Add analysis about amplified piezo
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docs/amplified_piezoelectric_stack.html
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<?xml version="1.0" encoding="utf-8"?>
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<head>
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<!-- 2020-05-20 mer. 15:49 -->
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<meta http-equiv="Content-Type" content="text/html;charset=utf-8" />
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<title>Amplified Piezoelectric Stack Actuator</title>
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<meta name="generator" content="Org mode" />
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<meta name="author" content="Dehaeze Thomas" />
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<div id="org-div-home-and-up">
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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">Amplified Piezoelectric Stack Actuator</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="#org996fd7c">1. Simplified Model</a>
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<ul>
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<li><a href="#org47cc3c4">1.1. Parameters</a></li>
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<li><a href="#org3b3c7ac">1.2. Identification</a></li>
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<li><a href="#org97f356d">1.3. Root Locus</a></li>
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</ul>
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</li>
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<li><a href="#orgf1a765f">2. Rotating X-Y platform</a>
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<ul>
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<li><a href="#org0dc544d">2.1. Parameters</a></li>
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<li><a href="#org08e3567">2.2. Identification</a></li>
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<li><a href="#orgbba342e">2.3. Root Locus</a></li>
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<li><a href="#org069f401">2.4. Analysis</a></li>
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</ul>
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</li>
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</ul>
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</div>
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</div>
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<p>
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The presented model is based on <a class='org-ref-reference' href="#souleille18_concep_activ_mount_space_applic">souleille18_concep_activ_mount_space_applic</a>.
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</p>
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<p>
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The model represents the amplified piezo APA100M from Cedrat-Technologies (Figure <a href="#orgb707bbd">1</a>).
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The parameters are shown in the table below.
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</p>
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<div id="orgb707bbd" class="figure">
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<p><img src="./figs/souleille18_model_piezo.png" alt="souleille18_model_piezo.png" />
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</p>
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<p><span class="figure-number">Figure 1: </span>Picture of an APA100M from Cedrat Technologies. Simplified model of a one DoF payload mounted on such isolator</p>
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</div>
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<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
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<caption class="t-above"><span class="table-number">Table 1:</span> Parameters used for the model of the APA 100M</caption>
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<colgroup>
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<col class="org-left" />
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<col class="org-left" />
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<col class="org-left" />
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</colgroup>
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<thead>
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<tr>
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<th scope="col" class="org-left"> </th>
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<th scope="col" class="org-left">Value</th>
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<th scope="col" class="org-left">Meaning</th>
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</tr>
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</thead>
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<tbody>
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<tr>
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<td class="org-left">\(m\)</td>
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<td class="org-left">\(1\,[kg]\)</td>
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<td class="org-left">Payload mass</td>
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</tr>
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<tr>
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<td class="org-left">\(k_e\)</td>
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<td class="org-left">\(4.8\,[N/\mu m]\)</td>
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<td class="org-left">Stiffness used to adjust the pole of the isolator</td>
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</tr>
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<tr>
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<td class="org-left">\(k_1\)</td>
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<td class="org-left">\(0.96\,[N/\mu m]\)</td>
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<td class="org-left">Stiffness of the metallic suspension when the stack is removed</td>
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</tr>
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<tr>
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<td class="org-left">\(k_a\)</td>
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<td class="org-left">\(65\,[N/\mu m]\)</td>
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<td class="org-left">Stiffness of the actuator</td>
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</tr>
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<tr>
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<td class="org-left">\(c_1\)</td>
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<td class="org-left">\(10\,[N/(m/s)]\)</td>
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<td class="org-left">Added viscous damping</td>
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</tr>
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</tbody>
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</table>
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<div id="outline-container-org996fd7c" class="outline-2">
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<h2 id="org996fd7c"><span class="section-number-2">1</span> Simplified Model</h2>
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<div class="outline-text-2" id="text-1">
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</div>
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<div id="outline-container-org47cc3c4" class="outline-3">
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<h3 id="org47cc3c4"><span class="section-number-3">1.1</span> Parameters</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">m = 1; % [kg]
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ke = 4.8e6; % [N/m]
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ce = 5; % [N/(m/s)]
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me = 0.001; % [kg]
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k1 = 0.96e6; % [N/m]
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c1 = 10; % [N/(m/s)]
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ka = 65e6; % [N/m]
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ca = 5; % [N/(m/s)]
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ma = 0.001; % [kg]
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h = 0.2; % [m]
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</pre>
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</div>
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<p>
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IFF Controller:
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</p>
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<div class="org-src-container">
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<pre class="src src-matlab">Kiff = -8000/s;
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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-org3b3c7ac" class="outline-3">
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<h3 id="org3b3c7ac"><span class="section-number-3">1.2</span> Identification</h3>
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<div class="outline-text-3" id="text-1-2">
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<p>
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Identification in open-loop.
