#+TITLE: Modal Analysis - Modal Parameter Extraction :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: #+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 * TODO Part to explain how to choose the modes frequencies - bro-band method used - Stabilization Chart - 21 modes * Obtained Modal Parameters From the modal analysis software, we can export the obtained modal parameters: - the resonance frequencies - the modes shapes - the modal damping - the residues These can be express as the *eigen matrices*: \[ \Omega = \begin{bmatrix} \omega_1^2 & & 0 \\ & \ddots & \\ 0 & & \omega_n^2 \end{bmatrix}; \quad \Psi = \begin{bmatrix} & & \\ \{\psi_1\} & \dots & \{\psi_n\} \\ & & \end{bmatrix} \] where $\bar{\omega}_r^2$ is the $r^\text{th}$ eigenvalue squared and $\{\phi\}_r$ is a description of the corresponding *mode shape*. The file containing the modal parameters is =mat/modes.asc=. Its first 20 lines as shown below. #+begin_src bash :results output :exports results :eval no-export sed 20q mat/modes.asc | sed $'s/\r//' #+end_src #+RESULTS: #+begin_example Created by N-Modal Estimator: bbfd 01-Jul-19 16:44:11 Mode 1 freq = 11.41275Hz damp = 8.72664% modal A = -4.50556e+003-9.41744e+003i modal B = -7.00928e+005+2.62922e+005i Mode matrix of local coordinate [DOF: Re IM] 1X-: -1.04114e-001 3.50664e-002 1Y-: 2.34008e-001 5.04273e-004 1Z+: -1.93303e-002 5.08614e-003 2X-: -8.38439e-002 3.45978e-002 2Y-: 2.42440e-001 0.00000e+000 2Z+: -7.40734e-003 5.17734e-003 3Y-: 2.17655e-001 6.10802e-003 3X+: 1.18685e-001 -3.54602e-002 3Z+: -2.37725e-002 -1.61649e-003 #+end_example We split this big =modes.asc= file into sub text files using =bash=. The obtained files are described one table [[tab:modes_files]]. #+begin_src bash :results none sed '/^\s*[0-9]*[XYZ][+-]:/!d' mat/modes.asc > mat/mode_shapes.txt sed '/freq/!d' mat/modes.asc | sed 's/.* = \(.*\)Hz/\1/' > mat/mode_freqs.txt sed '/damp/!d' mat/modes.asc | sed 's/.* = \(.*\)\%/\1/' > mat/mode_damps.txt sed '/modal A/!d' mat/modes.asc | sed 's/.* =\s\+\([-0-9.e]\++[0-9]\+\)\([-+0-9.e]\+\)i/\1 \2/' > mat/mode_modal_a.txt sed '/modal B/!d' mat/modes.asc | sed 's/.* =\s\+\([-0-9.e]\++[0-9]\+\)\([-+0-9.e]\+\)i/\1 \2/' > mat/mode_modal_b.txt #+end_src #+name: tab:modes_files #+caption: Split =modes.asc= file | Filename | Content | |------------------------+--------------------------------------------------| | =mat/mode_shapes.txt= | mode shapes | | =mat/mode_freqs.txt= | resonance frequencies | | =mat/mode_damps.txt= | modal damping | | =mat/mode_modal_a.txt= | modal residues at low frequency (to be checked) | | =mat/mode_modal_b.txt= | modal residues at high frequency (to be checked) | Then we import the obtained =.txt= files on Matlab using =readtable= function. #+begin_src matlab shapes = readtable('mat/mode_shapes.txt', 'ReadVariableNames', false); % [Sign / Real / Imag] freqs = table2array(readtable('mat/mode_freqs.txt', 'ReadVariableNames', false)); % in [Hz] damps = table2array(readtable('mat/mode_damps.txt', 'ReadVariableNames', false)); % in [%] modal_a = table2array(readtable('mat/mode_modal_a.txt', 'ReadVariableNames', false)); % [Real / Imag] modal_b = table2array(readtable('mat/mode_modal_b.txt', 'ReadVariableNames', false)); % [Real / Imag] modal_a = complex(modal_a(:, 1), modal_a(:, 2)); modal_b = complex(modal_b(:, 1), modal_b(:, 2)); #+end_src We guess the number of modes identified from the length of the imported data. #+begin_src matlab acc_n = 23; % Number of accelerometers dir_n = 3; % Number of directions dirs = 'XYZ'; mod_n = size(shapes,1)/acc_n/dir_n; % Number of modes #+end_src As the mode shapes are split into 3 parts (direction plus sign, real part and imaginary part), we aggregate them into one array of complex numbers. #+begin_src matlab T_sign = table2array(shapes(:, 1)); T_real = table2array(shapes(:, 2)); T_imag = table2array(shapes(:, 3)); modes = zeros(mod_n, acc_n, dir_n); for mod_i = 1:mod_n for acc_i = 1:acc_n % Get the correct section of the signs T = T_sign(acc_n*dir_n*(mod_i-1)+1:acc_n*dir_n*mod_i); for dir_i = 1:dir_n % Get the line corresponding to the sensor i = find(contains(T, sprintf('%i%s',acc_i, dirs(dir_i))), 1, 