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