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<div id="content">
<h1 class="title">Spindle Analysis</h1>
<div id="table-of-contents">
<h2>Table of Contents</h2>
<div id="text-table-of-contents">
<ul>
<li><a href="#org4ef1ca3">1. Notes</a></li>
<li><a href="#orge92ef42">2. Data Processing</a>
<ul>
<li><a href="#orgbf2f91a">2.1. Load Measurement Data</a></li>
<li><a href="#orge799cbd">2.2. Convert Signals from [deg] to [sec]</a></li>
<li><a href="#orge0e840d">2.3. Convert Signals</a></li>
<li><a href="#orgb79c23f">2.4. Ts and Fs for both measurements</a></li>
<li><a href="#org545b4dc">2.5. Find Noise of the ADC [m/sqrt(Hz)]</a></li>
<li><a href="#orgb7c25b7">2.6. Save all the data under spindle struct</a></li>
<li><a href="#org707a6a0">2.7. Compute Asynchronous data</a></li>
</ul>
</li>
<li><a href="#orga9ae55f">3. Time Domain Data</a>
<ul>
<li><a href="#orga3598c8">3.1. Plot X-Y-Z position with respect to Time - 1rpm</a></li>
<li><a href="#orgb1d7825">3.2. Plot X-Y-Z position with respect to Time - 60rpm</a></li>
<li><a href="#org487e41f">3.3. Plot Synchronous and Asynchronous - 1rpm</a></li>
<li><a href="#org5310667">3.4. Plot Synchronous and Asynchronous - 60rpm</a></li>
<li><a href="#org037b157">3.5. Plot X against Y</a></li>
<li><a href="#org067dc56">3.6. Plot X against Y - Asynchronous</a></li>
</ul>
</li>
<li><a href="#org590c4a3">4. Model of the spindle</a>
<ul>
<li><a href="#orgb7ea4e3">4.1. Parameters</a></li>
<li><a href="#orge4e720c">4.2. Compute Mass and Stiffness Matrices</a></li>
<li><a href="#org5d5795f">4.3. Compute resonance frequencies</a></li>
<li><a href="#org2f89d7b">4.4. From model<sub>damping</sub> compute the Damping Matrix</a></li>
<li><a href="#org31812e9">4.5. Define inputs, outputs and state names</a></li>
<li><a href="#org931a108">4.6. Define A, B and C matrices</a></li>
<li><a href="#org6246054">4.7. Generate the State Space Model</a></li>
<li><a href="#orgfed5d4b">4.8. Bode Plot</a></li>
</ul>
</li>
<li><a href="#org90551c6">5. Power Spectral Density</a>
<ul>
<li><a href="#org659372f">5.1. Compute the PSD</a></li>
<li><a href="#org3690c18">5.2. Plot the computed PSD</a></li>
<li><a href="#org0a6959c">5.3. Load the model of the spindle</a></li>
<li><a href="#orgb977795">5.4. Plot the PSD of the Force using the model</a></li>
<li><a href="#orgbd81578">5.5. Estimated Shape of the PSD of the force</a></li>
<li><a href="#orgebea7be">5.6. PSD in [N]</a></li>
<li><a href="#org4818470">5.7. PSD in [m]</a></li>
<li><a href="#org408aad1">5.8. Compute the resulting RMS value [m]</a></li>
<li><a href="#org97cda8b">5.9. Compute the resulting RMS value [m]</a></li>
</ul>
</li>
<li><a href="#orgb0ca16b">6. Functions</a>
<ul>
<li><a href="#org14006e0">6.1. getAsynchronousError</a></li>
</ul>
</li>
</ul>
</div>
</div>
<p>
<a href="../index.html">Back to main page</a>.
</p>
<p>
The report made by the PEL is accessible <a href="documents/Spindle_report_test.pdf">here</a>.
</p>
<div id="outline-container-org4ef1ca3" class="outline-2">
<h2 id="org4ef1ca3"><span class="section-number-2">1</span> Notes</h2>
<div class="outline-text-2" id="text-1">
<div id="org4241ebb" class="figure">
<p><img src="./figs/setup_spindle.png" alt="setup_spindle.png" />
</p>
<p><span class="figure-number">Figure 1: </span>Measurement setup at the PEL lab</p>
</div>
</div>
</div>
<div id="outline-container-orge92ef42" class="outline-2">
<h2 id="orge92ef42"><span class="section-number-2">2</span> Data Processing</h2>
<div class="outline-text-2" id="text-2">
</div>
<div id="outline-container-orgbf2f91a" class="outline-3">
<h3 id="orgbf2f91a"><span class="section-number-3">2.1</span> Load Measurement Data</h3>
<div class="outline-text-3" id="text-2-1">
<div class="org-src-container">
<pre class="src src-matlab">spindle_1rpm_table = readtable<span style="color: #DCDCCC;">(</span>'<span style="color: #7CB8BB;">./</span>data<span style="color: #7CB8BB;">/</span><span style="color: #BFEBBF;">10turns_1rpm_icepap</span>.txt'<span style="color: #DCDCCC;">)</span>;
spindle_60rpm_table = readtable<span style="color: #DCDCCC;">(</span>'<span style="color: #7CB8BB;">./</span>data<span style="color: #7CB8BB;">/</span><span style="color: #BFEBBF;">10turns_60rpm_IcepapFIR</span>.txt'<span style="color: #DCDCCC;">)</span>;
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab">spindle_1rpm_table<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">1</span>, <span style="color: #7CB8BB;">:</span><span style="color: #DCDCCC;">)</span>
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab">spindle_1rpm = table2array<span style="color: #DCDCCC;">(</span>spindle_1rpm_table<span style="color: #DCDCCC;">)</span>;
spindle_60rpm = table2array<span style="color: #DCDCCC;">(</span>spindle_60rpm_table<span style="color: #DCDCCC;">)</span>;
</pre>
</div>
</div>
</div>
<div id="outline-container-orge799cbd" class="outline-3">
<h3 id="orge799cbd"><span class="section-number-3">2.2</span> Convert Signals from [deg] to [sec]</h3>
<div class="outline-text-3" id="text-2-2">
<div class="org-src-container">
<pre class="src src-matlab">speed_1rpm = <span style="color: #BFEBBF;">360</span><span style="color: #7CB8BB;">/</span><span style="color: #BFEBBF;">60</span>; <span style="color: #7F9F7F;">% [deg/sec]</span>
spindle_1rpm<span style="color: #DCDCCC;">(</span><span style="color: #7CB8BB;">:</span>, <span style="color: #BFEBBF;">1</span><span style="color: #DCDCCC;">)</span> = spindle_1rpm<span style="color: #DCDCCC;">(</span><span style="color: #7CB8BB;">:</span>, <span style="color: #BFEBBF;">2</span><span style="color: #DCDCCC;">)</span><span style="color: #7CB8BB;">/</span>speed_1rpm; <span style="color: #7F9F7F;">% From position [deg] to time [s]</span>
speed_60rpm = <span style="color: #BFEBBF;">360</span><span style="color: #7CB8BB;">/</span><span style="color: #BFEBBF;">1</span>; <span style="color: #7F9F7F;">% [deg/sec]</span>
spindle_60rpm<span style="color: #DCDCCC;">(</span><span style="color: #7CB8BB;">:</span>, <span style="color: #BFEBBF;">1</span><span style="color: #DCDCCC;">)</span> = spindle_60rpm<span style="color: #DCDCCC;">(</span><span style="color: #7CB8BB;">:</span>, <span style="color: #BFEBBF;">2</span><span style="color: #DCDCCC;">)</span><span style="color: #7CB8BB;">/</span>speed_60rpm; <span style="color: #7F9F7F;">% From position [deg] to time [s]</span>
</pre>
</div>
</div>
</div>
<div id="outline-container-orge0e840d" class="outline-3">
<h3 id="orge0e840d"><span class="section-number-3">2.3</span> Convert Signals</h3>
<div class="outline-text-3" id="text-2-3">
<div class="org-src-container">
<pre class="src src-matlab">% scaling = <span style="color: #BFEBBF;">1</span><span style="color: #7CB8BB;">/</span><span style="color: #BFEBBF;">80000</span>; % <span style="color: #BFEBBF;">80</span> mV<span style="color: #7CB8BB;">/</span>um
scaling = <span style="color: #BFEBBF;">1e</span><span style="color: #7CB8BB;">-</span><span style="color: #BFEBBF;">6</span>; <span style="color: #7F9F7F;">% [um] to [m]</span>
spindle_1rpm<span style="color: #DCDCCC;">(</span><span style="color: #7CB8BB;">:</span>, <span style="color: #BFEBBF;">3</span><span style="color: #7CB8BB;">:</span>end<span style="color: #DCDCCC;">)</span> = scaling<span style="color: #7CB8BB;">*</span>spindle_1rpm<span style="color: #DCDCCC;">(</span><span style="color: #7CB8BB;">:</span>, <span style="color: #BFEBBF;">3</span><span style="color: #7CB8BB;">:</span>end<span style="color: #DCDCCC;">)</span>; <span style="color: #7F9F7F;">% [V] to [m]</span>
spindle_1rpm<span style="color: #DCDCCC;">(</span><span style="color: #7CB8BB;">:</span>, <span style="color: #BFEBBF;">3</span><span style="color: #7CB8BB;">:</span>end<span style="color: #DCDCCC;">)</span> = spindle_1rpm<span style="color: #DCDCCC;">(</span><span style="color: #7CB8BB;">:</span>, <span style="color: #BFEBBF;">3</span><span style="color: #7CB8BB;">:</span>end<span style="color: #DCDCCC;">)</span><span style="color: #7CB8BB;">-</span>mean<span style="color: #DCDCCC;">(</span>spindle_1rpm<span style="color: #BFEBBF;">(</span><span style="color: #7CB8BB;">:</span>, <span style="color: #BFEBBF;">3</span><span style="color: #7CB8BB;">:</span>end<span style="color: #BFEBBF;">)</span><span style="color: #DCDCCC;">)</span>; <span style="color: #7F9F7F;">% Remove mean</span>
spindle_60rpm<span style="color: #DCDCCC;">(</span><span style="color: #7CB8BB;">:</span>, <span style="color: #BFEBBF;">3</span><span style="color: #7CB8BB;">:</span>end<span style="color: #DCDCCC;">)</span> = scaling<span style="color: #7CB8BB;">*</span>spindle_60rpm<span style="color: #DCDCCC;">(</span><span style="color: #7CB8BB;">:</span>, <span style="color: #BFEBBF;">3</span><span style="color: #7CB8BB;">:</span>end<span style="color: #DCDCCC;">)</span>; <span style="color: #7F9F7F;">% [V] to [m]</span>
spindle_60rpm<span style="color: #DCDCCC;">(</span><span style="color: #7CB8BB;">:</span>, <span style="color: #BFEBBF;">3</span><span style="color: #7CB8BB;">:</span>end<span style="color: #DCDCCC;">)</span> = spindle_60rpm<span style="color: #DCDCCC;">(</span><span style="color: #7CB8BB;">:</span>, <span style="color: #BFEBBF;">3</span><span style="color: #7CB8BB;">:</span>end<span style="color: #DCDCCC;">)</span><span style="color: #7CB8BB;">-</span>mean<span style="color: #DCDCCC;">(</span>spindle_60rpm<span style="color: #BFEBBF;">(</span><span style="color: #7CB8BB;">:</span>, <span style="color: #BFEBBF;">3</span><span style="color: #7CB8BB;">:</span>end<span style="color: #BFEBBF;">)</span><span style="color: #DCDCCC;">)</span>; <span style="color: #7F9F7F;">% Remove mean</span>
</pre>
</div>
</div>
</div>
<div id="outline-container-orgb79c23f" class="outline-3">
<h3 id="orgb79c23f"><span class="section-number-3">2.4</span> Ts and Fs for both measurements</h3>
<div class="outline-text-3" id="text-2-4">
<div class="org-src-container">
<pre class="src src-matlab">Ts_1rpm = spindle_1rpm<span style="color: #DCDCCC;">(</span>end, <span style="color: #BFEBBF;">1</span><span style="color: #DCDCCC;">)</span><span style="color: #7CB8BB;">/</span><span style="color: #DCDCCC;">(</span>length<span style="color: #BFEBBF;">(</span>spindle_1rpm<span style="color: #D0BF8F;">(</span><span style="color: #7CB8BB;">:</span>, <span style="color: #BFEBBF;">1</span><span style="color: #D0BF8F;">)</span><span style="color: #BFEBBF;">)</span><span style="color: #7CB8BB;">-</span><span style="color: #BFEBBF;">1</span><span style="color: #DCDCCC;">)</span>;
