#+end_export
#+latex: \clearpage
* Build :noexport:
#+NAME: startblock
#+BEGIN_SRC emacs-lisp :results none :tangle no
(add-to-list 'org-latex-classes
'("scrreprt"
"\\documentclass{scrreprt}"
("\\chapter{%s}" . "\\chapter*{%s}")
("\\section{%s}" . "\\section*{%s}")
("\\subsection{%s}" . "\\subsection*{%s}")
("\\paragraph{%s}" . "\\paragraph*{%s}")
))
;; Remove automatic org heading labels
(defun my-latex-filter-removeOrgAutoLabels (text backend info)
"Org-mode automatically generates labels for headings despite explicit use of `#+LABEL`. This filter forcibly removes all automatically generated org-labels in headings."
(when (org-export-derived-backend-p backend 'latex)
(replace-regexp-in-string "\\\\label{sec:org[a-f0-9]+}\n" "" text)))
(add-to-list 'org-export-filter-headline-functions
'my-latex-filter-removeOrgAutoLabels)
;; Remove all org comments in the output LaTeX file
(defun delete-org-comments (backend)
(loop for comment in (reverse (org-element-map (org-element-parse-buffer)
'comment 'identity))
do
(setf (buffer-substring (org-element-property :begin comment)
(org-element-property :end comment))
"")))
(add-hook 'org-export-before-processing-hook 'delete-org-comments)
;; Use no package by default
(setq org-latex-packages-alist nil)
(setq org-latex-default-packages-alist nil)
;; Do not include the subtitle inside the title
(setq org-latex-subtitle-separate t)
(setq org-latex-subtitle-format "\\subtitle{%s}")
(setq org-export-before-parsing-hook '(org-ref-glossary-before-parsing
org-ref-acronyms-before-parsing))
#+END_SRC
* Introduction :ignore:
In order to properly make a multi-body model of the micro-station, an experimental modal-analysis is performed.
In fact, even though it is easy to estimate the inertia of each solid body from its geometry and its material density, it is much more difficult to properly estimate the stiffness and damping properties of the guiding elements connecting each solid body.
In this report, an experimental modal analysis is perform in order to ease the development of the multi-body model.
The measurement setup used for modal analysis is presented in Section ref:sec:modal_meas_setup.
This includes the instrumentation used (i.e. instrumented hammer, accelerometers and acquisition system), the test planing, and the first analysis of the obtained signals.
In Section ref:sec:modal_frf_processing, the obtained frequency response functions between the forces applied using the instrumented hammer and the various accelerometers fixed to the structure are computed.
These measurements are then projected at the center of mass of each solid body to ease the further use of the results.
The solid body assumption is also validated.
Finally, the modal analysis is performed in Section ref:sec:modal_analysis.
[[cite:&ewins00_modal]]
#+name: tab:modal_section_matlab_code
#+caption: Report sections and corresponding Matlab files
#+attr_latex: :environment tabularx :width 0.5\linewidth :align lX
#+attr_latex: :center t :booktabs t
| *Sections* | *Matlab File* |
|--------------------------------------+----------------------------|
| Section ref:sec:modal_meas_setup | =modal_1_meas_setup.m= |
| Section ref:sec:modal_frf_processing | =modal_2_frf_processing.m= |
| Section ref:sec:modal_analysis | =modal_3_analysis.m= |
* Measurement Setup
:PROPERTIES:
:HEADER-ARGS:matlab+: :tangle matlab/modal_1_meas_setup.m
:END:
<>
** Introduction :ignore:
ref:ssec:modal_instrumentation
ref:ssec:modal_test_preparation
ref:ssec:modal_accelerometers
ref:ssec:modal_hammer_impacts
ref:ssec:modal_measured_signals
** 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
#+begin_src matlab :tangle no :noweb yes
<>
#+end_src
#+begin_src matlab :eval no :noweb yes
<>
#+end_src
#+begin_src matlab :noweb yes
<>
#+end_src
** Used Instrumentation
<>
In order to perform to *Modal Analysis* and to obtain first a *Response Model*, the following devices are used:
- An *acquisition system* (OROS) with 24bits ADCs (figure ref:fig:modal_oros)
- 3 tri-axis *Accelerometers* (figure ref:fig:modal_accelero_M393B05) with parameters shown on table ref:tab:modal_accelero_M393B05
- An *Instrumented Hammer* with various Tips (figure ref:fig:modal_instrumented_hammer)
#+name: fig:modal_analysis_instrumentation
#+caption: Instrumentation used for the modal analysis
#+begin_figure
#+attr_latex: :caption \subcaption{\label{fig:modal_oros}OROS acquisition system}
#+attr_latex: :options {0.33\textwidth}
#+begin_subfigure
#+attr_latex: :height 6cm
[[file:figs/modal_oros.jpg]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:modal_accelero_M393B05}Accelerometer (M393B05)}
#+attr_latex: :options {0.33\textwidth}
#+begin_subfigure
#+attr_latex: :height 6cm
[[file:figs/modal_accelero_M393B05.jpg]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:modal_instrumented_hammer}Instrumented Hammer}
#+attr_latex: :options {0.33\textwidth}
#+begin_subfigure
#+attr_latex: :height 6cm
[[file:figs/modal_instrumented_hammer.jpg]]
#+end_subfigure
#+end_figure
The acquisition system permits to auto-range the inputs (probably using variable gain amplifiers) the obtain the maximum dynamic range.
This is done before each measurement.
Anti-aliasing filters are also included in the system.
#+name: tab:modal_accelero_M393B05
#+caption: 393B05 Accelerometer Data Sheet
#+attr_latex: :environment tabularx :width 0.4\linewidth :align lX
#+attr_latex: :center t :booktabs t
| Sensitivity | $10\,V/g$ |
| Measurement Range | 0.5 g pk |
| Broadband Resolution | $\SI{4}{\micro\g\rms}$ |
| Frequency Range | $0.7$ to $\SI{450}{Hz}$ |
| Resonance Frequency | $> \SI{2.5}{kHz}$ |
Tests have been conducted to determine the most suitable Hammer tip.
This has been found that the softer tip gives the best results.
It excites more the low frequency range where the coherence is low, the overall coherence was improved.
The accelerometers are glued on the structure.
** Structure Preparation and Test Planing
<>
All the stages are turned ON.
This is done for two reasons:
- Be closer to the real dynamic of the station in used
- If the control system of stages are turned OFF, this would results in very low frequency modes un-identifiable with the current setup, and this will also decouple the dynamics which would not be the case in practice
This is critical for the translation stage and the spindle as their is no stiffness in the free DOF (air-bearing for the spindle for instance).
The alternative would have been to mechanically block the stages with screws, but this may result in changing the modes.
The stages turned ON are:
- Translation Stage
- Tilt Stage
- Spindle and Slip-Ring
- Hexapod
The top part representing the NASS and the sample platform have been removed in order to reduce the complexity of the dynamics and also because this will be further added in the model inside Simscape.
All the stages are moved to their zero position (Ty, Ry, Rz, Slip-Ring, Hexapod).
All other elements have been remove from the granite such as another heavy positioning system.
The goal is to identify the full $N \times N$ FRF matrix $H$ (where $N$ is the number of degree of freedom of the system):
\begin{equation}
H_{jk} = \frac{X_j}{F_k}
\end{equation}
However, from only one column or one line of the matrix, we can compute the other terms thanks to the principle of reciprocity.
Either we choose to identify $\frac{X_k}{F_i}$ or $\frac{X_i}{F_k}$ for any chosen $k$ and for $i = 1,\ ...,\ N$.
We here choose to identify $\frac{X_i}{F_k}$ for practical reasons:
- it is easier to glue the accelerometers on all the stages and excite only a one particular point than doing the opposite
The measurement thus consists of:
- always excite the structure at the same location with the Hammer
- Move the accelerometers to measure all the DOF of the structure
We will measured 3 columns (3 impacts location) in order to have some redundancy and to make sure that all modes are excited.
** Location of the Accelerometers
<>
4 tri-axis accelerometers are used for each solid body.
Only 2 could have been used as only 6DOF have to be measured, however, we have chosen to have some *redundancy*.
# In reality 3 are needed as if only two are used, the rotation along the line connecting the two accelerometers will not be measured.
This could also help us identify measurement problems or flexible modes is present.
