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nass-micro-station-measurem.../modal-analysis/modal_extraction.org

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Reorganization of the modal-analysis folder
2019-07-03 13:54:33 +02:00
#+TITLE: Modal Analysis - Modal Parameter Extraction
Update the modal analysis
2019-07-02 17:48:34 +02:00
:DRAWER:
#+STARTUP: overview
#+LANGUAGE: en
#+EMAIL: dehaeze.thomas@gmail.com
#+AUTHOR: Dehaeze Thomas
#+HTML_LINK_HOME: ../index.html
Finish the file about measurements. Add raw measurements to git
2019-07-03 17:25:44 +02:00
#+HTML_LINK_UP: ./index.html
Update the modal analysis
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#+HTML_HEAD: <link rel="stylesheet" type="text/css" href="../css/htmlize.css"/>
#+HTML_HEAD: <link rel="stylesheet" type="text/css" href="../css/readtheorg.css"/>
#+HTML_HEAD: <link rel="stylesheet" type="text/css" href="../css/zenburn.css"/>
#+HTML_HEAD: <script type="text/javascript" src="../js/jquery.min.js"></script>
#+HTML_HEAD: <script type="text/javascript" src="../js/bootstrap.min.js"></script>
#+HTML_HEAD: <script type="text/javascript" src="../js/jquery.stickytableheaders.min.js"></script>
#+HTML_HEAD: <script type="text/javascript" src="../js/readtheorg.js"></script>
#+HTML_MATHJAX: align: center tagside: right font: TeX
#+PROPERTY: header-args:matlab :session *MATLAB*
#+PROPERTY: header-args:matlab+ :comments org
#+PROPERTY: header-args:matlab+ :results none
#+PROPERTY: header-args:matlab+ :exports both
#+PROPERTY: header-args:matlab+ :eval no-export
#+PROPERTY: header-args:matlab+ :output-dir figs
Update modal analysis, add .zip files (data and matlab files)
2019-07-05 11:20:02 +02:00
#+PROPERTY: header-args:matlab+ :tangle matlab/modal_extraction.m
#+PROPERTY: header-args:matlab+ :mkdirp yes
Update the modal analysis
2019-07-02 17:48:34 +02:00
#+PROPERTY: header-args:shell :eval no-export
#+PROPERTY: header-args:latex :headers '("\\usepackage{tikz}" "\\usepackage{import}" "\\import{$HOME/MEGA/These/LaTeX/}{config.tex}")
#+PROPERTY: header-args:latex+ :imagemagick t :fit yes
#+PROPERTY: header-args:latex+ :iminoptions -scale 100% -density 150
#+PROPERTY: header-args:latex+ :imoutoptions -quality 100
#+PROPERTY: header-args:latex+ :results raw replace :buffer no
#+PROPERTY: header-args:latex+ :eval no-export
#+PROPERTY: header-args:latex+ :exports both
#+PROPERTY: header-args:latex+ :mkdirp yes
#+PROPERTY: header-args:latex+ :output-dir figs
:END:
Update modal analysis, add .zip files (data and matlab files)
2019-07-05 11:20:02 +02:00
The goal here is to extract the modal parameters describing the modes of station being studied.
* ZIP file containing the data and matlab files :ignore:
#+begin_src bash :exports none :results none
if [ matlab/modal_extraction.m -nt data/modal_extraction.zip ]; then
cp matlab/modal_extraction.m modal_extraction.m;
zip data/modal_extraction \
mat/data.mat \
modal_extraction.m
rm modal_extraction.m;
fi
#+end_src
#+begin_note
All the files (data and Matlab scripts) are accessible [[file:data/modal_extraction.zip][here]].
#+end_note
Update the modal analysis
2019-07-02 17:48:34 +02:00
* Matlab Init :noexport:ignore:
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
<<matlab-dir>>
#+end_src
#+begin_src matlab :exports none :results silent :noweb yes
<<matlab-init>>
#+end_src
Update analysis => FRF in the global frame
2019-07-05 10:16:33 +02:00
* TODO Part to explain how to choose the modes frequencies
- bro-band method used
- Stabilization Chart
- 21 modes
Finish the file about measurements. Add raw measurements to git
2019-07-03 17:25:44 +02:00
* Obtained Modal Parameters
From the modal analysis software, we can export the obtained modal parameters:
- the resonance frequencies
- the modes shapes
- the modal damping
- the residues
These can be express as the *eigen matrices*:
\[ \Omega = \begin{bmatrix}
\omega_1^2 & & 0 \\
& \ddots & \\
0 & & \omega_n^2
\end{bmatrix}; \quad \Psi = \begin{bmatrix}
& & \\
\{\psi_1\} & \dots & \{\psi_n\} \\
& &
\end{bmatrix} \]
where $\bar{\omega}_r^2$ is the $r^\text{th}$ eigenvalue squared and $\{\phi\}_r$ is a description of the corresponding *mode shape*.