Fs_1rpm = <span style="color: #BFEBBF;">1</span><span style="color: #7CB8BB;">/</span>Ts_1rpm;
Ts_60rpm = spindle_60rpm<span style="color: #DCDCCC;">(</span>end, <span style="color: #BFEBBF;">1</span><span style="color: #DCDCCC;">)</span><span style="color: #7CB8BB;">/</span><span style="color: #DCDCCC;">(</span>length<span style="color: #BFEBBF;">(</span>spindle_60rpm<span style="color: #D0BF8F;">(</span><span style="color: #7CB8BB;">:</span>, <span style="color: #BFEBBF;">1</span><span style="color: #D0BF8F;">)</span><span style="color: #BFEBBF;">)</span><span style="color: #7CB8BB;">-</span><span style="color: #BFEBBF;">1</span><span style="color: #DCDCCC;">)</span>;
Fs_60rpm = <span style="color: #BFEBBF;">1</span><span style="color: #7CB8BB;">/</span>Ts_60rpm;
</pre>
</div>
</div>
</div>
<div id="outline-container-org545b4dc" class="outline-3">
<h3 id="org545b4dc"><span class="section-number-3">2.5</span> Find Noise of the ADC [m/sqrt(Hz)]</h3>
<div class="outline-text-3" id="text-2-5">
<div class="org-src-container">
<pre class="src src-matlab">data = spindle_1rpm<span style="color: #DCDCCC;">(</span><span style="color: #7CB8BB;">:</span>, <span style="color: #BFEBBF;">5</span><span style="color: #DCDCCC;">)</span>;
dV_1rpm = min<span style="color: #DCDCCC;">(</span>abs<span style="color: #BFEBBF;">(</span>data<span style="color: #D0BF8F;">(</span><span style="color: #BFEBBF;">1</span><span style="color: #D0BF8F;">)</span> <span style="color: #7CB8BB;">-</span> data<span style="color: #D0BF8F;">(</span>data <span style="color: #7CB8BB;">~=</span> data<span style="color: #93E0E3;">(</span><span style="color: #BFEBBF;">1</span><span style="color: #93E0E3;">)</span><span style="color: #D0BF8F;">)</span><span style="color: #BFEBBF;">)</span><span style="color: #DCDCCC;">)</span>;
noise_1rpm = dV_1rpm<span style="color: #7CB8BB;">/</span>sqrt<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">12</span><span style="color: #7CB8BB;">*</span>Fs_1rpm<span style="color: #7CB8BB;">/</span><span style="color: #BFEBBF;">2</span><span style="color: #DCDCCC;">)</span>;
data = spindle_60rpm<span style="color: #DCDCCC;">(</span><span style="color: #7CB8BB;">:</span>, <span style="color: #BFEBBF;">5</span><span style="color: #DCDCCC;">)</span>;
dV_60rpm = min<span style="color: #DCDCCC;">(</span>abs<span style="color: #BFEBBF;">(</span>data<span style="color: #D0BF8F;">(</span><span style="color: #BFEBBF;">50</span><span style="color: #D0BF8F;">)</span> <span style="color: #7CB8BB;">-</span> data<span style="color: #D0BF8F;">(</span>data <span style="color: #7CB8BB;">~=</span> data<span style="color: #93E0E3;">(</span><span style="color: #BFEBBF;">50</span><span style="color: #93E0E3;">)</span><span style="color: #D0BF8F;">)</span><span style="color: #BFEBBF;">)</span><span style="color: #DCDCCC;">)</span>;
noise_60rpm = dV_60rpm<span style="color: #7CB8BB;">/</span>sqrt<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">12</span><span style="color: #7CB8BB;">*</span>Fs_60rpm<span style="color: #7CB8BB;">/</span><span style="color: #BFEBBF;">2</span><span style="color: #DCDCCC;">)</span>;
</pre>
</div>
</div>
</div>
<div id="outline-container-orgb7c25b7" class="outline-3">
<h3 id="orgb7c25b7"><span class="section-number-3">2.6</span> Save all the data under spindle struct</h3>
<div class="outline-text-3" id="text-2-6">
<div class="org-src-container">
<pre class="src src-matlab">spindle.rpm1.time = spindle_1rpm<span style="color: #DCDCCC;">(</span><span style="color: #7CB8BB;">:</span>, <span style="color: #BFEBBF;">1</span><span style="color: #DCDCCC;">)</span>;
spindle.rpm1.deg = spindle_1rpm<span style="color: #DCDCCC;">(</span><span style="color: #7CB8BB;">:</span>, <span style="color: #BFEBBF;">2</span><span style="color: #DCDCCC;">)</span>;
spindle.rpm1.Ts = Ts_1rpm;
spindle.rpm1.Fs = <span style="color: #BFEBBF;">1</span><span style="color: #7CB8BB;">/</span>Ts_1rpm;
spindle.rpm1.x = spindle_1rpm<span style="color: #DCDCCC;">(</span><span style="color: #7CB8BB;">:</span>, <span style="color: #BFEBBF;">3</span><span style="color: #DCDCCC;">)</span>;
spindle.rpm1.y = spindle_1rpm<span style="color: #DCDCCC;">(</span><span style="color: #7CB8BB;">:</span>, <span style="color: #BFEBBF;">4</span><span style="color: #DCDCCC;">)</span>;
spindle.rpm1.z = spindle_1rpm<span style="color: #DCDCCC;">(</span><span style="color: #7CB8BB;">:</span>, <span style="color: #BFEBBF;">5</span><span style="color: #DCDCCC;">)</span>;
spindle.rpm1.adcn = noise_1rpm;
spindle.rpm60.time = spindle_60rpm<span style="color: #DCDCCC;">(</span><span style="color: #7CB8BB;">:</span>, <span style="color: #BFEBBF;">1</span><span style="color: #DCDCCC;">)</span>;
spindle.rpm60.deg = spindle_60rpm<span style="color: #DCDCCC;">(</span><span style="color: #7CB8BB;">:</span>, <span style="color: #BFEBBF;">2</span><span style="color: #DCDCCC;">)</span>;
spindle.rpm60.Ts = Ts_60rpm;
spindle.rpm60.Fs = <span style="color: #BFEBBF;">1</span><span style="color: #7CB8BB;">/</span>Ts_60rpm;
spindle.rpm60.x = spindle_60rpm<span style="color: #DCDCCC;">(</span><span style="color: #7CB8BB;">:</span>, <span style="color: #BFEBBF;">3</span><span style="color: #DCDCCC;">)</span>;
spindle.rpm60.y = spindle_60rpm<span style="color: #DCDCCC;">(</span><span style="color: #7CB8BB;">:</span>, <span style="color: #BFEBBF;">4</span><span style="color: #DCDCCC;">)</span>;
spindle.rpm60.z = spindle_60rpm<span style="color: #DCDCCC;">(</span><span style="color: #7CB8BB;">:</span>, <span style="color: #BFEBBF;">5</span><span style="color: #DCDCCC;">)</span>;
spindle.rpm60.adcn = noise_60rpm;
</pre>
</div>
</div>
</div>
<div id="outline-container-org707a6a0" class="outline-3">
<h3 id="org707a6a0"><span class="section-number-3">2.7</span> Compute Asynchronous data</h3>
<div class="outline-text-3" id="text-2-7">
<div class="org-src-container">
<pre class="src src-matlab"><span style="color: #F0DFAF; font-weight: bold;">for</span> <span style="color: #DFAF8F;">direction</span> = <span style="color: #DCDCCC;">{</span><span style="color: #CC9393;">'x'</span><span style="color: #CC9393;">, 'y', 'z'</span><span style="color: #DCDCCC;">}</span>
spindle.rpm1.<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">[</span>direction<span style="color: #D0BF8F;">{</span><span style="color: #BFEBBF;">1</span><span style="color: #D0BF8F;">}</span>, <span style="color: #CC9393;">'async'</span><span style="color: #BFEBBF;">]</span><span style="color: #DCDCCC;">)</span> = getAsynchronousError<span style="color: #DCDCCC;">(</span>spindle.rpm1.<span style="color: #BFEBBF;">(</span>direction<span style="color: #D0BF8F;">{</span><span style="color: #BFEBBF;">1</span><span style="color: #D0BF8F;">}</span><span style="color: #BFEBBF;">)</span>, <span style="color: #BFEBBF;">10</span><span style="color: #DCDCCC;">)</span>;
spindle.rpm60.<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">[</span>direction<span style="color: #D0BF8F;">{</span><span style="color: #BFEBBF;">1</span><span style="color: #D0BF8F;">}</span>, <span style="color: #CC9393;">'async'</span><span style="color: #BFEBBF;">]</span><span style="color: #DCDCCC;">)</span> = getAsynchronousError<span style="color: #DCDCCC;">(</span>spindle.rpm60.<span style="color: #BFEBBF;">(</span>direction<span style="color: #D0BF8F;">{</span><span style="color: #BFEBBF;">1</span><span style="color: #D0BF8F;">}</span><span style="color: #BFEBBF;">)</span>, <span style="color: #BFEBBF;">10</span><span style="color: #DCDCCC;">)</span>;
<span style="color: #F0DFAF; font-weight: bold;">end</span>
</pre>
</div>
</div>
</div>
</div>
<div id="outline-container-orga9ae55f" class="outline-2">
<h2 id="orga9ae55f"><span class="section-number-2">3</span> Time Domain Data</h2>
<div class="outline-text-2" id="text-3">
</div>
<div id="outline-container-orga3598c8" class="outline-3">
<h3 id="orga3598c8"><span class="section-number-3">3.1</span> Plot X-Y-Z position with respect to Time - 1rpm</h3>
<div class="outline-text-3" id="text-3-1">
<div class="org-src-container">
<pre class="src src-matlab"><span style="color: #7CB8BB;">figure</span>;
hold on;
plot<span style="color: #DCDCCC;">(</span>spindle.rpm1.time, spindle.rpm1.x<span style="color: #DCDCCC;">)</span>;
plot<span style="color: #DCDCCC;">(</span>spindle.rpm1.time, spindle.rpm1.y<span style="color: #DCDCCC;">)</span>;
plot<span style="color: #DCDCCC;">(</span>spindle.rpm1.time, spindle.rpm1.z<span style="color: #DCDCCC;">)</span>;
hold off;
xlabel<span style="color: #DCDCCC;">(</span><span style="color: #CC9393;">'Time </span><span style="color: #BFEBBF;">[</span><span style="color: #CC9393;">s</span><span style="color: #BFEBBF;">]</span><span style="color: #CC9393;">'</span><span style="color: #DCDCCC;">)</span><span style="color: #CC9393;">; ylabel</span><span style="color: #DCDCCC;">(</span><span style="color: #CC9393;">'Amplitude </span><span style="color: #BFEBBF;">[</span><span style="color: #CC9393;">m</span><span style="color: #BFEBBF;">]</span><span style="color: #CC9393;">'</span><span style="color: #DCDCCC;">)</span>;
legend<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">{</span>'tx <span style="color: #7CB8BB;">-</span> <span style="color: #BFEBBF;">1rpm</span>', 'ty <span style="color: #7CB8BB;">-</span> <span style="color: #BFEBBF;">1rpm</span>', 'tz <span style="color: #7CB8BB;">-</span> <span style="color: #BFEBBF;">1rpm</span>'<span style="color: #BFEBBF;">}</span><span style="color: #DCDCCC;">)</span>;
</pre>
</div>
<div id="org4532134" class="figure">
<p><img src="figs/spindle_xyz_1rpm.png" alt="spindle_xyz_1rpm.png" />
</p>
<p><span class="figure-number">Figure 2: </span>Raw time domain translation - 1rpm</p>
</div>
</div>
</div>
<div id="outline-container-orgb1d7825" class="outline-3">
<h3 id="orgb1d7825"><span class="section-number-3">3.2</span> Plot X-Y-Z position with respect to Time - 60rpm</h3>
<div class="outline-text-3" id="text-3-2">
<p>
The measurements for the spindle turning at 60rpm are shown figure <a href="#org624ade7">3</a>.
