%% Clear Workspace and Close figures clear; close all; clc; %% Intialize Laplace variable s = zpk('s'); % We then import that on =matlab=, and sort them. 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); % The positions of the sensors relative to the point of interest are shown below. data2orgtable([[1:23]', 1000*acc_pos], {}, {'ID', 'x [mm]', 'y [mm]', 'z [mm]'}, ' %.0f '); % Windowing % Windowing is used on the force and response signals. % A boxcar window (figure [[fig:windowing_force_signal]]) is used for the force signal as once the impact on the structure is done, the measured signal is meaningless. % The parameters are: % - *Start*: $3\%$ % - *Stop*: $7\%$ figure; plot(100*[0, 0.03, 0.03, 0.07, 0.07, 1], [0, 0, 1, 1, 0, 0]); xlabel('Time [%]'); ylabel('Amplitude'); xlim([0, 100]); ylim([0, 1]); % #+NAME: fig:windowing_force_signal % #+CAPTION: Window used for the force signal % [[file:figs/windowing_force_signal.png]] % An exponential window (figure [[fig:windowing_response_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. % The parameters are: % - FlatTop: % - *Start*: $3\%$ % - *Stop*: $2.96\%$ % - Decreasing point: % - *X*: $60.4\%$ % - *Y*: $14.7\%$ x0 = 0.296; xd = 0.604; yd = 0.147; alpha = log(yd)/(x0 - xd); t = x0:0.01:1.01; y = exp(-alpha*(t-x0)); figure; plot(100*[0, 0.03, 0.03, x0, t], [0, 0, 1, 1, y]); xlabel('Time [%]'); ylabel('Amplitude'); xlim([0, 100]); ylim([0, 1]); % Force and Response signals % Let's load some obtained data to look at the force and response signals. meas1_raw = load('mat/meas_raw_1.mat'); % Raw Force Data % The force input for the first measurement is shown on figure [[fig:raw_data_force]]. We can see 10 impacts, one zoom on one impact is shown on figure [[fig:raw_data_force_zoom]]. % The Fourier transform of the force is shown on figure [[fig:fourier_transfor_force_impact]]. This has been obtained without any windowing. time = linspace(0, meas1_raw.Track1_X_Resolution*length(meas1_raw.Track1), length(meas1_raw.Track1)); figure; plot(time, meas1_raw.Track1); xlabel('Time [s]'); ylabel('Force [N]'); % #+NAME: fig:raw_data_force % #+CAPTION: Raw Force Data from Hammer Blow % [[file:figs/raw_data_force.png]] figure; plot(time, meas1_raw.Track1); xlabel('Time [s]'); ylabel('Force [N]'); xlim([22.1, 22.3]); % #+NAME: fig:raw_data_force_zoom % #+CAPTION: Raw Force Data from Hammer Blow - Zoom % [[file:figs/raw_data_force_zoom.png]] Fs = 1/meas1_raw.Track1_X_Resolution; % Sampling Frequency [Hz] impacts = [5.9, 11.2, 16.6, 22.2, 27.3, 32.7, 38.1, 43.8, 50.4]; % Time just before the impact occurs [s] L = 8194; f = Fs*(0:(L/2))/L; % Frequency vector [Hz] F_fft = zeros((8193+1)/2+1, length(impacts)); for i = 1:length(impacts) t0 = impacts(i); [~, i_start] = min(abs(time-t0)); i_end = i_start + 8193; Y = fft(meas1_raw.Track1(i_start:i_end)); P2 = abs(Y/L); P1 = P2(1:L/2+1); P1(2:end-1) = 2*P1(2:end-1); F_fft(:, i) = P1; end figure; hold on; for i = 1:length(impacts) plot(f, F_fft(:, i), '-k'); end hold off; xlim([0, 200]); xlabel('Frequency [Hz]'); ylabel('Force [N]'); % Raw Response Data % The response signal for the first measurement is shown on figure [[fig:raw_data_acceleration]]. One zoom on one response is shown on figure [[fig:raw_data_acceleration_zoom]]. % The Fourier transform of the response signals is shown on figure [[fig:fourier_transform_response_signals]]. This