">'openinput'</span>, [], <span class="org-string">'Fty_x'</span>); io_i = io_i <span class="org-type">+</span> 1; <span class="org-comment">% Parasitic force Ty - X</span>
|
|
io(io_i) = linio([mdl, <span class="org-string">'/Disturbances'</span>], 1, <span class="org-string">'openinput'</span>, [], <span class="org-string">'Fty_z'</span>); io_i = io_i <span class="org-type">+</span> 1; <span class="org-comment">% Parasitic force Ty - Z</span>
|
|
io(io_i) = linio([mdl, <span class="org-string">'/Disturbances'</span>], 1, <span class="org-string">'openinput'</span>, [], <span class="org-string">'Frz_z'</span>); io_i = io_i <span class="org-type">+</span> 1; <span class="org-comment">% Parasitic force Rz - Z</span>
|
|
io(io_i) = linio([mdl, <span class="org-string">'/Disturbances'</span>], 1, <span class="org-string">'openinput'</span>, [], <span class="org-string">'Fd'</span>); io_i = io_i <span class="org-type">+</span> 1; <span class="org-comment">% Direct forces</span>
|
|
|
|
io(io_i) = linio([mdl, <span class="org-string">'/Tracking Error'</span>], 1, <span class="org-string">'openoutput'</span>, [], <span class="org-string">'En'</span>); io_i = io_i <span class="org-type">+</span> 1; <span class="org-comment">% Position Error</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>));
|
|
|
|
<span class="org-comment">% Run the linearization</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>};
|
|
Gd(<span class="org-constant">i</span>) = {minreal(G)};
|
|
<span class="org-keyword">end</span>
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org2b8e5e0" class="outline-3">
|
|
<h3 id="org2b8e5e0"><span class="section-number-3">2.3</span> Plots</h3>
|
|
<div class="outline-text-3" id="text-2-3">
|
|
<p>
|
|
Effect of Stages vibration (Filtering).
|
|
Effect of Ground motion (Transmissibility).
|
|
Direct Forces (Compliance).
|
|
</p>
|
|
</div>
|
|
</div>
|
|
<div id="outline-container-org71f73c5" class="outline-3">
|
|
<h3 id="org71f73c5"><span class="section-number-3">2.4</span> Save</h3>
|
|
<div class="outline-text-3" id="text-2-4">
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">save(<span class="org-string">'./mat/opt_stiffness_disturbances.mat'</span>, <span class="org-string">'Gd'</span>)
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org6527e58" class="outline-2">
|
|
<h2 id="org6527e58"><span class="section-number-2">3</span> Effect of granite stiffness</h2>
|
|
<div class="outline-text-2" id="text-3">
|
|
<p>
|
|
<a id="orgd4105b6"></a>
|
|
</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 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 1: </span>Figure caption</p>
|
|
</div>
|
|
|
|
<p>
|
|
If we write the equation of motion of the system in Figure <a href="#org8fb9606">1</a>, we obtain:
|
|
</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)
|
|
\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)}
|
|
\end{equation}
|
|
</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’s initialize a soft granite that will act as an isolation stage from ground motion.
