#+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:
#+HTML_HEAD:
#+HTML_HEAD:
#+HTML_HEAD:
#+HTML_HEAD:
#+HTML_HEAD:
#+HTML_HEAD:
#+HTML_MATHJAX: align: center tagside: right font: TeX
#+PROPERTY: header-args:matlab :session *MATLAB*
#+PROPERTY: header-args:matlab+ :comments org
#+PROPERTY: header-args:matlab+ :results none
#+PROPERTY: header-args:matlab+ :exports both
#+PROPERTY: header-args:matlab+ :eval no-export
#+PROPERTY: header-args:matlab+ :output-dir figs
#+PROPERTY: header-args:shell :eval no-export
:END:
* Goal
The goal is to experimentally extract a *Spatial Model* (mass, damping, stiffness) of the structure (shown on figure [[fig:nass_picture]]) in order to tune the Multi-Body model.
#+name: fig:nass_picture
#+caption: Picture of the ID31 Micro-Station. (1) Granite (2) Translation Stage (3) Tilt Stage (4) Hexapod (5) Dummy Mass
#+attr_html: :width 500px
[[file:img/nass_picture.png]]
The procedure is represented on figure [[fig:vibration_analysis_procedure]] where we go from left to right.
#+name: fig:vibration_analysis_procedure
#+caption: Vibration Analysis Procedure
#+attr_html: :width 400px
[[file:img/vibration_analysis_procedure.png]]
First, we obtain a *Response Model* (Frequency Response Functions) from measurements.
This is further converted into a *Modal Model* (Natural Frequencies and Mode Shapes).
Finally, this is converted into a *Spatial Model* with the Mass/Damping/Stiffness matrices.
Theses matrices will be used to tune the Simscape (multi-body) model.
The modes we want to identify are those in the frequency range between 0Hz and 150Hz.
* Type of Model
The model that we want to obtain is a *multi-body model*.
It is composed of several *solid bodies connected with springs and dampers*.
The solid bodies are represented with different colors on figure [[fig:nass_solidworks]].
In the simscape model, the solid bodies are:
- the granite (1 or 2 solids)
- the translation stage
- the tilt stage
- the spindle and slip-ring
- the hexapod
#+name: fig:nass_solidworks
#+caption: CAD view of the ID31 Micro-Station
#+attr_html: :width 800px
[[file:img/nass_solidworks.png]]
However, each of the DOF of the system may not be relevant for the modes present in the frequency band of interest.
For instance, the translation stage may not vibrate in the Z direction for all the modes identified. Then, we can block this DOF and this simplifies the model.
The modal identification done here will thus permit us to determine *which DOF can be neglected*.
* Instrumentation Used
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 [[fig:oros]])
- 3 tri-axis *Accelerometers* (figure [[fig:accelero_M393B05]]) with parameters shown on table [[tab:accelero_M393B05]]
- An *Instrumented Hammer* with various Tips (figure [[fig:instrumented_hammer]]) (figure [[fig:hammer_tips]])
#+name: fig:oros
#+caption: Acquisition system: OROS
#+attr_html: :width 500px
[[file:img/instrumentation/oros.png]]
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: fig:accelero_M393B05
#+caption: Accelerometer used: M393B05
#+attr_html: :width 500px
[[file:img/instrumentation/accelero_M393B05.png]]
#+name: tab:accelero_M393B05
#+caption: 393B05 Accelerometer Data Sheet
| Sensitivity | 10V/g |
| Measurement Range | 0.5 g pk |
| Broadband Resolution | 0.000004 g rms |
| Frequency Range | 0.7 to 450Hz |
| Resonance Frequency | > 2.5kHz |
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.
#+name: fig:instrumented_hammer
#+caption: Instrumented Hammer
#+attr_html: :width 500px
[[file:img/instrumentation/instrumented_hammer.png]]
#+name: fig:hammer_tips
#+caption: Hammer tips
#+attr_html: :width 500px
[[file:img/instrumentation/hammer_tips.png]]
The accelerometers are glued on the structure.
* Structure Preparation and Test Planning
** Structure Preparation
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.
** Test Planing
The goal is to identify the full $N \times N$ FRF matrix (where $N$ is the number of degree of freedom of the system).
However, the principle of reciprocity states that:
\[ H_{jk} = \frac{X_j}{F_k} = H_{kj} = \frac{X_k}{F_j} \]
Thus, only one column or one line of the matrix has to be identified.
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 some stages than to excite this particular stage with the Hammer
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
** 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*.
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 (figure [[fig:accelerometers_granite2_overview]])
- 4 on top of the translation stage (figure [[fig:accelerometers_ty_overview]])
- 4 on top of the tilt stage
- 4 on top of the spindle
- 4 on top of the hexapod (figure [[fig:accelerometers_hexa_overview]])
#+name: fig:accelerometers_granite2_overview
#+caption: Accelerometers located on the top granite
#+attr_html: :width 500px
[[file:img/accelerometers/accelerometers_granite2_overview.jpg]]
#+name: fig:accelerometers_ty_overview
#+caption: Accelerometers located on top of the translation stage
#+attr_html: :width 500px
[[file:img/accelerometers/accelerometers_ty_overview.jpg]]
#+name: fig:accelerometers_hexa_overview
#+caption: Accelerometers located on the Hexapod
#+attr_html: :width 500px
[[file:img/accelerometers/accelerometers_hexa_overview.jpg]]
** Hammer Impacts
Only 3 impact points are used.
The impact points are shown on figures [[fig:hammer_x]], [[fig:hammer_y]] and [[fig:hammer_z]].
#+name: fig:hammer_x
#+caption: Hammer Blow in the X direction
#+attr_html: :width 300px
[[file:img/impacts/hammer_x.gif]]
#+name: fig:hammer_y
#+caption: Hammer Blow in the Y direction
#+attr_html: :width 300px
[[file:img/impacts/hammer_y.gif]]
#+name: fig:hammer_z
#+caption: Hammer Blow in the Z direction
#+attr_html: :width 300px
[[file:img/impacts/hammer_z.gif]]
* Signal Processing
The measurements are averaged 10 times (figure [[fig:general_parameters]]) corresponding to 10 hammer impacts.
#+name: fig:general_parameters
#+caption: General Acquisition Settings
#+attr_html: :width 500px
[[file:img/parameters/general_parameters.jpg]]
Windowing is used on the force response signals.
A boxcar window (figure [[fig:window_force]]) is used for the force signal as once the impact on the structure is done, the measured signal is meaningless.
An exponential window (figure [[fig:window_response]]) 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.
#+name: fig:window_force
#+caption: Window used for the force signal
#+attr_html: :width 500px
[[file:img/parameters/window_force.jpg]]
#+name: fig:window_response
#+caption: Window used for the response signal
#+attr_html: :width 500px
[[file:img/parameters/window_response.jpg]]
* Frequency Response Functions and Coherence Results
** 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
** Load Data
#+begin_src matlab
meas1_raw = load('modal_analysis/raw_data/Measurement1.mat');
#+end_src
** Raw Force Data
#+begin_src matlab :exports none
time = linspace(0, meas1_raw.Track1_X_Resolution*length(meas1_raw.Track1), length(meas1_raw.Track1));
figure;
plot(time, meas1_raw.Track1);
xlabel('Time [s]');
ylabel('Force [N]');
#+end_src
#+HEADER: :tangle no :exports results :results none :noweb yes
#+begin_src matlab :var filepath="figs/raw_data_force.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
<>
#+end_src
#+NAME: fig:raw_data_force
#+CAPTION: Raw Force Data from Hammer Blow
[[file:figs/raw_data_force.png]]
#+begin_src matlab :exports none
xlim([22.1, 22.3]);
#+end_src
#+HEADER: :tangle no :exports results :results none :noweb yes
#+begin_src matlab :var filepath="figs/raw_data_force_zoom.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
<>
#+end_src
#+NAME: fig:raw_data_force_zoom
#+CAPTION: Raw Force Data from Hammer Blow - Zoom
[[file:figs/raw_data_force_zoom.png]]
** Raw Response Data
#+begin_src matlab :exports none
figure;
plot(time, meas1_raw.Track2);
xlabel('Time [s]');
ylabel('Acceleration [m/s2]');
#+end_src
#+HEADER: :tangle no :exports results :results none :noweb yes
#+begin_src matlab :var filepath="figs/raw_data_acceleration.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
<>
#+end_src
#+NAME: fig:raw_data_acceleration
#+CAPTION: Raw Acceleration Data from Accelerometer
[[file:figs/raw_data_acceleration.png]]
#+begin_src matlab :exports none
xlim([22.1, 22.5]);
#+end_src
#+HEADER: :tangle no :exports results :results none :noweb yes
#+begin_src matlab :var filepath="figs/raw_data_acceleration_zoom.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
<>
#+end_src
#+NAME: fig:raw_data_acceleration_zoom
#+CAPTION: Raw Acceleration Data from Accelerometer - Zoom
[[file:figs/raw_data_acceleration_zoom.png]]
** Load Data
#+begin_src matlab
meas1 = load('modal_analysis/frf_coh/Measurement1.mat');
#+end_src
** FRF and Coherence Results
#+begin_src matlab :exports none
figure;
ax1 = subplot(2, 1, 1);
plot(meas1.FFT1_AvSpc_2_RMS_X_Val, meas1.FFT1_AvXSpc_2_1_RMS_Y_Mod);
set(gca, 'XTickLabel',[]);
ylabel('Magnitude');
ax2 = subplot(2, 1, 2);
plot(meas1.FFT1_AvSpc_2_RMS_X_Val, meas1.FFT1_AvXSpc_2_1_RMS_Y_Phas);
ylim([-180, 180]);
yticks([-180, -90, 0, 90, 180]);
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
linkaxes([ax1,ax2],'x');
#+end_src
#+HEADER: :tangle no :exports results :results none :noweb yes
#+begin_src matlab :var filepath="figs/frf_result_example.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
<>
#+end_src
#+NAME: fig:frf_result_example
#+CAPTION: Example of one measured FRF
[[file:figs/frf_result_example.png]]
#+begin_src matlab :exports none
figure;
plot(meas1.FFT1_AvSpc_2_RMS_X_Val, meas1.FFT1_Coh_2_1_RMS_Y_Val);
xlabel('Frequency [Hz]');
ylabel('Coherence');
#+end_src
#+HEADER: :tangle no :exports results :results none :noweb yes
#+begin_src matlab :var filepath="figs/coh_result_example.pdf" :var figsize="wide-normal" :post pdf2svg(file=*this*, ext="png")
<>
#+end_src
#+NAME: fig:coh_result_example
#+CAPTION: Example of one measured Coherence
[[file:figs/coh_result_example.png]]
* Mode Shapes
Multiple modal extraction techniques can be used (SIMO, MIMO, narrow band, wide band, ...).