'first')+acc_n*dir_n*(mod_i-1); modes(mod_i, acc_i, dir_i) = str2num([T_sign{i}(end-1), '1'])*complex(T_real(i),T_imag(i)); end end end #+end_src The obtained mode frequencies and damping are shown below. #+begin_src matlab :exports both :results value table replace :post addhdr(*this*) data2orgtable([freqs, damps], {}, {'Frequency [Hz]', 'Damping [%]'}, ' %.1f '); #+end_src #+RESULTS: | Frequency [Hz] | Damping [%] | |----------------+-------------| | 11.4 | 8.7 | | 18.5 | 11.8 | | 37.6 | 6.4 | | 39.4 | 3.6 | | 54.0 | 0.2 | | 56.1 | 2.8 | | 69.7 | 4.6 | | 71.6 | 0.6 | | 72.4 | 1.6 | | 84.9 | 3.6 | | 90.6 | 0.3 | | 91.0 | 2.9 | | 95.8 | 3.3 | | 105.4 | 3.3 | | 106.8 | 1.9 | | 112.6 | 3.0 | | 116.8 | 2.7 | | 124.1 | 0.6 | | 145.4 | 1.6 | | 150.1 | 2.2 | | 164.7 | 1.4 | * Obtained Mode Shapes animations One all the FRFs are obtained, we can estimate the modal parameters (resonance frequencies, modal shapes and modal damping) within the modal software. For that, multiple modal extraction techniques can be used (SIMO, MIMO, narrow band, wide band, ...). Then, it is possible to show the modal shapes with an animation. Examples are shown on figures [[fig:mode1]] and [[fig:mode6]]. Animations of all the other modes are accessible using the following links: [[file:img/modes/mode1.gif][mode 1]], [[file:img/modes/mode2.gif][mode 2]], [[file:img/modes/mode3.gif][mode 3]], [[file:img/modes/mode4.gif][mode 4]], [[file:img/modes/mode5.gif][mode 5]], [[file:img/modes/mode6.gif][mode 6]], [[file:img/modes/mode7.gif][mode 7]], [[file:img/modes/mode8.gif][mode 8]], [[file:img/modes/mode9.gif][mode 9]], [[file:img/modes/mode10.gif][mode 10]], [[file:img/modes/mode11.gif][mode 11]], [[file:img/modes/mode12.gif][mode 12]], [[file:img/modes/mode13.gif][mode 13]], [[file:img/modes/mode14.gif][mode 14]], [[file:img/modes/mode15.gif][mode 15]], [[file:img/modes/mode16.gif][mode 16]], [[file:img/modes/mode17.gif][mode 17]], [[file:img/modes/mode18.gif][mode 18]], [[file:img/modes/mode19.gif][mode 19]], [[file:img/modes/mode20.gif][mode 20]], [[file:img/modes/mode21.gif][mode 21]]. #+name: fig:mode1 #+caption: Mode 1 [[file:img/modes/mode1.gif]] #+name: fig:mode6 #+caption: Mode 6 [[file:img/modes/mode6.gif]] * 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; 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-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, +3.70211e-003+j*2.02423e-002; -1.16798e-003-j*2.31638e-002, -4.75969e-004-j*2.23413e-002, -2.07031e-003-j*3.41651e-002; -1.99807e-004-j*2.09301e-002, -2.40560e-004-j*1.78264e-002, -9.91090e-003-j*3.12664e-002; +9.69966e-003+j*2.95222e-002, -1.19231e-002+j*5.23077e-003, -9.79113e-003-j*4.50715e-002; +1.36018e-002+j*2.24850e-002, -9.79427e-003+j*9.84679e-003, +2.18456e-002-j*1.57858e-002; +1.09214e-002+j*2.93115e-002, +2.43859e-002+j*3.35745e-002, -1.07735e-002+j*1.21178e-002]; mode_shapes(:, :, 7) = [... +4.18739e-001-j*2.21696e-002, +6.52636e-002-j*1.29498e-001, -1.90337e-002+j*9.61505e-003; +4.27024e-001-j*3.03566e-002, +4.22791e-002-j*1.32892e-001, -1.65941e-002+j*2.17638e-002; +4.66694e-001+j*1.38778e-017, +7.51273e-002-j*1.57468e-001, -2.91618e-002+j*4.11516e-002; +4.18867e-001+j*2.39590e-002, +7.51007e-002-j*1.13378e-001, -7.11238e-003+j*2.72692e-002; -3.26299e-002+j*2.12910e-002, -4.80710e-002+j*8.82799e-002, +5.82558e-002-j*9.16159e-002; -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 The mode shape of the first mode at 11Hz (figure [[fig:mode1]]) seems to indicate that this corresponds to a suspension mode. This could be due to the 4 Airloc Levelers that are used for the granite (figure [[fig:airloc]]). #+name: fig:airloc #+caption: AirLoc used for the granite (2120-KSKC) #+attr_html: :width 500px [[file:img/airloc/IMG_20190618_155522.jpg]] They are probably *not well leveled*, so the granite is supported only by two Airloc. * Setup #+name: fig:nass-modal-test #+caption: Position and orientation of the accelerometer used [[file:figs/nass-modal-test.png]] * Mode extraction and importation First, we split the big =modes.asc= files into sub text files using =bash=. #+begin_src bash :results none sed '/^\s*[0-9]*[XYZ][+-]:/!d' modal_analysis_updated/modes.asc > mat/mode_shapes.txt sed '/freq/!d' modal_analysis_updated/modes.asc | sed 's/.