318 lines
12 KiB
HTML
318 lines
12 KiB
HTML
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<!-- 2020-11-02 lun. 16:03 -->
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<meta http-equiv="Content-Type" content="text/html;charset=utf-8" />
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<title>Attocube - Test Bench</title>
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<meta name="author" content="Dehaeze Thomas" />
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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">Attocube - 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="#org301870a">1. Estimation of the Spectral Density of the Attocube Noise</a>
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<ul>
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<li><a href="#orga74fad8">1.1. Long and Slow measurement</a></li>
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<li><a href="#org594bfe8">1.2. Short and Fast measurement</a></li>
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<li><a href="#orgd9ca1ad">1.3. Obtained Amplitude Spectral Density of the measured displacement</a></li>
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</ul>
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</li>
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<li><a href="#orgacf938d">2. Effect of the “bubble sheet” and <b>Aluminium tube</b></a>
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<ul>
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<li><a href="#org30a0a1b">2.1. Aluminium Tube and Bubble Sheet</a></li>
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<li><a href="#orgfe4b4f1">2.2. Only Aluminium Tube</a></li>
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<li><a href="#org5d943ee">2.3. Nothing</a></li>
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<li><a href="#org6b21819">2.4. Comparison</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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<div id="outline-container-org301870a" class="outline-2">
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<h2 id="org301870a"><span class="section-number-2">1</span> Estimation of the Spectral Density of the Attocube Noise</h2>
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<div class="outline-text-2" id="text-1">
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<div id="org90d970d" class="figure">
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<p><img src="figs/test-bench-shematic.png" alt="test-bench-shematic.png" />
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</p>
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<p><span class="figure-number">Figure 1: </span>Test Bench Schematic</p>
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</div>
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<div id="org2b65a6c" class="figure">
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<p><img src="figs/IMG-7865.JPG" alt="IMG-7865.JPG" />
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</p>
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<p><span class="figure-number">Figure 2: </span>Picture of the test bench. The Attocube and mirror are covered by a “bubble sheet”</p>
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</div>
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</div>
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<div id="outline-container-orga74fad8" class="outline-3">
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<h3 id="orga74fad8"><span class="section-number-3">1.1</span> Long and Slow measurement</h3>
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<div class="outline-text-3" id="text-1-1">
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<p>
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The first measurement was made during ~17 hours with a sampling time of \(T_s = 0.1\,s\).
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</p>
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<div class="org-src-container">
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<pre class="src src-matlab">load(<span class="org-string">'./mat/long_test2.mat'</span>, <span class="org-string">'x'</span>, <span class="org-string">'t'</span>)
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Ts = 0.1; <span class="org-comment">% [s]</span>
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</pre>
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</div>
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<div id="org64c5513" class="figure">
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<p><img src="figs/long_meas_time_domain_full.png" alt="long_meas_time_domain_full.png" />
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</p>
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<p><span class="figure-number">Figure 3: </span>Long measurement time domain data</p>
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</div>
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<p>
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Let’s fit the data with a step response to a first order low pass filter (Figure <a href="#orgc356556">4</a>).
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</p>
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<div class="org-src-container">
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<pre class="src src-matlab">f = @(b,x) b(1)<span class="org-type">*</span>(1 <span class="org-type">-</span> exp(<span class="org-type">-</span>x<span class="org-type">/</span>b(2)));
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y_cur = x(t <span class="org-type"><</span> 17<span class="org-type">*</span>60<span class="org-type">*</span>60);
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t_cur = t(t <span class="org-type"><</span> 17<span class="org-type">*</span>60<span class="org-type">*</span>60);
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nrmrsd = @(b) norm(y_cur <span class="org-type">-</span> f(b,t_cur)); <span class="org-comment">% Residual Norm Cost Function</span>
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B0 = [400e<span class="org-type">-</span>9, 2<span class="org-type">*</span>60<span class="org-type">*</span>60]; <span class="org-comment">% Choose Appropriate Initial Estimates</span>
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[B,rnrm] = fminsearch(nrmrsd, B0); <span class="org-comment">% Estimate Parameters ‘B’</span>
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</pre>
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</div>
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<p>
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The corresponding time constant is (in [h]):
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</p>
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<pre class="example">
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2.0576
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</pre>
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<div id="orgc356556" class="figure">
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<p><img src="figs/long_meas_time_domain_fit.png" alt="long_meas_time_domain_fit.png" />
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</p>
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<p><span class="figure-number">Figure 4: </span>Fit of the measurement data with a step response of a first order low pass filter</p>
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</div>
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<p>
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We can see in Figure <a href="#org64c5513">3</a> that there is a transient period where the measured displacement experiences some drifts.
