Update the modal analysis

This commit is contained in:
Thomas Dehaeze 2019-07-02 17:48:34 +02:00
parent cceb3139de
commit b688d6300d
8 changed files with 1455 additions and 0 deletions

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@ -503,6 +503,983 @@ Mode matrix of local coordinate [DOF: Re IM]
23Y-: -1.90862e-001 -2.53414e-002
#+end_example
* 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;
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-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
4 Airloc Levelers are used for the granite (figure [[fig:airloc]]).

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#+TITLE: Modal Analysis
:DRAWER:
#+STARTUP: overview
#+LANGUAGE: en
#+EMAIL: dehaeze.thomas@gmail.com
#+AUTHOR: Dehaeze Thomas
#+HTML_LINK_HOME: ../index.html
#+HTML_LINK_UP: ../index.html
#+HTML_HEAD: <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
#+PROPERTY: header-args:shell :eval no-export
#+PROPERTY: header-args:latex :headers '("\\usepackage{tikz}" "\\usepackage{import}" "\\import{$HOME/MEGA/These/LaTeX/}{config.tex}")
#+PROPERTY: header-args:latex+ :imagemagick t :fit yes
#+PROPERTY: header-args:latex+ :iminoptions -scale 100% -density 150
#+PROPERTY: header-args:latex+ :imoutoptions -quality 100
#+PROPERTY: header-args:latex+ :results raw replace :buffer no
#+PROPERTY: header-args:latex+ :eval no-export
#+PROPERTY: header-args:latex+ :exports both
#+PROPERTY: header-args:latex+ :mkdirp yes
#+PROPERTY: header-args:latex+ :output-dir figs
:END:
* Matlab Init :noexport:ignore:
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
<<matlab-dir>>
#+end_src
#+begin_src matlab :exports none :results silent :noweb yes
<<matlab-init>>
#+end_src
* Setup
#+name: fig:nass-modal-test
#+caption: Position and orientation of the accelerometer used
[[file:figs/nass-modal-test.png]]
* Mode extraction and importation
First, we split the big =modes.asc= files into sub text files using =bash=.
#+begin_src bash :results none
sed '/^\s*[0-9]*[XYZ][+-]:/!d' modal_analysis_updated/modes.asc > mat/mode_shapes.txt
sed '/freq/!d' modal_analysis_updated/modes.asc | sed 's/.* = \(.*\)Hz/\1/' > mat/mode_freqs.txt
sed '/damp/!d' modal_analysis_updated/modes.asc | sed 's/.* = \(.*\)\%/\1/' > mat/mode_damps.txt
sed '/modal A/!d' modal_analysis_updated/modes.asc | sed 's/.* =\s\+\([-0-9.e]\++[0-9]\+\)\([-+0-9.e]\+\)i/\1 \2/' > mat/mode_modal_a.txt
sed '/modal B/!d' modal_analysis_updated/modes.asc | sed 's/.* =\s\+\([-0-9.e]\++[0-9]\+\)\([-+0-9.e]\+\)i/\1 \2/' > mat/mode_modal_b.txt
#+end_src
Then we import them on Matlab.
#+begin_src matlab
shapes = readtable('mat/mode_shapes.txt', 'ReadVariableNames', false); % [Sign / Real / Imag]
freqs = table2array(readtable('mat/mode_freqs.txt', 'ReadVariableNames', false)); % in [Hz]
damps = table2array(readtable('mat/mode_damps.txt', 'ReadVariableNames', false)); % in [%]
modal_a = table2array(readtable('mat/mode_modal_a.txt', 'ReadVariableNames', false)); % [Real / Imag]
modal_a = complex(modal_a(:, 1), modal_a(:, 2));
modal_b = table2array(readtable('mat/mode_modal_b.txt', 'ReadVariableNames', false)); % [Real / Imag]
modal_b = complex(modal_b(:, 1), modal_b(:, 2));
#+end_src
We guess the number of modes identified from the length of the imported data.
#+begin_src matlab
acc_n = 23; % Number of accelerometers
dir_n = 3; % Number of directions
dirs = 'XYZ';
mod_n = size(shapes,1)/acc_n/dir_n; % Number of modes
#+end_src
As the mode shapes are split into 3 parts (direction plus sign, real part and imaginary part), we aggregate them into one array of complex numbers.