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</p>
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<div class="org-src-container">
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<pre class="src src-matlab">%% Name of the Simulink File
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mdl = 'amplified_piezo_model';
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%% Input/Output definition
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clear io; io_i = 1;
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io(io_i) = linio([mdl, '/w'], 1, 'openinput'); io_i = io_i + 1; % Base Motion
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io(io_i) = linio([mdl, '/f'], 1, 'openinput'); io_i = io_i + 1; % Actuator Inputs
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io(io_i) = linio([mdl, '/F'], 1, 'openinput'); io_i = io_i + 1; % External Force
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io(io_i) = linio([mdl, '/Fs'], 3, 'openoutput'); io_i = io_i + 1; % Force Sensors
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io(io_i) = linio([mdl, '/x1'], 1, 'openoutput'); io_i = io_i + 1; % Mass displacement
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G = linearize(mdl, io, 0);
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G.InputName = {'w', 'f', 'F'};
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G.OutputName = {'Fs', 'x1'};
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</pre>
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</div>
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<p>
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Identification in closed-loop.
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</p>
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<div class="org-src-container">
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<pre class="src src-matlab">%% Name of the Simulink File
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mdl = 'amplified_piezo_model';
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%% Input/Output definition
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clear io; io_i = 1;
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io(io_i) = linio([mdl, '/w'], 1, 'input'); io_i = io_i + 1; % Base Motion
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io(io_i) = linio([mdl, '/f'], 1, 'input'); io_i = io_i + 1; % Actuator Inputs
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io(io_i) = linio([mdl, '/F'], 1, 'input'); io_i = io_i + 1; % External Force
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io(io_i) = linio([mdl, '/Fs'], 3, 'output'); io_i = io_i + 1; % Force Sensors
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io(io_i) = linio([mdl, '/x1'], 1, 'output'); io_i = io_i + 1; % Mass displacement
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Giff = linearize(mdl, io, 0);
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Giff.InputName = {'w', 'f', 'F'};
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Giff.OutputName = {'Fs', 'x1'};
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</pre>
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</div>
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<div id="org55d1535" class="figure">
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<p><img src="figs/amplified_piezo_tf_ol_and_cl.png" alt="amplified_piezo_tf_ol_and_cl.png" />
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</p>
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<p><span class="figure-number">Figure 2: </span>Matrix of transfer functions from input to output in open loop (blue) and closed loop (red)</p>
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</div>
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</div>
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</div>
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<div id="outline-container-org97f356d" class="outline-3">
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<h3 id="org97f356d"><span class="section-number-3">1.3</span> Root Locus</h3>
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<div class="outline-text-3" id="text-1-3">
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<div id="org85cd6e5" class="figure">
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<p><img src="figs/amplified_piezo_root_locus.png" alt="amplified_piezo_root_locus.png" />
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</p>
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<p><span class="figure-number">Figure 3: </span>Root Locus</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-orgf1a765f" class="outline-2">
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<h2 id="orgf1a765f"><span class="section-number-2">2</span> Rotating X-Y platform</h2>
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<div class="outline-text-2" id="text-2">
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</div>
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<div id="outline-container-org0dc544d" class="outline-3">
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<h3 id="org0dc544d"><span class="section-number-3">2.1</span> Parameters</h3>
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<div class="outline-text-3" id="text-2-1">
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<div class="org-src-container">
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<pre class="src src-matlab">m = 1; % [kg]
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ke = 4.8e6; % [N/m]
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ce = 5; % [N/(m/s)]
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me = 0.001; % [kg]
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k1 = 0.96e6; % [N/m]
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c1 = 10; % [N/(m/s)]
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ka = 65e6; % [N/m]
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ca = 5; % [N/(m/s)]
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ma = 0.001; % [kg]
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h = 0.2; % [m]
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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">Kiff = tf(0);
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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-org08e3567" class="outline-3">
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<h3 id="org08e3567"><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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Rotating speed in rad/s:
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</p>
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<div class="org-src-container">
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<pre class="src src-matlab">Ws = 2*pi*[0, 1, 10, 100];
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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">Gs = {zeros(length(Ws), 1)};
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</pre>
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</div>
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<p>
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Identification in open-loop.
