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< h1 class = "title" > Spindle Analysis< / h1 >
< div id = "table-of-contents" >
< h2 > Table of Contents< / h2 >
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< li > < a href = "#org926cca8" > 1. Notes< / a > < / li >
< li > < a href = "#org31b9342" > 2. Data Processing< / a >
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< ul >
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< li > < a href = "#org55318f9" > 2.1. Load Measurement Data< / a > < / li >
< li > < a href = "#orge988a52" > 2.2. Convert Signals from [deg] to [sec]< / a > < / li >
< li > < a href = "#orgb06c483" > 2.3. Convert Signals< / a > < / li >
< li > < a href = "#org56d1113" > 2.4. Ts and Fs for both measurements< / a > < / li >
< li > < a href = "#org535061e" > 2.5. Find Noise of the ADC [\(\frac{m}{\sqrt{Hz}}\)]< / a > < / li >
< li > < a href = "#orgd8739c6" > 2.6. Save all the data under spindle struct< / a > < / li >
< li > < a href = "#org7d5ef48" > 2.7. Compute Asynchronous data< / a > < / li >
< li > < a href = "#org4554fa6" > 2.8. Save data< / a > < / li >
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< li > < a href = "#org87d8df4" > 3. Time Domain Data< / a >
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< ul >
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< li > < a href = "#org97eb643" > 3.1. Load the processed data< / a > < / li >
< li > < a href = "#orge2f8b97" > 3.2. Plot X-Y-Z position with respect to Time - 1rpm< / a > < / li >
< li > < a href = "#orgca1fa1e" > 3.3. Plot X-Y-Z position with respect to Time - 60rpm< / a > < / li >
< li > < a href = "#org1f349e8" > 3.4. Plot Synchronous and Asynchronous - 1rpm< / a > < / li >
< li > < a href = "#orgd12a5c7" > 3.5. Plot Synchronous and Asynchronous - 60rpm< / a > < / li >
< li > < a href = "#orgc54777a" > 3.6. Plot X against Y< / a > < / li >
< li > < a href = "#orgc2dcbf0" > 3.7. Plot X against Y - Asynchronous< / a > < / li >
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< li > < a href = "#orgbb9b6e0" > 4. Model of the spindle< / a >
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< ul >
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< li > < a href = "#org6c8ee27" > 4.1. Schematic of the model< / a > < / li >
< li > < a href = "#orgcf466f7" > 4.2. Parameters< / a > < / li >
< li > < a href = "#org3d29eaa" > 4.3. Compute Mass and Stiffness Matrices< / a > < / li >
< li > < a href = "#org87a0e41" > 4.4. Compute resonance frequencies< / a > < / li >
< li > < a href = "#orgac905f5" > 4.5. From model_damping compute the Damping Matrix< / a > < / li >
< li > < a href = "#orgfb4ae66" > 4.6. Define inputs, outputs and state names< / a > < / li >
< li > < a href = "#org42fbd26" > 4.7. Define A, B and C matrices< / a > < / li >
< li > < a href = "#org4554e76" > 4.8. Generate the State Space Model< / a > < / li >
< li > < a href = "#org7b72401" > 4.9. Bode Plot< / a > < / li >
< li > < a href = "#org156efe8" > 4.10. Save the model< / a > < / li >
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< li > < a href = "#org055ab30" > 5. Frequency Domain Data< / a >
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< ul >
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< li > < a href = "#org4e59d5a" > 5.1. Load the processed data and the model< / a > < / li >
< li > < a href = "#org76b5b80" > 5.2. Compute the PSD< / a > < / li >
< li > < a href = "#org0a76f2c" > 5.3. Plot the computed PSD< / a > < / li >
< li > < a href = "#org85095d4" > 5.4. Compute the response of the model< / a > < / li >
< li > < a href = "#org6e1d56d" > 5.5. Plot the PSD of the Force using the model< / a > < / li >
< li > < a href = "#orgedd234c" > 5.6. Estimated Shape of the PSD of the force< / a > < / li >
< li > < a href = "#org0a01476" > 5.7. PSD in [N]< / a > < / li >
< li > < a href = "#orgb29a3ce" > 5.8. PSD in [m]< / a > < / li >
< li > < a href = "#org2ce75a0" > 5.9. Compute the resulting RMS value [m]< / a > < / li >
< li > < a href = "#org9bcf259" > 5.10. Compute the resulting RMS value [m]< / a > < / li >
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< li > < a href = "#orgc7eca20" > 6. Functions< / a >
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< ul >
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< li > < a href = "#orgb688035" > 6.1. getAsynchronousError< / a > < / li >
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< p >
The report made by the PEL is accessible < a href = "documents/Spindle_report_test.pdf" > here< / a > .
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< h2 id = "org926cca8" > < span class = "section-number-2" > 1< / span > Notes< / h2 >
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< div id = "org6208f54" class = "figure" >
< p > < img src = "./img/setup_spindle.png" alt = "setup_spindle.png" / >
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< p > < span class = "figure-number" > Figure 1: < / span > Measurement setup at the PEL lab< / p >
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< td class = "org-left" > < b > Date< / b > < / td >
< td class = "org-right" > 2017-04-25< / td >
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< td class = "org-left" > < b > Location< / b > < / td >
< td class = "org-right" > PEL Lab< / td >
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< p >
The goal is to estimate all the error motions induced by the Spindle
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< h2 id = "org31b9342" > < span class = "section-number-2" > 2< / span > Data Processing< / h2 >
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< a id = "org1273092" > < / a >
< / p >
< div class = "note" id = "org2ad3b15" >
< p >
All the files (data and Matlab scripts) are accessible < a href = "data/spindle_data_processing.zip" > here< / a > .