</p>
<div class="org-src-container">
<pre class="src src-matlab"><span style="color: #7CB8BB;">figure</span>;
hold on;
plot<span style="color: #DCDCCC;">(</span>spindle.rpm60.time, spindle.rpm60.x<span style="color: #DCDCCC;">)</span>;
plot<span style="color: #DCDCCC;">(</span>spindle.rpm60.time, spindle.rpm60.y<span style="color: #DCDCCC;">)</span>;
plot<span style="color: #DCDCCC;">(</span>spindle.rpm60.time, spindle.rpm60.z<span style="color: #DCDCCC;">)</span>;
hold off;
xlabel<span style="color: #DCDCCC;">(</span><span style="color: #CC9393;">'Time </span><span style="color: #BFEBBF;">[</span><span style="color: #CC9393;">s</span><span style="color: #BFEBBF;">]</span><span style="color: #CC9393;">'</span><span style="color: #DCDCCC;">)</span><span style="color: #CC9393;">; ylabel</span><span style="color: #DCDCCC;">(</span><span style="color: #CC9393;">'Amplitude </span><span style="color: #BFEBBF;">[</span><span style="color: #CC9393;">m</span><span style="color: #BFEBBF;">]</span><span style="color: #CC9393;">'</span><span style="color: #DCDCCC;">)</span>;
legend<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">{</span>'tx <span style="color: #7CB8BB;">-</span> <span style="color: #BFEBBF;">60rpm</span>', 'ty <span style="color: #7CB8BB;">-</span> <span style="color: #BFEBBF;">60rpm</span>', 'tz <span style="color: #7CB8BB;">-</span> <span style="color: #BFEBBF;">60rpm</span>'<span style="color: #BFEBBF;">}</span><span style="color: #DCDCCC;">)</span>;
</pre>
</div>
<div id="org624ade7" class="figure">
<p><img src="figs/spindle_xyz_60rpm.png" alt="spindle_xyz_60rpm.png" />
</p>
<p><span class="figure-number">Figure 3: </span>Raw time domain translation - 60rpm</p>
</div>
</div>
</div>
<div id="outline-container-org487e41f" class="outline-3">
<h3 id="org487e41f"><span class="section-number-3">3.3</span> Plot Synchronous and Asynchronous - 1rpm</h3>
<div class="outline-text-3" id="text-3-3">
<div class="org-src-container">
<pre class="src src-matlab"><span style="color: #7CB8BB;">figure</span>;
hold on;
plot<span style="color: #DCDCCC;">(</span>spindle.rpm1.time, spindle.rpm1.x<span style="color: #DCDCCC;">)</span>;
plot<span style="color: #DCDCCC;">(</span>spindle.rpm1.time, spindle.rpm1.xasync<span style="color: #DCDCCC;">)</span>;
hold off;
xlabel<span style="color: #DCDCCC;">(</span><span style="color: #CC9393;">'Time </span><span style="color: #BFEBBF;">[</span><span style="color: #CC9393;">s</span><span style="color: #BFEBBF;">]</span><span style="color: #CC9393;">'</span><span style="color: #DCDCCC;">)</span><span style="color: #CC9393;">; ylabel</span><span style="color: #DCDCCC;">(</span><span style="color: #CC9393;">'Amplitude </span><span style="color: #BFEBBF;">[</span><span style="color: #CC9393;">m</span><span style="color: #BFEBBF;">]</span><span style="color: #CC9393;">'</span><span style="color: #DCDCCC;">)</span>;
legend<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">{</span>'tx <span style="color: #7CB8BB;">-</span> <span style="color: #BFEBBF;">1rpm</span> <span style="color: #7CB8BB;">-</span> Sync', 'tx <span style="color: #7CB8BB;">-</span> <span style="color: #BFEBBF;">1rpm</span> <span style="color: #7CB8BB;">-</span> Async'<span style="color: #BFEBBF;">}</span><span style="color: #DCDCCC;">)</span>;
</pre>
</div>
<div id="orgaccc4b0" class="figure">
<p><img src="figs/spindle_1rpm_sync_async.png" alt="spindle_1rpm_sync_async.png" />
</p>
<p><span class="figure-number">Figure 4: </span>Comparison of the synchronous and asynchronous displacements - 1rpm</p>
</div>
</div>
</div>
<div id="outline-container-org5310667" class="outline-3">
<h3 id="org5310667"><span class="section-number-3">3.4</span> Plot Synchronous and Asynchronous - 60rpm</h3>
<div class="outline-text-3" id="text-3-4">
<p>
The data is split into its Synchronous and Asynchronous part (figure <a href="#orgdc342a6">5</a>). We then use the Asynchronous part for the analysis in the following sections as we suppose that we can deal with the synchronous part with feedforward control.
</p>
<div class="org-src-container">
<pre class="src src-matlab"><span style="color: #7CB8BB;">figure</span>;
hold on;
plot<span style="color: #DCDCCC;">(</span>spindle.rpm60.time, spindle.rpm60.x<span style="color: #DCDCCC;">)</span>;
plot<span style="color: #DCDCCC;">(</span>spindle.rpm60.time, spindle.rpm60.xasync<span style="color: #DCDCCC;">)</span>;
hold off;
xlabel<span style="color: #DCDCCC;">(</span><span style="color: #CC9393;">'Time </span><span style="color: #BFEBBF;">[</span><span style="color: #CC9393;">s</span><span style="color: #BFEBBF;">]</span><span style="color: #CC9393;">'</span><span style="color: #DCDCCC;">)</span><span style="color: #CC9393;">; ylabel</span><span style="color: #DCDCCC;">(</span><span style="color: #CC9393;">'Amplitude </span><span style="color: #BFEBBF;">[</span><span style="color: #CC9393;">m</span><span style="color: #BFEBBF;">]</span><span style="color: #CC9393;">'</span><span style="color: #DCDCCC;">)</span>;
legend<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">{</span>'tx <span style="color: #7CB8BB;">-</span> <span style="color: #BFEBBF;">60rpm</span> <span style="color: #7CB8BB;">-</span> Sync', 'tx <span style="color: #7CB8BB;">-</span> <span style="color: #BFEBBF;">60rpm</span> <span style="color: #7CB8BB;">-</span> Async'<span style="color: #BFEBBF;">}</span><span style="color: #DCDCCC;">)</span>;
</pre>
</div>
<div id="orgdc342a6" class="figure">
<p><img src="figs/spindle_60rpm_sync_async.png" alt="spindle_60rpm_sync_async.png" />
</p>
<p><span class="figure-number">Figure 5: </span>Comparison of the synchronous and asynchronous displacements - 60rpm</p>
</div>
</div>
</div>
<div id="outline-container-org037b157" class="outline-3">
<h3 id="org037b157"><span class="section-number-3">3.5</span> Plot X against Y</h3>
<div class="outline-text-3" id="text-3-5">
<div class="org-src-container">
<pre class="src src-matlab"><span style="color: #7CB8BB;">figure</span>;
hold on;
plot<span style="color: #DCDCCC;">(</span>spindle.rpm1.x, spindle.rpm1.y<span style="color: #DCDCCC;">)</span>;
plot<span style="color: #DCDCCC;">(</span>spindle.rpm60.x, spindle.rpm60.y<span style="color: #DCDCCC;">)</span>;
hold off;
xlabel<span style="color: #DCDCCC;">(</span><span style="color: #CC9393;">'X Amplitude </span><span style="color: #BFEBBF;">[</span><span style="color: #CC9393;">m</span><span style="color: #BFEBBF;">]</span><span style="color: #CC9393;">'</span><span style="color: #DCDCCC;">)</span><span style="color: #CC9393;">; ylabel</span><span style="color: #DCDCCC;">(</span><span style="color: #CC9393;">'Y Amplitude </span><span style="color: #BFEBBF;">[</span><span style="color: #CC9393;">m</span><span style="color: #BFEBBF;">]</span><span style="color: #CC9393;">'</span><span style="color: #DCDCCC;">)</span>;
legend<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">{</span>'<span style="color: #BFEBBF;">1rpm</span>', '<span style="color: #BFEBBF;">60rpm</span>'<span style="color: #BFEBBF;">}</span><span style="color: #DCDCCC;">)</span>;
</pre>
</div>
<div id="org6fbaabf" class="figure">
<p><img src="figs/spindle_xy_1_60rpm.png" alt="spindle_xy_1_60rpm.png" />
</p>
<p><span class="figure-number">Figure 6: </span>Synchronous x-y displacement</p>
</div>
</div>
</div>
<div id="outline-container-org067dc56" class="outline-3">
<h3 id="org067dc56"><span class="section-number-3">3.6</span> Plot X against Y - Asynchronous</h3>
<div class="outline-text-3" id="text-3-6">
<div class="org-src-container">
<pre class="src src-matlab"><span style="color: #7CB8BB;">figure</span>;
hold on;
plot<span style="color: #DCDCCC;">(</span>spindle.rpm1.xasync, spindle.rpm1.yasync<span style="color: #DCDCCC;">)</span>;
plot<span style="color: #DCDCCC;">(</span>spindle.rpm60.xasync, spindle.rpm60.yasync<span style="color: #DCDCCC;">)</span>;
hold off;
xlabel<span style="color: #DCDCCC;">(</span><span style="color: #CC9393;">'X Amplitude </span><span style="color: #BFEBBF;">[</span><span style="color: #CC9393;">m</span><span style="color: #BFEBBF;">]</span><span style="color: #CC9393;">'</span><span style="color: #DCDCCC;">)</span><span style="color: #CC9393;">; ylabel</span><span style="color: #DCDCCC;">(</span><span style="color: #CC9393;">'Y Amplitude </span><span style="color: #BFEBBF;">[</span><span style="color: #CC9393;">m</span><span style="color: #BFEBBF;">]</span><span style="color: #CC9393;">'</span><span style="color: #DCDCCC;">)</span>;
legend<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">{</span>'<span style="color: #BFEBBF;">1rpm</span>', '<span style="color: #BFEBBF;">60rpm</span>'<span style="color: #BFEBBF;">}</span><span style="color: #DCDCCC;">)</span>;
</pre>
</div>
<div id="org2cad174" class="figure">
<p><img src="figs/spindle_xy_1_60_rpm_async.png" alt="spindle_xy_1_60_rpm_async.png" />
</p>
<p><span class="figure-number">Figure 7: </span>Asynchronous x-y displacement</p>
</div>
</div>
</div>
</div>
<div id="outline-container-org590c4a3" class="outline-2">
<h2 id="org590c4a3"><span class="section-number-2">4</span> Model of the spindle</h2>
<div class="outline-text-2" id="text-4">
<p>
The model of the spindle used is shown figure <a href="#org5077262">8</a>.
</p>
<p>
\(f\) is the perturbation force of the spindle and \(d\) is the measured displacement.
</p>
<div id="org5077262" class="figure">
<p><img src="./figs/uniaxial-model-spindle.png" alt="uniaxial-model-spindle.png" />
</p>
<p><span class="figure-number">Figure 8: </span>Model of the Spindle</p>
</div>
</div>
<div id="outline-container-orgb7ea4e3" class="outline-3">
<h3 id="orgb7ea4e3"><span class="section-number-3">4.1</span> Parameters</h3>
<div class="outline-text-3" id="text-4-1">
<div class="org-src-container">
<pre class="src src-matlab">mg = <span style="color: #BFEBBF;">3000</span>; <span style="color: #7F9F7F;">% Mass of granite [kg]</span>
ms = <span style="color: #BFEBBF;">50</span>; <span style="color: #7F9F7F;">% Mass of Spindle [kg]</span>
kg = <span style="color: #BFEBBF;">1e8</span>; <span style="color: #7F9F7F;">% Stiffness of granite [N/m]</span>
ks = <span style="color: #BFEBBF;">5e7</span>; <span style="color: #7F9F7F;">% Stiffness of spindle [N/m]</span>
</pre>
</div>
</div>
</div>
<div id="outline-container-orge4e720c" class="outline-3">
<h3 id="orge4e720c"><span class="section-number-3">4.2</span> Compute Mass and Stiffness Matrices</h3>
<div class="outline-text-3" id="text-4-2">
<div class="org-src-container">
<pre class="src src-matlab">Mm = diag<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">[</span>ms, mg<span style="color: #BFEBBF;">]</span><span style="color: #DCDCCC;">)</span>;