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</p>
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<div class="org-src-container">
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<pre class="src src-matlab">%% Name of the Simulink File
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mdl = 'amplified_piezo_xy_rotating_stage';
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%% Input/Output definition
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clear io; io_i = 1;
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io(io_i) = linio([mdl, '/fx'], 1, 'openinput'); io_i = io_i + 1;
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io(io_i) = linio([mdl, '/fy'], 1, 'openinput'); io_i = io_i + 1;
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io(io_i) = linio([mdl, '/Fs'], 1, 'openoutput'); io_i = io_i + 1;
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io(io_i) = linio([mdl, '/Fs'], 2, 'openoutput'); io_i = io_i + 1;
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for i = 1:length(Ws)
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ws = Ws(i);
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G = linearize(mdl, io, 0);
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G.InputName = {'fx', 'fy'};
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G.OutputName = {'Fsx', 'Fsy'};
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Gs(i) = {G};
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end
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</pre>
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</div>
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<div id="orga4fc975" class="figure">
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<p><img src="figs/amplitifed_piezo_xy_rotation_plant_iff.png" alt="amplitifed_piezo_xy_rotation_plant_iff.png" />
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</p>
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<p><span class="figure-number">Figure 4: </span>Transfer function matrix from forces to force sensors for multiple rotation speed</p>
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</div>
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</div>
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</div>
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<div id="outline-container-orgbba342e" class="outline-3">
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<h3 id="orgbba342e"><span class="section-number-3">2.3</span> Root Locus</h3>
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<div class="outline-text-3" id="text-2-3">
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<div id="orgccd3396" class="figure">
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<p><img src="figs/amplified_piezo_xy_rotation_root_locus.png" alt="amplified_piezo_xy_rotation_root_locus.png" />
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</p>
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<p><span class="figure-number">Figure 5: </span>Root locus for 3 rotating speed</p>
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</div>
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</div>
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</div>
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<div id="outline-container-org069f401" class="outline-3">
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<h3 id="org069f401"><span class="section-number-3">2.4</span> Analysis</h3>
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<div class="outline-text-3" id="text-2-4">
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<p>
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The negative stiffness induced by the rotation is equal to \(m \omega_0^2\).
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Thus, the maximum rotation speed where IFF can be applied is:
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\[ \omega_\text{max} = \sqrt{\frac{k_1}{m}} \approx 156\,[Hz] \]
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</p>
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<p>
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Let’s verify that.
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</p>
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<div class="org-src-container">
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<pre class="src src-matlab">Ws = 2*pi*[140, 160];
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</pre>
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</div>
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<p>
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Identification
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</p>
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<div class="org-src-container">
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<pre class="src src-matlab">%% Name of the Simulink File
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mdl = 'amplified_piezo_xy_rotating_stage';
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%% Input/Output definition
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clear io; io_i = 1;
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io(io_i) = linio([mdl, '/fx'], 1, 'openinput'); io_i = io_i + 1;
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io(io_i) = linio([mdl, '/fy'], 1, 'openinput'); io_i = io_i + 1;
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io(io_i) = linio([mdl, '/Fs'], 1, 'openoutput'); io_i = io_i + 1;
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io(io_i) = linio([mdl, '/Fs'], 2, 'openoutput'); io_i = io_i + 1;
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for i = 1:length(Ws)
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ws = Ws(i);
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G = linearize(mdl, io, 0);
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G.InputName = {'fx', 'fy'};
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G.OutputName = {'Fsx', 'Fsy'};
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Gs(i) = {G};
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end
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</pre>
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</div>
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<div id="orgca23612" class="figure">
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<p><img src="figs/amplified_piezo_xy_rotating_unstable_root_locus.png" alt="amplified_piezo_xy_rotating_unstable_root_locus.png" />
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</p>
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<p><span class="figure-number">Figure 6: </span>Root Locus for the two considered rotation speed. For the red curve, the system is unstable.</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>
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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: 2020-05-20 mer. 15:49</p>
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</div>
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</body>
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</html>
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org/amplified_piezoelectric_stack.org
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org/amplified_piezoelectric_stack.org
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|
||||
#+TITLE: Amplified Piezoelectric Stack Actuator
|
||||
#+SETUPFILE: ./setup/org-setup-file.org
|
||||
|
||||
* Introduction :ignore:
|
||||
The presented model is based on cite:souleille18_concep_activ_mount_space_applic.