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This is probably due to thermal effects.
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We only select the data between <code>t1</code> and <code>t2</code>.
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The obtained displacement is shown in Figure <a href="#orgb851634">5</a>.
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</p>
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<div class="org-src-container">
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<pre class="src src-matlab">t1 = 11; t2 = 17; <span class="org-comment">% [h]</span>
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x = x(t <span class="org-type">></span> t1<span class="org-type">*</span>60<span class="org-type">*</span>60 <span class="org-type">&</span> t <span class="org-type"><</span> t2<span class="org-type">*</span>60<span class="org-type">*</span>60);
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x = x <span class="org-type">-</span> mean(x);
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t = t(t <span class="org-type">></span> t1<span class="org-type">*</span>60<span class="org-type">*</span>60 <span class="org-type">&</span> t <span class="org-type"><</span> t2<span class="org-type">*</span>60<span class="org-type">*</span>60);
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t = t <span class="org-type">-</span> t(1);
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</pre>
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</div>
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<div id="orgb851634" class="figure">
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<p><img src="figs/long_meas_time_domain_zoom.png" alt="long_meas_time_domain_zoom.png" />
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</p>
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<p><span class="figure-number">Figure 5: </span>Kept data (removed slow drifts during the first hours)</p>
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</div>
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<p>
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The Power Spectral Density of the measured displacement is computed
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</p>
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<div class="org-src-container">
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<pre class="src src-matlab">win = hann(ceil(length(x)<span class="org-type">/</span>20));
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[p_1, f_1] = pwelch(x, win, [], [], 1<span class="org-type">/</span>Ts);
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</pre>
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</div>
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<p>
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As a low pass filter was used in the measurement process, we multiply the PSD by the square of the inverse of the filter’s norm.
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</p>
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<div class="org-src-container">
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<pre class="src src-matlab">G_lpf = 1<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>);
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p_1 = p_1<span class="org-type">./</span>abs(squeeze(freqresp(G_lpf, f_1, <span class="org-string">'Hz'</span>)))<span class="org-type">.^</span>2;
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</pre>
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</div>
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<p>
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Only frequencies below 2Hz are taken into account (high frequency noise will be measured afterwards).
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</p>
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<div class="org-src-container">
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<pre class="src src-matlab">p_1 = p_1(f_1 <span class="org-type"><</span> 2);
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f_1 = f_1(f_1 <span class="org-type"><</span> 2);
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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-org594bfe8" class="outline-3">
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<h3 id="org594bfe8"><span class="section-number-3">1.2</span> Short and Fast measurement</h3>
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<div class="outline-text-3" id="text-1-2">
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<p>
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An second measurement is done in order to estimate the high frequency noise of the interferometer.
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The measurement is done with a sampling time of \(T_s = 0.1\,ms\) and a duration of ~100s.
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</p>
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<div class="org-src-container">
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<pre class="src src-matlab">load(<span class="org-string">'./mat/short_test_plastic.mat'</span>)
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Ts = 1e<span class="org-type">-</span>4; <span class="org-comment">% [s]</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">x = detrend(x, 0);
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</pre>
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</div>
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<p>
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The time domain measurement is shown in Figure <a href="#orged82baf">6</a>.