|
|
</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>
|
|
</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">
|
|
<p>
|
|
<a id="org5d05990"></a>
|
|
</p>
|
|
</div>
|
|
<div id="outline-container-org2ee8120" class="outline-3">
|
|
<h3 id="org2ee8120"><span class="section-number-3">4.1</span> Load of the identified disturbances and transfer functions</h3>
|
|
<div class="outline-text-3" id="text-4-1">
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">load(<span class="org-string">'./mat/dist_psd.mat'</span>, <span class="org-string">'dist_f'</span>);
|
|
load(<span class="org-string">'./mat/opt_stiffness_disturbances.mat'</span>, <span class="org-string">'Gd'</span>)
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orgb1a3177" class="outline-3">
|
|
<h3 id="orgb1a3177"><span class="section-number-3">4.2</span> Equations</h3>
|
|
</div>
|
|
|
|
<div id="outline-container-org63eade9" class="outline-3">
|
|
<h3 id="org63eade9"><span class="section-number-3">4.3</span> Results</h3>
|
|
<div class="outline-text-3" id="text-4-3">
|
|
<p>
|
|
Effect of all disturbances
|
|
</p>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">freqs = dist_f.f;
|
|
|
|
<span class="org-type">figure</span>;
|
|
hold on;
|
|
<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>
|
|
plot(freqs, sqrt(dist_f.psd_rz)<span class="org-type">.*</span>abs(squeeze(freqresp(Gd{<span class="org-constant">i</span>}(<span class="org-string">'Ez'</span>, <span class="org-string">'Frz_z'</span>), freqs, <span class="org-string">'Hz'</span>))));
|
|
<span class="org-keyword">end</span>
|
|
hold off;
|
|
<span class="org-type">set</span>(<span class="org-variable-name">gca</span>, <span class="org-string">'xscale'</span>, <span class="org-string">'log'</span>); <span class="org-type">set</span>(<span class="org-variable-name">gca</span>, <span class="org-string">'yscale'</span>, <span class="org-string">'log'</span>);
|
|
xlabel(<span class="org-string">'Frequency [Hz]'</span>); ylabel(<span class="org-string">'ASD $\left[\frac{m}{\sqrt{Hz}}\right]$'</span>)
|
|
legend(<span class="org-string">'Location'</span>, <span class="org-string">'southwest'</span>);
|
|
xlim([2, 500]);
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orgef96b89" class="outline-3">
|
|
<h3 id="orgef96b89"><span class="section-number-3">4.4</span> Cumulative Amplitude Spectrum</h3>
|
|
<div class="outline-text-3" id="text-4-4">
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">freqs = dist_f.f;
|
|
|
|
<span class="org-type">figure</span>;
|
|
hold on;
|
|
<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>
|
|
plot(freqs, sqrt(flip(<span class="org-type">-</span>cumtrapz(flip(freqs), flip(dist_f.psd_ty<span class="org-type">.*</span>abs(squeeze(freqresp(Gd{<span class="org-constant">i</span>}(<span class="org-string">'Ez'</span>, <span class="org-string">'Fty_z'</span>), freqs, <span class="org-string">'Hz'</span>)))<span class="org-type">.^</span>2)))), <span class="org-string">'-'</span>, ...
|
|
<span class="org-string">'DisplayName'</span>, sprintf(<span class="org-string">'$k = %.0g$ [N/m]'</span>, Ks(<span class="org-constant">i</span>)));
|
|
<span class="org-keyword">end</span>
|
|
plot([freqs(1) freqs(end)], [10e<span class="org-type">-</span>9 10e<span class="org-type">-</span>9], <span class="org-string">'k--'</span>, <span class="org-string">'HandleVisibility'</span>, <span class="org-string">'off'</span>);
|
|
hold off;
|
|
<span class="org-type">set</span>(<span class="org-variable-name">gca</span>, <span class="org-string">'xscale'</span>, <span class="org-string">'log'</span>); <span class="org-type">set</span>(<span class="org-variable-name">gca</span>, <span class="org-string">'yscale'</span>, <span class="org-string">'log'</span>);
|
|
xlabel(<span class="org-string">'Frequency [Hz]'</span>); ylabel(<span class="org-string">'CAS $[m]$'</span>)
|
|
legend(<span class="org-string">'Location'</span>, <span class="org-string">'southwest'</span>);
|
|
xlim([2, 500]); ylim([1e<span class="org-type">-</span>10 1e<span class="org-type">-</span>6]);
|
|
</pre>
|
|
</div>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">freqs = dist_f.f;
|
|
|
|
<span class="org-type">figure</span>;
|
|
hold on;
|
|
<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>
|
|
plot(freqs, sqrt(flip(<span class="org-type">-</span>cumtrapz(flip(freqs), flip(dist_f.psd_rz<span class="org-type">.*</span>abs(squeeze(freqresp(Gd{<span class="org-constant">i</span>}(<span class="org-string">'Ez'</span>, <span class="org-string">'Frz_z'</span>), freqs, <span class="org-string">'Hz'</span>)))<span class="org-type">.^</span>2)))), <span class="org-string">'-'</span>, ...
