Update the modal analysis
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@ -503,6 +503,983 @@ Mode matrix of local coordinate [DOF: Re IM]
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23Y-: -1.90862e-001 -2.53414e-002
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23Y-: -1.90862e-001 -2.53414e-002
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#+end_example
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#+end_example
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* Compute the Modal Model
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** Matlab Init :noexport:ignore:
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#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
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<<matlab-dir>>
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#+end_src
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#+begin_src matlab :exports none :results silent :noweb yes
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<<matlab-init>>
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#+end_src
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** Position of the accelerometers
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There are 23 accelerometers:
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- 4 on the bottom granite
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- 4 on the top granite
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- 4 on top of the translation stage
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- 4 on the tilt stage
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- 3 on top of the spindle
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- 4 on top of the hexapod
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The coordinates defined in the software are displayed below.
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#+begin_src bash :results output :exports results :eval no-export
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sed -n 18,40p modal_analysis/acc_coordinates.txt | tac --
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#+end_src
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#+RESULTS:
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#+begin_example
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1 1.0000e-001 1.0000e-001 1.1500e+000 0 Top
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2 1.0000e-001 -1.0000e-001 1.1500e+000 0 Top
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3 -1.0000e-001 -1.0000e-001 1.1500e+000 0 Top
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4 -1.0000e-001 1.0000e-001 1.1500e+000 0 Top
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5 4.0000e-001 4.0000e-001 9.5000e-001 0 inner
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6 4.0000e-001 -4.0000e-001 9.5000e-001 0 inner
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7 -4.0000e-001 -4.0000e-001 9.5000e-001 0 inner
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8 -4.0000e-001 4.0000e-001 9.5000e-001 0 inner
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9 5.0000e-001 5.0000e-001 9.0000e-001 0 outer
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10 5.0000e-001 -5.0000e-001 9.0000e-001 0 outer
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11 -5.0000e-001 -5.0000e-001 9.0000e-001 0 outer
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12 -5.0000e-001 5.0000e-001 9.0000e-001 0 outer
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13 5.5000e-001 5.5000e-001 5.5000e-001 0 top
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14 5.5000e-001 -5.5000e-001 5.5000e-001 0 top
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15 -5.5000e-001 -5.5000e-001 5.5000e-001 0 top
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16 -5.5000e-001 5.5000e-001 5.5000e-001 0 top
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17 9.5000e-001 9.5000e-001 4.0000e-001 0 low
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18 9.5000e-001 -9.5000e-001 4.0000e-001 0 low
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19 -9.5000e-001 -9.5000e-001 4.0000e-001 0 low
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20 -9.5000e-001 9.5000e-001 4.0000e-001 0 low
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21 2.0000e-001 2.0000e-001 8.5000e-001 0 bot
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22 0.0000e+000 -2.0000e-001 8.5000e-001 0 bot
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23 -2.0000e-001 2.0000e-001 8.5000e-001 0 bot
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#+end_example
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#+name: tab:acc_location
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#+caption: Location of each Accelerometer (using the normal coordinate frame with X aligned with the X ray)
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| *Node number* | *Solid Body* | *Location* | *X* | *Y* | *Z* |
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|---------------+-------------------+------------+-------+-------+------|
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| 1 | Hexapod - Top | -X/-Y | -0.10 | -0.10 | 1.15 |
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| 2 | | -X/+Y | -0.10 | 0.10 | 1.15 |
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| 3 | | +X/+Y | 0.10 | 0.10 | 1.15 |
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| 4 | | +X/-Y | 0.10 | -0.10 | 1.15 |
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|---------------+-------------------+------------+-------+-------+------|
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| 5 | Tilt - Top | -X/-Y | -0.40 | -0.40 | 0.95 |
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| 6 | | -X/+Y | -0.40 | 0.40 | 0.95 |
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| 7 | | +X/+Y | 0.40 | 0.40 | 0.95 |
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| 8 | | +X/-Y | 0.40 | -0.40 | 0.95 |
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|---------------+-------------------+------------+-------+-------+------|
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| 9 | Translation - Top | -X/-Y | -0.50 | -0.50 | 0.90 |
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| 10 | | -X/+Y | -0.50 | 0.50 | 0.90 |
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| 11 | | +X/+Y | 0.50 | 0.50 | 0.90 |
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| 12 | | +X/-Y | 0.50 | -0.50 | 0.90 |
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|---------------+-------------------+------------+-------+-------+------|
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| 13 | Top Granite | -X/-Y | -0.55 | -0.50 | 0.55 |
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| 14 | | -X/+Y | -0.55 | 0.50 | 0.55 |
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| 15 | | +X/+Y | 0.55 | 0.50 | 0.55 |
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| 16 | | +X/-Y | 0.55 | -0.50 | 0.55 |
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|---------------+-------------------+------------+-------+-------+------|
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| 17 | Bottom Granite | -X/-Y | -0.95 | -0.90 | 0.40 |
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| 18 | | -X/+Y | -0.95 | 0.90 | 0.40 |
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| 19 | | +X/+Y | 0.95 | 0.90 | 0.40 |
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| 20 | | +X/-Y | 0.95 | -0.90 | 0.40 |
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|---------------+-------------------+------------+-------+-------+------|
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| 21 | Spindle - Top | -X/-Y | -0.20 | -0.20 | 0.85 |
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| 22 | | +0/+Y | 0.00 | 0.20 | 0.85 |
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| 23 | | +X/-Y | 0.20 | -0.20 | 0.85 |
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** Define positions of the accelerometers on matlab
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We define the X-Y-Z position of each sensor.
