2013 lines
90 KiB
Org Mode
2013 lines
90 KiB
Org Mode
#+TITLE: Modal Analysis - Modal Parameter Extraction
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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:matlab+ :tangle matlab/modal_extraction.m
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#+PROPERTY: header-args:matlab+ :mkdirp yes
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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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The goal here is to extract the modal parameters describing the modes of station being studied.
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* ZIP file containing the data and matlab files :ignore:
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#+begin_src bash :exports none :results none
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if [ matlab/modal_extraction.m -nt data/modal_extraction.zip ]; then
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cp matlab/modal_extraction.m modal_extraction.m;
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zip data/modal_extraction \
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mat/data.mat \
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modal_extraction.m
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rm modal_extraction.m;
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fi
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#+end_src
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#+begin_note
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All the files (data and Matlab scripts) are accessible [[file:data/modal_extraction.zip][here]].
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#+end_note
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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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* TODO Part to explain how to choose the modes frequencies
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- bro-band method used
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- Stabilization Chart
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- 21 modes
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* Obtained Modal Parameters
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From the modal analysis software, we can export the obtained modal parameters:
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- the resonance frequencies
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- the modes shapes
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- the modal damping
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- the residues
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These can be express as the *eigen matrices*:
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\[ \Omega = \begin{bmatrix}
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\omega_1^2 & & 0 \\
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& \ddots & \\
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0 & & \omega_n^2
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\end{bmatrix}; \quad \Psi = \begin{bmatrix}
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& & \\
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\{\psi_1\} & \dots & \{\psi_n\} \\
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& &
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\end{bmatrix} \]
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where $\bar{\omega}_r^2$ is the $r^\text{th}$ eigenvalue squared and $\{\phi\}_r$ is a description of the corresponding *mode shape*.
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The file containing the modal parameters is =mat/modes.asc=. Its first 20 lines as shown below.
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#+begin_src bash :results output :exports results :eval no-export
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sed 20q mat/modes.asc | sed $'s/\r//'
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#+end_src
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#+RESULTS:
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#+begin_example
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Created by N-Modal
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Estimator: bbfd
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01-Jul-19 16:44:11
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Mode 1
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freq = 11.41275Hz
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damp = 8.72664%
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modal A = -4.50556e+003-9.41744e+003i
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modal B = -7.00928e+005+2.62922e+005i
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Mode matrix of local coordinate [DOF: Re IM]
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1X-: -1.04114e-001 3.50664e-002
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1Y-: 2.34008e-001 5.04273e-004
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1Z+: -1.93303e-002 5.08614e-003
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2X-: -8.38439e-002 3.45978e-002
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2Y-: 2.42440e-001 0.00000e+000
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2Z+: -7.40734e-003 5.17734e-003
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3Y-: 2.17655e-001 6.10802e-003
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3X+: 1.18685e-001 -3.54602e-002
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3Z+: -2.37725e-002 -1.61649e-003
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#+end_example
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We split this big =modes.asc= file into sub text files using =bash=. The obtained files are described one table [[tab:modes_files]].
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#+begin_src bash :results none
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sed '/^\s*[0-9]*[XYZ][+-]:/!d' mat/modes.asc > mat/mode_shapes.txt
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sed '/freq/!d' mat/modes.asc | sed 's/.* = \(.*\)Hz/\1/' > mat/mode_freqs.txt
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sed '/damp/!d' mat/modes.asc | sed 's/.* = \(.*\)\%/\1/' > mat/mode_damps.txt
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sed '/modal A/!d' mat/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' mat/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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#+name: tab:modes_files
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#+caption: Split =modes.asc= file
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| Filename | Content |
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|------------------------+--------------------------------------------------|
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| =mat/mode_shapes.txt= | mode shapes |
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| =mat/mode_freqs.txt= | resonance frequencies |
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| =mat/mode_damps.txt= | modal damping |
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| =mat/mode_modal_a.txt= | modal residues at low frequency (to be checked) |
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| =mat/mode_modal_b.txt= | modal residues at high frequency (to be checked) |
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Then we import the obtained =.txt= files on Matlab using =readtable= function.
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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_b = table2array(readtable('mat/mode_modal_b.txt', 'ReadVariableNames', false)); % [Real / Imag]
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modal_a = complex(modal_a(:, 1), modal_a(:, 2));
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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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* Obtained Mode Shapes animations
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One all the FRFs are obtained, we can estimate the modal parameters (resonance frequencies, modal shapes and modal damping) within the modal software.
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For that, multiple modal extraction techniques can be used (SIMO, MIMO, narrow band, wide band, ...).
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Then, it is possible to show the modal shapes with an animation.
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Examples are shown on figures [[fig:mode1]] and [[fig:mode6]].
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Animations of all the other modes are accessible using the following links: [[file:img/modes/mode1.gif][mode 1]], [[file:img/modes/mode2.gif][mode 2]], [[file:img/modes/mode3.gif][mode 3]], [[file:img/modes/mode4.gif][mode 4]], [[file:img/modes/mode5.gif][mode 5]], [[file:img/modes/mode6.gif][mode 6]], [[file:img/modes/mode7.gif][mode 7]], [[file:img/modes/mode8.gif][mode 8]], [[file:img/modes/mode9.gif][mode 9]], [[file:img/modes/mode10.gif][mode 10]], [[file:img/modes/mode11.gif][mode 11]], [[file:img/modes/mode12.gif][mode 12]], [[file:img/modes/mode13.gif][mode 13]], [[file:img/modes/mode14.gif][mode 14]], [[file:img/modes/mode15.gif][mode 15]], [[file:img/modes/mode16.gif][mode 16]], [[file:img/modes/mode17.gif][mode 17]], [[file:img/modes/mode18.gif][mode 18]], [[file:img/modes/mode19.gif][mode 19]], [[file:img/modes/mode20.gif][mode 20]], [[file:img/modes/mode21.gif][mode 21]].
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#+name: fig:mode1
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#+caption: Mode 1
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[[file:img/modes/mode1.gif]]
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#+name: fig:mode6
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#+caption: Mode 6
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[[file:img/modes/mode6.gif]]
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* Compute the Modal Model
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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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** Position of the accelerometers
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There are 23 accelerometers:
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- 4 on the bottom granite
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- 4 on the top granite
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- 4 on top of the translation stage
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- 4 on the tilt stage
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- 3 on top of the spindle
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- 4 on top of the hexapod
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The coordinates defined in the software are displayed below.
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#+begin_src bash :results output :exports results :eval no-export
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sed -n 18,40p modal_analysis/acc_coordinates.txt | tac --
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#+end_src
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#+RESULTS:
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#+begin_example
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1 1.0000e-001 1.0000e-001 1.1500e+000 0 Top
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2 1.0000e-001 -1.0000e-001 1.1500e+000 0 Top
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3 -1.0000e-001 -1.0000e-001 1.1500e+000 0 Top
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4 -1.0000e-001 1.0000e-001 1.1500e+000 0 Top
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5 4.0000e-001 4.0000e-001 9.5000e-001 0 inner
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6 4.0000e-001 -4.0000e-001 9.5000e-001 0 inner
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7 -4.0000e-001 -4.0000e-001 9.5000e-001 0 inner
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8 -4.0000e-001 4.0000e-001 9.5000e-001 0 inner
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9 5.0000e-001 5.0000e-001 9.0000e-001 0 outer
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10 5.0000e-001 -5.0000e-001 9.0000e-001 0 outer
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11 -5.0000e-001 -5.0000e-001 9.0000e-001 0 outer
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12 -5.0000e-001 5.0000e-001 9.0000e-001 0 outer
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13 5.5000e-001 5.5000e-001 5.5000e-001 0 top
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14 5.5000e-001 -5.5000e-001 5.5000e-001 0 top
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15 -5.5000e-001 -5.5000e-001 5.5000e-001 0 top
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16 -5.5000e-001 5.5000e-001 5.5000e-001 0 top
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17 9.5000e-001 9.5000e-001 4.0000e-001 0 low
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18 9.5000e-001 -9.5000e-001 4.0000e-001 0 low
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19 -9.5000e-001 -9.5000e-001 4.0000e-001 0 low
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20 -9.5000e-001 9.5000e-001 4.0000e-001 0 low
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21 2.0000e-001 2.0000e-001 8.5000e-001 0 bot
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22 0.0000e+000 -2.0000e-001 8.5000e-001 0 bot
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23 -2.0000e-001 2.0000e-001 8.5000e-001 0 bot
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#+end_example
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#+name: tab:acc_location
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#+caption: Location of each Accelerometer (using the normal coordinate frame with X aligned with the X ray)
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| *Node number* | *Solid Body* | *Location* | *X* | *Y* | *Z* |
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|---------------+-------------------+------------+-------+-------+------|
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| 1 | Hexapod - Top | -X/-Y | -0.10 | -0.10 | 1.15 |
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| 2 | | -X/+Y | -0.10 | 0.10 | 1.15 |
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| 3 | | +X/+Y | 0.10 | 0.10 | 1.15 |
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| 4 | | +X/-Y | 0.10 | -0.10 | 1.15 |
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|---------------+-------------------+------------+-------+-------+------|
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| 5 | Tilt - Top | -X/-Y | -0.40 | -0.40 | 0.95 |
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| 6 | | -X/+Y | -0.40 | 0.40 | 0.95 |
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| 7 | | +X/+Y | 0.40 | 0.40 | 0.95 |
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| 8 | | +X/-Y | 0.40 | -0.40 | 0.95 |
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|---------------+-------------------+------------+-------+-------+------|
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| 9 | Translation - Top | -X/-Y | -0.50 | -0.50 | 0.90 |
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| 10 | | -X/+Y | -0.50 | 0.50 | 0.90 |
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| 11 | | +X/+Y | 0.50 | 0.50 | 0.90 |
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| 12 | | +X/-Y | 0.50 | -0.50 | 0.90 |
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|---------------+-------------------+------------+-------+-------+------|
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| 13 | Top Granite | -X/-Y | -0.55 | -0.50 | 0.55 |
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| 14 | | -X/+Y | -0.55 | 0.50 | 0.55 |
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| 15 | | +X/+Y | 0.55 | 0.50 | 0.55 |
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| 16 | | +X/-Y | 0.55 | -0.50 | 0.55 |
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|---------------+-------------------+------------+-------+-------+------|
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| 17 | Bottom Granite | -X/-Y | -0.95 | -0.90 | 0.40 |
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| 18 | | -X/+Y | -0.95 | 0.90 | 0.40 |
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| 19 | | +X/+Y | 0.95 | 0.90 | 0.40 |
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| 20 | | +X/-Y | 0.95 | -0.90 | 0.40 |
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|---------------+-------------------+------------+-------+-------+------|
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| 21 | Spindle - Top | -X/-Y | -0.20 | -0.20 | 0.85 |
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| 22 | | +0/+Y | 0.00 | 0.20 | 0.85 |
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| 23 | | +X/-Y | 0.20 | -0.20 | 0.85 |
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** Define positions of the accelerometers on matlab
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We define the X-Y-Z position of each sensor.