|
||||
|
||||
The model represents the amplified piezo APA100M from Cedrat-Technologies (Figure [[fig:souleille18_model_piezo]]).
|
||||
The parameters are shown in the table below.
|
||||
|
||||
#+name: fig:souleille18_model_piezo
|
||||
#+caption: Picture of an APA100M from Cedrat Technologies. Simplified model of a one DoF payload mounted on such isolator
|
||||
[[file:./figs/souleille18_model_piezo.png]]
|
||||
|
||||
#+caption: Parameters used for the model of the APA 100M
|
||||
| | Value | Meaning |
|
||||
|-------+-------------------+----------------------------------------------------------------|
|
||||
| $m$ | $1\,[kg]$ | Payload mass |
|
||||
| $k_e$ | $4.8\,[N/\mu m]$ | Stiffness used to adjust the pole of the isolator |
|
||||
| $k_1$ | $0.96\,[N/\mu m]$ | Stiffness of the metallic suspension when the stack is removed |
|
||||
| $k_a$ | $65\,[N/\mu m]$ | Stiffness of the actuator |
|
||||
| $c_1$ | $10\,[N/(m/s)]$ | Added viscous damping |
|
||||
|
||||
* Simplified Model
|
||||
** Matlab Init :noexport:ignore:
|
||||
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
|
||||
<<matlab-dir>>
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab :exports none :results silent :noweb yes
|
||||
<<matlab-init>>
|
||||
#+end_src
|
||||
|
||||
#+BEGIN_SRC matlab
|
||||
simulinkproject('../');
|
||||
#+END_SRC
|
||||
|
||||
#+begin_src matlab
|
||||
open 'amplified_piezo_model.slx'
|
||||
#+end_src
|
||||
|
||||
** Parameters
|
||||
#+begin_src matlab
|
||||
m = 1; % [kg]
|
||||
|
||||
ke = 4.8e6; % [N/m]
|
||||
ce = 5; % [N/(m/s)]
|
||||
me = 0.001; % [kg]
|
||||
|
||||
k1 = 0.96e6; % [N/m]
|
||||
c1 = 10; % [N/(m/s)]
|
||||
|
||||
ka = 65e6; % [N/m]
|
||||
ca = 5; % [N/(m/s)]
|
||||
ma = 0.001; % [kg]
|
||||
|
||||
h = 0.2; % [m]
|
||||
#+end_src
|
||||
|
||||
IFF Controller:
|
||||
#+begin_src matlab
|
||||
Kiff = -8000/s;
|
||||
#+end_src
|
||||
|
||||
** Identification
|
||||
Identification in open-loop.
|
||||
#+begin_src matlab
|
||||
%% Name of the Simulink File
|
||||
mdl = 'amplified_piezo_model';
|
||||
|
||||
%% Input/Output definition
|
||||
clear io; io_i = 1;
|
||||
io(io_i) = linio([mdl, '/w'], 1, 'openinput'); io_i = io_i + 1; % Base Motion
|
||||
io(io_i) = linio([mdl, '/f'], 1, 'openinput'); io_i = io_i + 1; % Actuator Inputs
|
||||
io(io_i) = linio([mdl, '/F'], 1, 'openinput'); io_i = io_i + 1; % External Force
|
||||
|
||||
io(io_i) = linio([mdl, '/Fs'], 3, 'openoutput'); io_i = io_i + 1; % Force Sensors
|
||||
io(io_i) = linio([mdl, '/x1'], 1, 'openoutput'); io_i = io_i + 1; % Mass displacement
|
||||
|
||||
G = linearize(mdl, io, 0);
|
||||
G.InputName = {'w', 'f', 'F'};
|
||||
G.OutputName = {'Fs', 'x1'};
|
||||
#+end_src
|
||||
|
||||
Identification in closed-loop.