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Each line corresponds to one accelerometer, X-Y-Z position in meter.
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#+begin_src matlab
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positions = [...
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-0.10, -0.10, 1.15 ; ...
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-0.10, 0.10, 1.15 ; ...
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0.10, 0.10, 1.15 ; ...
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0.10, -0.10, 1.15 ; ...
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-0.40, -0.40, 0.95 ; ...
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-0.40, 0.40, 0.95 ; ...
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0.40, 0.40, 0.95 ; ...
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0.40, -0.40, 0.95 ; ...
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-0.50, -0.50, 0.90 ; ...
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-0.50, 0.50, 0.90 ; ...
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0.50, 0.50, 0.90 ; ...
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0.50, -0.50, 0.90 ; ...
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-0.55, -0.50, 0.55 ; ...
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-0.55, 0.50, 0.55 ; ...
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0.55, 0.50, 0.55 ; ...
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0.55, -0.50, 0.55 ; ...
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-0.95, -0.90, 0.40 ; ...
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-0.95, 0.90, 0.40 ; ...
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0.95, 0.90, 0.40 ; ...
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0.95, -0.90, 0.40 ; ...
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-0.20, -0.20, 0.85 ; ...
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0.00, 0.20, 0.85 ; ...
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0.20, -0.20, 0.85 ];
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#+end_src
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#+begin_src matlab
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figure;
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hold on;
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fill3(positions(1:4, 1), positions(1:4, 2), positions(1:4, 3), 'k', 'FaceAlpha', 0.5)
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fill3(positions(5:8, 1), positions(5:8, 2), positions(5:8, 3), 'k', 'FaceAlpha', 0.5)
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fill3(positions(9:12, 1), positions(9:12, 2), positions(9:12, 3), 'k', 'FaceAlpha', 0.5)
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fill3(positions(13:16, 1), positions(13:16, 2), positions(13:16, 3), 'k', 'FaceAlpha', 0.5)
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fill3(positions(17:20, 1), positions(17:20, 2), positions(17:20, 3), 'k', 'FaceAlpha', 0.5)
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fill3(positions(21:23, 1), positions(21:23, 2), positions(21:23, 3), 'k', 'FaceAlpha', 0.5)
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hold off;
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#+end_src
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** Import the modal vectors on matlab
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*** Mode1
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#+begin_src bash :results output :exports none :eval no-export
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sed -n 12,80p modal_analysis/modes_propres_narband.asc
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#+end_src
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#+RESULTS:
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#+begin_example
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1X+: -9.34637e-002 4.52445e-002
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1Y+: 2.33790e-001 1.41439e-003
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1Z+: -1.73754e-002 6.02449e-003
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2X+: -7.42108e-002 3.91543e-002
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2Y+: 2.41566e-001 -1.44869e-003
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2Z+: -5.99285e-003 2.10370e-003
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4X+: -1.02163e-001 2.79561e-002
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4Y+: 2.29048e-001 2.89782e-002
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4Z+: -2.85130e-002 1.77132e-004
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5X+: -8.77132e-002 3.34081e-002
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5Y+: 2.14182e-001 2.14655e-002
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5Z+: -1.54521e-002 1.26682e-002
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6X+: -7.90143e-002 2.42583e-002
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6Y+: 2.20669e-001 2.12738e-002
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6Z+: 4.60755e-002 4.96406e-003
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7X+: -7.79654e-002 2.58385e-002
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7Y+: 2.06861e-001 3.48019e-002
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7Z+: -1.78311e-002 -1.29704e-002
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8X+: -8.49357e-002 3.55200e-002
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8Y+: 2.07470e-001 3.59745e-002
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8Z+: -7.66974e-002 -3.19813e-003
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9X+: -7.38565e-002 1.95146e-002
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9Y+: 2.17403e-001 2.01550e-002
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9Z+: -1.77073e-002 -3.46414e-003
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10X+: -7.77587e-002 2.36700e-002
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10Y+: 2.35654e-001 -2.14540e-002
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10Z+: 7.94165e-002 -2.45897e-002
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11X+: -8.17972e-002 2.20583e-002
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11Y+: 2.20906e-001 -4.30164e-003
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11Z+: -5.60520e-003 3.10187e-003
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12X+: -8.64261e-002 3.66022e-002
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12Y+: 2.15000e-001 -5.74661e-003
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12Z+: -1.22622e-001 4.11767e-002
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13X+: -4.25169e-002 1.56602e-002
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13Y+: 5.31036e-002 -1.73951e-002
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13Z+: -4.07130e-002 1.26884e-002
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14X+: -3.85032e-002 1.29431e-002
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14Y+: 5.36716e-002 -1.80868e-002
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14Z+: 1.00367e-001 -3.48798e-002