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Each line corresponds to one accelerometer, X-Y-Z position in meter.
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#+begin_src matlab
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positions = [...
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-0.10, -0.10, 1.15 ; ...
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-0.10, 0.10, 1.15 ; ...
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0.10, 0.10, 1.15 ; ...
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0.10, -0.10, 1.15 ; ...
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-0.40, -0.40, 0.95 ; ...
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-0.40, 0.40, 0.95 ; ...
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0.40, 0.40, 0.95 ; ...
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0.40, -0.40, 0.95 ; ...
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-0.50, -0.50, 0.90 ; ...
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-0.50, 0.50, 0.90 ; ...
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0.50, 0.50, 0.90 ; ...
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0.50, -0.50, 0.90 ; ...
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-0.55, -0.50, 0.55 ; ...
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-0.55, 0.50, 0.55 ; ...
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0.55, 0.50, 0.55 ; ...
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0.55, -0.50, 0.55 ; ...
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-0.95, -0.90, 0.40 ; ...
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-0.95, 0.90, 0.40 ; ...
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0.95, 0.90, 0.40 ; ...
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0.95, -0.90, 0.40 ; ...
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-0.20, -0.20, 0.85 ; ...
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0.00, 0.20, 0.85 ; ...
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0.20, -0.20, 0.85 ];
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#+end_src
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#+begin_src matlab
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figure;
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hold on;
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fill3(positions(1:4, 1), positions(1:4, 2), positions(1:4, 3), 'k', 'FaceAlpha', 0.5)
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fill3(positions(5:8, 1), positions(5:8, 2), positions(5:8, 3), 'k', 'FaceAlpha', 0.5)
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fill3(positions(9:12, 1), positions(9:12, 2), positions(9:12, 3), 'k', 'FaceAlpha', 0.5)
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fill3(positions(13:16, 1), positions(13:16, 2), positions(13:16, 3), 'k', 'FaceAlpha', 0.5)
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fill3(positions(17:20, 1), positions(17:20, 2), positions(17:20, 3), 'k', 'FaceAlpha', 0.5)
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fill3(positions(21:23, 1), positions(21:23, 2), positions(21:23, 3), 'k', 'FaceAlpha', 0.5)
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hold off;
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#+end_src
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** Import the modal vectors on matlab
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*** Mode1
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#+begin_src bash :results output :exports none :eval no-export
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sed -n 12,80p modal_analysis/modes_propres_narband.asc
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#+end_src
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#+RESULTS:
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#+begin_example
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1X+: -9.34637e-002 4.52445e-002
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1Y+: 2.33790e-001 1.41439e-003
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1Z+: -1.73754e-002 6.02449e-003
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|
2X+: -7.42108e-002 3.91543e-002
|
|
2Y+: 2.41566e-001 -1.44869e-003
|
|
2Z+: -5.99285e-003 2.10370e-003
|
|
4X+: -1.02163e-001 2.79561e-002
|
|
4Y+: 2.29048e-001 2.89782e-002
|
|
4Z+: -2.85130e-002 1.77132e-004
|
|
5X+: -8.77132e-002 3.34081e-002
|
|
5Y+: 2.14182e-001 2.14655e-002
|
|
5Z+: -1.54521e-002 1.26682e-002
|
|
6X+: -7.90143e-002 2.42583e-002
|
|
6Y+: 2.20669e-001 2.12738e-002
|
|
6Z+: 4.60755e-002 4.96406e-003
|
|
7X+: -7.79654e-002 2.58385e-002
|
|
7Y+: 2.06861e-001 3.48019e-002
|
|
7Z+: -1.78311e-002 -1.29704e-002
|
|
8X+: -8.49357e-002 3.55200e-002
|
|
8Y+: 2.07470e-001 3.59745e-002
|
|
8Z+: -7.66974e-002 -3.19813e-003
|
|
9X+: -7.38565e-002 1.95146e-002
|
|
9Y+: 2.17403e-001 2.01550e-002
|
|
9Z+: -1.77073e-002 -3.46414e-003
|
|
10X+: -7.77587e-002 2.36700e-002
|
|
10Y+: 2.35654e-001 -2.14540e-002
|
|
10Z+: 7.94165e-002 -2.45897e-002
|
|
11X+: -8.17972e-002 2.20583e-002
|
|
11Y+: 2.20906e-001 -4.30164e-003
|
|
11Z+: -5.60520e-003 3.10187e-003
|
|
12X+: -8.64261e-002 3.66022e-002
|
|
12Y+: 2.15000e-001 -5.74661e-003
|
|
12Z+: -1.22622e-001 4.11767e-002
|
|
13X+: -4.25169e-002 1.56602e-002
|
|
13Y+: 5.31036e-002 -1.73951e-002
|
|
13Z+: -4.07130e-002 1.26884e-002
|
|
14X+: -3.85032e-002 1.29431e-002
|
|
14Y+: 5.36716e-002 -1.80868e-002
|
|
14Z+: 1.00367e-001 -3.48798e-002
|
|
15X+: -4.25524e-002 1.46363e-002
|
|
15Y+: 5.19668e-002 -1.69744e-002
|
|
15Z+: 5.89747e-003 -2.32428e-003
|
|
16X+: -4.31268e-002 1.38332e-002
|
|
16Y+: 5.07545e-002 -1.53045e-002
|
|
16Z+: -1.04172e-001 3.17984e-002
|
|
17X+: -2.69757e-002 9.07955e-003
|
|
17Y+: 3.07837e-002 -9.44663e-003
|
|
17Z+: -7.63502e-003 1.68203e-003
|
|
18X+: -3.00097e-002 9.23966e-003
|
|
18Y+: 2.83585e-002 -8.97747e-003
|
|
18Z+: 1.52467e-001 -4.78675e-002
|
|
19X+: -2.70223e-002 6.16478e-003
|
|
19Y+: 3.06149e-002 -6.25382e-003
|
|
19Z+: -4.84888e-003 1.93970e-003
|
|
20X+: -2.90976e-002 7.13184e-003
|
|
20Y+: 3.36738e-002 -7.30875e-003
|
|
20Z+: -1.66902e-001 3.93419e-002
|
|
3X+: -9.40720e-002 3.93724e-002
|
|
3Y+: 2.52307e-001 0.00000e+000
|
|
3Z+: -1.53864e-002 -9.25720e-004
|
|
21X+: -7.91940e-002 4.39648e-002
|
|
21Y+: 2.04567e-001 9.49987e-003
|
|
21Z+: -1.56087e-002 7.08838e-003
|
|
22X+: -1.01070e-001 3.13534e-002
|
|
22Y+: 1.92270e-001 1.80423e-002
|
|
22Z+: 2.93053e-003 -1.97308e-003
|
|
23X+: -8.86455e-002 4.29906e-002
|
|
23Z+: -3.38351e-002 1.81362e-003
|
|
23Y-: -1.90862e-001 -2.53414e-002
|
|
#+end_example
|
|
|
|
#+begin_src matlab
|
|
mode1 = [...
|
|
-9.34637e-002+j*4.52445e-002, +2.33790e-001+j*1.41439e-003, -1.73754e-002+j*6.02449e-003;
|
|
-7.42108e-002+j*3.91543e-002, +2.41566e-001-j*1.44869e-003, -5.99285e-003+j*2.10370e-003;
|
|
-9.40720e-002+j*3.93724e-002, +2.52307e-001+j*0.00000e+000, -1.53864e-002-j*9.25720e-004;
|
|
-1.02163e-001+j*2.79561e-002, +2.29048e-001+j*2.89782e-002, -2.85130e-002+j*1.77132e-004;
|
|
-8.77132e-002+j*3.34081e-002, +2.14182e-001+j*2.14655e-002, -1.54521e-002+j*1.26682e-002;
|
|
-7.90143e-002+j*2.42583e-002, +2.20669e-001+j*2.12738e-002, +4.60755e-002+j*4.96406e-003;
|
|
-7.79654e-002+j*2.58385e-002, +2.06861e-001+j*3.48019e-002, -1.78311e-002-j*1.29704e-002;
|
|
-8.49357e-002+j*3.55200e-002, +2.07470e-001+j*3.59745e-002, -7.66974e-002-j*3.19813e-003;
|
|
-7.38565e-002+j*1.95146e-002, +2.17403e-001+j*2.01550e-002, -1.77073e-002-j*3.46414e-003;
|
|
-7.77587e-002+j*2.36700e-002, +2.35654e-001-j*2.14540e-002, +7.94165e-002-j*2.45897e-002;
|
|
-8.17972e-002+j*2.20583e-002, +2.20906e-001-j*4.30164e-003, -5.60520e-003+j*3.10187e-003;
|
|
-8.64261e-002+j*3.66022e-002, +2.15000e-001-j*5.74661e-003, -1.22622e-001+j*4.11767e-002;
|
|
-4.25169e-002+j*1.56602e-002, +5.31036e-002-j*1.73951e-002, -4.07130e-002+j*1.26884e-002;
|
|
-3.85032e-002+j*1.29431e-002, +5.36716e-002-j*1.80868e-002, +1.00367e-001-j*3.48798e-002;
|
|
-4.25524e-002+j*1.46363e-002, +5.19668e-002-j*1.69744e-002, +5.89747e-003-j*2.32428e-003;
|
|
-4.31268e-002+j*1.38332e-002, +5.07545e-002-j*1.53045e-002, -1.04172e-001+j*3.17984e-002;
|
|
-2.69757e-002+j*9.07955e-003, +3.07837e-002-j*9.44663e-003, -7.63502e-003+j*1.68203e-003;
|
|