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< h3 id = "org55318f9" > < span class = "section-number-3" > 2.1< / span > Load Measurement Data< / h3 >
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< pre class = "src src-matlab" > spindle_1rpm_table = readtable(< span class = "org-string" > './mat/10turns_1rpm_icepap.txt'< / span > );
spindle_60rpm_table = readtable(< span class = "org-string" > './mat/10turns_60rpm_IcepapFIR.txt'< / span > );
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< pre class = "src src-matlab" > spindle_1rpm_table(1, < span class = "org-type" > :< / span > )
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< pre class = "src src-matlab" > spindle_1rpm = table2array(spindle_1rpm_table);
spindle_60rpm = table2array(spindle_60rpm_table);
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< h3 id = "orge988a52" > < span class = "section-number-3" > 2.2< / span > Convert Signals from [deg] to [sec]< / h3 >
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< pre class = "src src-matlab" > speed_1rpm = 360< span class = "org-type" > /< / span > 60; < span class = "org-comment" > % [deg/sec]< / span >
spindle_1rpm(< span class = "org-type" > :< / span > , 1) = spindle_1rpm(< span class = "org-type" > :< / span > , 2)< span class = "org-type" > /< / span > speed_1rpm; < span class = "org-comment" > % From position [deg] to time [s]< / span >
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speed_60rpm = 360< span class = "org-type" > /< / span > 1; < span class = "org-comment" > % [deg/sec]< / span >
spindle_60rpm(< span class = "org-type" > :< / span > , 1) = spindle_60rpm(< span class = "org-type" > :< / span > , 2)< span class = "org-type" > /< / span > speed_60rpm; < span class = "org-comment" > % From position [deg] to time [s]< / span >
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< h3 id = "orgb06c483" > < span class = "section-number-3" > 2.3< / span > Convert Signals< / h3 >
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< pre class = "src src-matlab" > < span class = "org-comment" > % scaling = 1/80000; % 80 mV/um< / span >
scaling = 1e< span class = "org-type" > -< / span > 6; < span class = "org-comment" > % [um] to [m]< / span >
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spindle_1rpm(< span class = "org-type" > :< / span > , 3< span class = "org-type" > :< / span > end) = scaling< span class = "org-type" > *< / span > spindle_1rpm(< span class = "org-type" > :< / span > , 3< span class = "org-type" > :< / span > end); < span class = "org-comment" > % [V] to [m]< / span >
spindle_1rpm(< span class = "org-type" > :< / span > , 3< span class = "org-type" > :< / span > end) = spindle_1rpm(< span class = "org-type" > :< / span > , 3< span class = "org-type" > :< / span > end)< span class = "org-type" > -< / span > mean(spindle_1rpm(< span class = "org-type" > :< / span > , 3< span class = "org-type" > :< / span > end)); < span class = "org-comment" > % Remove mean< / span >
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spindle_60rpm(< span class = "org-type" > :< / span > , 3< span class = "org-type" > :< / span > end) = scaling< span class = "org-type" > *< / span > spindle_60rpm(< span class = "org-type" > :< / span > , 3< span class = "org-type" > :< / span > end); < span class = "org-comment" > % [V] to [m]< / span >
spindle_60rpm(< span class = "org-type" > :< / span > , 3< span class = "org-type" > :< / span > end) = spindle_60rpm(< span class = "org-type" > :< / span > , 3< span class = "org-type" > :< / span > end)< span class = "org-type" > -< / span > mean(spindle_60rpm(< span class = "org-type" > :< / span > , 3< span class = "org-type" > :< / span > end)); < span class = "org-comment" > % Remove mean< / span >
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< h3 id = "org56d1113" > < span class = "section-number-3" > 2.4< / span > Ts and Fs for both measurements< / h3 >
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< pre class = "src src-matlab" > Ts_1rpm = spindle_1rpm(end, 1)< span class = "org-type" > /< / span > (length(spindle_1rpm(< span class = "org-type" > :< / span > , 1))< span class = "org-type" > -< / span > 1);
Fs_1rpm = 1< span class = "org-type" > /< / span > Ts_1rpm;
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Ts_60rpm = spindle_60rpm(end, 1)< span class = "org-type" > /< / span > (length(spindle_60rpm(< span class = "org-type" > :< / span > , 1))< span class = "org-type" > -< / span > 1);
Fs_60rpm = 1< span class = "org-type" > /< / span > Ts_60rpm;
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< h3 id = "org535061e" > < span class = "section-number-3" > 2.5< / span > Find Noise of the ADC [\(\frac{m}{\sqrt{Hz}}\)]< / h3 >
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< pre class = "src src-matlab" > data = spindle_1rpm(< span class = "org-type" > :< / span > , 5);
dV_1rpm = min(abs(data(1) < span class = "org-type" > -< / span > data(data < span class = "org-type" > ~=< / span > data(1))));
noise_1rpm = dV_1rpm< span class = "org-type" > /< / span > sqrt(12< span class = "org-type" > *< / span > Fs_1rpm< span class = "org-type" > /< / span > 2);
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data = spindle_60rpm(< span class = "org-type" > :< / span > , 5);
dV_60rpm = min(abs(data(50) < span class = "org-type" > -< / span > data(data < span class = "org-type" > ~=< / span > data(50))));
noise_60rpm = dV_60rpm< span class = "org-type" > /< / span > sqrt(12< span class = "org-type" > *< / span > Fs_60rpm< span class = "org-type" > /< / span > 2);
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< h3 id = "orgd8739c6" > < span class = "section-number-3" > 2.6< / span > Save all the data under spindle struct< / h3 >
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< pre class = "src src-matlab" > spindle.rpm1.time = spindle_1rpm(< span class = "org-type" > :< / span > , 1);
spindle.rpm1.deg = spindle_1rpm(< span class = "org-type" > :< / span > , 2);
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spindle.rpm1.Ts = Ts_1rpm;
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spindle.rpm1.Fs = 1< span class = "org-type" > /< / span > Ts_1rpm;
spindle.rpm1.x = spindle_1rpm(< span class = "org-type" > :< / span > , 3);
spindle.rpm1.y = spindle_1rpm(< span class = "org-type" > :< / span > , 4);
spindle.rpm1.z = spindle_1rpm(< span class = "org-type" > :< / span > , 5);
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spindle.rpm1.adcn = noise_1rpm;
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spindle.rpm60.time = spindle_60rpm(< span class = "org-type" > :< / span > , 1);
spindle.rpm60.deg = spindle_60rpm(< span class = "org-type" > :< / span > , 2);
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spindle.rpm60.Ts = Ts_60rpm;
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spindle.rpm60.Fs = 1< span class = "org-type" > /< / span > Ts_60rpm;
spindle.rpm60.x = spindle_60rpm(< span class = "org-type" > :< / span > , 3);
spindle.rpm60.y = spindle_60rpm(< span class = "org-type" > :< / span > , 4);
spindle.rpm60.z = spindle_60rpm(< span class = "org-type" > :< / span > , 5);
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spindle.rpm60.adcn = noise_60rpm;
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< h3 id = "org7d5ef48" > < span class = "section-number-3" > 2.7< / span > Compute Asynchronous data< / h3 >
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< pre class = "src src-matlab" > < span class = "org-keyword" > for< / span > < span class = "org-variable-name" > direction< / span > = < span class = "org-constant" > {< / span > < span class = "org-constant" > < span class = "org-string" > 'x'< / span > < / span > , < span class = "org-string" > 'y'< / span > , < span class = "org-string" > 'z'< / span > }
spindle.rpm1.([direction{1}, < span class = "org-string" > 'async'< / span > ]) = getAsynchronousError(spindle.rpm1.(direction{1}), 10);
spindle.rpm60.([direction{1}, < span class = "org-string" > 'async'< / span > ]) = getAsynchronousError(spindle.rpm60.(direction{1}), 10);
< span class = "org-keyword" > end< / span >
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< h3 id = "org4554fa6" > < span class = "section-number-3" > 2.8< / span > Save data< / h3 >
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< pre class = "src src-matlab" > save(< span class = "org-string" > './mat/spindle_data.mat'< / span > , < span class = "org-string" > 'spindle'< / span > );
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< h2 id = "org87d8df4" > < span class = "section-number-2" > 3< / span > Time Domain Data< / h2 >
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< a id = "org3ae8d1c" > < / a >
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All the files (data and Matlab scripts) are accessible < a href = "data/spindle_time_domain.zip" > here< / a > .