The position of the accelerometers are:
- 4 on the first granite
- 4 on the second granite
- 4 on top of the translation stage (figure ref:fig:modal_accelerometers_ty)
- 4 on top of the tilt stage
- 3 on top of the spindle
- 4 on top of the hexapod (figure ref:fig:modal_accelerometers_hexapod)
In total, 23 accelerometers are used: *69 DOFs are thus measured*.
The precise determination of the position of each accelerometer is done using the SolidWorks model (shown on figure ref:fig:modal_location_accelerometers).
#+name: fig:modal_accelerometer_pictures
#+caption: Accelerometers fixed on the micro-station
#+begin_figure
#+attr_latex: :caption \subcaption{\label{fig:modal_accelerometers_ty} $T_y$ stage}
#+attr_latex: :options {0.49\textwidth}
#+begin_subfigure
#+attr_latex: :height 6cm
[[file:figs/modal_accelerometers_ty.jpg]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:modal_accelerometers_hexapod} Micro-Hexapod}
#+attr_latex: :options {0.49\textwidth}
#+begin_subfigure
#+attr_latex: :height 6cm
[[file:figs/modal_accelerometers_hexapod.jpg]]
#+end_subfigure
#+end_figure
#+name: fig:modal_location_accelerometers
#+caption: Position of the accelerometers using SolidWorks
#+attr_latex: :width \linewidth
[[file:figs/modal_location_accelerometers.png]]
The precise position of all the 23 accelerometer with respect to a frame located at the point of interest (located 270mm above the top platform of the hexapod) are shown in table ref:tab:modal_position_accelerometers.
#+begin_src matlab
%% Load Accelerometer positions
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
#+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*)
data2orgtable([[1:23]', 1000*acc_pos], {'Hexapod', 'Hexapod', 'Hexapod', 'Hexapod', 'Tilt', 'Tilt', 'Tilt', 'Tilt', 'Translation', 'Translation', 'Translation', 'Translation', 'Upper Granite', 'Upper Granite', 'Upper Granite', 'Upper Granite', 'Lower Granite', 'Lower Granite', 'Lower Granite', 'Lower Granite', 'Spindle', 'Spindle', 'Spindle'}, {'Stage', 'ID', '$x$ [mm]', '$y$ [mm]', '$z$ [mm]'}, ' %.0f ');
#+end_src
#+name: tab:modal_position_accelerometers
#+caption: Position of the accelerometers
#+attr_latex: :environment tabularx :width 0.5\linewidth :align Xcccc
#+attr_latex: :center t :booktabs t :font \scriptsize
#+RESULTS:
| | ID | $x$ [mm] | $y$ [mm] | $z$ [mm] |
|---------------+----+----------+----------+----------|
| Hexapod | 1 | -64 | -64 | -270 |
| Hexapod | 2 | -64 | 64 | -270 |
| Hexapod | 3 | 64 | 64 | -270 |
| Hexapod | 4 | 64 | -64 | -270 |
| Tilt | 5 | -385 | -300 | -417 |
| Tilt | 6 | -420 | 280 | -417 |
| Tilt | 7 | 420 | 280 | -417 |
| Tilt | 8 | 380 | -300 | -417 |
| Translation | 9 | -475 | -414 | -427 |
| Translation | 10 | -465 | 407 | -427 |
| Translation | 11 | 475 | 424 | -427 |
| Translation | 12 | 475 | -419 | -427 |
| Upper Granite | 13 | -320 | -446 | -786 |
| Upper Granite | 14 | -480 | 534 | -786 |
| Upper Granite | 15 | 450 | 534 | -786 |
| Upper Granite | 16 | 295 | -481 | -786 |
| Lower Granite | 17 | -730 | -526 | -951 |
| Lower Granite | 18 | -735 | 814 | -951 |
| Lower Granite | 19 | 875 | 799 | -951 |
| Lower Granite | 20 | 865 | -506 | -951 |
| Spindle | 21 | -155 | -90 | -594 |
| Spindle | 22 | 0 | 180 | -594 |
| Spindle | 23 | 155 | -90 | -594 |
** Hammer Impacts
<>
Only 3 impact points are used.
The impact points are shown on figures ref:fig:modal_impact_x, ref:fig:modal_impact_y and ref:fig:modal_impact_z.
We chose this excitation point as it seems to excite all the modes in the frequency band of interest and because it provides good coherence for all the accelerometers.
# TODO - Explain that the excitation is at the location of acceleromter 11
# This is very important to measure the excitation point
From [[cite:ewins00_modal]]:
Most modal test require a point mobility measurement as one of the measured FRF.
This is hard to achieve as both force and response transducer should be at the same point on the structure.
#+name: fig:modal_hammer_impacts
#+caption: The three hammer impacts used for the modal analysis
#+begin_figure
#+attr_latex: :caption \subcaption{\label{fig:modal_impact_x} $X$ impact}
#+attr_latex: :options {0.33\textwidth}
#+begin_subfigure
#+attr_latex: :width \linewidth
[[file:figs/modal_impact_x.jpg]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:modal_impact_y} $Y$ impact}
#+attr_latex: :options {0.33\textwidth}
#+begin_subfigure
#+attr_latex: :width \linewidth
[[file:figs/modal_impact_y.jpg]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:modal_impact_z} $Z$ impact}
#+attr_latex: :options {0.33\textwidth}
#+begin_subfigure
#+attr_latex: :width \linewidth
[[file:figs/modal_impact_z.jpg]]
#+end_subfigure
#+end_figure
** Signal Processing :noexport:
<>
The measurements are averaged 10 times corresponding to 10 hammer impacts in order to reduce the effect of random noise.
Windowing is also used on the force and response signals.
A boxcar window (figure ref:fig:modal_windowing_force_signal) is used for the force signal as once the impact on the structure is done, the measured signal is meaningless.
The parameters are:
- *Start*: $3\%$
- *Stop*: $7\%$
#+begin_src matlab :exports none :results none
%% Boxcar window used for the force signal
figure;
plot(100*[0, 0.03, 0.03, 0.07, 0.07, 1], [0, 0, 1, 1, 0, 0]);
xlabel('Time [\%]'); ylabel('Amplitude');
xlim([0, 100]); ylim([0, 1]);
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/modal_windowing_force_signal.pdf', 'width', 'normal', 'height', 'normal');
#+end_src
#+name: fig:modal_windowing_force_signal
#+caption: Boxcar window used for the force signal
#+RESULTS:
[[file:figs/modal_windowing_force_signal.png]]
An exponential window (figure ref:fig:modal_windowing_acc_signal) is used for the response signal as we are measuring transient signals and most of the information is located at the beginning of the signal.
The parameters are:
- FlatTop:
- *Start*: $3\%$
- *Stop*: $2.96\%$
- Decreasing point:
- *X*: $60.4\%$
- *Y*: $14.7\%$
#+begin_src matlab :exports none :results none
%% Exponential window used for acceleration signal
x0 = 0.296;
xd = 0.604;
yd = 0.147;
alpha = log(yd)/(x0 - xd);
t = x0:0.01:1.01;
y = exp(-alpha*(t-x0));
figure;
plot(100*[0, 0.03, 0.03, x0, t], [0, 0, 1, 1, y]);
xlabel('Time [\%]'); ylabel('Amplitude');
xlim([0, 100]); ylim([0, 1]);
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/modal_windowing_acc_signal.pdf', 'width', 'normal', 'height', 'normal');
#+end_src
#+name: fig:modal_windowing_acc_signal
#+caption: Exponential window used for acceleration signal
#+RESULTS:
[[file:figs/modal_windowing_acc_signal.png]]
** Force and Response signals
<>
#+begin_src matlab
%% Load raw data
meas1_raw = load('mat/meas_raw_1.mat');
% Sampling Frequency [Hz]
Fs = 1/meas1_raw.Track1_X_Resolution;
% Time just before the impact occurs [s]
impacts = [5.937, 11.228, 16.681, 22.205, 27.350, 32.714, 38.115, 43.888, 50.407]-0.01;
% Time vector [s]
time = linspace(0, meas1_raw.Track1_X_Resolution*length(meas1_raw.Track1), length(meas1_raw.Track1));
#+end_src
The force sensor and the accelerometers signals are shown in the time domain in Figure ref:fig:modal_raw_meas.
Sharp "impacts" can be seen for the force sensor, indicating wide frequency band excitation.