The file containing the modal parameters is =mat/modes.asc=. Its first 20 lines as shown below.
#+begin_src bash :results output :exports results :eval no-export
sed 20q mat/modes.asc | sed $'s/\r//'
#+end_src
#+RESULTS:
#+begin_example
Created by N-Modal
Estimator: bbfd
01-Jul-19 16:44:11
Mode 1
freq = 11.41275Hz
damp = 8.72664%
modal A = -4.50556e+003-9.41744e+003i
modal B = -7.00928e+005+2.62922e+005i
Mode matrix of local coordinate [DOF: Re IM]
1X-: -1.04114e-001 3.50664e-002
1Y-: 2.34008e-001 5.04273e-004
1Z+: -1.93303e-002 5.08614e-003
2X-: -8.38439e-002 3.45978e-002
2Y-: 2.42440e-001 0.00000e+000
2Z+: -7.40734e-003 5.17734e-003
3Y-: 2.17655e-001 6.10802e-003
3X+: 1.18685e-001 -3.54602e-002
3Z+: -2.37725e-002 -1.61649e-003
#+end_example
We split this big =modes.asc= file into sub text files using =bash=. The obtained files are described one table [[tab:modes_files]].
#+begin_src bash :results none
sed '/^\s*[0-9]*[XYZ][+-]:/!d' mat/modes.asc > mat/mode_shapes.txt
sed '/freq/!d' mat/modes.asc | sed 's/.* = \(.*\)Hz/\1/' > mat/mode_freqs.txt
sed '/damp/!d' mat/modes.asc | sed 's/.* = \(.*\)\%/\1/' > mat/mode_damps.txt
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
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
#+end_src
#+name: tab:modes_files
#+caption: Split =modes.asc= file
| Filename | Content |
|------------------------+--------------------------------------------------|
| =mat/mode_shapes.txt= | mode shapes |
| =mat/mode_freqs.txt= | resonance frequencies |
| =mat/mode_damps.txt= | modal damping |
| =mat/mode_modal_a.txt= | modal residues at low frequency (to be checked) |
| =mat/mode_modal_b.txt= | modal residues at high frequency (to be checked) |
Then we import the obtained =.txt= files on Matlab using =readtable= function.
#+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_b = table2array(readtable('mat/mode_modal_b.txt', 'ReadVariableNames', false)); % [Real / Imag]
modal_a = complex(modal_a(:, 1), modal_a(:, 2));
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 |
Update analysis => FRF in the global frame
2019-07-05 10:16:33 +02:00
* Obtained Mode Shapes animations
One all the FRFs are obtained, we can estimate the modal parameters (resonance frequencies, modal shapes and modal damping) within the modal software.
For that, multiple modal extraction techniques can be used (SIMO, MIMO, narrow band, wide band, ...).
Then, it is possible to show the modal shapes with an animation.
Examples are shown on figures [[fig:mode1]] and [[fig:mode6]].
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]].
#+name: fig:mode1
#+caption: Mode 1
[[file:img/modes/mode1.gif]]
#+name: fig:mode6
#+caption: Mode 6
[[file:img/modes/mode6.gif]]
Finish the file about measurements. Add raw measurements to git
2019-07-03 17:25:44 +02:00
* Compute the Modal Model
** Matlab Init :noexport:ignore:
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
<<matlab-dir>>
#+end_src
#+begin_src matlab :exports none :results silent :noweb yes
<<matlab-init>>
#+end_src
** Position of the accelerometers
There are 23 accelerometers:
- 4 on the bottom granite
- 4 on the top granite
- 4 on top of the translation stage
- 4 on the tilt stage
- 3 on top of the spindle
- 4 on top of the hexapod
The coordinates defined in the software are displayed below.