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< h3 id = "org97eb643" > < span class = "section-number-3" > 3.1< / span > Load the processed data< / h3 >
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< div class = "outline-text-3" id = "text-3-1" >
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< div class = "org-src-container" >
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< pre class = "src src-matlab" > load(< span class = "org-string" > './mat/spindle_data.mat'< / span > , < span class = "org-string" > 'spindle'< / span > );
< / pre >
< / div >
< / div >
< / div >
< div id = "outline-container-orge2f8b97" class = "outline-3" >
< h3 id = "orge2f8b97" > < span class = "section-number-3" > 3.2< / span > Plot X-Y-Z position with respect to Time - 1rpm< / h3 >
< div class = "outline-text-3" id = "text-3-2" >
< div class = "org-src-container" >
< pre class = "src src-matlab" > < span class = "org-type" > figure< / span > ;
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hold on;
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plot(spindle.rpm1.time, spindle.rpm1.x);
plot(spindle.rpm1.time, spindle.rpm1.y);
plot(spindle.rpm1.time, spindle.rpm1.z);
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hold off;
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xlabel(< span class = "org-string" > 'Time [s]'< / span > ); ylabel(< span class = "org-string" > 'Amplitude [m]'< / span > );
legend({< span class = "org-string" > 'tx - 1rpm'< / span > , < span class = "org-string" > 'ty - 1rpm'< / span > , < span class = "org-string" > 'tz - 1rpm'< / span > });
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< / pre >
< / div >
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< div id = "org3945d8b" class = "figure" >
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< p > < img src = "figs/spindle_xyz_1rpm.png" alt = "spindle_xyz_1rpm.png" / >
< / p >
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< p > < span class = "figure-number" > Figure 2: < / span > Raw time domain translation - 1rpm< / p >
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< / div >
< / div >
< / div >
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< div id = "outline-container-orgca1fa1e" class = "outline-3" >
< h3 id = "orgca1fa1e" > < span class = "section-number-3" > 3.3< / span > Plot X-Y-Z position with respect to Time - 60rpm< / h3 >
< div class = "outline-text-3" id = "text-3-3" >
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< p >
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The measurements for the spindle turning at 60rpm are shown figure < a href = "#orgf769c8a" > 3< / a > .
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< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > < span class = "org-type" > figure< / span > ;
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hold on;
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plot(spindle.rpm60.time, spindle.rpm60.x);
plot(spindle.rpm60.time, spindle.rpm60.y);
plot(spindle.rpm60.time, spindle.rpm60.z);
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hold off;
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xlabel(< span class = "org-string" > 'Time [s]'< / span > ); ylabel(< span class = "org-string" > 'Amplitude [m]'< / span > );
legend({< span class = "org-string" > 'tx - 60rpm'< / span > , < span class = "org-string" > 'ty - 60rpm'< / span > , < span class = "org-string" > 'tz - 60rpm'< / span > });
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< / pre >
< / div >
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< div id = "orgf769c8a" class = "figure" >
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< p > < img src = "figs/spindle_xyz_60rpm.png" alt = "spindle_xyz_60rpm.png" / >
< / p >
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< p > < span class = "figure-number" > Figure 3: < / span > Raw time domain translation - 60rpm< / p >
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< / div >
< / div >
< / div >
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< div id = "outline-container-org1f349e8" class = "outline-3" >
< h3 id = "org1f349e8" > < span class = "section-number-3" > 3.4< / span > Plot Synchronous and Asynchronous - 1rpm< / h3 >
< div class = "outline-text-3" id = "text-3-4" >
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< div class = "org-src-container" >
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< pre class = "src src-matlab" > < span class = "org-type" > figure< / span > ;
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hold on;
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plot(spindle.rpm1.time, spindle.rpm1.x);
plot(spindle.rpm1.time, spindle.rpm1.xasync);
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hold off;
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xlabel(< span class = "org-string" > 'Time [s]'< / span > ); ylabel(< span class = "org-string" > 'Amplitude [m]'< / span > );
legend({< span class = "org-string" > 'tx - 1rpm - Sync'< / span > , < span class = "org-string" > 'tx - 1rpm - Async'< / span > });
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< / pre >
< / div >
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< div id = "orgb601205" class = "figure" >
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< p > < img src = "figs/spindle_1rpm_sync_async.png" alt = "spindle_1rpm_sync_async.png" / >
< / p >
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< p > < span class = "figure-number" > Figure 4: < / span > Comparison of the synchronous and asynchronous displacements - 1rpm< / p >
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< / div >
< / div >
< / div >
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< div id = "outline-container-orgd12a5c7" class = "outline-3" >
< h3 id = "orgd12a5c7" > < span class = "section-number-3" > 3.5< / span > Plot Synchronous and Asynchronous - 60rpm< / h3 >
< div class = "outline-text-3" id = "text-3-5" >
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< p >
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The data is split into its Synchronous and Asynchronous part (figure < a href = "#org4b7ac99" > 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.
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< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > < span class = "org-type" > figure< / span > ;
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hold on;
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plot(spindle.rpm60.time, spindle.rpm60.x);
plot(spindle.rpm60.time, spindle.rpm60.xasync);
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hold off;
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xlabel(< span class = "org-string" > 'Time [s]'< / span > ); ylabel(< span class = "org-string" > 'Amplitude [m]'< / span > );
legend({< span class = "org-string" > 'tx - 60rpm - Sync'< / span > , < span class = "org-string" > 'tx - 60rpm - Async'< / span > });
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< / pre >
< / div >
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< div id = "org4b7ac99" class = "figure" >
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< p > < img src = "figs/spindle_60rpm_sync_async.png" alt = "spindle_60rpm_sync_async.png" / >
< / p >
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< p > < span class = "figure-number" > Figure 5: < / span > Comparison of the synchronous and asynchronous displacements - 60rpm< / p >
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< / div >
< / div >
< / div >
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< div id = "outline-container-orgc54777a" class = "outline-3" >
< h3 id = "orgc54777a" > < span class = "section-number-3" > 3.6< / span > Plot X against Y< / h3 >
< div class = "outline-text-3" id = "text-3-6" >
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< div class = "org-src-container" >
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< pre class = "src src-matlab" > < span class = "org-type" > figure< / span > ;
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hold on;
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plot(spindle.rpm1.x, spindle.rpm1.y);
plot(spindle.rpm60.x, spindle.rpm60.y);
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hold off;
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xlabel(< span class = "org-string" > 'X Amplitude [m]'< / span > ); ylabel(< span class = "org-string" > 'Y Amplitude [m]'< / span > );
legend({< span class = "org-string" > '1rpm'< / span > , < span class = "org-string" > '60rpm'< / span > });
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< / pre >
< / div >
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< div id = "org476bb3d" class = "figure" >
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< p > < img src = "figs/spindle_xy_1_60rpm.png" alt = "spindle_xy_1_60rpm.png" / >
< / p >
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< p > < span class = "figure-number" > Figure 6: < / span > Synchronous x-y displacement< / p >
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< / div >
< / div >
< / div >
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< div id = "outline-container-orgc2dcbf0" class = "outline-3" >
< h3 id = "orgc2dcbf0" > < span class = "section-number-3" > 3.7< / span > Plot X against Y - Asynchronous< / h3 >
< div class = "outline-text-3" id = "text-3-7" >
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< div class = "org-src-container" >
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< pre class = "src src-matlab" > < span class = "org-type" > figure< / span > ;
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hold on;
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plot(spindle.rpm1.xasync, spindle.rpm1.yasync);
plot(spindle.rpm60.xasync, spindle.rpm60.yasync);
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hold off;
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xlabel(< span class = "org-string" > 'X Amplitude [m]'< / span > ); ylabel(< span class = "org-string" > 'Y Amplitude [m]'< / span > );
legend({< span class = "org-string" > '1rpm'< / span > , < span class = "org-string" > '60rpm'< / span > });
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< / pre >
< / div >
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< div id = "org258fc09" class = "figure" >
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< p > < img src = "figs/spindle_xy_1_60_rpm_async.png" alt = "spindle_xy_1_60_rpm_async.png" / >
< / p >
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< p > < span class = "figure-number" > Figure 7: < / span > Asynchronous x-y displacement< / p >
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< / div >
< / div >
< / div >
< / div >
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< div id = "outline-container-orgbb9b6e0" class = "outline-2" >
< h2 id = "orgbb9b6e0" > < span class = "section-number-2" > 4< / span > Model of the spindle< / h2 >
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< div class = "outline-text-2" id = "text-4" >
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< p >
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< a id = "orgf994efb" > < / a >
< / p >
< div class = "note" id = "orgbebec4b" >
< p >
All the files (data and Matlab scripts) are accessible < a href = "data/spindle_model.zip" > here< / a > .
