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figs/flexible_joint_geometry.pdf
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<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
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<!-- 2020-12-15 mar. 22:32 -->
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<meta http-equiv="Content-Type" content="text/html;charset=utf-8" />
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<title>Flexible Joint - Test Bench</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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macros: {bm: ["\\boldsymbol{#1}",1],}
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<body>
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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">Flexible Joint - Test Bench</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="#org3ec0fb6">1. Test Bench Description</a>
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<ul>
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<li><a href="#org1c35c31">1.1. Flexible joint Geometry</a></li>
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<li><a href="#org943ed6d">1.2. Required external applied force</a></li>
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<li><a href="#org866642a">1.3. Required actuator stroke and sensors range</a></li>
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<li><a href="#org4789077">1.4. First try with the APA95ML</a></li>
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</ul>
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</li>
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<li><a href="#org75cb5e5">2. Experimental measurement</a></li>
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</ul>
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</div>
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</div>
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<div id="outline-container-org3ec0fb6" class="outline-2">
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<h2 id="org3ec0fb6"><span class="section-number-2">1</span> Test Bench Description</h2>
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<div class="outline-text-2" id="text-1">
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<p>
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||||||
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The main characteristic of the flexible joint that we want to measure is its bending stiffness \(k_{R_x} \approx k_{R_y}\).
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</p>
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||||||
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||||||
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<p>
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||||||
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To do so, a test bench is used.
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||||||
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Specifications of the test bench to precisely measure the bending stiffness are described in this section.
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||||||
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</p>
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||||||
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||||||
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<p>
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||||||
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The basic idea is to measured the angular deflection of the flexible joint as a function of the applied torque.
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</p>
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<div id="org12e0ba4" class="figure">
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<p><img src="figs/test-bench-schematic.png" alt="test-bench-schematic.png" />
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</p>
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<p><span class="figure-number">Figure 1: </span>Schematic of the test bench to measure the bending stiffness of the flexible joints</p>
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</div>
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</div>
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<div id="outline-container-org1c35c31" class="outline-3">
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<h3 id="org1c35c31"><span class="section-number-3">1.1</span> Flexible joint Geometry</h3>
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<div class="outline-text-3" id="text-1-1">
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<p>
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||||||
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The flexible joint used for the Nano-Hexapod is shown in Figure <a href="#org907b319">2</a>.
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||||||
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Its bending stiffness is foreseen to be \(k_{R_y}\approx 20\,\frac{Nm}{rad}\) and its stroke \(\theta_{y,\text{max}}\approx 20\,mrad\).
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</p>
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<div id="org907b319" class="figure">
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<p><img src="figs/flexible_joint_geometry.png" alt="flexible_joint_geometry.png" />
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</p>
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||||||
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<p><span class="figure-number">Figure 2: </span>Geometry of the flexible joint</p>
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</div>
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||||||
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||||||
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<p>
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||||||
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The height between the flexible point (center of the joint) and the point where external forces are applied is \(h = 20\,mm\).
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</p>
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<p>
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Let’s define the parameters on Matlab.
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</p>
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<div class="org-src-container">
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<pre class="src src-matlab">kRx = 20; <span class="org-comment">% Bending Stiffness [Nm/rad]</span>
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Rxmax = 20e<span class="org-type">-</span>3; <span class="org-comment">% Bending Stroke [rad]</span>
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h = 20e<span class="org-type">-</span>3; <span class="org-comment">% Height [m]</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-org943ed6d" class="outline-3">
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<h3 id="org943ed6d"><span class="section-number-3">1.2</span> Required external applied force</h3>
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<div class="outline-text-3" id="text-1-2">
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<p>
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||||||
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The bending \(\theta_y\) of the flexible joint due to the force \(F_x\) is:
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</p>
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||||||
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\begin{equation}
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\theta_y = \frac{M_y}{k_{R_y}} = \frac{F_x h}{k_{R_y}}
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\end{equation}
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<p>
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||||||
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Therefore, the applied force to test the full range of the flexible joint is:
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</p>
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||||||
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\begin{equation}
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F_{x,\text{max}} = \frac{k_{R_y} \theta_{y,\text{max}}}{h}
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\end{equation}
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||||||
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<div class="org-src-container">
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<pre class="src src-matlab">Fxmax = kRx<span class="org-type">*</span>Rxmax<span class="org-type">/</span>h; <span class="org-comment">% Force to induce maximum stroke [N]</span>
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</pre>
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</div>
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<p>
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||||||
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And we obtain:
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</p>
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||||||
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\begin{equation} F_{max} = 20.0\, [N] \end{equation}
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||||||
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<p>
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||||||
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The measurement range of the force sensor should then be higher than \(20\,N\).
