272 lines
9.4 KiB
Mathematica
272 lines
9.4 KiB
Mathematica
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%% 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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% Singular Value Decomposition - Modal Indication Function
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% The Mode Indicator Functions are usually used on $n\times p$ FRF matrix where $n$ is a relatively large number of measurement DOFs and $p$ is the number of excitation DOFs, typically 3 or 4.
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% In these methods, the frequency dependent FRF matrix is subjected to a singular value decomposition analysis which thus yields a small number (3 or 4) of singular values, these also being frequency dependent.
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% These methods are used to *determine the number of modes* present in a given frequency range, to *identify repeated natural frequencies* and to pre-process the FRF data prior to modal analysis.
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% From the documentation of the modal software:
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% #+begin_quote
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% The MIF consist of the singular values of the Frequency response function matrix. The number of MIFs equals the number of excitations.
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% By the powerful singular value decomposition, the real signal space is separated from the noise space. Therefore, the MIFs exhibit the modes effectively.
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% A peak in the MIFs plot usually indicate the existence of a structural mode, and two peaks at the same frequency point means the existence of two repeated modes.
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% Moreover, the magnitude of the MIFs implies the strength of the a mode.
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% #+end_quote
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% #+begin_important
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% The *Complex Mode Indicator Function* is defined simply by the SVD of the FRF (sub) matrix:
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% \begin{align*}
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% [H(\omega)]_{n\times p} &= [U(\omega)]_{n\times n} [\Sigma(\omega)]_{n\times p} [V(\omega)]_{p\times p}^H\\
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% [CMIF(\omega)]_{p\times p} &= [\Sigma(\omega)]_{p\times n}^T [\Sigma(\omega)]_{n\times p}
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% \end{align*}
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% #+end_important
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% We compute the Complex Mode Indicator Function.
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% The result is shown on Figure ref:fig:modal_indication_function.
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%% Computation of the modal indication function
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MIF = zeros(size(frf, 2), size(frf, 2), size(frf, 3));
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for i = 1:length(freqs)
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[~,S,~] = svd(frf(:, :, i));
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MIF(:, :, i) = S'*S;
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end
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%% Modal Indication Function
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figure;
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hold on;
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for i = 1:size(MIF, 1)
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plot(freqs, squeeze(MIF(i, i, :)));
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end
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hold off;
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set(gca, 'Xscale', 'lin'); set(gca, 'Yscale', 'log');
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xlabel('Frequency [Hz]'); ylabel('CMIF Amplitude');
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xticks([0:20:200]);
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xlim([0, 200]);
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ylim([1e-6, 2e-2]);
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% Composite Response Function
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% An alternative is the Composite Response Function $HH(\omega)$ defined as the sum of all the measured FRF:
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% \begin{equation}
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% HH(\omega) = \sum_j\sum_kH_{jk}(\omega)
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% \end{equation}
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% Instead, we choose here to use the sum of the norms of the measured frf:
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% \begin{equation}
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% HH(\omega) = \sum_j\sum_k \left|H_{jk}(\omega) \right|
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% \end{equation}
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% The result is shown on figure ref:fig:modal_composite_reponse_function.
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%% Composite Response Function
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figure;
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hold on;
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plot(freqs, squeeze(sum(sum(abs(frf)))), '-k');
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hold off;
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xlabel('Frequency [Hz]'); ylabel('Amplitude');
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xlim([0, 200]);
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xticks([0:20:200]);
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% Importation of the modal parameters on Matlab
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% The obtained modal parameters are:
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% - Resonance frequencies in Hertz
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% - Modal damping ratio in percentage
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% - (complex) Modes shapes for each measured DoF
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% - Modal A and modal B which are parameters important for further normalization
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%% Load modal parameters
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shapes_m = readtable('mat/mode_shapes.txt', 'ReadVariableNames', false); % [Sign / Real / Imag]
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freqs_m = table2array(readtable('mat/mode_freqs.txt', 'ReadVariableNames', false)); % in [Hz]
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damps_m = table2array(readtable('mat/mode_damps.txt', 'ReadVariableNames', false)); % in [%]
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modal_a = table2array(readtable('mat/mode_modal_a.txt', 'ReadVariableNames', false)); % [Real / Imag]
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modal_b = table2array(readtable('mat/mode_modal_b.txt', 'ReadVariableNames', false)); % [Real / Imag]
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%% Guess the number of modes identified from the length of the imported data.