#+begin_src matlab
T_sign = table2array(shapes(:, 1));
T_real = table2array(shapes(:, 2));
T_imag = table2array(shapes(:, 3));
modes = zeros(mod_n, acc_n, dir_n);
for mod_i = 1:mod_n
for acc_i = 1:acc_n
% Get the correct section of the signs
T = T_sign(acc_n*dir_n*(mod_i-1)+1:acc_n*dir_n*mod_i);
for dir_i = 1:dir_n
% Get the line corresponding to the sensor
i = find(contains(T, sprintf('%i%s',acc_i, dirs(dir_i))), 1, 'first')+acc_n*dir_n*(mod_i-1);
modes(mod_i, acc_i, dir_i) = str2num([T_sign{i}(end-1), '1'])*complex(T_real(i),T_imag(i));
end
end
end
#+end_src
The obtained mode frequencies and damping are shown below.
#+begin_src matlab :exports both :results value table replace :post addhdr(*this*)
data2orgtable([freqs, damps], {}, {'Frequency [Hz]', 'Damping [%]'}, ' %.1f ');
#+end_src
#+RESULTS:
| Frequency [Hz] | Damping [%] |
|----------------+-------------|
| 11.4 | 8.7 |
| 18.5 | 11.8 |
| 37.6 | 6.4 |
| 39.4 | 3.6 |
| 54.0 | 0.2 |
| 56.1 | 2.8 |
| 69.7 | 4.6 |
| 71.6 | 0.6 |
| 72.4 | 1.6 |
| 84.9 | 3.6 |
| 90.6 | 0.3 |
| 91.0 | 2.9 |
| 95.8 | 3.3 |
| 105.4 | 3.3 |
| 106.8 | 1.9 |
| 112.6 | 3.0 |
| 116.8 | 2.7 |
| 124.1 | 0.6 |
| 145.4 | 1.6 |
| 150.1 | 2.2 |
| 164.7 | 1.4 |
* Positions of the sensors
We process the file exported from the =modal= software containing the positions of the sensors using =bash=.
#+begin_src bash :results none
cat modal_analysis_updated/id31_nanostation_modified.cfg | grep NODES -A 23 | sed '/\s\+[0-9]\+/!d' | sed 's/\(.*\)\s\+0\s\+.\+/\1/' > mat/acc_pos.txt
#+end_src
We then import that on =matlab=, and sort them.
#+begin_src matlab
acc_pos = readtable('mat/acc_pos.txt', 'ReadVariableNames', false);
acc_pos = table2array(acc_pos(:, 1:4));
[~, i] = sort(acc_pos(:, 1));
acc_pos = acc_pos(i, 2:4);
#+end_src
The positions of the sensors relative to the point of interest are shown below.
#+begin_src matlab :exports both :results value table replace :post addhdr(*this*)
data2orgtable(1000*acc_pos, {}, {'x [mm]', 'y [mm]', 'z [mm]'}, ' %.0f ');
#+end_src
#+RESULTS:
| x [mm] | y [mm] | z [mm] |
|--------+--------+--------|
| -64 | -64 | -296 |
| -64 | 64 | -296 |
| 64 | 64 | -296 |
| 64 | -64 | -296 |
| -385 | -300 | -417 |
| -420 | 280 | -417 |
| 420 | 280 | -417 |
| 380 | -300 | -417 |
| -475 | -414 | -427 |
| -465 | 407 | -427 |
| 475 | 424 | -427 |
| 475 | -419 | -427 |
| -320 | -446 | -786 |
| -480 | 534 | -786 |
| 450 | 534 | -786 |
| 295 | -481 | -786 |
| -730 | -526 | -951 |
| -735 | 814 | -951 |
| 875 | 799 | -951 |
| 865 | -506 | -951 |
| -155 | -90 | -594 |
| 0 | 180 | -594 |
| 155 | -90 | -594 |
* Solids
We consider the following solid bodies:
- Bottom Granite
- Top Granite
- Translation Stage
- Tilt Stage
- Spindle
- Hexapod
We create a structure =solids= that contains the accelerometer number of each solid bodies (as shown on figure [[fig:nass-modal-test]]).