#+begin_src bash :results output :exports results :eval no-export
sed -n 18,40p modal_analysis/acc_coordinates.txt | tac --
#+end_src
#+RESULTS:
#+begin_example
1 1.0000e-001 1.0000e-001 1.1500e+000 0 Top
2 1.0000e-001 -1.0000e-001 1.1500e+000 0 Top
3 -1.0000e-001 -1.0000e-001 1.1500e+000 0 Top
4 -1.0000e-001 1.0000e-001 1.1500e+000 0 Top
5 4.0000e-001 4.0000e-001 9.5000e-001 0 inner
6 4.0000e-001 -4.0000e-001 9.5000e-001 0 inner
7 -4.0000e-001 -4.0000e-001 9.5000e-001 0 inner
8 -4.0000e-001 4.0000e-001 9.5000e-001 0 inner
9 5.0000e-001 5.0000e-001 9.0000e-001 0 outer
10 5.0000e-001 -5.0000e-001 9.0000e-001 0 outer
11 -5.0000e-001 -5.0000e-001 9.0000e-001 0 outer
12 -5.0000e-001 5.0000e-001 9.0000e-001 0 outer
13 5.5000e-001 5.5000e-001 5.5000e-001 0 top
14 5.5000e-001 -5.5000e-001 5.5000e-001 0 top
15 -5.5000e-001 -5.5000e-001 5.5000e-001 0 top
16 -5.5000e-001 5.5000e-001 5.5000e-001 0 top
17 9.5000e-001 9.5000e-001 4.0000e-001 0 low
18 9.5000e-001 -9.5000e-001 4.0000e-001 0 low
19 -9.5000e-001 -9.5000e-001 4.0000e-001 0 low
20 -9.5000e-001 9.5000e-001 4.0000e-001 0 low
21 2.0000e-001 2.0000e-001 8.5000e-001 0 bot
22 0.0000e+000 -2.0000e-001 8.5000e-001 0 bot
23 -2.0000e-001 2.0000e-001 8.5000e-001 0 bot
#+end_example
#+name: tab:acc_location
#+caption: Location of each Accelerometer (using the normal coordinate frame with X aligned with the X ray)
| *Node number* | *Solid Body* | *Location* | *X* | *Y* | *Z* |
|---------------+-------------------+------------+-------+-------+------|
| 1 | Hexapod - Top | -X/-Y | -0.10 | -0.10 | 1.15 |
| 2 | | -X/+Y | -0.10 | 0.10 | 1.15 |
| 3 | | +X/+Y | 0.10 | 0.10 | 1.15 |
| 4 | | +X/-Y | 0.10 | -0.10 | 1.15 |
|---------------+-------------------+------------+-------+-------+------|
| 5 | Tilt - Top | -X/-Y | -0.40 | -0.40 | 0.95 |
| 6 | | -X/+Y | -0.40 | 0.40 | 0.95 |
| 7 | | +X/+Y | 0.40 | 0.40 | 0.95 |
| 8 | | +X/-Y | 0.40 | -0.40 | 0.95 |
|---------------+-------------------+------------+-------+-------+------|
| 9 | Translation - Top | -X/-Y | -0.50 | -0.50 | 0.90 |
| 10 | | -X/+Y | -0.50 | 0.50 | 0.90 |
| 11 | | +X/+Y | 0.50 | 0.50 | 0.90 |
| 12 | | +X/-Y | 0.50 | -0.50 | 0.90 |
|---------------+-------------------+------------+-------+-------+------|
| 13 | Top Granite | -X/-Y | -0.55 | -0.50 | 0.55 |
| 14 | | -X/+Y | -0.55 | 0.50 | 0.55 |
| 15 | | +X/+Y | 0.55 | 0.50 | 0.55 |
| 16 | | +X/-Y | 0.55 | -0.50 | 0.55 |
|---------------+-------------------+------------+-------+-------+------|
| 17 | Bottom Granite | -X/-Y | -0.95 | -0.90 | 0.40 |
| 18 | | -X/+Y | -0.95 | 0.90 | 0.40 |
| 19 | | +X/+Y | 0.95 | 0.90 | 0.40 |
| 20 | | +X/-Y | 0.95 | -0.90 | 0.40 |
|---------------+-------------------+------------+-------+-------+------|
| 21 | Spindle - Top | -X/-Y | -0.20 | -0.20 | 0.85 |
| 22 | | +0/+Y | 0.00 | 0.20 | 0.85 |
| 23 | | +X/-Y | 0.20 | -0.20 | 0.85 |
** Define positions of the accelerometers on matlab
We define the X-Y-Z position of each sensor.