* = \(.*\)Hz/\1/' > mat/mode_freqs.txt sed '/damp/!d' modal_analysis_updated/modes.asc | sed 's/.* = \(.*\)\%/\1/' > mat/mode_damps.txt sed '/modal A/!d' modal_analysis_updated/modes.asc | sed 's/.* =\s\+\([-0-9.e]\++[0-9]\+\)\([-+0-9.e]\+\)i/\1 \2/' > mat/mode_modal_a.txt sed '/modal B/!d' modal_analysis_updated/modes.asc | sed 's/.* =\s\+\([-0-9.e]\++[0-9]\+\)\([-+0-9.e]\+\)i/\1 \2/' > mat/mode_modal_b.txt #+end_src Then we import them on Matlab. #+begin_src matlab shapes = readtable('mat/mode_shapes.txt', 'ReadVariableNames', false); % [Sign / Real / Imag] freqs = table2array(readtable('mat/mode_freqs.txt', 'ReadVariableNames', false)); % in [Hz] damps = table2array(readtable('mat/mode_damps.txt', 'ReadVariableNames', false)); % in [%] modal_a = table2array(readtable('mat/mode_modal_a.txt', 'ReadVariableNames', false)); % [Real / Imag] modal_a = complex(modal_a(:, 1), modal_a(:, 2)); modal_b = table2array(readtable('mat/mode_modal_b.txt', 'ReadVariableNames', false)); % [Real / Imag] modal_b = complex(modal_b(:, 1), modal_b(:, 2)); #+end_src We guess the number of modes identified from the length of the imported data. #+begin_src matlab acc_n = 23; % Number of accelerometers dir_n = 3; % Number of directions dirs = 'XYZ'; mod_n = size(shapes,1)/acc_n/dir_n; % Number of modes #+end_src As the mode shapes are split into 3 parts (direction plus sign, real part and imaginary part), we aggregate them into one array of complex numbers. #+begin_src matlab T_sign = table2array(shapes(:, 1)); T_real = table2array(shapes(:, 2)); T_imag = table2array(shapes(:, 3)); modes = zeros(mod_n, acc_n, dir_n); for mod_i = 1:mod_n for acc_i = 1:acc_n % Get the correct section of the signs T = T_sign(acc_n*dir_n*(mod_i-1)+1:acc_n*dir_n*mod_i); for dir_i = 1:dir_n % Get the line corresponding to the sensor i = find(contains(T, sprintf('%i%s',acc_i, dirs(dir_i))), 1, 'first')+acc_n*dir_n*(mod_i-1); modes(mod_i, acc_i, dir_i) = str2num([T_sign{i}(end-1), '1'])*complex(T_real(i),T_imag(i)); end end end #+end_src The obtained mode frequencies and damping are shown below. #+begin_src matlab :exports both :results value table replace :post addhdr(*this*) data2orgtable([freqs, damps], {}, {'Frequency [Hz]', 'Damping [%]'}, ' %.1f '); #+end_src #+RESULTS: | Frequency [Hz] | Damping [%] | |----------------+-------------| | 11.4 | 8.7 | | 18.5 | 11.8 | | 37.6 | 6.4 | | 39.4 | 3.6 | | 54.0 | 0.2 | | 56.1 | 2.8 | | 69.7 | 4.6 | | 71.6 | 0.6 | | 72.4 | 1.6 | | 84.9 | 3.6 | | 90.6 | 0.3 | | 91.0 | 2.9 | | 95.8 | 3.3 | | 105.4 | 3.3 | | 106.8 | 1.9 | | 112.6 | 3.0 | | 116.8 | 2.7 | | 124.1 | 0.6 | | 145.4 | 1.6 | | 150.1 | 2.2 | | 164.7 | 1.4 | * Positions of the sensors We process the file exported from the =modal= software containing the positions of the sensors using =bash=. #+begin_src bash :results none cat modal_analysis_updated/id31_nanostation_modified.cfg | grep NODES -A 23 | sed '/\s\+[0-9]\+/!d' | sed 's/\(.*\)\s\+0\s\+.\+/\1/' > mat/acc_pos.txt #+end_src We then import that on =matlab=, and sort them. #+begin_src matlab acc_pos = readtable('mat/acc_pos.txt', 'ReadVariableNames', false); acc_pos = table2array(acc_pos(:, 1:4)); [~, i] = sort(acc_pos(:, 1)); acc_pos = acc_pos(i, 2:4); #+end_src The positions of the sensors relative to the point of interest are shown below. #+begin_src matlab :exports both :results value table replace :post addhdr(*this*) data2orgtable(1000*acc_pos, {}, {'x [mm]', 'y [mm]', 'z [mm]'}, ' %.0f '); #+end_src #+RESULTS: | x [mm] | y [mm] | z [mm] | |--------+--------+--------| | -64 | -64 | -296 | | -64 | 64 | -296 | | 64 | 64 | -296 | | 64 | -64 | -296 | | -385 | -300 | -417 | | -420 | 280 | -417 | | 420 | 280 | -417 | | 380 | -300 | -417 | | -475 | -414 | -427 | | -465 | 407 | -427 | | 475 | 424 | -427 | | 475 | -419 | -427 | | -320 | -446 | -786 | | -480 | 534 | -786 | | 450 | 534 | -786 | | 295 | -481 | -786 | | -730 | -526 | -951 | | -735 | 814 | -951 | | 875 | 799 | -951 | | 865 | -506 | -951 | | -155 | -90 | -594 | | 0 | 180 | -594 | | 155 | -90 | -594 | * Solids We consider the following solid bodies: - Bottom Granite - Top Granite - Translation Stage - Tilt Stage - Spindle - Hexapod We create a structure =solids= that contains the accelerometer number of each solid bodies (as shown on figure [[fig:nass-modal-test]]). #+begin_src matlab solids = {}; solids.granite_bot = [17, 18, 19, 20]; solids.granite_top = [13, 14, 15, 16]; solids.ty = [9, 10, 11, 12]; solids.ry = [5, 6, 7, 8]; solids.rz = [21, 22, 23]; solids.hexa = [1, 2, 3, 4]; solid_names = fields(solids); #+end_src * From local coordinates to global coordinates for the mode shapes #+begin_src latex :file local_to_global_coordinates.pdf :post pdf2svg(file=*this*, ext="png") :exports results \newcommand\irregularcircle[2]{% radius, irregularity \pgfextra {\pgfmathsetmacro\len{(#1)+rand*(#2)}} +(0:\len pt) \foreach \a in {10,20,...,350}{ \pgfextra {\pgfmathsetmacro\len{(#1)+rand*(#2)}} -- +(\a:\len pt) } -- cycle } \begin{tikzpicture} \draw[rounded corners=1mm] (0, 0) \irregularcircle{3cm}{1mm}; \node[] (origin) at (4, -1) {$\bullet$}; \begin{scope}[shift={(origin)}] \def\axissize{0.8cm} \draw[->] (0, 0) -- ++(\axissize, 0) node[above left]{$x$}; \draw[->] (0, 0) -- ++(0, \axissize) node[below right]{$y$}; \draw[fill, color=black] (0, 0) circle (0.05*\axissize); \node[draw, circle, inner sep=0pt, minimum size=0.4*\axissize, label=left:$z$] (yaxis) at (0, 0){}; \node[below right] at (0, 0){$\{O\}$}; \end{scope} \coordinate[] (p1) at (-1.5, -1.5); \coordinate[] (p2) at (-1.5, 1.5); \coordinate[] (p3) at ( 1.5, 1.5); \coordinate[] (p4) at ( 1.5, -1.5); \draw[->] (p1)node[]{$\bullet$}node[above]{$p_1$} -- ++(1, 0.5)node[right]{$v_1$}; \draw[->] (p2)node[]{$\bullet$}node[above]{$p_2$} -- ++(-0.5, 1)node[right]{$v_2$}; \draw[->] (p3)node[]{$\bullet$}node[above]{$p_3$} -- ++(1, 0.5)node[right]{$v_3$}; \draw[->] (p4)node[]{$\bullet$}node[above]{$p_4$} -- ++(0.5, 1)node[right]{$v_4$}; \end{tikzpicture} #+end_src #+RESULTS: [[file:figs/local_to_global_coordinates.png]] From the figure above, we can write: \begin{align*} \vec{v}_1 &= \vec{v} + \Omega \vec{p}_1\\ \vec{v}_2 &= \vec{v} + \Omega \vec{p}_2\\ \vec{v}_3 &= \vec{v} + \Omega \vec{p}_3\\ \vec{v}_4 &= \vec{v} + \Omega \vec{p}_4 \end{align*} With \begin{equation} \Omega = \begin{bmatrix} 0 & -\Omega_z & \Omega_y \\ \Omega_z & 0 & -\Omega_x \\ -\Omega_y & \Omega_x & 0 \end{bmatrix} \end{equation} $\vec{v}$ and $\Omega$ represent to velocity and rotation of the solid expressed in the frame $\{O\}$. We can rearrange the equations in a matrix form: \begin{equation} \left[\begin{array}{ccc|ccc} 1 & 0 & 0 & 0 & p_{1z} & -p_{1y} \\ 0 & 1 & 0 & -p_{1z} & 0 & p_{1x} \\ 0 & 0 & 1 & p_{1y} & -p_{1x} & 0 \\ \hline & \vdots & & & \vdots & \\ \hline 1 & 0 & 0 & 0 & p_{4z} & -p_{4y} \\ 0 & 1 & 0 & -p_{4z} & 0 & p_{4x} \\ 0 & 0 & 1 & p_{4y} & -p_{4x} & 0 \end{array}\right] \begin{bmatrix} v_x \\ v_y \\ v_z \\ \hline \Omega_x \\ \Omega_y \\ \Omega_z \end{bmatrix} = \begin{bmatrix} v_{1x} \\ v_{1y} \\ v_{1z} \\\hline \vdots \\\hline v_{4x} \\ v_{4y} \\ v_{4z} \end{bmatrix} \end{equation} and then we obtain the velocity and rotation of the solid in the wanted frame $\{O\}$: \begin{equation} \begin{bmatrix} v_x \\ v_y \\ v_z \\ \hline \Omega_x \\ \Omega_y \\ \Omega_z \end{bmatrix} = \left[\begin{array}{ccc|ccc} 1 & 0 & 0 & 0 & p_{1z} & -p_{1y} \\ 0 & 1 & 0 & -p_{1z} & 0 & p_{1x} \\ 0 & 0 & 1 & p_{1y} & -p_{1x} & 0 \\ \hline & \vdots & & & \vdots & \\ \hline 1 & 0 & 0 & 0 & p_{4z} & -p_{4y} \\ 0 & 1 & 0 & -p_{4z} & 0 & p_{4x} \\ 0 & 0 & 1 & p_{4y} & -p_{4x} & 0 \end{array}\right]^{-1} \begin{bmatrix} v_{1x} \\ v_{1y} \\ v_{1z} \\\hline \vdots \\\hline v_{4x} \\ v_{4y} \\ v_{4z} \end{bmatrix} \end{equation} This inversion is equivalent to a mean square problem. #+begin_src matlab mode_shapes_O = zeros(mod_n, length(solid_names), 6); for mod_i = 1:mod_n for solid_i = 1:length(solid_names) solids_i = solids.