Km = diag<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">[</span>ks, ks<span style="color: #7CB8BB;">+</span>kg<span style="color: #BFEBBF;">]</span><span style="color: #DCDCCC;">)</span> <span style="color: #7CB8BB;">-</span> diag<span style="color: #DCDCCC;">(</span>ks, <span style="color: #7CB8BB;">-</span><span style="color: #BFEBBF;">1</span><span style="color: #DCDCCC;">)</span> <span style="color: #7CB8BB;">-</span> diag<span style="color: #DCDCCC;">(</span>ks, <span style="color: #BFEBBF;">1</span><span style="color: #DCDCCC;">)</span>;
</pre>
</div>
</div>
</div>
<div id="outline-container-org5d5795f" class="outline-3">
<h3 id="org5d5795f"><span class="section-number-3">4.3</span> Compute resonance frequencies</h3>
<div class="outline-text-3" id="text-4-3">
<div class="org-src-container">
<pre class="src src-matlab">A = <span style="color: #DCDCCC;">[</span>zeros<span style="color: #BFEBBF;">(</span>size<span style="color: #D0BF8F;">(</span>Mm<span style="color: #D0BF8F;">)</span><span style="color: #BFEBBF;">)</span> eye<span style="color: #BFEBBF;">(</span>size<span style="color: #D0BF8F;">(</span>Mm<span style="color: #D0BF8F;">)</span><span style="color: #BFEBBF;">)</span> ; <span style="color: #7CB8BB;">-</span>Mm<span style="color: #7CB8BB;">\</span>Km zeros<span style="color: #BFEBBF;">(</span>size<span style="color: #D0BF8F;">(</span>Mm<span style="color: #D0BF8F;">)</span><span style="color: #BFEBBF;">)</span><span style="color: #DCDCCC;">]</span>;
eigA = imag<span style="color: #DCDCCC;">(</span>eigs<span style="color: #BFEBBF;">(</span>A<span style="color: #BFEBBF;">)</span><span style="color: #DCDCCC;">)</span><span style="color: #7CB8BB;">/</span><span style="color: #BFEBBF;">2</span><span style="color: #7CB8BB;">/</span><span style="color: #BFEBBF;">pi</span>;
eigA = eigA<span style="color: #DCDCCC;">(</span>eigA<span style="color: #7CB8BB;">></span><span style="color: #BFEBBF;">0</span><span style="color: #DCDCCC;">)</span>;
eigA = eigA<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">1</span><span style="color: #7CB8BB;">:</span><span style="color: #BFEBBF;">2</span><span style="color: #DCDCCC;">)</span>;
</pre>
</div>
</div>
</div>
<div id="outline-container-org2f89d7b" class="outline-3">
<h3 id="org2f89d7b"><span class="section-number-3">4.4</span> From model<sub>damping</sub> compute the Damping Matrix</h3>
<div class="outline-text-3" id="text-4-4">
<div class="org-src-container">
<pre class="src src-matlab">modal_damping = <span style="color: #BFEBBF;">1e</span><span style="color: #7CB8BB;">-</span><span style="color: #BFEBBF;">5</span>;
ab = <span style="color: #DCDCCC;">[</span><span style="color: #BFEBBF;">0</span>.<span style="color: #BFEBBF;">5</span><span style="color: #7CB8BB;">*</span>eigA<span style="color: #BFEBBF;">(</span><span style="color: #BFEBBF;">1</span><span style="color: #BFEBBF;">)</span> <span style="color: #BFEBBF;">0</span>.<span style="color: #BFEBBF;">5</span><span style="color: #7CB8BB;">/</span>eigA<span style="color: #BFEBBF;">(</span><span style="color: #BFEBBF;">1</span><span style="color: #BFEBBF;">)</span> ; <span style="color: #BFEBBF;">0</span>.<span style="color: #BFEBBF;">5</span><span style="color: #7CB8BB;">*</span>eigA<span style="color: #BFEBBF;">(</span><span style="color: #BFEBBF;">2</span><span style="color: #BFEBBF;">)</span> <span style="color: #BFEBBF;">0</span>.<span style="color: #BFEBBF;">5</span><span style="color: #7CB8BB;">/</span>eigA<span style="color: #BFEBBF;">(</span><span style="color: #BFEBBF;">2</span><span style="color: #BFEBBF;">)</span><span style="color: #DCDCCC;">]</span><span style="color: #7CB8BB;">\</span><span style="color: #DCDCCC;">[</span>modal_damping ; modal_damping<span style="color: #DCDCCC;">]</span>;
Cm = ab<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">1</span><span style="color: #DCDCCC;">)</span><span style="color: #7CB8BB;">*</span>Mm <span style="color: #7CB8BB;">+</span>ab<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">2</span><span style="color: #DCDCCC;">)</span><span style="color: #7CB8BB;">*</span>Km;
</pre>
</div>
</div>
</div>
<div id="outline-container-org31812e9" class="outline-3">
<h3 id="org31812e9"><span class="section-number-3">4.5</span> Define inputs, outputs and state names</h3>
<div class="outline-text-3" id="text-4-5">
<div class="org-src-container">
<pre class="src src-matlab">StateName = <span style="color: #DCDCCC;">{</span><span style="text-decoration: underline;">...</span>
<span style="color: #CC9393;">'xs'</span>, <span style="text-decoration: underline;">...</span> <span style="color: #7F9F7F;">% Displacement of Spindle [m]</span>
<span style="color: #CC9393;">'xg'</span>, <span style="text-decoration: underline;">...</span> <span style="color: #7F9F7F;">% Displacement of Granite [m]</span>
<span style="color: #CC9393;">'vs'</span>, <span style="text-decoration: underline;">...</span> <span style="color: #7F9F7F;">% Velocity of Spindle [m]</span>
<span style="color: #CC9393;">'vg'</span>, <span style="text-decoration: underline;">...</span> <span style="color: #7F9F7F;">% Velocity of Granite [m]</span>
<span style="color: #DCDCCC;">}</span>;
StateUnit = <span style="color: #DCDCCC;">{</span><span style="color: #CC9393;">'m', 'm', 'm/s', 'm/s'</span><span style="color: #DCDCCC;">}</span>;
InputName = <span style="color: #DCDCCC;">{</span><span style="text-decoration: underline;">...</span>
<span style="color: #CC9393;">'f'</span> <span style="text-decoration: underline;">...</span> <span style="color: #7F9F7F;">% Spindle Force [N]</span>
<span style="color: #DCDCCC;">}</span>;
InputUnit = <span style="color: #DCDCCC;">{</span><span style="color: #CC9393;">'N'</span><span style="color: #DCDCCC;">}</span>;
OutputName = <span style="color: #DCDCCC;">{</span><span style="text-decoration: underline;">...</span>
<span style="color: #CC9393;">'d'</span> <span style="text-decoration: underline;">...</span> <span style="color: #7F9F7F;">% Displacement [m]</span>
<span style="color: #DCDCCC;">}</span>;
OutputUnit = <span style="color: #DCDCCC;">{</span><span style="color: #CC9393;">'m'</span><span style="color: #DCDCCC;">}</span>;
</pre>
</div>
</div>
</div>
<div id="outline-container-org931a108" class="outline-3">
<h3 id="org931a108"><span class="section-number-3">4.6</span> Define A, B and C matrices</h3>
<div class="outline-text-3" id="text-4-6">
<div class="org-src-container">
<pre class="src src-matlab"><span style="color: #7F9F7F;">% A Matrix</span>
A = <span style="color: #DCDCCC;">[</span>zeros<span style="color: #BFEBBF;">(</span>size<span style="color: #D0BF8F;">(</span>Mm<span style="color: #D0BF8F;">)</span><span style="color: #BFEBBF;">)</span> eye<span style="color: #BFEBBF;">(</span>size<span style="color: #D0BF8F;">(</span>Mm<span style="color: #D0BF8F;">)</span><span style="color: #BFEBBF;">)</span> ; <span style="text-decoration: underline;">...</span>
<span style="color: #7CB8BB;">-</span>Mm<span style="color: #7CB8BB;">\</span>Km <span style="color: #7CB8BB;">-</span>Mm<span style="color: #7CB8BB;">\</span>Cm<span style="color: #DCDCCC;">]</span>;
<span style="color: #7F9F7F;">% B Matrix</span>
B_low = zeros<span style="color: #DCDCCC;">(</span>length<span style="color: #BFEBBF;">(</span>StateName<span style="color: #BFEBBF;">)</span>, length<span style="color: #BFEBBF;">(</span>InputName<span style="color: #BFEBBF;">)</span><span style="color: #DCDCCC;">)</span>;
B_low<span style="color: #DCDCCC;">(</span>strcmp<span style="color: #BFEBBF;">(</span>StateName,<span style="color: #CC9393;">'vs'</span><span style="color: #BFEBBF;">)</span><span style="color: #CC9393;">, strcmp</span><span style="color: #BFEBBF;">(</span><span style="color: #CC9393;">InputName,'f'</span><span style="color: #BFEBBF;">)</span><span style="color: #DCDCCC;">)</span> = <span style="color: #BFEBBF;">1</span>;
B_low<span style="color: #DCDCCC;">(</span>strcmp<span style="color: #BFEBBF;">(</span>StateName,<span style="color: #CC9393;">'vg'</span><span style="color: #BFEBBF;">)</span><span style="color: #CC9393;">, strcmp</span><span style="color: #BFEBBF;">(</span><span style="color: #CC9393;">InputName,'f'</span><span style="color: #BFEBBF;">)</span><span style="color: #DCDCCC;">)</span> = <span style="color: #7CB8BB;">-</span><span style="color: #BFEBBF;">1</span>;
B = blkdiag<span style="color: #DCDCCC;">(</span>zeros<span style="color: #BFEBBF;">(</span>length<span style="color: #D0BF8F;">(</span>StateName<span style="color: #D0BF8F;">)</span><span style="color: #7CB8BB;">/</span><span style="color: #BFEBBF;">2</span><span style="color: #BFEBBF;">)</span>, pinv<span style="color: #BFEBBF;">(</span>Mm<span style="color: #BFEBBF;">)</span><span style="color: #DCDCCC;">)</span><span style="color: #7CB8BB;">*</span>B_low;
<span style="color: #7F9F7F;">% C Matrix</span>
C = zeros<span style="color: #DCDCCC;">(</span>length<span style="color: #BFEBBF;">(</span>OutputName<span style="color: #BFEBBF;">)</span>, length<span style="color: #BFEBBF;">(</span>StateName<span style="color: #BFEBBF;">)</span><span style="color: #DCDCCC;">)</span>;
C<span style="color: #DCDCCC;">(</span>strcmp<span style="color: #BFEBBF;">(</span>OutputName,<span style="color: #CC9393;">'d'</span><span style="color: #BFEBBF;">)</span><span style="color: #CC9393;">, strcmp</span><span style="color: #BFEBBF;">(</span><span style="color: #CC9393;">StateName,'xs'</span><span style="color: #BFEBBF;">)</span><span style="color: #DCDCCC;">)</span> = <span style="color: #BFEBBF;">1</span>;
C<span style="color: #DCDCCC;">(</span>strcmp<span style="color: #BFEBBF;">(</span>OutputName,<span style="color: #CC9393;">'d'</span><span style="color: #BFEBBF;">)</span><span style="color: #CC9393;">, strcmp</span><span style="color: #BFEBBF;">(</span><span style="color: #CC9393;">StateName,'xg'</span><span style="color: #BFEBBF;">)</span><span style="color: #DCDCCC;">)</span> = <span style="color: #7CB8BB;">-</span><span style="color: #BFEBBF;">1</span>;
<span style="color: #7F9F7F;">% D Matrix</span>
D = zeros<span style="color: #DCDCCC;">(</span>length<span style="color: #BFEBBF;">(</span>OutputName<span style="color: #BFEBBF;">)</span>, length<span style="color: #BFEBBF;">(</span>InputName<span style="color: #BFEBBF;">)</span><span style="color: #DCDCCC;">)</span>;
</pre>
</div>
</div>
</div>
<div id="outline-container-org6246054" class="outline-3">
<h3 id="org6246054"><span class="section-number-3">4.7</span> Generate the State Space Model</h3>
<div class="outline-text-3" id="text-4-7">
<div class="org-src-container">
<pre class="src src-matlab">sys = ss<span style="color: #DCDCCC;">(</span>A, B, C, D<span style="color: #DCDCCC;">)</span>;
sys.StateName = StateName;
sys.StateUnit = StateUnit;
sys.InputName = InputName;
sys.InputUnit = InputUnit;
sys.OutputName = OutputName;
sys.OutputUnit = OutputUnit;
</pre>
</div>
</div>
</div>
<div id="outline-container-orgfed5d4b" class="outline-3">
<h3 id="orgfed5d4b"><span class="section-number-3">4.8</span> Bode Plot</h3>
<div class="outline-text-3" id="text-4-8">
<p>
The transfer function from a disturbance force \(f\) to the measured displacement \(d\) is shown figure <a href="#org2a58760">9</a>.