Each line corresponds to one accelerometer, X-Y-Z position in meter.
#+begin_src matlab
positions = [...
-0.10, -0.10, 1.15 ; ...
-0.10, 0.10, 1.15 ; ...
0.10, 0.10, 1.15 ; ...
0.10, -0.10, 1.15 ; ...
-0.40, -0.40, 0.95 ; ...
-0.40, 0.40, 0.95 ; ...
0.40, 0.40, 0.95 ; ...
0.40, -0.40, 0.95 ; ...
-0.50, -0.50, 0.90 ; ...
-0.50, 0.50, 0.90 ; ...
0.50, 0.50, 0.90 ; ...
0.50, -0.50, 0.90 ; ...
-0.55, -0.50, 0.55 ; ...
-0.55, 0.50, 0.55 ; ...
0.55, 0.50, 0.55 ; ...
0.55, -0.50, 0.55 ; ...
-0.95, -0.90, 0.40 ; ...
-0.95, 0.90, 0.40 ; ...
0.95, 0.90, 0.40 ; ...
0.95, -0.90, 0.40 ; ...
-0.20, -0.20, 0.85 ; ...
0.00, 0.20, 0.85 ; ...
0.20, -0.20, 0.85 ];
#+end_src
#+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
** Import the modal vectors on matlab
*** Mode1
#+begin_src bash :results output :exports none :eval no-export
sed -n 12,80p modal_analysis/modes_propres_narband.asc
#+end_src
#+RESULTS:
#+begin_example
1X+: -9.34637e-002 4.52445e-002
1Y+: 2.33790e-001 1.41439e-003
1Z+: -1.73754e-002 6.02449e-003
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;
+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, +1.63533e-002+j*6.09161e-002, +2.37693e-001+j*4.34204e-002];
mode_shapes(:, :, 3) = [...
+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, +3.57722e-003+j*3.64827e-003, -1.92038e-002-j*6.65895e-002];
mode_shapes(:, :, 4) = [...
-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, +2.30814e-001-j*8.33151e-002, -1.15217e-002-j*4.01143e-002];
mode_shapes(:, :, 5) = [...
+3.55328e-001+j*0.00000e+000, +6.67612e-002+j*5.48020e-002, +3.03237e-002+j*5.29473e-004;
+3.16372e-001-j*3.84091e-002, +5.27472e-002+j*5.88474e-002, +2.86305e-002-j*1.74805e-002;
+3.00803e-001-j*1.36309e-002, +7.04883e-002+j*1.24492e-001, +7.23329e-002+j*2.33738e-002;
+3.32527e-001-j*2.26876e-004, +9.82263e-002+j*1.20397e-001, +9.86580e-002+j*3.55048e-002;
+4.96498e-002+j*2.31008e-002, +9.79716e-002+j*1.42500e-002, -1.15121e-001-j*3.59085e-002;
+1.41924e-001+j*2.16209e-002, +8.76030e-002+j*6.39650e-003, -8.75727e-002-j*3.71261e-002;
+1.41522e-001+j*1.96964e-002, -1.01959e-001+j*4.10992e-004, +2.14744e-001+j*4.91249e-002;
+4.33170e-002+j*1.84481e-002, -8.24640e-002+j*3.42475e-003, +2.32281e-001+j*5.40699e-002;
+1.47782e-001+j*4.93091e-002, +8.75397e-002+j*7.75318e-004, -6.80833e-002-j*9.72902e-003;
+2.00055e-001+j*3.81689e-002, +8.06886e-002+j*1.19008e-002, -1.40810e-002-j*1.12625e-002;
+1.96526e-001+j*3.87737e-002, -8.42766e-002+j*9.20233e-003, +1.02951e-001+j*3.37680e-002;
+1.25035e-001+j*4.67796e-002, -8.81307e-002+j*5.81039e-004, +7.94320e-002+j*2.19736e-002;
+2.03946e-002+j*2.50162e-002, +7.93788e-002-j*1.40794e-002, -4.15470e-002+j*4.95855e-004;
+6.56876e-002-j*2.14826e-002, +8.21523e-002-j*1.94792e-002, +3.44089e-002+j*2.32727e-003;
+5.98960e-002-j*2.17160e-002, +4.74914e-002+j*2.31386e-002, +3.58704e-002+j*1.13591e-003;
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#+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.