#+begin_src matlab
solids = {};
solids.granite_bot = [17, 18, 19, 20];
solids.granite_top = [13, 14, 15, 16];
solids.ty = [9, 10, 11, 12];
solids.ry = [5, 6, 7, 8];
solids.rz = [21, 22, 23];
solids.hexa = [1, 2, 3, 4];
solid_names = fields(solids);
#+end_src
* From local coordinates to global coordinates
#+begin_src latex :file local_to_global_coordinates.pdf :post pdf2svg(file=*this*, ext="png") :exports results
\newcommand\irregularcircle[2]{% radius, irregularity
\pgfextra {\pgfmathsetmacro\len{(#1)+rand*(#2)}}
+(0:\len pt)
\foreach \a in {10,20,...,350}{
\pgfextra {\pgfmathsetmacro\len{(#1)+rand*(#2)}}
-- +(\a:\len pt)
} -- cycle
}
\begin{tikzpicture}
\draw[rounded corners=1mm] (0, 0) \irregularcircle{3cm}{1mm};
\node[] (origin) at (4, -1) {$\bullet$};
\begin{scope}[shift={(origin)}]
\def\axissize{0.8cm}
\draw[->] (0, 0) -- ++(\axissize, 0) node[above left]{$x$};
\draw[->] (0, 0) -- ++(0, \axissize) node[below right]{$y$};
\draw[fill, color=black] (0, 0) circle (0.05*\axissize);
\node[draw, circle, inner sep=0pt, minimum size=0.4*\axissize, label=left:$z$] (yaxis) at (0, 0){};
\node[below right] at (0, 0){$\{O\}$};
\end{scope}
\coordinate[] (p1) at (-1.5, -1.5);
\coordinate[] (p2) at (-1.5, 1.5);
\coordinate[] (p3) at ( 1.5, 1.5);
\coordinate[] (p4) at ( 1.5, -1.5);
\draw[->] (p1)node[]{$\bullet$}node[above]{$p_1$} -- ++(1, 0.5)node[right]{$v_1$};
\draw[->] (p2)node[]{$\bullet$}node[above]{$p_2$} -- ++(-0.5, 1)node[right]{$v_2$};
\draw[->] (p3)node[]{$\bullet$}node[above]{$p_3$} -- ++(1, 0.5)node[right]{$v_3$};
\draw[->] (p4)node[]{$\bullet$}node[above]{$p_4$} -- ++(0.5, 1)node[right]{$v_4$};
\end{tikzpicture}
#+end_src
#+RESULTS:
[[file:figs/local_to_global_coordinates.png]]
From the figure above, we can write:
\begin{align*}
\vec{v}_1 &= \vec{v} + \Omega \vec{p}_1\\
\vec{v}_2 &= \vec{v} + \Omega \vec{p}_2\\
\vec{v}_3 &= \vec{v} + \Omega \vec{p}_3\\
\vec{v}_4 &= \vec{v} + \Omega \vec{p}_4
\end{align*}
With
\begin{equation}
\Omega = \begin{bmatrix}
0 & -\Omega_z & \Omega_y \\
\Omega_z & 0 & -\Omega_x \\
-\Omega_y & \Omega_x & 0
\end{bmatrix}
\end{equation}
$\vec{v}$ and $\Omega$ represent to velocity and rotation of the solid expressed in the frame $\{O\}$.