|
|
<span class="org-string">'DisplayName'</span>, sprintf(<span class="org-string">'$k = %.0g$ [N/m]'</span>, Ks(<span class="org-constant">i</span>)));
|
|
<span class="org-keyword">end</span>
|
|
plot([freqs(1) freqs(end)], [10e<span class="org-type">-</span>9 10e<span class="org-type">-</span>9], <span class="org-string">'k--'</span>, <span class="org-string">'HandleVisibility'</span>, <span class="org-string">'off'</span>);
|
|
hold off;
|
|
<span class="org-type">set</span>(<span class="org-variable-name">gca</span>, <span class="org-string">'xscale'</span>, <span class="org-string">'log'</span>); <span class="org-type">set</span>(<span class="org-variable-name">gca</span>, <span class="org-string">'yscale'</span>, <span class="org-string">'log'</span>);
|
|
xlabel(<span class="org-string">'Frequency [Hz]'</span>); ylabel(<span class="org-string">'CAS $[m]$'</span>)
|
|
legend(<span class="org-string">'Location'</span>, <span class="org-string">'southwest'</span>);
|
|
xlim([2, 500]); ylim([1e<span class="org-type">-</span>10 1e<span class="org-type">-</span>6]);
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
Ground motion
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">freqs = dist_f.f;
|
|
|
|
<span class="org-type">figure</span>;
|
|
hold on;
|
|
<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>
|
|
plot(freqs, sqrt(flip(<span class="org-type">-</span>cumtrapz(flip(freqs), flip(dist_f.psd_gm<span class="org-type">.*</span>abs(squeeze(freqresp(Gd{<span class="org-constant">i</span>}(<span class="org-string">'Ez'</span>, <span class="org-string">'Dwz'</span>), freqs, <span class="org-string">'Hz'</span>)))<span class="org-type">.^</span>2)))), <span class="org-string">'-'</span>, ...
|
|
<span class="org-string">'DisplayName'</span>, sprintf(<span class="org-string">'$k = %.0g$ [N/m]'</span>, Ks(<span class="org-constant">i</span>)));
|
|
<span class="org-keyword">end</span>
|
|
plot([freqs(1) freqs(end)], [10e<span class="org-type">-</span>9 10e<span class="org-type">-</span>9], <span class="org-string">'k--'</span>, <span class="org-string">'HandleVisibility'</span>, <span class="org-string">'off'</span>);
|
|
hold off;
|
|
<span class="org-type">set</span>(<span class="org-variable-name">gca</span>, <span class="org-string">'xscale'</span>, <span class="org-string">'log'</span>); <span class="org-type">set</span>(<span class="org-variable-name">gca</span>, <span class="org-string">'yscale'</span>, <span class="org-string">'log'</span>);
|
|
xlabel(<span class="org-string">'Frequency [Hz]'</span>); ylabel(<span class="org-string">'CAS $E_y$ $[m]$'</span>)
|
|
legend(<span class="org-string">'Location'</span>, <span class="org-string">'northeast'</span>);
|
|
xlim([2, 500]); ylim([1e<span class="org-type">-</span>10 1e<span class="org-type">-</span>6]);
|
|
</pre>
|
|
</div>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">freqs = dist_f.f;
|
|
|
|
<span class="org-type">figure</span>;
|
|
hold on;
|
|
<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>
|
|
plot(freqs, sqrt(flip(<span class="org-type">-</span>cumtrapz(flip(freqs), flip(dist_f.psd_gm<span class="org-type">.*</span>abs(squeeze(freqresp(Gd{<span class="org-constant">i</span>}(<span class="org-string">'Ex'</span>, <span class="org-string">'Dwx'</span>), freqs, <span class="org-string">'Hz'</span>)))<span class="org-type">.^</span>2)))), <span class="org-string">'-'</span>, ...
