#+TITLE: Vibrations induced by the Slip-Ring and the Spindle
:DRAWER:
#+STARTUP: overview

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

#+HTML_LINK_HOME: ../index.html
#+HTML_LINK_UP:   ../index.html

#+HTML_HEAD: <link rel="stylesheet" type="text/css" href="https://research.tdehaeze.xyz/css/style.css"/>
#+HTML_HEAD: <script type="text/javascript" src="https://research.tdehaeze.xyz/js/script.js"></script>

#+HTML_MATHJAX: align: center tagside: right font: TeX

#+PROPERTY: header-args:matlab  :session *MATLAB*
#+PROPERTY: header-args:matlab+ :comments org
#+PROPERTY: header-args:matlab+ :results none
#+PROPERTY: header-args:matlab+ :exports both
#+PROPERTY: header-args:matlab+ :eval no-export
#+PROPERTY: header-args:matlab+ :output-dir figs

#+PROPERTY: header-args:shell  :eval no-export
:END:

* Experimental Setup
*Setup*:
All the stages are OFF.

Two geophone are use:
- One on the marble (corresponding to the first column in the data)
- One at the sample location (corresponding to the second column in the data)

Two voltage amplifiers are used, their setup is:
- gain of 60dB
- AC/DC switch on AC
- Low pass filter at 1kHz

A first order low pass filter is also added at the input of the voltage amplifiers.


*Goal*:
- Identify the vibrations induced by the rotation of the Slip-Ring and Spindle


*Measurements*:
Three measurements are done:
| Measurement File   | Description                                                          |
|--------------------+----------------------------------------------------------------------|
| =mat/data_024.mat= | All the stages are OFF                                               |
| =mat/data_025.mat= | The slip-ring is ON and rotates at 6rpm. The spindle is OFF          |
| =mat/data_026.mat= | The slip-ring and spindle are both ON. They are both turning at 6rpm |

Each of the measurement =mat= file contains one =data= array with 3 columns:
| Column number | Description       |
|---------------+-------------------|
|             1 | Geophone - Marble |
|             2 | Geophone - Sample |
|             3 | Time              |

A movie showing the experiment is shown on figure [[fig:exp_sl_sp_gif]].

#+name: fig:exp_sl_sp_gif
#+attr_html: :width 300px
#+caption: Movie of the experiment, rotation speed is 6rpm
[[file:./img/VID_20190510_155655.gif]]

* Data Analysis
  :PROPERTIES:
  :header-args:matlab+: :tangle matlab/spindle_slip_ring_vibrations.m
  :header-args:matlab+: :comments org :mkdirp yes
  :END:
  <<sec:spindle_slip_ring_vibrations>>

** ZIP file containing the data and matlab files                     :ignore:
#+begin_src bash :exports none :results none
  if [ matlab/spindle_slip_ring_vibrations.m -nt data/spindle_slip_ring_vibrations.zip ]; then
    cp matlab/spindle_slip_ring_vibrations.m spindle_slip_ring_vibrations.m;
    zip data/spindle_slip_ring_vibrations \
        mat/data_024.mat \
        mat/data_025.mat \
        mat/data_026.mat \
        spindle_slip_ring_vibrations.m
    rm spindle_slip_ring_vibrations.m;
  fi
#+end_src

#+begin_note
  All the files (data and Matlab scripts) are accessible [[file:data/spindle_slip_ring_vibrations.zip][here]].
#+end_note

** Matlab Init                                              :noexport:ignore:
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
  <<matlab-dir>>
#+end_src

#+begin_src matlab :exports none :results silent :noweb yes
  <<matlab-init>>
#+end_src

#+begin_src matlab
  addpath('../src');
#+end_src

** Load data
#+begin_src matlab
  of = load('mat/data_024.mat', 'data'); of = of.data; % OFF
  sr = load('mat/data_025.mat', 'data'); sr = sr.data; % Slip Ring
  sp = load('mat/data_026.mat', 'data'); sp = sp.data; % Spindle
#+end_src

#+begin_warning
There is a sign error for the Geophone located on top of the Hexapod.
The problem probably comes from the wiring in the Slip-Ring.
#+end_warning

#+begin_src matlab
  of(:, 2) = -of(:, 2);
  sr(:, 2) = -sr(:, 2);
  sp(:, 2) = -sp(:, 2);
#+end_src

** Voltage to Velocity
We convert the measured voltage to velocity using the function =voltageToVelocityL22= (accessible [[file:../src/index.org][here]]).