(solid_names{solid_i}); Y = reshape(squeeze(modes(mod_i, solids_i, :))', [], 1); A = zeros(3*length(solids_i), 6); for i = 1:length(solids_i) A(3*(i-1)+1:3*i, 1:3) = eye(3); A(3*(i-1)+1:3*i, 4:6) = [0 acc_pos(i, 3) -acc_pos(i, 2) ; -acc_pos(i, 3) 0 acc_pos(i, 1) ; acc_pos(i, 2) -acc_pos(i, 1) 0]; end mode_shapes_O(mod_i, solid_i, :) = A\Y; end end #+end_src * Modal Matrices We want to obtain the two following matrices: \[ \Omega = \begin{bmatrix} \omega_1^2 & & 0 \\ & \ddots & \\ 0 & & \omega_n^2 \end{bmatrix}; \quad \Psi = \begin{bmatrix} & & \\ \{\psi_1\} & \dots & \{\psi_n\} \\ & & \end{bmatrix} \] - [ ] How to add damping to the eigen value matrix? #+begin_src matlab eigen_value_M = diag(freqs*2*pi); eigen_vector_M = reshape(mode_shapes_O, [mod_n, 6*length(solid_names)])'; #+end_src \[ \{\psi_1\} = \begin{Bmatrix} \psi_{1_x} & \psi_{2_x} & \dots & \psi_{6_x} & \psi_{1_x} & \dots & \psi_{1\Omega_x} & \dots & \psi_{6\Omega_z} \end{Bmatrix}^T \] * Modal Complexity A method of displaying *modal complexity* is by plotting the elements of the eigenvector on an *Argand diagram*, such as the ones shown in figure [[fig:modal_complexity_small]]. To evaluate the complexity of the modes, we plot a polygon around the extremities of the individual vectors. The obtained area of this polygon is then compared with the area of the circle which is based on the length of the largest vector element. The resulting ratio is used as an indication of the complexity of the mode. A little complex mode is shown on figure [[fig:modal_complexity_small]] whereas an highly complex mode is shown on figure [[fig:modal_complexity_high]]. The complexity of all the modes are compared on figure [[fig:modal_complexities]]. #+begin_src matlab :exports none mod_i = 1; i_max = convhull(real(eigen_vector_M(:, mod_i)), imag(eigen_vector_M(:, mod_i))); radius = max(abs(eigen_vector_M(:, mod_i))); theta = linspace(0, 2*pi, 100); figure; hold on; plot(radius*cos(theta), radius*sin(theta), '-'); plot(real(eigen_vector_M(i_max, mod_i)), imag(eigen_vector_M(i_max, mod_i)), '-'); plot(real(eigen_vector_M(:, mod_i)), imag(eigen_vector_M(:, mod_i)), 'ko'); hold off; xlabel('Real Part'); ylabel('Imaginary Part'); title(sprintf('Mode %i', mod_i)); axis manual equal #+end_src #+HEADER: :tangle no :exports results :results none :noweb yes #+begin_src matlab :var filepath="figs/modal_complexity_small.pdf" :var figsize="normal-normal" :post pdf2svg(file=*this*, ext="png") <> #+end_src #+NAME: fig:modal_complexity_small #+CAPTION: Modal Complexity of one mode with small complexity [[file:figs/modal_complexity_small.png]] #+begin_src matlab :exports none mod_i = 8; i_max = convhull(real(eigen_vector_M(:, mod_i)), imag(eigen_vector_M(:, mod_i))); radius = max(abs(eigen_vector_M(:, mod_i))); theta = linspace(0, 2*pi, 100); figure; hold on; plot(radius*cos(theta), radius*sin(theta), '-'); plot(real(eigen_vector_M(i_max, mod_i)), imag(eigen_vector_M(i_max, mod_i)), '-'); plot(real(eigen_vector_M(:, mod_i)), imag(eigen_vector_M(:, mod_i)), 'ko'); hold off; xlabel('Real Part'); ylabel('Imaginary Part'); title(sprintf('Mode %i', mod_i)); axis manual equal #+end_src #+HEADER: :tangle no :exports results :results none :noweb yes #+begin_src matlab :var filepath="figs/modal_complexity_high.pdf" :var figsize="normal-normal" :post pdf2svg(file=*this*, ext="png") <> #+end_src #+NAME: fig:modal_complexity_high #+CAPTION: Modal Complexity of one higly complex mode [[file:figs/modal_complexity_high.png]] #+begin_src matlab :exports none modes_complexity = zeros(mod_n, 1); for mod_i = 1:mod_n i = convhull(real(eigen_vector_M(:, mod_i)), imag(eigen_vector_M(:, mod_i))); area_complex = polyarea(real(eigen_vector_M(i, mod_i)), imag(eigen_vector_M(i, mod_i))); area_circle = pi*max(abs(eigen_vector_M(:, mod_i)))^2; modes_complexity(mod_i) = area_complex/area_circle; end figure; plot(1:mod_n, modes_complexity, 'ok'); ylim([0, 1]); xlabel('Mode Number'); ylabel('Modal Complexity'); #+end_src #+HEADER: :tangle no :exports results :results none :noweb yes #+begin_src matlab :var filepath="figs/modal_complexities.pdf" :var figsize="wide-normal" :post pdf2svg(file=*this*, ext="png") <> #+end_src #+NAME: fig:modal_complexities #+CAPTION: Modal complexity for each mode [[file:figs/modal_complexities.png]] * Some notes about constraining the number of degrees of freedom We want to have the two eigen matrices. They should have the same size $n \times n$ where $n$ is the number of modes as well as the number of degrees of freedom. Thus, if