|
||||
#+begin_src matlab
|
||||
%% Name of the Simulink File
|
||||
mdl = 'amplified_piezo_model';
|
||||
|
||||
%% Input/Output definition
|
||||
clear io; io_i = 1;
|
||||
io(io_i) = linio([mdl, '/w'], 1, 'input'); io_i = io_i + 1; % Base Motion
|
||||
io(io_i) = linio([mdl, '/f'], 1, 'input'); io_i = io_i + 1; % Actuator Inputs
|
||||
io(io_i) = linio([mdl, '/F'], 1, 'input'); io_i = io_i + 1; % External Force
|
||||
|
||||
io(io_i) = linio([mdl, '/Fs'], 3, 'output'); io_i = io_i + 1; % Force Sensors
|
||||
io(io_i) = linio([mdl, '/x1'], 1, 'output'); io_i = io_i + 1; % Mass displacement
|
||||
|
||||
Giff = linearize(mdl, io, 0);
|
||||
Giff.InputName = {'w', 'f', 'F'};
|
||||
Giff.OutputName = {'Fs', 'x1'};
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab :exports none
|
||||
freqs = logspace(1, 3, 1000);
|
||||
|
||||
figure;
|
||||
|
||||
ax1 = subplot(2, 3, 1);
|
||||
title('$\displaystyle \frac{x_1}{w}$')
|
||||
hold on;
|
||||
plot(freqs, abs(squeeze(freqresp(G('x1', 'w'), freqs, 'Hz'))));
|
||||
plot(freqs, abs(squeeze(freqresp(Giff('x1', 'w'), freqs, 'Hz'))));
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
ylabel('Amplitude [m/m]');xlabel('Frequency [Hz]');
|
||||
|
||||
ax2 = subplot(2, 3, 2);
|
||||
title('$\displaystyle \frac{x_1}{f}$')
|
||||
hold on;
|
||||
plot(freqs, abs(squeeze(freqresp(G('x1', 'f'), freqs, 'Hz'))));
|
||||
plot(freqs, abs(squeeze(freqresp(Giff('x1', 'f'), freqs, 'Hz'))));
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
ylabel('Amplitude [m/N]');xlabel('Frequency [Hz]');
|
||||
|
||||
ax3 = subplot(2, 3, 3);
|
||||
title('$\displaystyle \frac{x_1}{F}$')
|
||||
hold on;
|
||||
plot(freqs, abs(squeeze(freqresp(G('x1', 'F'), freqs, 'Hz'))));
|
||||
plot(freqs, abs(squeeze(freqresp(Giff('x1', 'F'), freqs, 'Hz'))));
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
ylabel('Amplitude [m/N]');xlabel('Frequency [Hz]');
|
||||
|
||||
ax4 = subplot(2, 3, 4);
|
||||
title('$\displaystyle \frac{F_s}{w}$')
|
||||
hold on;
|
||||
plot(freqs, abs(squeeze(freqresp(G('Fs', 'w'), freqs, 'Hz'))));
|
||||
plot(freqs, abs(squeeze(freqresp(Giff('Fs', 'w'), freqs, 'Hz'))));
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
ylabel('Amplitude [m/m]');xlabel('Frequency [Hz]');
|
||||
|
||||
ax5 = subplot(2, 3, 5);
|
||||
title('$\displaystyle \frac{F_s}{f}$')
|
||||
hold on;
|
||||
plot(freqs, abs(squeeze(freqresp(G('Fs', 'f'), freqs, 'Hz'))));
|
||||
plot(freqs, abs(squeeze(freqresp(Giff('Fs', 'f'), freqs, 'Hz'))));
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
ylabel('Amplitude [m/N]');xlabel('Frequency [Hz]');
|
||||
|
||||
ax6 = subplot(2, 3, 6);
|
||||
title('$\displaystyle \frac{F_s}{F}$')
|
||||
hold on;
|
||||
plot(freqs, abs(squeeze(freqresp(G('Fs', 'F'), freqs, 'Hz'))));
|
||||