< / p >
< / div >
< / div >
< div id = "outline-container-org6c8ee27" class = "outline-3" >
< h3 id = "org6c8ee27" > < span class = "section-number-3" > 4.1< / span > Schematic of the model< / h3 >
< div class = "outline-text-3" id = "text-4-1" >
< p >
The model of the spindle used is shown figure < a href = "#org23eb2b8" > 8< / a > .
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< / p >
< p >
\(f\) is the perturbation force of the spindle and \(d\) is the measured displacement.
< / p >
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< div id = "org23eb2b8" class = "figure" >
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< p > < img src = "./figs/uniaxial-model-spindle.png" alt = "uniaxial-model-spindle.png" / >
< / p >
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< p > < span class = "figure-number" > Figure 8: < / span > Model of the Spindle< / p >
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< / div >
< / div >
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< / div >
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< div id = "outline-container-orgcf466f7" class = "outline-3" >
< h3 id = "orgcf466f7" > < span class = "section-number-3" > 4.2< / span > Parameters< / h3 >
< div class = "outline-text-3" id = "text-4-2" >
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< div class = "org-src-container" >
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< pre class = "src src-matlab" > mg = 3000; < span class = "org-comment" > % Mass of granite [kg]< / span >
ms = 50; < span class = "org-comment" > % Mass of Spindle [kg]< / span >
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kg = 1e8; < span class = "org-comment" > % Stiffness of granite [N/m]< / span >
ks = 5e7; < span class = "org-comment" > % Stiffness of spindle [N/m]< / span >
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< / pre >
< / div >
< / div >
< / div >
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< div id = "outline-container-org3d29eaa" class = "outline-3" >
< h3 id = "org3d29eaa" > < span class = "section-number-3" > 4.3< / span > Compute Mass and Stiffness Matrices< / h3 >
< div class = "outline-text-3" id = "text-4-3" >
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< div class = "org-src-container" >
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< pre class = "src src-matlab" > Mm = diag([ms, mg]);
Km = diag([ks, ks< span class = "org-type" > +< / span > kg]) < span class = "org-type" > -< / span > diag(ks, < span class = "org-type" > -< / span > 1) < span class = "org-type" > -< / span > diag(ks, 1);
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< / pre >
< / div >
< / div >
< / div >
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< div id = "outline-container-org87a0e41" class = "outline-3" >
< h3 id = "org87a0e41" > < span class = "section-number-3" > 4.4< / span > Compute resonance frequencies< / h3 >
< div class = "outline-text-3" id = "text-4-4" >
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< div class = "org-src-container" >
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< pre class = "src src-matlab" > A = [zeros(size(Mm)) eye(size(Mm)) ; < span class = "org-type" > -< / span > Mm< span class = "org-type" > \< / span > Km zeros(size(Mm))];
eigA = imag(eigs(A))< span class = "org-type" > /< / span > 2< span class = "org-type" > /< / span > < span class = "org-constant" > pi< / span > ;
eigA = eigA(eigA< span class = "org-type" > > < / span > 0);
eigA = eigA(1< span class = "org-type" > :< / span > 2);
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< / pre >
< / div >
< / div >
< / div >
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< div id = "outline-container-orgac905f5" class = "outline-3" >
< h3 id = "orgac905f5" > < span class = "section-number-3" > 4.5< / span > From model_damping compute the Damping Matrix< / h3 >
< div class = "outline-text-3" id = "text-4-5" >
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< div class = "org-src-container" >
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< pre class = "src src-matlab" > modal_damping = 1e< span class = "org-type" > -< / span > 5;
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ab = [0.5< span class = "org-type" > *< / span > eigA(1) 0.5< span class = "org-type" > /< / span > eigA(1) ; 0.5< span class = "org-type" > *< / span > eigA(2) 0.5< span class = "org-type" > /< / span > eigA(2)]< span class = "org-type" > \< / span > [modal_damping ; modal_damping];
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Cm = ab(1)< span class = "org-type" > *< / span > Mm < span class = "org-type" > +< / span > ab(2)< span class = "org-type" > *< / span > Km;
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< / pre >
< / div >
< / div >
< / div >
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< div id = "outline-container-orgfb4ae66" class = "outline-3" >
< h3 id = "orgfb4ae66" > < span class = "section-number-3" > 4.6< / span > Define inputs, outputs and state names< / h3 >
< div class = "outline-text-3" id = "text-4-6" >
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< div class = "org-src-container" >
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< pre class = "src src-matlab" > StateName = {...
< span class = "org-string" > 'xs'< / span > , ...< span class = "org-comment" > % Displacement of Spindle [m]< / span >
< span class = "org-string" > 'xg'< / span > , ...< span class = "org-comment" > % Displacement of Granite [m]< / span >
< span class = "org-string" > 'vs'< / span > , ...< span class = "org-comment" > % Velocity of Spindle [m]< / span >
< span class = "org-string" > 'vg'< / span > , ...< span class = "org-comment" > % Velocity of Granite [m]< / span >
};
StateUnit = {< span class = "org-string" > 'm'< / span > , < span class = "org-string" > 'm'< / span > , < span class = "org-string" > 'm/s'< / span > , < span class = "org-string" > 'm/s'< / span > };
InputName = {...
< span class = "org-string" > 'f'< / span > ...< span class = "org-comment" > % Spindle Force [N]< / span >
};
InputUnit = {< span class = "org-string" > 'N'< / span > };
OutputName = {...
< span class = "org-string" > 'd'< / span > ...< span class = "org-comment" > % Displacement [m]< / span >
};
OutputUnit = {< span class = "org-string" > 'm'< / span > };
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< / pre >
< / div >
< / div >
< / div >
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< div id = "outline-container-org42fbd26" class = "outline-3" >
< h3 id = "org42fbd26" > < span class = "section-number-3" > 4.7< / span > Define A, B and C matrices< / h3 >
< div class = "outline-text-3" id = "text-4-7" >
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< div class = "org-src-container" >
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< pre class = "src src-matlab" > < span class = "org-comment" > % A Matrix< / span >
A = [zeros(size(Mm)) eye(size(Mm)) ; ...