-3.00097e-002+j*9.23966e-003, +2.83585e-002-j*8.97747e-003, +1.52467e-001-j*4.78675e-002;
|
|
-2.70223e-002+j*6.16478e-003, +3.06149e-002-j*6.25382e-003, -4.84888e-003+j*1.93970e-003;
|
|
-2.90976e-002+j*7.13184e-003, +3.36738e-002-j*7.30875e-003, -1.66902e-001+j*3.93419e-002;
|
|
-7.91940e-002+j*4.39648e-002, +2.04567e-001+j*9.49987e-003, -1.56087e-002+j*7.08838e-003;
|
|
-1.01070e-001+j*3.13534e-002, +1.92270e-001+j*1.80423e-002, +2.93053e-003-j*1.97308e-003;
|
|
-8.86455e-002+j*4.29906e-002, +1.90862e-001+j*2.53414e-002, -3.38351e-002+j*1.81362e-003];
|
|
#+end_src
|
|
|
|
*** Mode2
|
|
#+begin_src bash :results output :exports none :eval no-export
|
|
sed -n 88,156p modal_analysis/modes_propres_narband.asc
|
|
#+end_src
|
|
|
|
#+RESULTS:
|
|
#+begin_example
|
|
1X+: 7.56931e-002 3.61548e-002
|
|
1Y+: 2.07574e-001 1.69205e-004
|
|
1Z+: 1.29733e-002 -6.78426e-004
|
|
2X+: 8.58732e-002 2.54470e-002
|
|
2Y+: 2.07117e-001 -1.31755e-003
|
|
2Z+: -2.13788e-003 -1.24974e-002
|
|
4X+: 7.09825e-002 3.66313e-002
|
|
4Y+: 2.09969e-001 1.11484e-002
|
|
4Z+: 9.19478e-003 3.47272e-002
|
|
5X+: 6.23935e-002 1.02488e-002
|
|
5Y+: 2.30687e-001 -3.58416e-003
|
|
5Z+: 3.27122e-002 -5.85468e-002
|
|
6X+: 7.61163e-002 -2.43630e-002
|
|
6Y+: 2.26743e-001 -1.15334e-002
|
|
6Z+: -6.20205e-003 -1.21742e-001
|
|
7X+: 8.01824e-002 -1.94769e-002
|
|
7Y+: 1.97485e-001 4.50105e-002
|
|
7Z+: -2.21170e-002 9.77052e-002
|
|
8X+: 6.19294e-002 8.15075e-003
|
|
8Y+: 2.03864e-001 4.45835e-002
|
|
8Z+: 2.55133e-002 1.36137e-001
|
|
9X+: 4.38135e-002 7.30537e-002
|
|
9Y+: 2.28426e-001 -6.58868e-003
|
|
9Z+: 1.16313e-002 5.09427e-004
|
|
10X+: 5.45770e-002 4.34251e-002
|
|
10Y+: 2.50823e-001 0.00000e+000
|
|
10Z+: -4.63460e-002 -4.76868e-002
|
|
11X+: 5.50987e-002 4.26178e-002
|
|
11Y+: 2.29394e-001 5.78236e-002
|
|
11Z+: 1.90158e-002 1.09139e-002
|
|
12X+: 4.98867e-002 7.30190e-002
|
|
12Y+: 2.07871e-001 4.57750e-002
|
|
12Z+: 6.69433e-002 9.00315e-002
|
|
13X+: 2.48819e-002 3.03222e-002
|
|
13Y+: -2.56046e-002 -3.34132e-002
|
|
13Z+: 2.13260e-002 2.58544e-002
|
|
14X+: 2.45706e-002 2.60221e-002
|
|
14Y+: -2.57723e-002 -3.35612e-002
|
|
14Z+: -5.71282e-002 -6.61562e-002
|
|
15X+: 2.68196e-002 2.83888e-002
|
|
15Y+: -2.57263e-002 -3.29627e-002
|
|
15Z+: -2.11722e-003 -3.37239e-003
|
|
16X+: 2.51442e-002 3.32558e-002
|
|
16Y+: -2.54372e-002 -3.25062e-002
|
|
16Z+: 5.65780e-002 7.64142e-002
|
|
17X+: 1.62437e-002 1.94534e-002
|
|
17Y+: -1.31293e-002 -2.05924e-002
|
|
17Z+: 1.05274e-003 3.59474e-003
|
|
18X+: 1.83431e-002 2.03836e-002
|
|
18Y+: -1.16818e-002 -1.86334e-002
|
|
18Z+: -8.66632e-002 -1.08216e-001
|
|
19X+: 1.62553e-002 1.79588e-002
|
|
19Y+: -1.28857e-002 -1.90512e-002
|
|
19Z+: 6.25653e-003 4.97733e-003
|
|
20X+: 1.63830e-002 2.03943e-002
|
|
20Y+: -1.48941e-002 -2.11717e-002
|
|
20Z+: 8.68045e-002 1.16491e-001
|
|
3X+: 8.17201e-002 2.36079e-002
|
|
3Y+: 2.15927e-001 1.61300e-002
|
|
3Z+: -5.48456e-004 2.55691e-002
|
|
21X+: 6.79204e-002 -5.55513e-002
|
|
21Y+: 2.32871e-001 2.33389e-002
|
|
21Z+: 1.34345e-002 -2.31815e-002
|
|
22X+: 4.02414e-002 -8.38957e-002
|
|
22Y+: 2.35273e-001 2.73256e-002
|
|
22Z+: -8.51632e-003 -7.49635e-003
|
|
23X+: 6.18293e-002 -5.99671e-002
|
|
23Z+: 1.63533e-002 6.09161e-002
|
|
23Y-: -2.37693e-001 -4.34204e-002
|
|
#+end_example
|
|
|
|
#+begin_src matlab
|
|
mode2 = [...
|
|
+7.56931e-002+j*3.61548e-002, +2.07574e-001+j*1.69205e-004, +1.29733e-002-j*6.78426e-004;
|
|
+8.58732e-002+j*2.54470e-002, +2.07117e-001-j*1.31755e-003, -2.13788e-003-j*1.24974e-002;
|
|
+8.17201e-002+j*2.36079e-002, +2.15927e-001+j*1.61300e-002, -5.48456e-004+j*2.55691e-002;
|
|
+7.09825e-002+j*3.66313e-002, +2.09969e-001+j*1.11484e-002, +9.19478e-003+j*3.47272e-002;
|
|
+6.23935e-002+j*1.02488e-002, +2.30687e-001-j*3.58416e-003, +3.27122e-002-j*5.85468e-002;
|
|
+7.61163e-002-j*2.43630e-002, +2.26743e-001-j*1.15334e-002, -6.20205e-003-j*1.21742e-001;
|
|
+8.01824e-002-j*1.94769e-002, +1.97485e-001+j*4.50105e-002, -2.21170e-002+j*9.77052e-002;
|
|
+6.19294e-002+j*8.15075e-003, +2.03864e-001+j*4.45835e-002, +2.55133e-002+j*1.36137e-001;
|
|
+4.38135e-002+j*7.30537e-002, +2.28426e-001-j*6.58868e-003, +1.16313e-002+j*5.09427e-004;
|
|
+5.45770e-002+j*4.34251e-002, +2.50823e-001+j*0.00000e+000, -4.63460e-002-j*4.76868e-002;
|
|
+5.50987e-002+j*4.26178e-002, +2.29394e-001+j*5.78236e-002, +1.90158e-002+j*1.09139e-002;
|
|
+4.98867e-002+j*7.30190e-002, +2.07871e-001+j*4.57750e-002, +6.69433e-002+j*9.00315e-002;
|
|
+2.48819e-002+j*3.03222e-002, -2.56046e-002-j*3.34132e-002, +2.13260e-002+j*2.58544e-002;
|
|
+2.45706e-002+j*2.60221e-002, -2.57723e-002-j*3.35612e-002, -5.71282e-002-j*6.61562e-002;
|
|
+2.68196e-002+j*2.83888e-002, -2.57263e-002-j*3.29627e-002, -2.11722e-003-j*3.37239e-003;
|
|
+2.51442e-002+j*3.32558e-002, -2.54372e-002-j*3.25062e-002, +5.65780e-002+j*7.64142e-002;
|
|
+1.62437e-002+j*1.94534e-002, -1.31293e-002-j*2.05924e-002, +1.05274e-003+j*3.59474e-003;
|
|
+1.83431e-002+j*2.03836e-002, -1.16818e-002-j*1.86334e-002, -8.66632e-002-j*1.08216e-001;
|
|
+1.62553e-002+j*1.79588e-002, -1.28857e-002-j*1.90512e-002, +6.25653e-003+j*4.97733e-003;
|
|
+1.63830e-002+j*2.03943e-002, -1.48941e-002-j*2.11717e-002, +8.68045e-002+j*1.16491e-001;
|
|
+6.79204e-002-j*5.55513e-002, +2.32871e-001+j*2.33389e-002, +1.34345e-002-j*2.31815e-002;
|
|
+4.02414e-002-j*8.38957e-002, +2.35273e-001+j*2.73256e-002, -8.51632e-003-j*7.49635e-003;
|
|
+6.18293e-002-j*5.99671e-002, +2.37693e-001+j*4.34204e-002, +1.63533e-002+j*6.09161e-002]
|
|
#+end_src
|
|
|
|
*** Mode3
|
|
#+begin_src bash :results output :exports none :eval no-export
|
|
sed -n 164,232p modal_analysis/modes_propres_narband.asc
|
|
#+end_src
|
|
|
|
#+RESULTS:
|
|
#+begin_example
|
|
1X+: 1.34688e-001 -6.65071e-002
|
|
1Y+: 1.55316e-002 1.01277e-002
|
|
1Z+: -5.88466e-002 1.14294e-002
|
|
2X+: 1.53934e-001 -9.76990e-003
|
|
2Y+: 7.17487e-003 1.11925e-002
|
|
2Z+: -4.57205e-002 7.26573e-003
|
|
4X+: 1.37298e-001 -5.24661e-002
|
|
4Y+: 1.19427e-003 -5.39240e-002
|
|
4Z+: -1.25915e-002 5.38133e-003
|
|
5X+: 2.43192e-001 -3.17374e-002
|
|
5Y+: -2.15730e-001 -7.69941e-004
|
|
5Z+: -1.56268e-001 1.44118e-002
|
|
6X+: -7.27705e-002 -3.54943e-003
|
|
6Y+: -2.47706e-001 2.66480e-003
|
|
6Z+: -1.21590e-001 1.06054e-002
|
|
7X+: -7.25870e-002 -4.62024e-003
|
|
7Y+: 2.27073e-001 -3.69315e-002
|
|
7Z+: 1.22611e-001 -6.67337e-003
|
|
8X+: 2.32731e-001 -2.85516e-002
|
|
8Y+: 2.35389e-001 -3.81905e-002
|
|
8Z+: 5.35574e-002 4.30394e-004
|
|
9X+: 2.64170e-001 -2.67367e-002
|
|
9Y+: -2.56227e-001 3.97957e-005
|
|
9Z+: -1.95398e-001 2.23549e-002
|
|
10X+: -1.66953e-002 -7.95698e-003
|
|
10Y+: -2.66547e-001 -2.17687e-002
|
|
10Z+: 1.56278e-002 2.23786e-003
|
|
11X+: -3.42364e-002 -9.30205e-003
|
|
11Y+: 2.52340e-001 -7.47237e-003
|
|
11Z+: -9.51643e-004 3.64798e-003
|
|
12X+: 2.97574e-001 0.00000e+000
|
|
12Y+: 2.23170e-001 -1.37831e-002
|
|
12Z+: 1.06266e-001 2.30324e-003
|
|
13X+: 2.67178e-002 -4.15723e-004
|
|
13Y+: 6.75423e-003 -2.18428e-003
|
|
13Z+: -1.69423e-002 3.12395e-003
|
|
14X+: -1.12283e-002 2.86316e-004
|
|
14Y+: 5.08225e-003 -2.14053e-003
|
|
14Z+: 2.18339e-002 -3.25204e-003
|
|
15X+: -1.17948e-002 6.82873e-004
|
|
15Y+: 1.94914e-002 -2.42151e-003
|
|
15Z+: 2.68660e-003 -2.92104e-004
|
|
16X+: 1.19490e-002 1.72236e-005
|
|
16Y+: 1.83552e-002 -2.71289e-003
|
|
16Z+: -2.70914e-002 4.84164e-003
|
|
17X+: 1.00173e-002 -5.80552e-005
|
|
17Y+: -3.87262e-003 -1.19607e-003
|
|
17Z+: -8.53809e-003 1.48424e-003
|
|
18X+: -1.22262e-002 5.13096e-004
|
|
18Y+: -5.73905e-003 -1.07659e-003
|
|
18Z+: 3.51730e-002 -6.13814e-003
|
|
19X+: -1.43735e-002 -4.78552e-004
|
|
19Y+: 2.31135e-002 -6.30554e-004
|
|
19Z+: 1.80171e-003 -1.98835e-004
|
|
20X+: 9.17792e-003 5.36661e-004
|
|
20Y+: 2.18969e-002 -5.81759e-004
|
|
20Z+: -3.72117e-002 5.35813e-003
|
|
3X+: 1.61551e-001 1.65478e-002
|
|
3Y+: -4.12527e-004 -5.60909e-002
|
|
3Z+: -9.00640e-003 3.50754e-003
|
|
21X+: 3.38754e-002 -3.38703e-002
|
|
21Y+: -2.20843e-002 2.78581e-002
|
|
21Z+: -8.79541e-002 -3.67473e-003
|
|
22X+: 3.93064e-002 4.69476e-002
|
|
22Y+: -1.69132e-002 -1.04606e-002
|
|
22Z+: -1.85351e-002 1.33750e-003
|
|
23X+: 3.60396e-002 -2.46238e-002
|
|
23Z+: 3.57722e-003 3.64827e-003
|
|
23Y-: 1.92038e-002 6.65895e-002
|
|
#+end_example
|
|
|
|
#+begin_src matlab
|
|
mode3 = [...