First preliminary results on 10 identified modes are presented here.
#+name: fig:mode1
#+caption: Mode 1
[[file:img/modes/mode1.gif]]
#+name: fig:mode2
#+caption: Mode 2
[[file:img/modes/mode2.gif]]
#+name: fig:mode3
#+caption: Mode 3
[[file:img/modes/mode3.gif]]
#+name: fig:mode4
#+caption: Mode 4
[[file:img/modes/mode4.gif]]
#+name: fig:mode5
#+caption: Mode 5
[[file:img/modes/mode5.gif]]
#+name: fig:mode6
#+caption: Mode 6
[[file:img/modes/mode6.gif]]
#+name: fig:mode7
#+caption: Mode 7
[[file:img/modes/mode7.gif]]
#+name: fig:mode8
#+caption: Mode 8
[[file:img/modes/mode8.gif]]
#+name: fig:mode9
#+caption: Mode 9
[[file:img/modes/mode9.gif]]
#+name: fig:mode10
#+caption: Mode 10
[[file:img/modes/mode10.gif]]
* Obtained Modal Matrices
From the modal analysis software, we can export the obtained *eigen matrices*:
\[ \begin{bmatrix}
\omega_1^2 & & 0 \\
& \ddots & \\
0 & & \omega_n^2
\end{bmatrix}; \quad \Psi = \begin{bmatrix}
& & \\
\{\psi_1\} & \dots & \{\psi_n\} \\
& &
\end{bmatrix} \]
where $\bar{\omega}_r^2$ is the $r^\text{th}$ eigenvalue squared and $\{\phi\}_r$ is a description of the corresponding *mode shape*.
The file containing the eigen frequencies and mode shapes are shown below (for the first mode).
#+begin_src bash :results output :exports results :eval no-export
sed 80q modal_analysis/modes_propres_narband.asc
#+end_src
#+RESULTS:
#+begin_example
Created by N-Modal
Estimator: cmif
18-Jun-19 16:31:25
Mode 1
freq = 11.11191Hz
damp = 10.51401%
modal A = 8.52879e+003-2.29043e+003i
modal B = -9.64203e+004-6.08978e+005i
Mode matrix of local coordinate [DOF: Re IM]
1X+: -9.34637e-002 4.52445e-002
1Y+: 2.33790e-001 1.41439e-003
1Z+: -1.73754e-002 6.02449e-003
2X+: -7.42108e-002 3.91543e-002
2Y+: 2.41566e-001 -1.44869e-003
2Z+: -5.99285e-003 2.10370e-003
4X+: -1.02163e-001 2.79561e-002
4Y+: 2.29048e-001 2.89782e-002
4Z+: -2.85130e-002 1.77132e-004
5X+: -8.77132e-002 3.34081e-002
5Y+: 2.14182e-001 2.14655e-002
5Z+: -1.54521e-002 1.26682e-002
6X+: -7.90143e-002 2.42583e-002
6Y+: 2.20669e-001 2.12738e-002
6Z+: 4.60755e-002 4.96406e-003
7X+: -7.79654e-002 2.58385e-002
7Y+: 2.06861e-001 3.48019e-002
7Z+: -1.78311e-002 -1.29704e-002
8X+: -8.49357e-002 3.55200e-002
8Y+: 2.07470e-001 3.59745e-002
8Z+: -7.66974e-002 -3.19813e-003
9X+: -7.38565e-002 1.95146e-002
9Y+: 2.17403e-001 2.01550e-002
9Z+: -1.77073e-002 -3.46414e-003
10X+: -7.77587e-002 2.36700e-002
10Y+: 2.35654e-001 -2.14540e-002
10Z+: 7.94165e-002 -2.45897e-002
11X+: -8.17972e-002 2.20583e-002
11Y+: 2.20906e-001 -4.30164e-003
11Z+: -5.60520e-003 3.10187e-003
12X+: -8.64261e-002 3.66022e-002
12Y+: 2.15000e-001 -5.74661e-003
12Z+: -1.22622e-001 4.11767e-002
13X+: -4.25169e-002 1.56602e-002
13Y+: 5.31036e-002 -1.73951e-002
13Z+: -4.07130e-002 1.26884e-002
14X+: -3.85032e-002 1.29431e-002
14Y+: 5.36716e-002 -1.80868e-002
14Z+: 1.00367e-001 -3.48798e-002
15X+: -4.25524e-002 1.46363e-002
15Y+: 5.19668e-002 -1.69744e-002
15Z+: 5.89747e-003 -2.32428e-003
16X+: -4.31268e-002 1.38332e-002
16Y+: 5.07545e-002 -1.53045e-002
16Z+: -1.04172e-001 3.17984e-002
17X+: -2.69757e-002 9.07955e-003
17Y+: 3.07837e-002 -9.44663e-003
17Z+: -7.63502e-003 1.68203e-003
18X+: -3.00097e-002 9.23966e-003
18Y+: 2.83585e-002 -8.97747e-003
18Z+: 1.52467e-001 -4.78675e-002
19X+: -2.70223e-002 6.16478e-003
19Y+: 3.06149e-002 -6.25382e-003
19Z+: -4.84888e-003 1.93970e-003
20X+: -2.90976e-002 7.13184e-003
20Y+: 3.36738e-002 -7.30875e-003
20Z+: -1.66902e-001 3.93419e-002
3X+: -9.40720e-002 3.93724e-002
3Y+: 2.52307e-001 0.00000e+000
3Z+: -1.53864e-002 -9.25720e-004
21X+: -7.91940e-002 4.39648e-002
21Y+: 2.04567e-001 9.49987e-003
21Z+: -1.56087e-002 7.08838e-003
22X+: -1.01070e-001 3.13534e-002
22Y+: 1.92270e-001 1.80423e-002
22Z+: 2.93053e-003 -1.97308e-003
23X+: -8.86455e-002 4.29906e-002
23Z+: -3.38351e-002 1.81362e-003
23Y-: -1.90862e-001 -2.53414e-002
#+end_example
* Compute the Modal Model
** Matlab Init :noexport:ignore:
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
<>
#+end_src
#+begin_src matlab :exports none :results silent :noweb yes
<>
#+end_src
** Position of the accelerometers
There are 23 accelerometers:
- 4 on the bottom granite
- 4 on the top granite
- 4 on top of the translation stage
- 4 on the tilt stage
- 3 on top of the spindle
- 4 on top of the hexapod
The coordinates defined in the software are displayed below.
