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Add analysis about Cedrat Piezo actuators
2019-10-29 10:52:59 +01:00
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<h1 class="title">Cedrat Actuator</h1>
<div id="table-of-contents">
<h2>Table of Contents</h2>
<div id="text-table-of-contents">
<ul>
<li><a href="#org001e432">1. Documentation</a></li>
<li><a href="#orgc953781">2. Parameters</a></li>
<li><a href="#org168aa38">3. Identification</a></li>
<li><a href="#org0681d7e">4. Integral Force Feedback</a></li>
<li><a href="#org364fbd0">5. Damped System</a></li>
</ul>
</div>
</div>
<div id="outline-container-org001e432" class="outline-2">
<h2 id="org001e432"><span class="section-number-2">1</span> Documentation</h2>
<div class="outline-text-2" id="text-1">
<ul class="org-ul">
<li>Blocked force: \(1400\ N\)</li>
<li>Stiffness: \(10.8\ N/\mu m\)</li>
<li>Resonance (free-free): \(6450\ Hz\)</li>
<li>Resonance (blocked-free): \(1750\ Hz\)</li>
<li>Height: \(45\ mm\)</li>
<li>Length: \(80\ mm\)</li>
<li>Width: \(22\ mm\)</li>
<li>Mass: \(160\ g\)</li>
</ul>
</div>
</div>
<div id="outline-container-orgc953781" class="outline-2">
<h2 id="orgc953781"><span class="section-number-2">2</span> Parameters</h2>
<div class="outline-text-2" id="text-2">
<div class="org-src-container">
<pre class="src src-matlab">cedrat.L = <span class="org-highlight-numbers-number">80</span>; <span class="org-comment">% Total Width of the Actuator[mm]</span>
cedrat.H = <span class="org-highlight-numbers-number">45</span>; <span class="org-comment">% Total Height of the Actuator [mm]</span>
cedrat.L2 = sqrt<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbow-delimiters-depth-2">(</span>cedrat.L<span class="org-type">/</span><span class="org-highlight-numbers-number">2</span><span class="org-rainbow-delimiters-depth-2">)</span><span class="org-type">^</span><span class="org-highlight-numbers-number">2</span> <span class="org-type">+</span> <span class="org-rainbow-delimiters-depth-2">(</span>cedrat.H<span class="org-type">/</span><span class="org-highlight-numbers-number">2</span><span class="org-rainbow-delimiters-depth-2">)</span><span class="org-type">^</span><span class="org-highlight-numbers-number">2</span><span class="org-rainbow-delimiters-depth-1">)</span>; <span class="org-comment">% Length of the elipsoidal sections [mm]</span>
cedrat.alpha = <span class="org-highlight-numbers-number">180</span><span class="org-type">/</span><span class="org-constant">pi</span><span class="org-type">*</span>atan2<span class="org-rainbow-delimiters-depth-1">(</span>cedrat.L<span class="org-type">/</span><span class="org-highlight-numbers-number">2</span>, cedrat.H<span class="org-type">/</span><span class="org-highlight-numbers-number">2</span><span class="org-rainbow-delimiters-depth-1">)</span>; <span class="org-comment">% [deg]</span>
cedrat.mtot = <span class="org-highlight-numbers-number">0</span>.<span class="org-highlight-numbers-number">160</span>; <span class="org-comment">% Total mass of the Actuator [kg]</span>
cedrat.m = cedrat.mtot<span class="org-type">/</span><span class="org-highlight-numbers-number">6</span>; <span class="org-comment">% Mass of each single element [kg]</span>
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab">cedrat.k = <span class="org-highlight-numbers-number">1e9</span>; <span class="org-comment">% Linear Stiffness of each "blade" [N/m]</span>
cedrat.c = <span class="org-highlight-numbers-number">0</span>.<span class="org-highlight-numbers-number">1</span><span class="org-type">*</span>sqrt<span class="org-rainbow-delimiters-depth-1">(</span>cedrat.mtot<span class="org-type">*</span>cedrat.k<span class="org-rainbow-delimiters-depth-1">)</span>; <span class="org-comment">% [N/(m/s)]</span>
cedrat.ka = <span class="org-highlight-numbers-number">5e7</span>; <span class="org-comment">% Linear Stiffness of the stack [N/m]</span>
cedrat.ca = <span class="org-highlight-numbers-number">0</span>.<span class="org-highlight-numbers-number">1</span><span class="org-type">*</span>sqrt<span class="org-rainbow-delimiters-depth-1">(</span>cedrat.mtot<span class="org-type">*</span>cedrat.ka<span class="org-rainbow-delimiters-depth-1">)</span>; <span class="org-comment">% [N/(m/s)]</span>
cedrat.kr = <span class="org-highlight-numbers-number">10</span>; <span class="org-comment">% Rotation Stiffness [N*m/(deg)]</span>
cedrat.cr = <span class="org-highlight-numbers-number">0</span>.<span class="org-highlight-numbers-number">0001</span>; <span class="org-comment">% Rotation Damping [N*m/(deg/s)]</span>
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab">K_iff = tf<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">0</span><span class="org-rainbow-delimiters-depth-1">)</span>;
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab">% dummy_mass = <span class="org-highlight-numbers-number">140</span>; % [kg]
dummy_mass = <span class="org-highlight-numbers-number">1</span>; <span class="org-comment">% [kg]</span>
</pre>
</div>
</div>
</div>
<div id="outline-container-org168aa38" class="outline-2">
<h2 id="org168aa38"><span class="section-number-2">3</span> Identification</h2>
<div class="outline-text-2" id="text-3">
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Options for Linearized</span></span>