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15X+: -4.25524e-002 1.46363e-002
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15Y+: 5.19668e-002 -1.69744e-002
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15Z+: 5.89747e-003 -2.32428e-003
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16X+: -4.31268e-002 1.38332e-002
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16Y+: 5.07545e-002 -1.53045e-002
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16Z+: -1.04172e-001 3.17984e-002
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17X+: -2.69757e-002 9.07955e-003
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17Y+: 3.07837e-002 -9.44663e-003
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17Z+: -7.63502e-003 1.68203e-003
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18X+: -3.00097e-002 9.23966e-003
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18Y+: 2.83585e-002 -8.97747e-003
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18Z+: 1.52467e-001 -4.78675e-002
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19X+: -2.70223e-002 6.16478e-003
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19Y+: 3.06149e-002 -6.25382e-003
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19Z+: -4.84888e-003 1.93970e-003
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20X+: -2.90976e-002 7.13184e-003
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20Y+: 3.36738e-002 -7.30875e-003
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20Z+: -1.66902e-001 3.93419e-002
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3X+: -9.40720e-002 3.93724e-002
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3Y+: 2.52307e-001 0.00000e+000
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3Z+: -1.53864e-002 -9.25720e-004
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21X+: -7.91940e-002 4.39648e-002
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21Y+: 2.04567e-001 9.49987e-003
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21Z+: -1.56087e-002 7.08838e-003
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22X+: -1.01070e-001 3.13534e-002
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22Y+: 1.92270e-001 1.80423e-002
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22Z+: 2.93053e-003 -1.97308e-003
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23X+: -8.86455e-002 4.29906e-002
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23Z+: -3.38351e-002 1.81362e-003
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23Y-: -1.90862e-001 -2.53414e-002
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#+end_example
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#+begin_src matlab
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mode1 = [...
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-9.34637e-002+j*4.52445e-002, +2.33790e-001+j*1.41439e-003, -1.73754e-002+j*6.02449e-003;
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-7.42108e-002+j*3.91543e-002, +2.41566e-001-j*1.44869e-003, -5.99285e-003+j*2.10370e-003;
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-9.40720e-002+j*3.93724e-002, +2.52307e-001+j*0.00000e+000, -1.53864e-002-j*9.25720e-004;
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-1.02163e-001+j*2.79561e-002, +2.29048e-001+j*2.89782e-002, -2.85130e-002+j*1.77132e-004;
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-8.77132e-002+j*3.34081e-002, +2.14182e-001+j*2.14655e-002, -1.54521e-002+j*1.26682e-002;
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-7.90143e-002+j*2.42583e-002, +2.20669e-001+j*2.12738e-002, +4.60755e-002+j*4.96406e-003;
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-7.79654e-002+j*2.58385e-002, +2.06861e-001+j*3.48019e-002, -1.78311e-002-j*1.29704e-002;
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-8.49357e-002+j*3.55200e-002, +2.07470e-001+j*3.59745e-002, -7.66974e-002-j*3.19813e-003;
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-7.38565e-002+j*1.95146e-002, +2.17403e-001+j*2.01550e-002, -1.77073e-002-j*3.46414e-003;
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-7.77587e-002+j*2.36700e-002, +2.35654e-001-j*2.14540e-002, +7.94165e-002-j*2.45897e-002;
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-8.17972e-002+j*2.20583e-002, +2.20906e-001-j*4.30164e-003, -5.60520e-003+j*3.10187e-003;
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-8.64261e-002+j*3.66022e-002, +2.15000e-001-j*5.74661e-003, -1.22622e-001+j*4.11767e-002;
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-4.25169e-002+j*1.56602e-002, +5.31036e-002-j*1.73951e-002, -4.07130e-002+j*1.26884e-002;
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-3.85032e-002+j*1.29431e-002, +5.36716e-002-j*1.80868e-002, +1.00367e-001-j*3.48798e-002;
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-4.25524e-002+j*1.46363e-002, +5.19668e-002-j*1.69744e-002, +5.89747e-003-j*2.32428e-003;
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-4.31268e-002+j*1.38332e-002, +5.07545e-002-j*1.53045e-002, -1.04172e-001+j*3.17984e-002;
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-2.69757e-002+j*9.07955e-003, +3.07837e-002-j*9.44663e-003, -7.63502e-003+j*1.68203e-003;
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-3.00097e-002+j*9.23966e-003, +2.83585e-002-j*8.97747e-003, +1.52467e-001-j*4.78675e-002;
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-2.70223e-002+j*6.16478e-003, +3.06149e-002-j*6.25382e-003, -4.84888e-003+j*1.93970e-003;
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-2.90976e-002+j*7.13184e-003, +3.36738e-002-j*7.30875e-003, -1.66902e-001+j*3.93419e-002;
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-7.91940e-002+j*4.39648e-002, +2.04567e-001+j*9.49987e-003, -1.56087e-002+j*7.08838e-003;
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-1.01070e-001+j*3.13534e-002, +1.92270e-001+j*1.80423e-002, +2.93053e-003-j*1.97308e-003;
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-8.86455e-002+j*4.29906e-002, +1.90862e-001+j*2.53414e-002, -3.38351e-002+j*1.81362e-003];
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#+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;
|
||||||
|
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||||||
|
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||||||
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|
||||||
|
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|
||||||
|
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|
||||||
|
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|
||||||
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|
||||||
|
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|
||||||
|
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|
||||||
|
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|
||||||
|
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|
||||||
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|
||||||
|
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||||||
|
|
||||||
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mode_shapes(:, :, 4) = [...