#+begin_src bash :results output :exports results :eval no-export
sed -n 18,40p modal_analysis/acc_coordinates.txt | tac --
#+end_src
#+RESULTS:
#+begin_example
1 1.0000e-001 1.0000e-001 1.1500e+000 0 Top
2 1.0000e-001 -1.0000e-001 1.1500e+000 0 Top
3 -1.0000e-001 -1.0000e-001 1.1500e+000 0 Top
4 -1.0000e-001 1.0000e-001 1.1500e+000 0 Top
5 4.0000e-001 4.0000e-001 9.5000e-001 0 inner
6 4.0000e-001 -4.0000e-001 9.5000e-001 0 inner
7 -4.0000e-001 -4.0000e-001 9.5000e-001 0 inner
8 -4.0000e-001 4.0000e-001 9.5000e-001 0 inner
9 5.0000e-001 5.0000e-001 9.0000e-001 0 outer
10 5.0000e-001 -5.0000e-001 9.0000e-001 0 outer
11 -5.0000e-001 -5.0000e-001 9.0000e-001 0 outer
12 -5.0000e-001 5.0000e-001 9.0000e-001 0 outer
13 5.5000e-001 5.5000e-001 5.5000e-001 0 top
14 5.5000e-001 -5.5000e-001 5.5000e-001 0 top
15 -5.5000e-001 -5.5000e-001 5.5000e-001 0 top
16 -5.5000e-001 5.5000e-001 5.5000e-001 0 top
17 9.5000e-001 9.5000e-001 4.0000e-001 0 low
18 9.5000e-001 -9.5000e-001 4.0000e-001 0 low
19 -9.5000e-001 -9.5000e-001 4.0000e-001 0 low
20 -9.5000e-001 9.5000e-001 4.0000e-001 0 low
21 2.0000e-001 2.0000e-001 8.5000e-001 0 bot
22 0.0000e+000 -2.0000e-001 8.5000e-001 0 bot
23 -2.0000e-001 2.0000e-001 8.5000e-001 0 bot
#+end_example
#+name: tab:acc_location
#+caption: Location of each Accelerometer (using the normal coordinate frame with X aligned with the X ray)
| *Node number* | *Solid Body* | *Location* | *X* | *Y* | *Z* |
|---------------+-------------------+------------+-------+-------+------|
| 1 | Hexapod - Top | -X/-Y | -0.10 | -0.10 | 1.15 |
| 2 | | -X/+Y | -0.10 | 0.10 | 1.15 |
| 3 | | +X/+Y | 0.10 | 0.10 | 1.15 |
| 4 | | +X/-Y | 0.10 | -0.10 | 1.15 |
|---------------+-------------------+------------+-------+-------+------|
| 5 | Tilt - Top | -X/-Y | -0.40 | -0.40 | 0.95 |
| 6 | | -X/+Y | -0.40 | 0.40 | 0.95 |
| 7 | | +X/+Y | 0.40 | 0.40 | 0.95 |
| 8 | | +X/-Y | 0.40 | -0.40 | 0.95 |
|---------------+-------------------+------------+-------+-------+------|
| 9 | Translation - Top | -X/-Y | -0.50 | -0.50 | 0.90 |
| 10 | | -X/+Y | -0.50 | 0.50 | 0.90 |
| 11 | | +X/+Y | 0.50 | 0.50 | 0.90 |
| 12 | | +X/-Y | 0.50 | -0.50 | 0.90 |
|---------------+-------------------+------------+-------+-------+------|
| 13 | Top Granite | -X/-Y | -0.55 | -0.50 | 0.55 |
| 14 | | -X/+Y | -0.55 | 0.50 | 0.55 |
| 15 | | +X/+Y | 0.55 | 0.50 | 0.55 |
| 16 | | +X/-Y | 0.55 | -0.50 | 0.55 |
|---------------+-------------------+------------+-------+-------+------|
| 17 | Bottom Granite | -X/-Y | -0.95 | -0.90 | 0.40 |
| 18 | | -X/+Y | -0.95 | 0.90 | 0.40 |
| 19 | | +X/+Y | 0.95 | 0.90 | 0.40 |
| 20 | | +X/-Y | 0.95 | -0.90 | 0.40 |
|---------------+-------------------+------------+-------+-------+------|
| 21 | Spindle - Top | -X/-Y | -0.20 | -0.20 | 0.85 |
| 22 | | +0/+Y | 0.00 | 0.20 | 0.85 |
| 23 | | +X/-Y | 0.20 | -0.20 | 0.85 |
** Define positions of the accelerometers on matlab
We define the X-Y-Z position of each sensor.
Each line corresponds to one accelerometer, X-Y-Z position in meter.
#+begin_src matlab
positions = [...
-0.10, -0.10, 1.15 ; ...
-0.10, 0.10, 1.15 ; ...
0.10, 0.10, 1.15 ; ...
0.10, -0.10, 1.15 ; ...
-0.40, -0.40, 0.95 ; ...
-0.40, 0.40, 0.95 ; ...
0.40, 0.40, 0.95 ; ...
0.40, -0.40, 0.95 ; ...
-0.50, -0.50, 0.90 ; ...
-0.50, 0.50, 0.90 ; ...
0.50, 0.50, 0.90 ; ...
0.50, -0.50, 0.90 ; ...
-0.55, -0.50, 0.55 ; ...
-0.55, 0.50, 0.55 ; ...
0.55, 0.50, 0.55 ; ...
0.55, -0.50, 0.55 ; ...
-0.95, -0.90, 0.40 ; ...
-0.95, 0.90, 0.40 ; ...
0.95, 0.90, 0.40 ; ...
0.95, -0.90, 0.40 ; ...
-0.20, -0.20, 0.85 ; ...
0.00, 0.20, 0.85 ; ...
0.20, -0.20, 0.85 ];
#+end_src
#+begin_src matlab
figure;
hold on;
fill3(positions(1:4, 1), positions(1:4, 2), positions(1:4, 3), 'k', 'FaceAlpha', 0.5)
fill3(positions(5:8, 1), positions(5:8, 2), positions(5:8, 3), 'k', 'FaceAlpha', 0.5)
fill3(positions(9:12, 1), positions(9:12, 2), positions(9:12, 3), 'k', 'FaceAlpha', 0.5)
fill3(positions(13:16, 1), positions(13:16, 2), positions(13:16, 3), 'k', 'FaceAlpha', 0.5)
fill3(positions(17:20, 1), positions(17:20, 2), positions(17:20, 3), 'k', 'FaceAlpha', 0.5)
fill3(positions(21:23, 1), positions(21:23, 2), positions(21:23, 3), 'k', 'FaceAlpha', 0.5)
hold off;
#+end_src
** Import the modal vectors on matlab
*** Mode1
#+begin_src bash :results output :exports none :eval no-export
sed -n 12,80p modal_analysis/modes_propres_narband.asc
#+end_src
#+RESULTS:
#+begin_example
1X+: -9.34637e-002 4.52445e-002
1Y+: 2.33790e-001 1.41439e-003
1Z+: -1.73754e-002 6.02449e-003
2X+: -7.42108e-002 3.91543e-002
2Y+: 2.41566e-001 -1.44869e-003
2Z+: -5.99285e-003 2.10370e-003
4X+: -1.02163e-001 2.79561e-002
4Y+: 2.29048e-001 2.89782e-002
4Z+: -2.85130e-002 1.77132e-004
5X+: -8.77132e-002 3.34081e-002
5Y+: 2.14182e-001 2.14655e-002
5Z+: -1.54521e-002 1.26682e-002
6X+: -7.90143e-002 2.42583e-002
6Y+: 2.20669e-001 2.12738e-002
6Z+: 4.60755e-002 4.96406e-003
7X+: -7.79654e-002 2.58385e-002
7Y+: 2.06861e-001 3.48019e-002
7Z+: -1.78311e-002 -1.29704e-002
8X+: -8.49357e-002 3.55200e-002
8Y+: 2.07470e-001 3.59745e-002
8Z+: -7.66974e-002 -3.19813e-003
9X+: -7.38565e-002 1.95146e-002
9Y+: 2.17403e-001 2.01550e-002
9Z+: -1.77073e-002 -3.46414e-003
10X+: -7.77587e-002 2.36700e-002
10Y+: 2.35654e-001 -2.14540e-002
10Z+: 7.94165e-002 -2.45897e-002
11X+: -8.17972e-002 2.20583e-002
11Y+: 2.20906e-001 -4.30164e-003
11Z+: -5.60520e-003 3.10187e-003
12X+: -8.64261e-002 3.66022e-002
12Y+: 2.15000e-001 -5.74661e-003
12Z+: -1.22622e-001 4.11767e-002
13X+: -4.25169e-002 1.56602e-002
13Y+: 5.31036e-002 -1.73951e-002
13Z+: -4.07130e-002 1.26884e-002
14X+: -3.85032e-002 1.29431e-002
14Y+: 5.36716e-002 -1.80868e-002
14Z+: 1.00367e-001 -3.48798e-002
15X+: -4.25524e-002 1.46363e-002
15Y+: 5.19668e-002 -1.69744e-002
15Z+: 5.89747e-003 -2.32428e-003
16X+: -4.31268e-002 1.38332e-002
16Y+: 5.07545e-002 -1.53045e-002
16Z+: -1.04172e-001 3.17984e-002
17X+: -2.69757e-002 9.07955e-003
17Y+: 3.07837e-002 -9.44663e-003
17Z+: -7.63502e-003 1.68203e-003
18X+: -3.00097e-002 9.23966e-003
18Y+: 2.83585e-002 -8.97747e-003
18Z+: 1.52467e-001 -4.78675e-002
19X+: -2.70223e-002 6.16478e-003
19Y+: 3.06149e-002 -6.25382e-003
19Z+: -4.84888e-003 1.93970e-003
20X+: -2.90976e-002 7.13184e-003
20Y+: 3.36738e-002 -7.30875e-003
20Z+: -1.66902e-001 3.93419e-002
3X+: -9.40720e-002 3.93724e-002
3Y+: 2.52307e-001 0.00000e+000
3Z+: -1.53864e-002 -9.25720e-004
21X+: -7.91940e-002 4.39648e-002
21Y+: 2.04567e-001 9.49987e-003
21Z+: -1.56087e-002 7.08838e-003
22X+: -1.01070e-001 3.13534e-002
22Y+: 1.92270e-001 1.80423e-002
22Z+: 2.93053e-003 -1.97308e-003
23X+: -8.86455e-002 4.29906e-002
23Z+: -3.38351e-002 1.81362e-003
23Y-: -1.90862e-001 -2.53414e-002
#+end_example
#+begin_src matlab
mode1 = [...