|
|
+1.34688e-001-j*6.65071e-002, +1.55316e-002+j*1.01277e-002, -5.88466e-002+j*1.14294e-002;
|
|
+1.53934e-001-j*9.76990e-003, +7.17487e-003+j*1.11925e-002, -4.57205e-002+j*7.26573e-003;
|
|
+1.61551e-001+j*1.65478e-002, -4.12527e-004-j*5.60909e-002, -9.00640e-003+j*3.50754e-003;
|
|
+1.37298e-001-j*5.24661e-002, +1.19427e-003-j*5.39240e-002, -1.25915e-002+j*5.38133e-003;
|
|
+2.43192e-001-j*3.17374e-002, -2.15730e-001-j*7.69941e-004, -1.56268e-001+j*1.44118e-002;
|
|
-7.27705e-002-j*3.54943e-003, -2.47706e-001+j*2.66480e-003, -1.21590e-001+j*1.06054e-002;
|
|
-7.25870e-002-j*4.62024e-003, +2.27073e-001-j*3.69315e-002, +1.22611e-001-j*6.67337e-003;
|
|
+2.32731e-001-j*2.85516e-002, +2.35389e-001-j*3.81905e-002, +5.35574e-002+j*4.30394e-004;
|
|
+2.64170e-001-j*2.67367e-002, -2.56227e-001+j*3.97957e-005, -1.95398e-001+j*2.23549e-002;
|
|
-1.66953e-002-j*7.95698e-003, -2.66547e-001-j*2.17687e-002, +1.56278e-002+j*2.23786e-003;
|
|
-3.42364e-002-j*9.30205e-003, +2.52340e-001-j*7.47237e-003, -9.51643e-004+j*3.64798e-003;
|
|
+2.97574e-001+j*0.00000e+000, +2.23170e-001-j*1.37831e-002, +1.06266e-001+j*2.30324e-003;
|
|
+2.67178e-002-j*4.15723e-004, +6.75423e-003-j*2.18428e-003, -1.69423e-002+j*3.12395e-003;
|
|
-1.12283e-002+j*2.86316e-004, +5.08225e-003-j*2.14053e-003, +2.18339e-002-j*3.25204e-003;
|
|
-1.17948e-002+j*6.82873e-004, +1.94914e-002-j*2.42151e-003, +2.68660e-003-j*2.92104e-004;
|
|
+1.19490e-002+j*1.72236e-005, +1.83552e-002-j*2.71289e-003, -2.70914e-002+j*4.84164e-003;
|
|
+1.00173e-002-j*5.80552e-005, -3.87262e-003-j*1.19607e-003, -8.53809e-003+j*1.48424e-003;
|
|
-1.22262e-002+j*5.13096e-004, -5.73905e-003-j*1.07659e-003, +3.51730e-002-j*6.13814e-003;
|
|
-1.43735e-002-j*4.78552e-004, +2.31135e-002-j*6.30554e-004, +1.80171e-003-j*1.98835e-004;
|
|
+9.17792e-003+j*5.36661e-004, +2.18969e-002-j*5.81759e-004, -3.72117e-002+j*5.35813e-003;
|
|
+3.38754e-002-j*3.38703e-002, -2.20843e-002+j*2.78581e-002, -8.79541e-002-j*3.67473e-003;
|
|
+3.93064e-002+j*4.69476e-002, -1.69132e-002-j*1.04606e-002, -1.85351e-002+j*1.33750e-003;
|
|
+3.60396e-002-j*2.46238e-002, -1.92038e-002-j*6.65895e-002, +3.57722e-003+j*3.64827e-003];
|
|
#+end_src
|
|
|
|
*** Mode4
|
|
#+begin_src bash :results output :exports none :eval no-export
|
|
sed -n 240,308p modal_analysis/modes_propres_narband.asc
|
|
#+end_src
|
|
|
|
#+RESULTS:
|
|
#+begin_example
|
|
1X+: -1.02501e-001 -1.43802e-001
|
|
1Y+: -1.07971e-001 5.61418e-004
|
|
1Z+: 1.87145e-001 -1.03605e-001
|
|
2X+: -9.44764e-002 -1.36856e-001
|
|
2Y+: -1.04428e-001 5.27790e-003
|
|
2Z+: 1.60710e-001 -7.74212e-002
|
|
4X+: -9.17242e-002 -1.36656e-001
|
|
4Y+: -1.34249e-001 -1.03884e-002
|
|
4Z+: 1.92123e-001 -1.25627e-001
|
|
5X+: 1.05875e-002 -1.03886e-001
|
|
5Y+: -8.26338e-002 3.58498e-002
|
|
5Z+: 2.55819e-001 -6.94290e-003
|
|
6X+: -4.58970e-002 -1.33904e-002
|
|
6Y+: -9.41660e-002 4.99682e-002
|
|
6Z+: 1.28276e-001 4.59685e-002
|
|
7X+: -6.01521e-002 -1.30165e-002
|
|
7Y+: 2.56439e-003 -6.78141e-002
|
|
7Z+: 5.03428e-002 -1.59420e-001
|
|
8X+: -1.00895e-002 -8.80550e-002
|
|
8Y+: 1.26327e-002 -8.14444e-002
|
|
8Z+: 1.59506e-001 -2.05360e-001
|
|
9X+: -3.04658e-003 -1.57921e-001
|
|
9Y+: -8.23501e-002 4.82748e-002
|
|
9Z+: 1.69315e-001 1.22804e-002
|
|
10X+: -8.25875e-002 -7.31038e-002
|
|
10Y+: -1.08668e-001 3.56364e-002
|
|
10Z+: 8.28567e-002 -4.49596e-003
|
|
11X+: -1.06792e-001 -6.95394e-002
|
|
11Y+: 3.77195e-002 -7.65410e-002
|
|
11Z+: 8.00590e-003 -2.32461e-002
|
|
12X+: -4.84292e-002 -1.45790e-001
|
|
12Y+: 1.03862e-002 -7.31212e-002
|
|
12Z+: 1.78122e-001 -1.00939e-001
|
|
13X+: -3.49891e-002 -6.20969e-003
|
|
13Y+: -1.18504e-002 -1.94225e-002
|
|
13Z+: 4.13007e-002 7.67087e-003
|
|
14X+: -3.55795e-002 1.16708e-003
|
|
14Y+: -1.68128e-002 -1.82344e-002
|
|
14Z+: 3.92416e-002 -3.64434e-002
|
|
15X+: -3.45304e-002 3.78185e-003
|
|
15Y+: -7.62559e-003 -2.24241e-002
|
|
15Z+: 6.28286e-003 -1.32711e-002
|
|
16X+: -9.95646e-003 -6.04395e-003
|
|
16Y+: -8.73465e-003 -2.20807e-002
|
|
16Z+: 3.56946e-002 1.69231e-002
|
|
17X+: -9.32661e-003 -5.51944e-003
|
|
17Y+: -1.91087e-002 -9.09191e-003
|
|
17Z+: 4.04981e-002 8.38685e-004
|
|
18X+: -2.84456e-002 4.02762e-003
|
|
18Y+: -2.20044e-002 -8.86197e-003
|
|
18Z+: 4.43051e-002 -5.21033e-002
|
|
19X+: -3.27019e-002 3.59765e-003
|
|
19Y+: 2.93163e-003 -2.05064e-002
|
|
19Z+: -1.77289e-002 -1.29477e-002
|
|
20X+: -1.08474e-002 -5.78419e-003
|
|
20Y+: 3.86759e-003 -1.91642e-002
|
|
20Z+: 2.10135e-002 3.18051e-002
|
|
3X+: -9.11657e-002 -1.36611e-001
|
|
3Y+: -1.78165e-001 -3.47193e-002
|
|
3Z+: 2.37121e-001 -4.96494e-002
|
|
21X+: -1.34808e-002 -9.69121e-003
|
|
21Y+: 1.25218e-002 -2.71411e-002
|
|
21Z+: 2.76673e-001 0.00000e+000
|
|
22X+: 1.96744e-003 4.90797e-003
|
|
22Y+: -9.82609e-004 -3.31065e-002
|
|
22Z+: 1.79246e-001 -3.33238e-002
|
|
23X+: -1.08728e-002 -8.80278e-003
|
|
23Z+: 2.30814e-001 -8.33151e-002
|
|
23Y-: 1.15217e-002 4.01143e-002
|
|
#+end_example
|
|
|
|
#+begin_src matlab
|
|
mode4 = [...