For the accelerometer, many resonances can be seen on the right, indicating complex dynamics
#+begin_src matlab :exports none :results none
%% Raw measurement of the Accelerometer
figure;
tiledlayout(1, 3, 'TileSpacing', 'Compact', 'Padding', 'None');
ax1 = nexttile([1,2]);
hold on;
plot(time, meas1_raw.Track2, 'DisplayName', 'Acceleration [$m/s^2$]');
plot(time, 1e-3*meas1_raw.Track1, 'DisplayName', 'Force [kN]');
hold off;
xlabel('Time [s]');
ylabel('Amplitude');
xlim([0, time(end)]);
legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 1);
ax2 = nexttile();
hold on;
plot(time, meas1_raw.Track2);
plot(time, 1e-3*meas1_raw.Track1);
hold off;
xlabel('Time [s]');
set(gca, 'YTickLabel',[]);
xlim([22.19, 22.4]);
linkaxes([ax1,ax2],'y');
ylim([-2, 2]);
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/modal_raw_meas.pdf', 'width', 'full', 'height', 'normal');
#+end_src
#+name: fig:modal_raw_meas
#+caption: Raw measurement of the acceleromter (blue) and of the force sensor at the Hammer tip (red). Zoom on one impact is shown on the right.
#+RESULTS:
[[file:figs/modal_raw_meas.png]]
#+begin_src matlab
%% Frequency Analysis
Nfft = floor(5.0*Fs); % Number of frequency points
win = hanning(Nfft); % Windowing
Noverlap = floor(Nfft/2); % Overlap for frequency analysis
%% Comnpute the power spectral density of the force and acceleration
[pxx_force, f] = pwelch(meas1_raw.Track1, win, Noverlap, Nfft, Fs);
[pxx_acc, ~] = pwelch(meas1_raw.Track2, win, Noverlap, Nfft, Fs);
#+end_src
The "normalized" amplitude spectral density of the two signals are computed and shown in Figure ref:fig:modal_asd_acc_force.
Conclusions based on the time domain signals can be clearly seen in the frequency domain (wide frequency content for the force signal and complex dynamics for the accelerometer).
#+begin_src matlab :exports none :results none
%% Normalized Amplitude Spectral Density of the measured force and acceleration
figure;
hold on;
plot(f, sqrt(pxx_force./max(pxx_force(f<200))), 'DisplayName', 'Force');
plot(f, sqrt(pxx_acc./max(pxx_acc(f<200))), 'DisplayName', 'Acceleration');
hold off;
set(gca, 'XScale', 'lin'); set(gca, 'YScale', 'lin');
xlabel('Frequency [Hz]'); ylabel('Normalized Spectral Density');
xlim([0, 200]);
xticks([0:20:200]);
ylim([0, 1])
legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 1);
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/modal_asd_acc_force.pdf', 'width', 'wide', 'height', 'normal');
#+end_src
#+name: fig:modal_asd_acc_force
#+caption: Normalized Amplitude Spectral Density of the measured force and acceleration
#+RESULTS:
[[file:figs/modal_asd_acc_force.png]]
The frequency response function from the applied force to the measured acceleration can then be computed (Figure ref:fig:modal_frf_acc_force).
#+begin_src matlab
%% Compute the transfer function and Coherence
[G1, f] = tfestimate(meas1_raw.Track1, meas1_raw.Track2, win, Noverlap, Nfft, Fs);
[coh1, ~] = mscohere( meas1_raw.Track1, meas1_raw.Track2, win, Noverlap, Nfft, Fs);
#+end_src
#+begin_src matlab :exports none :results none
%% Frequency Response Function between the force and the acceleration
figure;
plot(f, abs(G1));
xlabel('Frequency [Hz]'); ylabel('FRF [$m/s^2/N$]')
set(gca, 'XScale', 'lin'); set(gca, 'YScale', 'log');
xlim([0, 200]);
xticks([0:20:200]);
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/modal_frf_acc_force.pdf', 'width', 'wide', 'height', 'normal');
#+end_src
#+name: fig:modal_frf_acc_force
#+caption: Frequency Response Function between the measured force and acceleration
#+RESULTS:
[[file:figs/modal_frf_acc_force.png]]
The coherence between the input and output signals is also computed and found to be good between 20 and 200Hz (Figure ref:fig:modal_coh_acc_force).
#+begin_src matlab :exports none :results none
%% Frequency Response Function between the force and the acceleration
figure;
plot(f, coh1);
xlabel('Frequency [Hz]'); ylabel('Coherence [-]')
set(gca, 'XScale', 'lin'); set(gca, 'YScale', 'lin');
xlim([0, 200]); ylim([0,1]);
xticks([0:20:200]);
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/modal_coh_acc_force.pdf', 'width', 'wide', 'height', 'normal');
#+end_src
#+name: fig:modal_coh_acc_force
#+caption: Coherence between the measured force and acceleration
#+RESULTS:
[[file:figs/modal_coh_acc_force.png]]
* Frequency Analysis
:PROPERTIES:
:HEADER-ARGS:matlab+: :tangle matlab/modal_2_frf_processing.m
:END:
<>
** Introduction :ignore:
The measurements have been conducted and a $n \times p \times q$ Frequency Response Functions Matrix has been computed with:
- $n$: the number of output measured accelerations: $23 \times 3 = 69$ (23 accelerometers measuring 3 directions each)
- $p$: the number of input force excitations: $3$
- $q$: the number of frequency points $\omega_i$
Thus, the FRF matrix is an $69 \times 3 \times 801$ matrix.
For each frequency point $\omega_i$, a 2D matrix is obtained that links the 3 force inputs to the 69 output accelerations:
\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}
However, for the multi-body model being developed, only 6 solid bodies are considered, namely: the bottom granite, the top granite, the translation stage, the tilt stage, the spindle and the hexapod.
Therefore, only $6 \times 6 = 36$ degrees of freedom are of interest.
The objective in this section is therefore to process the Frequency Response Matrix to reduce the number of measured DoFs from 69 to 36.
In order to be able to perform this reduction of measured DoFs, the measures stages have to behave as rigid bodies in the frequency band of interest.
This
ref:ssec:modal_solid_body_first_check
To go from the accelerometers DoFs to the solid body 6 DoFs (three translations and three rotations), some computations have to be performed.
This is explained in Section ref:ssec:modal_acc_to_solid_dof.
Finally, the $69 \times 3 \times 801$ frequency response matrix is reduced to a $36 \times 3 \times 801$ frequency response matrix where the motion of each solid body is expressed with respect to the CoM of the solid body (Section ref:ssec:modal_frf_com).
To validate this reduction of DoF and the solid body assumption, the frequency response function at the accelerometer location are synthesized from the reduced frequency response matrix and are compared with the initial measurements in Section ref:ssec:modal_solid_body_assumption.
** 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
#+begin_src matlab :tangle no :noweb yes
<>
#+end_src
#+begin_src matlab :eval no :noweb yes
<>
#+end_src
#+begin_src matlab :noweb yes
<>
#+end_src
** First verification of the solid body assumption
<>
In this section, it is shown that two accelerometers fixed to a rigid body at positions $\vec{p}_1$ and $\vec{p}_2$ in such a way that $\vec{p}_2 = \vec{p}_1 + \alpha \vec{x}$ will measure the same value in the $\vec{x}$ direction.
Such situation is illustrated on Figure ref:fig:modal_aligned_accelerometers and is the case for many accelerometers as shown in Figure ref:fig:modal_location_accelerometers.
#+begin_src latex :file modal_aligned_accelerometers.pdf :results file raw silent
\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, fill=blue!30!white] (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);
\draw[->] (p1)node[]{$\bullet$}node[above]{$p_1$} -- ++(1, 0)node[above]{$\delta p_{x1}$};
\draw[->] (p2)node[]{$\bullet$}node[above]{$p_2$} -- ++(1, 0)node[above]{$\delta p_{x2}$};
\draw[dashed] ($(p1)+(-1, 0)$) -- ($(p2)+(2, 0)$);
\end{tikzpicture}
#+end_src
#+name: fig:modal_aligned_accelerometers
#+caption: Aligned measurement of the motion of a solid body
#+RESULTS:
[[file:figs/modal_aligned_accelerometers.png]]
The motion of the rigid body of figure ref:fig:modal_aligned_accelerometers can be defined by its displacement $\delta \vec{p}$ and rotation $\vec{\Omega}$ with respect to a reference frame $\{O\}$.