</p>
<div class="org-src-container">
<pre class="src src-matlab">freqs = logspace<span style="color: #DCDCCC;">(</span><span style="color: #7CB8BB;">-</span><span style="color: #BFEBBF;">1</span>, <span style="color: #BFEBBF;">3</span>, <span style="color: #BFEBBF;">1000</span><span style="color: #DCDCCC;">)</span>;
<span style="color: #7CB8BB;">figure</span>;
plot<span style="color: #DCDCCC;">(</span>freqs, abs<span style="color: #BFEBBF;">(</span>squeeze<span style="color: #D0BF8F;">(</span>freqresp<span style="color: #93E0E3;">(</span>sys<span style="color: #9FC59F;">(</span><span style="color: #CC9393;">'d', 'f'</span><span style="color: #9FC59F;">)</span><span style="color: #CC9393;">, freqs, 'Hz'</span><span style="color: #93E0E3;">)</span><span style="color: #D0BF8F;">)</span><span style="color: #BFEBBF;">)</span><span style="color: #DCDCCC;">)</span>;
<span style="color: #7CB8BB;">set</span><span style="color: #DCDCCC;">(</span><span style="color: #DFAF8F;">gca</span>, <span style="color: #CC9393;">'XScale', 'log'</span><span style="color: #DCDCCC;">)</span><span style="color: #CC9393;">; set</span><span style="color: #DCDCCC;">(</span><span style="color: #CC9393;">gca, 'YScale', 'log'</span><span style="color: #DCDCCC;">)</span>;
xlabel<span style="color: #DCDCCC;">(</span><span style="color: #CC9393;">'Frequency </span><span style="color: #BFEBBF;">[</span><span style="color: #CC9393;">Hz</span><span style="color: #BFEBBF;">]</span><span style="color: #CC9393;">'</span><span style="color: #DCDCCC;">)</span><span style="color: #CC9393;">; ylabel</span><span style="color: #DCDCCC;">(</span><span style="color: #CC9393;">'Amplitude </span><span style="color: #BFEBBF;">[</span><span style="color: #CC9393;">m/N</span><span style="color: #BFEBBF;">]</span><span style="color: #CC9393;">'</span><span style="color: #DCDCCC;">)</span>;
</pre>
</div>
<div id="org2a58760" class="figure">
<p><img src="figs/spindle_f_to_d.png" alt="spindle_f_to_d.png" />
</p>
<p><span class="figure-number">Figure 9: </span>Bode plot of the transfer function from \(f\) to \(d\)</p>
</div>
</div>
</div>
</div>
<div id="outline-container-org90551c6" class="outline-2">
<h2 id="org90551c6"><span class="section-number-2">5</span> Power Spectral Density</h2>
<div class="outline-text-2" id="text-5">
</div>
<div id="outline-container-org659372f" class="outline-3">
<h3 id="org659372f"><span class="section-number-3">5.1</span> Compute the PSD</h3>
<div class="outline-text-3" id="text-5-1">
<div class="org-src-container">
<pre class="src src-matlab">n_av = <span style="color: #BFEBBF;">4</span>; <span style="color: #7F9F7F;">% Number of average</span>
<span style="color: #DCDCCC;">[</span>pxx_1rpm, f_1rpm<span style="color: #DCDCCC;">]</span> = pwelch<span style="color: #DCDCCC;">(</span>spindle.rpm1.xasync, hanning<span style="color: #BFEBBF;">(</span>ceil<span style="color: #D0BF8F;">(</span>length<span style="color: #93E0E3;">(</span>spindle.rpm1.xasync<span style="color: #93E0E3;">)</span><span style="color: #7CB8BB;">/</span>n_av<span style="color: #D0BF8F;">)</span><span style="color: #BFEBBF;">)</span>, <span style="color: #BFEBBF;">[]</span>, <span style="color: #BFEBBF;">[]</span>, spindle.rpm1.Fs<span style="color: #DCDCCC;">)</span>;
<span style="color: #DCDCCC;">[</span>pyy_1rpm, <span style="color: #7CB8BB;">~</span><span style="color: #DCDCCC;">]</span> = pwelch<span style="color: #DCDCCC;">(</span>spindle.rpm1.yasync, hanning<span style="color: #BFEBBF;">(</span>ceil<span style="color: #D0BF8F;">(</span>length<span style="color: #93E0E3;">(</span>spindle.rpm1.yasync<span style="color: #93E0E3;">)</span><span style="color: #7CB8BB;">/</span>n_av<span style="color: #D0BF8F;">)</span><span style="color: #BFEBBF;">)</span>, <span style="color: #BFEBBF;">[]</span>, <span style="color: #BFEBBF;">[]</span>, spindle.rpm1.Fs<span style="color: #DCDCCC;">)</span>;
<span style="color: #DCDCCC;">[</span>pzz_1rpm, <span style="color: #7CB8BB;">~</span><span style="color: #DCDCCC;">]</span> = pwelch<span style="color: #DCDCCC;">(</span>spindle.rpm1.zasync, hanning<span style="color: #BFEBBF;">(</span>ceil<span style="color: #D0BF8F;">(</span>length<span style="color: #93E0E3;">(</span>spindle.rpm1.zasync<span style="color: #93E0E3;">)</span><span style="color: #7CB8BB;">/</span>n_av<span style="color: #D0BF8F;">)</span><span style="color: #BFEBBF;">)</span>, <span style="color: #BFEBBF;">[]</span>, <span style="color: #BFEBBF;">[]</span>, spindle.rpm1.Fs<span style="color: #DCDCCC;">)</span>;
<span style="color: #DCDCCC;">[</span>pxx_60rpm, f_60rpm<span style="color: #DCDCCC;">]</span> = pwelch<span style="color: #DCDCCC;">(</span>spindle.rpm60.xasync, hanning<span style="color: #BFEBBF;">(</span>ceil<span style="color: #D0BF8F;">(</span>length<span style="color: #93E0E3;">(</span>spindle.rpm60.xasync<span style="color: #93E0E3;">)</span><span style="color: #7CB8BB;">/</span>n_av<span style="color: #D0BF8F;">)</span><span style="color: #BFEBBF;">)</span>, <span style="color: #BFEBBF;">[]</span>, <span style="color: #BFEBBF;">[]</span>, spindle.rpm60.Fs<span style="color: #DCDCCC;">)</span>;
<span style="color: #DCDCCC;">[</span>pyy_60rpm, <span style="color: #7CB8BB;">~</span><span style="color: #DCDCCC;">]</span> = pwelch<span style="color: #DCDCCC;">(</span>spindle.rpm60.yasync, hanning<span style="color: #BFEBBF;">(</span>ceil<span style="color: #D0BF8F;">(</span>length<span style="color: #93E0E3;">(</span>spindle.rpm60.yasync<span style="color: #93E0E3;">)</span><span style="color: #7CB8BB;">/</span>n_av<span style="color: #D0BF8F;">)</span><span style="color: #BFEBBF;">)</span>, <span style="color: #BFEBBF;">[]</span>, <span style="color: #BFEBBF;">[]</span>, spindle.rpm60.Fs<span style="color: #DCDCCC;">)</span>;
<span style="color: #DCDCCC;">[</span>pzz_60rpm, <span style="color: #7CB8BB;">~</span><span style="color: #DCDCCC;">]</span> = pwelch<span style="color: #DCDCCC;">(</span>spindle.rpm60.zasync, hanning<span style="color: #BFEBBF;">(</span>ceil<span style="color: #D0BF8F;">(</span>length<span style="color: #93E0E3;">(</span>spindle.rpm60.zasync<span style="color: #93E0E3;">)</span><span style="color: #7CB8BB;">/</span>n_av<span style="color: #D0BF8F;">)</span><span style="color: #BFEBBF;">)</span>, <span style="color: #BFEBBF;">[]</span>, <span style="color: #BFEBBF;">[]</span>, spindle.rpm60.Fs<span style="color: #DCDCCC;">)</span>;
</pre>
</div>
</div>
</div>
<div id="outline-container-org3690c18" class="outline-3">
<h3 id="org3690c18"><span class="section-number-3">5.2</span> Plot the computed PSD</h3>
<div class="outline-text-3" id="text-5-2">
<p>
The Amplitude Spectral Densities of the displacement of the spindle for the \(x\), \(y\) and \(z\) directions are shown figure <a href="#org94523dd">11</a>. They correspond to the Asynchronous part shown figure <a href="#orgdc342a6">5</a>.
</p>
<div class="org-src-container">
<pre class="src src-matlab"><span style="color: #7CB8BB;">figure</span>;
hold on;
plot<span style="color: #DCDCCC;">(</span>f_1rpm, <span style="color: #BFEBBF;">(</span>pxx_1rpm<span style="color: #BFEBBF;">)</span><span style="color: #7CB8BB;">.^</span>.<span style="color: #BFEBBF;">5</span>, 'DisplayName', '$P_<span style="color: #BFEBBF;">{</span>xx<span style="color: #BFEBBF;">}</span>$ <span style="color: #7CB8BB;">-</span> <span style="color: #BFEBBF;">1rpm</span>'<span style="color: #DCDCCC;">)</span>;
plot<span style="color: #DCDCCC;">(</span>f_1rpm, <span style="color: #BFEBBF;">(</span>pyy_1rpm<span style="color: #BFEBBF;">)</span><span style="color: #7CB8BB;">.^</span>.<span style="color: #BFEBBF;">5</span>, 'DisplayName', '$P_<span style="color: #BFEBBF;">{</span>yy<span style="color: #BFEBBF;">}</span>$ <span style="color: #7CB8BB;">-</span> <span style="color: #BFEBBF;">1rpm</span>'<span style="color: #DCDCCC;">)</span>;
plot<span style="color: #DCDCCC;">(</span>f_1rpm, <span style="color: #BFEBBF;">(</span>pzz_1rpm<span style="color: #BFEBBF;">)</span><span style="color: #7CB8BB;">.^</span>.<span style="color: #BFEBBF;">5</span>, 'DisplayName', '$P_<span style="color: #BFEBBF;">{</span>zz<span style="color: #BFEBBF;">}</span>$ <span style="color: #7CB8BB;">-</span> <span style="color: #BFEBBF;">1rpm</span>'<span style="color: #DCDCCC;">)</span>;
% plot(f_1rpm, spindle.rpm1.adcn<span style="color: #7CB8BB;">*</span>ones(size(f_1rpm)), '<span style="color: #7CB8BB;">--</span>k', 'DisplayName', 'ADC <span style="color: #7CB8BB;">-</span> <span style="color: #BFEBBF;">1rpm</span>');
hold off;
<span style="color: #7CB8BB;">set</span><span style="color: #DCDCCC;">(</span><span style="color: #DFAF8F;">gca</span>, <span style="color: #CC9393;">'XScale', 'log'</span><span style="color: #DCDCCC;">)</span><span style="color: #CC9393;">; set</span><span style="color: #DCDCCC;">(</span><span style="color: #CC9393;">gca, 'YScale', 'log'</span><span style="color: #DCDCCC;">)</span>;
xlabel<span style="color: #DCDCCC;">(</span><span style="color: #CC9393;">'Frequency </span><span style="color: #BFEBBF;">[</span><span style="color: #CC9393;">Hz</span><span style="color: #BFEBBF;">]</span><span style="color: #CC9393;">'</span><span style="color: #DCDCCC;">)</span><span style="color: #CC9393;">; ylabel</span><span style="color: #DCDCCC;">(</span><span style="color: #CC9393;">'</span>ASD <span style="color: #BFEBBF;">[</span>$m<span style="color: #7CB8BB;">/\</span>sqrt<span style="color: #D0BF8F;">{</span>Hz<span style="color: #D0BF8F;">}</span>$<span style="color: #BFEBBF;">]</span>'<span style="color: #DCDCCC;">)</span>;
legend<span style="color: #DCDCCC;">(</span><span style="color: #CC9393;">'Location', 'northeast'</span><span style="color: #DCDCCC;">)</span>;
xlim<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">[</span>f_1rpm<span style="color: #D0BF8F;">(</span><span style="color: #BFEBBF;">2</span><span style="color: #D0BF8F;">)</span>, f_1rpm<span style="color: #D0BF8F;">(</span>end<span style="color: #D0BF8F;">)</span><span style="color: #BFEBBF;">]</span><span style="color: #DCDCCC;">)</span>;
</pre>
</div>
<div id="orgd766415" class="figure">
<p><img src="figs/spindle_psd_xyz_1rpm.png" alt="spindle_psd_xyz_1rpm.png" />
</p>
<p><span class="figure-number">Figure 10: </span>Power spectral density of the Asynchronous displacement - 1rpm</p>
</div>
<div class="org-src-container">
<pre class="src src-matlab"><span style="color: #7CB8BB;">figure</span>;
hold on;