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acc_n = 23; % Number of accelerometers
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dir_n = 3; % Number of directions
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dirs = 'XYZ';
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mod_n = size(shapes_m,1)/acc_n/dir_n; % Number of modes
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%% Mode shapes are split into 3 parts (direction plus sign, real part and imaginary part)
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% we aggregate them into one array of complex numbers
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T_sign = table2array(shapes_m(:, 1));
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T_real = table2array(shapes_m(:, 2));
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T_imag = table2array(shapes_m(:, 3));
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mode_shapes = zeros(mod_n, dir_n, acc_n);
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for mod_i = 1:mod_n
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for acc_i = 1:acc_n
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% Get the correct section of the signs
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T = T_sign(acc_n*dir_n*(mod_i-1)+1:acc_n*dir_n*mod_i);
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for dir_i = 1:dir_n
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% Get the line corresponding to the sensor
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i = find(contains(T, sprintf('%i%s',acc_i, dirs(dir_i))), 1, 'first')+acc_n*dir_n*(mod_i-1);
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mode_shapes(mod_i, dir_i, acc_i) = str2num([T_sign{i}(end-1), '1'])*complex(T_real(i),T_imag(i));
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end
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end
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end
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% Modal Matrices
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% We would like to arrange the obtained modal parameters into two modal matrices:
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% \[ \Lambda = \begin{bmatrix}
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% s_1 & & 0 \\
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% & \ddots & \\
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% 0 & & s_N
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% \end{bmatrix}_{N \times N}; \quad \Psi = \begin{bmatrix}
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% & & \\
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% \{\psi_1\} & \dots & \{\psi_N\} \\
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% & &
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% \end{bmatrix}_{M \times N} \]
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% \[ \{\psi_i\} = \begin{Bmatrix} \psi_{i, 1_x} & \psi_{i, 1_y} & \psi_{i, 1_z} & \psi_{i, 2_x} & \dots & \psi_{i, 23_z} \end{Bmatrix}^T \]
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% $M$ is the number of DoF: here it is $23 \times 3 = 69$.
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% $N$ is the number of mode
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eigen_val_M = diag(2*pi*freqs_m.*(-damps_m/100 + j*sqrt(1 - (damps_m/100).^2)));
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eigen_vec_M = reshape(mode_shapes, [mod_n, acc_n*dir_n]).';
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% Each eigen vector is normalized: $\| \{\psi_i\} \|_2 = 1$
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% However, the eigen values and eigen vectors appears as complex conjugates:
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% \[ s_r, s_r^*, \{\psi\}_r, \{\psi\}_r^*, \quad r = 1, N \]
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% In the end, they are $2N$ eigen values.
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% We then build two extended eigen matrices as follow:
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% \[ \mathcal{S} = \begin{bmatrix}
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% s_1 & & & & & \\
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% & \ddots & & & 0 & \\
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% & & s_N & & & \\
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% & & & s_1^* & & \\
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% & 0 & & & \ddots & \\
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% & & & & & s_N^*
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% \end{bmatrix}_{2N \times 2N}; \quad \Phi = \begin{bmatrix}
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% & & & & &\\
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% \{\psi_1\} & \dots & \{\psi_N\} & \{\psi_1^*\} & \dots & \{\psi_N^*\} \\
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% & & & & &
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% \end{bmatrix}_{M \times 2N} \]
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eigen_val_ext_M = blkdiag(eigen_val_M, conj(eigen_val_M));
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eigen_vec_ext_M = [eigen_vec_M, conj(eigen_vec_M)];
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% We also build the Modal A and Modal B matrices:
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% \begin{equation}
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% A = \begin{bmatrix}
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% a_1 & & 0 \\
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% & \ddots & \\
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% 0 & & a_N
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% \end{bmatrix}_{N \times N}; \quad B = \begin{bmatrix}
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% b_1 & & 0 \\
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% & \ddots & \\
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% 0 & & b_N
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% \end{bmatrix}_{N \times N}
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% \end{equation}
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% With $a_i$ is the "Modal A" parameter linked to mode i.