|
|
-1.02501e-001-j*1.43802e-001, -1.07971e-001+j*5.61418e-004, +1.87145e-001-j*1.03605e-001;
|
|
-9.44764e-002-j*1.36856e-001, -1.04428e-001+j*5.27790e-003, +1.60710e-001-j*7.74212e-002;
|
|
-9.11657e-002-j*1.36611e-001, -1.78165e-001-j*3.47193e-002, +2.37121e-001-j*4.96494e-002;
|
|
-9.17242e-002-j*1.36656e-001, -1.34249e-001-j*1.03884e-002, +1.92123e-001-j*1.25627e-001;
|
|
+1.05875e-002-j*1.03886e-001, -8.26338e-002+j*3.58498e-002, +2.55819e-001-j*6.94290e-003;
|
|
-4.58970e-002-j*1.33904e-002, -9.41660e-002+j*4.99682e-002, +1.28276e-001+j*4.59685e-002;
|
|
-6.01521e-002-j*1.30165e-002, +2.56439e-003-j*6.78141e-002, +5.03428e-002-j*1.59420e-001;
|
|
-1.00895e-002-j*8.80550e-002, +1.26327e-002-j*8.14444e-002, +1.59506e-001-j*2.05360e-001;
|
|
-3.04658e-003-j*1.57921e-001, -8.23501e-002+j*4.82748e-002, +1.69315e-001+j*1.22804e-002;
|
|
-8.25875e-002-j*7.31038e-002, -1.08668e-001+j*3.56364e-002, +8.28567e-002-j*4.49596e-003;
|
|
-1.06792e-001-j*6.95394e-002, +3.77195e-002-j*7.65410e-002, +8.00590e-003-j*2.32461e-002;
|
|
-4.84292e-002-j*1.45790e-001, +1.03862e-002-j*7.31212e-002, +1.78122e-001-j*1.00939e-001;
|
|
-3.49891e-002-j*6.20969e-003, -1.18504e-002-j*1.94225e-002, +4.13007e-002+j*7.67087e-003;
|
|
-3.55795e-002+j*1.16708e-003, -1.68128e-002-j*1.82344e-002, +3.92416e-002-j*3.64434e-002;
|
|
-3.45304e-002+j*3.78185e-003, -7.62559e-003-j*2.24241e-002, +6.28286e-003-j*1.32711e-002;
|
|
-9.95646e-003-j*6.04395e-003, -8.73465e-003-j*2.20807e-002, +3.56946e-002+j*1.69231e-002;
|
|
-9.32661e-003-j*5.51944e-003, -1.91087e-002-j*9.09191e-003, +4.04981e-002+j*8.38685e-004;
|
|
-2.84456e-002+j*4.02762e-003, -2.20044e-002-j*8.86197e-003, +4.43051e-002-j*5.21033e-002;
|
|
-3.27019e-002+j*3.59765e-003, +2.93163e-003-j*2.05064e-002, -1.77289e-002-j*1.29477e-002;
|
|
-1.08474e-002-j*5.78419e-003, +3.86759e-003-j*1.91642e-002, +2.10135e-002+j*3.18051e-002;
|
|
-1.34808e-002-j*9.69121e-003, +1.25218e-002-j*2.71411e-002, +2.76673e-001+j*0.00000e+000;
|
|
+1.96744e-003+j*4.90797e-003, -9.82609e-004-j*3.31065e-002, +1.79246e-001-j*3.33238e-002;
|
|
-1.08728e-002-j*8.80278e-003, -1.15217e-002-j*4.01143e-002, +2.30814e-001-j*8.33151e-002];
|
|
#+end_src
|
|
|
|
*** All modes
|
|
#+begin_src matlab
|
|
mode_shapes = zeros(23, 3, 10);
|
|
|
|
mode_shapes(:, :, 1) = [...
|
|
-9.34637e-002+j*4.52445e-002, +2.33790e-001+j*1.41439e-003, -1.73754e-002+j*6.02449e-003;
|
|
-7.42108e-002+j*3.91543e-002, +2.41566e-001-j*1.44869e-003, -5.99285e-003+j*2.10370e-003;
|
|
-9.40720e-002+j*3.93724e-002, +2.52307e-001+j*0.00000e+000, -1.53864e-002-j*9.25720e-004;
|
|
-1.02163e-001+j*2.79561e-002, +2.29048e-001+j*2.89782e-002, -2.85130e-002+j*1.77132e-004;
|
|
-8.77132e-002+j*3.34081e-002, +2.14182e-001+j*2.14655e-002, -1.54521e-002+j*1.26682e-002;
|
|
-7.90143e-002+j*2.42583e-002, +2.20669e-001+j*2.12738e-002, +4.60755e-002+j*4.96406e-003;
|
|
-7.79654e-002+j*2.58385e-002, +2.06861e-001+j*3.48019e-002, -1.78311e-002-j*1.29704e-002;
|
|
-8.49357e-002+j*3.55200e-002, +2.07470e-001+j*3.59745e-002, -7.66974e-002-j*3.19813e-003;
|
|
-7.38565e-002+j*1.95146e-002, +2.17403e-001+j*2.01550e-002, -1.77073e-002-j*3.46414e-003;
|
|
-7.77587e-002+j*2.36700e-002, +2.35654e-001-j*2.14540e-002, +7.94165e-002-j*2.45897e-002;
|
|
-8.17972e-002+j*2.20583e-002, +2.20906e-001-j*4.30164e-003, -5.60520e-003+j*3.10187e-003;
|
|
-8.64261e-002+j*3.66022e-002, +2.15000e-001-j*5.74661e-003, -1.22622e-001+j*4.11767e-002;
|
|
-4.25169e-002+j*1.56602e-002, +5.31036e-002-j*1.73951e-002, -4.07130e-002+j*1.26884e-002;
|
|
-3.85032e-002+j*1.29431e-002, +5.36716e-002-j*1.80868e-002, +1.00367e-001-j*3.48798e-002;
|
|
-4.25524e-002+j*1.46363e-002, +5.19668e-002-j*1.69744e-002, +5.89747e-003-j*2.32428e-003;
|
|
-4.31268e-002+j*1.38332e-002, +5.07545e-002-j*1.53045e-002, -1.04172e-001+j*3.17984e-002;
|
|
-2.69757e-002+j*9.07955e-003, +3.07837e-002-j*9.44663e-003, -7.63502e-003+j*1.68203e-003;
|
|
-3.00097e-002+j*9.23966e-003, +2.83585e-002-j*8.97747e-003, +1.52467e-001-j*4.78675e-002;
|
|
-2.70223e-002+j*6.16478e-003, +3.06149e-002-j*6.25382e-003, -4.84888e-003+j*1.93970e-003;
|
|
-2.90976e-002+j*7.13184e-003, +3.36738e-002-j*7.30875e-003, -1.66902e-001+j*3.93419e-002;
|
|
-7.91940e-002+j*4.39648e-002, +2.04567e-001+j*9.49987e-003, -1.56087e-002+j*7.08838e-003;
|
|
-1.01070e-001+j*3.13534e-002, +1.92270e-001+j*1.80423e-002, +2.93053e-003-j*1.97308e-003;
|
|
-8.86455e-002+j*4.29906e-002, -3.38351e-002+j*1.81362e-003, +1.90862e-001+j*2.53414e-002];
|
|
|
|
mode_shapes(:, :, 2) = [...
|
|
+7.56931e-002+j*3.61548e-002, +2.07574e-001+j*1.69205e-004, +1.29733e-002-j*6.78426e-004;
|
|
+8.58732e-002+j*2.54470e-002, +2.07117e-001-j*1.31755e-003, -2.13788e-003-j*1.24974e-002;
|
|
+8.17201e-002+j*2.36079e-002, +2.15927e-001+j*1.61300e-002, -5.48456e-004+j*2.55691e-002;
|
|
+7.09825e-002+j*3.66313e-002, +2.09969e-001+j*1.11484e-002, +9.19478e-003+j*3.47272e-002;
|
|
+6.23935e-002+j*1.02488e-002, +2.30687e-001-j*3.58416e-003, +3.27122e-002-j*5.85468e-002;
|
|
+7.61163e-002-j*2.43630e-002, +2.26743e-001-j*1.15334e-002, -6.20205e-003-j*1.21742e-001;
|
|
+8.01824e-002-j*1.94769e-002, +1.97485e-001+j*4.50105e-002, -2.21170e-002+j*9.77052e-002;
|
|
+6.19294e-002+j*8.15075e-003, +2.03864e-001+j*4.45835e-002, +2.55133e-002+j*1.36137e-001;
|
|
+4.38135e-002+j*7.30537e-002, +2.28426e-001-j*6.58868e-003, +1.16313e-002+j*5.09427e-004;
|
|
+5.45770e-002+j*4.34251e-002, +2.50823e-001+j*0.00000e+000, -4.63460e-002-j*4.76868e-002;
|
|
+5.50987e-002+j*4.26178e-002, +2.29394e-001+j*5.78236e-002, +1.90158e-002+j*1.09139e-002;
|
|
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|
|
+2.28149e-002-j*8.22905e-003, -4.83167e-003-j*3.10509e-003, -2.10958e-002-j*1.33421e-002;
|
|
-1.83145e-002+j*2.76844e-002, +5.61668e-003-j*1.41226e-002, +2.20876e-002-j*2.27446e-002;
|
|
-5.42112e-003+j*2.22444e-002, -4.20426e-005-j*8.78901e-003, +5.54714e-003+j*2.71564e-002;
|
|
-1.70108e-002+j*2.83751e-002, -1.44473e-002+j*4.50880e-002, -5.26736e-003-j*5.75716e-003];
|
|
|
|
mode_shapes(:, :, 8) = [...
