From b688d6300d39b99b7781586e55fa6c50517ec7f2 Mon Sep 17 00:00:00 2001 From: Thomas Dehaeze Date: Tue, 2 Jul 2019 17:48:34 +0200 Subject: [PATCH] Update the modal analysis --- .../figs/local_to_global_coordinates.png | Bin 0 -> 12760 bytes modal-analysis/figs/modal_complexities.png | Bin 0 -> 25036 bytes modal-analysis/figs/modal_complexity.png | Bin 0 -> 32115 bytes modal-analysis/figs/nass-modal-test.png | Bin 0 -> 102355 bytes modal-analysis/index.html | Bin 34546 -> 595496 bytes modal-analysis/index.org | 977 ++++++++++++++++++ modal-analysis/modes_analysis.html | Bin 0 -> 63436 bytes modal-analysis/modes_analysis.org | 478 +++++++++ 8 files changed, 1455 insertions(+) create mode 100644 modal-analysis/figs/local_to_global_coordinates.png create mode 100644 modal-analysis/figs/modal_complexities.png create mode 100644 modal-analysis/figs/modal_complexity.png create mode 100644 modal-analysis/figs/nass-modal-test.png create mode 100644 modal-analysis/modes_analysis.html create mode 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zMr~s70Z*=xdqFyD7>Wk%&^OT_$)ucWpwAA}C7#kyS%-`A5!vTxTEqE&>l5ei{aNML I8WIct0snL{X#fBK diff --git a/modal-analysis/index.org b/modal-analysis/index.org index 19ad5e3..8b8b6f8 100644 --- a/modal-analysis/index.org +++ b/modal-analysis/index.org @@ -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) + <> +#+end_src + +#+begin_src matlab :exports none :results silent :noweb yes + <> +#+end_src + +** Position of the accelerometers +There are 23 accelerometers: +- 4 on the bottom granite +- 4 on the top granite +- 4 on top of the translation stage +- 4 on the tilt stage +- 3 on top of the spindle +- 4 on top of the hexapod + +The coordinates defined in the software are displayed below. + +#+begin_src bash :results output :exports results :eval no-export + sed -n 18,40p modal_analysis/acc_coordinates.txt | tac -- +#+end_src + +#+RESULTS: +#+begin_example + 1 1.0000e-001 1.0000e-001 1.1500e+000 0 Top + 2 1.0000e-001 -1.0000e-001 1.1500e+000 0 Top + 3 -1.0000e-001 -1.0000e-001 1.1500e+000 0 Top + 4 -1.0000e-001 1.0000e-001 1.1500e+000 0 Top + 5 4.0000e-001 4.0000e-001 9.5000e-001 0 inner + 6 4.0000e-001 -4.0000e-001 9.5000e-001 0 inner + 7 -4.0000e-001 -4.0000e-001 9.5000e-001 0 inner + 8 -4.0000e-001 4.0000e-001 9.5000e-001 0 inner + 9 5.0000e-001 5.0000e-001 9.0000e-001 0 outer + 10 5.0000e-001 -5.0000e-001 9.0000e-001 0 outer + 11 -5.0000e-001 -5.0000e-001 9.0000e-001 0 outer + 12 -5.0000e-001 5.0000e-001 9.0000e-001 0 outer + 13 5.5000e-001 5.5000e-001 5.5000e-001 0 top + 14 5.5000e-001 -5.5000e-001 5.5000e-001 0 top + 15 -5.5000e-001 -5.5000e-001 5.5000e-001 0 top + 16 -5.5000e-001 5.5000e-001 5.5000e-001 0 top + 17 9.5000e-001 9.5000e-001 4.0000e-001 0 low + 18 9.5000e-001 -9.5000e-001 4.0000e-001 0 low + 19 -9.5000e-001 -9.5000e-001 4.0000e-001 0 low + 20 -9.5000e-001 9.5000e-001 4.0000e-001 0 low + 21 2.0000e-001 2.0000e-001 8.5000e-001 0 bot + 22 0.0000e+000 -2.0000e-001 8.5000e-001 0 bot + 23 -2.0000e-001 2.0000e-001 8.5000e-001 0 bot +#+end_example + +#+name: tab:acc_location +#+caption: Location of each Accelerometer (using the normal coordinate frame with X aligned with the X ray) +| *Node number* | *Solid Body* | *Location* | *X* | *Y* | *Z* | +|---------------+-------------------+------------+-------+-------+------| +| 1 | Hexapod - Top | -X/-Y | -0.10 | -0.10 | 1.15 | +| 2 | | -X/+Y | -0.10 | 0.10 | 1.15 | +| 3 | | +X/+Y | 0.10 | 0.10 | 1.15 | +| 4 | | +X/-Y | 0.10 | -0.10 | 1.15 | +|---------------+-------------------+------------+-------+-------+------| +| 5 | Tilt - Top | -X/-Y | -0.40 | -0.40 | 0.95 | +| 6 | | -X/+Y | -0.40 | 0.40 | 0.95 | +| 7 | | +X/+Y | 0.40 | 0.40 | 0.95 | +| 8 | | +X/-Y | 0.40 | -0.40 | 0.95 | +|---------------+-------------------+------------+-------+-------+------| +| 9 | Translation - Top | -X/-Y | -0.50 | -0.50 | 0.90 | +| 10 | | -X/+Y | -0.50 | 0.50 | 0.90 | +| 11 | | +X/+Y | 0.50 | 0.50 | 0.90 | +| 12 | | +X/-Y | 0.50 | -0.50 | 0.90 | +|---------------+-------------------+------------+-------+-------+------| +| 13 | Top Granite | -X/-Y | -0.55 | -0.50 | 0.55 | +| 14 | | -X/+Y | -0.55 | 0.50 | 0.55 | +| 15 | | +X/+Y | 0.55 | 0.50 | 0.55 | +| 16 | | +X/-Y | 0.55 | -0.50 | 0.55 | +|---------------+-------------------+------------+-------+-------+------| +| 17 | Bottom Granite | -X/-Y | -0.95 | -0.90 | 0.40 | +| 18 | | -X/+Y | -0.95 | 0.90 | 0.40 | +| 19 | | +X/+Y | 0.95 | 0.90 | 0.40 | +| 20 | | +X/-Y | 0.95 | -0.90 | 0.40 | +|---------------+-------------------+------------+-------+-------+------| +| 21 | Spindle - Top | -X/-Y | -0.20 | -0.20 | 0.85 | +| 22 | | +0/+Y | 0.00 | 0.20 | 0.85 | +| 23 | | +X/-Y | 0.20 | -0.20 | 0.85 | + +** Define positions of the accelerometers on matlab +We define the X-Y-Z position of each sensor. +Each line corresponds to one accelerometer, X-Y-Z position in meter. +#+begin_src matlab + positions = [... + -0.10, -0.10, 1.15 ; ... + -0.10, 0.10, 1.15 ; ... + 0.10, 0.10, 1.15 ; ... + 0.10, -0.10, 1.15 ; ... + -0.40, -0.40, 0.95 ; ... + -0.40, 0.40, 0.95 ; ... + 0.40, 0.40, 0.95 ; ... + 0.40, -0.40, 0.95 ; ... + -0.50, -0.50, 0.90 ; ... + -0.50, 0.50, 0.90 ; ... + 0.50, 0.50, 0.90 ; ... + 0.50, -0.50, 0.90 ; ... + -0.55, -0.50, 0.55 ; ... + -0.55, 0.50, 0.55 ; ... + 0.55, 0.50, 0.55 ; ... + 0.55, -0.50, 0.55 ; ... + -0.95, -0.90, 0.40 ; ... + -0.95, 0.90, 0.40 ; ... + 0.95, 0.90, 0.40 ; ... + 0.95, -0.90, 0.40 ; ... + -0.20, -0.20, 0.85 ; ... + 0.00, 0.20, 0.85 ; ... + 0.20, -0.20, 0.85 ]; +#+end_src + + +#+begin_src matlab + figure; + hold on; + fill3(positions(1:4, 1), positions(1:4, 2), positions(1:4, 3), 'k', 'FaceAlpha', 0.5) + fill3(positions(5:8, 1), positions(5:8, 2), positions(5:8, 3), 'k', 'FaceAlpha', 0.5) + fill3(positions(9:12, 1), positions(9:12, 2), positions(9:12, 3), 'k', 'FaceAlpha', 0.5) + fill3(positions(13:16, 1), positions(13:16, 2), positions(13:16, 3), 'k', 'FaceAlpha', 0.5) + fill3(positions(17:20, 1), positions(17:20, 2), positions(17:20, 3), 'k', 'FaceAlpha', 0.5) + fill3(positions(21:23, 1), positions(21:23, 2), positions(21:23, 3), 'k', 'FaceAlpha', 0.5) + hold off; +#+end_src + +** Import the modal vectors on matlab +*** Mode1 +#+begin_src bash :results output :exports none :eval no-export + sed -n 12,80p modal_analysis/modes_propres_narband.asc +#+end_src + +#+RESULTS: +#+begin_example + 1X+: -9.34637e-002 4.52445e-002 + 1Y+: 2.33790e-001 1.41439e-003 + 1Z+: -1.73754e-002 6.02449e-003 + 2X+: -7.42108e-002 3.91543e-002 + 2Y+: 2.41566e-001 -1.44869e-003 + 2Z+: -5.99285e-003 2.10370e-003 + 4X+: -1.02163e-001 2.79561e-002 + 4Y+: 2.29048e-001 2.89782e-002 + 4Z+: -2.85130e-002 1.77132e-004 + 5X+: -8.77132e-002 3.34081e-002 + 5Y+: 2.14182e-001 2.14655e-002 + 5Z+: -1.54521e-002 1.26682e-002 + 6X+: -7.90143e-002 2.42583e-002 + 6Y+: 2.20669e-001 2.12738e-002 + 6Z+: 4.60755e-002 4.96406e-003 + 7X+: -7.79654e-002 2.58385e-002 + 7Y+: 2.06861e-001 3.48019e-002 + 7Z+: -1.78311e-002 -1.29704e-002 + 8X+: -8.49357e-002 3.55200e-002 + 8Y+: 2.07470e-001 3.59745e-002 + 8Z+: -7.66974e-002 -3.19813e-003 + 9X+: -7.38565e-002 1.95146e-002 + 9Y+: 2.17403e-001 2.01550e-002 + 9Z+: -1.77073e-002 -3.46414e-003 + 10X+: -7.77587e-002 2.36700e-002 + 10Y+: 2.35654e-001 -2.14540e-002 + 10Z+: 7.94165e-002 -2.45897e-002 + 11X+: -8.17972e-002 2.20583e-002 + 11Y+: 2.20906e-001 -4.30164e-003 + 11Z+: -5.60520e-003 3.10187e-003 + 12X+: -8.64261e-002 3.66022e-002 + 12Y+: 2.15000e-001 -5.74661e-003 + 12Z+: -1.22622e-001 4.11767e-002 + 13X+: -4.25169e-002 1.56602e-002 + 13Y+: 5.31036e-002 -1.73951e-002 + 13Z+: -4.07130e-002 1.26884e-002 + 14X+: -3.85032e-002 1.29431e-002 + 14Y+: 5.36716e-002 -1.80868e-002 + 14Z+: 1.00367e-001 -3.48798e-002 + 15X+: -4.25524e-002 1.46363e-002 + 15Y+: 5.19668e-002 -1.69744e-002 + 15Z+: 5.89747e-003 -2.32428e-003 + 16X+: -4.31268e-002 1.38332e-002 + 16Y+: 5.07545e-002 -1.53045e-002 + 16Z+: -1.04172e-001 3.17984e-002 + 17X+: -2.69757e-002 9.07955e-003 + 17Y+: 3.07837e-002 -9.44663e-003 + 17Z+: -7.63502e-003 1.68203e-003 + 18X+: -3.00097e-002 9.23966e-003 + 18Y+: 2.83585e-002 -8.97747e-003 + 18Z+: 1.52467e-001 -4.78675e-002 + 19X+: -2.70223e-002 6.16478e-003 + 19Y+: 3.06149e-002 -6.25382e-003 + 19Z+: -4.84888e-003 1.93970e-003 + 20X+: -2.90976e-002 7.13184e-003 + 20Y+: 3.36738e-002 -7.30875e-003 + 20Z+: -1.66902e-001 3.93419e-002 + 3X+: -9.40720e-002 3.93724e-002 + 3Y+: 2.52307e-001 0.00000e+000 + 3Z+: -1.53864e-002 -9.25720e-004 + 21X+: -7.91940e-002 4.39648e-002 + 21Y+: 2.04567e-001 9.49987e-003 + 21Z+: -1.56087e-002 7.08838e-003 + 22X+: -1.01070e-001 3.13534e-002 + 22Y+: 1.92270e-001 1.80423e-002 + 22Z+: 2.93053e-003 -1.97308e-003 + 23X+: -8.86455e-002 4.29906e-002 + 23Z+: -3.38351e-002 1.81362e-003 + 23Y-: -1.90862e-001 -2.53414e-002 +#+end_example + +#+begin_src matlab + mode1 = [... + -9.34637e-002+j*4.52445e-002, +2.33790e-001+j*1.41439e-003, -1.73754e-002+j*6.02449e-003; + -7.42108e-002+j*3.91543e-002, +2.41566e-001-j*1.44869e-003, -5.99285e-003+j*2.10370e-003; + -9.40720e-002+j*3.93724e-002, +2.52307e-001+j*0.00000e+000, -1.53864e-002-j*9.25720e-004; + -1.02163e-001+j*2.79561e-002, +2.29048e-001+j*2.89782e-002, -2.85130e-002+j*1.77132e-004; + -8.77132e-002+j*3.34081e-002, +2.14182e-001+j*2.14655e-002, -1.54521e-002+j*1.26682e-002; + -7.90143e-002+j*2.42583e-002, +2.20669e-001+j*2.12738e-002, +4.60755e-002+j*4.96406e-003; + -7.79654e-002+j*2.58385e-002, +2.06861e-001+j*3.48019e-002, -1.78311e-002-j*1.29704e-002; + -8.49357e-002+j*3.55200e-002, +2.07470e-001+j*3.59745e-002, -7.66974e-002-j*3.19813e-003; + -7.38565e-002+j*1.95146e-002, +2.17403e-001+j*2.01550e-002, -1.77073e-002-j*3.46414e-003; + -7.77587e-002+j*2.36700e-002, +2.35654e-001-j*2.14540e-002, +7.94165e-002-j*2.45897e-002; + -8.17972e-002+j*2.20583e-002, +2.20906e-001-j*4.30164e-003, -5.60520e-003+j*3.10187e-003; + -8.64261e-002+j*3.66022e-002, +2.15000e-001-j*5.74661e-003, -1.22622e-001+j*4.11767e-002; + -4.25169e-002+j*1.56602e-002, +5.31036e-002-j*1.73951e-002, -4.07130e-002+j*1.26884e-002; + -3.85032e-002+j*1.29431e-002, +5.36716e-002-j*1.80868e-002, +1.00367e-001-j*3.48798e-002; + -4.25524e-002+j*1.46363e-002, +5.19668e-002-j*1.69744e-002, +5.89747e-003-j*2.32428e-003; + -4.31268e-002+j*1.38332e-002, +5.07545e-002-j*1.53045e-002, -1.04172e-001+j*3.17984e-002; + -2.69757e-002+j*9.07955e-003, +3.07837e-002-j*9.44663e-003, -7.63502e-003+j*1.68203e-003; + -3.00097e-002+j*9.23966e-003, +2.83585e-002-j*8.97747e-003, +1.52467e-001-j*4.78675e-002; + -2.70223e-002+j*6.16478e-003, +3.06149e-002-j*6.25382e-003, -4.84888e-003+j*1.93970e-003; + -2.90976e-002+j*7.13184e-003, +3.36738e-002-j*7.30875e-003, -1.66902e-001+j*3.93419e-002; + -7.91940e-002+j*4.39648e-002, +2.04567e-001+j*9.49987e-003, -1.56087e-002+j*7.08838e-003; + -1.01070e-001+j*3.13534e-002, +1.92270e-001+j*1.80423e-002, +2.93053e-003-j*1.97308e-003; + -8.86455e-002+j*4.29906e-002, +1.90862e-001+j*2.53414e-002, -3.38351e-002+j*1.81362e-003]; +#+end_src + +*** Mode2 +#+begin_src bash :results output :exports none :eval no-export + sed -n 88,156p modal_analysis/modes_propres_narband.asc +#+end_src + +#+RESULTS: +#+begin_example + 1X+: 7.56931e-002 3.61548e-002 + 1Y+: 2.07574e-001 1.69205e-004 + 1Z+: 1.29733e-002 -6.78426e-004 + 2X+: 8.58732e-002 2.54470e-002 + 2Y+: 2.07117e-001 -1.31755e-003 + 2Z+: -2.13788e-003 -1.24974e-002 + 4X+: 7.09825e-002 3.66313e-002 + 4Y+: 2.09969e-001 1.11484e-002 + 4Z+: 9.19478e-003 3.47272e-002 + 5X+: 6.23935e-002 1.02488e-002 + 5Y+: 2.30687e-001 -3.58416e-003 + 5Z+: 3.27122e-002 -5.85468e-002 + 6X+: 7.61163e-002 -2.43630e-002 + 6Y+: 2.26743e-001 -1.15334e-002 + 6Z+: -6.20205e-003 -1.21742e-001 + 7X+: 8.01824e-002 -1.94769e-002 + 7Y+: 1.97485e-001 4.50105e-002 + 7Z+: -2.21170e-002 9.77052e-002 + 8X+: 6.19294e-002 8.15075e-003 + 8Y+: 2.03864e-001 4.45835e-002 + 8Z+: 2.55133e-002 1.36137e-001 + 9X+: 4.38135e-002 7.30537e-002 + 9Y+: 2.28426e-001 -6.58868e-003 + 9Z+: 1.16313e-002 5.09427e-004 + 10X+: 5.45770e-002 4.34251e-002 + 10Y+: 2.50823e-001 0.00000e+000 + 10Z+: -4.63460e-002 -4.76868e-002 + 11X+: 5.50987e-002 4.26178e-002 + 11Y+: 2.29394e-001 5.78236e-002 + 11Z+: 1.90158e-002 1.09139e-002 + 12X+: 4.98867e-002 7.30190e-002 + 12Y+: 2.07871e-001 4.57750e-002 + 12Z+: 6.69433e-002 9.00315e-002 + 13X+: 2.48819e-002 3.03222e-002 + 13Y+: -2.56046e-002 -3.34132e-002 + 13Z+: 2.13260e-002 2.58544e-002 + 14X+: 2.45706e-002 2.60221e-002 + 14Y+: -2.57723e-002 -3.35612e-002 + 14Z+: -5.71282e-002 -6.61562e-002 + 15X+: 2.68196e-002 2.83888e-002 + 15Y+: -2.57263e-002 -3.29627e-002 + 15Z+: -2.11722e-003 -3.37239e-003 + 16X+: 2.51442e-002 3.32558e-002 + 16Y+: -2.54372e-002 -3.25062e-002 + 16Z+: 5.65780e-002 7.64142e-002 + 17X+: 1.62437e-002 1.94534e-002 + 17Y+: -1.31293e-002 -2.05924e-002 + 17Z+: 1.05274e-003 3.59474e-003 + 18X+: 1.83431e-002 2.03836e-002 + 18Y+: -1.16818e-002 -1.86334e-002 + 18Z+: -8.66632e-002 -1.08216e-001 + 19X+: 1.62553e-002 1.79588e-002 + 19Y+: -1.28857e-002 -1.90512e-002 + 19Z+: 6.25653e-003 4.97733e-003 + 20X+: 1.63830e-002 2.03943e-002 + 20Y+: -1.48941e-002 -2.11717e-002 + 20Z+: 8.68045e-002 1.16491e-001 + 3X+: 8.17201e-002 2.36079e-002 + 3Y+: 2.15927e-001 1.61300e-002 + 3Z+: -5.48456e-004 2.55691e-002 + 21X+: 6.79204e-002 -5.55513e-002 + 21Y+: 2.32871e-001 2.33389e-002 + 21Z+: 1.34345e-002 -2.31815e-002 + 22X+: 4.02414e-002 -8.38957e-002 + 22Y+: 2.35273e-001 2.73256e-002 + 22Z+: -8.51632e-003 -7.49635e-003 + 23X+: 