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modal_a_M = diag(complex(modal_a(:, 1), modal_a(:, 2)));
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modal_b_M = diag(complex(modal_b(:, 1), modal_b(:, 2)));
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modal_a_ext_M = blkdiag(modal_a_M, conj(modal_a_M));
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modal_b_ext_M = blkdiag(modal_b_M, conj(modal_b_M));
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% Matlab Implementation
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Hsyn = zeros(acc_n*dir_n, acc_n*dir_n, length(freqs));
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for i = 1:length(freqs)
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Hsyn(:, :, i) = eigen_vec_ext_M*((j*2*pi*freqs(i)).^2*inv(modal_a_ext_M)/(diag(j*2*pi*freqs(i) - diag(eigen_val_ext_M))))*eigen_vec_ext_M.';
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end
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% Because the synthesize frequency response functions are representing the displacement response in $[m/N]$, we multiply each element of the FRF matrix by $(j \omega)^2$ in order to obtain the acceleration response in $[m/s^2/N]$.
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for i = 1:size(Hsyn, 1)
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Hsyn(i, :, :) = squeeze(Hsyn(i, :, :)).*(j*2*pi*freqs).^2;
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end
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% Original and Synthesize FRF matrix comparison
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acc_o = 1; dir_o = 1; dir_i = 1;
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figure;
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ax1 = subplot(2, 1, 1);
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hold on;
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plot(freqs, abs(squeeze(frf(3*(acc_o-1)+dir_o, dir_i, :))), 'DisplayName', 'Original');
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plot(freqs, abs(squeeze(Hsyn(3*(acc_o-1)+dir_o, 3*(11-1)+dir_i, :))), 'DisplayName', 'Synthesize');
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hold off;
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set(gca, 'yscale', 'log');
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set(gca, 'XTickLabel',[]);
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ylabel('Magnitude [$\frac{m/s^2}{N}$]');
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title(sprintf('From acc %i %s to acc %i %s', 11, dirs(dir_i), acc_o, dirs(dir_o)))
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legend('location', 'northwest');
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ax2 = subplot(2, 1, 2);
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hold on;
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plot(freqs, mod(180/pi*phase(squeeze(frf(3*(acc_o-1)+dir_o, dir_i, :)))+180, 360)-180);
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plot(freqs, mod(180/pi*phase(squeeze(Hsyn(3*(acc_o-1)+dir_o, 3*(11-1)+dir_i, :)))+180, 360)-180);
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hold off;
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yticks(-360:90:360); ylim([-180, 180]);
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xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
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linkaxes([ax1,ax2],'x');
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xlim([1, 200]);
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% Synthesize FRF that has not yet been measured
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accs = [1]; dirs = [1:3];
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figure;
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ax1 = subplot(2, 1, 1);
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hold on;
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for acc_i = accs
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for dir_i = dirs
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plot(freqs, abs((1./(j*2*pi*freqs').^2).*squeeze(Hsyn(3*(acc_i-1)+dir_i, 3*(acc_i-1)+dir_i, :))), 'DisplayName', sprintf('Acc %i - %s', acc_i, dirs(dir_i)));
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end
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end
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hold off;
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set(gca, 'xscale', 'log'); set(gca, 'yscale', 'log');
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set(gca, 'XTickLabel',[]);
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ylabel('Magnitude [$\frac{m}{N}$]');
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legend('location', 'southwest');
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ax2 = subplot(2, 1, 2);
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hold on;
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for acc_i = accs
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for dir_i = dirs
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plot(freqs, mod(180/pi*phase((1./(j*2*pi*freqs').^2).*squeeze(Hsyn(3*(acc_i-1)+dir_i, 3*(acc_i-1)+dir_i, :)))+180, 360)-180);
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end
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end
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hold off;
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yticks(-360:90:360); ylim([-180, 180]);
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set(gca, 'xscale', 'log');
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xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
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linkaxes([ax1,ax2],'x');
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xlim([1, 200]);
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