|
|
<span class="org-string">'DisplayName'</span>, sprintf(<span class="org-string">'$k = %.0g$ [N/m]'</span>, Ks(<span class="org-constant">i</span>)));
|
|
<span class="org-keyword">end</span>
|
|
plot([freqs(1) freqs(end)], [10e<span class="org-type">-</span>9 10e<span class="org-type">-</span>9], <span class="org-string">'k--'</span>, <span class="org-string">'HandleVisibility'</span>, <span class="org-string">'off'</span>);
|
|
hold off;
|
|
<span class="org-type">set</span>(<span class="org-variable-name">gca</span>, <span class="org-string">'xscale'</span>, <span class="org-string">'log'</span>); <span class="org-type">set</span>(<span class="org-variable-name">gca</span>, <span class="org-string">'yscale'</span>, <span class="org-string">'lin'</span>);
|
|
xlabel(<span class="org-string">'Frequency [Hz]'</span>); ylabel(<span class="org-string">'CAS $E_y$ $[m]$'</span>)
|
|
legend(<span class="org-string">'Location'</span>, <span class="org-string">'northeast'</span>);
|
|
xlim([2, 500]);
|
|
</pre>
|
|
</div>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">freqs = dist_f.f;
|
|
|
|
<span class="org-type">figure</span>;
|
|
hold on;
|
|
<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>
|
|
plot(freqs, sqrt(flip(<span class="org-type">-</span>cumtrapz(flip(freqs), flip(dist_f.psd_gm<span class="org-type">.*</span>abs(squeeze(freqresp(Gd{<span class="org-constant">i</span>}(<span class="org-string">'Ey'</span>, <span class="org-string">'Dwy'</span>), freqs, <span class="org-string">'Hz'</span>)))<span class="org-type">.^</span>2)))), <span class="org-string">'-'</span>, ...
|
|
<span class="org-string">'DisplayName'</span>, sprintf(<span class="org-string">'$k = %.0g$ [N/m]'</span>, Ks(<span class="org-constant">i</span>)));
|
|
<span class="org-keyword">end</span>
|
|
plot([freqs(1) freqs(end)], [10e<span class="org-type">-</span>9 10e<span class="org-type">-</span>9], <span class="org-string">'k--'</span>, <span class="org-string">'HandleVisibility'</span>, <span class="org-string">'off'</span>);
|
|
hold off;
|
|
<span class="org-type">set</span>(<span class="org-variable-name">gca</span>, <span class="org-string">'xscale'</span>, <span class="org-string">'log'</span>); <span class="org-type">set</span>(<span class="org-variable-name">gca</span>, <span class="org-string">'yscale'</span>, <span class="org-string">'lin'</span>);
|
|
xlabel(<span class="org-string">'Frequency [Hz]'</span>); ylabel(<span class="org-string">'CAS $E_y$ $[m]$'</span>)
|
|
legend(<span class="org-string">'Location'</span>, <span class="org-string">'northeast'</span>);
|
|
xlim([2, 500]);
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
Sum of all perturbations
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">psd_tot = zeros(length(freqs), length(Ks));
|
|
|
|
<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>
|
|
psd_tot(<span class="org-type">:</span>,<span class="org-constant">i</span>) = dist_f.psd_gm<span class="org-type">.*</span>abs(squeeze(freqresp(Gd{<span class="org-constant">i</span>}(<span class="org-string">'Ez'</span>, <span class="org-string">'Dwz'</span> ), freqs, <span class="org-string">'Hz'</span>)))<span class="org-type">.^</span>2 <span class="org-type">+</span> ...