-9.34637e-002+j*4.52445e-002, +2.33790e-001+j*1.41439e-003, -1.73754e-002+j*6.02449e-003;
-7.42108e-002+j*3.91543e-002, +2.41566e-001-j*1.44869e-003, -5.99285e-003+j*2.10370e-003;
-9.40720e-002+j*3.93724e-002, +2.52307e-001+j*0.00000e+000, -1.53864e-002-j*9.25720e-004;
-1.02163e-001+j*2.79561e-002, +2.29048e-001+j*2.89782e-002, -2.85130e-002+j*1.77132e-004;
-8.77132e-002+j*3.34081e-002, +2.14182e-001+j*2.14655e-002, -1.54521e-002+j*1.26682e-002;
-7.90143e-002+j*2.42583e-002, +2.20669e-001+j*2.12738e-002, +4.60755e-002+j*4.96406e-003;
-7.79654e-002+j*2.58385e-002, +2.06861e-001+j*3.48019e-002, -1.78311e-002-j*1.29704e-002;
-8.49357e-002+j*3.55200e-002, +2.07470e-001+j*3.59745e-002, -7.66974e-002-j*3.19813e-003;
-7.38565e-002+j*1.95146e-002, +2.17403e-001+j*2.01550e-002, -1.77073e-002-j*3.46414e-003;
-7.77587e-002+j*2.36700e-002, +2.35654e-001-j*2.14540e-002, +7.94165e-002-j*2.45897e-002;
-8.17972e-002+j*2.20583e-002, +2.20906e-001-j*4.30164e-003, -5.60520e-003+j*3.10187e-003;
-8.64261e-002+j*3.66022e-002, +2.15000e-001-j*5.74661e-003, -1.22622e-001+j*4.11767e-002;
-4.25169e-002+j*1.56602e-002, +5.31036e-002-j*1.73951e-002, -4.07130e-002+j*1.26884e-002;
-3.85032e-002+j*1.29431e-002, +5.36716e-002-j*1.80868e-002, +1.00367e-001-j*3.48798e-002;
-4.25524e-002+j*1.46363e-002, +5.19668e-002-j*1.69744e-002, +5.89747e-003-j*2.32428e-003;
-4.31268e-002+j*1.38332e-002, +5.07545e-002-j*1.53045e-002, -1.04172e-001+j*3.17984e-002;
-2.69757e-002+j*9.07955e-003, +3.07837e-002-j*9.44663e-003, -7.63502e-003+j*1.68203e-003;
-3.00097e-002+j*9.23966e-003, +2.83585e-002-j*8.97747e-003, +1.52467e-001-j*4.78675e-002;
-2.70223e-002+j*6.16478e-003, +3.06149e-002-j*6.25382e-003, -4.84888e-003+j*1.93970e-003;
-2.90976e-002+j*7.13184e-003, +3.36738e-002-j*7.30875e-003, -1.66902e-001+j*3.93419e-002;
-7.91940e-002+j*4.39648e-002, +2.04567e-001+j*9.49987e-003, -1.56087e-002+j*7.08838e-003;
-1.01070e-001+j*3.13534e-002, +1.92270e-001+j*1.80423e-002, +2.93053e-003-j*1.97308e-003;
-8.86455e-002+j*4.29906e-002, +1.90862e-001+j*2.53414e-002, -3.38351e-002+j*1.81362e-003];
#+end_src
*** Mode2
#+begin_src bash :results output :exports none :eval no-export
sed -n 88,156p modal_analysis/modes_propres_narband.asc
#+end_src
#+RESULTS:
#+begin_example
1X+: 7.56931e-002 3.61548e-002
1Y+: 2.07574e-001 1.69205e-004
1Z+: 1.29733e-002 -6.78426e-004
2X+: 8.58732e-002 2.54470e-002
2Y+: 2.07117e-001 -1.31755e-003
2Z+: -2.13788e-003 -1.24974e-002
4X+: 7.09825e-002 3.66313e-002
4Y+: 2.09969e-001 1.11484e-002
4Z+: 9.19478e-003 3.47272e-002
5X+: 6.23935e-002 1.02488e-002
5Y+: 2.30687e-001 -3.58416e-003
5Z+: 3.27122e-002 -5.85468e-002
6X+: 7.61163e-002 -2.43630e-002
6Y+: 2.26743e-001 -1.15334e-002
6Z+: -6.20205e-003 -1.21742e-001
7X+: 8.01824e-002 -1.94769e-002
7Y+: 1.97485e-001 4.50105e-002
7Z+: -2.21170e-002 9.77052e-002
8X+: 6.19294e-002 8.15075e-003
8Y+: 2.03864e-001 4.45835e-002
8Z+: 2.55133e-002 1.36137e-001
9X+: 4.38135e-002 7.30537e-002
9Y+: 2.28426e-001 -6.58868e-003
9Z+: 1.16313e-002 5.09427e-004
10X+: 5.45770e-002 4.34251e-002
10Y+: 2.50823e-001 0.00000e+000
10Z+: -4.63460e-002 -4.76868e-002
11X+: 5.50987e-002 4.26178e-002
11Y+: 2.29394e-001 5.78236e-002
11Z+: 1.90158e-002 1.09139e-002
12X+: 4.98867e-002 7.30190e-002
12Y+: 2.07871e-001 4.57750e-002
12Z+: 6.69433e-002 9.00315e-002
13X+: 2.48819e-002 3.03222e-002
13Y+: -2.56046e-002 -3.34132e-002
13Z+: 2.13260e-002 2.58544e-002
14X+: 2.45706e-002 2.60221e-002
14Y+: -2.57723e-002 -3.35612e-002
14Z+: -5.71282e-002 -6.61562e-002
15X+: 2.68196e-002 2.83888e-002
15Y+: -2.57263e-002 -3.29627e-002
15Z+: -2.11722e-003 -3.37239e-003
16X+: 2.51442e-002 3.32558e-002
16Y+: -2.54372e-002 -3.25062e-002
16Z+: 5.65780e-002 7.64142e-002
17X+: 1.62437e-002 1.94534e-002
17Y+: -1.31293e-002 -2.05924e-002
17Z+: 1.05274e-003 3.59474e-003
18X+: 1.83431e-002 2.03836e-002
18Y+: -1.16818e-002 -1.86334e-002
18Z+: -8.66632e-002 -1.08216e-001
19X+: 1.62553e-002 1.79588e-002
19Y+: -1.28857e-002 -1.90512e-002
19Z+: 6.25653e-003 4.97733e-003
20X+: 1.63830e-002 2.03943e-002
20Y+: -1.48941e-002 -2.11717e-002
20Z+: 8.68045e-002 1.16491e-001
3X+: 8.17201e-002 2.36079e-002
3Y+: 2.15927e-001 1.61300e-002
3Z+: -5.48456e-004 2.55691e-002
21X+: 6.79204e-002 -5.55513e-002
21Y+: 2.32871e-001 2.33389e-002
21Z+: 1.34345e-002 -2.31815e-002
22X+: 4.02414e-002 -8.38957e-002
22Y+: 2.35273e-001 2.73256e-002
22Z+: -8.51632e-003 -7.49635e-003
23X+: 6.18293e-002 -5.99671e-002
23Z+: 1.63533e-002 6.09161e-002
23Y-: -2.37693e-001 -4.34204e-002
#+end_example
#+begin_src matlab
mode2 = [...
+7.56931e-002+j*3.61548e-002, +2.07574e-001+j*1.69205e-004, +1.29733e-002-j*6.78426e-004;
+8.58732e-002+j*2.54470e-002, +2.07117e-001-j*1.31755e-003, -2.13788e-003-j*1.24974e-002;
+8.17201e-002+j*2.36079e-002, +2.15927e-001+j*1.61300e-002, -5.48456e-004+j*2.55691e-002;
+7.09825e-002+j*3.66313e-002, +2.09969e-001+j*1.11484e-002, +9.19478e-003+j*3.47272e-002;
+6.23935e-002+j*1.02488e-002, +2.30687e-001-j*3.58416e-003, +3.27122e-002-j*5.85468e-002;
+7.61163e-002-j*2.43630e-002, +2.26743e-001-j*1.15334e-002, -6.20205e-003-j*1.21742e-001;
+8.01824e-002-j*1.94769e-002, +1.97485e-001+j*4.50105e-002, -2.21170e-002+j*9.77052e-002;
+6.19294e-002+j*8.15075e-003, +2.03864e-001+j*4.45835e-002, +2.55133e-002+j*1.36137e-001;
+4.38135e-002+j*7.30537e-002, +2.28426e-001-j*6.58868e-003, +1.16313e-002+j*5.09427e-004;
+5.45770e-002+j*4.34251e-002, +2.50823e-001+j*0.00000e+000, -4.63460e-002-j*4.76868e-002;
+5.50987e-002+j*4.26178e-002, +2.29394e-001+j*5.78236e-002, +1.90158e-002+j*1.09139e-002;
+4.98867e-002+j*7.30190e-002, +2.07871e-001+j*4.57750e-002, +6.69433e-002+j*9.00315e-002;
+2.48819e-002+j*3.03222e-002, -2.56046e-002-j*3.34132e-002, +2.13260e-002+j*2.58544e-002;
+2.45706e-002+j*2.60221e-002, -2.57723e-002-j*3.35612e-002, -5.71282e-002-j*6.61562e-002;
+2.68196e-002+j*2.83888e-002, -2.57263e-002-j*3.29627e-002, -2.11722e-003-j*3.37239e-003;
+2.51442e-002+j*3.32558e-002, -2.54372e-002-j*3.25062e-002, +5.65780e-002+j*7.64142e-002;
+1.62437e-002+j*1.94534e-002, -1.31293e-002-j*2.05924e-002, +1.05274e-003+j*3.59474e-003;
+1.83431e-002+j*2.03836e-002, -1.16818e-002-j*1.86334e-002, -8.66632e-002-j*1.08216e-001;
+1.62553e-002+j*1.79588e-002, -1.28857e-002-j*1.90512e-002, +6.25653e-003+j*4.97733e-003;
+1.63830e-002+j*2.03943e-002, -1.48941e-002-j*2.11717e-002, +8.68045e-002+j*1.16491e-001;
+6.79204e-002-j*5.55513e-002, +2.32871e-001+j*2.33389e-002, +1.34345e-002-j*2.31815e-002;
+4.02414e-002-j*8.38957e-002, +2.35273e-001+j*2.73256e-002, -8.51632e-003-j*7.49635e-003;
+6.18293e-002-j*5.99671e-002, +2.37693e-001+j*4.34204e-002, +1.63533e-002+j*6.09161e-002]
#+end_src
*** Mode3
#+begin_src bash :results output :exports none :eval no-export
sed -n 164,232p modal_analysis/modes_propres_narband.asc
#+end_src
#+RESULTS:
#+begin_example
1X+: 1.34688e-001 -6.65071e-002