options = linearizeOptions;
options.SampleTime = <span class="org-highlight-numbers-number">0</span>;
<span class="org-matlab-cellbreak"><span class="org-comment">%% Name of the Simulink File</span></span>
mdl = <span class="org-string">'cedrat_piezo'</span>;
<span class="org-matlab-cellbreak"><span class="org-comment">%% Input/Output definition</span></span>
io<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">1</span><span class="org-rainbow-delimiters-depth-1">)</span> = linio<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbow-delimiters-depth-2">[</span>mdl, <span class="org-string">'/F'</span><span class="org-rainbow-delimiters-depth-2">]</span>, <span class="org-highlight-numbers-number">1</span>, <span class="org-string">'input'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
io<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">2</span><span class="org-rainbow-delimiters-depth-1">)</span> = linio<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbow-delimiters-depth-2">[</span>mdl, <span class="org-string">'/Fz'</span><span class="org-rainbow-delimiters-depth-2">]</span>, <span class="org-highlight-numbers-number">1</span>, <span class="org-string">'input'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
io<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">3</span><span class="org-rainbow-delimiters-depth-1">)</span> = linio<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbow-delimiters-depth-2">[</span>mdl, <span class="org-string">'/Dw'</span><span class="org-rainbow-delimiters-depth-2">]</span>, <span class="org-highlight-numbers-number">1</span>, <span class="org-string">'input'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
io<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">4</span><span class="org-rainbow-delimiters-depth-1">)</span> = linio<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbow-delimiters-depth-2">[</span>mdl, <span class="org-string">'/Dz'</span><span class="org-rainbow-delimiters-depth-2">]</span>, <span class="org-highlight-numbers-number">1</span>, <span class="org-string">'output'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
io<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">5</span><span class="org-rainbow-delimiters-depth-1">)</span> = linio<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbow-delimiters-depth-2">[</span>mdl, <span class="org-string">'/Fm'</span><span class="org-rainbow-delimiters-depth-2">]</span>, <span class="org-highlight-numbers-number">1</span>, <span class="org-string">'output'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
<span class="org-matlab-cellbreak"><span class="org-comment">%% Run the linearization</span></span>
G = linearize<span class="org-rainbow-delimiters-depth-1">(</span>mdl, io, options<span class="org-rainbow-delimiters-depth-1">)</span>;
G.InputName = <span class="org-rainbow-delimiters-depth-1">{</span><span class="org-string">'F'</span>, <span class="org-string">'Fz'</span>, <span class="org-string">'Dw'</span><span class="org-rainbow-delimiters-depth-1">}</span>;
G.OutputName = <span class="org-rainbow-delimiters-depth-1">{</span><span class="org-string">'Dz'</span>, <span class="org-string">'Fm'</span><span class="org-rainbow-delimiters-depth-1">}</span>;
</pre>
</div>
<div id="org829aed3" class="figure">
<p><img src="figs/cedrat_piezo_identified_tf.png" alt="cedrat_piezo_identified_tf.png" />
</p>
<p><span class="figure-number">Figure 1: </span>Identified Transfer function (<a href="./figs/cedrat_piezo_identified_tf.png">png</a>, <a href="./figs/cedrat_piezo_identified_tf.pdf">pdf</a>)</p>
</div>
</div>
</div>
<div id="outline-container-org0681d7e" class="outline-2">
<h2 id="org0681d7e"><span class="section-number-2">4</span> Integral Force Feedback</h2>
<div class="outline-text-2" id="text-4">
<div id="orgf87cbd6" class="figure">
<p><img src="figs/iff_plant_cedrat.png" alt="iff_plant_cedrat.png" />
</p>
<p><span class="figure-number">Figure 2: </span>Transfer function from \(F\) to \(F_m\) (<a href="./figs/iff_plant_cedrat.png">png</a>, <a href="./figs/iff_plant_cedrat.pdf">pdf</a>)</p>
</div>
<div class="org-src-container">
<pre class="src src-matlab">K_iff = <span class="org-type">-</span><span class="org-highlight-numbers-number">100000</span><span class="org-type">/</span>s<span class="org-type">*</span><span class="org-rainbow-delimiters-depth-1">(</span>s<span class="org-type">/</span><span class="org-highlight-numbers-number">2</span><span class="org-type">/</span><span class="org-constant">pi</span><span class="org-type">/</span><span class="org-highlight-numbers-number">100</span><span class="org-rainbow-delimiters-depth-1">)</span><span class="org-type">/</span><span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">1</span> <span class="org-type">+</span> s<span class="org-type">/</span><span class="org-highlight-numbers-number">2</span><span class="org-type">/</span><span class="org-constant">pi</span><span class="org-type">/</span><span class="org-highlight-numbers-number">100</span><span class="org-rainbow-delimiters-depth-1">)</span><span class="org-type">*</span><span class="org-rainbow-delimiters-depth-1">(</span>s<span class="org-type">/</span><span class="org-highlight-numbers-number">2</span><span class="org-type">/</span><span class="org-constant">pi</span><span class="org-type">/</span><span class="org-highlight-numbers-number">100</span><span class="org-rainbow-delimiters-depth-1">)</span><span class="org-type">/</span><span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">1</span> <span class="org-type">+</span> s<span class="org-type">/</span><span class="org-highlight-numbers-number">2</span><span class="org-type">/</span><span class="org-constant">pi</span><span class="org-type">/</span><span class="org-highlight-numbers-number">100</span><span class="org-rainbow-delimiters-depth-1">)</span>;