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</p>
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<div id="orged82baf" class="figure">
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<p><img src="figs/short_meas_time_domain.png" alt="short_meas_time_domain.png" />
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</p>
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<p><span class="figure-number">Figure 6: </span>Time domain measurement with the high sampling rate</p>
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</div>
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<p>
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The Power Spectral Density of the measured displacement is computed
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</p>
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<div class="org-src-container">
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<pre class="src src-matlab">win = hann(ceil(length(x)<span class="org-type">/</span>10));
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[p_2, f_2] = pwelch(x, win, [], [], 1<span class="org-type">/</span>Ts);
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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-orgd9ca1ad" class="outline-3">
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<h3 id="orgd9ca1ad"><span class="section-number-3">1.3</span> Obtained Amplitude Spectral Density of the measured displacement</h3>
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<div class="outline-text-3" id="text-1-3">
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<p>
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The computed ASD of the two measurements are combined in Figure <a href="#org5032549">7</a>.
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</p>
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<div id="org5032549" class="figure">
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<p><img src="figs/psd_combined.png" alt="psd_combined.png" />
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</p>
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<p><span class="figure-number">Figure 7: </span>Obtained Amplitude Spectral Density of the measured displacement</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-orgacf938d" class="outline-2">
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<h2 id="orgacf938d"><span class="section-number-2">2</span> Effect of the “bubble sheet” and <b>Aluminium tube</b></h2>
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<div class="outline-text-2" id="text-2">
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<div id="orgd503177" class="figure">
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<p><img src="figs/IMG-7864.JPG" alt="IMG-7864.JPG" />
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</p>
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<p><span class="figure-number">Figure 8: </span>Aluminium tube used to protect the beam path from disturbances</p>
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</div>
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</div>
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<div id="outline-container-org30a0a1b" class="outline-3">
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<h3 id="org30a0a1b"><span class="section-number-3">2.1</span> Aluminium Tube and Bubble Sheet</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">load(<span class="org-string">'./mat/long_test_plastic.mat'</span>);
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Ts = 1e<span class="org-type">-</span>4; <span class="org-comment">% [s]</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">x = detrend(x, 0);
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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">win = hann(ceil(length(x)<span class="org-type">/</span>10));
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[p_1, f_1] = pwelch(x, win, [], [], 1<span class="org-type">/</span>Ts);
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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-orgfe4b4f1" class="outline-3">
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<h3 id="orgfe4b4f1"><span class="section-number-3">2.2</span> Only Aluminium Tube</h3>
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<div class="outline-text-3" id="text-2-2">
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<div class="org-src-container">
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<pre class="src src-matlab">load(<span class="org-string">'./mat/long_test_alu_tube.mat'</span>);
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Ts = 1e<span class="org-type">-</span>4; <span class="org-comment">% [s]</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">x = detrend(x, 0);
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</pre>
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</div>
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<p>
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The time domain measurement is shown in Figure <a href="#orged82baf">6</a>.
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</p>
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<div class="org-src-container">
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<pre class="src src-matlab">win = hann(ceil(length(x)<span class="org-type">/</span>10));
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[p_2, f_2] = pwelch(x, win, [], [], 1<span class="org-type">/</span>Ts);
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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-org5d943ee" class="outline-3">
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<h3 id="org5d943ee"><span class="section-number-3">2.3</span> Nothing</h3>
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</div>
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<div id="outline-container-org6b21819" class="outline-3">
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<h3 id="org6b21819"><span class="section-number-3">2.4</span> Comparison</h3>
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<div class="outline-text-3" id="text-2-4">
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<div id="org2d3dd04" class="figure">
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<p><img src="figs/asd_noise_comp_bubble_aluminium.png" alt="asd_noise_comp_bubble_aluminium.png" />
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</p>
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<p><span class="figure-number">Figure 9: </span>Comparison of the noise ASD with and without bubble sheet</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: 2020-11-02 lun. 16:03</p>
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</div>
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</body>
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