1Y+: 1.55316e-002 1.01277e-002
1Z+: -5.88466e-002 1.14294e-002
2X+: 1.53934e-001 -9.76990e-003
2Y+: 7.17487e-003 1.11925e-002
2Z+: -4.57205e-002 7.26573e-003
4X+: 1.37298e-001 -5.24661e-002
4Y+: 1.19427e-003 -5.39240e-002
4Z+: -1.25915e-002 5.38133e-003
5X+: 2.43192e-001 -3.17374e-002
5Y+: -2.15730e-001 -7.69941e-004
5Z+: -1.56268e-001 1.44118e-002
6X+: -7.27705e-002 -3.54943e-003
6Y+: -2.47706e-001 2.66480e-003
6Z+: -1.21590e-001 1.06054e-002
7X+: -7.25870e-002 -4.62024e-003
7Y+: 2.27073e-001 -3.69315e-002
7Z+: 1.22611e-001 -6.67337e-003
8X+: 2.32731e-001 -2.85516e-002
8Y+: 2.35389e-001 -3.81905e-002
8Z+: 5.35574e-002 4.30394e-004
9X+: 2.64170e-001 -2.67367e-002
9Y+: -2.56227e-001 3.97957e-005
9Z+: -1.95398e-001 2.23549e-002
10X+: -1.66953e-002 -7.95698e-003
10Y+: -2.66547e-001 -2.17687e-002
10Z+: 1.56278e-002 2.23786e-003
11X+: -3.42364e-002 -9.30205e-003
11Y+: 2.52340e-001 -7.47237e-003
11Z+: -9.51643e-004 3.64798e-003
12X+: 2.97574e-001 0.00000e+000
12Y+: 2.23170e-001 -1.37831e-002
12Z+: 1.06266e-001 2.30324e-003
13X+: 2.67178e-002 -4.15723e-004
13Y+: 6.75423e-003 -2.18428e-003
13Z+: -1.69423e-002 3.12395e-003
14X+: -1.12283e-002 2.86316e-004
14Y+: 5.08225e-003 -2.14053e-003
14Z+: 2.18339e-002 -3.25204e-003
15X+: -1.17948e-002 6.82873e-004
15Y+: 1.94914e-002 -2.42151e-003
15Z+: 2.68660e-003 -2.92104e-004
16X+: 1.19490e-002 1.72236e-005
16Y+: 1.83552e-002 -2.71289e-003
16Z+: -2.70914e-002 4.84164e-003
17X+: 1.00173e-002 -5.80552e-005
17Y+: -3.87262e-003 -1.19607e-003
17Z+: -8.53809e-003 1.48424e-003
18X+: -1.22262e-002 5.13096e-004
18Y+: -5.73905e-003 -1.07659e-003
18Z+: 3.51730e-002 -6.13814e-003
19X+: -1.43735e-002 -4.78552e-004
19Y+: 2.31135e-002 -6.30554e-004
19Z+: 1.80171e-003 -1.98835e-004
20X+: 9.17792e-003 5.36661e-004
20Y+: 2.18969e-002 -5.81759e-004
20Z+: -3.72117e-002 5.35813e-003
3X+: 1.61551e-001 1.65478e-002
3Y+: -4.12527e-004 -5.60909e-002
3Z+: -9.00640e-003 3.50754e-003
21X+: 3.38754e-002 -3.38703e-002
21Y+: -2.20843e-002 2.78581e-002
21Z+: -8.79541e-002 -3.67473e-003
22X+: 3.93064e-002 4.69476e-002
22Y+: -1.69132e-002 -1.04606e-002
22Z+: -1.85351e-002 1.33750e-003
23X+: 3.60396e-002 -2.46238e-002
23Z+: 3.57722e-003 3.64827e-003
23Y-: 1.92038e-002 6.65895e-002
#+end_example
#+begin_src matlab
mode3 = [...
+1.34688e-001-j*6.65071e-002, +1.55316e-002+j*1.01277e-002, -5.88466e-002+j*1.14294e-002;
+1.53934e-001-j*9.76990e-003, +7.17487e-003+j*1.11925e-002, -4.57205e-002+j*7.26573e-003;
+1.61551e-001+j*1.65478e-002, -4.12527e-004-j*5.60909e-002, -9.00640e-003+j*3.50754e-003;
+1.37298e-001-j*5.24661e-002, +1.19427e-003-j*5.39240e-002, -1.25915e-002+j*5.38133e-003;
+2.43192e-001-j*3.17374e-002, -2.15730e-001-j*7.69941e-004, -1.56268e-001+j*1.44118e-002;
-7.27705e-002-j*3.54943e-003, -2.47706e-001+j*2.66480e-003, -1.21590e-001+j*1.06054e-002;
-7.25870e-002-j*4.62024e-003, +2.27073e-001-j*3.69315e-002, +1.22611e-001-j*6.67337e-003;
+2.32731e-001-j*2.85516e-002, +2.35389e-001-j*3.81905e-002, +5.35574e-002+j*4.30394e-004;
+2.64170e-001-j*2.67367e-002, -2.56227e-001+j*3.97957e-005, -1.95398e-001+j*2.23549e-002;
-1.66953e-002-j*7.95698e-003, -2.66547e-001-j*2.17687e-002, +1.56278e-002+j*2.23786e-003;
-3.42364e-002-j*9.30205e-003, +2.52340e-001-j*7.47237e-003, -9.51643e-004+j*3.64798e-003;
+2.97574e-001+j*0.00000e+000, +2.23170e-001-j*1.37831e-002, +1.06266e-001+j*2.30324e-003;
+2.67178e-002-j*4.15723e-004, +6.75423e-003-j*2.18428e-003, -1.69423e-002+j*3.12395e-003;
-1.12283e-002+j*2.86316e-004, +5.08225e-003-j*2.14053e-003, +2.18339e-002-j*3.25204e-003;
-1.17948e-002+j*6.82873e-004, +1.94914e-002-j*2.42151e-003, +2.68660e-003-j*2.92104e-004;
+1.19490e-002+j*1.72236e-005, +1.83552e-002-j*2.71289e-003, -2.70914e-002+j*4.84164e-003;
+1.00173e-002-j*5.80552e-005, -3.87262e-003-j*1.19607e-003, -8.53809e-003+j*1.48424e-003;
-1.22262e-002+j*5.13096e-004, -5.73905e-003-j*1.07659e-003, +3.51730e-002-j*6.13814e-003;
-1.43735e-002-j*4.78552e-004, +2.31135e-002-j*6.30554e-004, +1.80171e-003-j*1.98835e-004;
+9.17792e-003+j*5.36661e-004, +2.18969e-002-j*5.81759e-004, -3.72117e-002+j*5.35813e-003;
+3.38754e-002-j*3.38703e-002, -2.20843e-002+j*2.78581e-002, -8.79541e-002-j*3.67473e-003;
+3.93064e-002+j*4.69476e-002, -1.69132e-002-j*1.04606e-002, -1.85351e-002+j*1.33750e-003;
+3.60396e-002-j*2.46238e-002, -1.92038e-002-j*6.65895e-002, +3.57722e-003+j*3.64827e-003];
#+end_src
*** Mode4
#+begin_src bash :results output :exports none :eval no-export
sed -n 240,308p modal_analysis/modes_propres_narband.asc
#+end_src
#+RESULTS:
#+begin_example
1X+: -1.02501e-001 -1.43802e-001
1Y+: -1.07971e-001 5.61418e-004
1Z+: 1.87145e-001 -1.03605e-001
2X+: -9.44764e-002 -1.36856e-001
2Y+: -1.04428e-001 5.27790e-003
2Z+: 1.60710e-001 -7.74212e-002
4X+: -9.17242e-002 -1.36656e-001
4Y+: -1.34249e-001 -1.03884e-002
4Z+: 1.92123e-001 -1.25627e-001
5X+: 1.05875e-002 -1.03886e-001
5Y+: -8.26338e-002 3.58498e-002
5Z+: 2.55819e-001 -6.94290e-003
6X+: -4.58970e-002 -1.33904e-002
6Y+: -9.41660e-002 4.99682e-002
6Z+: 1.28276e-001 4.59685e-002
7X+: -6.01521e-002 -1.30165e-002
7Y+: 2.56439e-003 -6.78141e-002
7Z+: 5.03428e-002 -1.59420e-001
8X+: -1.00895e-002 -8.80550e-002
8Y+: 1.26327e-002 -8.14444e-002
8Z+: 1.59506e-001 -2.05360e-001
9X+: -3.04658e-003 -1.57921e-001
9Y+: -8.23501e-002 4.82748e-002
9Z+: 1.69315e-001 1.22804e-002
10X+: -8.25875e-002 -7.31038e-002
10Y+: -1.08668e-001 3.56364e-002
10Z+: 8.28567e-002 -4.49596e-003
11X+: -1.06792e-001 -6.95394e-002
11Y+: 3.77195e-002 -7.65410e-002
11Z+: 8.00590e-003 -2.32461e-002
12X+: -4.84292e-002 -1.45790e-001
12Y+: 1.03862e-002 -7.31212e-002
12Z+: 1.78122e-001 -1.00939e-001
13X+: -3.49891e-002 -6.20969e-003
13Y+: -1.18504e-002 -1.94225e-002
13Z+: 4.13007e-002 7.67087e-003
14X+: -3.55795e-002 1.16708e-003
14Y+: -1.68128e-002 -1.82344e-002
14Z+: 3.92416e-002 -3.64434e-002
15X+: -3.45304e-002 3.78185e-003
15Y+: -7.62559e-003 -2.24241e-002
15Z+: 6.28286e-003 -1.32711e-002
16X+: -9.95646e-003 -6.04395e-003
16Y+: -8.73465e-003 -2.20807e-002
16Z+: 3.56946e-002 1.69231e-002
17X+: -9.32661e-003 -5.51944e-003
17Y+: -1.91087e-002 -9.09191e-003
17Z+: 4.04981e-002 8.38685e-004
18X+: -2.84456e-002 4.02762e-003
18Y+: -2.20044e-002 -8.86197e-003
18Z+: 4.43051e-002 -5.21033e-002
19X+: -3.27019e-002 3.59765e-003
19Y+: 2.93163e-003 -2.05064e-002
19Z+: -1.77289e-002 -1.29477e-002
20X+: -1.08474e-002 -5.78419e-003
20Y+: 3.86759e-003 -1.91642e-002
20Z+: 2.10135e-002 3.18051e-002
3X+: -9.11657e-002 -1.36611e-001
3Y+: -1.78165e-001 -3.47193e-002
3Z+: 2.37121e-001 -4.96494e-002
21X+: -1.34808e-002 -9.69121e-003
21Y+: 1.25218e-002 -2.71411e-002
21Z+: 2.76673e-001 0.00000e+000
22X+: 1.96744e-003 4.90797e-003
22Y+: -9.82609e-004 -3.31065e-002
22Z+: 1.79246e-001 -3.33238e-002
23X+: -1.08728e-002 -8.80278e-003
23Z+: 2.30814e-001 -8.33151e-002
23Y-: 1.15217e-002 4.01143e-002
#+end_example
#+begin_src matlab
mode4 = [...