The motions at points $1$ and $2$ are:
\begin{align*}
\delta \vec{p}_1 &= \delta \vec{p} + \vec{\Omega} \times \vec{p}_1 \\
\delta \vec{p}_2 &= \delta \vec{p} + \vec{\Omega} \times \vec{p}_2
\end{align*}
Taking only the $x$ direction:
\begin{align*}
\delta p_{x1} &= \delta p_x + \Omega_y p_{z1} - \Omega_z p_{y1} \\
\delta p_{x2} &= \delta p_x + \Omega_y p_{z2} - \Omega_z p_{y2}
\end{align*}
However, we have $p_{1y} = p_{2y}$ and $p_{1z} = p_{2z}$ because of the co-linearity of the two sensors in the $x$ direction, and thus we obtain:
\begin{equation}
\boxed{\delta p_{x1} = \delta p_{x2}}
\end{equation}
It is therefore concluded that two position sensors fixed to a rigid body will measure the same quantity in the direction "linking" the two sensors.
Such property can be used to verify that the considered stages are indeed behaving as rigid body in the frequency band of interest.
From Table ref:tab:modal_position_accelerometers, we can identify which pair of accelerometers are aligned in the X and Y directions.
The response in the X direction of pairs of sensors aligned in the X direction are compared in Figure ref:fig:modal_solid_body_comp_x_dir.
Good match is observed up to 200Hz.
Similar result is obtained for the Y direction.
#+begin_src matlab
%% Load frequency response matrix
load('frf_matrix.mat', 'freqs', 'frf');
#+end_src
#+begin_src matlab
%% Accelerometers ID connected to each solid body
solids = {};
solids.gbot = [17, 18, 19, 20]; % bottom granite
solids.gtop = [13, 14, 15, 16]; % top granite
solids.ty = [9, 10, 11, 12]; % Ty stage
solids.ry = [5, 6, 7, 8]; % Ry stage
solids.rz = [21, 22, 23]; % Rz stage
solids.hexa = [1, 2, 3, 4]; % Hexapod
% Names of the solid bodies
solid_names = fields(solids);
#+end_src
#+begin_src matlab :eval no
%% Save the acceleromter positions are well as the solid bodies
save('mat/geometry.mat', 'solids', 'solid_names', 'acc_pos');
#+end_src
#+begin_src matlab :tangle no
%% Save the acceleromter positions are well as the solid bodies
save('matlab/mat/geometry.mat', 'solids', 'solid_names', 'acc_pos');
#+end_src
#+begin_src matlab :exports none
meas_dir = 1; % X
exc_dir = 1; % X
% Pair of accelerometers aligned in the X direction
acc_i = [1 , 4 ;
2 , 3 ;
5 , 8 ;
6 , 7 ;
9 , 12;
10, 11;
14, 15;
18, 19;
21, 23];
#+end_src
#+begin_src matlab :exports none :results none
%% Comparaison of measured frequency response function for in the X directions for accelerometers aligned along X
figure;
tiledlayout(3, 3, 'TileSpacing', 'Compact', 'Padding', 'None');
for i = 1:size(acc_i, 1)
nexttile();
hold on;
plot(freqs, abs(squeeze(frf(meas_dir+3*(acc_i(i, 1)-1), exc_dir, :))), ...
'DisplayName', sprintf('%i', acc_i(i, 1)))
plot(freqs, abs(squeeze(frf(meas_dir+3*(acc_i(i, 2)-1), exc_dir, :))), ...
'DisplayName', sprintf('%i', acc_i(i, 2)))
hold off;
set(gca, 'XScale', 'lin'); set(gca, 'YScale', 'log');
if i > 6
xlabel('Frequency [Hz]');
legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 1);
else
set(gca, 'XTickLabel',[]);
legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 1);
end
if rem(i, 3) == 1
ylabel('Amplitude');
else
set(gca, 'YTickLabel',[]);
end
xlim([0, 200]); ylim([1e-5, 2e-2]);
xticks([0:50:200]);
yticks([1e-5, 1e-4, 1e-3, 1e-2]);
end
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/modal_solid_body_comp_x_dir.pdf', 'width', 'full', 'height', 'tall');
#+end_src
#+name: fig:modal_solid_body_comp_x_dir
#+caption: Comparaison of measured frequency response function for in the X directions for accelerometers aligned along X. Amplitude is in $\frac{m/s^2}{N}$. Accelerometer number is shown in the legend.
#+RESULTS:
[[file:figs/modal_solid_body_comp_x_dir.png]]
** From accelerometer DOFs to solid body DOFs
<>
Let's consider the schematic shown in Figure ref:fig:modal_local_to_global_coordinates where the motion of a solid body is measured at 4 distinct locations (in $x$, $y$ and $z$ directions).
The goal here is to link these $4 \times 3 = 12$ measurements to the 6 DOFs of the solid body expressed in the frame $\{O\}$.
#+begin_src latex :file modal_local_to_global_coordinates.pdf
\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, fill=blue!30!white] (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]{$\vec{p}_1$} -- ++( 1 , 0.5)node[right]{$\delta \vec{p}_1$};
\draw[->] (p2)node[]{$\bullet$}node[above]{$\vec{p}_2$} -- ++(-0.5, 1 )node[right]{$\delta \vec{p}_2$};
\draw[->] (p3)node[]{$\bullet$}node[above]{$\vec{p}_3$} -- ++( 1 , 0.5)node[right]{$\delta \vec{p}_3$};
\draw[->] (p4)node[]{$\bullet$}node[above]{$\vec{p}_4$} -- ++( 0.5, 1 )node[right]{$\delta \vec{p}_4$};
\end{tikzpicture}
#+end_src
#+name: fig:modal_local_to_global_coordinates
#+caption: Schematic of the measured motions of a solid body
#+RESULTS:
[[file:figs/modal_local_to_global_coordinates.png]]
Let's consider the motion of the rigid body defined by its displacement $\delta \vec{p}$ and rotation $\delta\vec{\Omega}$ with respect to the reference frame $\{O\}$.
From the figure ref:fig:modal_local_to_global_coordinates, we can write:
\begin{align*}
\delta \vec{p}_1 &= \delta \vec{p} + \bm{\delta \Omega} \vec{p}_1\\
\delta \vec{p}_2 &= \delta \vec{p} + \bm{\delta \Omega} \vec{p}_2\\
\delta \vec{p}_3 &= \delta \vec{p} + \bm{\delta \Omega} \vec{p}_3\\
\delta \vec{p}_4 &= \delta \vec{p} + \bm{\delta \Omega} \vec{p}_4
\end{align*}
With
\begin{equation}
\bm{\delta\Omega} = \begin{bmatrix}
0 & -\delta\Omega_z & \delta\Omega_y \\
\delta\Omega_z & 0 & -\delta\Omega_x \\
-\delta\Omega_y & \delta\Omega_x & 0
\end{bmatrix}
\end{equation}
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] \left[\begin{array}{c}
\delta p_x \\ \delta p_y \\ \delta p_z \\ \hline \delta\Omega_x \\ \delta\Omega_y \\ \delta\Omega_z
\end{array}\right] = \left[\begin{array}{c}
\delta p_{1x} \\ \delta p_{1y} \\ \delta p_{1z} \\\hline \vdots \\\hline \delta p_{4x} \\ \delta p_{4y} \\ \delta p_{4z}
\end{array}\right]
\end{equation}
and then we obtain the velocity and rotation of the solid in the wanted frame $\{O\}$:
\begin{equation}
\left[\begin{array}{c}
\delta p_x \\ \delta p_y \\ \delta p_z \\ \hline \delta\Omega_x \\ \delta\Omega_y \\ \delta\Omega_z
\end{array}\right] =
\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} \left[\begin{array}{c}
\delta p_{1x} \\ \delta p_{1y} \\ \delta p_{1z} \\\hline \vdots \\\hline \delta p_{4x} \\ \delta p_{4y} \\ \delta p_{4z}
\end{array}\right] \label{eq:modal_determine_global_disp}
\end{equation}
#+begin_important
Using equation eqref:eq:modal_determine_global_disp, we can determine the motion of the solid body expressed in a chosen frame $\{O\}$ using the accelerometers attached to it.
The inversion is equivalent to resolving a mean square problem.