plot(freqs, abs(squeeze(freqresp(Giff('Fs', 'F'), freqs, 'Hz'))));
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
ylabel('Amplitude [m/N]'); xlabel('Frequency [Hz]');
|
||||
|
||||
linkaxes([ax1,ax2,ax3,ax4,ax5,ax6],'x');
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab :tangle no :exports results :results file replace
|
||||
exportFig('figs/amplified_piezo_tf_ol_and_cl.pdf', 'width', 'full', 'height', 'full');
|
||||
#+end_src
|
||||
|
||||
#+name: fig:amplified_piezo_tf_ol_and_cl
|
||||
#+caption: Matrix of transfer functions from input to output in open loop (blue) and closed loop (red)
|
||||
#+RESULTS:
|
||||
[[file:figs/amplified_piezo_tf_ol_and_cl.png]]
|
||||
|
||||
** Root Locus
|
||||
#+begin_src matlab :exports none :post
|
||||
figure;
|
||||
|
||||
gains = logspace(1, 6, 500);
|
||||
|
||||
hold on;
|
||||
plot(real(pole(G('Fs', 'f'))), imag(pole(G('Fs', 'f'))), 'kx');
|
||||
plot(real(tzero(G('Fs', 'f'))), imag(tzero(G('Fs', 'f'))), 'ko');
|
||||
for k = 1:length(gains)
|
||||
cl_poles = pole(feedback(G('Fs', 'f'), -gains(k)/s));
|
||||
plot(real(cl_poles), imag(cl_poles), 'k.');
|
||||
end
|
||||
hold off;
|
||||
axis square;
|
||||
xlim([-2500, 100]); ylim([0, 2600]);
|
||||
|
||||
xlabel('Real Part'); ylabel('Imaginary Part');
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab :tangle no :exports results :results file replace
|
||||
exportFig('figs/amplified_piezo_root_locus.pdf', 'width', 'wide', 'height', 'tall');
|
||||
#+end_src
|
||||
|
||||
#+name: fig:amplified_piezo_root_locus
|
||||
#+caption: Root Locus
|
||||
#+RESULTS:
|
||||
[[file:figs/amplified_piezo_root_locus.png]]
|
||||
|
||||
* Rotating X-Y platform
|
||||
** Matlab Init :noexport:ignore:
|
||||
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
|
||||
<<matlab-dir>>
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab :exports none :results silent :noweb yes
|
||||
<<matlab-init>>
|
||||
#+end_src
|
||||
|
||||
#+BEGIN_SRC matlab
|
||||
simulinkproject('../');
|
||||
#+END_SRC
|
||||
|
||||
#+begin_src matlab
|
||||
open 'amplified_piezo_xy_rotating_stage.slx'
|
||||
#+end_src
|
||||
|
||||
** Parameters
|
||||
#+begin_src matlab
|
||||
m = 1; % [kg]
|
||||
|
||||
ke = 4.8e6; % [N/m]
|
||||
ce = 5; % [N/(m/s)]
|
||||
me = 0.001; % [kg]
|
||||
|
||||
k1 = 0.96e6; % [N/m]
|
||||
c1 = 10; % [N/(m/s)]
|
||||
|
||||
ka = 65e6; % [N/m]
|
||||
ca = 5; % [N/(m/s)]
|
||||
ma = 0.001; % [kg]
|
||||
|
||||
h = 0.2; % [m]
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab
|
||||
Kiff = tf(0);
|
||||
#+end_src
|
||||
|
||||
** Identification
|
||||
Rotating speed in rad/s:
|
||||
#+begin_src matlab
|
||||
Ws = 2*pi*[0, 1, 10, 100];
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab
|
||||
Gs = {zeros(length(Ws), 1)};
|
||||
#+end_src
|
||||
|
||||
Identification in open-loop.