has been obtained without any windowing. figure; plot(time, meas1_raw.Track2); xlabel('Time [s]'); ylabel('Acceleration [m/s2]'); % #+NAME: fig:raw_data_acceleration % #+CAPTION: Raw Acceleration Data from Accelerometer % [[file:figs/raw_data_acceleration.png]] figure; plot(time, meas1_raw.Track2); xlabel('Time [s]'); ylabel('Acceleration [m/s2]'); xlim([22.1, 22.5]); % #+NAME: fig:raw_data_acceleration_zoom % #+CAPTION: Raw Acceleration Data from Accelerometer - Zoom % [[file:figs/raw_data_acceleration_zoom.png]] X_fft = zeros((8193+1)/2+1, length(impacts)); for i = 1:length(impacts) t0 = impacts(i); [~, i_start] = min(abs(time-t0)); i_end = i_start + 8193; Y = fft(meas1_raw.Track2(i_start:i_end)); P2 = abs(Y/L); P1 = P2(1:L/2+1); P1(2:end-1) = 2*P1(2:end-1); X_fft(:, i) = P1; end figure; hold on; for i = 1:length(impacts) plot(f, X_fft(:, i), '-k'); end hold off; xlim([0, 200]); set(gca, 'Yscale', 'log'); xlabel('Frequency [Hz]'); ylabel('Acceleration [$m/s^2$]'); % Computation of one Frequency Response Function % Now that we have obtained the Fourier transform of both the force input and the response signal, we can compute the Frequency Response Function from the force to the acceleration. % We then compare the result obtained with the FRF computed by the modal software (figure [[fig:frf_comparison_software]]). % The slight difference can probably be explained by the use of windows. % In the following analysis, FRF computed from the software will be used. meas1 = load('mat/meas_frf_coh_1.mat'); figure; hold on; for i = 1:length(impacts) plot(f, X_fft(:, i)./F_fft(:, i), '-k'); end plot(meas1.FFT1_AvSpc_2_RMS_X_Val, meas1.FFT1_AvSpc_2_RMS_Y_Val) hold off; xlim([0, 200]); set(gca, 'Yscale', 'log'); xlabel('Frequency [Hz]'); ylabel('Acceleration/Force [$m/s^2/N$]'); % Frequency Response Functions and Coherence Results % Let's see one computed Frequency Response Function and one coherence in order to attest the quality of the measurement. % First, we load the data. meas1 = load('mat/meas_frf_coh_1.mat'); % And we plot on figure [[fig:frf_result_example]] the frequency response function from the force applied in the $X$ direction at the location of the accelerometer number 11 to the acceleration in the $X$ direction as measured by the first accelerometer located on the top platform of the hexapod. % The coherence associated is shown on figure [[fig:frf_result_example]]. figure; ax1 = subplot(2, 1, 1); plot(meas1.FFT1_AvSpc_2_RMS_X_Val, meas1.FFT1_AvXSpc_2_1_RMS_Y_Mod); set(gca, 'Yscale', 'log'); set(gca, 'XTickLabel',[]); ylabel('Magnitude'); ax2 = subplot(2, 1, 2); plot(meas1.FFT1_AvSpc_2_RMS_X_Val, meas1.FFT1_AvXSpc_2_1_RMS_Y_Phas); ylim([-180, 180]); yticks([-180, -90, 0, 90, 180]); xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); linkaxes([ax1,ax2],'x'); % #+NAME: fig:frf_result_example % #+CAPTION: Example of one measured FRF % [[file:figs/frf_result_example.png]] figure; plot(meas1.FFT1_AvSpc_2_RMS_X_Val, meas1.FFT1_Coh_2_1_RMS_Y_Val); xlabel('Frequency [Hz]'); ylabel('Coherence'); % Generation of a FRF matrix and a Coherence matrix from the measurements % We want here to combine all the Frequency Response Functions measured into one big array called the *Frequency Response Matrix*. % The frequency response matrix is an $n \times p \times q$: % - $n$ is the number of