we consider 21 modes, we should restrict our system to have only 21 DOFs. Actually, we are measured 6 DOFs of 6 solids, thus we have 36 DOFs. From the mode shapes animations, it seems that in the frequency range of interest, the two marbles can be considered as one solid. We thus have 5 solids and 30 DOFs. In order to determine which DOF can be neglected, two solutions seems possible: - compare the mode shapes - compare the FRFs The question is: in which base (frame) should be express the modes shapes and FRFs? Is it meaningful to compare mode shapes as they give no information about the amplitudes of vibration? | Stage | Motion DOFs | Parasitic DOF | Total DOF | Description of DOF | |---------+-------------+---------------+-----------+--------------------| | Granite | 0 | 3 | 3 | | | Ty | 1 | 2 | 3 | Ty, Rz | | Ry | 1 | 2 | 3 | Ry, | | Rz | 1 | 2 | 3 | Rz, Rx, Ry | | Hexapod | 6 | 0 | 6 | Txyz, Rxyz | |---------+-------------+---------------+-----------+--------------------| | | 9 | 9 | 18 | | #+TBLFM: $4=vsum($2..$3) #+TBLFM: @>$2..$>=vsum(@I..@II) * TODO Normalization of mode shapes? We normalize each column of the eigen vector matrix. Then, each eigenvector as a norm of 1. #+begin_src matlab eigen_vector_M = eigen_vector_M./vecnorm(eigen_vector_M); #+end_src * Compare Mode Shapes Let's say we want to see for the first mode which DOFs can be neglected. In order to do so, we should estimate the motion of each stage in particular directions. If we look at the z motion for instance, we will find that we cannot neglect that motion (because of the tilt causing z motion). #+begin_src matlab mode_i = 3; dof_i = 6; mode = eigen_vector_M(dof_i:6:end, mode_i); figure; hold on; for i=1:length(mode) plot([0, real(mode(i))], [0, imag(mode(i))], '-', 'DisplayName', solid_names{i}); end hold off; legend(); #+end_src #+begin_src matlab figure; subplot(2, 1, 1); hold on; for i=1:length(mode) plot(1, norm(mode(i)), 'o'); end hold off; ylabel('Amplitude'); subplot(2, 1, 2); hold on; for i=1:length(mode) plot(1, 180/pi*angle(mode(i)), 'o', 'DisplayName', solid_names{i}); end hold off; ylim([-180, 180]); yticks([-180:90:180]); ylabel('Phase [deg]'); legend(); #+end_src #+begin_src matlab test = mode_shapes_O(10, 1, :)/norm(squeeze(mode_shapes_O(10, 1, :))); test = mode_shapes_O(10, 2, :)/norm(squeeze(mode_shapes_O(10, 2, :))); #+end_src * Importation of measured FRF curves There are 24 measurements files corresponding to 24 series of impacts: - 3 directions, 8 sets of 3 accelerometers For each measurement file, the FRF and coherence between the impact and the 9 accelerations measured. In reality: 4 sets of 10 things #+begin_src matlab a = load('mat/meas_frf_coh_1.mat'); #+end_src #+begin_src matlab figure; ax1 = subplot(2, 1, 1); hold on; plot(a.FFT1_AvXSpc_2_1_RMS_X_Val, a.FFT1_AvXSpc_2_1_RMS_Y_Mod) hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); set(gca, 'XTickLabel',[]); ylabel('Amplitude'); title(sprintf('From %s, to %s', FFT1_AvXSpc_2_1_RfName, FFT1_AvXSpc_2_1_RpName)) ax2 = subplot(2, 1, 2); hold on; plot(a.FFT1_AvXSpc_2_1_RMS_X_Val, a.FFT1_AvXSpc_2_1_RMS_Y_Phas) hold off; ylim([-180, 180]); yticks(-180:90:180); xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); set(gca, 'xscale', 'log'); linkaxes([ax1,ax2],'x'); xlim([1, 200]); #+end_src * Importation of measured FRF curves to global FRF matrix FRF matrix $n \times p$: - $n$ is the number of measurements: $3 \times 24$ - $p$ is the number of excitation inputs: 3 23 measurements: 3 accelerometers \begin{equation} \text{FRF}(\omega_i) = \begin{bmatrix} \frac{D_{1_x}}{F_x}(\omega_i) & \frac{D_{1_x}}{F_y}(\omega_i) & \frac{D_{1_x}}{F_z}(\omega_i) \\ \frac{D_{1_y}}{F_x}(\omega_i) & \frac{D_{1_y}}{F_y}(\omega_i) & \frac{D_{1_y}}{F_z}(\omega_i) \\ \frac{D_{1_z}}{F_x}(\omega_i) & \frac{D_{1_z}}{F_y}(\omega_i) & \frac{D_{1_z}}{F_z}(\omega_i) \\ \frac{D_{2_x}}{F_x}(\omega_i) & \frac{D_{2_x}}{F_y}(\omega_i) & \frac{D_{2_x}}{F_z}(\omega_i) \\ \vdots & \vdots & \vdots \\ \frac{D_{23_z}}{F_x}(\omega_i) & \frac{D_{23_z}}{F_y}(\omega_i) & \frac{D_{23_z}}{F_z}(\omega_i) \\ \end{bmatrix} \end{equation} #+begin_src matlab n_meas = 24; n_acc = 23; dirs = 'XYZ'; % Number