6.18293e-002 -5.99671e-002 + 23Z+: 1.63533e-002 6.09161e-002 + 23Y-: -2.37693e-001 -4.34204e-002 +#+end_example + +#+begin_src matlab + mode2 = [... + +7.56931e-002+j*3.61548e-002, +2.07574e-001+j*1.69205e-004, +1.29733e-002-j*6.78426e-004; + +8.58732e-002+j*2.54470e-002, +2.07117e-001-j*1.31755e-003, -2.13788e-003-j*1.24974e-002; + +8.17201e-002+j*2.36079e-002, +2.15927e-001+j*1.61300e-002, -5.48456e-004+j*2.55691e-002; + +7.09825e-002+j*3.66313e-002, +2.09969e-001+j*1.11484e-002, +9.19478e-003+j*3.47272e-002; + +6.23935e-002+j*1.02488e-002, +2.30687e-001-j*3.58416e-003, +3.27122e-002-j*5.85468e-002; + +7.61163e-002-j*2.43630e-002, +2.26743e-001-j*1.15334e-002, -6.20205e-003-j*1.21742e-001; + +8.01824e-002-j*1.94769e-002, +1.97485e-001+j*4.50105e-002, -2.21170e-002+j*9.77052e-002; + +6.19294e-002+j*8.15075e-003, +2.03864e-001+j*4.45835e-002, +2.55133e-002+j*1.36137e-001; + +4.38135e-002+j*7.30537e-002, +2.28426e-001-j*6.58868e-003, +1.16313e-002+j*5.09427e-004; + +5.45770e-002+j*4.34251e-002, +2.50823e-001+j*0.00000e+000, -4.63460e-002-j*4.76868e-002; + +5.50987e-002+j*4.26178e-002, +2.29394e-001+j*5.78236e-002, +1.90158e-002+j*1.09139e-002; + +4.98867e-002+j*7.30190e-002, +2.07871e-001+j*4.57750e-002, +6.69433e-002+j*9.00315e-002; + +2.48819e-002+j*3.03222e-002, -2.56046e-002-j*3.34132e-002, +2.13260e-002+j*2.58544e-002; + +2.45706e-002+j*2.60221e-002, -2.57723e-002-j*3.35612e-002, -5.71282e-002-j*6.61562e-002; + +2.68196e-002+j*2.83888e-002, -2.57263e-002-j*3.29627e-002, -2.11722e-003-j*3.37239e-003; + +2.51442e-002+j*3.32558e-002, -2.54372e-002-j*3.25062e-002, +5.65780e-002+j*7.64142e-002; + +1.62437e-002+j*1.94534e-002, -1.31293e-002-j*2.05924e-002, +1.05274e-003+j*3.59474e-003; + +1.83431e-002+j*2.03836e-002, -1.16818e-002-j*1.86334e-002, -8.66632e-002-j*1.08216e-001; + +1.62553e-002+j*1.79588e-002, -1.28857e-002-j*1.90512e-002, +6.25653e-003+j*4.97733e-003; + +1.63830e-002+j*2.03943e-002, -1.48941e-002-j*2.11717e-002, +8.68045e-002+j*1.16491e-001; + +6.79204e-002-j*5.55513e-002, +2.32871e-001+j*2.33389e-002, +1.34345e-002-j*2.31815e-002; + +4.02414e-002-j*8.38957e-002, +2.35273e-001+j*2.73256e-002, -8.51632e-003-j*7.49635e-003; + +6.18293e-002-j*5.99671e-002, +2.37693e-001+j*4.34204e-002, +1.63533e-002+j*6.09161e-002] +#+end_src + +*** Mode3 +#+begin_src bash :results output :exports none :eval no-export + sed -n 164,232p modal_analysis/modes_propres_narband.asc +#+end_src + +#+RESULTS: +#+begin_example + 1X+: 1.34688e-001 -6.65071e-002 + 1Y+: 1.55316e-002 1.01277e-002 + 1Z+: -5.88466e-002 1.14294e-002 + 2X+: 1.53934e-001 -9.76990e-003 + 2Y+: 7.17487e-003 1.11925e-002 + 2Z+: -4.57205e-002 7.26573e-003 + 4X+: 1.37298e-001 -5.24661e-002 + 4Y+: 1.19427e-003 -5.39240e-002 + 4Z+: -1.25915e-002 5.38133e-003 + 5X+: 2.43192e-001 -3.17374e-002 + 5Y+: -2.15730e-001 -7.69941e-004 + 5Z+: -1.56268e-001 1.44118e-002 + 6X+: -7.27705e-002 -3.54943e-003 + 6Y+: -2.47706e-001 2.66480e-003 + 6Z+: -1.21590e-001 1.06054e-002 + 7X+: -7.25870e-002 -4.62024e-003 + 7Y+: 2.27073e-001 -3.69315e-002 + 7Z+: 1.22611e-001 -6.67337e-003 + 8X+: 2.32731e-001 -2.85516e-002 + 8Y+: 2.35389e-001 -3.81905e-002 + 8Z+: 5.35574e-002 4.30394e-004 + 9X+: 2.64170e-001 -2.67367e-002 + 9Y+: -2.56227e-001 3.97957e-005 + 9Z+: -1.95398e-001 2.23549e-002 + 10X+: -1.66953e-002 -7.95698e-003 + 10Y+: -2.66547e-001 -2.17687e-002 + 10Z+: 1.56278e-002 2.23786e-003 + 11X+: -3.42364e-002 -9.30205e-003 + 11Y+: 2.52340e-001 -7.47237e-003 + 11Z+: -9.51643e-004 3.64798e-003 + 12X+: 2.97574e-001 0.00000e+000 + 12Y+: 2.23170e-001 -1.37831e-002 + 12Z+: 1.06266e-001 2.30324e-003 + 13X+: 2.67178e-002 -4.15723e-004 + 13Y+: 6.75423e-003 -2.18428e-003 + 13Z+: -1.69423e-002 3.12395e-003 + 14X+: -1.12283e-002 2.86316e-004 + 14Y+: 5.08225e-003 -2.14053e-003 + 14Z+: 2.18339e-002 -3.25204e-003 + 15X+: -1.17948e-002 6.82873e-004 + 15Y+: 1.94914e-002 -2.42151e-003 + 15Z+: 2.68660e-003 -2.92104e-004 + 16X+: 1.19490e-002 1.72236e-005 + 16Y+: 1.83552e-002 -2.71289e-003 + 16Z+: -2.70914e-002 4.84164e-003 + 17X+: 1.00173e-002 -5.80552e-005 + 17Y+: -3.87262e-003 -1.19607e-003 + 17Z+: -8.53809e-003 1.48424e-003 + 18X+: -1.22262e-002 5.13096e-004 + 18Y+: -5.73905e-003 -1.07659e-003 + 18Z+: 3.51730e-002 -6.13814e-003 + 19X+: -1.43735e-002 -4.78552e-004 + 19Y+: 2.31135e-002 -6.30554e-004 + 19Z+: 1.80171e-003 -1.98835e-004 + 20X+: 9.17792e-003 5.36661e-004 + 20Y+: 2.18969e-002 -5.81759e-004 + 20Z+: -3.72117e-002 5.35813e-003 + 3X+: 1.61551e-001 1.65478e-002 + 3Y+: -4.12527e-004 -5.60909e-002 + 3Z+: -9.00640e-003 3.50754e-003 + 21X+: 3.38754e-002 -3.38703e-002 + 21Y+: -2.20843e-002 2.78581e-002 + 21Z+: -8.79541e-002 -3.67473e-003 + 22X+: 3.93064e-002 4.69476e-002 + 22Y+: -1.69132e-002 -1.04606e-002 + 22Z+: -1.85351e-002 1.33750e-003 + 23X+: 3.60396e-002 -2.46238e-002 + 23Z+: 3.57722e-003 3.64827e-003 + 23Y-: 1.92038e-002 6.65895e-002 +#+end_example + +#+begin_src matlab + mode3 = [... + +1.34688e-001-j*6.65071e-002, +1.55316e-002+j*1.01277e-002, -5.88466e-002+j*1.14294e-002; + +1.53934e-001-j*9.76990e-003, +7.17487e-003+j*1.11925e-002, -4.57205e-002+j*7.26573e-003; + +1.61551e-001+j*1.65478e-002, -4.12527e-004-j*5.60909e-002, -9.00640e-003+j*3.50754e-003; + +1.37298e-001-j*5.24661e-002, +1.19427e-003-j*5.39240e-002, -1.25915e-002+j*5.38133e-003; + +2.43192e-001-j*3.17374e-002, -2.15730e-001-j*7.69941e-004, -1.56268e-001+j*1.44118e-002; + -7.27705e-002-j*3.54943e-003, -2.47706e-001+j*2.66480e-003, -1.21590e-001+j*1.06054e-002; + -7.25870e-002-j*4.62024e-003, +2.27073e-001-j*3.69315e-002, +1.22611e-001-j*6.67337e-003; + +2.32731e-001-j*2.85516e-002, +2.35389e-001-j*3.81905e-002, +5.35574e-002+j*4.30394e-004; + +2.64170e-001-j*2.67367e-002, -2.56227e-001+j*3.97957e-005, -1.95398e-001+j*2.23549e-002; + -1.66953e-002-j*7.95698e-003, -2.66547e-001-j*2.17687e-002, +1.56278e-002+j*2.23786e-003; + -3.42364e-002-j*9.30205e-003, +2.52340e-001-j*7.47237e-003, -9.51643e-004+j*3.64798e-003; + +2.97574e-001+j*0.00000e+000, +2.23170e-001-j*1.37831e-002, +1.06266e-001+j*2.30324e-003; + +2.67178e-002-j*4.15723e-004, +6.75423e-003-j*2.18428e-003, -1.69423e-002+j*3.12395e-003; + -1.12283e-002+j*2.86316e-004, +5.08225e-003-j*2.14053e-003, +2.18339e-002-j*3.25204e-003; + -1.17948e-002+j*6.82873e-004, +1.94914e-002-j*2.42151e-003, +2.68660e-003-j*2.92104e-004; + +1.19490e-002+j*1.72236e-005, +1.83552e-002-j*2.71289e-003, -2.70914e-002+j*4.84164e-003; + +1.00173e-002-j*5.80552e-005, -3.87262e-003-j*1.19607e-003, -8.53809e-003+j*1.48424e-003; + -1.22262e-002+j*5.13096e-004, -5.73905e-003-j*1.07659e-003, +3.51730e-002-j*6.13814e-003; + -1.43735e-002-j*4.78552e-004, +2.31135e-002-j*6.30554e-004, +1.80171e-003-j*1.98835e-004; + +9.17792e-003+j*5.36661e-004, +2.18969e-002-j*5.81759e-004, -3.72117e-002+j*5.35813e-003; + +3.38754e-002-j*3.38703e-002, -2.20843e-002+j*2.78581e-002, -8.79541e-002-j*3.67473e-003; + +3.93064e-002+j*4.69476e-002, -1.69132e-002-j*1.04606e-002, -1.85351e-002+j*1.33750e-003; + +3.60396e-002-j*2.46238e-002, -1.92038e-002-j*6.65895e-002, +3.57722e-003+j*3.64827e-003]; +#+end_src + +*** Mode4 +#+begin_src bash :results output :exports none :eval no-export + sed -n 240,308p modal_analysis/modes_propres_narband.asc +#+end_src + +#+RESULTS: +#+begin_example + 1X+: -1.02501e-001 -1.43802e-001 + 1Y+: -1.07971e-001 5.61418e-004 + 1Z+: 1.87145e-001 -1.03605e-001 + 2X+: -9.44764e-002 -1.36856e-001 + 2Y+: -1.04428e-001 5.27790e-003 + 2Z+: 1.60710e-001 -7.74212e-002 + 4X+: -9.17242e-002 -1.36656e-001 + 4Y+: -1.34249e-001 -1.03884e-002 + 4Z+: 1.92123e-001 -1.25627e-001 + 5X+: 1.05875e-002 -1.03886e-001 + 5Y+: -8.26338e-002 3.58498e-002 + 5Z+: 2.55819e-001 -6.94290e-003 + 6X+: -4.58970e-002 -1.33904e-002 + 6Y+: -9.41660e-002 4.99682e-002 + 6Z+: 1.28276e-001 4.59685e-002 + 7X+: -6.01521e-002 -1.30165e-002 + 7Y+: 2.56439e-003 -6.78141e-002 + 7Z+: 5.03428e-002 -1.59420e-001 + 8X+: -1.00895e-002 -8.80550e-002 + 8Y+: 1.26327e-002 -8.14444e-002 + 8Z+: 1.59506e-001 -2.05360e-001 + 9X+: -3.04658e-003 -1.57921e-001 + 9Y+: -8.23501e-002 4.82748e-002 + 9Z+: 1.69315e-001 1.22804e-002 + 10X+: -8.25875e-002 -7.31038e-002 + 10Y+: -1.08668e-001 3.56364e-002 + 10Z+: 8.28567e-002 -4.49596e-003 + 11X+: -1.06792e-001 -6.95394e-002 + 11Y+: 3.77195e-002 -7.65410e-002 + 11Z+: 8.00590e-003 -2.32461e-002 + 12X+: -4.84292e-002 -1.45790e-001 + 12Y+: 1.03862e-002 -7.31212e-002 + 12Z+: 1.78122e-001 -1.00939e-001 + 13X+: -3.49891e-002 -6.20969e-003 + 13Y+: -1.18504e-002 -1.94225e-002 + 13Z+: 4.13007e-002 7.67087e-003 + 14X+: -3.55795e-002 1.16708e-003 + 14Y+: -1.68128e-002 -1.82344e-002 + 14Z+: 3.92416e-002 -3.64434e-002 + 15X+: -3.45304e-002 3.78185e-003 + 15Y+: -7.62559e-003 -2.24241e-002 + 15Z+: 6.28286e-003 -1.32711e-002 + 16X+: -9.95646e-003 -6.04395e-003 + 16Y+: -8.73465e-003 -2.20807e-002 + 16Z+: 3.56946e-002 1.69231e-002 + 17X+: -9.32661e-003 -5.51944e-003 + 17Y+: -1.91087e-002 -9.09191e-003 + 17Z+: 4.04981e-002 8.38685e-004 + 18X+: -2.84456e-002 4.02762e-003 + 18Y+: -2.20044e-002 -8.86197e-003 + 18Z+: 4.43051e-002 -5.21033e-002 + 19X+: -3.27019e-002 3.59765e-003 + 19Y+: 2.93163e-003 -2.05064e-002 + 19Z+: -1.77289e-002 -1.29477e-002 + 20X+: -1.08474e-002 -5.78419e-003 + 20Y+: 3.86759e-003 -1.91642e-002 + 20Z+: 2.10135e-002 3.18051e-002 + 3X+: -9.11657e-002 -1.36611e-001 + 3Y+: -1.78165e-001 -3.47193e-002 + 3Z+: 2.37121e-001 -4.96494e-002 + 21X+: -1.34808e-002 -9.69121e-003 + 21Y+: 1.25218e-002 -2.71411e-002 + 21Z+: 2.76673e-001 0.00000e+000 + 22X+: 1.96744e-003 4.90797e-003 + 22Y+: -9.82609e-004 -3.31065e-002 + 22Z+: 1.79246e-001 -3.33238e-002 + 23X+: -1.08728e-002 -8.80278e-003 + 23Z+: 2.30814e-001 -8.33151e-002 + 23Y-: 1.15217e-002 4.01143e-002 +#+end_example + +#+begin_src matlab + mode4 = [... + -1.02501e-001-j*1.43802e-001, -1.07971e-001+j*5.61418e-004, +1.87145e-001-j*1.03605e-001; + -9.44764e-002-j*1.36856e-001, -1.04428e-001+j*5.27790e-003, +1.60710e-001-j*7.74212e-002; + -9.11657e-002-j*1.36611e-001, -1.78165e-001-j*3.47193e-002, +2.37121e-001-j*4.96494e-002; + -9.17242e-002-j*1.36656e-001, -1.34249e-001-j*1.03884e-002, +1.92123e-001-j*1.25627e-001; + +1.05875e-002-j*1.03886e-001, -8.26338e-002+j*3.58498e-002, +2.55819e-001-j*6.94290e-003; + -4.58970e-002-j*1.33904e-002, -9.41660e-002+j*4.99682e-002, +1.28276e-001+j*4.59685e-002; + -6.01521e-002-j*1.30165e-002, +2.56439e-003-j*6.78141e-002, +5.03428e-002-j*1.59420e-001; + -1.00895e-002-j*8.80550e-002, +1.26327e-002-j*8.14444e-002, +1.59506e-001-j*2.05360e-001; + -3.04658e-003-j*1.57921e-001, -8.23501e-002+j*4.82748e-002, +1.69315e-001+j*1.22804e-002; + -8.25875e-002-j*7.31038e-002, -1.08668e-001+j*3.56364e-002, +8.28567e-002-j*4.49596e-003; + -1.06792e-001-j*6.95394e-002, +3.77195e-002-j*7.65410e-002, +8.00590e-003-j*2.32461e-002; + -4.84292e-002-j*1.45790e-001, +1.03862e-002-j*7.31212e-002, +1.78122e-001-j*1.00939e-001; + -3.49891e-002-j*6.20969e-003, -1.18504e-002-j*1.94225e-002, +4.13007e-002+j*7.67087e-003; + -3.55795e-002+j*1.16708e-003, -1.68128e-002-j*1.82344e-002, +3.92416e-002-j*3.64434e-002; + -3.45304e-002+j*3.78185e-003, -7.62559e-003-j*2.24241e-002, +6.28286e-003-j*1.32711e-002; + -9.95646e-003-j*6.04395e-003, -8.73465e-003-j*2.20807e-002, +3.56946e-002+j*1.69231e-002; + -9.32661e-003-j*5.51944e-003, -1.91087e-002-j*9.09191e-003, +4.04981e-002+j*8.38685e-004; + -2.84456e-002+j*4.02762e-003, -2.20044e-002-j*8.86197e-003, +4.43051e-002-j*5.21033e-002; + -3.27019e-002+j*3.59765e-003, +2.93163e-003-j*2.05064e-002, -1.77289e-002-j*1.29477e-002; + -1.08474e-002-j*5.78419e-003, +3.86759e-003-j*1.91642e-002, +2.10135e-002+j*3.18051e-002; + -1.34808e-002-j*9.69121e-003, +1.25218e-002-j*2.71411e-002, +2.76673e-001+j*0.00000e+000; + +1.96744e-003+j*4.90797e-003, -9.82609e-004-j*3.31065e-002, +1.79246e-001-j*3.33238e-002; + -1.08728e-002-j*8.80278e-003, -1.15217e-002-j*4.01143e-002, +2.30814e-001-j*8.33151e-002]; +#+end_src + +*** All modes +#+begin_src matlab +mode_shapes = zeros(23, 3, 10); + +mode_shapes(:, :, 1) = [... +-9.34637e-002+j*4.52445e-002, +2.33790e-001+j*1.41439e-003, -1.73754e-002+j*6.02449e-003; +-7.42108e-002+j*3.91543e-002, +2.41566e-001-j*1.44869e-003, -5.99285e-003+j*2.10370e-003; +-9.40720e-002+j*3.93724e-002, +2.52307e-001+j*0.00000e+000, -1.53864e-002-j*9.25720e-004; +-1.02163e-001+j*2.79561e-002, +2.29048e-001+j*2.89782e-002, -2.85130e-002+j*1.77132e-004; +-8.77132e-002+j*3.34081e-002, +2.14182e-001+j*2.14655e-002, -1.54521e-002+j*1.26682e-002; +-7.90143e-002+j*2.42583e-002, +2.20669e-001+j*2.12738e-002, +4.60755e-002+j*4.96406e-003; +-7.79654e-002+j*2.58385e-002, +2.06861e-001+j*3.48019e-002, -1.78311e-002-j*1.29704e-002; +-8.49357e-002+j*3.55200e-002, +2.07470e-001+j*3.59745e-002, -7.66974e-002-j*3.19813e-003; +-7.38565e-002+j*1.95146e-002, +2.17403e-001+j*2.01550e-002, -1.77073e-002-j*3.46414e-003; +-7.77587e-002+j*2.36700e-002, +2.35654e-001-j*2.14540e-002, +7.94165e-002-j*2.45897e-002; +-8.17972e-002+j*2.20583e-002, +2.20906e-001-j*4.30164e-003, -5.60520e-003+j*3.10187e-003; +-8.64261e-002+j*3.66022e-002, +2.15000e-001-j*5.74661e-003, -1.22622e-001+j*4.11767e-002; +-4.25169e-002+j*1.56602e-002, +5.31036e-002-j*1.73951e-002, -4.07130e-002+j*1.26884e-002; +-3.85032e-002+j*1.29431e-002, +5.36716e-002-j*1.80868e-002, +1.00367e-001-j*3.48798e-002; +-4.25524e-002+j*1.46363e-002, +5.19668e-002-j*1.69744e-002, +5.89747e-003-j*2.32428e-003; +-4.31268e-002+j*1.38332e-002, +5.07545e-002-j*1.53045e-002, -1.04172e-001+j*3.17984e-002; +-2.69757e-002+j*9.07955e-003, +3.07837e-002-j*9.44663e-003, -7.63502e-003+j*1.68203e-003; +-3.00097e-002+j*9.23966e-003, +2.83585e-002-j*8.97747e-003, +1.52467e-001-j*4.78675e-002; +-2.70223e-002+j*6.16478e-003, +3.06149e-002-j*6.25382e-003, -4.84888e-003+j*1.93970e-003; +-2.90976e-002+j*7.13184e-003, +3.36738e-002-j*7.30875e-003, -1.66902e-001+j*3.93419e-002; +-7.91940e-002+j*4.39648e-002, +2.04567e-001+j*9.49987e-003, -1.56087e-002+j*7.08838e-003; +-1.01070e-001+j*3.13534e-002, +1.92270e-001+j*1.80423e-002, +2.93053e-003-j*1.97308e-003; +-8.86455e-002+j*4.29906e-002, -3.38351e-002+j*1.81362e-003, +1.90862e-001+j*2.53414e-002]; + +mode_shapes(:, :, 