#+TITLE: Modal Analysis :DRAWER: #+STARTUP: overview #+LANGUAGE: en #+EMAIL: dehaeze.thomas@gmail.com #+AUTHOR: Dehaeze Thomas #+HTML_LINK_HOME: ../index.html #+HTML_LINK_UP: ../index.html #+HTML_HEAD: #+HTML_HEAD: #+HTML_HEAD: #+HTML_HEAD: #+HTML_HEAD: #+HTML_HEAD: #+HTML_HEAD: #+HTML_MATHJAX: align: center tagside: right font: TeX #+PROPERTY: header-args:matlab :session *MATLAB* #+PROPERTY: header-args:matlab+ :comments org #+PROPERTY: header-args:matlab+ :results none #+PROPERTY: header-args:matlab+ :exports both #+PROPERTY: header-args:matlab+ :eval no-export #+PROPERTY: header-args:matlab+ :output-dir figs #+PROPERTY: header-args:shell :eval no-export #+PROPERTY: header-args:latex :headers '("\\usepackage{tikz}" "\\usepackage{import}" "\\import{$HOME/MEGA/These/LaTeX/}{config.tex}") #+PROPERTY: header-args:latex+ :imagemagick t :fit yes #+PROPERTY: header-args:latex+ :iminoptions -scale 100% -density 150 #+PROPERTY: header-args:latex+ :imoutoptions -quality 100 #+PROPERTY: header-args:latex+ :results raw replace :buffer no #+PROPERTY: header-args:latex+ :eval no-export #+PROPERTY: header-args:latex+ :exports both #+PROPERTY: header-args:latex+ :mkdirp yes #+PROPERTY: header-args:latex+ :output-dir figs :END: * Matlab Init :noexport:ignore: #+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name) <> #+end_src #+begin_src matlab :exports none :results silent :noweb yes <> #+end_src * Setup #+name: fig:nass-modal-test #+caption: Position and orientation of the accelerometer used [[file:figs/nass-modal-test.png]] * Mode extraction and importation First, we split the big =modes.asc= files into sub text files using =bash=. #+begin_src bash :results none sed '/^\s*[0-9]*[XYZ][+-]:/!d' modal_analysis_updated/modes.asc > mat/mode_shapes.txt sed '/freq/!d' modal_analysis_updated/modes.asc | sed 's/.* = \(.*\)Hz/\1/' > mat/mode_freqs.txt sed '/damp/!d' modal_analysis_updated/modes.asc | sed 's/.* = \(.*\)\%/\1/' > mat/mode_damps.txt sed '/modal A/!d' modal_analysis_updated/modes.asc | sed 's/.* =\s\+\([-0-9.e]\++[0-9]\+\)\([-+0-9.e]\+\)i/\1 \2/' > mat/mode_modal_a.txt sed '/modal B/!d' modal_analysis_updated/modes.asc | sed 's/.* =\s\+\([-0-9.e]\++[0-9]\+\)\([-+0-9.e]\+\)i/\1 \2/' > mat/mode_modal_b.txt #+end_src Then we import them on Matlab. #+begin_src matlab shapes = readtable('mat/mode_shapes.txt', 'ReadVariableNames', false); % [Sign / Real / Imag] freqs = table2array(readtable('mat/mode_freqs.txt', 'ReadVariableNames', false)); % in [Hz] damps = table2array(readtable('mat/mode_damps.txt', 'ReadVariableNames', false)); % in [%] modal_a = table2array(readtable('mat/mode_modal_a.txt', 'ReadVariableNames', false)); % [Real / Imag] modal_a = complex(modal_a(:, 1), modal_a(:, 2)); modal_b = table2array(readtable('mat/mode_modal_b.txt', 'ReadVariableNames', false)); % [Real / Imag] modal_b = complex(modal_b(:, 1), modal_b(:, 2)); #+end_src We guess the number of modes identified from the length of the imported data. #+begin_src matlab acc_n = 23; % Number of accelerometers dir_n = 3; % Number of directions dirs = 'XYZ'; mod_n = size(shapes,1)/acc_n/dir_n; % Number of modes #+end_src As the mode shapes are split into 3 parts (direction plus sign, real part and imaginary part), we aggregate them into one array of complex