</pre>
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<div id="org97388b4" class="figure">
<p><img src="figs/iff_open_loop_cedrat.png" alt="iff_open_loop_cedrat.png" />
</p>
<p><span class="figure-number">Figure 3: </span>Open Loop Gain Transfer Function - Cedrat (<a href="./figs/iff_open_loop_cedrat.png">png</a>, <a href="./figs/iff_open_loop_cedrat.pdf">pdf</a>)</p>
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<div id="outline-container-org364fbd0" class="outline-2">
<h2 id="org364fbd0"><span class="section-number-2">5</span> Damped System</h2>
<div class="outline-text-2" id="text-5">
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Options for Linearized</span></span>
options = linearizeOptions;
options.SampleTime = <span class="org-highlight-numbers-number">0</span>;
<span class="org-matlab-cellbreak"><span class="org-comment">%% Name of the Simulink File</span></span>
mdl = <span class="org-string">'cedrat_piezo'</span>;
<span class="org-matlab-cellbreak"><span class="org-comment">%% Input/Output definition</span></span>
io<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">1</span><span class="org-rainbow-delimiters-depth-1">)</span> = linio<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbow-delimiters-depth-2">[</span>mdl, <span class="org-string">'/F'</span><span class="org-rainbow-delimiters-depth-2">]</span>, <span class="org-highlight-numbers-number">1</span>, <span class="org-string">'input'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
io<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">2</span><span class="org-rainbow-delimiters-depth-1">)</span> = linio<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbow-delimiters-depth-2">[</span>mdl, <span class="org-string">'/Fz'</span><span class="org-rainbow-delimiters-depth-2">]</span>, <span class="org-highlight-numbers-number">1</span>, <span class="org-string">'input'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
io<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">3</span><span class="org-rainbow-delimiters-depth-1">)</span> = linio<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbow-delimiters-depth-2">[</span>mdl, <span class="org-string">'/Dw'</span><span class="org-rainbow-delimiters-depth-2">]</span>, <span class="org-highlight-numbers-number">1</span>, <span class="org-string">'input'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
io<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">4</span><span class="org-rainbow-delimiters-depth-1">)</span> = linio<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbow-delimiters-depth-2">[</span>mdl, <span class="org-string">'/Dz'</span><span class="org-rainbow-delimiters-depth-2">]</span>, <span class="org-highlight-numbers-number">1</span>, <span class="org-string">'output'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
io<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">5</span><span class="org-rainbow-delimiters-depth-1">)</span> = linio<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbow-delimiters-depth-2">[</span>mdl, <span class="org-string">'/Fm'</span><span class="org-rainbow-delimiters-depth-2">]</span>, <span class="org-highlight-numbers-number">1</span>, <span class="org-string">'output'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
<span class="org-matlab-cellbreak"><span class="org-comment">%% Run the linearization</span></span>
G_iff = linearize<span class="org-rainbow-delimiters-depth-1">(</span>mdl, io, options<span class="org-rainbow-delimiters-depth-1">)</span>;
G_iff.InputName = <span class="org-rainbow-delimiters-depth-1">{</span><span class="org-string">'F'</span>, <span class="org-string">'Fz'</span>, <span class="org-string">'Dw'</span><span class="org-rainbow-delimiters-depth-1">}</span>;
G_iff.OutputName = <span class="org-rainbow-delimiters-depth-1">{</span><span class="org-string">'Dz'</span>, <span class="org-string">'Fm'</span><span class="org-rainbow-delimiters-depth-1">}</span>;
</pre>
</div>
<div id="orgd56e05f" class="figure">
<p><img src="figs/cedrat_iff_piezo_identified_tf.png" alt="cedrat_iff_piezo_identified_tf.png" />
</p>
<p><span class="figure-number">Figure 4: </span>Identified Transfer function (<a href="./figs/cedrat_iff_piezo_identified_tf.png">png</a>, <a href="./figs/cedrat_iff_piezo_identified_tf.pdf">pdf</a>)</p>
</div>
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<div id="postamble" class="status">
<p class="author">Author: Dehaeze Thomas</p>
<p class="date">Created: 2019-10-29 mar. 10:51</p>
<p class="validation"><a href="http://validator.w3.org/check?uri=referer">Validate</a></p>
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