of Accelerometer * DOF for each acccelerometer / Number of excitation / frequency points FRFs = zeros(3*n_acc, 3, 801); COHs = zeros(3*n_acc, 3, 801); % Loop through measurements for i = 1:n_meas % Load the measurement file meas = load(sprintf('mat/meas_frf_coh_%i.mat', i)); % First: determine what is the exitation (direction and sign) exc_dir = meas.FFT1_AvXSpc_2_1_RMS_RfName(end); exc_sign = meas.FFT1_AvXSpc_2_1_RMS_RfName(end-1); % Determine what is the correct excitation sign exc_factor = str2num([exc_sign, '1']); if exc_dir ~= 'Z' exc_factor = exc_factor*(-1); end % Then: loop through the nine measurements and store them at the correct location for j = 2:10 % Determine what is the accelerometer and direction [indices_acc_i] = strfind(meas.(sprintf('FFT1_H1_%i_1_RpName', j)), '.'); acc_i = str2num(meas.(sprintf('FFT1_H1_%i_1_RpName', j))(indices_acc_i(1)+1:indices_acc_i(2)-1)); meas_dir = meas.(sprintf('FFT1_H1_%i_1_RpName', j))(end); meas_sign = meas.(sprintf('FFT1_H1_%i_1_RpName', j))(end-1); % Determine what is the correct measurement sign meas_factor = str2num([meas_sign, '1']); if meas_dir ~= 'Z' meas_factor = meas_factor*(-1); end % FRFs(acc_i+n_acc*(find(dirs==meas_dir)-1), find(dirs==exc_dir), :) = exc_factor*meas_factor*meas.(sprintf('FFT1_H1_%i_1_Y_ReIm', j)); % COHs(acc_i+n_acc*(find(dirs==meas_dir)-1), find(dirs==exc_dir), :) = meas.(sprintf('FFT1_Coh_%i_1_RMS_Y_Val', j)); FRFs(find(dirs==meas_dir)+3*(acc_i-1), find(dirs==exc_dir), :) = exc_factor*meas_factor*meas.(sprintf('FFT1_H1_%i_1_Y_ReIm', j)); COHs(find(dirs==meas_dir)+3*(acc_i-1), find(dirs==exc_dir), :) = meas.(sprintf('FFT1_Coh_%i_1_RMS_Y_Val', j)); end end freqs = meas.FFT1_Coh_10_1_RMS_X_Val; #+end_src * Analysis of some FRFs #+begin_src matlab acc_i = 3; acc_dir = 1; exc_dir = 1; figure; ax1 = subplot(2, 1, 1); hold on; plot(freqs, abs(squeeze(FRFs(acc_dir+3*(acc_i-1), exc_dir, :)))); hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); set(gca, 'XTickLabel',[]); ylabel('Amplitude'); ax2 = subplot(2, 1, 2); hold on; plot(freqs, mod(180+180/pi*phase(squeeze(FRFs(acc_dir+3*(acc_i-1), exc_dir, :))), 360)-180); hold off; ylim([-180, 180]); yticks(-180:90:180); xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); set(gca, 'xscale', 'log'); linkaxes([ax1,ax2],'x'); xlim([1, 200]); #+end_src * Composite Response Function We here sum the norm instead of the complex numbers. #+begin_src matlab HHx = squeeze(sum(abs(FRFs(:, 1, :)))); HHy = squeeze(sum(abs(FRFs(:, 2, :)))); HHz = squeeze(sum(abs(FRFs(:, 3, :)))); HH = squeeze(sum([HHx, HHy, HHz], 2)); #+end_src #+begin_src matlab exc_dir = 3; figure; hold on; for i = 1:3*n_acc plot(freqs, abs(squeeze(FRFs(i, exc_dir, :))), 'color', [0, 0, 0, 0.2]); end plot(freqs, abs(HHx)); plot(freqs, abs(HHy)); plot(freqs, abs(HHz)); plot(freqs, abs(HH), 'k'); hold off; set(gca, 'XScale', 'lin'); set(gca, 'YScale', 'lin'); xlabel('Frequency [Hz]'); ylabel('Amplitude'); xlim([1, 200]); #+end_src #+HEADER: :tangle no :exports results :results none :noweb yes #+begin_src matlab :var filepath="figs/composite_response_function.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png") <> #+end_src #+NAME: fig:composite_response_function #+CAPTION: Composite Response Function [[file:figs/composite_response_function.png]] * TODO Singular Value Decomposition - Modal Indication Function Show the same plot as in the modal software. This helps to identify double modes. From the documentation of the modal software: #+begin_quote The MIF consist of the singular values of the Frequency response function matrix. The number of MIFs equals the number of excitations. By the powerful singular value decomposition, the real signal space is separated from the noise space. Therefore, the MIFs exhibit the modes effectively. A peak in the MIFs plot usually indicate the existence of a structural mode, and two peaks at the same frequency point means the existence of two repeated modes. Moreover, the magnitude of the MIFs implies the strength of the a mode. #+end_quote * From local coordinates to global coordinates with the FRFs #+begin_src matlab % Number of Solids * DOF for each solid / Number of excitation / frequency points FRFs_O = zeros(length(solid_names)*6, 3, 801); for exc_dir = 1:3 for solid_i = 1:length(solid_names) solids_i = solids.