< span class = "org-type" > -< / span > Mm< span class = "org-type" > \< / span > Km < span class = "org-type" > -< / span > Mm< span class = "org-type" > \< / span > Cm];
< span class = "org-comment" > % B Matrix< / span >
B_low = zeros(length(StateName), length(InputName));
B_low(strcmp(StateName,< span class = "org-string" > 'vs'< / span > ), strcmp(InputName,< span class = "org-string" > 'f'< / span > )) = 1;
B_low(strcmp(StateName,< span class = "org-string" > 'vg'< / span > ), strcmp(InputName,< span class = "org-string" > 'f'< / span > )) = < span class = "org-type" > -< / span > 1;
B = blkdiag(zeros(length(StateName)< span class = "org-type" > /< / span > 2), pinv(Mm))< span class = "org-type" > *< / span > B_low;
< span class = "org-comment" > % C Matrix< / span >
C = zeros(length(OutputName), length(StateName));
C(strcmp(OutputName,< span class = "org-string" > 'd'< / span > ), strcmp(StateName,< span class = "org-string" > 'xs'< / span > )) = 1;
C(strcmp(OutputName,< span class = "org-string" > 'd'< / span > ), strcmp(StateName,< span class = "org-string" > 'xg'< / span > )) = < span class = "org-type" > -< / span > 1;
< span class = "org-comment" > % D Matrix< / span >
D = zeros(length(OutputName), length(InputName));
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< / pre >
< / div >
< / div >
< / div >
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< div id = "outline-container-org4554e76" class = "outline-3" >
< h3 id = "org4554e76" > < span class = "section-number-3" > 4.8< / span > Generate the State Space Model< / h3 >
< div class = "outline-text-3" id = "text-4-8" >
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< div class = "org-src-container" >
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< pre class = "src src-matlab" > sys = ss(A, B, C, D);
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sys.StateName = StateName;
sys.StateUnit = StateUnit;
sys.InputName = InputName;
sys.InputUnit = InputUnit;
sys.OutputName = OutputName;
sys.OutputUnit = OutputUnit;
< / pre >
< / div >
< / div >
< / div >
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< div id = "outline-container-org7b72401" class = "outline-3" >
< h3 id = "org7b72401" > < span class = "section-number-3" > 4.9< / span > Bode Plot< / h3 >
< div class = "outline-text-3" id = "text-4-9" >
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< p >
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The transfer function from a disturbance force \(f\) to the measured displacement \(d\) is shown figure < a href = "#orgc179dd0" > 9< / a > .
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< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > freqs = logspace(< span class = "org-type" > -< / span > 1, 3, 1000);
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< span class = "org-type" > figure< / span > ;
plot(freqs, abs(squeeze(freqresp(sys(< span class = "org-string" > 'd'< / span > , < span class = "org-string" > 'f'< / span > ), freqs, < span class = "org-string" > 'Hz'< / span > ))));
< span class = "org-type" > set< / span > (< span class = "org-variable-name" > gca< / span > , < span class = "org-string" > 'XScale'< / span > , < span class = "org-string" > 'log'< / span > ); < span class = "org-type" > set< / span > (< span class = "org-variable-name" > gca< / span > , < span class = "org-string" > 'YScale'< / span > , < span class = "org-string" > 'log'< / span > );
xlabel(< span class = "org-string" > 'Frequency [Hz]'< / span > ); ylabel(< span class = "org-string" > 'Amplitude [m/N]'< / span > );
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< / pre >
< / div >
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< div id = "orgc179dd0" class = "figure" >
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< p > < img src = "figs/spindle_f_to_d.png" alt = "spindle_f_to_d.png" / >
< / p >
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< p > < span class = "figure-number" > Figure 9: < / span > Bode plot of the transfer function from \(f\) to \(d\)< / p >
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< / div >
< / div >
< / div >
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< div id = "outline-container-org156efe8" class = "outline-3" >
< h3 id = "org156efe8" > < span class = "section-number-3" > 4.10< / span > Save the model< / h3 >
< div class = "outline-text-3" id = "text-4-10" >
< div class = "org-src-container" >
< pre class = "src src-matlab" > save(< span class = "org-string" > './mat/spindle_model.mat'< / span > , < span class = "org-string" > 'sys'< / span > );
< / pre >
< / div >
< / div >
< / div >
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< / div >
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< div id = "outline-container-org055ab30" class = "outline-2" >
< h2 id = "org055ab30" > < span class = "section-number-2" > 5< / span > Frequency Domain Data< / h2 >
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< div class = "outline-text-2" id = "text-5" >
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< p >
< a id = "orgbaa269a" > < / a >
< / p >
< div class = "note" id = "org76ead94" >
< p >
All the files (data and Matlab scripts) are accessible < a href = "data/spindle_psd.zip" > here< / a > .
< / p >
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< / div >
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< / div >
< div id = "outline-container-org4e59d5a" class = "outline-3" >
< h3 id = "org4e59d5a" > < span class = "section-number-3" > 5.1< / span > Load the processed data and the model< / h3 >
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< div class = "outline-text-3" id = "text-5-1" >
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< div class = "org-src-container" >
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< pre class = "src src-matlab" > load(< span class = "org-string" > './mat/spindle_data.mat'< / span > , < span class = "org-string" > 'spindle'< / span > );
load(< span class = "org-string" > './mat/spindle_model.mat'< / span > , < span class = "org-string" > 'sys'< / span > );
< / pre >
< / div >
< / div >
< / div >
< div id = "outline-container-org76b5b80" class = "outline-3" >
< h3 id = "org76b5b80" > < span class = "section-number-3" > 5.2< / span > Compute the PSD< / h3 >
< div class = "outline-text-3" id = "text-5-2" >
< div class = "org-src-container" >
< pre class = "src src-matlab" > n_av = 4; < span class = "org-comment" > % Number of average< / span >
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[pxx_1rpm, f_1rpm] = pwelch(spindle.rpm1.xasync, hanning(ceil(length(spindle.rpm1.xasync)< span class = "org-type" > /< / span > n_av)), [], [], spindle.rpm1.Fs);
[pyy_1rpm, < span class = "org-type" > ~< / span > ] = pwelch(spindle.rpm1.yasync, hanning(ceil(length(spindle.rpm1.yasync)< span class = "org-type" > /< / span > n_av)), [], [], spindle.rpm1.Fs);
[pzz_1rpm, < span class = "org-type" > ~< / span > ] = pwelch(spindle.rpm1.zasync, hanning(ceil(length(spindle.rpm1.zasync)< span class = "org-type" > /< / span > n_av)), [], [], spindle.rpm1.Fs);
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[pxx_60rpm, f_60rpm] = pwelch(spindle.rpm60.xasync, hanning(ceil(length(spindle.rpm60.xasync)< span class = "org-type" > /< / span > n_av)), [], [], spindle.rpm60.Fs);
[pyy_60rpm, < span class = "org-type" > ~< / span > ] = pwelch(spindle.rpm60.yasync, hanning(ceil(length(spindle.rpm60.yasync)< span class = "org-type" > /< / span > n_av)), [], [], spindle.rpm60.Fs);
[pzz_60rpm, < span class = "org-type" > ~< / span > ] = pwelch(spindle.rpm60.zasync, hanning(ceil(length(spindle.rpm60.zasync)< span class = "org-type" > /< / span > n_av)), [], [], spindle.rpm60.Fs);
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< / pre >
< / div >
< / div >
< / div >
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< div id = "outline-container-org0a76f2c" class = "outline-3" >
< h3 id = "org0a76f2c" > < span class = "section-number-3" > 5.3< / span > Plot the computed PSD< / h3 >
< div class = "outline-text-3" id = "text-5-3" >
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< p >
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The Amplitude Spectral Densities of the displacement of the spindle for the \(x\), \(y\) and \(z\) directions are shown figure < a href = "#org0bf401d" > 11< / a > . They correspond to the Asynchronous part shown figure < a href = "#org4b7ac99" > 5< / a > .