2) = [... ++7.56931e-002+j*3.61548e-002, +2.07574e-001+j*1.69205e-004, +1.29733e-002-j*6.78426e-004; ++8.58732e-002+j*2.54470e-002, +2.07117e-001-j*1.31755e-003, -2.13788e-003-j*1.24974e-002; ++8.17201e-002+j*2.36079e-002, +2.15927e-001+j*1.61300e-002, -5.48456e-004+j*2.55691e-002; ++7.09825e-002+j*3.66313e-002, +2.09969e-001+j*1.11484e-002, +9.19478e-003+j*3.47272e-002; ++6.23935e-002+j*1.02488e-002, +2.30687e-001-j*3.58416e-003, +3.27122e-002-j*5.85468e-002; ++7.61163e-002-j*2.43630e-002, +2.26743e-001-j*1.15334e-002, -6.20205e-003-j*1.21742e-001; ++8.01824e-002-j*1.94769e-002, +1.97485e-001+j*4.50105e-002, -2.21170e-002+j*9.77052e-002; ++6.19294e-002+j*8.15075e-003, +2.03864e-001+j*4.45835e-002, +2.55133e-002+j*1.36137e-001; ++4.38135e-002+j*7.30537e-002, +2.28426e-001-j*6.58868e-003, +1.16313e-002+j*5.09427e-004; ++5.45770e-002+j*4.34251e-002, +2.50823e-001+j*0.00000e+000, -4.63460e-002-j*4.76868e-002; ++5.50987e-002+j*4.26178e-002, +2.29394e-001+j*5.78236e-002, +1.90158e-002+j*1.09139e-002; ++4.98867e-002+j*7.30190e-002, +2.07871e-001+j*4.57750e-002, +6.69433e-002+j*9.00315e-002; ++2.48819e-002+j*3.03222e-002, -2.56046e-002-j*3.34132e-002, +2.13260e-002+j*2.58544e-002; ++2.45706e-002+j*2.60221e-002, -2.57723e-002-j*3.35612e-002, -5.71282e-002-j*6.61562e-002; ++2.68196e-002+j*2.83888e-002, -2.57263e-002-j*3.29627e-002, -2.11722e-003-j*3.37239e-003; ++2.51442e-002+j*3.32558e-002, -2.54372e-002-j*3.25062e-002, +5.65780e-002+j*7.64142e-002; ++1.62437e-002+j*1.94534e-002, -1.31293e-002-j*2.05924e-002, +1.05274e-003+j*3.59474e-003; ++1.83431e-002+j*2.03836e-002, -1.16818e-002-j*1.86334e-002, -8.66632e-002-j*1.08216e-001; ++1.62553e-002+j*1.79588e-002, -1.28857e-002-j*1.90512e-002, +6.25653e-003+j*4.97733e-003; ++1.63830e-002+j*2.03943e-002, -1.48941e-002-j*2.11717e-002, +8.68045e-002+j*1.16491e-001; ++6.79204e-002-j*5.55513e-002, +2.32871e-001+j*2.33389e-002, +1.34345e-002-j*2.31815e-002; ++4.02414e-002-j*8.38957e-002, +2.35273e-001+j*2.73256e-002, -8.51632e-003-j*7.49635e-003; ++6.18293e-002-j*5.99671e-002, +1.63533e-002+j*6.09161e-002, +2.37693e-001+j*4.34204e-002]; + +mode_shapes(:, :, 3) = [... ++1.34688e-001-j*6.65071e-002, +1.55316e-002+j*1.01277e-002, -5.88466e-002+j*1.14294e-002; ++1.53934e-001-j*9.76990e-003, +7.17487e-003+j*1.11925e-002, -4.57205e-002+j*7.26573e-003; ++1.61551e-001+j*1.65478e-002, -4.12527e-004-j*5.60909e-002, -9.00640e-003+j*3.50754e-003; ++1.37298e-001-j*5.24661e-002, +1.19427e-003-j*5.39240e-002, -1.25915e-002+j*5.38133e-003; ++2.43192e-001-j*3.17374e-002, -2.15730e-001-j*7.69941e-004, -1.56268e-001+j*1.44118e-002; +-7.27705e-002-j*3.54943e-003, -2.47706e-001+j*2.66480e-003, -1.21590e-001+j*1.06054e-002; +-7.25870e-002-j*4.62024e-003, +2.27073e-001-j*3.69315e-002, +1.22611e-001-j*6.67337e-003; ++2.32731e-001-j*2.85516e-002, +2.35389e-001-j*3.81905e-002, +5.35574e-002+j*4.30394e-004; ++2.64170e-001-j*2.67367e-002, -2.56227e-001+j*3.97957e-005, -1.95398e-001+j*2.23549e-002; +-1.66953e-002-j*7.95698e-003, -2.66547e-001-j*2.17687e-002, +1.56278e-002+j*2.23786e-003; +-3.42364e-002-j*9.30205e-003, +2.52340e-001-j*7.47237e-003, -9.51643e-004+j*3.64798e-003; ++2.97574e-001+j*0.00000e+000, +2.23170e-001-j*1.37831e-002, +1.06266e-001+j*2.30324e-003; ++2.67178e-002-j*4.15723e-004, +6.75423e-003-j*2.18428e-003, -1.69423e-002+j*3.12395e-003; +-1.12283e-002+j*2.86316e-004, +5.08225e-003-j*2.14053e-003, +2.18339e-002-j*3.25204e-003; +-1.17948e-002+j*6.82873e-004, +1.94914e-002-j*2.42151e-003, +2.68660e-003-j*2.92104e-004; ++1.19490e-002+j*1.72236e-005, +1.83552e-002-j*2.71289e-003, -2.70914e-002+j*4.84164e-003; ++1.00173e-002-j*5.80552e-005, -3.87262e-003-j*1.19607e-003, -8.53809e-003+j*1.48424e-003; +-1.22262e-002+j*5.13096e-004, -5.73905e-003-j*1.07659e-003, +3.51730e-002-j*6.13814e-003; +-1.43735e-002-j*4.78552e-004, +2.31135e-002-j*6.30554e-004, +1.80171e-003-j*1.98835e-004; ++9.17792e-003+j*5.36661e-004, +2.18969e-002-j*5.81759e-004, -3.72117e-002+j*5.35813e-003; ++3.38754e-002-j*3.38703e-002, -2.20843e-002+j*2.78581e-002, -8.79541e-002-j*3.67473e-003; ++3.93064e-002+j*4.69476e-002, -1.69132e-002-j*1.04606e-002, -1.85351e-002+j*1.33750e-003; ++3.60396e-002-j*2.46238e-002, +3.57722e-003+j*3.64827e-003, -1.92038e-002-j*6.65895e-002]; + +mode_shapes(:, :, 4) = [... +-1.02501e-001-j*1.43802e-001, -1.07971e-001+j*5.61418e-004, +1.87145e-001-j*1.03605e-001; +-9.44764e-002-j*1.36856e-001, -1.04428e-001+j*5.27790e-003, +1.60710e-001-j*7.74212e-002; +-9.11657e-002-j*1.36611e-001, -1.78165e-001-j*3.47193e-002, +2.37121e-001-j*4.96494e-002; +-9.17242e-002-j*1.36656e-001, -1.34249e-001-j*1.03884e-002, +1.92123e-001-j*1.25627e-001; ++1.05875e-002-j*1.03886e-001, -8.26338e-002+j*3.58498e-002, +2.55819e-001-j*6.94290e-003; +-4.58970e-002-j*1.33904e-002, -9.41660e-002+j*4.99682e-002, +1.28276e-001+j*4.59685e-002; +-6.01521e-002-j*1.30165e-002, +2.56439e-003-j*6.78141e-002, +5.03428e-002-j*1.59420e-001; +-1.00895e-002-j*8.80550e-002, +1.26327e-002-j*8.14444e-002, +1.59506e-001-j*2.05360e-001; +-3.04658e-003-j*1.57921e-001, -8.23501e-002+j*4.82748e-002, +1.69315e-001+j*1.22804e-002; +-8.25875e-002-j*7.31038e-002, -1.08668e-001+j*3.56364e-002, +8.28567e-002-j*4.49596e-003; +-1.06792e-001-j*6.95394e-002, +3.77195e-002-j*7.65410e-002, +8.00590e-003-j*2.32461e-002; +-4.84292e-002-j*1.45790e-001, +1.03862e-002-j*7.31212e-002, +1.78122e-001-j*1.00939e-001; +-3.49891e-002-j*6.20969e-003, -1.18504e-002-j*1.94225e-002, +4.13007e-002+j*7.67087e-003; +-3.55795e-002+j*1.16708e-003, -1.68128e-002-j*1.82344e-002, +3.92416e-002-j*3.64434e-002; +-3.45304e-002+j*3.78185e-003, -7.62559e-003-j*2.24241e-002, +6.28286e-003-j*1.32711e-002; +-9.95646e-003-j*6.04395e-003, -8.73465e-003-j*2.20807e-002, +3.56946e-002+j*1.69231e-002; +-9.32661e-003-j*5.51944e-003, -1.91087e-002-j*9.09191e-003, +4.04981e-002+j*8.38685e-004; +-2.84456e-002+j*4.02762e-003, -2.20044e-002-j*8.86197e-003, +4.43051e-002-j*5.21033e-002; +-3.27019e-002+j*3.59765e-003, +2.93163e-003-j*2.05064e-002, -1.77289e-002-j*1.29477e-002; +-1.08474e-002-j*5.78419e-003, +3.86759e-003-j*1.91642e-002, +2.10135e-002+j*3.18051e-002; +-1.34808e-002-j*9.69121e-003, +1.25218e-002-j*2.71411e-002, +2.76673e-001+j*0.00000e+000; ++1.96744e-003+j*4.90797e-003, -9.82609e-004-j*3.31065e-002, +1.79246e-001-j*3.33238e-002; +-1.08728e-002-j*8.80278e-003, +2.30814e-001-j*8.33151e-002, -1.15217e-002-j*4.01143e-002]; + +mode_shapes(:, :, 