We can rearrange the equations in a matrix form:
\begin{equation}
\left[\begin{array}{ccc|ccc}
1 & 0 & 0 & 0 & p_{1z} & -p_{1y} \\
0 & 1 & 0 & -p_{1z} & 0 & p_{1x} \\
0 & 0 & 1 & p_{1y} & -p_{1x} & 0 \\ \hline
& \vdots & & & \vdots & \\ \hline
1 & 0 & 0 & 0 & p_{4z} & -p_{4y} \\
0 & 1 & 0 & -p_{4z} & 0 & p_{4x} \\
0 & 0 & 1 & p_{4y} & -p_{4x} & 0
\end{array}\right] \begin{bmatrix}
v_x \\ v_y \\ v_z \\ \hline \Omega_x \\ \Omega_y \\ \Omega_z
\end{bmatrix} = \begin{bmatrix}
v_{1x} \\ v_{1y} \\ v_{1z} \\\hline \vdots \\\hline v_{4x} \\ v_{4y} \\ v_{4z}
\end{bmatrix}
\end{equation}
and then we obtain the velocity and rotation of the solid in the wanted frame $\{O\}$:
\begin{equation}
\begin{bmatrix}
v_x \\ v_y \\ v_z \\ \hline \Omega_x \\ \Omega_y \\ \Omega_z
\end{bmatrix} =
\left[\begin{array}{ccc|ccc}
1 & 0 & 0 & 0 & p_{1z} & -p_{1y} \\
0 & 1 & 0 & -p_{1z} & 0 & p_{1x} \\
0 & 0 & 1 & p_{1y} & -p_{1x} & 0 \\ \hline
& \vdots & & & \vdots & \\ \hline
1 & 0 & 0 & 0 & p_{4z} & -p_{4y} \\
0 & 1 & 0 & -p_{4z} & 0 & p_{4x} \\
0 & 0 & 1 & p_{4y} & -p_{4x} & 0
\end{array}\right]^{-1} \begin{bmatrix}
v_{1x} \\ v_{1y} \\ v_{1z} \\\hline \vdots \\\hline v_{4x} \\ v_{4y} \\ v_{4z}
\end{bmatrix}
\end{equation}
This inversion is equivalent to a mean square problem.
#+begin_src matlab
mode_shapes_O = zeros(mod_n, length(solid_names), 6);
for mod_i = 1:mod_n
for solid_i = 1:length(solid_names)
solids_i = solids.(solid_names{solid_i});
Y = reshape(squeeze(modes(mod_i, solids_i, :))', [], 1);
A = zeros(3*length(solids_i), 6);
for i = 1:length(solids_i)
A(3*(i-1)+1:3*i, 1:3) = eye(3);
A(3*(i-1)+1:3*i, 4:6) = [0 acc_pos(i, 3) -acc_pos(i, 2) ; -acc_pos(i, 3) 0 acc_pos(i, 1) ; acc_pos(i, 2) -acc_pos(i, 1) 0];
end
mode_shapes_O(mod_i, solid_i, :) = A\Y;
end
end
#+end_src
* Modal Matrices
We want to obtain the two following matrices:
\[ \Omega = \begin{bmatrix}
\omega_1^2 & & 0 \\
& \ddots & \\
0 & & \omega_n^2
\end{bmatrix}; \quad \Psi = \begin{bmatrix}
& & \\
\{\psi_1\} & \dots & \{\psi_n\} \\
& &
\end{bmatrix} \]
- [ ] How to add damping to the eigen value matrix?
#+begin_src matlab
eigen_value_M = diag(freqs*2*pi);
eigen_vector_M = reshape(mode_shapes_O, [mod_n, 6*length(solid_names)])';
#+end_src
\[ \{\psi_1\} = \begin{Bmatrix} \psi_{1_x} & \psi_{2_x} & \dots & \psi_{6_x} & \psi_{1_x} & \dots & \psi_{1\Omega_x} & \dots & \psi_{6\Omega_z} \end{Bmatrix}^T \]
* Modal Complexity
Complexity of one mode
#+begin_src matlab
mod_i = 1;
i_max = convhull(real(eigen_vector_M(:, mod_i)), imag(eigen_vector_M(:, mod_i)));
radius = max(abs(eigen_vector_M(:, mod_i)));
theta = linspace(0, 2*pi, 100);
figure;
hold on;
plot(radius*cos(theta), radius*sin(theta), '-');
plot(real(eigen_vector_M(i_max, mod_i)), imag(eigen_vector_M(i_max, mod_i)), '-');
plot(real(eigen_vector_M(:, mod_i)), imag(eigen_vector_M(:, mod_i)), 'ko');
hold off;
xlabel('Real Part'); ylabel('Imaginary Part');
axis manual equal
#+end_src
#+HEADER: :tangle no :exports results :results none :noweb yes
#+begin_src matlab :var filepath="figs/modal_complexity.pdf" :var figsize="normal-normal" :post pdf2svg(file=*this*, ext="png")
<<plt-matlab>>
#+end_src
#+NAME: fig:modal_complexity
#+CAPTION: Modal Complexity of one mode
[[file:figs/modal_complexity.png]]
Complexity function of the mode order.