measurements: $23 \times 3$ (23 accelerometers measuring 3 directions each) % - $p$ is the number of excitation inputs: $3$ % - $q$ is the number of frequency points $\omega_i$ % Thus, the FRF matrix is an $69 \times 3 \times 801$ matrix. % #+begin_important % For each frequency point $\omega_i$, we obtain a 2D matrix: % \begin{equation} % \text{FRF}(\omega_i) = \begin{bmatrix} % \frac{D_{1_x}}{F_x}(\omega_i) & \frac{D_{1_x}}{F_y}(\omega_i) & \frac{D_{1_x}}{F_z}(\omega_i) \\ % \frac{D_{1_y}}{F_x}(\omega_i) & \frac{D_{1_y}}{F_y}(\omega_i) & \frac{D_{1_y}}{F_z}(\omega_i) \\ % \frac{D_{1_z}}{F_x}(\omega_i) & \frac{D_{1_z}}{F_y}(\omega_i) & \frac{D_{1_z}}{F_z}(\omega_i) \\ % \frac{D_{2_x}}{F_x}(\omega_i) & \frac{D_{2_x}}{F_y}(\omega_i) & \frac{D_{2_x}}{F_z}(\omega_i) \\ % \vdots & \vdots & \vdots \\ % \frac{D_{23_z}}{F_x}(\omega_i) & \frac{D_{23_z}}{F_y}(\omega_i) & \frac{D_{23_z}}{F_z}(\omega_i) \\ % \end{bmatrix} % \end{equation} % #+end_important % We generate such FRF matrix from the measurements using the following script. n_meas = 24; n_acc = 23; dirs = 'XYZ'; % Number of Accelerometer * DOF for each acccelerometer / Number of excitation / frequency points FRFs = zeros(3*n_acc, 3, 801); COHs = zeros(3*n_acc, 3, 801); % Loop through measurements for i = 1:n_meas % Load the measurement file meas = load(sprintf('mat/meas_frf_coh_%i.mat', i)); % First: determine what is the exitation (direction and sign) exc_dir = meas.FFT1_AvXSpc_2_1_RMS_RfName(end); exc_sign = meas.FFT1_AvXSpc_2_1_RMS_RfName(end-1); % Determine what is the correct excitation sign exc_factor = str2num([exc_sign, '1']); if exc_dir ~= 'Z' exc_factor = exc_factor*(-1); end % Then: loop through the nine measurements and store them at the correct location for j = 2:10 % Determine what is the accelerometer and direction [indices_acc_i] = strfind(meas.(sprintf('FFT1_H1_%i_1_RpName', j)), '.'); acc_i = str2num(meas.(sprintf('FFT1_H1_%i_1_RpName', j))(indices_acc_i(1)+1:indices_acc_i(2)-1)); meas_dir = meas.(sprintf('FFT1_H1_%i_1_RpName', j))(end); meas_sign = meas.(sprintf('FFT1_H1_%i_1_RpName', j))(end-1); % Determine what is the correct measurement sign meas_factor = str2num([meas_sign, '1']); if meas_dir ~= 'Z' meas_factor = meas_factor*(-1); end FRFs(find(dirs==meas_dir)+3*(acc_i-1), find(dirs==exc_dir), :) = exc_factor*meas_factor*meas.(sprintf('FFT1_H1_%i_1_Y_ReIm', j)); COHs(find(dirs==meas_dir)+3*(acc_i-1), find(dirs==exc_dir), :) = meas.(sprintf('FFT1_Coh_%i_1_RMS_Y_Val', j)); end end freqs = meas.FFT1_Coh_10_1_RMS_X_Val; % And we save the obtained FRF matrix and Coherence matrix in a =.mat= file. save('./mat/frf_coh_matrices.mat', 'FRFs', 'COHs', 'freqs'); % Solid Bodies considered for further analysis % We consider the following solid bodies for further analysis: % - Bottom Granite % - Top Granite % - Translation Stage % - Tilt Stage % - Spindle % - Hexapod % We create a =matlab= structure =solids= that contains the accelerometers ID connected to each solid bodies (as shown on figure [[fig:nass-modal-test]]). solids = {}; solids.granite_bot = [17, 18, 19, 20]; solids.granite_top = [13, 14, 15, 16]; solids.ty = [9, 10, 11, 12]; solids.ry = [5, 6, 7, 8]; solids.rz = [21, 22, 23]; solids.hexa = [1, 2, 3, 4]; solid_names = fields(solids); % Finally, we save that into a =.mat= file. save('mat/geometry.mat', 'solids', 'solid_names', 'acc_pos');