#+TITLE: Modal Analysis
:DRAWER:
#+STARTUP: overview

#+LANGUAGE: en
#+EMAIL: dehaeze.thomas@gmail.com
#+AUTHOR: Dehaeze Thomas

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#+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