#+end_important
** Frequency Response Matrix expressed at the Center of Mass
<>
**** What reference frame to choose?
The question we wish here to answer is how to choose the reference frame $\{O\}$ in which the DOFs of the solid bodies are defined.
The possibles choices are:
- *One frame for each solid body* which is located at its center of mass
- *One common frame*, for instance located at the point of interest ($270mm$ above the Hexapod)
- *Base located at the joint position*: this is where we want to see the motion and estimate stiffness
#+name: tab:modal_frame_comparison
#+caption: Advantages and disadvantages for the choice of reference frame
#+attr_latex: :environment tabularx :width \linewidth :align XXX
#+attr_latex: :center t :booktabs t :font \scriptsize
| Chosen Frame | Advantages | Disadvantages |
|--------------------------+-----------------------------------------------------+------------------------------------------------------|
| Frames at CoM | Physically, it makes more sense | How to compare the motion of the solid bodies? |
| Common Frame | We can compare the motion of each solid body | Small $\theta_{x, y}$ may result in large $T_{x, y}$ |
| Frames at joint position | Directly gives which joint direction can be blocked | How to choose the joint position? |
The choice of the frame depends of what we want to do with the data.
One of the goals is to compare the motion of each solid body to see which relative DOFs between solid bodies can be neglected, that is to say, which joint between solid bodies can be regarded as perfect (and this in all the frequency range of interest).
Ideally, we would like to have the same number of degrees of freedom than the number of identified modes.
In the next sections, we will express the FRF matrix in the different frames.
**** Center of Mass of each solid body
From solidworks, we can export the position of the center of mass of each solid body considered.
These are summarized in Table ref:tab:modal_com_solid_bodies
#+begin_src matlab
%% Extract the CoM of considered solid bodies
model_com = reshape(table2array(readtable('mat/model_solidworks_com.txt', 'ReadVariableNames', false)), [3, 6]);
#+end_src
#+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*)
data2orgtable(1000*model_com', {'Bottom Granite', 'Top granite', 'Translation stage', 'Tilt Stage', 'Spindle', 'Hexapod'}, {'$X$ [mm]', '$Y$ [mm]', '$Z$ [mm]'}, ' %.0f ');
#+end_src
#+name: tab:modal_com_solid_bodies
#+caption: Center of mass of considered solid bodies
#+attr_latex: :environment tabularx :width 0.6\linewidth :align lXXX
#+attr_latex: :center t :booktabs t
#+RESULTS:
| | $X$ [mm] | $Y$ [mm] | $Z$ [mm] |
|-------------------+----------+----------+----------|
| Bottom Granite | 45 | 144 | -1251 |
| Top granite | 52 | 258 | -778 |
| Translation stage | 0 | 14 | -600 |
| Tilt Stage | 0 | -5 | -628 |
| Spindle | 0 | 0 | -580 |
| Hexapod | -4 | 6 | -319 |
**** Computation
First, we initialize a new FRF matrix which is an $n \times p \times q$ with:
- $n$ is the number of DOFs of the considered 6 solid-bodies: $6 \times 6 = 36$
- $p$ is the number of excitation inputs: $3$
- $q$ is the number of frequency points $\omega_i$
#+begin_important
For each frequency point $\omega_i$, the FRF matrix is a $n\times p$ matrix:
\begin{equation}
\text{FRF}_\text{CoM}(\omega_i) = \begin{bmatrix}
\frac{D_{1,T_x}}{F_x}(\omega_i) & \frac{D_{1,T_x}}{F_y}(\omega_i) & \frac{D_{1,T_x}}{F_z}(\omega_i) \\
\frac{D_{1,T_y}}{F_x}(\omega_i) & \frac{D_{1,T_y}}{F_y}(\omega_i) & \frac{D_{1,T_y}}{F_z}(\omega_i) \\
\frac{D_{1,T_z}}{F_x}(\omega_i) & \frac{D_{1,T_z}}{F_y}(\omega_i) & \frac{D_{1,T_z}}{F_z}(\omega_i) \\
\frac{D_{1,R_x}}{F_x}(\omega_i) & \frac{D_{1,R_x}}{F_y}(\omega_i) & \frac{D_{1,R_x}}{F_z}(\omega_i) \\
\frac{D_{1,R_y}}{F_x}(\omega_i) & \frac{D_{1,R_y}}{F_y}(\omega_i) & \frac{D_{1,R_y}}{F_z}(\omega_i) \\
\frac{D_{1,R_z}}{F_x}(\omega_i) & \frac{D_{1,R_z}}{F_y}(\omega_i) & \frac{D_{1,R_z}}{F_z}(\omega_i) \\
\frac{D_{2,T_x}}{F_x}(\omega_i) & \frac{D_{2,T_x}}{F_y}(\omega_i) & \frac{D_{2,T_x}}{F_z}(\omega_i) \\
\vdots & \vdots & \vdots \\
\frac{D_{6,R_z}}{F_x}(\omega_i) & \frac{D_{6,R_z}}{F_y}(\omega_i) & \frac{D_{6,R_z}}{F_z}(\omega_i)
\end{bmatrix}
\end{equation}
where 1, 2, ..., 6 corresponds to the 6 solid bodies.
#+end_important
Then, as we know the positions of the accelerometers on each solid body, and we have the response of those accelerometers, we can use the equations derived in the previous section to determine the response of each solid body expressed in their center of mass.
#+begin_src matlab
%% Frequency Response Matrix - Response expressed at the CoM of the solid bodies
FRFs_CoM = zeros(length(solid_names)*6, 3, 801);
for solid_i = 1:length(solid_names)
% Number of accelerometers fixed to this solid body
solids_i = solids.(solid_names{solid_i});
% "Jacobian" matrix to go from accelerometer frame to CoM frame
A = zeros(3*length(solids_i), 6);
for i = 1:length(solids_i)
acc_i = solids_i(i);
acc_pos_com = acc_pos(acc_i, :).' - model_com(:, solid_i);
A(3*(i-1)+1:3*i, 1:3) = eye(3);
A(3*(i-1)+1:3*i, 4:6) = [ 0 acc_pos_com(3) -acc_pos_com(2) ;
-acc_pos_com(3) 0 acc_pos_com(1) ;
acc_pos_com(2) -acc_pos_com(1) 0];
end
for exc_dir = 1:3
FRFs_CoM((solid_i-1)*6+1:solid_i*6, exc_dir, :) = A\squeeze(frf((solids_i(1)-1)*3+1:solids_i(end)*3, exc_dir, :));
end
end
#+end_src
#+begin_src matlab :eval no
%% Save the computed FRF at the CoM
save('mat/frf_com.mat', 'FRFs_CoM');
#+end_src
#+begin_src matlab :tangle no
%% Save the computed FRF at the CoM
save('matlab/mat/frf_com.mat', 'FRFs_CoM');
#+end_src
** Verify that we find the original FRF from the FRF in the global coordinates
<>
We have computed the Frequency Response Functions Matrix representing the response of the 6 solid bodies in their 6 DOFs with respect to their center of mass.
From the response of one body in its 6 DOFs, we should be able to compute the FRF of each of its accelerometer fixed to it during the measurement, supposing that this stage is a solid body.
We can then compare the result with the original measurements.
This will help us to determine if:
- the previous inversion used is correct
- the solid body assumption is correct in the frequency band of interest
From the translation $\delta p$ and rotation $\delta \Omega$ of a solid body and the positions $p_i$ of the accelerometers attached to it, we can compute the response that would have been measured by the accelerometers using the following formula:
\begin{align*}
\delta p_1 &= \delta p + \delta\Omega p_1\\
\delta p_2 &= \delta p + \delta\Omega p_2\\
\delta p_3 &= \delta p + \delta\Omega p_3\\
\delta p_4 &= \delta p + \delta\Omega p_4
\end{align*}
Thus, we can obtain the FRF matrix =FRFs_A= that gives the responses of the accelerometers to the forces applied by the hammer.