#+begin_src matlab
  gain = 60; % [dB]

  of(:, 1) = voltageToVelocityL22(of(:, 1), of(:, 3), gain);
  sr(:, 1) = voltageToVelocityL22(sr(:, 1), sr(:, 3), gain);
  sp(:, 1) = voltageToVelocityL22(sp(:, 1), sp(:, 3), gain);

  of(:, 2) = voltageToVelocityL22(of(:, 2), of(:, 3), gain);
  sr(:, 2) = voltageToVelocityL22(sr(:, 2), sr(:, 3), gain);
  sp(:, 2) = voltageToVelocityL22(sp(:, 2), sp(:, 3), gain);
#+end_src

** Time domain plots
#+begin_src matlab
  figure;
  hold on;
  plot(sp(:, 3), sp(:, 1), 'DisplayName', 'Spindle - 6rpm');
  plot(sr(:, 3), sr(:, 1), 'DisplayName', 'Slip-Ring - 6rpm');
  plot(of(:, 3), of(:, 1), 'DisplayName', 'OFF');
  hold off;
  xlabel('Time [s]'); ylabel('Velocity [m/s]');
  xlim([0, 100]);
  legend('Location', 'northeast');
#+end_src

#+NAME: fig:slip_ring_spindle_marble_time
#+HEADER: :tangle no :exports results :results value raw replace :noweb yes
#+begin_src matlab :var filepath="figs/slip_ring_spindle_marble_time.pdf" :var figsize="wide-normal" :post pdf2svg(file=*this*, ext="png")
  <<plt-matlab>>
#+end_src

#+NAME: fig:slip_ring_spindle_marble_time
#+CAPTION: Velocity as measured by the geophone located on the marble - Time domain
#+RESULTS: fig:slip_ring_spindle_marble_time
[[file:figs/slip_ring_spindle_marble_time.png]]

#+begin_src matlab
  figure;
  hold on;
  plot(sp(:, 3), sp(:, 2), 'DisplayName', 'Spindle and Slip-Ring');
  plot(sr(:, 3), sr(:, 2), 'DisplayName', 'Only Slip-Ring');
  plot(of(:, 3), of(:, 2), 'DisplayName', 'OFF');
  hold off;
  xlabel('Time [s]'); ylabel('Velocity [m/s]');
  xlim([0, 100]);
  legend('Location', 'northeast');
#+end_src

#+NAME: fig:slip_ring_spindle_sample_time
#+HEADER: :tangle no :exports results :results value raw replace :noweb yes
#+begin_src matlab :var filepath="figs/slip_ring_spindle_sample_time.pdf" :var figsize="wide-normal" :post pdf2svg(file=*this*, ext="png")
  <<plt-matlab>>
#+end_src

#+NAME: fig:slip_ring_spindle_sample_time
#+CAPTION: Velocity as measured by the geophone at the sample location - Time domain
#+RESULTS: fig:slip_ring_spindle_sample_time
[[file:figs/slip_ring_spindle_sample_time.png]]

#+begin_src matlab :exports none :tangle no
  xlim([0, 1]);
#+end_src

#+NAME: fig:slip_ring_spindle_sample_zoom
#+HEADER: :tangle no :exports results :results value raw replace :noweb yes
#+begin_src matlab :var filepath="figs/slip_ring_spindle_sample_zoom.pdf" :var figsize="wide-normal" :post pdf2svg(file=*this*, ext="png")
  <<plt-matlab>>
#+end_src

#+NAME: fig:slip_ring_spindle_sample_zoom
#+CAPTION: Velocity as measured by the geophone at the sample location - Time domain
#+RESULTS: fig:slip_ring_spindle_sample_zoom
[[file:figs/slip_ring_spindle_sample_zoom.png]]

** Frequency Domain
We first compute some parameters that will be used for the PSD computation.
#+begin_src matlab :results none
  dt = of(2, 3)-of(1, 3); % [s]

  Fs = 1/dt; % [Hz]

  win = hanning(ceil(10*Fs)); % Window used
#+end_src

Then we compute the Power Spectral Density using =pwelch= function.

First for the geophone located on the marble
#+begin_src matlab
  [pxof_m, f] = pwelch(of(:, 1), win, [], [], Fs);
  [pxsr_m, ~] = pwelch(sr(:, 1), win, [], [], Fs);
  [pxsp_m, ~] = pwelch(sp(:, 1), win, [], [], Fs);
#+end_src

And for the geophone located at the sample position.
#+begin_src matlab
  [pxof_s, ~] = pwelch(of(:, 2), win, [], [], Fs);
  [pxsr_s, ~] = pwelch(sr(:, 2), win, [], [], Fs);
  [pxsp_s, ~] = pwelch(sp(:, 2), win, [], [], Fs);
#+end_src

And we plot the ASD of the measured velocities:
- figure [[fig:sr_sp_psd_marble_compare]] for the geophone located on the marble
- figure [[fig:sr_sp_psd_sample_compare]] for the geophone at the sample position

#+begin_src matlab :results none
  figure;
  hold on;
  plot(f, sqrt(pxsp_m), 'DisplayName', 'Spindle - 6rpm');
  plot(f, sqrt(pxsr_m), 'DisplayName', 'Slip-Ring - 6rpm');
  plot(f, sqrt(pxof_m), 'DisplayName', 'OFF');
  hold off;
  set(gca, 'xscale', 'log');
  set(gca, 'yscale', 'log');
  xlabel('Frequency [Hz]'); ylabel('ASD of the measured velocity $\left[\frac{m/s}{\sqrt{Hz}}\right]$')
  legend('Location', 'southwest');
  xlim([2, 500]);
#+end_src