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||||||
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</p>
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||||||
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</div>
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||||||
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</div>
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||||||
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||||||
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<div id="outline-container-org866642a" class="outline-3">
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||||||
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<h3 id="org866642a"><span class="section-number-3">1.3</span> Required actuator stroke and sensors range</h3>
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||||||
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<div class="outline-text-3" id="text-1-3">
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||||||
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<p>
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||||||
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The flexible joint is designed to allow a bending motion of \(\pm 20\,mrad\).
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||||||
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The corresponding actuator stroke to impose such motion is:
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||||||
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</p>
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||||||
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||||||
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<p>
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||||||
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\[ d_{x,\text{max}} = h \tan(R_{x,\text{max}}) \]
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</p>
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<div class="org-src-container">
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<pre class="src src-matlab">dxmax = h<span class="org-type">*</span>tan(Rxmax);
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||||||
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</pre>
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||||||
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</div>
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||||||
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||||||
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\begin{equation} d_{max} = 0.4\, [mm] \end{equation}
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||||||
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||||||
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<p>
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||||||
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In order to test the full range of the flexible joint, the stroke of the actuator should be higher than \(0.4\,mm\).
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||||||
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The measurement range of the displacement sensor should also be higher than \(0.4\,mm\).
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||||||
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</p>
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||||||
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</div>
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||||||
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</div>
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||||||
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||||||
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<div id="outline-container-org4789077" class="outline-3">
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||||||
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<h3 id="org4789077"><span class="section-number-3">1.4</span> First try with the APA95ML</h3>
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||||||
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<div class="outline-text-3" id="text-1-4">
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||||||
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<p>
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||||||
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The APA95ML as a stroke of \(100\,\mu m\) and the encoder in parallel can easily measure the required stroke.
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||||||
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</p>
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||||||
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||||||
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<p>
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||||||
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Suppose the full stroke of the APA can be used to bend the flexible joint (ideal case), the measured force will be:
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||||||
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</p>
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||||||
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<div class="org-src-container">
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||||||
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<pre class="src src-matlab">Fxmax = kRx<span class="org-type">*</span>100e<span class="org-type">-</span>6<span class="org-type">/</span>h<span class="org-type">^</span>2; <span class="org-comment">% Force at maximum stroke [N]</span>
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||||||
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</pre>
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||||||
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</div>
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||||||
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||||||
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\begin{equation} F_{max} = 5.0\, [N] \end{equation}
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||||||
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<p>
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||||||
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And the tested angular range is:
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||||||
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</p>
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||||||
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<div class="org-src-container">
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||||||
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<pre class="src src-matlab">Rmax = tan(100e<span class="org-type">-</span>6<span class="org-type">/</span>h);
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||||||
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</pre>
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||||||
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</div>
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||||||
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||||||