|
|
dist_f.psd_ty<span class="org-type">.*</span>abs(squeeze(freqresp(Gd{<span class="org-constant">i</span>}(<span class="org-string">'Ez'</span>, <span class="org-string">'Fty_z'</span>), freqs, <span class="org-string">'Hz'</span>)))<span class="org-type">.^</span>2 <span class="org-type">+</span> ...
|
|
dist_f.psd_rz<span class="org-type">.*</span>abs(squeeze(freqresp(Gd{<span class="org-constant">i</span>}(<span class="org-string">'Ez'</span>, <span class="org-string">'Frz_z'</span>), freqs, <span class="org-string">'Hz'</span>)))<span class="org-type">.^</span>2;
|
|
<span class="org-keyword">end</span>
|
|
</pre>
|
|
</div>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">freqs = dist_f.f;
|
|
|
|
<span class="org-type">figure</span>;
|
|
hold on;
|
|
<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>
|
|
plot(freqs, sqrt(flip(<span class="org-type">-</span>cumtrapz(flip(freqs), flip(psd_tot(<span class="org-type">:</span>,<span class="org-constant">i</span>))))), <span class="org-string">'-'</span>, ...
|
|
<span class="org-string">'DisplayName'</span>, sprintf(<span class="org-string">'$k = %.0g$ [N/m]'</span>, Ks(<span class="org-constant">i</span>)));
|
|
<span class="org-keyword">end</span>
|
|
plot([freqs(1) freqs(end)], [10e<span class="org-type">-</span>9 10e<span class="org-type">-</span>9], <span class="org-string">'k--'</span>, <span class="org-string">'HandleVisibility'</span>, <span class="org-string">'off'</span>);
|
|
hold off;
|
|
<span class="org-type">set</span>(<span class="org-variable-name">gca</span>, <span class="org-string">'xscale'</span>, <span class="org-string">'log'</span>); <span class="org-type">set</span>(<span class="org-variable-name">gca</span>, <span class="org-string">'yscale'</span>, <span class="org-string">'log'</span>);
|
|
xlabel(<span class="org-string">'Frequency [Hz]'</span>); ylabel(<span class="org-string">'CAS $E_z$ $[m]$'</span>)
|
|
legend(<span class="org-string">'Location'</span>, <span class="org-string">'northeast'</span>);
|
|
xlim([1, 500]); ylim([1e<span class="org-type">-</span>10 1e<span class="org-type">-</span>6]);
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org34c0f38" class="outline-2">
|
|
<h2 id="org34c0f38"><span class="section-number-2">5</span> Closed Loop Budget Error</h2>
|
|
<div class="outline-text-2" id="text-5">
|
|
<p>
|
|
<a id="orgd3503fb"></a>
|
|
</p>
|
|
</div>
|
|
<div id="outline-container-orgf2d36a1" class="outline-3">
|
|
<h3 id="orgf2d36a1"><span class="section-number-3">5.1</span> Reduction thanks to feedback - Required bandwidth</h3>
|
|
<div class="outline-text-3" id="text-5-1">
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">wc = 1<span class="org-type">*</span>2<span class="org-type">*</span><span class="org-constant">pi</span>; <span class="org-comment">% [rad/s]</span>
|
|
xic = 0.5;
|
|
|
|
S = (s<span class="org-type">/</span>wc)<span class="org-type">/</span>(1 <span class="org-type">+</span> s<span class="org-type">/</span>wc);
|
|
|
|
bodeFig({S}, logspace(<span class="org-type">-</span>1,2,1000))
|
|
</pre>
|
|
</div>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">wc = [1, 5, 10, 20, 50, 100, 200];
|
|
|
|
S1 = {zeros(length(wc), 1)};
|
|
S2 = {zeros(length(wc), 1)};
|
|
|
|
<span class="org-keyword">for</span> <span class="org-variable-name"><span class="org-constant">j</span></span> = <span class="org-constant">1:length(wc)</span>
|
|
L = (2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>wc(<span class="org-constant">j</span>))<span class="org-type">/</span>s; <span class="org-comment">% Simple integrator</span>