5) = [... ++3.55328e-001+j*0.00000e+000, +6.67612e-002+j*5.48020e-002, +3.03237e-002+j*5.29473e-004; ++3.16372e-001-j*3.84091e-002, +5.27472e-002+j*5.88474e-002, +2.86305e-002-j*1.74805e-002; ++3.00803e-001-j*1.36309e-002, +7.04883e-002+j*1.24492e-001, +7.23329e-002+j*2.33738e-002; ++3.32527e-001-j*2.26876e-004, +9.82263e-002+j*1.20397e-001, +9.86580e-002+j*3.55048e-002; ++4.96498e-002+j*2.31008e-002, +9.79716e-002+j*1.42500e-002, -1.15121e-001-j*3.59085e-002; ++1.41924e-001+j*2.16209e-002, +8.76030e-002+j*6.39650e-003, -8.75727e-002-j*3.71261e-002; ++1.41522e-001+j*1.96964e-002, -1.01959e-001+j*4.10992e-004, +2.14744e-001+j*4.91249e-002; ++4.33170e-002+j*1.84481e-002, -8.24640e-002+j*3.42475e-003, +2.32281e-001+j*5.40699e-002; ++1.47782e-001+j*4.93091e-002, +8.75397e-002+j*7.75318e-004, -6.80833e-002-j*9.72902e-003; ++2.00055e-001+j*3.81689e-002, +8.06886e-002+j*1.19008e-002, -1.40810e-002-j*1.12625e-002; ++1.96526e-001+j*3.87737e-002, -8.42766e-002+j*9.20233e-003, +1.02951e-001+j*3.37680e-002; ++1.25035e-001+j*4.67796e-002, -8.81307e-002+j*5.81039e-004, +7.94320e-002+j*2.19736e-002; ++2.03946e-002+j*2.50162e-002, +7.93788e-002-j*1.40794e-002, -4.15470e-002+j*4.95855e-004; ++6.56876e-002-j*2.14826e-002, +8.21523e-002-j*1.94792e-002, +3.44089e-002+j*2.32727e-003; ++5.98960e-002-j*2.17160e-002, +4.74914e-002+j*2.31386e-002, +3.58704e-002+j*1.13591e-003; ++1.91580e-002+j*2.15329e-002, +5.14631e-002+j*1.70019e-002, -3.90820e-002-j*9.20853e-003; ++8.98876e-003+j*2.56390e-002, +7.93497e-002-j*2.34846e-002, -5.61039e-002-j*3.03271e-003; ++5.72051e-002-j*2.29477e-002, +8.76985e-002-j*2.73606e-002, +5.13896e-002+j*8.58341e-003; ++6.71099e-002-j*2.67997e-002, +1.80119e-002+j*4.02601e-002, +6.40092e-002+j*7.11273e-003; ++1.48349e-002+j*2.57533e-002, +1.35189e-002+j*3.69351e-002, -3.31024e-002-j*1.65471e-002; ++2.52156e-002-j*4.65138e-003, +7.94380e-004+j*1.56790e-002, +1.01067e-002-j*3.35468e-003; ++3.30841e-002+j*6.73531e-003, +1.69827e-004+j*9.48996e-003, +8.16746e-002+j*1.39885e-002; ++2.91404e-002-j*2.43138e-003, +1.38761e-001+j*3.67410e-002, -4.17833e-003+j*8.35629e-004]; + +mode_shapes(:, :, 6) = [... ++3.76055e-001-j*2.00477e-002, -2.18528e-002-j*2.46738e-001, -2.01369e-002-j*2.48481e-002; ++3.73870e-001-j*6.46353e-002, -3.66940e-002-j*2.49113e-001, -9.20410e-002-j*2.72520e-002; ++3.71070e-001-j*4.42411e-003, +5.79596e-002-j*2.58311e-001, +2.62275e-003+j*9.26708e-003; ++3.82440e-001+j*0.00000e+000, +3.30266e-002-j*2.08256e-001, +7.22558e-002+j*1.47204e-002; +-3.73785e-003+j*3.81592e-002, +4.74482e-002+j*5.41354e-002, -5.78227e-002-j*8.70215e-002; ++6.08721e-002+j*7.20835e-002, +5.73258e-002+j*5.38176e-002, -3.34121e-002-j*8.54509e-002; ++5.96968e-002+j*8.31733e-002, -7.64618e-002-j*4.19504e-002, +3.48953e-002+j*4.79635e-002; +-1.41532e-003+j*3.79318e-002, -6.79853e-002-j*3.88324e-002, +6.64542e-002+j*9.20955e-002; ++2.96044e-002+j*7.83613e-002, +4.91378e-002+j*4.74080e-002, -2.15758e-002-j*5.51292e-002; ++3.52566e-002+j*1.00976e-001, +4.16302e-002+j*5.95793e-002, -1.83944e-002-j*5.09313e-002; ++3.49287e-002+j*1.10404e-001, -5.46487e-002-j*3.95054e-002, +1.88545e-002+j*3.82156e-002; ++1.70173e-002+j*8.68762e-002, -5.52511e-002-j*4.46179e-002, +7.63555e-003+j*6.17646e-002; +-8.22469e-004-j*1.81402e-002, -1.96193e-003-j*2.76646e-002, -1.97033e-003+j*1.06615e-002; +-1.16438e-003-j*2.68725e-002, -1.11909e-003-j*2.38894e-002, +1.50332e-003+j*1.31644e-002; +-1.77060e-003-j*2.77807e-002, -1.01121e-003-j*2.37147e-002, -2.45798e-003-j*1.41886e-002; +-1.85500e-003-j*2.20304e-002, -4.77462e-004-j*2.19932e-002, -6.85097e-003-j*7.04903e-003; +-6.77197e-004-j*1.56812e-002, -2.19412e-003-j*2.65284e-002, +3.96653e-004+j*2.95178e-002; +-1.14513e-003-j*1.99551e-002, -2.35828e-003-j*2.88909e-002, 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+-4.61031e-002+j*8.08300e-002, -2.80650e-002+j*7.37925e-002, +3.54903e-002-j*3.02341e-002; +-5.14543e-002+j*7.23863e-002, +1.91387e-002-j*7.65710e-002, -3.92182e-002+j*5.06089e-002; +-3.35647e-002+j*2.16695e-002, +1.88798e-002-j*6.19822e-002, -5.08508e-002+j*8.78653e-002; +-4.93941e-002+j*6.46940e-002, -2.90637e-002+j*7.66360e-002, +5.24626e-002-j*6.16359e-002; +-5.85364e-002+j*5.59044e-002, -2.63581e-002+j*5.43466e-002, +2.90174e-002-j*3.98049e-002; +-6.07978e-002+j*5.54585e-002, +3.16828e-002-j*8.54810e-002, -3.44914e-002-j*4.03684e-003; +-6.18120e-002+j*7.90821e-002, +2.38807e-002-j*5.62399e-002, -5.04609e-002+j*3.27008e-002; ++1.57670e-002-j*3.20728e-003, -5.98007e-003-j*3.10416e-003, -1.67104e-003-j*1.59089e-003; ++1.79638e-002-j*7.40235e-003, -8.20109e-003-j*2.36675e-004, -1.69812e-003+j*1.67596e-003; ++1.87809e-002-j*8.55724e-003, -2.81069e-003-j*3.03393e-003, -5.71319e-003-j*2.26161e-003; ++1.88829e-002-j*8.50228e-003, -5.28373e-003-j*2.52386e-003, -1.48737e-002-j*5.08140e-003; ++2.28380e-002-j*7.51769e-003, -7.57170e-003-j*2.75553e-003, -6.10380e-003+j*5.15562e-003; ++2.08600e-002-j*6.11732e-003, -4.73105e-003-j*3.30979e-003, -2.51369e-003+j*5.42921e-003; ++2.00008e-002-j*5.92617e-003, -3.50988e-003-j*4.55853e-003, -5.43701e-003-j*6.57229e-003; ++2.28149e-002-j*8.22905e-003, -4.83167e-003-j*3.10509e-003, -2.10958e-002-j*1.33421e-002; +-1.83145e-002+j*2.76844e-002, +5.61668e-003-j*1.41226e-002, +2.20876e-002-j*2.27446e-002; +-5.42112e-003+j*2.22444e-002, -4.20426e-005-j*8.78901e-003, +5.54714e-003+j*2.71564e-002; +-1.70108e-002+j*2.83751e-002, -1.44473e-002+j*4.50880e-002, -5.26736e-003-j*5.75716e-003]; + +mode_shapes(:, :, 