237 lines
7.7 KiB
Matlab
237 lines
7.7 KiB
Matlab
%% Clear Workspace and Close figures
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clear; close all; clc;
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%% Intialize Laplace variable
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s = zpk('s');
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%% Path for functions, data and scripts
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addpath('./mat/'); % Path for data
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%% Colors for the figures
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colors = colororder;
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% Location of the Accelerometers
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% <<ssec:modal_accelerometers>>
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% 4 tri-axis accelerometers are used for each solid body.
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% Only 2 could have been used as only 6DOF have to be measured, however, we have chosen to have some *redundancy*.
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% This could also help us identify measurement problems or flexible modes is present.
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% The position of the accelerometers are:
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% - 4 on the first granite
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% - 4 on the second granite
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% - 4 on top of the translation stage (figure ref:fig:accelerometers_ty)
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% - 4 on top of the tilt stage
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% - 3 on top of the spindle
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% - 4 on top of the hexapod (figure ref:fig:accelerometers_hexapod)
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% In total, 23 accelerometers are used: *69 DOFs are thus measured*.
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% The precise determination of the position of each accelerometer is done using the SolidWorks model (shown on figure ref:fig:location_accelerometers).
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% #+name: fig:accelerometer_pictures
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% #+caption: Accelerometers fixed on the micro-station
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% #+begin_figure
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% #+attr_latex: :caption \subcaption{\label{fig:accelerometers_ty}$T_y$ stage}
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% #+attr_latex: :options {0.49\textwidth}
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% #+begin_subfigure
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% #+attr_latex: :height 6cm
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% [[file:figs/accelerometers_ty.jpg]]
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% #+end_subfigure
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% #+attr_latex: :caption \subcaption{\label{fig:accelerometers_hexapod}Micro-Hexapod}
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% #+attr_latex: :options {0.49\textwidth}
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% #+begin_subfigure
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% #+attr_latex: :height 6cm
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% [[file:figs/accelerometers_hexapod.jpg]]
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% #+end_subfigure
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% #+end_figure
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% #+name: fig:location_accelerometers
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% #+caption: Position of the accelerometers using SolidWorks
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% #+attr_latex: :width \linewidth
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% [[file:figs/location_accelerometers.png]]
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% The precise position of all the 23 accelerometer with respect to a frame located at the point of interest (located 270mm above the top platform of the hexapod) are shown in table ref:tab:position_accelerometers.
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%% Load Accelerometer positions
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acc_pos = readtable('mat/acc_pos.txt', 'ReadVariableNames', false);
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acc_pos = table2array(acc_pos(:, 1:4));
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[~, i] = sort(acc_pos(:, 1));
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acc_pos = acc_pos(i, 2:4);
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% Signal Processing :noexport:
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% <<ssec:modal_signal_processing>>
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% The measurements are averaged 10 times corresponding to 10 hammer impacts in order to reduce the effect of random noise.
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% Windowing is also used on the force and response signals.
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% A boxcar window (figure ref:fig:modal_windowing_force_signal) is used for the force signal as once the impact on the structure is done, the measured signal is meaningless.
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% The parameters are:
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% - *Start*: $3\%$
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% - *Stop*: $7\%$
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%% Boxcar window used for the force signal
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figure;
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plot(100*[0, 0.03, 0.03, 0.07, 0.07, 1], [0, 0, 1, 1, 0, 0]);
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xlabel('Time [\%]'); ylabel('Amplitude');
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xlim([0, 100]); ylim([0, 1]);
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% #+name: fig:modal_windowing_force_signal
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% #+caption: Boxcar window used for the force signal
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% #+RESULTS:
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% [[file:figs/modal_windowing_force_signal.png]]
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% An exponential window (figure ref:fig:modal_windowing_acc_signal) is used for the response signal as we are measuring transient signals and most of the information is located at the beginning of the signal.
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% The parameters are:
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% - FlatTop:
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% - *Start*: $3\%$
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% - *Stop*: $2.96\%$
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% - Decreasing point:
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% - *X*: $60.4\%$
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% - *Y*: $14.7\%$
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%% Exponential window used for acceleration signal
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x0 = 0.296;
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xd = 0.604;
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yd = 0.147;
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alpha = log(yd)/(x0 - xd);
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t = x0:0.01:1.01;
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y = exp(-alpha*(t-x0));
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figure;
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plot(100*[0, 0.03, 0.03, x0, t], [0, 0, 1, 1, y]);
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xlabel('Time [\%]'); ylabel('Amplitude');
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xlim([0, 100]); ylim([0, 1]);
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% Force and Response signals
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% <<ssec:modal_measured_signals>>
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%% Load raw data
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meas1_raw = load('mat/meas_raw_1.mat');
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% Sampling Frequency [Hz]
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Fs = 1/meas1_raw.Track1_X_Resolution;
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% Time just before the impact occurs [s]
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impacts = [5.937, 11.228, 16.681, 22.205, 27.350, 32.714, 38.115, 43.888, 50.407]-0.01;
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% Time vector [s]
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time = linspace(0, meas1_raw.Track1_X_Resolution*length(meas1_raw.Track1), length(meas1_raw.Track1));
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% The force sensor and the accelerometers signals are shown in the time domain in Figure ref:fig:modal_raw_meas.