Update the modal analysis
2019-07-02 17:48:34 +02:00
* 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
Add few notes about modal complexity
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cat modal_analysis_updated/id31_nanostation_modified.cfg | grep NODES -A 23 | sed '/\s\+[0-9]\+/!d' | sed 's/\(.*\)\s\+0\s\+.\+/\1/' > mat/acc_pos.txt
Update the modal analysis
2019-07-02 17:48:34 +02:00
#+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
Update analysis => FRF in the global frame
2019-07-05 10:16:33 +02:00
* From local coordinates to global coordinates for the mode shapes
Update the modal analysis
2019-07-02 17:48:34 +02:00
#+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
Add few notes about modal complexity
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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]].
Add importation of FRF and COH
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#+begin_src matlab :exports none
Update the modal analysis
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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');
Add few notes about modal complexity
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title(sprintf('Mode %i', mod_i));
Update the modal analysis
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axis manual equal
#+end_src
#+HEADER: :tangle no :exports results :results none :noweb yes
Add few notes about modal complexity
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#+begin_src matlab :var filepath="figs/modal_complexity_small.pdf" :var figsize="normal-normal" :post pdf2svg(file=*this*, ext="png")
Update the modal analysis
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<<plt-matlab>>
#+end_src
Add few notes about modal complexity
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#+NAME: fig:modal_complexity_small
#+CAPTION: Modal Complexity of one mode with small complexity
[[file:figs/modal_complexity_small.png]]
Update the modal analysis
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Add importation of FRF and COH
2019-07-03 11:57:59 +02:00
#+begin_src matlab :exports none
Add few notes about modal complexity
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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]]
Add importation of FRF and COH
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#+begin_src matlab :exports none
Update the modal analysis
2019-07-02 17:48:34 +02:00
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
Add importation of FRF and COH
2019-07-03 11:57:59 +02:00
* 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
Finish the file about measurements. Add raw measurements to git
2019-07-03 17:25:44 +02:00
a = load('mat/meas_frf_coh_1.mat');
Add importation of FRF and COH
2019-07-03 11:57:59 +02:00
#+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
Finish the file about measurements. Add raw measurements to git
2019-07-03 17:25:44 +02:00
meas = load(sprintf('mat/meas_frf_coh_%i.mat', i));
Add importation of FRF and COH
2019-07-03 11:57:59 +02:00
% 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
Update analysis => FRF in the global frame
2019-07-05 10:16:33 +02:00
* Composite Response Function
Add importation of FRF and COH
2019-07-03 11:57:59 +02:00
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
Update analysis => FRF in the global frame
2019-07-05 10:16:33 +02:00
#+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
Add importation of FRF and COH
2019-07-03 11:57:59 +02:00
* 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;
FRF from accelerometers to global cartesian and comparison
2019-07-03 13:40:02 +02:00
acc_dir_O = 6;
Add importation of FRF and COH
2019-07-03 11:57:59 +02:00
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
FRF from accelerometers to global cartesian and comparison
2019-07-03 13:40:02 +02:00
plot(freqs, abs(squeeze(FRFs_O((solid_i-1)*6+acc_dir_O, exc_dir, :))), '-k');
Add importation of FRF and COH
2019-07-03 11:57:59 +02:00
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
FRF from accelerometers to global cartesian and comparison
2019-07-03 13:40:02 +02:00
plot(freqs, mod(180+180/pi*phase(squeeze(FRFs_O((solid_i-1)*6+acc_dir_O, exc_dir, :))), 360)-180, '-k');
Add importation of FRF and COH
2019-07-03 11:57:59 +02:00
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
FRF from accelerometers to global cartesian and comparison
2019-07-03 13:40:02 +02:00
* 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.
Finish the file about measurements. Add raw measurements to git
2019-07-03 17:25:44 +02:00
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.
FRF from accelerometers to global cartesian and comparison
2019-07-03 13:40:02 +02:00
#+end_important
Add importation of FRF and COH
2019-07-03 11:57:59 +02:00
Update the modal analysis
2019-07-02 17:48:34 +02:00
* TODO Synthesis of FRF curves
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