numbers. #+begin_src matlab T_sign = table2array(shapes(:, 1)); T_real = table2array(shapes(:, 2)); T_imag = table2array(shapes(:, 3)); modes = zeros(mod_n, acc_n, dir_n); for mod_i = 1:mod_n for acc_i = 1:acc_n % Get the correct section of the signs T = T_sign(acc_n*dir_n*(mod_i-1)+1:acc_n*dir_n*mod_i); for dir_i = 1:dir_n % Get the line corresponding to the sensor i = find(contains(T, sprintf('%i%s',acc_i, dirs(dir_i))), 1, 'first')+acc_n*dir_n*(mod_i-1); modes(mod_i, acc_i, dir_i) = str2num([T_sign{i}(end-1), '1'])*complex(T_real(i),T_imag(i)); end end end #+end_src The obtained mode frequencies and damping are shown below. #+begin_src matlab :exports both :results value table replace :post addhdr(*this*) data2orgtable([freqs, damps], {}, {'Frequency [Hz]', 'Damping [%]'}, ' %.1f '); #+end_src #+RESULTS: | Frequency [Hz] | Damping [%] | |----------------+-------------| | 11.4 | 8.7 | | 18.5 | 11.8 | | 37.6 | 6.4 | | 39.4 | 3.6 | | 54.0 | 0.2 | | 56.1 | 2.8 | | 69.7 | 4.6 | | 71.6 | 0.6 | | 72.4 | 1.6 | | 84.9 | 3.6 | | 90.6 | 0.3 | | 91.0 | 2.9 | | 95.8 | 3.3 | | 105.4 | 3.3 | | 106.8 | 1.9 | | 112.6 | 3.0 | | 116.8 | 2.7 | | 124.1 | 0.6 | | 145.4 | 1.6 | | 150.1 | 2.2 | | 164.7 | 1.4 | * Positions of the sensors We process the file exported from the =modal= software containing the positions of the sensors using =bash=. #+begin_src bash :results none cat modal_analysis_updated/id31_nanostation_modified.cfg | grep NODES -A 23 | sed '/\s\+[0-9]\+/!d' | sed 's/\(.*\)\s\+0\s\+.\+/\1/' > mat/acc_pos.txt #+end_src We then import that on =matlab=, and sort them. #+begin_src matlab 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); #+end_src The positions of the sensors relative to the point of interest are shown below. #+begin_src matlab :exports both :results value table replace :post addhdr(*this*) data2orgtable(1000*acc_pos, {}, {'x [mm]', 'y [mm]', 'z [mm]'}, ' %.0f '); #+end_src #+RESULTS: | x [mm] | y [mm] | z [mm] | |--------+--------+--------| | -64 | -64 | -296 | | -64 | 64 | -296 | | 64 | 64 | -296 | | 64 | -64 | -296 | | -385 | -300 | -417 | | -420 | 280 | -417 | | 420 | 280 | -417 | | 380 | -300 | -417 | | -475 | -414 | -427 | | -465 | 407 | -427 | | 475 | 424 | -427 | | 475 | -419 | -427 | | -320 | -446 | -786 | | -480 | 534 | -786 | | 450 | 534 | -786 | | 295 | -481 | -786 | | -730 | -526 | -951 | | -735 | 814 | -951 | | 875 | 799 | -951 | | 865 | -506 | -951 | | -155 | -90 | -594 | | 0 | 180 | -594 | | 155 | -90 | -594 | * Solids We consider the following solid bodies: - Bottom Granite - Top Granite - Translation Stage - Tilt Stage - Spindle - Hexapod We create a structure =solids= that contains the accelerometer number of each solid bodies (as shown on figure [[fig:nass-modal-test]]). #+begin_src matlab 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); #+end_src * From local coordinates to global coordinates #+begin_src latex :file local_to_global_coordinates.pdf :post pdf2svg(file=*this*, ext="png") :exports results \newcommand\irregularcircle[2]{% radius, irregularity \pgfextra {\pgfmathsetmacro\len{(#1)+rand*(#2)}} +(0:\len pt) \foreach \a in {10,20,...,350}{ \pgfextra {\pgfmathsetmacro\len{(#1)+rand*(#2)}} -- +(\a:\len pt) } -- cycle } \begin{tikzpicture} \draw[rounded corners=1mm] (0, 