plot<span style="color: #DCDCCC;">(</span>f_60rpm, <span style="color: #BFEBBF;">(</span>pxx_60rpm<span style="color: #BFEBBF;">)</span><span style="color: #7CB8BB;">.^</span>.<span style="color: #BFEBBF;">5</span>, 'DisplayName', '$P_<span style="color: #BFEBBF;">{</span>xx<span style="color: #BFEBBF;">}</span>$ <span style="color: #7CB8BB;">-</span> <span style="color: #BFEBBF;">60rpm</span>'<span style="color: #DCDCCC;">)</span>;
plot<span style="color: #DCDCCC;">(</span>f_60rpm, <span style="color: #BFEBBF;">(</span>pyy_60rpm<span style="color: #BFEBBF;">)</span><span style="color: #7CB8BB;">.^</span>.<span style="color: #BFEBBF;">5</span>, 'DisplayName', '$P_<span style="color: #BFEBBF;">{</span>yy<span style="color: #BFEBBF;">}</span>$ <span style="color: #7CB8BB;">-</span> <span style="color: #BFEBBF;">60rpm</span>'<span style="color: #DCDCCC;">)</span>;
plot<span style="color: #DCDCCC;">(</span>f_60rpm, <span style="color: #BFEBBF;">(</span>pzz_60rpm<span style="color: #BFEBBF;">)</span><span style="color: #7CB8BB;">.^</span>.<span style="color: #BFEBBF;">5</span>, 'DisplayName', '$P_<span style="color: #BFEBBF;">{</span>zz<span style="color: #BFEBBF;">}</span>$ <span style="color: #7CB8BB;">-</span> <span style="color: #BFEBBF;">60rpm</span>'<span style="color: #DCDCCC;">)</span>;
% plot(f_60rpm, spindle.rpm60.adcn<span style="color: #7CB8BB;">*</span>ones(size(f_60rpm)), '<span style="color: #7CB8BB;">--</span>k', 'DisplayName', 'ADC <span style="color: #7CB8BB;">-</span> <span style="color: #BFEBBF;">60rpm</span>');
hold off;
<span style="color: #7CB8BB;">set</span><span style="color: #DCDCCC;">(</span><span style="color: #DFAF8F;">gca</span>, <span style="color: #CC9393;">'XScale', 'log'</span><span style="color: #DCDCCC;">)</span><span style="color: #CC9393;">; set</span><span style="color: #DCDCCC;">(</span><span style="color: #CC9393;">gca, 'YScale', 'log'</span><span style="color: #DCDCCC;">)</span>;
xlabel<span style="color: #DCDCCC;">(</span><span style="color: #CC9393;">'Frequency </span><span style="color: #BFEBBF;">[</span><span style="color: #CC9393;">Hz</span><span style="color: #BFEBBF;">]</span><span style="color: #CC9393;">'</span><span style="color: #DCDCCC;">)</span><span style="color: #CC9393;">; ylabel</span><span style="color: #DCDCCC;">(</span><span style="color: #CC9393;">'</span>ASD <span style="color: #BFEBBF;">[</span>$m<span style="color: #7CB8BB;">/\</span>sqrt<span style="color: #D0BF8F;">{</span>Hz<span style="color: #D0BF8F;">}</span>$<span style="color: #BFEBBF;">]</span>'<span style="color: #DCDCCC;">)</span>;
legend<span style="color: #DCDCCC;">(</span><span style="color: #CC9393;">'Location', 'northeast'</span><span style="color: #DCDCCC;">)</span>;
xlim<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">[</span>f_60rpm<span style="color: #D0BF8F;">(</span><span style="color: #BFEBBF;">2</span><span style="color: #D0BF8F;">)</span>, f_60rpm<span style="color: #D0BF8F;">(</span>end<span style="color: #D0BF8F;">)</span><span style="color: #BFEBBF;">]</span><span style="color: #DCDCCC;">)</span>;
</pre>
</div>
<div id="org94523dd" class="figure">
<p><img src="figs/spindle_psd_xyz_60rpm.png" alt="spindle_psd_xyz_60rpm.png" />
</p>
<p><span class="figure-number">Figure 11: </span>Power spectral density of the Asynchronous displacement - 60rpm</p>
</div>
</div>
</div>
<div id="outline-container-org0a6959c" class="outline-3">
<h3 id="org0a6959c"><span class="section-number-3">5.3</span> Load the model of the spindle</h3>
<div class="outline-text-3" id="text-5-3">
<div class="org-src-container">
<pre class="src src-matlab">load<span style="color: #DCDCCC;">(</span><span style="color: #CC9393;">'./mat/spindle_model.mat', 'sys'</span><span style="color: #DCDCCC;">)</span>;
Tfd = abs<span style="color: #DCDCCC;">(</span>squeeze<span style="color: #BFEBBF;">(</span>freqresp<span style="color: #D0BF8F;">(</span>sys<span style="color: #93E0E3;">(</span><span style="color: #CC9393;">'d', 'f'</span><span style="color: #93E0E3;">)</span><span style="color: #CC9393;">, f_60rpm, 'Hz'</span><span style="color: #D0BF8F;">)</span><span style="color: #BFEBBF;">)</span><span style="color: #DCDCCC;">)</span>;
</pre>
</div>
</div>
</div>
<div id="outline-container-orgb977795" class="outline-3">
<h3 id="orgb977795"><span class="section-number-3">5.4</span> Plot the PSD of the Force using the model</h3>
<div class="outline-text-3" id="text-5-4">
<div class="org-src-container">
<pre class="src src-matlab"><span style="color: #7CB8BB;">figure</span>;
plot<span style="color: #DCDCCC;">(</span>f_60rpm, <span style="color: #BFEBBF;">(</span>pxx_60rpm<span style="color: #7CB8BB;">.^</span>.<span style="color: #BFEBBF;">5</span><span style="color: #BFEBBF;">)</span><span style="color: #7CB8BB;">./</span>Tfd, <span style="color: #CC9393;">'DisplayName', '$P_</span><span style="color: #BFEBBF;">{</span><span style="color: #CC9393;">xx</span><span style="color: #BFEBBF;">}</span><span style="color: #CC9393;">$'</span><span style="color: #DCDCCC;">)</span>;
<span style="color: #7CB8BB;">set</span><span style="color: #DCDCCC;">(</span><span style="color: #DFAF8F;">gca</span>, <span style="color: #CC9393;">'XScale', 'log'</span><span style="color: #DCDCCC;">)</span><span style="color: #CC9393;">; set</span><span style="color: #DCDCCC;">(</span><span style="color: #CC9393;">gca, 'YScale', 'log'</span><span style="color: #DCDCCC;">)</span>;
xlabel<span style="color: #DCDCCC;">(</span><span style="color: #CC9393;">'Frequency </span><span style="color: #BFEBBF;">[</span><span style="color: #CC9393;">Hz</span><span style="color: #BFEBBF;">]</span><span style="color: #CC9393;">'</span><span style="color: #DCDCCC;">)</span><span style="color: #CC9393;">; ylabel</span><span style="color: #DCDCCC;">(</span><span style="color: #CC9393;">'</span>ASD <span style="color: #BFEBBF;">[</span>$N<span style="color: #7CB8BB;">/\</span>sqrt<span style="color: #D0BF8F;">{</span>Hz<span style="color: #D0BF8F;">}</span>$<span style="color: #BFEBBF;">]</span>'<span style="color: #DCDCCC;">)</span>;
xlim<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">[</span>f_60rpm<span style="color: #D0BF8F;">(</span><span style="color: #BFEBBF;">2</span><span style="color: #D0BF8F;">)</span>, f_60rpm<span style="color: #D0BF8F;">(</span>end<span style="color: #D0BF8F;">)</span><span style="color: #BFEBBF;">]</span><span style="color: #DCDCCC;">)</span>;
</pre>
</div>
<div id="orgfe4f821" class="figure">
<p><img src="figs/spindle_psd_f_60rpm.png" alt="spindle_psd_f_60rpm.png" />
</p>
<p><span class="figure-number">Figure 12: </span>Power spectral density of the force - 60rpm</p>
</div>
</div>
</div>
<div id="outline-container-orgbd81578" class="outline-3">
<h3 id="orgbd81578"><span class="section-number-3">5.5</span> Estimated Shape of the PSD of the force</h3>
<div class="outline-text-3" id="text-5-5">
<div class="org-src-container">
<pre class="src src-matlab">s = tf<span style="color: #DCDCCC;">(</span><span style="color: #CC9393;">'s'</span><span style="color: #DCDCCC;">)</span>;
Wd_simple = <span style="color: #BFEBBF;">1e</span><span style="color: #7CB8BB;">-</span><span style="color: #BFEBBF;">8</span><span style="color: #7CB8BB;">/</span><span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">1</span><span style="color: #7CB8BB;">+</span>s<span style="color: #7CB8BB;">/</span><span style="color: #BFEBBF;">2</span><span style="color: #7CB8BB;">/</span><span style="color: #BFEBBF;">pi</span><span style="color: #7CB8BB;">/</span><span style="color: #BFEBBF;">0</span>.<span style="color: #BFEBBF;">5</span><span style="color: #DCDCCC;">)</span><span style="color: #7CB8BB;">/</span><span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">1</span><span style="color: #7CB8BB;">+</span>s<span style="color: #7CB8BB;">/</span><span style="color: #BFEBBF;">2</span><span style="color: #7CB8BB;">/</span><span style="color: #BFEBBF;">pi</span><span style="color: #7CB8BB;">/</span><span style="color: #BFEBBF;">100</span><span style="color: #DCDCCC;">)</span>;
Wf_simple = Wd_simple<span style="color: #7CB8BB;">/</span>tf<span style="color: #DCDCCC;">(</span>sys<span style="color: #BFEBBF;">(</span><span style="color: #CC9393;">'d', 'f'</span><span style="color: #BFEBBF;">)</span><span style="color: #DCDCCC;">)</span>;
TWf_simple = abs<span style="color: #DCDCCC;">(</span>squeeze<span style="color: #BFEBBF;">(</span>freqresp<span style="color: #D0BF8F;">(</span>Wf_simple, f_60rpm, <span style="color: #CC9393;">'Hz'</span><span style="color: #D0BF8F;">)</span><span style="color: #BFEBBF;">)</span><span style="color: #DCDCCC;">)</span>;
% Wf = <span style="color: #BFEBBF;">0</span>.<span style="color: #BFEBBF;">48902</span><span style="color: #7CB8BB;">*</span>(s<span style="color: #7CB8BB;">+</span><span style="color: #BFEBBF;">327</span>.<span style="color: #BFEBBF;">9</span>)<span style="color: #7CB8BB;">*</span>(s<span style="color: #7CB8BB;">^</span><span style="color: #BFEBBF;">2</span> <span style="color: #7CB8BB;">+</span> <span style="color: #BFEBBF;">109</span>.<span style="color: #BFEBBF;">6</span><span style="color: #7CB8BB;">*</span>s <span style="color: #7CB8BB;">+</span> <span style="color: #BFEBBF;">1</span>.<span style="color: #BFEBBF;">687e04</span>)<span style="color: #7CB8BB;">/</span>((s<span style="color: #7CB8BB;">^</span><span style="color: #BFEBBF;">2</span> <span style="color: #7CB8BB;">+</span> <span style="color: #BFEBBF;">30</span>.<span style="color: #BFEBBF;">59</span><span style="color: #7CB8BB;">*</span>s <span style="color: #7CB8BB;">+</span> <span style="color: #BFEBBF;">8541</span>)<span style="color: #7CB8BB;">*</span>(s<span style="color: #7CB8BB;">^</span><span style="color: #BFEBBF;">2</span> <span style="color: #7CB8BB;">+</span> <span style="color: #BFEBBF;">29</span>.<span style="color: #BFEBBF;">11</span><span style="color: #7CB8BB;">*</span>s <span style="color: #7CB8BB;">+</span> <span style="color: #BFEBBF;">3</span>.<span style="color: #BFEBBF;">268e04</span>));