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< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > < span class = "org-type" > figure< / span > ;
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hold on;
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plot(f_1rpm, (pxx_1rpm)< span class = "org-type" > .^< / span > .5, < span class = "org-string" > 'DisplayName'< / span > , < span class = "org-string" > '$P_{xx}$ - 1rpm'< / span > );
plot(f_1rpm, (pyy_1rpm)< span class = "org-type" > .^< / span > .5, < span class = "org-string" > 'DisplayName'< / span > , < span class = "org-string" > '$P_{yy}$ - 1rpm'< / span > );
plot(f_1rpm, (pzz_1rpm)< span class = "org-type" > .^< / span > .5, < span class = "org-string" > 'DisplayName'< / span > , < span class = "org-string" > '$P_{zz}$ - 1rpm'< / span > );
< span class = "org-comment" > % plot(f_1rpm, spindle.rpm1.adcn*ones(size(f_1rpm)), '--k', 'DisplayName', 'ADC - 1rpm');< / span >
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hold off;
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< span class = "org-type" > set< / span > (< span class = "org-variable-name" > gca< / span > , < span class = "org-string" > 'XScale'< / span > , < span class = "org-string" > 'log'< / span > ); < span class = "org-type" > set< / span > (< span class = "org-variable-name" > gca< / span > , < span class = "org-string" > 'YScale'< / span > , < span class = "org-string" > 'log'< / span > );
xlabel(< span class = "org-string" > 'Frequency [Hz]'< / span > ); ylabel(< span class = "org-string" > 'ASD [$m/\sqrt{Hz}$]'< / span > );
legend(< span class = "org-string" > 'Location'< / span > , < span class = "org-string" > 'northeast'< / span > );
xlim([f_1rpm(2), f_1rpm(end)]);
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< / pre >
< / div >
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< div id = "org618a78b" class = "figure" >
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< p > < img src = "figs/spindle_psd_xyz_1rpm.png" alt = "spindle_psd_xyz_1rpm.png" / >
< / p >
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< p > < span class = "figure-number" > Figure 10: < / span > Power spectral density of the Asynchronous displacement - 1rpm< / p >
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< / div >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > < span class = "org-type" > figure< / span > ;
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hold on;
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plot(f_60rpm, (pxx_60rpm)< span class = "org-type" > .^< / span > .5, < span class = "org-string" > 'DisplayName'< / span > , < span class = "org-string" > '$P_{xx}$ - 60rpm'< / span > );
plot(f_60rpm, (pyy_60rpm)< span class = "org-type" > .^< / span > .5, < span class = "org-string" > 'DisplayName'< / span > , < span class = "org-string" > '$P_{yy}$ - 60rpm'< / span > );
plot(f_60rpm, (pzz_60rpm)< span class = "org-type" > .^< / span > .5, < span class = "org-string" > 'DisplayName'< / span > , < span class = "org-string" > '$P_{zz}$ - 60rpm'< / span > );
< span class = "org-comment" > % plot(f_60rpm, spindle.rpm60.adcn*ones(size(f_60rpm)), '--k', 'DisplayName', 'ADC - 60rpm');< / span >
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hold off;
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< span class = "org-type" > set< / span > (< span class = "org-variable-name" > gca< / span > , < span class = "org-string" > 'XScale'< / span > , < span class = "org-string" > 'log'< / span > ); < span class = "org-type" > set< / span > (< span class = "org-variable-name" > gca< / span > , < span class = "org-string" > 'YScale'< / span > , < span class = "org-string" > 'log'< / span > );
xlabel(< span class = "org-string" > 'Frequency [Hz]'< / span > ); ylabel(< span class = "org-string" > 'ASD [$m/\sqrt{Hz}$]'< / span > );
legend(< span class = "org-string" > 'Location'< / span > , < span class = "org-string" > 'northeast'< / span > );
xlim([f_60rpm(2), f_60rpm(end)]);
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< / pre >
< / div >
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< div id = "org0bf401d" class = "figure" >
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< p > < img src = "figs/spindle_psd_xyz_60rpm.png" alt = "spindle_psd_xyz_60rpm.png" / >
< / p >
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< p > < span class = "figure-number" > Figure 11: < / span > Power spectral density of the Asynchronous displacement - 60rpm< / p >
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< / div >
< / div >
< / div >
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< div id = "outline-container-org85095d4" class = "outline-3" >
< h3 id = "org85095d4" > < span class = "section-number-3" > 5.4< / span > Compute the response of the model< / h3 >
< div class = "outline-text-3" id = "text-5-4" >
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< div class = "org-src-container" >
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< pre class = "src src-matlab" > Tfd = abs(squeeze(freqresp(sys(< span class = "org-string" > 'd'< / span > , < span class = "org-string" > 'f'< / span > ), f_60rpm, < span class = "org-string" > 'Hz'< / span > )));
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< / pre >
< / div >
< / div >
< / div >
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< div id = "outline-container-org6e1d56d" class = "outline-3" >
< h3 id = "org6e1d56d" > < span class = "section-number-3" > 5.5< / span > Plot the PSD of the Force using the model< / h3 >
< div class = "outline-text-3" id = "text-5-5" >
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< div class = "org-src-container" >
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< pre class = "src src-matlab" > < span class = "org-type" > figure< / span > ;
plot(f_60rpm, (pxx_60rpm< span class = "org-type" > .^< / span > .5)< span class = "org-type" > ./< / span > Tfd, < span class = "org-string" > 'DisplayName'< / span > , < span class = "org-string" > '$P_{xx}$'< / span > );
< span class = "org-type" > set< / span > (< span class = "org-variable-name" > gca< / span > , < span class = "org-string" > 'XScale'< / span > , < span class = "org-string" > 'log'< / span > ); < span class = "org-type" > set< / span > (< span class = "org-variable-name" > gca< / span > , < span class = "org-string" > 'YScale'< / span > , < span class = "org-string" > 'log'< / span > );
xlabel(< span class = "org-string" > 'Frequency [Hz]'< / span > ); ylabel(< span class = "org-string" > 'ASD [$N/\sqrt{Hz}$]'< / span > );
xlim([f_60rpm(2), f_60rpm(end)]);
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< / pre >
< / div >
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< div id = "orgeb951fe" class = "figure" >
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< p > < img src = "figs/spindle_psd_f_60rpm.png" alt = "spindle_psd_f_60rpm.png" / >
< / p >
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< p > < span class = "figure-number" > Figure 12: < / span > Power spectral density of the force - 60rpm< / p >
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< / div >
< / div >
< / div >
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< div id = "outline-container-orgedd234c" class = "outline-3" >
< h3 id = "orgedd234c" > < span class = "section-number-3" > 5.6< / span > Estimated Shape of the PSD of the force< / h3 >
< div class = "outline-text-3" id = "text-5-6" >
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< div class = "org-src-container" >
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< pre class = "src src-matlab" > s = tf(< span class = "org-string" > 's'< / span > );
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Wd_simple = 1e< span class = "org-type" > -< / span > 8< 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 > < span class = "org-type" > /< / span > 0.5)< 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 > < span class = "org-type" > /< / span > 100);
Wf_simple = Wd_simple< span class = "org-type" > /< / span > tf(sys(< span class = "org-string" > 'd'< / span > , < span class = "org-string" > 'f'< / span > ));
TWf_simple = abs(squeeze(freqresp(Wf_simple, f_60rpm, < span class = "org-string" > 'Hz'< / span > )));