0) \irregularcircle{3cm}{1mm}; \node[] (origin) at (4, -1) {$\bullet$}; \begin{scope}[shift={(origin)}] \def\axissize{0.8cm} \draw[->] (0, 0) -- ++(\axissize, 0) node[above left]{$x$}; \draw[->] (0, 0) -- ++(0, \axissize) node[below right]{$y$}; \draw[fill, color=black] (0, 0) circle (0.05*\axissize); \node[draw, circle, inner sep=0pt, minimum size=0.4*\axissize, label=left:$z$] (yaxis) at (0, 0){}; \node[below right] at (0, 0){$\{O\}$}; \end{scope} \coordinate[] (p1) at (-1.5, -1.5); \coordinate[] (p2) at (-1.5, 1.5); \coordinate[] (p3) at ( 1.5, 1.5); \coordinate[] (p4) at ( 1.5, -1.5); \draw[->] (p1)node[]{$\bullet$}node[above]{$p_1$} -- ++(1, 0.5)node[right]{$v_1$}; \draw[->] (p2)node[]{$\bullet$}node[above]{$p_2$} -- ++(-0.5, 1)node[right]{$v_2$}; \draw[->] (p3)node[]{$\bullet$}node[above]{$p_3$} -- ++(1, 0.5)node[right]{$v_3$}; \draw[->] (p4)node[]{$\bullet$}node[above]{$p_4$} -- ++(0.5, 1)node[right]{$v_4$}; \end{tikzpicture} #+end_src #+RESULTS: [[file:figs/local_to_global_coordinates.png]] From the figure above, we can write: \begin{align*} \vec{v}_1 &= \vec{v} + \Omega \vec{p}_1\\ \vec{v}_2 &= \vec{v} + \Omega \vec{p}_2\\ \vec{v}_3 &= \vec{v} + \Omega \vec{p}_3\\ \vec{v}_4 &= \vec{v} + \Omega \vec{p}_4 \end{align*} With \begin{equation} \Omega = \begin{bmatrix} 0 & -\Omega_z & \Omega_y \\ \Omega_z & 0 & -\Omega_x \\ -\Omega_y & \Omega_x & 0 \end{bmatrix} \end{equation} $\vec{v}$ and $\Omega$ represent to velocity and rotation of the solid expressed in the frame $\{O\}$. We can rearrange the equations in a matrix form: \begin{equation} \left[\begin{array}{ccc|ccc} 1 & 0 & 0 & 0 & p_{1z} & -p_{1y} \\ 0 & 1 & 0 & -p_{1z} & 0 & p_{1x} \\ 0 & 0 & 1 & p_{1y} & -p_{1x} & 0 \\ \hline & \vdots & & & \vdots & \\ \hline 1 & 0 & 0 & 0 & p_{4z} & -p_{4y} \\ 0 & 1 & 0 & -p_{4z} & 0 & p_{4x} \\ 0 & 0 & 1 & p_{4y} & -p_{4x} & 0 \end{array}\right] \begin{bmatrix} v_x \\ v_y \\ v_z \\ \hline \Omega_x \\ \Omega_y \\ \Omega_z \end{bmatrix} = \begin{bmatrix} v_{1x} \\ v_{1y} \\ v_{1z} \\\hline \vdots \\\hline v_{4x} \\ v_{4y} \\ v_{4z} \end{bmatrix} \end{equation} and then we obtain the velocity and rotation of the solid in the wanted frame $\{O\}$: \begin{equation} \begin{bmatrix} v_x \\ v_y \\ v_z \\ \hline \Omega_x \\ \Omega_y \\ \Omega_z \end{bmatrix} = \left[\begin{array}{ccc|ccc} 1 & 0 & 0 & 0 & p_{1z} & -p_{1y} \\ 0 & 1 & 0 & -p_{1z} & 0 & p_{1x} \\ 0 & 0 & 1 & p_{1y} & -p_{1x} & 0 \\ \hline & \vdots & & & \vdots & \\ \hline 1 & 0 & 0 & 0 & p_{4z} & -p_{4y} \\ 0 & 1 & 0 & -p_{4z} & 0 & p_{4x} \\ 0 & 0 & 1 & p_{4y} & -p_{4x} & 0 \end{array}\right]^{-1} \begin{bmatrix} v_{1x} \\ v_{1y} \\ v_{1z} \\\hline \vdots \\\hline v_{4x} \\ v_{4y} \\ v_{4z} \end{bmatrix} \end{equation} This inversion is equivalent to a mean square problem. #+begin_src matlab mode_shapes_O = zeros(mod_n, length(solid_names), 6); for mod_i = 1:mod_n for solid_i = 1:length(solid_names) solids_i = solids.