% Wf = <span style="color: #BFEBBF;">0</span>.<span style="color: #BFEBBF;">15788</span><span style="color: #7CB8BB;">*</span>(s<span style="color: #7CB8BB;">+</span><span style="color: #BFEBBF;">418</span>.<span style="color: #BFEBBF;">6</span>)<span style="color: #7CB8BB;">*</span>(s<span style="color: #7CB8BB;">+</span><span style="color: #BFEBBF;">1697</span>)<span style="color: #7CB8BB;">^</span><span style="color: #BFEBBF;">2</span><span style="color: #7CB8BB;">*</span>(s<span style="color: #7CB8BB;">^</span><span style="color: #BFEBBF;">2</span> <span style="color: #7CB8BB;">+</span> <span style="color: #BFEBBF;">124</span>.<span style="color: #BFEBBF;">3</span><span style="color: #7CB8BB;">*</span>s <span style="color: #7CB8BB;">+</span> <span style="color: #BFEBBF;">2</span>.<span style="color: #BFEBBF;">529e04</span>)<span style="color: #7CB8BB;">*</span>(s<span style="color: #7CB8BB;">^</span><span style="color: #BFEBBF;">2</span> <span style="color: #7CB8BB;">+</span> <span style="color: #BFEBBF;">681</span>.<span style="color: #BFEBBF;">3</span><span style="color: #7CB8BB;">*</span>s <span style="color: #7CB8BB;">+</span> <span style="color: #BFEBBF;">9</span>.<span style="color: #BFEBBF;">018e05</span>)<span style="color: #7CB8BB;">/</span>((s<span style="color: #7CB8BB;">^</span><span style="color: #BFEBBF;">2</span> <span style="color: #7CB8BB;">+</span> <span style="color: #BFEBBF;">23</span>.<span style="color: #BFEBBF;">03</span><span style="color: #7CB8BB;">*</span>s <span style="color: #7CB8BB;">+</span> <span style="color: #BFEBBF;">8916</span>)<span style="color: #7CB8BB;">*</span>(s<span style="color: #7CB8BB;">^</span><span style="color: #BFEBBF;">2</span> <span style="color: #7CB8BB;">+</span> <span style="color: #BFEBBF;">33</span>.<span style="color: #BFEBBF;">85</span><span style="color: #7CB8BB;">*</span>s <span style="color: #7CB8BB;">+</span> <span style="color: #BFEBBF;">6</span>.<span style="color: #BFEBBF;">559e04</span>)<span style="color: #7CB8BB;">*</span>(s<span style="color: #7CB8BB;">^</span><span style="color: #BFEBBF;">2</span> <span style="color: #7CB8BB;">+</span> <span style="color: #BFEBBF;">71</span>.<span style="color: #BFEBBF;">43</span><span style="color: #7CB8BB;">*</span>s <span style="color: #7CB8BB;">+</span> <span style="color: #BFEBBF;">4</span>.<span style="color: #BFEBBF;">283e05</span>)<span style="color: #7CB8BB;">*</span>(s<span style="color: #7CB8BB;">^</span><span style="color: #BFEBBF;">2</span> <span style="color: #7CB8BB;">+</span> <span style="color: #BFEBBF;">40</span>.<span style="color: #BFEBBF;">64</span><span style="color: #7CB8BB;">*</span>s <span style="color: #7CB8BB;">+</span> <span style="color: #BFEBBF;">1</span>.<span style="color: #BFEBBF;">789e06</span>));
Wf = <span style="color: #DCDCCC;">(</span>s<span style="color: #7CB8BB;">+</span><span style="color: #BFEBBF;">1697</span><span style="color: #DCDCCC;">)</span><span style="color: #7CB8BB;">^</span><span style="color: #BFEBBF;">2</span><span style="color: #7CB8BB;">*</span><span style="color: #DCDCCC;">(</span>s<span style="color: #7CB8BB;">^</span><span style="color: #BFEBBF;">2</span> <span style="color: #7CB8BB;">+</span> <span style="color: #BFEBBF;">114</span>.<span style="color: #BFEBBF;">5</span><span style="color: #7CB8BB;">*</span>s <span style="color: #7CB8BB;">+</span> <span style="color: #BFEBBF;">2</span>.<span style="color: #BFEBBF;">278e04</span><span style="color: #DCDCCC;">)</span><span style="color: #7CB8BB;">*</span><span style="color: #DCDCCC;">(</span>s<span style="color: #7CB8BB;">^</span><span style="color: #BFEBBF;">2</span> <span style="color: #7CB8BB;">+</span> <span style="color: #BFEBBF;">205</span>.<span style="color: #BFEBBF;">1</span><span style="color: #7CB8BB;">*</span>s <span style="color: #7CB8BB;">+</span> <span style="color: #BFEBBF;">1</span>.<span style="color: #BFEBBF;">627e05</span><span style="color: #DCDCCC;">)</span><span style="color: #7CB8BB;">*</span><span style="color: #DCDCCC;">(</span>s<span style="color: #7CB8BB;">^</span><span style="color: #BFEBBF;">2</span> <span style="color: #7CB8BB;">+</span> <span style="color: #BFEBBF;">285</span>.<span style="color: #BFEBBF;">8</span><span style="color: #7CB8BB;">*</span>s <span style="color: #7CB8BB;">+</span> <span style="color: #BFEBBF;">8</span>.<span style="color: #BFEBBF;">624e05</span><span style="color: #DCDCCC;">)</span><span style="color: #7CB8BB;">*</span><span style="color: #DCDCCC;">(</span>s<span style="color: #7CB8BB;">+</span><span style="color: #BFEBBF;">100</span><span style="color: #DCDCCC;">)</span><span style="color: #7CB8BB;">/</span><span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">(</span>s<span style="color: #7CB8BB;">+</span><span style="color: #BFEBBF;">0</span>.<span style="color: #BFEBBF;">5</span><span style="color: #BFEBBF;">)</span><span style="color: #7CB8BB;">*</span><span style="color: #BFEBBF;">3012</span><span style="color: #7CB8BB;">*</span><span style="color: #BFEBBF;">(</span>s<span style="color: #7CB8BB;">^</span><span style="color: #BFEBBF;">2</span> <span style="color: #7CB8BB;">+</span> <span style="color: #BFEBBF;">23</span>.<span style="color: #BFEBBF;">03</span><span style="color: #7CB8BB;">*</span>s <span style="color: #7CB8BB;">+</span> <span style="color: #BFEBBF;">8916</span><span style="color: #BFEBBF;">)</span><span style="color: #7CB8BB;">*</span><span style="color: #BFEBBF;">(</span>s<span style="color: #7CB8BB;">^</span><span style="color: #BFEBBF;">2</span> <span style="color: #7CB8BB;">+</span> <span style="color: #BFEBBF;">17</span>.<span style="color: #BFEBBF;">07</span><span style="color: #7CB8BB;">*</span>s <span style="color: #7CB8BB;">+</span> <span style="color: #BFEBBF;">4</span>.<span style="color: #BFEBBF;">798e04</span><span style="color: #BFEBBF;">)</span><span style="color: #7CB8BB;">*</span><span style="color: #BFEBBF;">(</span>s<span style="color: #7CB8BB;">^</span><span style="color: #BFEBBF;">2</span> <span style="color: #7CB8BB;">+</span> <span style="color: #BFEBBF;">41</span>.<span style="color: #BFEBBF;">17</span><span style="color: #7CB8BB;">*</span>s <span style="color: #7CB8BB;">+</span> <span style="color: #BFEBBF;">4</span>.<span style="color: #BFEBBF;">347e05</span><span style="color: #BFEBBF;">)</span><span style="color: #7CB8BB;">*</span><span style="color: #BFEBBF;">(</span>s<span style="color: #7CB8BB;">^</span><span style="color: #BFEBBF;">2</span> <span style="color: #7CB8BB;">+</span> <span style="color: #BFEBBF;">78</span>.<span style="color: #BFEBBF;">99</span><span style="color: #7CB8BB;">*</span>s <span style="color: #7CB8BB;">+</span> <span style="color: #BFEBBF;">1</span>.<span style="color: #BFEBBF;">789e06</span><span style="color: #BFEBBF;">)</span><span style="color: #DCDCCC;">)</span>;
TWf = abs<span style="color: #DCDCCC;">(</span>squeeze<span style="color: #BFEBBF;">(</span>freqresp<span style="color: #D0BF8F;">(</span>Wf, f_60rpm, <span style="color: #CC9393;">'Hz'</span><span style="color: #D0BF8F;">)</span><span style="color: #BFEBBF;">)</span><span style="color: #DCDCCC;">)</span>;
</pre>
</div>
</div>
</div>
<div id="outline-container-orgebea7be" class="outline-3">
<h3 id="orgebea7be"><span class="section-number-3">5.6</span> PSD in [N]</h3>
<div class="outline-text-3" id="text-5-6">
<p>
Above \(200Hz\), the Amplitude Spectral Density seems dominated by noise coming from the electronics (charge amplifier, ADC, …). So we don't know what is the frequency content of the force above that frequency. However, we assume that \(P_{xx}\) is decreasing with \(1/f\) as it seems so be the case below \(100Hz\) (figure <a href="#org94523dd">11</a>).
</p>
<p>
We then fit the PSD of the displacement with a transfer function (figure <a href="#org54db7f7">14</a>).
</p>
<div class="org-src-container">
<pre class="src src-matlab"><span style="color: #7CB8BB;">figure</span>;
hold on;
plot<span style="color: #DCDCCC;">(</span>f_60rpm, <span style="color: #BFEBBF;">(</span>pxx_60rpm<span style="color: #7CB8BB;">.^</span>.<span style="color: #BFEBBF;">5</span><span style="color: #BFEBBF;">)</span><span style="color: #7CB8BB;">./</span>Tfd, <span style="color: #CC9393;">'DisplayName', '</span>$<span style="color: #7CB8BB;">\</span>sqrt<span style="color: #BFEBBF;">{</span>P_<span style="color: #D0BF8F;">{</span>xx<span style="color: #D0BF8F;">}</span><span style="color: #BFEBBF;">}</span><span style="color: #7CB8BB;">/|</span>T_<span style="color: #BFEBBF;">{</span>d<span style="color: #7CB8BB;">/</span>f<span style="color: #BFEBBF;">}</span><span style="color: #7CB8BB;">|</span>$'<span style="color: #DCDCCC;">)</span>;
plot<span style="color: #DCDCCC;">(</span>f_60rpm, TWf, <span style="color: #CC9393;">'DisplayName', 'Wf'</span><span style="color: #DCDCCC;">)</span>;
plot<span style="color: #DCDCCC;">(</span>f_60rpm, TWf_simple, <span style="color: #CC9393;">'-k', 'DisplayName', 'Wfs'</span><span style="color: #DCDCCC;">)</span>;
<span style="color: #7CB8BB;">set</span><span style="color: #DCDCCC;">(</span><span style="color: #DFAF8F;">gca</span>, <span style="color: #CC9393;">'XScale', 'log'</span><span style="color: #DCDCCC;">)</span><span style="color: #CC9393;">; set</span><span style="color: #DCDCCC;">(</span><span style="color: #CC9393;">gca, 'YScale', 'log'</span><span style="color: #DCDCCC;">)</span>;
xlabel<span style="color: #DCDCCC;">(</span><span style="color: #CC9393;">'Frequency </span><span style="color: #BFEBBF;">[</span><span style="color: #CC9393;">Hz</span><span style="color: #BFEBBF;">]</span><span style="color: #CC9393;">'</span><span style="color: #DCDCCC;">)</span><span style="color: #CC9393;">; ylabel</span><span style="color: #DCDCCC;">(</span><span style="color: #CC9393;">'</span>ASD <span style="color: #BFEBBF;">[</span>$N<span style="color: #7CB8BB;">/\</span>sqrt<span style="color: #D0BF8F;">{</span>Hz<span style="color: #D0BF8F;">}</span>$<span style="color: #BFEBBF;">]</span>'<span style="color: #DCDCCC;">)</span>;
xlim<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">[</span>f_60rpm<span style="color: #D0BF8F;">(</span><span style="color: #BFEBBF;">2</span><span style="color: #D0BF8F;">)</span>, f_60rpm<span style="color: #D0BF8F;">(</span>end<span style="color: #D0BF8F;">)</span><span style="color: #BFEBBF;">]</span><span style="color: #DCDCCC;">)</span>;
legend<span style="color: #DCDCCC;">(</span><span style="color: #CC9393;">'Location', 'northeast'</span><span style="color: #DCDCCC;">)</span>;
</pre>
</div>
<div id="org6eac4af" class="figure">
<p><img src="figs/spindle_psd_f_comp_60rpm.png" alt="spindle_psd_f_comp_60rpm.png" />
</p>
<p><span class="figure-number">Figure 13: </span>Power spectral density of the force - 60rpm</p>
</div>
</div>
</div>
<div id="outline-container-org4818470" class="outline-3">
<h3 id="org4818470"><span class="section-number-3">5.7</span> PSD in [m]</h3>
<div class="outline-text-3" id="text-5-7">
<p>
To obtain the PSD of the force \(f\) that induce such displacement, we use the following formula:
\[ \sqrt{PSD(d)} = |T_{d/f}| \sqrt{PSD(f)} \]
</p>
<p>
And so we have:
\[ \sqrt{PSD(f)} = |T_{d/f}|^{-1} \sqrt{PSD(d)} \]
</p>
<p>
The obtain Power Spectral Density of the force is displayed figure <a href="#org6eac4af">13</a>.
