479 lines
17 KiB
Org Mode
479 lines
17 KiB
Org Mode
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#+TITLE: Modal Analysis
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:DRAWER:
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#+STARTUP: overview
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#+LANGUAGE: en
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#+EMAIL: dehaeze.thomas@gmail.com
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#+AUTHOR: Dehaeze Thomas
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#+HTML_LINK_HOME: ../index.html
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#+HTML_LINK_UP: ../index.html
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#+HTML_HEAD: <link rel="stylesheet" type="text/css" href="../css/htmlize.css"/>
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#+HTML_HEAD: <link rel="stylesheet" type="text/css" href="../css/readtheorg.css"/>
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#+HTML_HEAD: <link rel="stylesheet" type="text/css" href="../css/zenburn.css"/>
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#+HTML_HEAD: <script type="text/javascript" src="../js/jquery.min.js"></script>
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#+HTML_HEAD: <script type="text/javascript" src="../js/bootstrap.min.js"></script>
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#+HTML_HEAD: <script type="text/javascript" src="../js/jquery.stickytableheaders.min.js"></script>
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#+HTML_HEAD: <script type="text/javascript" src="../js/readtheorg.js"></script>
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#+HTML_MATHJAX: align: center tagside: right font: TeX
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#+PROPERTY: header-args:matlab :session *MATLAB*
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#+PROPERTY: header-args:matlab+ :comments org
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#+PROPERTY: header-args:matlab+ :results none
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#+PROPERTY: header-args:matlab+ :exports both
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#+PROPERTY: header-args:matlab+ :eval no-export
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#+PROPERTY: header-args:matlab+ :output-dir figs
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#+PROPERTY: header-args:shell :eval no-export
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#+PROPERTY: header-args:latex :headers '("\\usepackage{tikz}" "\\usepackage{import}" "\\import{$HOME/MEGA/These/LaTeX/}{config.tex}")
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#+PROPERTY: header-args:latex+ :imagemagick t :fit yes
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#+PROPERTY: header-args:latex+ :iminoptions -scale 100% -density 150
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#+PROPERTY: header-args:latex+ :imoutoptions -quality 100
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#+PROPERTY: header-args:latex+ :results raw replace :buffer no
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#+PROPERTY: header-args:latex+ :eval no-export
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#+PROPERTY: header-args:latex+ :exports both
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#+PROPERTY: header-args:latex+ :mkdirp yes
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#+PROPERTY: header-args:latex+ :output-dir figs
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:END:
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* Matlab Init :noexport:ignore:
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#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
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<<matlab-dir>>
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#+end_src
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#+begin_src matlab :exports none :results silent :noweb yes
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<<matlab-init>>
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#+end_src
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* Setup
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#+name: fig:nass-modal-test
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#+caption: Position and orientation of the accelerometer used
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[[file:figs/nass-modal-test.png]]
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* Mode extraction and importation
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First, we split the big =modes.asc= files into sub text files using =bash=.
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#+begin_src bash :results none
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sed '/^\s*[0-9]*[XYZ][+-]:/!d' modal_analysis_updated/modes.asc > mat/mode_shapes.txt
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sed '/freq/!d' modal_analysis_updated/modes.asc | sed 's/.* = \(.*\)Hz/\1/' > mat/mode_freqs.txt
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sed '/damp/!d' modal_analysis_updated/modes.asc | sed 's/.* = \(.*\)\%/\1/' > mat/mode_damps.txt
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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
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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
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#+end_src
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Then we import them on Matlab.
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#+begin_src matlab
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shapes = readtable('mat/mode_shapes.txt', 'ReadVariableNames', false); % [Sign / Real / Imag]
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freqs = table2array(readtable('mat/mode_freqs.txt', 'ReadVariableNames', false)); % in [Hz]
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damps = 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_a = complex(modal_a(:, 1), modal_a(:, 2));
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modal_b = table2array(readtable('mat/mode_modal_b.txt', 'ReadVariableNames', false)); % [Real / Imag]
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modal_b = complex(modal_b(:, 1), modal_b(:, 2));
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#+end_src
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We guess the number of modes identified from the length of the imported data.
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#+begin_src matlab
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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,1)/acc_n/dir_n; % Number of modes
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#+end_src
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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.
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#+begin_src matlab
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T_sign = table2array(shapes(:, 1));
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T_real = table2array(shapes(:, 2));
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T_imag = table2array(shapes(:, 3));
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modes = zeros(mod_n, acc_n, dir_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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modes(mod_i, acc_i, dir_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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#+end_src
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The obtained mode frequencies and damping are shown below.