(solid_names{solid_i}); Y = reshape(squeeze(modes(mod_i, solids_i, :))', [], 1); A = zeros(3*length(solids_i), 6); for i = 1:length(solids_i) A(3*(i-1)+1:3*i, 1:3) = eye(3); A(3*(i-1)+1:3*i, 4:6) = [0 acc_pos(i, 3) -acc_pos(i, 2) ; -acc_pos(i, 3) 0 acc_pos(i, 1) ; acc_pos(i, 2) -acc_pos(i, 1) 0]; end mode_shapes_O(mod_i, solid_i, :) = A\Y; end end #+end_src * Modal Matrices We want to obtain the two following matrices: \[ \Omega = \begin{bmatrix} \omega_1^2 & & 0 \\ & \ddots & \\ 0 & & \omega_n^2 \end{bmatrix}; \quad \Psi = \begin{bmatrix} & & \\ \{\psi_1\} & \dots & \{\psi_n\} \\ & & \end{bmatrix} \] - [ ] How to add damping to the eigen value matrix? #+begin_src matlab eigen_value_M = diag(freqs*2*pi); eigen_vector_M = reshape(mode_shapes_O, [mod_n, 6*length(solid_names)])'; #+end_src \[ \{\psi_1\} = \begin{Bmatrix} \psi_{1_x} & \psi_{2_x} & \dots & \psi_{6_x} & \psi_{1_x} & \dots & \psi_{1\Omega_x} & \dots & \psi_{6\Omega_z} \end{Bmatrix}^T \] * Modal Complexity A method of displaying *modal complexity* is by plotting the elements of the eigenvector on an *Argand diagram*, such as the ones shown in figure [[fig:modal_complexity_small]]. To evaluate the complexity of the modes, we plot a polygon around the extremities of the individual vectors. The obtained area of this polygon is then compared with the area of the circle which is based on the length of the largest vector element. The resulting ratio is used as an indication of the complexity of the mode. A little complex mode is shown on figure [[fig:modal_complexity_small]] whereas an highly complex mode is shown on figure [[fig:modal_complexity_high]]. The complexity of all the modes are compared on figure [[fig:modal_complexities]]. #+begin_src matlab :export none mod_i = 1; i_max = convhull(real(eigen_vector_M(:, mod_i)), imag(eigen_vector_M(:, mod_i))); radius = max(abs(eigen_vector_M(:, mod_i))); theta = linspace(0, 2*pi, 100); figure; hold on; plot(radius*cos(theta), radius*sin(theta), '-'); plot(real(eigen_vector_M(i_max, mod_i)), imag(eigen_vector_M(i_max, mod_i)), '-'); plot(real(eigen_vector_M(:, mod_i)), imag(eigen_vector_M(:, mod_i)), 'ko'); hold off; xlabel('Real Part'); ylabel('Imaginary Part'); title(sprintf('Mode %i', mod_i)); axis manual equal #+end_src #+HEADER: :tangle no :exports results :results none :noweb yes #+begin_src matlab :var filepath="figs/modal_complexity_small.pdf" :var figsize="normal-normal" :post pdf2svg(file=*this*, ext="png") <> #+end_src #+NAME: fig:modal_complexity_small #+CAPTION: Modal Complexity of one mode with small complexity [[file:figs/modal_complexity_small.png]] #+begin_src matlab :export none mod_i = 8; i_max = convhull(real(eigen_vector_M(:, mod_i)), imag(eigen_vector_M(:, mod_i))); radius = max(abs(eigen_vector_M(:, mod_i))); theta = linspace(0, 2*pi, 100); figure; hold on; plot(radius*cos(theta), radius*sin(theta), '-'); plot(real(eigen_vector_M(i_max, mod_i)), imag(eigen_vector_M(i_max, mod_i)), '-'); plot(real(eigen_vector_M(:, mod_i)), imag(eigen_vector_M(:, mod_i)), 'ko'); hold off; xlabel('Real Part'); ylabel('Imaginary Part'); title(sprintf('Mode %i', mod_i)); axis manual equal #+end_src #+HEADER: :tangle no :exports results :results none :noweb yes #+begin_src matlab :var filepath="figs/modal_complexity_high.pdf" :var figsize="normal-normal" :post pdf2svg(file=*this*, ext="png") <> #+end_src #+NAME: fig:modal_complexity_high #+CAPTION: Modal Complexity of one higly complex mode [[file:figs/modal_complexity_high.png]] #+begin_src matlab :export none modes_complexity = zeros(mod_n, 1); for mod_i = 1:mod_n i = convhull(real(eigen_vector_M(:, mod_i)), imag(eigen_vector_M(:, mod_i))); area_complex = polyarea(real(eigen_vector_M(i, mod_i)), imag(eigen_vector_M(i, mod_i))); area_circle = pi*max(abs(eigen_vector_M(:, mod_i)))^2; modes_complexity(mod_i) = area_complex/area_circle; end figure; plot(1:mod_n, modes_complexity, 'ok'); ylim([0, 1]); xlabel('Mode Number'); ylabel('Modal Complexity'); #+end_src #+HEADER: :tangle no :exports results :results none :noweb yes #+begin_src matlab :var filepath="figs/modal_complexities.pdf" :var figsize="wide-normal" :post pdf2svg(file=*this*, ext="png") <> #+end_src #+NAME: fig:modal_complexities #+CAPTION: Modal complexity for each mode [[file:figs/modal_complexities.png]] * Some notes about constraining the number of degrees of freedom We want to have the two eigen matrices. They should have the same size $n \times n$ where $n$ is the number of modes as well as the number of degrees of freedom. Thus, if we consider 21 modes, we should restrict our system to have only 21 DOFs. Actually, we are measured 6 DOFs of 6 solids, thus we have 36 DOFs. From the mode shapes animations, it seems that in the frequency range of interest, the two marbles can be considered as one solid. We thus have 5 solids and 30 DOFs. In order to determine which DOF can be neglected, two solutions seems possible: - compare the mode shapes - compare the FRFs The question is: in which base (frame) should be express the modes shapes and FRFs? Is it meaningful to compare mode shapes as they give no information about the amplitudes of vibration? | Stage | Motion DOFs | Parasitic DOF | Total DOF | Description of DOF | |---------+-------------+---------------+-----------+--------------------| | Granite | 0 | 3 | 3 | | | Ty | 1 | 2 | 3 | Ty, Rz | | Ry | 1 | 2 | 3 | Ry, | | Rz | 1 | 2 | 3 | Rz, Rx, Ry | | Hexapod | 6 | 0 | 6 | Txyz, Rxyz | |---------+-------------+---------------+-----------+--------------------| | | 9 | 9 | 18 | | #+TBLFM: $4=vsum($2..$3) #+TBLFM: @>$2..$>=vsum(@I..@II) * TODO Normalization of mode shapes? We normalize each column of the eigen vector matrix. Then, each eigenvector as a norm of 1. #+begin_src matlab eigen_vector_M = eigen_vector_M./vecnorm(eigen_vector_M); #+end_src * Compare Mode Shapes Let's say we want to see for the first mode which DOFs can be neglected. In order to do so, we should estimate the motion of each stage in particular directions. If we look at the z motion for instance, we will find that we cannot neglect that motion (because of the tilt causing z motion). #+begin_src matlab mode_i = 3; dof_i = 6; mode = eigen_vector_M(dof_i:6:end, mode_i); figure; hold on; for i=1:length(mode) plot([0, real(mode(i))], [0, imag(mode(i))], '-', 'DisplayName', solid_names{i}); end hold off; legend(); #+end_src #+begin_src matlab figure; subplot(2, 1, 1); hold on; for i=1:length(mode) plot(1, norm(mode(i)), 'o'); end hold off; ylabel('Amplitude'); subplot(2, 1, 2); hold on; for i=1:length(mode) plot(1, 180/pi*angle(mode(i)), 'o', 'DisplayName', solid_names{i}); end hold off; ylim([-180, 180]); yticks([-180:90:180]); ylabel('Phase [deg]'); legend(); #+end_src #+begin_src matlab test = mode_shapes_O(10, 1, :)/norm(squeeze(mode_shapes_O(10, 1, :))); test = mode_shapes_O(10, 2, :)/norm(squeeze(mode_shapes_O(10, 2, :))); #+end_src * TODO Synthesis of FRF curves