It is implemented in matlab as follow:
#+begin_src matlab
FRFs_A = zeros(size(frf));
% For each excitation direction
for exc_dir = 1:3
% For each solid
for solid_i = 1:length(solid_names)
v0 = squeeze(FRFs_CoM((solid_i-1)*6+1:(solid_i-1)*6+3, exc_dir, :));
W0 = squeeze(FRFs_CoM((solid_i-1)*6+4:(solid_i-1)*6+6, exc_dir, :));
% For each accelerometer attached to the current solid
for acc_i = solids.(solid_names{solid_i})
% We get the position of the accelerometer expressed in frame O
pos = acc_pos(acc_i, :).' - model_com(:, solid_i);
% pos = acc_pos(acc_i, :).';
posX = [0 pos(3) -pos(2); -pos(3) 0 pos(1) ; pos(2) -pos(1) 0];
FRFs_A(3*(acc_i-1)+1:3*(acc_i-1)+3, exc_dir, :) = v0 + posX*W0;
end
end
end
#+end_src
We then compare the original FRF measured for each accelerometer =FRFs= with the "recovered" FRF =FRFs_A= from the global FRF matrix in the common frame.
The FRF for the 4 accelerometers on the Hexapod are compared in Figure ref:fig:modal_comp_acc_solid_body_frf.
All the FRF are matching very well in all the frequency range displayed.
# The FRF for accelerometers located on the translation stage are compared on figure ref:fig:modal_recovered_frf_comparison_ty.
# The FRF are matching well until 100Hz.
#+begin_src matlab :exports none :results none
%% Comparaison of the original accelerometer response and reconstructed response from the solid body response
exc_names = {'$F_x$', '$F_y$', '$F_z$'};
DOFs = {'x', 'y', 'z', '\theta_x', '\theta_y', '\theta_z'};
solid_i = 6; % Considered solid body
exc_dir = 1; % Excited direction
accs_i = solids.(solid_names{solid_i}); % Accelerometers fixed to this solid body
figure;
tiledlayout(2, 2, 'TileSpacing', 'Compact', 'Padding', 'None');
for i = 1:length(accs_i)
acc_i = accs_i(i);
nexttile();
hold on;
for dir_i = 1:3
plot(freqs, abs(squeeze(frf(3*(acc_i-1)+dir_i, exc_dir, :))), '-', 'DisplayName', sprintf('$a_{%i,%s}$', acc_i, DOFs{dir_i}));
end
set(gca,'ColorOrderIndex',1)
for dir_i = 1:3
plot(freqs, abs(squeeze(FRFs_A(3*(acc_i-1)+dir_i, exc_dir, :))), '--', 'HandleVisibility', 'off');
end
hold off;
if i > 2
xlabel('Frequency [Hz]');
else
set(gca, 'XTickLabel',[]);
end
if rem(i, 2) == 1
ylabel('Amplitude [$\frac{m/s^2}{N}$]');
else
set(gca, 'YTickLabel',[]);
end
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlim([5, 200]); ylim([1e-6, 1e-1]);
legend('location', 'northwest', 'FontSize', 8, 'NumColumns', 1);
end
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/modal_comp_acc_solid_body_frf.pdf', 'width', 'full', 'height', 'tall');
#+end_src
#+name: fig:modal_comp_acc_solid_body_frf
#+caption: Comparaison of the original accelerometer response and reconstructed response from the solid body response. For accelerometers 1 to 4 corresponding to the hexapod.
#+RESULTS:
[[file:figs/modal_comp_acc_solid_body_frf.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 the stages are indeed behaving as a solid body in the frequency band of interest.
This valid the fact that a multi-body model can be used to represent the dynamics of the micro-station.
#+end_important
* Modal Analysis
:PROPERTIES:
:HEADER-ARGS:matlab+: :tangle matlab/modal_3_analysis.m
:END:
<>
** Introduction :ignore:
The goal here is to extract the modal parameters describing the modes of station being studied, namely:
- the eigen frequencies and the modal damping (eigen values)
- the mode shapes (eigen vectors)
This is done from the FRF matrix previously extracted from the measurements.
In order to do the modal parameter extraction, we first have to estimate the order of the modal model we want to obtain.
This corresponds to how many modes are present in the frequency band of interest.
In section ref:ssec:modal_number_of_modes, we will use the Singular Value Decomposition and the Modal Indication Function to estimate the number of modes.
The modal parameter extraction methods generally consists of *curve-fitting a theoretical expression for an individual FRF to the actual measured data*.
However, there are multiple level of complexity:
- works on a part of a single FRF curve
- works on a complete curve encompassing several resonances
- works on a set of many FRF plots all obtained from the same structure
The third method is the most complex but gives better results. This is the one we will use in section ref:ssec:modal_parameter_extraction.
From the modal model, it is possible to obtain a graphic display of the mode shapes (section ref:ssec:modal_mode_shapes).
In order to validate the quality of the modal model, we will synthesize the FRF matrix from the modal model and compare it with the FRF measured (section ref:ssec:modal_model_validity).
The modes of the structure are expected to be complex, however real modes are easier to work with when it comes to obtain a spatial model from the modal parameters.
** 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
#+begin_src matlab :tangle no :noweb yes
<>
#+end_src
#+begin_src matlab :eval no :noweb yes
<>
#+end_src
#+begin_src matlab :noweb yes
<>
#+end_src
** Determine the number of modes
<>
**** Singular Value Decomposition - Modal Indication Function
The Mode Indicator Functions are usually used on $n\times p$ FRF matrix where $n$ is a relatively large number of measurement DOFs and $p$ is the number of excitation DOFs, typically 3 or 4.
In these methods, the frequency dependent FRF matrix is subjected to a singular value decomposition analysis which thus yields a small number (3 or 4) of singular values, these also being frequency dependent.
These methods are used to *determine the number of modes* present in a given frequency range, to *identify repeated natural frequencies* and to pre-process the FRF data prior to modal analysis.
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
#+begin_important
The *Complex Mode Indicator Function* is defined simply by the SVD of the FRF (sub) matrix:
\begin{align*}
[H(\omega)]_{n\times p} &= [U(\omega)]_{n\times n} [\Sigma(\omega)]_{n\times p} [V(\omega)]_{p\times p}^H\\
[CMIF(\omega)]_{p\times p} &= [\Sigma(\omega)]_{p\times n}^T [\Sigma(\omega)]_{n\times p}
\end{align*}
#+end_important
We compute the Complex Mode Indicator Function.
The result is shown on Figure ref:fig:modal_indication_function.
#+begin_src matlab
%% Computation of the modal indication function
MIF = zeros(size(frf, 2), size(frf, 2), size(frf, 3));
for i = 1:length(freqs)
[~,S,~] = svd(frf(:, :, i));
MIF(:, :, i) = S'*S;
end
#+end_src
#+begin_src matlab :exports none :results none
%% Modal Indication Function
figure;
hold on;
for i = 1:size(MIF, 1)
plot(freqs, squeeze(MIF(i, i, :)));
end
hold off;
set(gca, 'Xscale', 'lin'); set(gca, 'Yscale', 'log');
xlabel('Frequency [Hz]'); ylabel('CMIF Amplitude');
xticks([0:20:200]);
xlim([0, 200]);
ylim([1e-6, 2e-2]);
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/modal_indication_function.pdf', 'width', 'wide', 'height', 'normal');
#+end_src
#+name: fig:modal_indication_function
#+caption: Modal Indication Function
#+RESULTS:
[[file:figs/modal_indication_function.png]]
**** Composite Response Function
An alternative is the Composite Response Function $HH(\omega)$ defined as the sum of all the measured FRF:
\begin{equation}
HH(\omega) = \sum_j\sum_kH_{jk}(\omega)
\end{equation}
Instead, we choose here to use the sum of the norms of the measured frf:
\begin{equation}
HH(\omega) = \sum_j\sum_k \left|H_{jk}(\omega) \right|
\end{equation}
The result is shown on figure ref:fig:modal_composite_reponse_function.
#+begin_src matlab :exports none :results none
%% Composite Response Function
figure;
hold on;
plot(freqs, squeeze(sum(sum(abs(frf)))), '-k');
hold off;
xlabel('Frequency [Hz]'); ylabel('Amplitude');
xlim([0, 200]);
xticks([0:20:200]);
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/modal_composite_reponse_function.pdf', 'width', 'wide', 'height', 'normal');
#+end_src
#+name: fig:modal_composite_reponse_function
#+caption: Composite Response Function
#+RESULTS:
[[file:figs/modal_composite_reponse_function.png]]
** Modal parameter extraction
<>
**** OROS - Modal software
Modal identification are done within the Modal software of OROS.
Several modal parameter extraction methods are available.