8) = [... +-1.40928e-001+j*1.28570e-001, +2.95471e-001-j*1.35692e-001, -6.61656e-002+j*2.95705e-002; +-1.56673e-001+j*5.19030e-002, +3.08231e-001-j*1.41453e-001, -9.70918e-002+j*4.95018e-002; +-1.72505e-001+j*2.26273e-002, +3.97224e-001-j*2.77556e-017, -1.29223e-001+j*4.47412e-002; +-1.69978e-001+j*1.16284e-001, +3.37516e-001+j*7.69873e-003, -8.49480e-002+j*2.17071e-002; ++5.38303e-004-j*3.59916e-003, -6.72455e-002-j*2.06230e-002, +2.66448e-002+j*4.10505e-002; +-7.96526e-003-j*7.76851e-002, -2.63530e-002-j*3.75474e-002, -1.05984e-001+j*9.32474e-002; ++4.72518e-002-j*1.00199e-001, -5.50664e-002+j*1.50246e-001, -8.50976e-002+j*2.78531e-002; ++7.58419e-003-j*8.61594e-003, -9.02101e-002+j*1.58224e-001, +6.03081e-002-j*1.24162e-001; ++2.99027e-002-j*5.27128e-002, -8.80464e-002-j*2.99113e-004, +9.02851e-002-j*3.99771e-002; ++3.23132e-002-j*5.87278e-002, +3.81174e-002-j*3.69992e-002, +1.16643e-002+j*6.21068e-002; ++5.79795e-002-j*8.33565e-002, -1.22448e-003+j*8.81473e-002, +8.40150e-002-j*1.16264e-002; ++1.35399e-002-j*3.80303e-002, -9.58200e-002+j*1.46531e-001, +1.06769e-001-j*8.97034e-002; ++5.34299e-004+j*1.35179e-002, +8.71327e-004-j*6.41448e-003, +3.33208e-002-j*2.12545e-002; +-2.79263e-004+j*5.08578e-003, -1.45476e-003-j*7.65161e-003, +6.98235e-002-j*2.45395e-002; +-9.22822e-005+j*7.03205e-003, -5.62836e-003-j*2.79991e-003, +3.99717e-002-j*8.30891e-003; ++1.87833e-002+j*3.26772e-003, -4.86774e-003-j*4.32297e-003, +5.97375e-002-j*1.77542e-002; ++1.14169e-002+j*5.70930e-003, -8.23489e-003-j*4.53684e-003, +3.14016e-002-j*2.50637e-002; ++1.15995e-003+j*5.79180e-003, -6.69740e-003-j*4.66433e-003, +8.17695e-002-j*2.78384e-002; ++5.23838e-004+j*6.46432e-003, +3.36104e-003-j*4.42572e-003, +3.64589e-002+j*5.74796e-004; ++1.57042e-002+j*5.94177e-003, -1.49670e-003-j*4.22955e-003, +8.68520e-002-j*1.43981e-002; ++8.00706e-004-j*2.91734e-002, +1.20708e-002+j*4.24081e-002, -5.91796e-002+j*4.00346e-002; +-1.91799e-003-j*1.37294e-002, +9.85285e-003+j*3.17934e-002, -1.78010e-001+j*7.91267e-002; ++3.57271e-003-j*3.09959e-002, -5.77781e-002-j*1.27957e-002, +1.31025e-002+j*1.92303e-002]; + +mode_shapes(:, :, 9) = [... ++1.58897e-002+j*3.23763e-002, -1.23332e-001-j*3.20376e-002, +6.78860e-002-j*1.28743e-002; ++5.67179e-003+j*4.26539e-002, -1.20726e-001-j*4.15603e-002, +7.66846e-002-j*1.24290e-002; ++2.24198e-002+j*3.45953e-002, -1.03213e-001-j*4.97049e-002, +5.49175e-002-j*5.50883e-003; ++2.68792e-002+j*2.97222e-002, -1.17598e-001-j*3.13791e-002, +5.59736e-002-j*1.71122e-002; ++1.39733e-002-j*1.56260e-002, +1.43952e-001+j*2.28119e-002, -4.56377e-003+j*4.88790e-002; +-2.18507e-002-j*1.25664e-002, +6.15387e-002-j*6.31793e-003, +3.05342e-002+j*3.24595e-002; +-7.85412e-003-j*1.85600e-002, +1.25733e-001+j*1.77063e-002, +5.34635e-002+j*4.72260e-003; +-1.10454e-002-j*2.13217e-002, +1.26440e-001+j*1.96001e-002, -5.62624e-002-j*1.07192e-002; ++5.84467e-003-j*4.07134e-002, +5.16711e-003+j*4.70857e-002, -1.93010e-001+j*6.79213e-003; ++5.31962e-002-j*1.11322e-002, +3.23294e-001-j*2.12981e-002, +2.14310e-001+j*4.85898e-003; +-2.44580e-002-j*1.33220e-002, +4.08800e-001+j*2.09082e-002, +5.11454e-001+j*0.00000e+000; +-2.51332e-002-j*1.42719e-003, +8.97105e-002+j*4.85852e-002, -1.27426e-001-j*9.59723e-003; +-2.08176e-003+j*1.37185e-002, -3.99530e-002+j*2.11895e-002, -1.17813e-001+j*7.60972e-002; +-1.34824e-002+j*7.11258e-003, -4.19473e-002+j*1.83590e-002, -2.07198e-002-j*2.51991e-002; +-6.84747e-003+j*8.45921e-003, -3.33872e-002+j*1.71496e-002, +6.99867e-002-j*6.93158e-002; +-2.64313e-002+j*5.08903e-003, -3.03569e-002+j*1.29946e-002, -1.39115e-001+j*5.72459e-002; +-2.90186e-002+j*1.48257e-002, -5.55429e-002+j*2.74156e-002, -1.62035e-001+j*8.04187e-002; +-2.05855e-002+j*1.11922e-002, -6.58789e-002+j*3.20524e-002, -1.02263e-002-j*4.24087e-002; +-1.40204e-002+j*8.01102e-003, -5.72647e-002+j*2.37484e-002, +1.75053e-001-j*9.63667e-002; +-3.50818e-002+j*1.41152e-002, -5.19701e-002+j*2.31951e-002, -1.15951e-001+j*2.91582e-002; +-5.78005e-003-j*7.05841e-003, +8.29016e-002+j*1.36984e-002, +4.03470e-003+j*4.03325e-002; +-1.39928e-002-j*1.14088e-002, +8.05288e-002+j*1.51031e-002, +1.12255e-002+j*3.21224e-002; +-1.02276e-002-j*8.35724e-003, -4.89246e-003+j*1.67800e-002, +7.80514e-002+j*1.53467e-002]; + +mode_shapes(:, :, 10) = [... ++3.33349e-002-j*4.89606e-003, -8.67138e-002-j*1.69402e-002, +2.87366e-002-j*1.66842e-002; ++2.95730e-002-j*6.10477e-004, -9.24590e-002-j*1.92562e-002, +5.21162e-002-j*1.31811e-002; ++3.26966e-002+j*1.03975e-002, -8.55682e-002-j*4.71847e-002, +3.99404e-002+j*8.59358e-003; ++3.45452e-002-j*3.05951e-003, -7.73823e-002-j*2.32199e-002, +2.30960e-002-j*7.49928e-003; +-3.72461e-003-j*5.40336e-003, +1.80151e-001-j*1.42898e-002, -8.63921e-003+j*1.95638e-002; +-1.11285e-002-j*2.22175e-003, +6.92355e-002-j*1.24144e-002, +8.02097e-002-j*6.81531e-003; +-2.99885e-002-j*4.21951e-004, +8.19709e-002+j*1.22484e-002, +3.97531e-002+j*1.33874e-002; +-1.59231e-002-j*3.96929e-003, +9.31295e-002+j*9.75532e-003, -4.01947e-002-j*5.17841e-003; +-7.32828e-003-j*2.93496e-002, +4.39909e-002+j*1.25298e-002, -8.87525e-002+j*2.05359e-002; ++8.20167e-002+j*6.86693e-004, +4.25475e-001-j*2.36494e-002, +3.37034e-001-j*2.44199e-002; +-2.68694e-002+j*3.82921e-003, +4.79292e-001+j*1.35903e-002, +5.06762e-001+j*0.00000e+000; ++1.40565e-002-j*5.41957e-003, +1.17563e-001+j*2.50398e-002, -5.97219e-002+j*5.92813e-004; ++2.30939e-002+j*1.16755e-002, +4.11136e-003+j*1.65726e-002, +2.41863e-002+j*5.06658e-002; +-4.92960e-003+j*9.24082e-003, -5.44667e-003+j*1.41983e-002, -3.73537e-002-j*1.87263e-002; ++9.68682e-003+j*1.11791e-002, -5.13436e-003+j*1.29205e-002, -4.88765e-002-j*4.97821e-002; +-1.66711e-003+j*1.03280e-002, +9.57955e-004+j*1.28350e-002, +3.30268e-002+j*5.44211e-002; ++5.90540e-003+j*1.67113e-002, +1.34280e-002+j*2.62111e-002, +4.38613e-002+j*7.17028e-002; ++4.97752e-003+j*1.47634e-002, +1.56773e-002+j*3.16026e-002, -5.27493e-002-j*3.38315e-002; ++2.83485e-003+j*1.14816e-002, +9.80676e-003+j*2.51504e-002, -6.78645e-002-j*9.90875e-002; ++5.62294e-003+j*1.84035e-002, +1.03717e-002+j*2.37801e-002, +2.58497e-002+j*3.75352e-002; +-8.75236e-003-j*4.71723e-003, +6.99107e-002+j*1.51894e-002, +3.46273e-002+j*1.48547e-002; +-1.72822e-002-j*8.75192e-003, +6.96759e-002+j*1.59783e-002, +8.07917e-002+j*1.97809e-002; +-1.29601e-002-j*5.61834e-003, +1.76126e-002+j*4.62761e-003, +6.64667e-002+j*1.61199e-002]; +#+end_src + +** Define a point for each solid body +We define accelerometer indices used to define the motion of each solid body (2 3-axis accelerometer are enough). +#+begin_src matlab + stages = [17, 19; % Bottom Granite + 13, 15; % Top Granite + 9, 11; % Ty + 5, 7; % Ry + 21, 22; % Spindle + 1, 3]; % Hexapod +#+end_src + +We define the origin point ${}^AO_B$ of the solid body $\{B\}$. +Here we choose the middle point between the two accelerometers. +This could be define differently (for instance by choosing the center of mass). +#+begin_src matlab + AOB = zeros(3, size(stages, 1)); + + for i = 1:size(stages, 1) + AOB(:, i) = mean(positions(stages(i, :), 1:3))'; + end +#+end_src + +Then we compute the positions of the sensors with respect to the previously defined origin for the frame $\{B\}$: ${}^BP_1$ and ${}^BP_2$. +#+begin_src matlab + BP1 = zeros(3, size(stages, 1)); + BP2 = zeros(3, size(stages, 1)); + + for i = 1:size(stages, 1) + BP1(:, i) = positions(stages(i, 1), 1:3)' - AOB(:, i); + BP2(:, i) = positions(stages(i, 2), 1:3)' - AOB(:, i); + end +#+end_src + +Let's define one absolute frame $\{A\}$ and one frame $\{B\}$ fixed w.r.t. the solid body. +We