</p>
<div class="org-src-container">
<pre class="src src-matlab"><span style="color: #7CB8BB;">figure</span>;
hold on;
plot<span style="color: #DCDCCC;">(</span>f_60rpm, pxx_60rpm<span style="color: #7CB8BB;">.^</span>.<span style="color: #BFEBBF;">5</span>, <span style="color: #CC9393;">'DisplayName', '</span>$<span style="color: #7CB8BB;">\</span>sqrt<span style="color: #BFEBBF;">{</span>P_<span style="color: #D0BF8F;">{</span>xx<span style="color: #D0BF8F;">}</span><span style="color: #BFEBBF;">}</span>$'<span style="color: #DCDCCC;">)</span>;
plot<span style="color: #DCDCCC;">(</span>f_60rpm, TWf<span style="color: #7CB8BB;">.*</span>Tfd, <span style="color: #CC9393;">'DisplayName', '$|W_f|*|T_</span><span style="color: #BFEBBF;">{</span><span style="color: #CC9393;">d/f</span><span style="color: #BFEBBF;">}</span><span style="color: #CC9393;">|$'</span><span style="color: #DCDCCC;">)</span>;
<span style="color: #7CB8BB;">set</span><span style="color: #DCDCCC;">(</span><span style="color: #DFAF8F;">gca</span>, <span style="color: #CC9393;">'XScale', 'log'</span><span style="color: #DCDCCC;">)</span><span style="color: #CC9393;">; set</span><span style="color: #DCDCCC;">(</span><span style="color: #CC9393;">gca, 'YScale', 'log'</span><span style="color: #DCDCCC;">)</span>;
xlabel<span style="color: #DCDCCC;">(</span><span style="color: #CC9393;">'Frequency </span><span style="color: #BFEBBF;">[</span><span style="color: #CC9393;">Hz</span><span style="color: #BFEBBF;">]</span><span style="color: #CC9393;">'</span><span style="color: #DCDCCC;">)</span><span style="color: #CC9393;">; ylabel</span><span style="color: #DCDCCC;">(</span><span style="color: #CC9393;">'</span>ASD <span style="color: #BFEBBF;">[</span>$m<span style="color: #7CB8BB;">/\</span>sqrt<span style="color: #D0BF8F;">{</span>Hz<span style="color: #D0BF8F;">}</span>$<span style="color: #BFEBBF;">]</span>'<span style="color: #DCDCCC;">)</span>;
xlim<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">[</span>f_60rpm<span style="color: #D0BF8F;">(</span><span style="color: #BFEBBF;">2</span><span style="color: #D0BF8F;">)</span>, f_60rpm<span style="color: #D0BF8F;">(</span>end<span style="color: #D0BF8F;">)</span><span style="color: #BFEBBF;">]</span><span style="color: #DCDCCC;">)</span>;
legend<span style="color: #DCDCCC;">(</span><span style="color: #CC9393;">'Location', 'northeast'</span><span style="color: #DCDCCC;">)</span>;
</pre>
</div>
<div id="org54db7f7" class="figure">
<p><img src="figs/spindle_psd_d_comp_60rpm.png" alt="spindle_psd_d_comp_60rpm.png" />
</p>
<p><span class="figure-number">Figure 14: </span>Comparison of the power spectral density of the measured displacement and of the model</p>
</div>
</div>
</div>
<div id="outline-container-org408aad1" class="outline-3">
<h3 id="org408aad1"><span class="section-number-3">5.8</span> Compute the resulting RMS value [m]</h3>
<div class="outline-text-3" id="text-5-8">
<div class="org-src-container">
<pre class="src src-matlab"><span style="color: #7CB8BB;">figure</span>;
hold on;
plot<span style="color: #DCDCCC;">(</span>f_60rpm, <span style="color: #BFEBBF;">1e9</span><span style="color: #7CB8BB;">*</span>cumtrapz<span style="color: #BFEBBF;">(</span>f_60rpm, <span style="color: #D0BF8F;">(</span>pxx_60rpm<span style="color: #D0BF8F;">)</span><span style="color: #BFEBBF;">)</span><span style="color: #7CB8BB;">.^</span>.<span style="color: #BFEBBF;">5</span>, <span style="color: #CC9393;">'--', 'DisplayName', 'Exp. Data'</span><span style="color: #DCDCCC;">)</span>;
plot<span style="color: #DCDCCC;">(</span>f_60rpm, <span style="color: #BFEBBF;">1e9</span><span style="color: #7CB8BB;">*</span>cumtrapz<span style="color: #BFEBBF;">(</span>f_60rpm, <span style="color: #D0BF8F;">(</span><span style="color: #93E0E3;">(</span>TWf<span style="color: #7CB8BB;">.*</span>Tfd<span style="color: #93E0E3;">)</span><span style="color: #7CB8BB;">.^</span><span style="color: #BFEBBF;">2</span><span style="color: #D0BF8F;">)</span><span style="color: #BFEBBF;">)</span><span style="color: #7CB8BB;">.^</span>.<span style="color: #BFEBBF;">5</span>, <span style="color: #CC9393;">'--', 'DisplayName', 'Estimated'</span><span style="color: #DCDCCC;">)</span>;
hold off;
<span style="color: #7CB8BB;">set</span><span style="color: #DCDCCC;">(</span><span style="color: #DFAF8F;">gca</span>, <span style="color: #CC9393;">'XScale', 'log'</span><span style="color: #DCDCCC;">)</span>;
xlabel<span style="color: #DCDCCC;">(</span><span style="color: #CC9393;">'Frequency </span><span style="color: #BFEBBF;">[</span><span style="color: #CC9393;">Hz</span><span style="color: #BFEBBF;">]</span><span style="color: #CC9393;">'</span><span style="color: #DCDCCC;">)</span><span style="color: #CC9393;">; ylabel</span><span style="color: #DCDCCC;">(</span><span style="color: #CC9393;">'CPS </span><span style="color: #BFEBBF;">[</span><span style="color: #CC9393;">$nm$ rms</span><span style="color: #BFEBBF;">]</span><span style="color: #CC9393;">'</span><span style="color: #DCDCCC;">)</span>;
xlim<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">[</span>f_60rpm<span style="color: #D0BF8F;">(</span><span style="color: #BFEBBF;">2</span><span style="color: #D0BF8F;">)</span>, f_60rpm<span style="color: #D0BF8F;">(</span>end<span style="color: #D0BF8F;">)</span><span style="color: #BFEBBF;">]</span><span style="color: #DCDCCC;">)</span>;
legend<span style="color: #DCDCCC;">(</span><span style="color: #CC9393;">'Location', 'southeast'</span><span style="color: #DCDCCC;">)</span>;
</pre>
</div>
<div id="orgf1eaaa3" class="figure">
<p><img src="figs/spindle_cps_d_comp_60rpm.png" alt="spindle_cps_d_comp_60rpm.png" />
</p>
<p><span class="figure-number">Figure 15: </span>Cumulative Power Spectrum - 60rpm</p>
</div>
</div>
</div>
<div id="outline-container-org97cda8b" class="outline-3">
<h3 id="org97cda8b"><span class="section-number-3">5.9</span> Compute the resulting RMS value [m]</h3>
<div class="outline-text-3" id="text-5-9">
<div class="org-src-container">
<pre class="src src-matlab"><span style="color: #7CB8BB;">figure</span>;
hold on;
plot<span style="color: #DCDCCC;">(</span>f_1rpm, <span style="color: #BFEBBF;">1e9</span><span style="color: #7CB8BB;">*</span>cumtrapz<span style="color: #BFEBBF;">(</span>f_1rpm, <span style="color: #D0BF8F;">(</span>pxx_1rpm<span style="color: #D0BF8F;">)</span><span style="color: #BFEBBF;">)</span>, <span style="color: #CC9393;">'--', 'DisplayName', 'Exp. Data'</span><span style="color: #DCDCCC;">)</span>;
plot<span style="color: #DCDCCC;">(</span>f_1rpm, <span style="color: #BFEBBF;">1e9</span><span style="color: #7CB8BB;">*</span><span style="color: #BFEBBF;">(</span>f_1rpm<span style="color: #D0BF8F;">(</span>end<span style="color: #D0BF8F;">)</span><span style="color: #7CB8BB;">-</span>f_1rpm<span style="color: #D0BF8F;">(</span><span style="color: #BFEBBF;">1</span><span style="color: #D0BF8F;">)</span><span style="color: #BFEBBF;">)</span><span style="color: #7CB8BB;">/</span><span style="color: #BFEBBF;">(</span>length<span style="color: #D0BF8F;">(</span>f_1rpm<span style="color: #D0BF8F;">)</span><span style="color: #7CB8BB;">-</span><span style="color: #BFEBBF;">1</span><span style="color: #BFEBBF;">)</span><span style="color: #7CB8BB;">*</span>cumsum<span style="color: #BFEBBF;">(</span>pxx_1rpm<span style="color: #BFEBBF;">)</span>, <span style="color: #CC9393;">'--', 'DisplayName', 'Exp. Data'</span><span style="color: #DCDCCC;">)</span>;
hold off;
<span style="color: #7CB8BB;">set</span><span style="color: #DCDCCC;">(</span><span style="color: #DFAF8F;">gca</span>, <span style="color: #CC9393;">'XScale', 'log'</span><span style="color: #DCDCCC;">)</span>;
xlabel<span style="color: #DCDCCC;">(</span><span style="color: #CC9393;">'Frequency </span><span style="color: #BFEBBF;">[</span><span style="color: #CC9393;">Hz</span><span style="color: #BFEBBF;">]</span><span style="color: #CC9393;">'</span><span style="color: #DCDCCC;">)</span><span style="color: #CC9393;">; ylabel</span><span style="color: #DCDCCC;">(</span><span style="color: #CC9393;">'CPS </span><span style="color: #BFEBBF;">[</span><span style="color: #CC9393;">$nm$ rms</span><span style="color: #BFEBBF;">]</span><span style="color: #CC9393;">'</span><span style="color: #DCDCCC;">)</span>;
xlim<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">[</span>f_1rpm<span style="color: #D0BF8F;">(</span><span style="color: #BFEBBF;">2</span><span style="color: #D0BF8F;">)</span>, f_1rpm<span style="color: #D0BF8F;">(</span>end<span style="color: #D0BF8F;">)</span><span style="color: #BFEBBF;">]</span><span style="color: #DCDCCC;">)</span>;
legend<span style="color: #DCDCCC;">(</span><span style="color: #CC9393;">'Location', 'southeast'</span><span style="color: #DCDCCC;">)</span>;
</pre>
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<div id="org4fdbcb3" class="figure">
<p><img src="figs/spindle_cps_d_comp_1rpm.png" alt="spindle_cps_d_comp_1rpm.png" />
</p>
<p><span class="figure-number">Figure 16: </span>Cumulative Power Spectrum - 1rpm</p>
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<div id="outline-container-orgb0ca16b" class="outline-2">
<h2 id="orgb0ca16b"><span class="section-number-2">6</span> Functions</h2>
<div class="outline-text-2" id="text-6">
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<h3 id="org14006e0"><span class="section-number-3">6.1</span> getAsynchronousError</h3>
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<pre class="src src-matlab"><span style="color: #F0DFAF; font-weight: bold;">function</span> <span style="color: #DCDCCC;">[</span><span style="color: #DFAF8F;">Wxdec</span><span style="color: #DCDCCC;">]</span> = <span style="color: #93E0E3;">getAsynchronousError</span><span style="color: #DCDCCC;">(</span><span style="color: #DFAF8F;">data</span>, <span style="color: #DFAF8F;">NbTurn</span><span style="color: #DCDCCC;">)</span>
<span style="color: #7F9F7F; font-weight: bold; text-decoration: overline;">%%</span>
L = length<span style="color: #DCDCCC;">(</span>data<span style="color: #DCDCCC;">)</span>;
res_per_rev = L<span style="color: #7CB8BB;">/</span>NbTurn;
P = <span style="color: #BFEBBF;">0</span><span style="color: #7CB8BB;">:</span><span style="color: #DCDCCC;">(</span>res_per_rev<span style="color: #7CB8BB;">*</span>NbTurn<span style="color: #7CB8BB;">-</span><span style="color: #BFEBBF;">1</span><span style="color: #DCDCCC;">)</span>;
Pos = P' <span style="color: #7CB8BB;">*</span> <span style="color: #BFEBBF;">360</span><span style="color: #7CB8BB;">/</span>res_per_rev;
<span style="color: #7F9F7F;">% Temperature correction</span>
x1 = myfit2<span style="color: #DCDCCC;">(</span>Pos, data<span style="color: #DCDCCC;">)</span>;
<span style="color: #7F9F7F;">% Convert data to frequency domain and scale accordingly</span>
X2 = <span style="color: #BFEBBF;">2</span><span style="color: #7CB8BB;">/</span><span style="color: #DCDCCC;">(</span>res_per_rev<span style="color: #7CB8BB;">*</span>NbTurn<span style="color: #DCDCCC;">)</span><span style="color: #7CB8BB;">*</span>fft<span style="color: #DCDCCC;">(</span>x1<span style="color: #DCDCCC;">)</span>;
f2 = <span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">0</span><span style="color: #7CB8BB;">:</span>L<span style="color: #7CB8BB;">-</span><span style="color: #BFEBBF;">1</span><span style="color: #DCDCCC;">)</span><span style="color: #7CB8BB;">./</span>NbTurn; <span style="color: #7F9F7F;">%upr -> once per revolution</span>
<span style="color: #7F9F7F; font-weight: bold; text-decoration: overline;">%%</span>
X2dec = zeros<span style="color: #DCDCCC;">(</span>size<span style="color: #BFEBBF;">(</span>X2<span style="color: #BFEBBF;">)</span><span style="color: #DCDCCC;">)</span>;
<span style="color: #7F9F7F;">% Get only the non integer data</span>
X2dec<span style="color: #DCDCCC;">(</span>mod<span style="color: #BFEBBF;">(</span>f2<span style="color: #D0BF8F;">(</span><span style="color: #7CB8BB;">:</span><span style="color: #D0BF8F;">)</span>, <span style="color: #BFEBBF;">1</span><span style="color: #BFEBBF;">)</span> <span style="color: #7CB8BB;">~=</span> <span style="color: #BFEBBF;">0</span><span style="color: #DCDCCC;">)</span> = X2<span style="color: #DCDCCC;">(</span>mod<span style="color: #BFEBBF;">(</span>f2<span style="color: #D0BF8F;">(</span><span style="color: #7CB8BB;">:</span><span style="color: #D0BF8F;">)</span>, <span style="color: #BFEBBF;">1</span><span style="color: #BFEBBF;">)</span> <span style="color: #7CB8BB;">~=</span> <span style="color: #BFEBBF;">0</span><span style="color: #DCDCCC;">)</span>;
Wxdec = real<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">(</span>res_per_rev<span style="color: #7CB8BB;">*</span>NbTurn<span style="color: #BFEBBF;">)</span><span style="color: #7CB8BB;">/</span><span style="color: #BFEBBF;">2</span> <span style="color: #7CB8BB;">*</span> ifft<span style="color: #BFEBBF;">(</span>X2dec<span style="color: #BFEBBF;">)</span><span style="color: #DCDCCC;">)</span>;
<span style="color: #7F9F7F; font-weight: bold; text-decoration: overline;">%%</span>
<span style="color: #F0DFAF; font-weight: bold;">function</span> <span style="color: #DFAF8F;">Y</span> = <span style="color: #93E0E3;">myfit2</span><span style="color: #DCDCCC;">(</span><span style="color: #DFAF8F;">x</span>,<span style="color: #DFAF8F;">y</span><span style="color: #DCDCCC;">)</span>
A = <span style="color: #DCDCCC;">[</span>x ones<span style="color: #BFEBBF;">(</span>size<span style="color: #D0BF8F;">(</span>x<span style="color: #D0BF8F;">)</span><span style="color: #BFEBBF;">)</span><span style="color: #DCDCCC;">]</span><span style="color: #7CB8BB;">\</span>y;
a = A<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">1</span><span style="color: #DCDCCC;">)</span>; b = A<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">2</span><span style="color: #DCDCCC;">)</span>;
Y = y <span style="color: #7CB8BB;">-</span> <span style="color: #DCDCCC;">(</span>a<span style="color: #7CB8BB;">*</span>x <span style="color: #7CB8BB;">+</span> b<span style="color: #DCDCCC;">)</span>;
<span style="color: #F0DFAF; font-weight: bold;">end</span>
<span style="color: #F0DFAF; font-weight: bold;">end</span>
</pre>
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<div id="postamble" class="status">
<p class="author">Author: Thomas Dehaeze</p>
<p class="date">Created: 2019-03-20 mer. 15:17</p>
<p class="validation"><a href="http://validator.w3.org/check?uri=referer">Validate</a></p>
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