We choose to use the "broad band" method as it permits to identify the modal parameters using all the FRF curves at the same time.
It takes into account the fact the the properties of all the individual curves are related by being from the same structure: all FRF plots on a given structure should indicate the same values for the natural frequencies and damping factor of each mode.
Such method also have the advantage of producing a *unique and consistent model* as direct output.
In order to apply this method, we select the frequency range of interest and we give an estimate of how many modes are present.
Then, it shows a stabilization charts, such as the one shown on figure ref:fig:modal_stabilization_chart, where we have to manually select which modes to take into account in the modal model.
#+name: fig:modal_stabilization_chart
#+caption: Stabilization Chart
#+attr_latex: :width \linewidth
[[file:img/modal_software/stabilisation_chart.jpg]]
We can then run the modal analysis, and the software will identify the modal damping and mode shapes at the selected frequency modes.
**** Importation of the modal parameters on Matlab
The obtained modal parameters are:
- Resonance frequencies in Hertz
- Modal damping ratio in percentage
- (complex) Modes shapes for each measured DoF
- Modal A and modal B which are parameters important for further normalization
#+begin_src matlab
%% Load modal parameters
shapes_m = readtable('mat/mode_shapes.txt', 'ReadVariableNames', false); % [Sign / Real / Imag]
freqs_m = table2array(readtable('mat/mode_freqs.txt', 'ReadVariableNames', false)); % in [Hz]
damps_m = 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]
#+end_src
#+begin_src matlab
%% Guess the number of modes identified from the length of the imported data.
acc_n = 23; % Number of accelerometers
dir_n = 3; % Number of directions
dirs = 'XYZ';
mod_n = size(shapes_m,1)/acc_n/dir_n; % Number of modes
#+end_src
#+begin_src matlab
%% Mode shapes are split into 3 parts (direction plus sign, real part and imaginary part)
% we aggregate them into one array of complex numbers
T_sign = table2array(shapes_m(:, 1));
T_real = table2array(shapes_m(:, 2));
T_imag = table2array(shapes_m(:, 3));
mode_shapes = zeros(mod_n, dir_n, acc_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);
mode_shapes(mod_i, dir_i, acc_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 in Table ref:tab:modal_obtained_modes_freqs_damps.
#+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*)
data2orgtable([(1:length(freqs_m))', freqs_m, damps_m], {}, {'Mode', 'Frequency [Hz]', 'Damping [%]'}, ' %.1f ');
#+end_src
#+name: tab:modal_obtained_modes_freqs_damps
#+caption: Obtained eigen frequencies and modal damping
#+attr_latex: :environment tabularx :width 0.35\linewidth :align ccc
#+attr_latex: :center t :booktabs t :font \scriptsize
#+RESULTS:
| Mode | Frequency [Hz] | Damping [%] |
|------+----------------+-------------|
| 1 | 11.9 | 12.2 |
| 2 | 18.6 | 11.7 |
| 3 | 37.8 | 6.2 |
| 4 | 39.1 | 2.8 |
| 5 | 56.3 | 2.8 |
| 6 | 69.8 | 4.3 |
| 7 | 72.5 | 1.3 |
| 8 | 84.8 | 3.7 |
| 9 | 91.3 | 2.9 |
| 10 | 105.5 | 3.2 |
| 11 | 106.6 | 1.6 |
| 12 | 112.7 | 3.1 |
| 13 | 124.2 | 2.8 |
| 14 | 145.3 | 1.3 |
| 15 | 150.5 | 2.4 |
| 16 | 165.4 | 1.4 |
**** Theory
It seems that the modal analysis software makes the *assumption* of viscous damping for the model with which it tries to fit the FRF measurements.
If we note $N$ the number of modes identified, then there are $2N$ eigenvalues and eigenvectors given by the software:
\begin{align}
s_r &= \omega_r (-\xi_r + i \sqrt{1 - \xi_r^2}),\quad s_r^* \\
\{\psi_r\} &= \begin{Bmatrix} \psi_{1_x} & \psi_{2_x} & \dots & \psi_{23_x} & \psi_{1_y} & \dots & \psi_{1_z} & \dots & \psi_{23_z} \end{Bmatrix}^T, \quad \{\psi_r\}^*
\end{align}
for $r = 1, \dots, N$ where $\omega_r$ is the natural frequency and $\xi_r$ is the critical damping ratio for that mode.
**** Modal Matrices
We would like to arrange the obtained modal parameters into two modal matrices:
\[ \Lambda = \begin{bmatrix}
s_1 & & 0 \\
& \ddots & \\
0 & & s_N
\end{bmatrix}_{N \times N}; \quad \Psi = \begin{bmatrix}
& & \\
\{\psi_1\} & \dots & \{\psi_N\} \\
& &
\end{bmatrix}_{M \times N} \]
\[ \{\psi_i\} = \begin{Bmatrix} \psi_{i, 1_x} & \psi_{i, 1_y} & \psi_{i, 1_z} & \psi_{i, 2_x} & \dots & \psi_{i, 23_z} \end{Bmatrix}^T \]
$M$ is the number of DoF: here it is $23 \times 3 = 69$.
$N$ is the number of mode
#+begin_src matlab
eigen_val_M = diag(2*pi*freqs_m.*(-damps_m/100 + j*sqrt(1 - (damps_m/100).^2)));
eigen_vec_M = reshape(mode_shapes, [mod_n, acc_n*dir_n]).';
#+end_src
Each eigen vector is normalized: $\| \{\psi_i\} \|_2 = 1$
However, the eigen values and eigen vectors appears as complex conjugates:
\[ s_r, s_r^*, \{\psi\}_r, \{\psi\}_r^*, \quad r = 1, N \]
In the end, they are $2N$ eigen values.
We then build two extended eigen matrices as follow:
\[ \mathcal{S} = \begin{bmatrix}
s_1 & & & & & \\
& \ddots & & & 0 & \\
& & s_N & & & \\
& & & s_1^* & & \\
& 0 & & & \ddots & \\
& & & & & s_N^*
\end{bmatrix}_{2N \times 2N}; \quad \Phi = \begin{bmatrix}
& & & & &\\
\{\psi_1\} & \dots & \{\psi_N\} & \{\psi_1^*\} & \dots & \{\psi_N^*\} \\
& & & & &
\end{bmatrix}_{M \times 2N} \]
#+begin_src matlab
eigen_val_ext_M = blkdiag(eigen_val_M, conj(eigen_val_M));
eigen_vec_ext_M = [eigen_vec_M, conj(eigen_vec_M)];
#+end_src
We also build the Modal A and Modal B matrices:
\begin{equation}
A = \begin{bmatrix}
a_1 & & 0 \\
& \ddots & \\
0 & & a_N
\end{bmatrix}_{N \times N}; \quad B = \begin{bmatrix}
b_1 & & 0 \\
& \ddots & \\
0 & & b_N
\end{bmatrix}_{N \times N}
\end{equation}
With $a_i$ is the "Modal A" parameter linked to mode i.
#+begin_src matlab
modal_a_M = diag(complex(modal_a(:, 1), modal_a(:, 2)));
modal_b_M = diag(complex(modal_b(:, 1), modal_b(:, 2)));
modal_a_ext_M = blkdiag(modal_a_M, conj(modal_a_M));
modal_b_ext_M = blkdiag(modal_b_M, conj(modal_b_M));
#+end_src
"Modal A" and "modal B" are linked through the following formula:
\[ B = - A \Lambda \]
** Obtained Mode Shapes animations
<>
From the modal parameters, it is possible to show the modal shapes with an animation.
Examples are shown in Figures ref:fig:modal_mode_animations.
#+name: fig:modal_mode_animations
#+caption: Instrumentation used for the modal analysis
#+begin_figure
#+attr_latex: :caption \subcaption{\label{fig:modal_mode1_animation}Mode 1}
#+attr_latex: :options {\textwidth}
#+begin_subfigure
#+attr_latex: :width \linewidth
[[file:figs/modal_mode1_animation.jpg]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:modal_mode6_animation}Mode 6}
#+attr_latex: :options {\textwidth}
#+begin_subfigure
#+attr_latex: :width \linewidth
[[file:figs/modal_mode6_animation.jpg]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:modal_mode13_animation}Mode 13}
#+attr_latex: :options {\textwidth}
#+begin_subfigure
#+attr_latex: :width \linewidth
[[file:figs/modal_mode13_animation.jpg]]
#+end_subfigure
#+end_figure
We can learn quite a lot from these mode shape animations.