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\begin{equation} \theta_{max} = 5.0\, [mrad] \end{equation}
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</div>
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</div>
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</div>
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||||||
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<div id="outline-container-org75cb5e5" class="outline-2">
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<h2 id="org75cb5e5"><span class="section-number-2">2</span> Experimental measurement</h2>
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</div>
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</div>
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<div id="postamble" class="status">
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<p class="author">Author: Dehaeze Thomas</p>
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<p class="date">Created: 2020-12-15 mar. 22:32</p>
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</div>
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</body>
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</html>
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index.org
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#+TITLE: Flexible Joint - Test Bench
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:DRAWER:
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#+LANGUAGE: en
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#+EMAIL: dehaeze.thomas@gmail.com
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#+AUTHOR: Dehaeze Thomas
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#+HTML_LINK_HOME: ../index.html
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#+HTML_LINK_UP: ../index.html
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#+HTML_HEAD: <link rel="stylesheet" type="text/css" href="https://research.tdehaeze.xyz/css/style.css"/>
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#+HTML_HEAD: <script type="text/javascript" src="https://research.tdehaeze.xyz/js/script.js"></script>
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#+BIND: org-latex-image-default-option "scale=1"
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#+LaTeX_CLASS: scrreprt
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#+LaTeX_CLASS_OPTIONS: [a4paper, 10pt, DIV=12, parskip=full]
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#+LaTeX_HEADER_EXTRA: \input{preamble.tex}
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#+PROPERTY: header-args:matlab :session *MATLAB*
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#+PROPERTY: header-args:matlab+ :comments org
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#+PROPERTY: header-args:matlab+ :exports both
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#+PROPERTY: header-args:matlab+ :results none
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#+PROPERTY: header-args:matlab+ :eval no-export
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#+PROPERTY: header-args:matlab+ :noweb yes
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||||||
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#+PROPERTY: header-args:matlab+ :mkdirp yes
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#+PROPERTY: header-args:matlab+ :output-dir figs
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#+PROPERTY: header-args:latex :headers '("\\usepackage{tikz}" "\\usepackage{import}" "\\import{$HOME/Cloud/tikz/org/}{config.tex}")
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#+PROPERTY: header-args:latex+ :imagemagick t :fit yes
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#+PROPERTY: header-args:latex+ :iminoptions -scale 100% -density 150
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#+PROPERTY: header-args:latex+ :imoutoptions -quality 100
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#+PROPERTY: header-args:latex+ :results file raw replace
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#+PROPERTY: header-args:latex+ :buffer no
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#+PROPERTY: header-args:latex+ :tangle no
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#+PROPERTY: header-args:latex+ :eval no-export
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#+PROPERTY: header-args:latex+ :exports results
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#+PROPERTY: header-args:latex+ :mkdirp yes
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#+PROPERTY: header-args:latex+ :output-dir figs
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#+PROPERTY: header-args:latex+ :post pdf2svg(file=*this*, ext="png")
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:END:
|
||||||
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|
||||||
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* Test Bench Description
|
||||||
|
** Introduction :ignore:
|
||||||
|
|
||||||
|
The main characteristic of the flexible joint that we want to measure is its bending stiffness $k_{R_x} \approx k_{R_y}$.
|
||||||
|
|
||||||
|
To do so, a test bench is used.
|
||||||
|
Specifications of the test bench to precisely measure the bending stiffness are described in this section.
|
||||||
|
|
||||||
|
The basic idea is to measured the angular deflection of the flexible joint as a function of the applied torque.
|
||||||
|
|
||||||
|
#+name: fig:test-bench-schematic
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||||||
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#+caption: Schematic of the test bench to measure the bending stiffness of the flexible joints
|
||||||
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[[file:figs/test-bench-schematic.png]]
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||||||
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||||||
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** Matlab Init :noexport:ignore:
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#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
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<<matlab-dir>>
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#+end_src
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#+begin_src matlab :exports none :results silent :noweb yes
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<<matlab-init>>
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||||||
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#+end_src
|
||||||
|
|
||||||
|
** Flexible joint Geometry
|
||||||
|
The flexible joint used for the Nano-Hexapod is shown in Figure [[fig:flexible_joint_geometry]].
|
||||||
|
Its bending stiffness is foreseen to be $k_{R_y}\approx 20\,\frac{Nm}{rad}$ and its stroke $\theta_{y,\text{max}}\approx 20\,mrad$.
|
||||||
|
|
||||||
|
#+name: fig:flexible_joint_geometry
|
||||||
|
#+caption: Geometry of the flexible joint
|
||||||
|
[[file:figs/flexible_joint_geometry.png]]
|
||||||
|
|
||||||
|
The height between the flexible point (center of the joint) and the point where external forces are applied is $h = 20\,mm$.
|
||||||
|
|
||||||
|
Let's define the parameters on Matlab.