-1.02501e-001-j*1.43802e-001, -1.07971e-001+j*5.61418e-004, +1.87145e-001-j*1.03605e-001;
-9.44764e-002-j*1.36856e-001, -1.04428e-001+j*5.27790e-003, +1.60710e-001-j*7.74212e-002;
-9.11657e-002-j*1.36611e-001, -1.78165e-001-j*3.47193e-002, +2.37121e-001-j*4.96494e-002;
-9.17242e-002-j*1.36656e-001, -1.34249e-001-j*1.03884e-002, +1.92123e-001-j*1.25627e-001;
+1.05875e-002-j*1.03886e-001, -8.26338e-002+j*3.58498e-002, +2.55819e-001-j*6.94290e-003;
-4.58970e-002-j*1.33904e-002, -9.41660e-002+j*4.99682e-002, +1.28276e-001+j*4.59685e-002;
-6.01521e-002-j*1.30165e-002, +2.56439e-003-j*6.78141e-002, +5.03428e-002-j*1.59420e-001;
-1.00895e-002-j*8.80550e-002, +1.26327e-002-j*8.14444e-002, +1.59506e-001-j*2.05360e-001;
-3.04658e-003-j*1.57921e-001, -8.23501e-002+j*4.82748e-002, +1.69315e-001+j*1.22804e-002;
-8.25875e-002-j*7.31038e-002, -1.08668e-001+j*3.56364e-002, +8.28567e-002-j*4.49596e-003;
-1.06792e-001-j*6.95394e-002, +3.77195e-002-j*7.65410e-002, +8.00590e-003-j*2.32461e-002;
-4.84292e-002-j*1.45790e-001, +1.03862e-002-j*7.31212e-002, +1.78122e-001-j*1.00939e-001;
-3.49891e-002-j*6.20969e-003, -1.18504e-002-j*1.94225e-002, +4.13007e-002+j*7.67087e-003;
-3.55795e-002+j*1.16708e-003, -1.68128e-002-j*1.82344e-002, +3.92416e-002-j*3.64434e-002;
-3.45304e-002+j*3.78185e-003, -7.62559e-003-j*2.24241e-002, +6.28286e-003-j*1.32711e-002;
-9.95646e-003-j*6.04395e-003, -8.73465e-003-j*2.20807e-002, +3.56946e-002+j*1.69231e-002;
-9.32661e-003-j*5.51944e-003, -1.91087e-002-j*9.09191e-003, +4.04981e-002+j*8.38685e-004;
-2.84456e-002+j*4.02762e-003, -2.20044e-002-j*8.86197e-003, +4.43051e-002-j*5.21033e-002;
-3.27019e-002+j*3.59765e-003, +2.93163e-003-j*2.05064e-002, -1.77289e-002-j*1.29477e-002;
-1.08474e-002-j*5.78419e-003, +3.86759e-003-j*1.91642e-002, +2.10135e-002+j*3.18051e-002;
-1.34808e-002-j*9.69121e-003, +1.25218e-002-j*2.71411e-002, +2.76673e-001+j*0.00000e+000;
+1.96744e-003+j*4.90797e-003, -9.82609e-004-j*3.31065e-002, +1.79246e-001-j*3.33238e-002;
-1.08728e-002-j*8.80278e-003, -1.15217e-002-j*4.01143e-002, +2.30814e-001-j*8.33151e-002];
#+end_src
*** All modes
#+begin_src matlab
mode_shapes = zeros(23, 3, 10);
mode_shapes(:, :, 1) = [...
-9.34637e-002+j*4.52445e-002, +2.33790e-001+j*1.41439e-003, -1.73754e-002+j*6.02449e-003;
-7.42108e-002+j*3.91543e-002, +2.41566e-001-j*1.44869e-003, -5.99285e-003+j*2.10370e-003;
-9.40720e-002+j*3.93724e-002, +2.52307e-001+j*0.00000e+000, -1.53864e-002-j*9.25720e-004;
-1.02163e-001+j*2.79561e-002, +2.29048e-001+j*2.89782e-002, -2.85130e-002+j*1.77132e-004;
-8.77132e-002+j*3.34081e-002, +2.14182e-001+j*2.14655e-002, -1.54521e-002+j*1.26682e-002;
-7.90143e-002+j*2.42583e-002, +2.20669e-001+j*2.12738e-002, +4.60755e-002+j*4.96406e-003;
-7.79654e-002+j*2.58385e-002, +2.06861e-001+j*3.48019e-002, -1.78311e-002-j*1.29704e-002;
-8.49357e-002+j*3.55200e-002, +2.07470e-001+j*3.59745e-002, -7.66974e-002-j*3.19813e-003;
-7.38565e-002+j*1.95146e-002, +2.17403e-001+j*2.01550e-002, -1.77073e-002-j*3.46414e-003;
-7.77587e-002+j*2.36700e-002, +2.35654e-001-j*2.14540e-002, +7.94165e-002-j*2.45897e-002;
-8.17972e-002+j*2.20583e-002, +2.20906e-001-j*4.30164e-003, -5.60520e-003+j*3.10187e-003;
-8.64261e-002+j*3.66022e-002, +2.15000e-001-j*5.74661e-003, -1.22622e-001+j*4.11767e-002;
-4.25169e-002+j*1.56602e-002, +5.31036e-002-j*1.73951e-002, -4.07130e-002+j*1.26884e-002;
-3.85032e-002+j*1.29431e-002, +5.36716e-002-j*1.80868e-002, +1.00367e-001-j*3.48798e-002;
-4.25524e-002+j*1.46363e-002, +5.19668e-002-j*1.69744e-002, +5.89747e-003-j*2.32428e-003;
-4.31268e-002+j*1.38332e-002, +5.07545e-002-j*1.53045e-002, -1.04172e-001+j*3.17984e-002;
-2.69757e-002+j*9.07955e-003, +3.07837e-002-j*9.44663e-003, -7.63502e-003+j*1.68203e-003;
-3.00097e-002+j*9.23966e-003, +2.83585e-002-j*8.97747e-003, +1.52467e-001-j*4.78675e-002;
-2.70223e-002+j*6.16478e-003, +3.06149e-002-j*6.25382e-003, -4.84888e-003+j*1.93970e-003;
-2.90976e-002+j*7.13184e-003, +3.36738e-002-j*7.30875e-003, -1.66902e-001+j*3.93419e-002;
-7.91940e-002+j*4.39648e-002, +2.04567e-001+j*9.49987e-003, -1.56087e-002+j*7.08838e-003;
-1.01070e-001+j*3.13534e-002, +1.92270e-001+j*1.80423e-002, +2.93053e-003-j*1.97308e-003;
-8.86455e-002+j*4.29906e-002, -3.38351e-002+j*1.81362e-003, +1.90862e-001+j*2.53414e-002];
mode_shapes(:, :, 2) = [...
