195 lines
8.0 KiB
Matlab
195 lines
8.0 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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% The location of the accelerometers fixed to the micro-station is essential as it defines where the dynamics is measured.
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% A total of 23 accelerometers are fixed to the six key stages of the micro station: the lower and upper granites, the translation stage, the tilt stage, the spindle and the micro hexapod.
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% The position of the accelerometers are visually shown on a CAD model in Figure ref:fig:modal_location_accelerometers and their precise locations with respect to a frame located at the point of interest are summarized in Table ref:tab:modal_position_accelerometers.
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% Pictures of the accelerometers fixed to the translation stage and to the micro-hexapod are shown in Figure ref:fig:modal_accelerometer_pictures.
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% As all key stages of the micro-station are foreseen to behave as solid bodies, only 6 acrshort:dof can be considered per solid body.
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% However, it was chosen to use four 3-axis accelerometers (i.e. 12 measured acrshort:dof) for each considered solid body to have some redundancy and to be able to verify the solid body assumption (see Section ref:ssec:modal_solid_body_assumption).
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% #+attr_latex: :options [t]{0.60\linewidth}
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% #+begin_minipage
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% #+name: fig:modal_location_accelerometers
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% #+caption: Position of the accelerometers
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% #+attr_latex: :width 0.99\linewidth :float nil
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% [[file:figs/modal_location_accelerometers.png]]
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% #+end_minipage
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% \hfill
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% #+attr_latex: :options [b]{0.38\linewidth}
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% #+begin_minipage
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% #+begin_scriptsize
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% #+name: tab:modal_position_accelerometers
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% #+caption: Positions in mm
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% #+attr_latex: :environment tabularx :width \linewidth :placement [b] :align Xccc
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% #+attr_latex: :booktabs t :float nil :center nil
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% #+RESULTS:
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% | | $x$ | $y$ | $z$ |
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% |--------------+------+------+------|
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% | Low. Granite | -730 | -526 | -951 |
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% | Low. Granite | -735 | 814 | -951 |
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% | Low. Granite | 875 | 799 | -951 |
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% | Low. Granite | 865 | -506 | -951 |
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% | Up. Granite | -320 | -446 | -786 |
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% | Up. Granite | -480 | 534 | -786 |
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% | Up. Granite | 450 | 534 | -786 |
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% | Up. Granite | 295 | -481 | -786 |
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% | Translation | -475 | -414 | -427 |
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% | Translation | -465 | 407 | -427 |
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% | Translation | 475 | 424 | -427 |
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% | Translation | 475 | -419 | -427 |
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% | Tilt | -385 | -300 | -417 |
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% | Tilt | -420 | 280 | -417 |
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% | Tilt | 420 | 280 | -417 |
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% | Tilt | 380 | -300 | -417 |
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% | Spindle | -155 | -90 | -594 |
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% | Spindle | 0 | 180 | -594 |
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% | Spindle | 155 | -90 | -594 |
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% | Hexapod | -64 | -64 | -270 |
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% | Hexapod | -64 | 64 | -270 |
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% | Hexapod | 64 | 64 | -270 |
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% | Hexapod | 64 | -64 | -270 |
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% #+end_scriptsize
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% #+end_minipage
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% #+name: fig:modal_accelerometer_pictures
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% #+caption: Accelerometers fixed on the micro-station stages
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% #+attr_latex: :options [htbp]
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% #+begin_figure
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% #+attr_latex: :caption \subcaption{\label{fig:modal_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/modal_accelerometers_ty.jpg]]
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% #+end_subfigure
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% #+attr_latex: :caption \subcaption{\label{fig:modal_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/modal_accelerometers_hexapod.jpg]]
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% #+end_subfigure
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% #+end_figure
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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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% Force and Response signals
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% <<ssec:modal_measured_signals>>
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% The force sensor of the instrumented hammer 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, a much more complex signal can be observed, indicating complex dynamics.
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% The "normalized" acrfull:asd 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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% Similar results are obtained for all the measured frequency response functions.
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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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%% Raw measurement of the Accelerometer
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figure;
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hold on;
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plot(time-22.2, meas1_raw.Track2, 'DisplayName', '$X_{j}$ [$m/s^2$]');
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plot(time-22.2, 1e-3*meas1_raw.Track1, 'DisplayName', '$F_{k}$ [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, 0.2]);
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ylim([-2, 2]);
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legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 1);
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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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%% 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_acc./max(pxx_acc(f<200))), 'DisplayName', '$X_{j}$');
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plot(f, sqrt(pxx_force./max(pxx_force(f<200))), 'DisplayName', '$F_{k}$');
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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_raw_meas_asd
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% #+caption: Raw measurement of the acceleromter (blue) and of the force sensor at the Hammer tip (red) (\subref{fig:modal_raw_meas}). Computed Amplitude Spectral Density of the two signals (normalized) (\subref{fig:modal_asd_acc_force})
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% #+attr_latex: :options [htbp]
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% #+begin_figure
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% #+attr_latex: :caption \subcaption{\label{fig:modal_raw_meas}Time domain signals}
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% #+attr_latex: :options {0.49\textwidth}
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% #+begin_subfigure
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% #+attr_latex: :width 0.95\linewidth
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% [[file:figs/modal_raw_meas.png]]
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% #+end_subfigure
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% #+attr_latex: :caption \subcaption{\label{fig:modal_asd_acc_force}Amplitude Spectral Density (normalized)}
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% #+attr_latex: :options {0.49\textwidth}
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% #+begin_subfigure
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% #+attr_latex: :width 0.95\linewidth
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% [[file:figs/modal_asd_acc_force.png]]
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% #+end_subfigure
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% #+end_figure
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% The frequency response function $H_{jk}$ from the applied force $F_{k}$ to the measured acceleration $X_j$ is then computed and shown Figure ref:fig:modal_frf_acc_force.
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% The quality of the obtained data can be estimated using the /coherence/ function, which is shown in Figure ref:fig:modal_coh_acc_force.
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% Good coherence is obtained from $20\,\text{Hz}$ to $200\,\text{Hz}$ which corresponds to the frequency range of interest.
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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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%% 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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