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#+begin_src matlab :exports both :results value table replace :post addhdr(*this*)
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data2orgtable([freqs, damps], {}, {'Frequency [Hz]', 'Damping [%]'}, ' %.1f ');
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#+end_src
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#+RESULTS:
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| Frequency [Hz] | Damping [%] |
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|----------------+-------------|
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| 11.4 | 8.7 |
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| 18.5 | 11.8 |
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| 37.6 | 6.4 |
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| 39.4 | 3.6 |
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| 54.0 | 0.2 |
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| 56.1 | 2.8 |
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| 69.7 | 4.6 |
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| 71.6 | 0.6 |
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| 72.4 | 1.6 |
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| 84.9 | 3.6 |
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| 90.6 | 0.3 |
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| 91.0 | 2.9 |
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| 95.8 | 3.3 |
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| 105.4 | 3.3 |
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| 106.8 | 1.9 |
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| 112.6 | 3.0 |
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| 116.8 | 2.7 |
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| 124.1 | 0.6 |
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| 145.4 | 1.6 |
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| 150.1 | 2.2 |
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| 164.7 | 1.4 |
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* Positions of the sensors
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We process the file exported from the =modal= software containing the positions of the sensors using =bash=.
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#+begin_src bash :results none
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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
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#+end_src
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We then import that on =matlab=, and sort them.
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#+begin_src matlab
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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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#+end_src
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The positions of the sensors relative to the point of interest are shown below.
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#+begin_src matlab :exports both :results value table replace :post addhdr(*this*)
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data2orgtable(1000*acc_pos, {}, {'x [mm]', 'y [mm]', 'z [mm]'}, ' %.0f ');
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#+end_src
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#+RESULTS:
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| x [mm] | y [mm] | z [mm] |
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|--------+--------+--------|
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| -64 | -64 | -296 |
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| -64 | 64 | -296 |
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| 64 | 64 | -296 |
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| 64 | -64 | -296 |
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| -385 | -300 | -417 |
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| -420 | 280 | -417 |
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| 420 | 280 | -417 |
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| 380 | -300 | -417 |
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| -475 | -414 | -427 |
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| -465 | 407 | -427 |
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| 475 | 424 | -427 |
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| 475 | -419 | -427 |
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| -320 | -446 | -786 |
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| -480 | 534 | -786 |
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| 450 | 534 | -786 |
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| 295 | -481 | -786 |
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| -730 | -526 | -951 |
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| -735 | 814 | -951 |
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| 875 | 799 | -951 |
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| 865 | -506 | -951 |
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| -155 | -90 | -594 |
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| 0 | 180 | -594 |
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| 155 | -90 | -594 |
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* Solids
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We consider the following solid bodies:
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- Bottom Granite
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- Top Granite
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- Translation Stage
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- Tilt Stage
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- Spindle
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- Hexapod
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We create a structure =solids= that contains the accelerometer number of each solid bodies (as shown on figure [[fig:nass-modal-test]]).
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#+begin_src matlab
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solids = {};
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solids.granite_bot = [17, 18, 19, 20];
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solids.granite_top = [13, 14, 15, 16];
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solids.ty = [9, 10, 11, 12];
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solids.ry = [5, 6, 7, 8];
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solids.rz = [21, 22, 23];
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solids.hexa = [1, 2, 3, 4];
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solid_names = fields(solids);
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#+end_src
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* From local coordinates to global coordinates
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#+begin_src latex :file local_to_global_coordinates.pdf :post pdf2svg(file=*this*, ext="png") :exports results
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\newcommand\irregularcircle[2]{% radius, irregularity
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\pgfextra {\pgfmathsetmacro\len{(#1)+rand*(#2)}}
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+(0:\len pt)
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\foreach \a in {10,20,...,350}{
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\pgfextra {\pgfmathsetmacro\len{(#1)+rand*(#2)}}
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-- +(\a:\len pt)
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} -- cycle
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}
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\begin{tikzpicture}
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\draw[rounded corners=1mm] (0, 0) \irregularcircle{3cm}{1mm};
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\node[] (origin) at (4, -1) {$\bullet$};
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\begin{scope}[shift={(origin)}]
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\def\axissize{0.8cm}
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\draw[->] (0, 0) -- ++(\axissize, 0) node[above left]{$x$};
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\draw[->] (0, 0) -- ++(0, \axissize) node[below right]{$y$};
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\draw[fill, color=black] (0, 0) circle (0.05*\axissize);
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\node[draw, circle, inner sep=0pt, minimum size=0.4*\axissize, label=left:$z$] (yaxis) at (0, 0){};
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\node[below right] at (0, 0){$\{O\}$};
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\end{scope}
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\coordinate[] (p1) at (-1.5, -1.5);
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\coordinate[] (p2) at (-1.5, 1.5);
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\coordinate[] (p3) at ( 1.5, 1.5);
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\coordinate[] (p4) at ( 1.5, -1.5);
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\draw[->] (p1)node[]{$\bullet$}node[above]{$p_1$} -- ++(1, 0.5)node[right]{$v_1$};
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\draw[->] (p2)node[]{$\bullet$}node[above]{$p_2$} -- ++(-0.5, 1)node[right]{$v_2$};
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\draw[->] (p3)node[]{$\bullet$}node[above]{$p_3$} -- ++(1, 0.5)node[right]{$v_3$};
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\draw[->] (p4)node[]{$\bullet$}node[above]{$p_4$} -- ++(0.5, 1)node[right]{$v_4$};
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\end{tikzpicture}
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#+end_src
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#+RESULTS:
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[[file:figs/local_to_global_coordinates.png]]
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From the figure above, we can write:
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\begin{align*}
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\vec{v}_1 &= \vec{v} + \Omega \vec{p}_1\\
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\vec{v}_2 &= \vec{v} + \Omega \vec{p}_2\\
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\vec{v}_3 &= \vec{v} + \Omega \vec{p}_3\\
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\vec{v}_4 &= \vec{v} + \Omega \vec{p}_4
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\end{align*}
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With
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\begin{equation}
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\Omega = \begin{bmatrix}
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0 & -\Omega_z & \Omega_y \\
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\Omega_z & 0 & -\Omega_x \\
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-\Omega_y & \Omega_x & 0
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\end{bmatrix}
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\end{equation}
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$\vec{v}$ and $\Omega$ represent to velocity and rotation of the solid expressed in the frame $\{O\}$.