note ${}^AO_B$ the position of origin of $\{B\}$ expressed in $\{A\}$. + +We are measuring with the accelerometers the absolute motion of points $P_1$ and $P_2$: ${}^Av_{P_1}$ and ${}^Av_{P_2}$. + +Let's note ${}^BP_1$ and ${}^BP_2$ the (known) coordinates of $P_1$ and $P_2$ expressed in the frame $\{B\}$. + +Then we have: +\begin{align} + {}^Av_{P_1} &= {}^Av_{O_B} + {}^A\Omega^\times {}^AR_B {}^BP_1 \\ + {}^Av_{P_2} &= {}^Av_{O_B} + {}^A\Omega^\times {}^AR_B {}^BP_2 +\end{align} + +And we obtain +\begin{align} + {}^A\Omega^\times {}^AR_B &= \left( {}^Av_{P_2} - {}^Av_{P_1} \right) \left( {}^BP_2 - {}^BP_1 \right)^{-1}\\ + {}^Av_{O_B} &= {}^Av_{P_1} - \left( {}^Av_{P_2} - {}^Av_{P_1} \right) \left( {}^BP_2 - {}^BP_1 \right)^{-1} {}^BP_1 +\end{align} + +#+begin_src matlab + AVOB = zeros(3, size(stages, 1)); + ARB = zeros(3, 3, size(stages, 1)); + for i = 1:size(stages, 1) + AVOB(:, i) = mode1(stages(i, 1), :)' - (mode1(stages(i, 2), :)' - mode1(stages(i, 1), :)')*pinv(BP2(:, i) - BP1(:, i))*BP1(:, i); + ARB(:, :, i) = (mode1(stages(i, 2), :)' - mode1(stages(i, 1), :)')*pinv(BP2(:, i) - BP1(:, i)); + end +#+end_src + +** Argand Diagram +For mode 1 + +#+begin_src matlab + figure; + hold on; + for i=1:size(mode1, 1) + plot([0, real(mode1(i, 1))], [0, imag(mode1(i, 1))], '-k') + plot([0, real(mode1(i, 2))], [0, imag(mode1(i, 2))], '-k') + plot([0, real(mode1(i, 3))], [0, imag(mode1(i, 3))], '-k') + + % plot([0, real(mode2(i, 1))], [0, imag(mode2(i, 1))], '-r') + % plot([0, real(mode2(i, 2))], [0, imag(mode2(i, 2))], '-r') + % plot([0, real(mode2(i, 3))], [0, imag(mode2(i, 3))], '-r') + + % plot([0, real(mode3(i, 1))], [0, imag(mode3(i, 1))], '-b') + % plot([0, real(mode3(i, 2))], [0, imag(mode3(i, 2))], '-b') + % plot([0, real(mode3(i, 3))], [0, imag(mode3(i, 3))], '-b') + end + + for i=1:size(AVOB, 2) + plot([0, real(AVOB(1, i))], [0, imag(AVOB(1, i))], '-r') + plot([0, real(AVOB(2, i))], [0, imag(AVOB(2, i))], '-r') + plot([0, real(AVOB(3, i))], [0, imag(AVOB(3, i))], '-r') + end + + % ang=0:0.01:2*pi; + % radius1 = max(max(sqrt(real(mode1).^2+imag(mode1).^2))); + % plot(radius1*cos(ang), radius1*sin(ang), '-k'); + + % radius2 = max(max(sqrt(real(mode2).^2+imag(mode2).^2))); + % plot(radius2*cos(ang), radius2*sin(ang), '-r'); + + % radius3 = max(max(sqrt(real(mode3).^2+imag(mode3).^2))); + % plot(radius3*cos(ang), radius3*sin(ang), '-b'); + + hold off; + axis manual equal +#+end_src + +** TEST: animate first mode +#+begin_src matlab + figure; + hold on; + fill3(positions(1:4, 1), positions(1:4, 2), positions(1:4, 3), 'k', 'FaceAlpha', 0.5) + fill3(positions(5:8, 1), positions(5:8, 2), positions(5:8, 3), 'k', 'FaceAlpha', 0.5) + fill3(positions(9:12, 1), positions(9:12, 2), positions(9:12, 3), 'k', 'FaceAlpha', 0.5) + fill3(positions(13:16, 1), positions(13:16, 2), positions(13:16, 3), 'k', 'FaceAlpha', 0.5) + fill3(positions(17:20, 1), positions(17:20, 2), positions(17:20, 3), 'k', 'FaceAlpha', 0.5) + fill3(positions(21:23, 1), positions(21:23, 2), positions(21:23, 3), 'k', 'FaceAlpha', 0.5) + hold off; +#+end_src + +#+NAME: fig:mode_shapes +#+HEADER: :tangle no :exports results :results value file raw replace :noweb yes +#+begin_src matlab + rec = polyshape([-2 -2 2 2],[-3 3 3 -3]); + h = figure; + filename = 'figs/mode_shapes.gif'; + n = 20; + + for i = 1:n + axis manual equal + + Dm = real(V(1:3, 5)*cos(2*pi*i/n)); + rec_i = rotate(rec, 180/pi*Dm(3)); + rec_i = translate(rec_i, 10*Dm(1), 10*Dm(2)); + plot(rec_i); + + xlim([-3, 3]); ylim([-4, 4]); + + set(h, 'visible', 'off'); + set(h, 'pos', [0, 0, 500, 500]); + drawnow; + + % Capture the plot as an image + frame = getframe(h); + im = frame2im(frame); + [imind,cm] = rgb2ind(im,256); + + % Write to the GIF File + if i == 1 + imwrite(imind,cm,filename,'gif','DelayTime',0.1,'Loopcount',inf); + else + imwrite(imind,cm,filename,'gif','DelayTime',0.1,'WriteMode','append'); + end + end + + set(h, 'visible', 'on'); + ans = filename; +#+end_src + +** From 6 translations to translation + rotation +Let's define one absolute frame $\{A\}$ and one frame $\{B\}$ fixed w.r.t. the solid body. +We note ${}^AO_B$ the position of origin of $\{B\}$ expressed in $\{A\}$. + +We are measuring with the accelerometers the absolute motion of points $P_1$ and $P_2$: ${}^AP_1$ and ${}^AP_2$. + +Let's note ${}^BP_1$ and ${}^BP_2$ the (known) coordinates of $P_1$ and $P_2$ expressed in the frame $\{B\}$. + +Then we have: +\begin{align} + {}^AP_1 &= {}^AO_B + {}^AR_B {}^BP_1 \\ + {}^AP_2 &= {}^AO_B + {}^AR_B {}^BP_2 +\end{align} + +And we obtain +\begin{align} + {}^AR_B &= \left( {}^AP_2 - {}^AP_1 \right) \left( {}^BP_2 - {}^BP_1 \right)^{-1}\\ + {}^AO_B &= {}^Av_{P_1} - \left( {}^AP_2 - {}^AP_1 \right) \left( {}^BP_2 - {}^BP_1 \right)^{-1} {}^BP_1 +\end{align} + * Problem with AirLoc System 4 Airloc Levelers are used for the granite (figure [[fig:airloc]]). diff --git a/modal-analysis/modes_analysis.html b/modal-analysis/modes_analysis.html new file mode 100644 index 0000000000000000000000000000000000000000..56156896876f59f3f1f3d4ac5b6b3319ad0c4411 GIT binary patch literal 63436 zcmeHwdwbhPuJ`}?6tn8nIu}c_WIMja?&~^DlYMeKvDocf_iN@7BhDoNRvvw1)J z`C%>;$ss9Ha+wmO|(H%~g9M`w@Nf1aE@f5y-OJBZMD~Cg5FqA-hdCW| z_xAQgyC!oD!qfJe4B-3mZ`6lvScG`1#d;gvz1GG~Yoo`;M$l&6o&BwC)@tGP*p3X$ zDVntGKc>#*Ve`cIBHN2vXS0djWTyD-uo>CcQHRhvFh@oZ+R>q)4eO{*JY#GhHZL9f zYT^e`qUn`mMWaK@zI05x#eZ36j^{*<;kH84aP7lxlXnw2k!v44_btO^$DZNNLMMFK z5l&?ldU;Mrw0H=M{om{ifcO8p;W16OrhrWN^3P*ieBd;2sW(n1-h)ED%Y_|}Bzo;IFeQp01oR-0D6m}+>(UlO&AYpF zLI;Erk0nkm5?<^yq(jo(?)UA19}~cG!ij4@ls$;6>>(b|=z}KHNIutv>uThvmwdQZ zml%vsOfuK}K};)Q@4JxSEV?6_XpA(A5q3Ocr2`^Z=N@|obucxCcE~1nz__9} 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z^EXsA31b?K{GfSs%>UhIkL;0wON`jr$R8W%AGy8{Ckog?B1ZG*1OZB9An&u@Mt85Z zvD4b +#+HTML_HEAD: +#+HTML_HEAD: +#+HTML_HEAD: +#+HTML_HEAD: +#+HTML_HEAD: +#+HTML_HEAD: + +#+HTML_MATHJAX: align: center tagside: right font: TeX + +#+PROPERTY: header-args:matlab :session *MATLAB* +#+PROPERTY: header-args:matlab+ :comments org +#+PROPERTY: header-args:matlab+ :results none +#+PROPERTY: header-args:matlab+ :exports both +#+PROPERTY: header-args:matlab+ :eval no-export +#+PROPERTY: header-args:matlab+ :output-dir figs + +#+PROPERTY: header-args:shell :eval no-export + +#+PROPERTY: header-args:latex :headers '("\\usepackage{tikz}" "\\usepackage{import}" "\\import{$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) + <> +#+end_src + +#+begin_src matlab :exports none :results silent :noweb yes + <> +#+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") + <> +#+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") + <> +#+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