|
||||
#+begin_src matlab
|
||||
%% Name of the Simulink File
|
||||
mdl = 'amplified_piezo_xy_rotating_stage';
|
||||
|
||||
%% Input/Output definition
|
||||
clear io; io_i = 1;
|
||||
io(io_i) = linio([mdl, '/fx'], 1, 'openinput'); io_i = io_i + 1;
|
||||
io(io_i) = linio([mdl, '/fy'], 1, 'openinput'); io_i = io_i + 1;
|
||||
|
||||
io(io_i) = linio([mdl, '/Fs'], 1, 'openoutput'); io_i = io_i + 1;
|
||||
io(io_i) = linio([mdl, '/Fs'], 2, 'openoutput'); io_i = io_i + 1;
|
||||
|
||||
for i = 1:length(Ws)
|
||||
ws = Ws(i);
|
||||
G = linearize(mdl, io, 0);
|
||||
G.InputName = {'fx', 'fy'};
|
||||
G.OutputName = {'Fsx', 'Fsy'};
|
||||
Gs(i) = {G};
|
||||
end
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab :exports none
|
||||
freqs = logspace(1, 3, 1000);
|
||||
|
||||
figure;
|
||||
|
||||
ax1 = subplot(2, 2, 1);
|
||||
title('$\displaystyle \frac{F_{s,x}}{f_x}$')
|
||||
hold on;
|
||||
for i = 1:length(Ws)
|
||||
plot(freqs, abs(squeeze(freqresp(Gs{i}('Fsx', 'fx'), freqs, 'Hz'))));
|
||||
end
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
ylabel('Amplitude [m/m]');xlabel('Frequency [Hz]');
|
||||
|
||||
ax2 = subplot(2, 2, 2);
|
||||
title('$\displaystyle \frac{F_{s,y}}{f_x}$')
|
||||
hold on;
|
||||
for i = 1:length(Ws)
|
||||
plot(freqs, abs(squeeze(freqresp(Gs{i}('Fsy', 'fx'), freqs, 'Hz'))));
|
||||
end
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
ylabel('Amplitude [m/N]');xlabel('Frequency [Hz]');
|
||||
|
||||
ax3 = subplot(2, 2, 3);
|
||||
title('$\displaystyle \frac{F_{s,x}}{f_y}$')
|
||||
hold on;
|
||||
for i = 1:length(Ws)
|
||||
plot(freqs, abs(squeeze(freqresp(Gs{i}('Fsx', 'fy'), freqs, 'Hz'))));
|
||||
end
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
ylabel('Amplitude [m/N]');xlabel('Frequency [Hz]');
|
||||
|
||||
ax4 = subplot(2, 2, 4);
|
||||
title('$\displaystyle \frac{F_{s,y}}{f_y}$')
|
||||
hold on;
|
||||
for i = 1:length(Ws)
|
||||
plot(freqs, abs(squeeze(freqresp(Gs{i}('Fsy', 'fy'), freqs, 'Hz'))));
|
||||
end
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
ylabel('Amplitude [m/m]');xlabel('Frequency [Hz]');
|
||||
|
||||
linkaxes([ax1,ax2,ax3,ax4],'x');
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab :tangle no :exports results :results file replace
|
||||
exportFig('figs/amplitifed_piezo_xy_rotation_plant_iff.pdf', 'width', 'full', 'height', 'full');
|
||||
#+end_src
|
||||
|
||||
#+name: fig:amplitifed_piezo_xy_rotation_plant_iff
|
||||
#+caption: Transfer function matrix from forces to force sensors for multiple rotation speed
|
||||
#+RESULTS:
|
||||
[[file:figs/amplitifed_piezo_xy_rotation_plant_iff.png]]
|
||||
|
||||
** Root Locus
|
||||
#+begin_src matlab :exports none :post
|
||||
figure;
|
||||
|
||||
gains = logspace(1, 6, 500);
|
||||
|
||||
hold on;
|
||||
for i = 1:length(Ws)
|
||||
set(gca,'ColorOrderIndex',i);
|
||||
plot(real(pole(Gs{i})), imag(pole(Gs{i})), 'x');
|
||||
set(gca,'ColorOrderIndex',i);
|
||||
plot(real(tzero(Gs{i})), imag(tzero(Gs{i})), 'o');
|
||||
for k = 1:length(gains)
|
||||
set(gca,'ColorOrderIndex',i);
|
||||
cl_poles = pole(feedback(Gs{i}, -gains(k)/s*eye(2)));
|
||||
plot(real(cl_poles), imag(cl_poles), '.');
|
||||
end
|
||||
end
|
||||
hold off;
|
||||
axis square;
|
||||
xlim([-2900, 100]); ylim([0, 3000]);
|
||||
|
||||
xlabel('Real Part'); ylabel('Imaginary Part');
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab :tangle no :exports results :results file replace
|
||||
exportFig('figs/amplified_piezo_xy_rotation_root_locus.pdf', 'width', 'tall', 'height', 'wide');
|
||||
#+end_src
|
||||
|
||||
#+name: fig:amplified_piezo_xy_rotation_root_locus
|
||||
#+caption: Root locus for 3 rotating speed
|
||||
#+RESULTS:
|
||||
[[file:figs/amplified_piezo_xy_rotation_root_locus.png]]
|
||||
|
||||
** Analysis
|
||||
The negative stiffness induced by the rotation is equal to $m \omega_0^2$.