+7.56931e-002+j*3.61548e-002, +2.07574e-001+j*1.69205e-004, +1.29733e-002-j*6.78426e-004;
+8.58732e-002+j*2.54470e-002, +2.07117e-001-j*1.31755e-003, -2.13788e-003-j*1.24974e-002;
+8.17201e-002+j*2.36079e-002, +2.15927e-001+j*1.61300e-002, -5.48456e-004+j*2.55691e-002;
+7.09825e-002+j*3.66313e-002, +2.09969e-001+j*1.11484e-002, +9.19478e-003+j*3.47272e-002;
+6.23935e-002+j*1.02488e-002, +2.30687e-001-j*3.58416e-003, +3.27122e-002-j*5.85468e-002;
+7.61163e-002-j*2.43630e-002, +2.26743e-001-j*1.15334e-002, -6.20205e-003-j*1.21742e-001;
+8.01824e-002-j*1.94769e-002, +1.97485e-001+j*4.50105e-002, -2.21170e-002+j*9.77052e-002;
+6.19294e-002+j*8.15075e-003, +2.03864e-001+j*4.45835e-002, +2.55133e-002+j*1.36137e-001;
+4.38135e-002+j*7.30537e-002, +2.28426e-001-j*6.58868e-003, +1.16313e-002+j*5.09427e-004;
+5.45770e-002+j*4.34251e-002, +2.50823e-001+j*0.00000e+000, -4.63460e-002-j*4.76868e-002;
+5.50987e-002+j*4.26178e-002, +2.29394e-001+j*5.78236e-002, +1.90158e-002+j*1.09139e-002;
+4.98867e-002+j*7.30190e-002, +2.07871e-001+j*4.57750e-002, +6.69433e-002+j*9.00315e-002;
+2.48819e-002+j*3.03222e-002, -2.56046e-002-j*3.34132e-002, +2.13260e-002+j*2.58544e-002;
+2.45706e-002+j*2.60221e-002, -2.57723e-002-j*3.35612e-002, -5.71282e-002-j*6.61562e-002;
+2.68196e-002+j*2.83888e-002, -2.57263e-002-j*3.29627e-002, -2.11722e-003-j*3.37239e-003;
+2.51442e-002+j*3.32558e-002, -2.54372e-002-j*3.25062e-002, +5.65780e-002+j*7.64142e-002;
+1.62437e-002+j*1.94534e-002, -1.31293e-002-j*2.05924e-002, +1.05274e-003+j*3.59474e-003;
+1.83431e-002+j*2.03836e-002, -1.16818e-002-j*1.86334e-002, -8.66632e-002-j*1.08216e-001;
+1.62553e-002+j*1.79588e-002, -1.28857e-002-j*1.90512e-002, +6.25653e-003+j*4.97733e-003;
+1.63830e-002+j*2.03943e-002, -1.48941e-002-j*2.11717e-002, +8.68045e-002+j*1.16491e-001;
+6.79204e-002-j*5.55513e-002, +2.32871e-001+j*2.33389e-002, +1.34345e-002-j*2.31815e-002;
+4.02414e-002-j*8.38957e-002, +2.35273e-001+j*2.73256e-002, -8.51632e-003-j*7.49635e-003;
+6.18293e-002-j*5.99671e-002, +1.63533e-002+j*6.09161e-002, +2.37693e-001+j*4.34204e-002];
mode_shapes(:, :, 3) = [...
+1.34688e-001-j*6.65071e-002, +1.55316e-002+j*1.01277e-002, -5.88466e-002+j*1.14294e-002;
+1.53934e-001-j*9.76990e-003, +7.17487e-003+j*1.11925e-002, -4.57205e-002+j*7.26573e-003;
+1.61551e-001+j*1.65478e-002, -4.12527e-004-j*5.60909e-002, -9.00640e-003+j*3.50754e-003;
+1.37298e-001-j*5.24661e-002, +1.19427e-003-j*5.39240e-002, -1.25915e-002+j*5.38133e-003;
+2.43192e-001-j*3.17374e-002, -2.15730e-001-j*7.69941e-004, -1.56268e-001+j*1.44118e-002;
-7.27705e-002-j*3.54943e-003, -2.47706e-001+j*2.66480e-003, -1.21590e-001+j*1.06054e-002;
-7.25870e-002-j*4.62024e-003, +2.27073e-001-j*3.69315e-002, +1.22611e-001-j*6.67337e-003;
+2.32731e-001-j*2.85516e-002, +2.35389e-001-j*3.81905e-002, +5.35574e-002+j*4.30394e-004;
+2.64170e-001-j*2.67367e-002, -2.56227e-001+j*3.97957e-005, -1.95398e-001+j*2.23549e-002;
-1.66953e-002-j*7.95698e-003, -2.66547e-001-j*2.17687e-002, +1.56278e-002+j*2.23786e-003;
-3.42364e-002-j*9.30205e-003, +2.52340e-001-j*7.47237e-003, -9.51643e-004+j*3.64798e-003;
+2.97574e-001+j*0.00000e+000, +2.23170e-001-j*1.37831e-002, +1.06266e-001+j*2.30324e-003;
+2.67178e-002-j*4.15723e-004, +6.75423e-003-j*2.18428e-003, -1.69423e-002+j*3.12395e-003;
-1.12283e-002+j*2.86316e-004, +5.08225e-003-j*2.14053e-003, +2.18339e-002-j*3.25204e-003;
-1.17948e-002+j*6.82873e-004, +1.94914e-002-j*2.42151e-003, +2.68660e-003-j*2.92104e-004;
+1.19490e-002+j*1.72236e-005, +1.83552e-002-j*2.71289e-003, -2.70914e-002+j*4.84164e-003;
+1.00173e-002-j*5.80552e-005, -3.87262e-003-j*1.19607e-003, -8.53809e-003+j*1.48424e-003;
-1.22262e-002+j*5.13096e-004, -5.73905e-003-j*1.07659e-003, +3.51730e-002-j*6.13814e-003;
-1.43735e-002-j*4.78552e-004, +2.31135e-002-j*6.30554e-004, +1.80171e-003-j*1.98835e-004;
+9.17792e-003+j*5.36661e-004, +2.18969e-002-j*5.81759e-004, -3.72117e-002+j*5.35813e-003;
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-6.84747e-003+j*8.45921e-003, -3.33872e-002+j*1.71496e-002, +6.99867e-002-j*6.93158e-002;
-2.64313e-002+j*5.08903e-003, -3.03569e-002+j*1.29946e-002, -1.39115e-001+j*5.72459e-002;
-2.90186e-002+j*1.48257e-002, -5.55429e-002+j*2.74156e-002, -1.62035e-001+j*8.04187e-002;
-2.05855e-002+j*1.11922e-002, -6.58789e-002+j*3.20524e-002, -1.02263e-002-j*4.24087e-002;
-1.40204e-002+j*8.01102e-003, -5.72647e-002+j*2.37484e-002, +1.75053e-001-j*9.63667e-002;
-3.50818e-002+j*1.41152e-002, -5.19701e-002+j*2.31951e-002, -1.15951e-001+j*2.91582e-002;
-5.78005e-003-j*7.05841e-003, +8.29016e-002+j*1.36984e-002, +4.03470e-003+j*4.03325e-002;
-1.39928e-002-j*1.14088e-002, +8.05288e-002+j*1.51031e-002, +1.12255e-002+j*3.21224e-002;
-1.02276e-002-j*8.35724e-003, -4.89246e-003+j*1.67800e-002, +7.80514e-002+j*1.53467e-002];
mode_shapes(:, :, 10) = [...
+3.33349e-002-j*4.89606e-003, -8.67138e-002-j*1.69402e-002, +2.87366e-002-j*1.66842e-002;
+2.95730e-002-j*6.10477e-004, -9.24590e-002-j*1.92562e-002, +5.21162e-002-j*1.31811e-002;
+3.26966e-002+j*1.03975e-002, -8.55682e-002-j*4.71847e-002, +3.99404e-002+j*8.59358e-003;
+3.45452e-002-j*3.05951e-003, -7.73823e-002-j*2.32199e-002, +2.30960e-002-j*7.49928e-003;
-3.72461e-003-j*5.40336e-003, +1.80151e-001-j*1.42898e-002, -8.63921e-003+j*1.95638e-002;
-1.11285e-002-j*2.22175e-003, +6.92355e-002-j*1.24144e-002, +8.02097e-002-j*6.81531e-003;
-2.99885e-002-j*4.21951e-004, +8.19709e-002+j*1.22484e-002, +3.97531e-002+j*1.33874e-002;
-1.59231e-002-j*3.96929e-003, +9.31295e-002+j*9.75532e-003, -4.01947e-002-j*5.17841e-003;
-7.32828e-003-j*2.93496e-002, +4.39909e-002+j*1.25298e-002, -8.87525e-002+j*2.05359e-002;
+8.20167e-002+j*6.86693e-004, +4.25475e-001-j*2.36494e-002, +3.37034e-001-j*2.44199e-002;
-2.68694e-002+j*3.82921e-003, +4.79292e-001+j*1.35903e-002, +5.06762e-001+j*0.00000e+000;
+1.40565e-002-j*5.41957e-003, +1.17563e-001+j*2.50398e-002, -5.97219e-002+j*5.92813e-004;
+2.30939e-002+j*1.16755e-002, +4.11136e-003+j*1.65726e-002, +2.41863e-002+j*5.06658e-002;
-4.92960e-003+j*9.24082e-003, -5.44667e-003+j*1.41983e-002, -3.73537e-002-j*1.87263e-002;
+9.68682e-003+j*1.11791e-002, -5.13436e-003+j*1.29205e-002, -4.88765e-002-j*4.97821e-002;
-1.66711e-003+j*1.03280e-002, +9.57955e-004+j*1.28350e-002, +3.30268e-002+j*5.44211e-002;
+5.90540e-003+j*1.67113e-002, +1.34280e-002+j*2.62111e-002, +4.38613e-002+j*7.17028e-002;
+4.97752e-003+j*1.47634e-002, +1.56773e-002+j*3.16026e-002, -5.27493e-002-j*3.38315e-002;
+2.83485e-003+j*1.14816e-002, +9.80676e-003+j*2.51504e-002, -6.78645e-002-j*9.90875e-002;
+5.62294e-003+j*1.84035e-002, +1.03717e-002+j*2.37801e-002, +2.58497e-002+j*3.75352e-002;
-8.75236e-003-j*4.71723e-003, +6.99107e-002+j*1.51894e-002, +3.46273e-002+j*1.48547e-002;
-1.72822e-002-j*8.75192e-003, +6.96759e-002+j*1.59783e-002, +8.07917e-002+j*1.97809e-002;
-1.29601e-002-j*5.61834e-003, +1.76126e-002+j*4.62761e-003, +6.64667e-002+j*1.61199e-002];
#+end_src
** Define a point for each solid body
We define accelerometer indices used to define the motion of each solid body (2 3-axis accelerometer are enough).