For instance, the mode shape of the first mode at 11Hz (figure ref:fig:modal_mode1_animation) 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 ref:fig:modal_airloc).
#+name: fig:modal_airloc
#+caption: AirLoc used for the granite (2120-KSKC)
#+attr_latex: :width 0.6\linewidth
[[file:figs/modal_airlock_picture.jpg]]
They are probably *not well leveled*, so the granite is supported only by two Airloc.
** Verify the validity of the Modal Model
<>
There are two main ways to verify the validity of the modal model
- Synthesize FRF measurements that has been used to generate the modal model and compare
- Synthesize FRF that has not yet been measured. Then measure that FRF and compare
From the modal model, we want to synthesize the Frequency Response Functions that has been used to build the modal model.
Let's recall that:
- $M$ is the number of measured DOFs ($3 \times n_\text{acc}$)
- $N$ is the number of modes identified
We then have that the FRF matrix $[H_{\text{syn}}]$ can be synthesize using the following formula:
#+begin_important
\begin{equation}
[H_{\text{syn}}(\omega)]_{M\times M} = [\Phi]_{M\times2N} \left[\frac{Q_r}{j\omega - s_r}\right]_{2N\times2N} [\Phi]_{2N\times M}^T
\end{equation}
with $Q_r = 1/M_{A_r}$
#+end_important
An alternative formulation is:
\[ H_{pq}(s_i) = \sum_{r=1}^N \frac{A_{pqr}}{s_i - \lambda_r} + \frac{A_{pqr}^*}{s_i - \lambda_r^*} \]
with:
- $A_{pqr} = \frac{\psi_{pr}\psi_{qr}}{M_{A_r}}$, $M_{A_r}$ is called "Modal A"
- $\psi_{pr}$: scaled modal coefficient for output DOF $p$, mode $r$
- $\lambda_r$: complex modal frequency
From the modal software documentation:
#+begin_quote
*Modal A*
Scaling constant for a complex mode. It has the same properties as modal mass for normal modes (undamped or proportionally damped cases). Assuming
- $\psi_{pr}$ = Modal coefficient for measured degree of freedom p and mode r
- $\psi_{qr}$ = Modal coefficient for measured degree of freedom q and mode r
- $A_{pqr}$ = Residue for measured degree of freedom p, measured degree of q and mode r
- $M_{Ar}$ = Modal A of mode r
Then
\[ A_{pqr} = \frac{\psi_{pr}\psi_{qr}}{M_{Ar}} \]
*Modal B*
Scaling constant for a complex mode. It has the same properties as modal stiffness for normal modes (undamped or proportionally damped cases). Assuming
- $M_{Ar}$ = Modal A of mode r
- $\lambda_r$ = System pole of mode r
Then
\[ M_{Br} = - \lambda_r M_{Ar} \]
#+end_quote
#+begin_src matlab
%% Synthesize FRF from the modal model
Hsyn = zeros(acc_n*dir_n, acc_n*dir_n, length(freqs));
for i = 1:length(freqs)
Hsyn(:, :, i) = eigen_vec_ext_M*((j*2*pi*freqs(i)).^2*inv(modal_a_ext_M)/(diag(j*2*pi*freqs(i) - diag(eigen_val_ext_M))))*eigen_vec_ext_M.';
end
%% Derivate two times to to have the acceleration response
for i = 1:size(Hsyn, 1)
Hsyn(i, :, :) = squeeze(Hsyn(i, :, :)).*(j*2*pi*freqs).^2;
end
#+end_src
The comparison between the original measured frequency response function and the synthesized one from the modal model is done in Figure ref:fig:modal_comp_acc_frf_modal.
# Check if I have the compliance of the micro-station, could be useful to compare and verify the quality of the model
# TODO - say it will be useful when validating the multi-body model
#+begin_src matlab :exports none
acc_o = 1; dir_o = 1; dir_i = 1;
figure;
ax1 = subplot(2, 1, 1);
hold on;
plot(freqs, abs(squeeze(frf(3*(acc_o-1)+dir_o, dir_i, :))), 'DisplayName', 'Original');
plot(freqs, abs(squeeze(Hsyn(3*(acc_o-1)+dir_o, 3*(11-1)+dir_i, :))), 'DisplayName', 'Synthesize');
hold off;
set(gca, 'yscale', 'log');
set(gca, 'XTickLabel',[]);
ylabel('Magnitude [$\frac{m/s^2}{N}$]');
title(sprintf('From acc %i %s to acc %i %s', 11, dirs(dir_i), acc_o, dirs(dir_o)))
legend('location', 'northwest');
ax2 = subplot(2, 1, 2);
hold on;
plot(freqs, mod(180/pi*phase(squeeze(frf(3*(acc_o-1)+dir_o, dir_i, :)))+180, 360)-180);
plot(freqs, mod(180/pi*phase(squeeze(Hsyn(3*(acc_o-1)+dir_o, 3*(11-1)+dir_i, :)))+180, 360)-180);
hold off;
yticks(-360:90:360); ylim([-180, 180]);
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
linkaxes([ax1,ax2],'x');
xlim([1, 200]);
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/modal_comp_acc_frf_modal.pdf', 'width', 'wide', 'height', 'tall');
#+end_src
#+name: fig:modal_comp_acc_frf_modal
#+caption: description
#+RESULTS:
[[file:figs/modal_comp_acc_frf_modal.png]]
# #+name: fig:from11xto1x
# #+caption: Original FRF and synthesize FRF using the modal software. From force applied in the X direction to the acceleration of accelerometer 11 in the X direction
# #+attr_latex: :width \linewidth
# [[file:img/modal_software/from11xto1x.jpg]]
# #+begin_warning
# The synthesize frf from the modal software do not match the synthesize frf computed here.
# It is possible that it uses residues that are not exported? Nothing is said about that in the documentation.
# #+end_warning
Frequency response functions that has not been measured can be synthesized as shown in Figure ref:fig:modal_synthesized_frf.
#+begin_src matlab :exports none :results none
%% description
accs = [1]; dirs = [1:3];
figure;
ax1 = subplot(2, 1, 1);
hold on;
for acc_i = accs
for dir_i = dirs
plot(freqs, abs((1./(j*2*pi*freqs').^2).*squeeze(Hsyn(3*(acc_i-1)+dir_i, 3*(acc_i-1)+dir_i, :))), 'DisplayName', sprintf('Acc %i - %s', acc_i, dirs(dir_i)));
end
end
hold off;
set(gca, 'xscale', 'log'); set(gca, 'yscale', 'log');
set(gca, 'XTickLabel',[]);
ylabel('Magnitude [$\frac{m}{N}$]');
legend('location', 'southwest');
ax2 = subplot(2, 1, 2);
hold on;
for acc_i = accs
for dir_i = dirs
plot(freqs, mod(180/pi*phase((1./(j*2*pi*freqs').^2).*squeeze(Hsyn(3*(acc_i-1)+dir_i, 3*(acc_i-1)+dir_i, :)))+180, 360)-180);
end
end
hold off;
yticks(-360:90:360); ylim([-180, 180]);
set(gca, 'xscale', 'log');
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
linkaxes([ax1,ax2],'x');
xlim([1, 200]);
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/modal_synthesized_frf.pdf', 'width', 'wide', 'height', 'tall');
#+end_src
#+name: fig:modal_synthesized_frf
#+caption: description
#+RESULTS:
[[file:figs/modal_synthesized_frf.png]]
* Conclusion
<>
Validation of solid body model.
Further step: go from modal model to parameters of the solid body model.
* Bibliography :ignore:
#+latex: \printbibliography[heading=bibintoc,title={Bibliography}]
* Helping Functions :noexport:
** Initialize Path
#+NAME: m-init-path
#+BEGIN_SRC matlab
%% Path for functions, data and scripts
addpath('./matlab/mat/'); % Path for data
addpath('./matlab/'); % Path for scripts
#+END_SRC
#+NAME: m-init-path-tangle
#+BEGIN_SRC matlab
%% Path for functions, data and scripts
addpath('./mat/'); % Path for data
#+END_SRC
** Initialize other elements
#+NAME: m-init-other
#+BEGIN_SRC matlab
%% Colors for the figures
colors = colororder;
#+END_SRC