|
||||
Thus, the maximum rotation speed where IFF can be applied is:
|
||||
\[ \omega_\text{max} = \sqrt{\frac{k_1}{m}} \approx 156\,[Hz] \]
|
||||
|
||||
Let's verify that.
|
||||
#+begin_src matlab
|
||||
Ws = 2*pi*[140, 160];
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab :exports none
|
||||
Gs = {zeros(length(Ws), 1)};
|
||||
#+end_src
|
||||
|
||||
Identification
|
||||
#+begin_src matlab
|
||||
%% Name of the Simulink File
|
||||
mdl = 'amplified_piezo_xy_rotating_stage';
|
||||
|
||||
%% Input/Output definition
|
||||
clear io; io_i = 1;
|
||||
io(io_i) = linio([mdl, '/fx'], 1, 'openinput'); io_i = io_i + 1;
|
||||
io(io_i) = linio([mdl, '/fy'], 1, 'openinput'); io_i = io_i + 1;
|
||||
|
||||
io(io_i) = linio([mdl, '/Fs'], 1, 'openoutput'); io_i = io_i + 1;
|
||||
io(io_i) = linio([mdl, '/Fs'], 2, 'openoutput'); io_i = io_i + 1;
|
||||
|
||||
for i = 1:length(Ws)
|
||||
ws = Ws(i);
|
||||
G = linearize(mdl, io, 0);
|
||||
G.InputName = {'fx', 'fy'};
|
||||
G.OutputName = {'Fsx', 'Fsy'};
|
||||
Gs(i) = {G};
|
||||
end
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab :exports none
|
||||
figure;
|
||||
|
||||
gains = logspace(1, 6, 500);
|
||||
|
||||
hold on;
|
||||
for i = 1:length(Ws)
|
||||
set(gca,'ColorOrderIndex',i);
|
||||
plot(real(pole(Gs{i})), imag(pole(Gs{i})), 'x');
|
||||
set(gca,'ColorOrderIndex',i);
|
||||
plot(real(tzero(Gs{i})), imag(tzero(Gs{i})), 'o');
|
||||
for k = 1:length(gains)
|
||||
set(gca,'ColorOrderIndex',i);
|
||||
cl_poles = pole(feedback(Gs{i}, -gains(k)/s*eye(2)));
|
||||
plot(real(cl_poles), imag(cl_poles), '.');
|
||||
end
|
||||
end
|
||||
hold off;
|
||||
axis square;
|
||||
xlim([-100, 50]); ylim([0, 150]);
|
||||
|
||||
xlabel('Real Part'); ylabel('Imaginary Part');
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab :tangle no :exports results :results file replace
|
||||
exportFig('figs/amplified_piezo_xy_rotating_unstable_root_locus.pdf', 'width', 'wide', 'height', 'tall');
|
||||
#+end_src
|
||||
|
||||
#+name: fig:amplified_piezo_xy_rotating_unstable_root_locus
|
||||
#+caption: Root Locus for the two considered rotation speed. For the red curve, the system is unstable.
|
||||
#+RESULTS:
|
||||
[[file:figs/amplified_piezo_xy_rotating_unstable_root_locus.png]]
|
||||
Loading…
Reference in New Issue
Block a user