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< span class = "org-comment" > % Wf = 0.48902*(s+327.9)*(s^2 + 109.6*s + 1.687e04)/((s^2 + 30.59*s + 8541)*(s^2 + 29.11*s + 3.268e04));< / span >
< span class = "org-comment" > % Wf = 0.15788*(s+418.6)*(s+1697)^2*(s^2 + 124.3*s + 2.529e04)*(s^2 + 681.3*s + 9.018e05)/((s^2 + 23.03*s + 8916)*(s^2 + 33.85*s + 6.559e04)*(s^2 + 71.43*s + 4.283e05)*(s^2 + 40.64*s + 1.789e06));< / span >
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Wf = (s< span class = "org-type" > +< / span > 1697)< span class = "org-type" > ^< / span > 2< span class = "org-type" > *< / span > (s< span class = "org-type" > ^< / span > 2 < span class = "org-type" > +< / span > 114.5< span class = "org-type" > *< / span > s < span class = "org-type" > +< / span > 2.278e04)< span class = "org-type" > *< / span > (s< span class = "org-type" > ^< / span > 2 < span class = "org-type" > +< / span > 205.1< span class = "org-type" > *< / span > s < span class = "org-type" > +< / span > 1.627e05)< span class = "org-type" > *< / span > (s< span class = "org-type" > ^< / span > 2 < span class = "org-type" > +< / span > 285.8< span class = "org-type" > *< / span > s < span class = "org-type" > +< / span > 8.624e05)< span class = "org-type" > *< / span > (s< span class = "org-type" > +< / span > 100)< span class = "org-type" > /< / span > ((s< span class = "org-type" > +< / span > 0.5)< span class = "org-type" > *< / span > 3012< span class = "org-type" > *< / span > (s< span class = "org-type" > ^< / span > 2 < span class = "org-type" > +< / span > 23.03< span class = "org-type" > *< / span > s < span class = "org-type" > +< / span > 8916)< span class = "org-type" > *< / span > (s< span class = "org-type" > ^< / span > 2 < span class = "org-type" > +< / span > 17.07< span class = "org-type" > *< / span > s < span class = "org-type" > +< / span > 4.798e04)< span class = "org-type" > *< / span > (s< span class = "org-type" > ^< / span > 2 < span class = "org-type" > +< / span > 41.17< span class = "org-type" > *< / span > s < span class = "org-type" > +< / span > 4.347e05)< span class = "org-type" > *< / span > (s< span class = "org-type" > ^< / span > 2 < span class = "org-type" > +< / span > 78.99< span class = "org-type" > *< / span > s < span class = "org-type" > +< / span > 1.789e06));
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TWf = abs(squeeze(freqresp(Wf, f_60rpm, < span class = "org-string" > 'Hz'< / span > )));
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< / pre >
< / div >
< / div >
< / div >
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< div id = "outline-container-org0a01476" class = "outline-3" >
< h3 id = "org0a01476" > < span class = "section-number-3" > 5.7< / span > PSD in [N]< / h3 >
< div class = "outline-text-3" id = "text-5-7" >
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< p >
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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 = "#org0bf401d" > 11< / a > ).
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< / p >
< p >
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We then fit the PSD of the displacement with a transfer function (figure < a href = "#orgeeae78e" > 14< / a > ).
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< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > < span class = "org-type" > figure< / span > ;
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hold on;
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plot(f_60rpm, (pxx_60rpm< span class = "org-type" > .^< / span > .5)< span class = "org-type" > ./< / span > Tfd, < span class = "org-string" > 'DisplayName'< / span > , < span class = "org-string" > '$\sqrt{P_{xx}}/|T_{d/f}|$'< / span > );
plot(f_60rpm, TWf, < span class = "org-string" > 'DisplayName'< / span > , < span class = "org-string" > 'Wf'< / span > );
plot(f_60rpm, TWf_simple, < span class = "org-string" > '-k'< / span > , < span class = "org-string" > 'DisplayName'< / span > , < span class = "org-string" > 'Wfs'< / span > );
< span class = "org-type" > set< / span > (< span class = "org-variable-name" > gca< / span > , < span class = "org-string" > 'XScale'< / span > , < span class = "org-string" > 'log'< / span > ); < span class = "org-type" > set< / span > (< span class = "org-variable-name" > gca< / span > , < span class = "org-string" > 'YScale'< / span > , < span class = "org-string" > 'log'< / span > );
xlabel(< span class = "org-string" > 'Frequency [Hz]'< / span > ); ylabel(< span class = "org-string" > 'ASD [$N/\sqrt{Hz}$]'< / span > );
xlim([f_60rpm(2), f_60rpm(end)]);
legend(< span class = "org-string" > 'Location'< / span > , < span class = "org-string" > 'northeast'< / span > );
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< / pre >
< / div >
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< div id = "org2c48007" class = "figure" >
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< p > < img src = "figs/spindle_psd_f_comp_60rpm.png" alt = "spindle_psd_f_comp_60rpm.png" / >
< / p >
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< p > < span class = "figure-number" > Figure 13: < / span > Power spectral density of the force - 60rpm< / p >
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< / div >
< / div >
< / div >
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< div id = "outline-container-orgb29a3ce" class = "outline-3" >
< h3 id = "orgb29a3ce" > < span class = "section-number-3" > 5.8< / span > PSD in [m]< / h3 >
< div class = "outline-text-3" id = "text-5-8" >
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< 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 >
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The obtain Power Spectral Density of the force is displayed figure < a href = "#org2c48007" > 13< / a > .
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< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > < span class = "org-type" > figure< / span > ;
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hold on;
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plot(f_60rpm, pxx_60rpm< span class = "org-type" > .^< / span > .5, < span class = "org-string" > 'DisplayName'< / span > , < span class = "org-string" > '$\sqrt{P_{xx}}$'< / span > );
plot(f_60rpm, TWf< span class = "org-type" > .*< / span > Tfd, < span class = "org-string" > 'DisplayName'< / span > , < span class = "org-string" > '$|W_f|*|T_{d/f}|$'< / span > );
< span class = "org-type" > set< / span > (< span class = "org-variable-name" > gca< / span > , < span class = "org-string" > 'XScale'< / span > , < span class = "org-string" > 'log'< / span > ); < span class = "org-type" > set< / span > (< span class = "org-variable-name" > gca< / span > , < span class = "org-string" > 'YScale'< / span > , < span class = "org-string" > 'log'< / span > );
xlabel(< span class = "org-string" > 'Frequency [Hz]'< / span > ); ylabel(< span class = "org-string" > 'ASD [$m/\sqrt{Hz}$]'< / span > );
xlim([f_60rpm(2), f_60rpm(end)]);
legend(< span class = "org-string" > 'Location'< / span > , < span class = "org-string" > 'northeast'< / span > );
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< / pre >
< / div >
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< div id = "orgeeae78e" class = "figure" >
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< p > < img src = "figs/spindle_psd_d_comp_60rpm.png" alt = "spindle_psd_d_comp_60rpm.png" / >
< / p >
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< p > < span class = "figure-number" > Figure 14: < / span > Comparison of the power spectral density of the measured displacement and of the model< / p >
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< / div >
< / div >
< / div >
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< div id = "outline-container-org2ce75a0" class = "outline-3" >
< h3 id = "org2ce75a0" > < span class = "section-number-3" > 5.9< / span > Compute the resulting RMS value [m]< / h3 >
< div class = "outline-text-3" id = "text-5-9" >
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< div class = "org-src-container" >
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< pre class = "src src-matlab" > < span class = "org-type" > figure< / span > ;
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hold on;