|
|
-1.40928e-001+j*1.28570e-001, +2.95471e-001-j*1.35692e-001, -6.61656e-002+j*2.95705e-002;
|
|
-1.56673e-001+j*5.19030e-002, +3.08231e-001-j*1.41453e-001, -9.70918e-002+j*4.95018e-002;
|
|
-1.72505e-001+j*2.26273e-002, +3.97224e-001-j*2.77556e-017, -1.29223e-001+j*4.47412e-002;
|
|
-1.69978e-001+j*1.16284e-001, +3.37516e-001+j*7.69873e-003, -8.49480e-002+j*2.17071e-002;
|
|
+5.38303e-004-j*3.59916e-003, -6.72455e-002-j*2.06230e-002, +2.66448e-002+j*4.10505e-002;
|
|
-7.96526e-003-j*7.76851e-002, -2.63530e-002-j*3.75474e-002, -1.05984e-001+j*9.32474e-002;
|
|
+4.72518e-002-j*1.00199e-001, -5.50664e-002+j*1.50246e-001, -8.50976e-002+j*2.78531e-002;
|
|
+7.58419e-003-j*8.61594e-003, -9.02101e-002+j*1.58224e-001, +6.03081e-002-j*1.24162e-001;
|
|
+2.99027e-002-j*5.27128e-002, -8.80464e-002-j*2.99113e-004, +9.02851e-002-j*3.99771e-002;
|
|
+3.23132e-002-j*5.87278e-002, +3.81174e-002-j*3.69992e-002, +1.16643e-002+j*6.21068e-002;
|
|
+5.79795e-002-j*8.33565e-002, -1.22448e-003+j*8.81473e-002, +8.40150e-002-j*1.16264e-002;
|
|
+1.35399e-002-j*3.80303e-002, -9.58200e-002+j*1.46531e-001, +1.06769e-001-j*8.97034e-002;
|
|
+5.34299e-004+j*1.35179e-002, +8.71327e-004-j*6.41448e-003, +3.33208e-002-j*2.12545e-002;
|
|
-2.79263e-004+j*5.08578e-003, -1.45476e-003-j*7.65161e-003, +6.98235e-002-j*2.45395e-002;
|
|
-9.22822e-005+j*7.03205e-003, -5.62836e-003-j*2.79991e-003, +3.99717e-002-j*8.30891e-003;
|
|
+1.87833e-002+j*3.26772e-003, -4.86774e-003-j*4.32297e-003, +5.97375e-002-j*1.77542e-002;
|
|
+1.14169e-002+j*5.70930e-003, -8.23489e-003-j*4.53684e-003, +3.14016e-002-j*2.50637e-002;
|
|
+1.15995e-003+j*5.79180e-003, -6.69740e-003-j*4.66433e-003, +8.17695e-002-j*2.78384e-002;
|
|
+5.23838e-004+j*6.46432e-003, +3.36104e-003-j*4.42572e-003, +3.64589e-002+j*5.74796e-004;
|
|
+1.57042e-002+j*5.94177e-003, -1.49670e-003-j*4.22955e-003, +8.68520e-002-j*1.43981e-002;
|
|
+8.00706e-004-j*2.91734e-002, +1.20708e-002+j*4.24081e-002, -5.91796e-002+j*4.00346e-002;
|
|
-1.91799e-003-j*1.37294e-002, +9.85285e-003+j*3.17934e-002, -1.78010e-001+j*7.91267e-002;
|
|
+3.57271e-003-j*3.09959e-002, -5.77781e-002-j*1.27957e-002, +1.31025e-002+j*1.92303e-002];
|
|
|
|
mode_shapes(:, :, 9) = [...
|
|
+1.58897e-002+j*3.23763e-002, -1.23332e-001-j*3.20376e-002, +6.78860e-002-j*1.28743e-002;
|
|
+5.67179e-003+j*4.26539e-002, -1.20726e-001-j*4.15603e-002, +7.66846e-002-j*1.24290e-002;
|
|
+2.24198e-002+j*3.45953e-002, -1.03213e-001-j*4.97049e-002, +5.49175e-002-j*5.50883e-003;
|
|
+2.68792e-002+j*2.97222e-002, -1.17598e-001-j*3.13791e-002, +5.59736e-002-j*1.71122e-002;
|
|
+1.39733e-002-j*1.56260e-002, +1.43952e-001+j*2.28119e-002, -4.56377e-003+j*4.88790e-002;
|
|
-2.18507e-002-j*1.25664e-002, +6.15387e-002-j*6.31793e-003, +3.05342e-002+j*3.24595e-002;
|
|
-7.85412e-003-j*1.85600e-002, +1.25733e-001+j*1.77063e-002, +5.34635e-002+j*4.72260e-003;
|
|
-1.10454e-002-j*2.13217e-002, +1.26440e-001+j*1.96001e-002, -5.62624e-002-j*1.07192e-002;
|
|
+5.84467e-003-j*4.07134e-002, +5.16711e-003+j*4.70857e-002, -1.93010e-001+j*6.79213e-003;
|
|
+5.31962e-002-j*1.11322e-002, +3.23294e-001-j*2.12981e-002, +2.14310e-001+j*4.85898e-003;
|
|
-2.44580e-002-j*1.33220e-002, +4.08800e-001+j*2.09082e-002, +5.11454e-001+j*0.00000e+000;
|
|
-2.51332e-002-j*1.42719e-003, +8.97105e-002+j*4.85852e-002, -1.27426e-001-j*9.59723e-003;
|
|
-2.08176e-003+j*1.37185e-002, -3.99530e-002+j*2.11895e-002, -1.17813e-001+j*7.60972e-002;
|
|
-1.34824e-002+j*7.11258e-003, -4.19473e-002+j*1.83590e-002, -2.07198e-002-j*2.51991e-002;
|
|
-6.84747e-003+j*8.45921e-003, -3.33872e-002+j*1.71496e-002, +6.99867e-002-j*6.93158e-002;
|
|
-2.64313e-002+j*5.08903e-003, -3.03569e-002+j*1.29946e-002, -1.39115e-001+j*5.72459e-002;
|
|
-2.90186e-002+j*1.48257e-002, -5.55429e-002+j*2.74156e-002, -1.62035e-001+j*8.04187e-002;
|
|
-2.05855e-002+j*1.11922e-002, -6.58789e-002+j*3.20524e-002, -1.02263e-002-j*4.24087e-002;
|
|
-1.40204e-002+j*8.01102e-003, -5.72647e-002+j*2.37484e-002, +1.75053e-001-j*9.63667e-002;
|
|
-3.50818e-002+j*1.41152e-002, -5.19701e-002+j*2.31951e-002, -1.15951e-001+j*2.91582e-002;
|
|
-5.78005e-003-j*7.05841e-003, +8.29016e-002+j*1.36984e-002, +4.03470e-003+j*4.03325e-002;
|
|
-1.39928e-002-j*1.14088e-002, +8.05288e-002+j*1.51031e-002, +1.12255e-002+j*3.21224e-002;
|
|
-1.02276e-002-j*8.35724e-003, -4.89246e-003+j*1.67800e-002, +7.80514e-002+j*1.53467e-002];
|
|
|
|
mode_shapes(:, :, 10) = [...
|
|
+3.33349e-002-j*4.89606e-003, -8.67138e-002-j*1.69402e-002, +2.87366e-002-j*1.66842e-002;
|
|
+2.95730e-002-j*6.10477e-004, -9.24590e-002-j*1.92562e-002, +5.21162e-002-j*1.31811e-002;
|
|
+3.26966e-002+j*1.03975e-002, -8.55682e-002-j*4.71847e-002, +3.99404e-002+j*8.59358e-003;
|
|
+3.45452e-002-j*3.05951e-003, -7.73823e-002-j*2.32199e-002, +2.30960e-002-j*7.49928e-003;
|
|
-3.72461e-003-j*5.40336e-003, +1.80151e-001-j*1.42898e-002, -8.63921e-003+j*1.95638e-002;
|
|
-1.11285e-002-j*2.22175e-003, +6.92355e-002-j*1.24144e-002, +8.02097e-002-j*6.81531e-003;
|
|
-2.99885e-002-j*4.21951e-004, +8.19709e-002+j*1.22484e-002, +3.97531e-002+j*1.33874e-002;
|
|
-1.59231e-002-j*3.96929e-003, +9.31295e-002+j*9.75532e-003, -4.01947e-002-j*5.17841e-003;
|
|
-7.32828e-003-j*2.93496e-002, +4.39909e-002+j*1.25298e-002, -8.87525e-002+j*2.05359e-002;
|
|
+8.20167e-002+j*6.86693e-004, +4.25475e-001-j*2.36494e-002, +3.37034e-001-j*2.44199e-002;
|
|
-2.68694e-002+j*3.82921e-003, +4.79292e-001+j*1.35903e-002, +5.06762e-001+j*0.00000e+000;
|
|
+1.40565e-002-j*5.41957e-003, +1.17563e-001+j*2.50398e-002, -5.97219e-002+j*5.92813e-004;
|
|
+2.30939e-002+j*1.16755e-002, +4.11136e-003+j*1.65726e-002, +2.41863e-002+j*5.06658e-002;
|
|
-4.92960e-003+j*9.24082e-003, -5.44667e-003+j*1.41983e-002, -3.73537e-002-j*1.87263e-002;
|
|
+9.68682e-003+j*1.11791e-002, -5.13436e-003+j*1.29205e-002, -4.88765e-002-j*4.97821e-002;
|
|
-1.66711e-003+j*1.03280e-002, +9.57955e-004+j*1.28350e-002, +3.30268e-002+j*5.44211e-002;
|
|
+5.90540e-003+j*1.67113e-002, +1.34280e-002+j*2.62111e-002, +4.38613e-002+j*7.17028e-002;
|
|
+4.97752e-003+j*1.47634e-002, +1.56773e-002+j*3.16026e-002, -5.27493e-002-j*3.38315e-002;
|
|
+2.83485e-003+j*1.14816e-002, +9.80676e-003+j*2.51504e-002, -6.78645e-002-j*9.90875e-002;
|
|
+5.62294e-003+j*1.84035e-002, +1.03717e-002+j*2.37801e-002, +2.58497e-002+j*3.75352e-002;
|
|
-8.75236e-003-j*4.71723e-003, +6.99107e-002+j*1.51894e-002, +3.46273e-002+j*1.48547e-002;
|
|
-1.72822e-002-j*8.75192e-003, +6.96759e-002+j*1.59783e-002, +8.07917e-002+j*1.97809e-002;
|
|
-1.29601e-002-j*5.61834e-003, +1.76126e-002+j*4.62761e-003, +6.64667e-002+j*1.61199e-002];
|
|
#+end_src
|
|
|
|
** Define a point for each solid body
|
|
We define accelerometer indices used to define the motion of each solid body (2 3-axis accelerometer are enough).
|
|
#+begin_src matlab
|
|
stages = [17, 19; % Bottom Granite
|
|
13, 15; % Top Granite
|
|
9, 11; % Ty
|
|
5, 7; % Ry
|
|
21, 22; % Spindle
|
|
1, 3]; % Hexapod
|
|
#+end_src
|
|
|
|
We define the origin point ${}^AO_B$ of the solid body $\{B\}$.
|
|
Here we choose the middle point between the two accelerometers.
|
|
This could be define differently (for instance by choosing the center of mass).
|
|
#+begin_src matlab
|
|
AOB = zeros(3, size(stages, 1));
|
|
|
|
for i = 1:size(stages, 1)
|
|
AOB(:, i) = mean(positions(stages(i, :), 1:3))';
|
|
end
|
|
#+end_src
|
|
|
|
Then we compute the positions of the sensors with respect to the previously defined origin for the frame $\{B\}$: ${}^BP_1$ and ${}^BP_2$.
|
|
#+begin_src matlab
|
|
BP1 = zeros(3, size(stages, 1));
|
|
BP2 = zeros(3, size(stages, 1));
|
|
|
|
for i = 1:size(stages, 1)
|
|
BP1(:, i) = positions(stages(i, 1), 1:3)' - AOB(:, i);
|
|
BP2(:, i) = positions(stages(i, 2), 1:3)' - AOB(:, i);
|
|
end
|
|
#+end_src
|
|
|
|
Let's define one absolute frame $\{A\}$ and one frame $\{B\}$ fixed w.r.t. the solid body.