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||||||
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||||||
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||||||
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||||||
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||||||
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||||||
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||||||
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||||||
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||||||
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||||||
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||||||
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||||||
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||||||
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||||||
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||||||
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||||||
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||||||
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||||||
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||||||
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||||||
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||||||
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||||||
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|
||||||
|
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|
||||||
|
|
||||||
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mode_shapes(:, :, 5) = [...
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||||||
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||||||
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||||||
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||||||
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||||||
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||||||
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||||||
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||||||
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||||||
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||||||
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||||||
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||||||
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||||||
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||||||
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||||||
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||||||
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||||||
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||||||
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||||||
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||||||
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||||||
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|
||||||
|
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|
||||||
|
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|
||||||
|
|
||||||
|
mode_shapes(:, :, 6) = [...
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||||||
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||||||
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|
||||||
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||||||
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||||||
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||||||
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||||||
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||||||
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|
||||||
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|
||||||
|
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|
||||||
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|
||||||
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||||||
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|
||||||
|
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|
||||||
|
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|
||||||
|
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|
||||||
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|
||||||
|
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|
||||||
|
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|
||||||
|
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|
||||||
|
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|
||||||
|
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|
||||||
|
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|
||||||
|
|
||||||
|
mode_shapes(:, :, 7) = [...
|
||||||
|
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|
||||||
|
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||||||
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||||||
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||||||
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||||||
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||||||
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||||||
|
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|
||||||
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|
||||||
|
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|
||||||
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|
||||||
|
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|
||||||
|
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|
||||||
|
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|
||||||
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|
||||||
|
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|
||||||
|
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|
||||||
|
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|
||||||
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|
||||||
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|
||||||
|
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|
||||||
|
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|
||||||
|
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|
||||||
|
|
||||||
|
mode_shapes(:, :, 8) = [...
|
||||||
|
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|
||||||
|
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||||||
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||||||
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|
||||||
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||||||
|
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||||||
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||||||
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|
||||||
|
+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
|
* Problem with AirLoc System
|
||||||
4 Airloc Levelers are used for the granite (figure [[fig:airloc]]).
|
4 Airloc Levelers are used for the granite (figure [[fig:airloc]]).
|
||||||
|
|
||||||
|
|||||||
BIN
modal-analysis/modes_analysis.html
Normal file
BIN
modal-analysis/modes_analysis.html
Normal file
Binary file not shown.
478
modal-analysis/modes_analysis.org
Normal file
478
modal-analysis/modes_analysis.org
Normal file
@ -0,0 +1,478 @@
|
|||||||
|
#+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
|
||||||
Loading…
Reference in New Issue
Block a user