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% Sharp "impacts" can be seen for the force sensor, indicating wide frequency band excitation.
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% For the accelerometer, many resonances can be seen on the right, indicating complex dynamics
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%% Raw measurement of the Accelerometer
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figure;
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tiledlayout(1, 3, 'TileSpacing', 'Compact', 'Padding', 'None');
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ax1 = nexttile([1,2]);
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hold on;
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plot(time, meas1_raw.Track2, 'DisplayName', 'Acceleration [$m/s^2$]');
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plot(time, 1e-3*meas1_raw.Track1, 'DisplayName', 'Force [kN]');
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hold off;
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xlabel('Time [s]');
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ylabel('Amplitude');
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xlim([0, time(end)]);
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legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 1);
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ax2 = nexttile();
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hold on;
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plot(time, meas1_raw.Track2);
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plot(time, 1e-3*meas1_raw.Track1);
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hold off;
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xlabel('Time [s]');
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set(gca, 'YTickLabel',[]);
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xlim([22.19, 22.4]);
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linkaxes([ax1,ax2],'y');
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ylim([-2, 2]);
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% #+name: fig:modal_raw_meas
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% #+caption: Raw measurement of the acceleromter (blue) and of the force sensor at the Hammer tip (red). Zoom on one impact is shown on the right.
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% #+RESULTS:
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% [[file:figs/modal_raw_meas.png]]
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%% Frequency Analysis
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Nfft = floor(5.0*Fs); % Number of frequency points
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win = hanning(Nfft); % Windowing
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Noverlap = floor(Nfft/2); % Overlap for frequency analysis
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%% Comnpute the power spectral density of the force and acceleration
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[pxx_force, f] = pwelch(meas1_raw.Track1, win, Noverlap, Nfft, Fs);
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[pxx_acc, ~] = pwelch(meas1_raw.Track2, win, Noverlap, Nfft, Fs);
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% The "normalized" amplitude spectral density of the two signals are computed and shown in Figure ref:fig:modal_asd_acc_force.
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% Conclusions based on the time domain signals can be clearly seen in the frequency domain (wide frequency content for the force signal and complex dynamics for the accelerometer).
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%% Normalized Amplitude Spectral Density of the measured force and acceleration
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figure;
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hold on;
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plot(f, sqrt(pxx_force./max(pxx_force(f<200))), 'DisplayName', 'Force');
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plot(f, sqrt(pxx_acc./max(pxx_acc(f<200))), 'DisplayName', 'Acceleration');
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hold off;
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set(gca, 'XScale', 'lin'); set(gca, 'YScale', 'lin');
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xlabel('Frequency [Hz]'); ylabel('Normalized Spectral Density');
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xlim([0, 200]);
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xticks([0:20:200]);
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ylim([0, 1])
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legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 1);
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% #+name: fig:modal_asd_acc_force
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% #+caption: Normalized Amplitude Spectral Density of the measured force and acceleration
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% #+RESULTS:
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% [[file:figs/modal_asd_acc_force.png]]
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% The frequency response function from the applied force to the measured acceleration can then be computed (Figure ref:fig:modal_frf_acc_force).
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%% Compute the transfer function and Coherence
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[G1, f] = tfestimate(meas1_raw.Track1, meas1_raw.Track2, win, Noverlap, Nfft, Fs);
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[coh1, ~] = mscohere( meas1_raw.Track1, meas1_raw.Track2, win, Noverlap, Nfft, Fs);
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%% Frequency Response Function between the force and the acceleration
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figure;
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plot(f, abs(G1));
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xlabel('Frequency [Hz]'); ylabel('FRF [$m/s^2/N$]')
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set(gca, 'XScale', 'lin'); set(gca, 'YScale', 'log');
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xlim([0, 200]);
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xticks([0:20:200]);
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% #+name: fig:modal_frf_acc_force
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% #+caption: Frequency Response Function between the measured force and acceleration
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% #+RESULTS:
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% [[file:figs/modal_frf_acc_force.png]]
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% The coherence between the input and output signals is also computed and found to be good between 20 and 200Hz (Figure ref:fig:modal_coh_acc_force).
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%% Frequency Response Function between the force and the acceleration
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figure;
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plot(f, coh1);
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xlabel('Frequency [Hz]'); ylabel('Coherence [-]')
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set(gca, 'XScale', 'lin'); set(gca, 'YScale', 'lin');
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xlim([0, 200]); ylim([0,1]);
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xticks([0:20:200]);
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