(solid_names{solid_i}); A = zeros(3*length(solids_i), 6); for i = 1:length(solids_i) A(3*(i-1)+1:3*i, 1:3) = eye(3); A(3*(i-1)+1:3*i, 4:6) = [0 acc_pos(i, 3) -acc_pos(i, 2) ; -acc_pos(i, 3) 0 acc_pos(i, 1) ; acc_pos(i, 2) -acc_pos(i, 1) 0]; end for i = 1:801 FRFs_O((solid_i-1)*6+1:solid_i*6, exc_dir, i) = A\FRFs((solids_i(1)-1)*3+1:solids_i(end)*3, exc_dir, i); end end end #+end_src * Analysis of some FRF in the global coordinates #+begin_src matlab solid_i = 6; dir_i = 1; exc_dir = 1; figure; ax1 = subplot(2, 1, 1); hold on; plot(freqs, abs(squeeze(FRFs_O((solid_i-1)*6+dir_i, exc_dir, :)))); hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); set(gca, 'XTickLabel',[]); ylabel('Amplitude'); ax2 = subplot(2, 1, 2); hold on; plot(freqs, mod(180+180/pi*phase(squeeze(FRFs_O((solid_i-1)*6+dir_i, exc_dir, :))), 360)-180); hold off; ylim([-180, 180]); yticks(-180:90:180); xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); set(gca, 'xscale', 'log'); linkaxes([ax1,ax2],'x'); xlim([1, 200]); #+end_src * Compare global coordinates to local coordinates #+begin_src matlab solid_i = 1; acc_dir_O = 6; acc_dir = 3; exc_dir = 3; figure; ax1 = subplot(2, 1, 1); hold on; for i = solids.(solid_names{solid_i}) plot(freqs, abs(squeeze(FRFs(acc_dir+3*(i-1), exc_dir, :)))); end plot(freqs, abs(squeeze(FRFs_O((solid_i-1)*6+acc_dir_O, exc_dir, :))), '-k'); hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); set(gca, 'XTickLabel',[]); ylabel('Amplitude'); ax2 = subplot(2, 1, 2); hold on; for i = solids.(solid_names{solid_i}) plot(freqs, mod(180+180/pi*phase(squeeze(FRFs(acc_dir+3*(i-1), exc_dir, :))), 360)-180); end plot(freqs, mod(180+180/pi*phase(squeeze(FRFs_O((solid_i-1)*6+acc_dir_O, exc_dir, :))), 360)-180, '-k'); hold off; ylim([-180, 180]); yticks(-180:90:180); xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); set(gca, 'xscale', 'log'); linkaxes([ax1,ax2],'x'); xlim([1, 200]); #+end_src * Verify that we find the original FRF from the FRF in the global coordinates From the computed FRF of the Hexapod in its 6 DOFs, compute the FRF of the accelerometer 1 fixed to the Hexapod during the measurement. #+begin_src matlab FRF_test = zeros(801, 3); for i = 1:801 FRF_test(i, :) = FRFs_O(31:33, 1, i) + cross(FRFs_O(34:36, 1, i), acc_pos(1, :)'); end #+end_src #+begin_src matlab :exports none figure; ax1 = subplot(3, 1, 1); hold on; plot(freqs, abs(squeeze(FRFs(1, 1, :)))); plot(freqs, abs(squeeze(FRF_test(:, 1))), '--k'); hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); set(gca, 'XTickLabel',[]); xlim([1, 200]); title('FRF $\frac{D_{1x}}{F_x}$'); ax2 = subplot(3, 1, 2); hold on; plot(freqs, abs(squeeze(FRFs(2, 1, :)))); plot(freqs, abs(squeeze(FRF_test(:, 2))), '--k'); hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); set(gca, 'XTickLabel',[]); ylabel('Amplitude'); xlim([1, 200]); title('FRF $\frac{D_{1y}}{F_x}$'); ax3 = subplot(3, 1, 3); hold on; plot(freqs, abs(squeeze(FRFs(3, 1, :)))); plot(freqs, abs(squeeze(FRF_test(:, 3))), '--k'); hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); xlabel('Frequency [Hz]'); xlim([1, 200]); legend({'Original Measurement', 'Recovered Measurement'}, 'Location', 'southeast'); title('FRF $\frac{D_{1z}}{F_x}$'); #+end_src #+HEADER: :tangle no :exports results :results none :noweb yes #+begin_src matlab :var filepath="figs/compare_original_meas_with_recovered.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png") <> #+end_src #+NAME: fig:compare_original_meas_with_recovered #+CAPTION: Comparison of the original measured FRFs with the recovered FRF from the FRF in the common cartesian frame [[file:figs/compare_original_meas_with_recovered.png]] #+begin_important The reduction of the number of degrees of freedom from 69 (23 accelerometers with each 3DOF) to 36 (6 solid bodies with 6 DOF) seems to work well. This confirms the fact that this stage, for that mode is indeed behaving as a solid body. This should be verified for all the stages for modes with high resonance frequencies. #+end_important * TODO Synthesis of FRF curves