|
|
We note ${}^AO_B$ the position of origin of $\{B\}$ expressed in $\{A\}$.
|
|
|
|
We are measuring with the accelerometers the absolute motion of points $P_1$ and $P_2$: ${}^Av_{P_1}$ and ${}^Av_{P_2}$.
|
|
|
|
Let's note ${}^BP_1$ and ${}^BP_2$ the (known) coordinates of $P_1$ and $P_2$ expressed in the frame $\{B\}$.
|
|
|
|
Then we have:
|
|
\begin{align}
|
|
{}^Av_{P_1} &= {}^Av_{O_B} + {}^A\Omega^\times {}^AR_B {}^BP_1 \\
|
|
{}^Av_{P_2} &= {}^Av_{O_B} + {}^A\Omega^\times {}^AR_B {}^BP_2
|
|
\end{align}
|
|
|
|
And we obtain
|
|
\begin{align}
|
|
{}^A\Omega^\times {}^AR_B &= \left( {}^Av_{P_2} - {}^Av_{P_1} \right) \left( {}^BP_2 - {}^BP_1 \right)^{-1}\\
|
|
{}^Av_{O_B} &= {}^Av_{P_1} - \left( {}^Av_{P_2} - {}^Av_{P_1} \right) \left( {}^BP_2 - {}^BP_1 \right)^{-1} {}^BP_1
|
|
\end{align}
|
|
|
|
#+begin_src matlab
|
|
AVOB = zeros(3, size(stages, 1));
|
|
ARB = zeros(3, 3, size(stages, 1));
|
|
for i = 1:size(stages, 1)
|
|
AVOB(:, i) = mode1(stages(i, 1), :)' - (mode1(stages(i, 2), :)' - mode1(stages(i, 1), :)')*pinv(BP2(:, i) - BP1(:, i))*BP1(:, i);
|
|
ARB(:, :, i) = (mode1(stages(i, 2), :)' - mode1(stages(i, 1), :)')*pinv(BP2(:, i) - BP1(:, i));
|
|
end
|
|
#+end_src
|
|
|
|
** Argand Diagram
|
|
For mode 1
|
|
|
|
#+begin_src matlab
|
|
figure;
|
|
hold on;
|
|
for i=1:size(mode1, 1)
|
|
plot([0, real(mode1(i, 1))], [0, imag(mode1(i, 1))], '-k')
|
|
plot([0, real(mode1(i, 2))], [0, imag(mode1(i, 2))], '-k')
|
|
plot([0, real(mode1(i, 3))], [0, imag(mode1(i, 3))], '-k')
|
|
|
|
% plot([0, real(mode2(i, 1))], [0, imag(mode2(i, 1))], '-r')
|
|
% plot([0, real(mode2(i, 2))], [0, imag(mode2(i, 2))], '-r')
|
|
% plot([0, real(mode2(i, 3))], [0, imag(mode2(i, 3))], '-r')
|
|
|
|
% plot([0, real(mode3(i, 1))], [0, imag(mode3(i, 1))], '-b')
|
|
% plot([0, real(mode3(i, 2))], [0, imag(mode3(i, 2))], '-b')
|
|
% plot([0, real(mode3(i, 3))], [0, imag(mode3(i, 3))], '-b')
|
|
end
|
|
|
|
for i=1:size(AVOB, 2)
|
|
plot([0, real(AVOB(1, i))], [0, imag(AVOB(1, i))], '-r')
|
|
plot([0, real(AVOB(2, i))], [0, imag(AVOB(2, i))], '-r')
|
|
plot([0, real(AVOB(3, i))], [0, imag(AVOB(3, i))], '-r')
|
|
end
|
|
|
|
% ang=0:0.01:2*pi;
|
|
% radius1 = max(max(sqrt(real(mode1).^2+imag(mode1).^2)));
|
|
% plot(radius1*cos(ang), radius1*sin(ang), '-k');
|
|
|
|
% radius2 = max(max(sqrt(real(mode2).^2+imag(mode2).^2)));
|
|
% plot(radius2*cos(ang), radius2*sin(ang), '-r');
|
|
|
|
% radius3 = max(max(sqrt(real(mode3).^2+imag(mode3).^2)));
|
|
% plot(radius3*cos(ang), radius3*sin(ang), '-b');
|
|
|
|
hold off;
|
|
axis manual equal
|
|
#+end_src
|
|
|
|
** TEST: animate first mode
|
|
#+begin_src matlab
|
|
figure;
|
|
hold on;
|
|
fill3(positions(1:4, 1), positions(1:4, 2), positions(1:4, 3), 'k', 'FaceAlpha', 0.5)
|
|
fill3(positions(5:8, 1), positions(5:8, 2), positions(5:8, 3), 'k', 'FaceAlpha', 0.5)
|
|
fill3(positions(9:12, 1), positions(9:12, 2), positions(9:12, 3), 'k', 'FaceAlpha', 0.5)
|
|
fill3(positions(13:16, 1), positions(13:16, 2), positions(13:16, 3), 'k', 'FaceAlpha', 0.5)
|
|
fill3(positions(17:20, 1), positions(17:20, 2), positions(17:20, 3), 'k', 'FaceAlpha', 0.5)
|
|
fill3(positions(21:23, 1), positions(21:23, 2), positions(21:23, 3), 'k', 'FaceAlpha', 0.5)
|
|
hold off;
|
|
#+end_src
|
|
|
|
#+NAME: fig:mode_shapes
|
|
#+HEADER: :tangle no :exports results :results value file raw replace :noweb yes
|
|
#+begin_src matlab
|
|
rec = polyshape([-2 -2 2 2],[-3 3 3 -3]);
|
|
h = figure;
|
|
filename = 'figs/mode_shapes.gif';
|
|
n = 20;
|
|
|
|
for i = 1:n
|
|
axis manual equal
|
|
|
|
Dm = real(V(1:3, 5)*cos(2*pi*i/n));
|
|
rec_i = rotate(rec, 180/pi*Dm(3));
|
|
rec_i = translate(rec_i, 10*Dm(1), 10*Dm(2));
|
|
plot(rec_i);
|
|
|
|
xlim([-3, 3]); ylim([-4, 4]);
|
|
|
|
set(h, 'visible', 'off');
|
|
set(h, 'pos', [0, 0, 500, 500]);
|
|
drawnow;
|
|
|
|
% Capture the plot as an image
|
|
frame = getframe(h);
|
|
im = frame2im(frame);
|
|
[imind,cm] = rgb2ind(im,256);
|
|
|
|
% Write to the GIF File
|
|
if i == 1
|
|
imwrite(imind,cm,filename,'gif','DelayTime',0.1,'Loopcount',inf);
|
|
else
|
|
imwrite(imind,cm,filename,'gif','DelayTime',0.1,'WriteMode','append');
|
|
end
|
|
end
|
|
|
|
set(h, 'visible', 'on');
|
|
ans = filename;
|
|
#+end_src
|
|
|
|
** From 6 translations to translation + rotation
|
|
Let's define one absolute frame $\{A\}$ and one frame $\{B\}$ fixed w.r.t. the solid body.
|
|
We note ${}^AO_B$ the position of origin of $\{B\}$ expressed in $\{A\}$.
|
|
|
|
We are measuring with the accelerometers the absolute motion of points $P_1$ and $P_2$: ${}^AP_1$ and ${}^AP_2$.
|
|
|
|
Let's note ${}^BP_1$ and ${}^BP_2$ the (known) coordinates of $P_1$ and $P_2$ expressed in the frame $\{B\}$.
|
|
|
|
Then we have:
|
|
\begin{align}
|
|
{}^AP_1 &= {}^AO_B + {}^AR_B {}^BP_1 \\
|
|
{}^AP_2 &= {}^AO_B + {}^AR_B {}^BP_2
|
|
\end{align}
|
|
|
|
And we obtain
|
|
\begin{align}
|
|
{}^AR_B &= \left( {}^AP_2 - {}^AP_1 \right) \left( {}^BP_2 - {}^BP_1 \right)^{-1}\\
|
|
{}^AO_B &= {}^Av_{P_1} - \left( {}^AP_2 - {}^AP_1 \right) \left( {}^BP_2 - {}^BP_1 \right)^{-1} {}^BP_1
|
|
\end{align}
|
|
|
|
* Problem with AirLoc System
|
|
The mode shape of the first mode at 11Hz (figure [[fig:mode1]]) seems to indicate that this corresponds to a suspension mode.
|
|
|
|
This could be due to the 4 Airloc Levelers that are used for the granite (figure [[fig:airloc]]).
|
|
|
|
#+name: fig:airloc
|
|
#+caption: AirLoc used for the granite (2120-KSKC)
|
|
#+attr_html: :width 500px
|
|
[[file:img/airloc/IMG_20190618_155522.jpg]]
|
|
|
|
They are probably *not well leveled*, so the granite is supported only by two Airloc.
|
|
|
|
* 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 for the mode shapes
|
|
#+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 :exports 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")
|
|
<<plt-matlab>>
|
|
#+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 :exports 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")
|
|
<<plt-matlab>>
|
|
#+end_src
|
|
|
|
#+NAME: fig:modal_complexity_high
|
|
#+CAPTION: Modal Complexity of one higly complex mode
|
|
[[file:figs/modal_complexity_high.png]]
|
|
|
|
#+begin_src matlab :exports 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")
|
|
<<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
|
|
|
|
* Importation of measured FRF curves
|
|
There are 24 measurements files corresponding to 24 series of impacts:
|
|
- 3 directions, 8 sets of 3 accelerometers
|
|
For each measurement file, the FRF and coherence between the impact and the 9 accelerations measured.
