%% Clear Workspace and Close figures clear; close all; clc; %% Intialize Laplace variable s = zpk('s'); %% Path for functions, data and scripts addpath('./mat/'); % Path for data %% Colors for the figures colors = colororder; % Location of the Accelerometers % <> % The location of the accelerometers fixed to the micro-station is essential as it defines where the dynamics is measured. % 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. % 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. % Pictures of the accelerometers fixed to the translation stage and to the micro-hexapod are shown in Figure ref:fig:modal_accelerometer_pictures. % 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. % 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). % #+attr_latex: :options [t]{0.60\linewidth} % #+begin_minipage % #+name: fig:modal_location_accelerometers % #+caption: Position of the accelerometers % #+attr_latex: :width 0.99\linewidth :float nil % [[file:figs/modal_location_accelerometers.png]] % #+end_minipage % \hfill % #+attr_latex: :options [b]{0.38\linewidth} % #+begin_minipage % #+begin_scriptsize % #+name: tab:modal_position_accelerometers % #+caption: Positions in mm % #+attr_latex: :environment tabularx :width \linewidth :placement [b] :align Xccc % #+attr_latex: :booktabs t :float nil :center nil % #+RESULTS: % | | $x$ | $y$ | $z$ | % |--------------+------+------+------| % | Low. Granite | -730 | -526 | -951 | % | Low. Granite | -735 | 814 | -951 | % | Low. Granite | 875 | 799 | -951 | % | Low. Granite | 865 | -506 | -951 | % | Up. Granite | -320 | -446 | -786 | % | Up. Granite | -480 | 534 | -786 | % | Up. Granite | 450 | 534 | -786 | % | Up. Granite | 295 | -481 | -786 | % | Translation | -475 | -414 | -427 | % | Translation | -465 | 407 | -427 | % | Translation | 475 | 424 | -427 | % | Translation | 475 | -419 | -427 | % | Tilt | -385 | -300 | -417 | % | Tilt | -420 | 280 | -417 | % | Tilt | 420 | 280 | -417 | % | Tilt | 380 | -300 | -417 | % | Spindle | -155 | -90 | -594 | % | Spindle | 0 | 180 | -594 | % | Spindle | 155 | -90 | -594 | % | Hexapod | -64 | -64 | -270 | % | Hexapod | -64 | 64 | -270 | % | Hexapod | 64 | 64 | -270 | % | Hexapod | 64 | -64 | -270 | % #+end_scriptsize % #+end_minipage % #+name: fig:modal_accelerometer_pictures % #+caption: Accelerometers fixed on the micro-station stages % #+attr_latex: :options [htbp] % #+begin_figure % #+attr_latex: :caption \subcaption{\label{fig:modal_accelerometers_ty} $T_y$ stage} % #+attr_latex: :options {0.49\textwidth} % #+begin_subfigure % #+attr_latex: :height 6cm % [[file:figs/modal_accelerometers_ty.jpg]] % #+end_subfigure % #+attr_latex: :caption \subcaption{\label{fig:modal_accelerometers_hexapod} Micro-Hexapod} % #+attr_latex: :options {0.49\textwidth} % #+begin_subfigure % #+attr_latex: :height 6cm % [[file:figs/modal_accelerometers_hexapod.jpg]] % #+end_subfigure % #+end_figure %% Load Accelerometer positions acc_pos = readtable('mat/acc_pos.txt', 'ReadVariableNames', false); acc_pos = table2array(acc_pos(:, 1:4)); [~, i] = sort(acc_pos(:, 1)); acc_pos = acc_pos(i, 2:4); % Force and Response signals % <> % The force sensor of the instrumented hammer and the accelerometers signals are shown in the time domain in Figure ref:fig:modal_raw_meas. % Sharp "impacts" can be seen for the force sensor, indicating wide frequency band excitation. % For the accelerometer, a much more complex signal can be observed, indicating complex dynamics. % The "normalized" acrfull:asd of the two signals are computed and shown in Figure ref:fig:modal_asd_acc_force. % 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). % Similar results are obtained for all the measured frequency response functions. %% Load raw data meas1_raw = load('mat/meas_raw_1.mat'); % Sampling Frequency [Hz] Fs = 1/meas1_raw.Track1_X_Resolution; % Time just before the impact occurs [s] impacts = [5.937, 11.228, 16.681, 22.205, 27.350, 32.714, 38.115, 43.888, 50.407]-0.01; % Time vector [s] time = linspace(0, meas1_raw.Track1_X_Resolution*length(meas1_raw.Track1), length(meas1_raw.Track1)); %% Raw measurement of the Accelerometer figure; hold on; plot(time-22.2, meas1_raw.Track2, 'DisplayName', '$X_{j}$ [$m/s^2$]'); plot(time-22.2, 1e-3*meas1_raw.Track1, 'DisplayName', '$F_{k}$ [kN]'); hold off; xlabel('Time [s]'); ylabel('Amplitude'); xlim([0, 0.2]); ylim([-2, 2]); legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 1); %% Frequency Analysis Nfft = floor(5.0*Fs); % Number of frequency points win = hanning(Nfft); % Windowing Noverlap = floor(Nfft/2); % Overlap for frequency analysis %% Comnpute the power spectral density of the force and acceleration [pxx_force, f] = pwelch(meas1_raw.Track1, win, Noverlap, Nfft, Fs); [pxx_acc, ~] = pwelch(meas1_raw.Track2, win, Noverlap, Nfft, Fs); %% Normalized Amplitude Spectral Density of the measured force and acceleration figure; hold on; plot(f, sqrt(pxx_acc./max(pxx_acc(f<200))), 'DisplayName', '$X_{j}$'); plot(f, sqrt(pxx_force./max(pxx_force(f<200))), 'DisplayName', '$F_{k}$'); hold off; set(gca, 'XScale', 'lin'); set(gca, 'YScale', 'lin'); xlabel('Frequency [Hz]'); ylabel('Normalized Spectral Density'); xlim([0, 200]); xticks([0:20:200]); ylim([0, 1]) legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 1); % #+name: fig:modal_raw_meas_asd % #+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}) % #+attr_latex: :options [htbp] % #+begin_figure % #+attr_latex: :caption \subcaption{\label{fig:modal_raw_meas}Time domain signals} % #+attr_latex: :options {0.49\textwidth} % #+begin_subfigure % #+attr_latex: :width 0.95\linewidth % [[file:figs/modal_raw_meas.png]] % #+end_subfigure % #+attr_latex: :caption \subcaption{\label{fig:modal_asd_acc_force}Amplitude Spectral Density (normalized)} % #+attr_latex: :options {0.49\textwidth} % #+begin_subfigure % #+attr_latex: :width 0.95\linewidth % [[file:figs/modal_asd_acc_force.png]] % #+end_subfigure % #+end_figure % 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. % The quality of the obtained data can be estimated using the /coherence/ function, which is shown in Figure ref:fig:modal_coh_acc_force. % Good coherence is obtained from $20\,\text{Hz}$ to $200\,\text{Hz}$ which corresponds to the frequency range of interest. %% Compute the transfer function and Coherence [G1, f] = tfestimate(meas1_raw.Track1, meas1_raw.Track2, win, Noverlap, Nfft, Fs); [coh1, ~] = mscohere( meas1_raw.Track1, meas1_raw.Track2, win, Noverlap, Nfft, Fs); %% Frequency Response Function between the force and the acceleration figure; plot(f, abs(G1)); xlabel('Frequency [Hz]'); ylabel('FRF [$m/s^2/N$]') set(gca, 'XScale', 'lin'); set(gca, 'YScale', 'log'); xlim([0, 200]); xticks([0:20:200]); %% Frequency Response Function between the force and the acceleration figure; plot(f, coh1); xlabel('Frequency [Hz]'); ylabel('Coherence [-]') set(gca, 'XScale', 'lin'); set(gca, 'YScale', 'lin'); xlim([0, 200]); ylim([0,1]); xticks([0:20:200]);