|
|
S1{<span class="org-constant">j</span>} = 1<span class="org-type">/</span>(1 <span class="org-type">+</span> L);
|
|
L = ((2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>wc(<span class="org-constant">j</span>))<span class="org-type">/</span>s)<span class="org-type">^</span>2<span class="org-type">*</span>(1 <span class="org-type">+</span> s<span class="org-type">/</span>(2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>wc(<span class="org-constant">j</span>)<span class="org-type">/</span>2))<span class="org-type">/</span>(1 <span class="org-type">+</span> s<span class="org-type">/</span>(2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>wc(<span class="org-constant">j</span>)<span class="org-type">*</span>2));
|
|
S2{<span class="org-constant">j</span>} = 1<span class="org-type">/</span>(1 <span class="org-type">+</span> L);
|
|
<span class="org-keyword">end</span>
|
|
</pre>
|
|
</div>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">freqs = dist_f.f;
|
|
|
|
<span class="org-type">figure</span>;
|
|
hold on;
|
|
<span class="org-constant">i</span> = 6;
|
|
<span class="org-keyword">for</span> <span class="org-variable-name"><span class="org-constant">j</span></span> = <span class="org-constant">1:length(wc)</span>
|
|
<span class="org-type">set</span>(<span class="org-variable-name">gca</span>,<span class="org-string">'ColorOrderIndex'</span>,<span class="org-constant">j</span>);
|
|
plot(freqs, sqrt(flip(<span class="org-type">-</span>cumtrapz(flip(freqs), flip(abs(squeeze(freqresp(S1{<span class="org-constant">j</span>}, freqs, <span class="org-string">'Hz'</span>)))<span class="org-type">.^</span>2<span class="org-type">.*</span>psd_tot(<span class="org-type">:</span>,<span class="org-constant">i</span>))))), <span class="org-string">'-'</span>, ...
|
|
<span class="org-string">'DisplayName'</span>, sprintf(<span class="org-string">'$\\omega_c = %.0f$ [Hz]'</span>, wc(<span class="org-constant">j</span>)));
|
|
<span class="org-keyword">end</span>
|
|
plot(freqs, sqrt(flip(<span class="org-type">-</span>cumtrapz(flip(freqs), flip(psd_tot(<span class="org-type">:</span>,<span class="org-constant">i</span>))))), <span class="org-string">'k-'</span>, ...
|
|
<span class="org-string">'DisplayName'</span>, <span class="org-string">'Open-Loop'</span>);
|
|
plot([freqs(1) freqs(end)], [10e<span class="org-type">-</span>9 10e<span class="org-type">-</span>9], <span class="org-string">'k--'</span>, <span class="org-string">'HandleVisibility'</span>, <span class="org-string">'off'</span>);
|
|
hold off;
|
|
<span class="org-type">set</span>(<span class="org-variable-name">gca</span>, <span class="org-string">'xscale'</span>, <span class="org-string">'log'</span>); <span class="org-type">set</span>(<span class="org-variable-name">gca</span>, <span class="org-string">'yscale'</span>, <span class="org-string">'log'</span>);
|
|
xlabel(<span class="org-string">'Frequency [Hz]'</span>); ylabel(<span class="org-string">'CAS $E_y$ $[m]$'</span>)
|
|
legend(<span class="org-string">'Location'</span>, <span class="org-string">'northeast'</span>);
|
|
xlim([0.5, 500]); ylim([1e<span class="org-type">-</span>10 1e<span class="org-type">-</span>6]);
|
|
</pre>
|
|
</div>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">wc = logspace(0, 3, 100);
|
|
|
|
Dz1_rms = zeros(length(Ks), length(wc));
|
|
Dz2_rms = zeros(length(Ks), length(wc));
|
|
<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>
|
|
<span class="org-keyword">for</span> <span class="org-variable-name"><span class="org-constant">j</span></span> = <span class="org-constant">1:length(wc)</span>