#+begin_src matlab
stages = [17, 19; % Bottom Granite
13, 15; % Top Granite
9, 11; % Ty
5, 7; % Ry
21, 22; % Spindle
1, 3]; % Hexapod
#+end_src
We define the origin point ${}^AO_B$ of the solid body $\{B\}$.
Here we choose the middle point between the two accelerometers.
This could be define differently (for instance by choosing the center of mass).
#+begin_src matlab
AOB = zeros(3, size(stages, 1));
for i = 1:size(stages, 1)
AOB(:, i) = mean(positions(stages(i, :), 1:3))';
end
#+end_src
Then we compute the positions of the sensors with respect to the previously defined origin for the frame $\{B\}$: ${}^BP_1$ and ${}^BP_2$.
#+begin_src matlab
BP1 = zeros(3, size(stages, 1));
BP2 = zeros(3, size(stages, 1));
for i = 1:size(stages, 1)
BP1(:, i) = positions(stages(i, 1), 1:3)' - AOB(:, i);
BP2(:, i) = positions(stages(i, 2), 1:3)' - AOB(:, i);
end
#+end_src
Let's define one absolute frame $\{A\}$ and one frame $\{B\}$ fixed w.r.t. the solid body.
We note ${}^AO_B$ the position of origin of $\{B\}$ expressed in $\{A\}$.
We are measuring with the accelerometers the absolute motion of points $P_1$ and $P_2$: ${}^Av_{P_1}$ and ${}^Av_{P_2}$.
Let's note ${}^BP_1$ and ${}^BP_2$ the (known) coordinates of $P_1$ and $P_2$ expressed in the frame $\{B\}$.
Then we have:
\begin{align}
{}^Av_{P_1} &= {}^Av_{O_B} + {}^A\Omega^\times {}^AR_B {}^BP_1 \\
{}^Av_{P_2} &= {}^Av_{O_B} + {}^A\Omega^\times {}^AR_B {}^BP_2
\end{align}
And we obtain
\begin{align}
{}^A\Omega^\times {}^AR_B &= \left( {}^Av_{P_2} - {}^Av_{P_1} \right) \left( {}^BP_2 - {}^BP_1 \right)^{-1}\\
{}^Av_{O_B} &= {}^Av_{P_1} - \left( {}^Av_{P_2} - {}^Av_{P_1} \right) \left( {}^BP_2 - {}^BP_1 \right)^{-1} {}^BP_1
\end{align}
#+begin_src matlab
AVOB = zeros(3, size(stages, 1));
ARB = zeros(3, 3, size(stages, 1));
for i = 1:size(stages, 1)
AVOB(:, i) = mode1(stages(i, 1), :)' - (mode1(stages(i, 2), :)' - mode1(stages(i, 1), :)')*pinv(BP2(:, i) - BP1(:, i))*BP1(:, i);
ARB(:, :, i) = (mode1(stages(i, 2), :)' - mode1(stages(i, 1), :)')*pinv(BP2(:, i) - BP1(:, i));
end
#+end_src
** Argand Diagram
For mode 1
#+begin_src matlab
figure;
hold on;
for i=1:size(mode1, 1)
plot([0, real(mode1(i, 1))], [0, imag(mode1(i, 1))], '-k')
plot([0, real(mode1(i, 2))], [0, imag(mode1(i, 2))], '-k')
plot([0, real(mode1(i, 3))], [0, imag(mode1(i, 3))], '-k')
% plot([0, real(mode2(i, 1))], [0, imag(mode2(i, 1))], '-r')
% plot([0, real(mode2(i, 2))], [0, imag(mode2(i, 2))], '-r')
% plot([0, real(mode2(i, 3))], [0, imag(mode2(i, 3))], '-r')
% plot([0, real(mode3(i, 1))], [0, imag(mode3(i, 1))], '-b')
% plot([0, real(mode3(i, 2))], [0, imag(mode3(i, 2))], '-b')
% plot([0, real(mode3(i, 3))], [0, imag(mode3(i, 3))], '-b')
end
for i=1:size(AVOB, 2)
plot([0, real(AVOB(1, i))], [0, imag(AVOB(1, i))], '-r')
plot([0, real(AVOB(2, i))], [0, imag(AVOB(2, i))], '-r')
plot([0, real(AVOB(3, i))], [0, imag(AVOB(3, i))], '-r')
end
% ang=0:0.01:2*pi;
% radius1 = max(max(sqrt(real(mode1).^2+imag(mode1).^2)));
% plot(radius1*cos(ang), radius1*sin(ang), '-k');
% radius2 = max(max(sqrt(real(mode2).^2+imag(mode2).^2)));
% plot(radius2*cos(ang), radius2*sin(ang), '-r');
% radius3 = max(max(sqrt(real(mode3).^2+imag(mode3).^2)));
% plot(radius3*cos(ang), radius3*sin(ang), '-b');
hold off;
axis manual equal
#+end_src
** TEST: animate first mode
#+begin_src matlab
figure;
hold on;
fill3(positions(1:4, 1), positions(1:4, 2), positions(1:4, 3), 'k', 'FaceAlpha', 0.5)
fill3(positions(5:8, 1), positions(5:8, 2), positions(5:8, 3), 'k', 'FaceAlpha', 0.5)
fill3(positions(9:12, 1), positions(9:12, 2), positions(9:12, 3), 'k', 'FaceAlpha', 0.5)
fill3(positions(13:16, 1), positions(13:16, 2), positions(13:16, 3), 'k', 'FaceAlpha', 0.5)
fill3(positions(17:20, 1), positions(17:20, 2), positions(17:20, 3), 'k', 'FaceAlpha', 0.5)
fill3(positions(21:23, 1), positions(21:23, 2), positions(21:23, 3), 'k', 'FaceAlpha', 0.5)
hold off;
#+end_src
#+NAME: fig:mode_shapes
#+HEADER: :tangle no :exports results :results value file raw replace :noweb yes
#+begin_src matlab
rec = polyshape([-2 -2 2 2],[-3 3 3 -3]);
h = figure;
filename = 'figs/mode_shapes.gif';
n = 20;
for i = 1:n
axis manual equal
Dm = real(V(1:3, 5)*cos(2*pi*i/n));
rec_i = rotate(rec, 180/pi*Dm(3));
rec_i = translate(rec_i, 10*Dm(1), 10*Dm(2));
plot(rec_i);
xlim([-3, 3]); ylim([-4, 4]);
set(h, 'visible', 'off');
set(h, 'pos', [0, 0, 500, 500]);
drawnow;
% Capture the plot as an image
frame = getframe(h);
im = frame2im(frame);
[imind,cm] = rgb2ind(im,256);
% Write to the GIF File
if i == 1
imwrite(imind,cm,filename,'gif','DelayTime',0.1,'Loopcount',inf);
else
imwrite(imind,cm,filename,'gif','DelayTime',0.1,'WriteMode','append');
end
end
set(h, 'visible', 'on');
ans = filename;
#+end_src
** From 6 translations to translation + rotation
Let's define one absolute frame $\{A\}$ and one frame $\{B\}$ fixed w.r.t. the solid body.
We note ${}^AO_B$ the position of origin of $\{B\}$ expressed in $\{A\}$.
We are measuring with the accelerometers the absolute motion of points $P_1$ and $P_2$: ${}^AP_1$ and ${}^AP_2$.
Let's note ${}^BP_1$ and ${}^BP_2$ the (known) coordinates of $P_1$ and $P_2$ expressed in the frame $\{B\}$.
Then we have:
\begin{align}
{}^AP_1 &= {}^AO_B + {}^AR_B {}^BP_1 \\
{}^AP_2 &= {}^AO_B + {}^AR_B {}^BP_2
\end{align}
And we obtain
\begin{align}
{}^AR_B &= \left( {}^AP_2 - {}^AP_1 \right) \left( {}^BP_2 - {}^BP_1 \right)^{-1}\\
{}^AO_B &= {}^Av_{P_1} - \left( {}^AP_2 - {}^AP_1 \right) \left( {}^BP_2 - {}^BP_1 \right)^{-1} {}^BP_1
\end{align}
* Problem with AirLoc System
4 Airloc Levelers are used for the granite (figure [[fig:airloc]]).
#+name: fig:airloc
#+caption: AirLoc used for the granite (2120-KSKC)
#+attr_html: :width 500px
[[file:img/airloc/IMG_20190618_155522.jpg]]
They are probably *not well leveled* so that could explain the first modes at 11Hz and 17Hz.
* Spatial Mode Extraction