|
|
|
|
In reality: 4 sets of 10 things
|
|
|
|
#+begin_src matlab
|
|
a = load('mat/meas_frf_coh_1.mat');
|
|
#+end_src
|
|
|
|
#+begin_src matlab
|
|
figure;
|
|
|
|
ax1 = subplot(2, 1, 1);
|
|
hold on;
|
|
plot(a.FFT1_AvXSpc_2_1_RMS_X_Val, a.FFT1_AvXSpc_2_1_RMS_Y_Mod)
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
set(gca, 'XTickLabel',[]);
|
|
ylabel('Amplitude');
|
|
title(sprintf('From %s, to %s', FFT1_AvXSpc_2_1_RfName, FFT1_AvXSpc_2_1_RpName))
|
|
|
|
ax2 = subplot(2, 1, 2);
|
|
hold on;
|
|
plot(a.FFT1_AvXSpc_2_1_RMS_X_Val, a.FFT1_AvXSpc_2_1_RMS_Y_Phas)
|
|
hold off;
|
|
ylim([-180, 180]); yticks(-180:90:180);
|
|
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
|
|
set(gca, 'xscale', 'log');
|
|
|
|
linkaxes([ax1,ax2],'x');
|
|
xlim([1, 200]);
|
|
#+end_src
|
|
|
|
* Importation of measured FRF curves to global FRF matrix
|
|
FRF matrix $n \times p$:
|
|
- $n$ is the number of measurements: $3 \times 24$
|
|
- $p$ is the number of excitation inputs: 3
|
|
|
|
23 measurements: 3 accelerometers
|
|
|
|
\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}
|
|
|
|
#+begin_src matlab
|
|
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(acc_i+n_acc*(find(dirs==meas_dir)-1), find(dirs==exc_dir), :) = exc_factor*meas_factor*meas.(sprintf('FFT1_H1_%i_1_Y_ReIm', j));
|
|
% COHs(acc_i+n_acc*(find(dirs==meas_dir)-1), find(dirs==exc_dir), :) = meas.(sprintf('FFT1_Coh_%i_1_RMS_Y_Val', j));
|
|
|
|
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;
|
|
#+end_src
|
|
|
|
* Analysis of some FRFs
|
|
#+begin_src matlab
|
|
acc_i = 3;
|
|
acc_dir = 1;
|
|
exc_dir = 1;
|
|
|
|
figure;
|
|
|
|
ax1 = subplot(2, 1, 1);
|
|
hold on;
|
|
plot(freqs, abs(squeeze(FRFs(acc_dir+3*(acc_i-1), exc_dir, :))));
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
set(gca, 'XTickLabel',[]);
|
|
ylabel('Amplitude');
|
|
|
|
ax2 = subplot(2, 1, 2);
|
|
hold on;
|
|
plot(freqs, mod(180+180/pi*phase(squeeze(FRFs(acc_dir+3*(acc_i-1), exc_dir, :))), 360)-180);
|
|
hold off;
|
|
ylim([-180, 180]); yticks(-180:90:180);
|
|
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
|
|
set(gca, 'xscale', 'log');
|
|
|
|
linkaxes([ax1,ax2],'x');
|
|
xlim([1, 200]);
|
|
#+end_src
|
|
|
|
* Composite Response Function
|
|
We here sum the norm instead of the complex numbers.
|
|
|
|
#+begin_src matlab
|
|
HHx = squeeze(sum(abs(FRFs(:, 1, :))));
|
|
HHy = squeeze(sum(abs(FRFs(:, 2, :))));
|
|
HHz = squeeze(sum(abs(FRFs(:, 3, :))));
|
|
HH = squeeze(sum([HHx, HHy, HHz], 2));
|
|
#+end_src
|
|
|
|
#+begin_src matlab
|
|
exc_dir = 3;
|
|
|
|
figure;
|
|
hold on;
|
|
for i = 1:3*n_acc
|
|
plot(freqs, abs(squeeze(FRFs(i, exc_dir, :))), 'color', [0, 0, 0, 0.2]);
|
|
end
|
|
plot(freqs, abs(HHx));
|
|
plot(freqs, abs(HHy));
|
|
plot(freqs, abs(HHz));
|
|
plot(freqs, abs(HH), 'k');
|
|
hold off;
|
|
set(gca, 'XScale', 'lin'); set(gca, 'YScale', 'lin');
|
|
xlabel('Frequency [Hz]'); ylabel('Amplitude');
|
|
xlim([1, 200]);
|
|
#+end_src
|
|
|
|
#+HEADER: :tangle no :exports results :results none :noweb yes
|
|
#+begin_src matlab :var filepath="figs/composite_response_function.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
|
|
<<plt-matlab>>
|
|
#+end_src
|
|
|
|
#+NAME: fig:composite_response_function
|
|
#+CAPTION: Composite Response Function
|
|
[[file:figs/composite_response_function.png]]
|
|
|
|
* TODO Singular Value Decomposition - Modal Indication Function
|
|
Show the same plot as in the modal software.
|
|
This helps to identify double modes.
|
|
|
|
From the documentation of the modal software:
|
|
#+begin_quote
|
|
The MIF consist of the singular values of the Frequency response function matrix.
|
|
The number of MIFs equals the number of excitations.
|
|
By the powerful singular value decomposition, the real signal space is separated from the noise space.
|
|
Therefore, the MIFs exhibit the modes effectively.
|
|
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.
|
|
Moreover, the magnitude of the MIFs implies the strength of the a mode.
|
|
#+end_quote
|
|
|
|
* From local coordinates to global coordinates with the FRFs
|
|
#+begin_src matlab
|
|
% Number of Solids * DOF for each solid / Number of excitation / frequency points
|
|
FRFs_O = zeros(length(solid_names)*6, 3, 801);
|
|
|
|
for exc_dir = 1:3
|
|
for solid_i = 1:length(solid_names)
|
|
solids_i = solids.(solid_names{solid_i});
|
|
|
|
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
|
|
|
|
for i = 1:801
|
|
FRFs_O((solid_i-1)*6+1:solid_i*6, exc_dir, i) = A\FRFs((solids_i(1)-1)*3+1:solids_i(end)*3, exc_dir, i);
|
|
end
|
|
end
|
|
end
|
|
#+end_src
|
|
|
|
* Analysis of some FRF in the global coordinates
|
|
#+begin_src matlab
|
|
solid_i = 6;
|
|
dir_i = 1;
|
|
exc_dir = 1;
|
|
|
|
figure;
|
|
|
|
ax1 = subplot(2, 1, 1);
|
|
hold on;
|
|
plot(freqs, abs(squeeze(FRFs_O((solid_i-1)*6+dir_i, exc_dir, :))));
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
set(gca, 'XTickLabel',[]);
|
|
ylabel('Amplitude');
|
|
|
|
ax2 = subplot(2, 1, 2);
|
|
hold on;
|
|
plot(freqs, mod(180+180/pi*phase(squeeze(FRFs_O((solid_i-1)*6+dir_i, exc_dir, :))), 360)-180);
|
|
hold off;
|
|
ylim([-180, 180]); yticks(-180:90:180);
|
|
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
|
|
set(gca, 'xscale', 'log');
|
|
|
|
linkaxes([ax1,ax2],'x');
|
|
xlim([1, 200]);
|
|
#+end_src
|
|
|
|
* Compare global coordinates to local coordinates
|
|
#+begin_src matlab
|
|
solid_i = 1;
|
|
acc_dir_O = 6;
|
|
acc_dir = 3;
|
|
exc_dir = 3;
|
|
|
|
figure;
|
|
|
|
ax1 = subplot(2, 1, 1);
|
|
hold on;
|
|
for i = solids.(solid_names{solid_i})
|
|
plot(freqs, abs(squeeze(FRFs(acc_dir+3*(i-1), exc_dir, :))));
|
|
end
|
|
plot(freqs, abs(squeeze(FRFs_O((solid_i-1)*6+acc_dir_O, exc_dir, :))), '-k');
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
set(gca, 'XTickLabel',[]);
|
|
ylabel('Amplitude');
|
|
|
|
ax2 = subplot(2, 1, 2);
|
|
hold on;
|
|
for i = solids.(solid_names{solid_i})
|
|
plot(freqs, mod(180+180/pi*phase(squeeze(FRFs(acc_dir+3*(i-1), exc_dir, :))), 360)-180);
|
|
end
|
|
plot(freqs, mod(180+180/pi*phase(squeeze(FRFs_O((solid_i-1)*6+acc_dir_O, exc_dir, :))), 360)-180, '-k');
|
|
hold off;
|
|
ylim([-180, 180]); yticks(-180:90:180);
|
|
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
|
|
set(gca, 'xscale', 'log');
|
|
|
|
linkaxes([ax1,ax2],'x');
|
|
xlim([1, 200]);
|
|
#+end_src
|
|
|
|
* Verify that we find the original FRF from the FRF in the global coordinates
|
|
From the computed FRF of the Hexapod in its 6 DOFs, compute the FRF of the accelerometer 1 fixed to the Hexapod during the measurement.
|
|
|
|
#+begin_src matlab
|
|
FRF_test = zeros(801, 3);
|
|
for i = 1:801
|
|
FRF_test(i, :) = FRFs_O(31:33, 1, i) + cross(FRFs_O(34:36, 1, i), acc_pos(1, :)');
|
|
end
|
|
#+end_src
|
|
|
|
#+begin_src matlab :exports none
|
|
figure;
|
|
|
|
ax1 = subplot(3, 1, 1);
|
|
hold on;
|
|
plot(freqs, abs(squeeze(FRFs(1, 1, :))));
|
|
plot(freqs, abs(squeeze(FRF_test(:, 1))), '--k');
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
set(gca, 'XTickLabel',[]);
|
|
xlim([1, 200]);
|
|
title('FRF $\frac{D_{1x}}{F_x}$');
|
|
|
|
ax2 = subplot(3, 1, 2);
|
|
hold on;
|
|
plot(freqs, abs(squeeze(FRFs(2, 1, :))));
|
|
plot(freqs, abs(squeeze(FRF_test(:, 2))), '--k');
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
set(gca, 'XTickLabel',[]);
|
|
ylabel('Amplitude');
|
|
xlim([1, 200]);
|
|
title('FRF $\frac{D_{1y}}{F_x}$');
|
|
|
|
ax3 = subplot(3, 1, 3);
|
|
hold on;
|
|
plot(freqs, abs(squeeze(FRFs(3, 1, :))));
|
|
plot(freqs, abs(squeeze(FRF_test(:, 3))), '--k');
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
xlabel('Frequency [Hz]');
|
|
xlim([1, 200]);
|
|
legend({'Original Measurement', 'Recovered Measurement'}, 'Location', 'southeast');
|
|
title('FRF $\frac{D_{1z}}{F_x}$');
|
|
#+end_src
|
|
|
|
#+HEADER: :tangle no :exports results :results none :noweb yes
|
|
#+begin_src matlab :var filepath="figs/compare_original_meas_with_recovered.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
|
|
<<plt-matlab>>
|
|
#+end_src
|
|
|
|
#+NAME: fig:compare_original_meas_with_recovered
|
|
#+CAPTION: Comparison of the original measured FRFs with the recovered FRF from the FRF in the common cartesian frame
|
|
[[file:figs/compare_original_meas_with_recovered.png]]
|
|
|
|
#+begin_important
|
|
The reduction of the number of degrees of freedom from 69 (23 accelerometers with each 3DOF) to 36 (6 solid bodies with 6 DOF) seems to work well.
|
|
|
|
This confirms the fact that this stage, for that mode is indeed behaving as a solid body.
|
|
This should be verified for all the stages for modes with high resonance frequencies.
|
|
#+end_important
|
|
|
|
* TODO Synthesis of FRF curves
|