|
|
L = (2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>wc(<span class="org-constant">j</span>))<span class="org-type">/</span>s;
|
|
Dz1_rms(<span class="org-constant">i</span>, <span class="org-constant">j</span>) = sqrt(trapz(freqs, abs(squeeze(freqresp(1<span class="org-type">/</span>(1 <span class="org-type">+</span> L), freqs, <span class="org-string">'Hz'</span>)))<span class="org-type">.^</span>2<span class="org-type">.*</span>psd_tot(<span class="org-type">:</span>,<span class="org-constant">i</span>)));
|
|
|
|
L = ((2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>wc(<span class="org-constant">j</span>))<span class="org-type">/</span>s)<span class="org-type">^</span>2<span class="org-type">*</span>(1 <span class="org-type">+</span> s<span class="org-type">/</span>(2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>wc(<span class="org-constant">j</span>)<span class="org-type">/</span>2))<span class="org-type">/</span>(1 <span class="org-type">+</span> s<span class="org-type">/</span>(2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>wc(<span class="org-constant">j</span>)<span class="org-type">*</span>2));
|
|
Dz2_rms(<span class="org-constant">i</span>, <span class="org-constant">j</span>) = sqrt(trapz(freqs, abs(squeeze(freqresp(1<span class="org-type">/</span>(1 <span class="org-type">+</span> L), freqs, <span class="org-string">'Hz'</span>)))<span class="org-type">.^</span>2<span class="org-type">.*</span>psd_tot(<span class="org-type">:</span>,<span class="org-constant">i</span>)));
|
|
<span class="org-keyword">end</span>
|
|
<span class="org-keyword">end</span>
|
|
</pre>
|
|
</div>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">freqs = dist_f.f;
|
|
|
|
<span class="org-type">figure</span>;
|
|
hold on;
|
|
<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>
|
|
<span class="org-type">set</span>(<span class="org-variable-name">gca</span>,<span class="org-string">'ColorOrderIndex'</span>,<span class="org-constant">i</span>);
|
|
plot(wc, Dz1_rms(<span class="org-constant">i</span>, <span class="org-type">:</span>), <span class="org-string">'-'</span>, ...
|
|
<span class="org-string">'DisplayName'</span>, sprintf(<span class="org-string">'$k = %.0g$ [N/m]'</span>, Ks(<span class="org-constant">i</span>)))
|
|
|
|
<span class="org-type">set</span>(<span class="org-variable-name">gca</span>,<span class="org-string">'ColorOrderIndex'</span>,<span class="org-constant">i</span>);
|
|
plot(wc, Dz2_rms(<span class="org-constant">i</span>, <span class="org-type">:</span>), <span class="org-string">'--'</span>, ...
|
|
<span class="org-string">'HandleVisibility'</span>, <span class="org-string">'off'</span>)
|
|
<span class="org-keyword">end</span>
|
|
hold off;
|
|
<span class="org-type">set</span>(<span class="org-variable-name">gca</span>, <span class="org-string">'xscale'</span>, <span class="org-string">'log'</span>); <span class="org-type">set</span>(<span class="org-variable-name">gca</span>, <span class="org-string">'yscale'</span>, <span class="org-string">'log'</span>);
|
|
xlabel(<span class="org-string">'Control Bandwidth [Hz]'</span>); ylabel(<span class="org-string">'$E_z\ [m, rms]$'</span>)
|
|
legend(<span class="org-string">'Location'</span>, <span class="org-string">'southwest'</span>);
|
|
xlim([1, 500]);
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
<div id="outline-container-orgea74617" class="outline-2">
|
|
<h2 id="orgea74617"><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. 14:57</p>
|
|
</div>
|
|
</body>
|
|
</html>
|