|
||||||
|
#+begin_src matlab
|
||||||
|
kRx = 20; % Bending Stiffness [Nm/rad]
|
||||||
|
Rxmax = 20e-3; % Bending Stroke [rad]
|
||||||
|
h = 20e-3; % Height [m]
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
** Required external applied force
|
||||||
|
|
||||||
|
The bending $\theta_y$ of the flexible joint due to the force $F_x$ is:
|
||||||
|
\begin{equation}
|
||||||
|
\theta_y = \frac{M_y}{k_{R_y}} = \frac{F_x h}{k_{R_y}}
|
||||||
|
\end{equation}
|
||||||
|
|
||||||
|
Therefore, the applied force to test the full range of the flexible joint is:
|
||||||
|
\begin{equation}
|
||||||
|
F_{x,\text{max}} = \frac{k_{R_y} \theta_{y,\text{max}}}{h}
|
||||||
|
\end{equation}
|
||||||
|
|
||||||
|
#+begin_src matlab
|
||||||
|
Fxmax = kRx*Rxmax/h; % Force to induce maximum stroke [N]
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
And we obtain:
|
||||||
|
#+begin_src matlab :results value raw replace :exports results
|
||||||
|
sprintf('\\begin{equation} F_{max} = %.1f\\, [N] \\end{equation}', Fxmax)
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
#+RESULTS:
|
||||||
|
\begin{equation} F_{max} = 20.0\, [N] \end{equation}
|
||||||
|
|
||||||
|
The measurement range of the force sensor should then be higher than $20\,N$.
|
||||||
|
|
||||||
|
** Required actuator stroke and sensors range
|
||||||
|
|
||||||
|
The flexible joint is designed to allow a bending motion of $\pm 20\,mrad$.
|
||||||
|
The corresponding actuator stroke to impose such motion is:
|
||||||
|
|
||||||
|
\[ d_{x,\text{max}} = h \tan(R_{x,\text{max}}) \]
|
||||||
|
|
||||||
|
#+begin_src matlab
|
||||||
|
dxmax = h*tan(Rxmax);
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
#+begin_src matlab :results value raw replace :exports results
|
||||||
|
sprintf('\\begin{equation} d_{max} = %.1f\\, [mm] \\end{equation}', 1e3*dxmax)
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
#+RESULTS:
|
||||||
|
\begin{equation} d_{max} = 0.4\, [mm] \end{equation}
|
||||||
|
|
||||||
|
In order to test the full range of the flexible joint, the stroke of the actuator should be higher than $0.4\,mm$.
|
||||||
|
The measurement range of the displacement sensor should also be higher than $0.4\,mm$.
|
||||||
|
|
||||||
|
** First try with the APA95ML
|
||||||
|
|
||||||
|
The APA95ML as a stroke of $100\,\mu m$ and the encoder in parallel can easily measure the required stroke.
|
||||||
|
|
||||||
|
Suppose the full stroke of the APA can be used to bend the flexible joint (ideal case), the measured force will be:
|
||||||
|
#+begin_src matlab
|
||||||
|
Fxmax = kRx*100e-6/h^2; % Force at maximum stroke [N]
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
#+begin_src matlab :results value raw replace :exports results
|
||||||
|
sprintf('\\begin{equation} F_{max} = %.1f\\, [N] \\end{equation}', Fxmax)
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
#+RESULTS:
|
||||||
|
\begin{equation} F_{max} = 5.0\, [N] \end{equation}
|
||||||
|
|
||||||
|
And the tested angular range is:
|
||||||
|
#+begin_src matlab
|
||||||
|
Rmax = tan(100e-6/h);
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
#+begin_src matlab :results value raw replace :exports results
|
||||||
|
sprintf('\\begin{equation} \\theta_{max} = %.1f\\, [mrad] \\end{equation}', 1e3*Rmax)
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
#+RESULTS:
|
||||||
|
\begin{equation} \theta_{max} = 5.0\, [mrad] \end{equation}
|
||||||
|
|
||||||
|
* Experimental measurement
|
||||||
Loading…
Reference in New Issue
Block a user