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We can rearrange the equations in a matrix form:
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\begin{equation}
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\left[\begin{array}{ccc|ccc}
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1 & 0 & 0 & 0 & p_{1z} & -p_{1y} \\
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0 & 1 & 0 & -p_{1z} & 0 & p_{1x} \\
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0 & 0 & 1 & p_{1y} & -p_{1x} & 0 \\ \hline
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& \vdots & & & \vdots & \\ \hline
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1 & 0 & 0 & 0 & p_{4z} & -p_{4y} \\
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0 & 1 & 0 & -p_{4z} & 0 & p_{4x} \\
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0 & 0 & 1 & p_{4y} & -p_{4x} & 0
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\end{array}\right] \begin{bmatrix}
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v_x \\ v_y \\ v_z \\ \hline \Omega_x \\ \Omega_y \\ \Omega_z
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\end{bmatrix} = \begin{bmatrix}
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v_{1x} \\ v_{1y} \\ v_{1z} \\\hline \vdots \\\hline v_{4x} \\ v_{4y} \\ v_{4z}
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\end{bmatrix}
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\end{equation}
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and then we obtain the velocity and rotation of the solid in the wanted frame $\{O\}$:
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\begin{equation}
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\begin{bmatrix}
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v_x \\ v_y \\ v_z \\ \hline \Omega_x \\ \Omega_y \\ \Omega_z
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\end{bmatrix} =
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\left[\begin{array}{ccc|ccc}
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1 & 0 & 0 & 0 & p_{1z} & -p_{1y} \\
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0 & 1 & 0 & -p_{1z} & 0 & p_{1x} \\
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0 & 0 & 1 & p_{1y} & -p_{1x} & 0 \\ \hline
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& \vdots & & & \vdots & \\ \hline
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1 & 0 & 0 & 0 & p_{4z} & -p_{4y} \\
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0 & 1 & 0 & -p_{4z} & 0 & p_{4x} \\
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0 & 0 & 1 & p_{4y} & -p_{4x} & 0
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\end{array}\right]^{-1} \begin{bmatrix}
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v_{1x} \\ v_{1y} \\ v_{1z} \\\hline \vdots \\\hline v_{4x} \\ v_{4y} \\ v_{4z}
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\end{bmatrix}
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\end{equation}
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This inversion is equivalent to a mean square problem.
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#+begin_src matlab
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mode_shapes_O = zeros(mod_n, length(solid_names), 6);
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for mod_i = 1:mod_n
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for solid_i = 1:length(solid_names)
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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
|
||
|
|
Complexity of one mode
|
||
|
|
#+begin_src matlab
|
||
|
|
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');
|
||
|
|
axis manual equal
|
||
|
|
#+end_src
|
||
|
|
|
||
|
|
#+HEADER: :tangle no :exports results :results none :noweb yes
|
||
|
|
#+begin_src matlab :var filepath="figs/modal_complexity.pdf" :var figsize="normal-normal" :post pdf2svg(file=*this*, ext="png")
|
||
|
|
<<plt-matlab>>
|
||
|
|
#+end_src
|
||
|
|
|
||
|
|
#+NAME: fig:modal_complexity
|
||
|
|
#+CAPTION: Modal Complexity of one mode
|
||
|
|
[[file:figs/modal_complexity.png]]
|
||
|
|
|
||
|
|
Complexity function of the mode order.
|
||
|
|
#+begin_src matlab
|
||
|
|
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")
|
||
|
|
<<plt-matlab>>
|
||
|
|
#+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
|