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plot(f_60rpm, 1e9< span class = "org-type" > *< / span > cumtrapz(f_60rpm, (pxx_60rpm))< span class = "org-type" > .^< / span > .5, < span class = "org-string" > '--'< / span > , < span class = "org-string" > 'DisplayName'< / span > , < span class = "org-string" > 'Exp. Data'< / span > );
plot(f_60rpm, 1e9< span class = "org-type" > *< / span > cumtrapz(f_60rpm, ((TWf< span class = "org-type" > .*< / span > Tfd)< span class = "org-type" > .^< / span > 2))< span class = "org-type" > .^< / span > .5, < span class = "org-string" > '--'< / span > , < span class = "org-string" > 'DisplayName'< / span > , < span class = "org-string" > 'Estimated'< / span > );
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hold off;
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< span class = "org-type" > set< / span > (< span class = "org-variable-name" > gca< / span > , < span class = "org-string" > 'XScale'< / span > , < span class = "org-string" > 'log'< / span > );
xlabel(< span class = "org-string" > 'Frequency [Hz]'< / span > ); ylabel(< span class = "org-string" > 'CPS [$nm$ rms]'< / span > );
xlim([f_60rpm(2), f_60rpm(end)]);
legend(< span class = "org-string" > 'Location'< / span > , < span class = "org-string" > 'southeast'< / span > );
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< / pre >
< / div >
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< div id = "org96369c9" class = "figure" >
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< p > < img src = "figs/spindle_cps_d_comp_60rpm.png" alt = "spindle_cps_d_comp_60rpm.png" / >
< / p >
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< p > < span class = "figure-number" > Figure 15: < / span > Cumulative Power Spectrum - 60rpm< / p >
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< / div >
< / div >
< / div >
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< div id = "outline-container-org9bcf259" class = "outline-3" >
< h3 id = "org9bcf259" > < span class = "section-number-3" > 5.10< / span > Compute the resulting RMS value [m]< / h3 >
< div class = "outline-text-3" id = "text-5-10" >
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< div class = "org-src-container" >
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< pre class = "src src-matlab" > < span class = "org-type" > figure< / span > ;
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hold on;
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plot(f_1rpm, 1e9< span class = "org-type" > *< / span > cumtrapz(f_1rpm, (pxx_1rpm)), < span class = "org-string" > '--'< / span > , < span class = "org-string" > 'DisplayName'< / span > , < span class = "org-string" > 'Exp. Data'< / span > );
plot(f_1rpm, 1e9< span class = "org-type" > *< / span > (f_1rpm(end)< span class = "org-type" > -< / span > f_1rpm(1))< span class = "org-type" > /< / span > (length(f_1rpm)< span class = "org-type" > -< / span > 1)< span class = "org-type" > *< / span > cumsum(pxx_1rpm), < span class = "org-string" > '--'< / span > , < span class = "org-string" > 'DisplayName'< / span > , < span class = "org-string" > 'Exp. Data'< / span > );
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hold off;
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< span class = "org-type" > set< / span > (< span class = "org-variable-name" > gca< / span > , < span class = "org-string" > 'XScale'< / span > , < span class = "org-string" > 'log'< / span > );
xlabel(< span class = "org-string" > 'Frequency [Hz]'< / span > ); ylabel(< span class = "org-string" > 'CPS [$nm$ rms]'< / span > );
xlim([f_1rpm(2), f_1rpm(end)]);
legend(< span class = "org-string" > 'Location'< / span > , < span class = "org-string" > 'southeast'< / span > );
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< / pre >
< / div >
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< div id = "org48f5cad" class = "figure" >
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< p > < img src = "figs/spindle_cps_d_comp_1rpm.png" alt = "spindle_cps_d_comp_1rpm.png" / >
< / p >
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< p > < span class = "figure-number" > Figure 16: < / span > Cumulative Power Spectrum - 1rpm< / p >
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< / div >
< / div >
< / div >
< / div >
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< div id = "outline-container-orgc7eca20" class = "outline-2" >
< h2 id = "orgc7eca20" > < span class = "section-number-2" > 6< / span > Functions< / h2 >
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< div class = "outline-text-2" id = "text-6" >
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< / div >
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< div id = "outline-container-orgb688035" class = "outline-3" >
< h3 id = "orgb688035" > < span class = "section-number-3" > 6.1< / span > getAsynchronousError< / h3 >
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< div class = "outline-text-3" id = "text-6-1" >
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< p >
< a id = "org34058ab" > < / a >
< / p >
< p >
This Matlab function is accessible < a href = "src/getAsynchronousError.m" > here< / a > .
< / p >
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< div class = "org-src-container" >
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< pre class = "src src-matlab" > < span class = "org-keyword" > function< / span > < span class = "org-variable-name" > [Wxdec]< / span > = < span class = "org-function-name" > getAsynchronousError< / span > (< span class = "org-variable-name" > data< / span > , < span class = "org-variable-name" > NbTurn< / span > )
< span class = "org-matlab-cellbreak" > < span class = "org-comment" > %%< / span > < / span >
L = length(data);
res_per_rev = L< span class = "org-type" > /< / span > NbTurn;
P = 0< span class = "org-type" > :< / span > (res_per_rev< span class = "org-type" > *< / span > NbTurn< span class = "org-type" > -< / span > 1);
Pos = P< span class = "org-type" > '< / span > < span class = "org-type" > *< / span > 360< span class = "org-type" > /< / span > res_per_rev;
< span class = "org-comment" > % Temperature correction< / span >
x1 = myfit2(Pos, data);
< span class = "org-comment" > % Convert data to frequency domain and scale accordingly< / span >
X2 = 2< span class = "org-type" > /< / span > (res_per_rev< span class = "org-type" > *< / span > NbTurn)< span class = "org-type" > *< / span > fft(x1);
f2 = (0< span class = "org-type" > :< / span > L< span class = "org-type" > -< / span > 1)< span class = "org-type" > ./< / span > NbTurn; < span class = "org-comment" > %upr -> once per revolution< / span >
< span class = "org-matlab-cellbreak" > < span class = "org-comment" > %%< / span > < / span >
X2dec = zeros(size(X2));
< span class = "org-comment" > % Get only the non integer data< / span >
X2dec(mod(f2(< span class = "org-type" > :< / span > ), 1) < span class = "org-type" > ~=< / span > 0) = X2(mod(f2(< span class = "org-type" > :< / span > ), 1) < span class = "org-type" > ~=< / span > 0);
Wxdec = real((res_per_rev< span class = "org-type" > *< / span > NbTurn)< span class = "org-type" > /< / span > 2 < span class = "org-type" > *< / span > ifft(X2dec));
< span class = "org-matlab-cellbreak" > < span class = "org-comment" > %%< / span > < / span >
< span class = "org-keyword" > function< / span > < span class = "org-variable-name" > Y< / span > = < span class = "org-function-name" > myfit2< / span > (< span class = "org-variable-name" > x< / span > ,< span class = "org-variable-name" > y< / span > )
A = [x ones(size(x))]< span class = "org-type" > \< / span > y;
a = A(1); b = A(2);
Y = y < span class = "org-type" > -< / span > (a< span class = "org-type" > *< / span > x < span class = "org-type" > +< / span > b);
< span class = "org-keyword" > end< / span >
< span class = "org-keyword" > end< / span >
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< / pre >
< / div >
< / div >
< / div >
< / div >
< / div >
< div id = "postamble" class = "status" >
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< p class = "author" > Author: Dehaeze Thomas< / p >
< p class = "date" > Created: 2020-11-12 jeu. 10:30< / p >
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< / div >
< / body >
< / html >