#+begin_src matlab
modes_complexity = zeros(mod_n, 1);
for mod_i = 1:mod_n
i = convhull(real(eigen_vector_M(:, mod_i)), imag(eigen_vector_M(:, mod_i)));
area_complex = polyarea(real(eigen_vector_M(i, mod_i)), imag(eigen_vector_M(i, mod_i)));
area_circle = pi*max(abs(eigen_vector_M(:, mod_i)))^2;
modes_complexity(mod_i) = area_complex/area_circle;
end
figure;
plot(1:mod_n, modes_complexity, 'ok');
ylim([0, 1]);
xlabel('Mode Number'); ylabel('Modal Complexity');
#+end_src
#+HEADER: :tangle no :exports results :results none :noweb yes
#+begin_src matlab :var filepath="figs/modal_complexities.pdf" :var figsize="wide-normal" :post pdf2svg(file=*this*, ext="png")
<<plt-matlab>>
#+end_src
#+NAME: fig:modal_complexities
#+CAPTION: Modal complexity for each mode
[[file:figs/modal_complexities.png]]
* Some notes about constraining the number of degrees of freedom
We want to have the two eigen matrices.
They should have the same size $n \times n$ where $n$ is the number of modes as well as the number of degrees of freedom.
Thus, if we consider 21 modes, we should restrict our system to have only 21 DOFs.
Actually, we are measured 6 DOFs of 6 solids, thus we have 36 DOFs.
From the mode shapes animations, it seems that in the frequency range of interest, the two marbles can be considered as one solid.
We thus have 5 solids and 30 DOFs.
In order to determine which DOF can be neglected, two solutions seems possible:
- compare the mode shapes
- compare the FRFs
The question is: in which base (frame) should be express the modes shapes and FRFs?
Is it meaningful to compare mode shapes as they give no information about the amplitudes of vibration?
| Stage | Motion DOFs | Parasitic DOF | Total DOF | Description of DOF |
|---------+-------------+---------------+-----------+--------------------|
| Granite | 0 | 3 | 3 | |
| Ty | 1 | 2 | 3 | Ty, Rz |
| Ry | 1 | 2 | 3 | Ry, |
| Rz | 1 | 2 | 3 | Rz, Rx, Ry |
| Hexapod | 6 | 0 | 6 | Txyz, Rxyz |
|---------+-------------+---------------+-----------+--------------------|
| | 9 | 9 | 18 | |
#+TBLFM: $4=vsum($2..$3)
#+TBLFM: @>$2..$>=vsum(@I..@II)
* TODO Normalization of mode shapes?
We normalize each column of the eigen vector matrix.
Then, each eigenvector as a norm of 1.
#+begin_src matlab
eigen_vector_M = eigen_vector_M./vecnorm(eigen_vector_M);
#+end_src
* Compare Mode Shapes
Let's say we want to see for the first mode which DOFs can be neglected.
In order to do so, we should estimate the motion of each stage in particular directions.
If we look at the z motion for instance, we will find that we cannot neglect that motion (because of the tilt causing z motion).
#+begin_src matlab
mode_i = 3;
dof_i = 6;
mode = eigen_vector_M(dof_i:6:end, mode_i);
figure;
hold on;
for i=1:length(mode)
plot([0, real(mode(i))], [0, imag(mode(i))], '-', 'DisplayName', solid_names{i});
end
hold off;
legend();
#+end_src
#+begin_src matlab
figure;
subplot(2, 1, 1);
hold on;
for i=1:length(mode)
plot(1, norm(mode(i)), 'o');
end
hold off;
ylabel('Amplitude');
subplot(2, 1, 2);
hold on;
for i=1:length(mode)
plot(1, 180/pi*angle(mode(i)), 'o', 'DisplayName', solid_names{i});
end
hold off;
ylim([-180, 180]); yticks([-180:90:180]);
ylabel('Phase [deg]');
legend();
#+end_src
#+begin_src matlab
test = mode_shapes_O(10, 1, :)/norm(squeeze(mode_shapes_O(10, 1, :)));
test = mode_shapes_O(10, 2, :)/norm(squeeze(mode_shapes_O(10, 2, :)));
#+end_src
* TODO Synthesis of FRF curves