#+NAME: fig:sr_sp_psd_marble_compare
#+HEADER: :tangle no :exports results :results value raw replace :noweb yes
#+begin_src matlab :var filepath="figs/sr_sp_psd_marble_compare.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
  <<plt-matlab>>
#+end_src

#+NAME: fig:sr_sp_psd_marble_compare
#+CAPTION: Comparison of the ASD of the measured velocities from the Geophone on the marble
#+RESULTS: fig:sr_sp_psd_marble_compare
[[file:figs/sr_sp_psd_marble_compare.png]]

#+begin_src matlab :results none
  figure;
  hold on;
  plot(f, sqrt(pxsp_s), 'DisplayName', 'Spindle - 6rpm');
  plot(f, sqrt(pxsr_s), 'DisplayName', 'Slip-Ring - 6rpm');
  plot(f, sqrt(pxof_s), 'DisplayName', 'OFF');
  hold off;
  set(gca, 'xscale', 'log');
  set(gca, 'yscale', 'log');
  xlabel('Frequency [Hz]'); ylabel('ASD of the measured velocity $\left[\frac{m/s}{\sqrt{Hz}}\right]$')
  legend('Location', 'southwest');
  xlim([2, 500]);
#+end_src

#+NAME: fig:sr_sp_psd_sample_compare
#+HEADER: :tangle no :exports results :results value raw replace :noweb yes
#+begin_src matlab :var filepath="figs/sr_sp_psd_sample_compare.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
  <<plt-matlab>>
#+end_src

#+NAME: fig:sr_sp_psd_sample_compare
#+CAPTION: Comparison of the ASD of the measured velocities from the Geophone at the sample location
#+RESULTS: fig:sr_sp_psd_sample_compare
[[file:figs/sr_sp_psd_sample_compare.png]]

We load the ground motion to compare with the measurements (Fig. [[fig:ty_comp_gm]]).
We see that the motion is dominated by the ground motion below 20Hz.
#+begin_src matlab
  gm = load('../ground-motion/mat/psd_gm.mat', 'f', 'psd_gv');
#+end_src

#+begin_src matlab :results none
  figure;
  hold on;
  plot(f, sqrt(pxsp_m), 'DisplayName', 'Spindle - 6rpm');
  plot(f, sqrt(pxsr_m), 'DisplayName', 'Slip-Ring - 6rpm');
  plot(f, sqrt(pxof_m), 'DisplayName', 'OFF');
  plot(gm.f, sqrt(gm.psd_gv), 'k-', 'DisplayName', 'Ground Motion');
  hold off;
  set(gca, 'xscale', 'log');
  set(gca, 'yscale', 'log');
  xlabel('Frequency [Hz]'); ylabel('ASD of the measured velocity $\left[\frac{m/s}{\sqrt{Hz}}\right]$')
  legend('Location', 'southwest');
  xlim([2, 500]);
#+end_src

#+HEADER: :tangle no :exports results :results none :noweb yes
#+begin_src matlab :var filepath="figs/ty_comp_gm.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
  <<plt-matlab>>
#+end_src

#+NAME: fig:ty_comp_gm
#+CAPTION: Comparison of the ground velocity with the measured velocity ([[./figs/ty_comp_gm.png][png]], [[./figs/ty_comp_gm.pdf][pdf]])
[[file:figs/ty_comp_gm.png]]


** Relative Motion
The relative velocity between the sample and the marble is shown in Fig. [[fig:rz_relative_velocity]].
The velocity is integrated to have the relative displacement in Fig. [[fig:rz_relative_motion]].

#+begin_src matlab :exports results
  figure;
  hold on;
  plot(sp(:, 3), sp(:, 2)-sp(:, 1), 'DisplayName', 'Spindle - 6rpm');
  plot(sr(:, 3), sr(:, 2)-sr(:, 1), 'DisplayName', 'Slip-Ring - 6rpm');
  plot(of(:, 3), of(:, 2)-of(:, 1), 'DisplayName', 'OFF');
  hold off;
  xlabel('Time [s]'); ylabel('Velocity [m/s]');
  xlim([0, 100]); ylim([-1e-4, 1e-4]);
  legend('Location', 'northeast');
#+end_src

#+HEADER: :tangle no :exports results :results none :noweb yes
#+begin_src matlab :var filepath="figs/rz_relative_velocity.pdf" :var figsize="wide-normal" :post pdf2svg(file=*this*, ext="png")
  <<plt-matlab>>
#+end_src

#+NAME: fig:rz_relative_velocity
#+CAPTION: Relative velocity between the hexapod and the marble ([[./figs/rz_relative_velocity.png][png]], [[./figs/rz_relative_velocity.pdf][pdf]])
[[file:figs/rz_relative_velocity.png]]

Time domain: Integration to have the displacement
#+begin_src matlab :exports results
  figure;
  hold on;
  plot(sp(:, 3), lsim(1/s, sp(:, 2)-sp(:, 1), sp(:, 3)), 'DisplayName', 'Spindle - 6rpm');
  plot(sr(:, 3), lsim(1/s, sr(:, 2)-sr(:, 1), sr(:, 3)), 'DisplayName', 'Slip-Ring - 6rpm');
  plot(of(:, 3), lsim(1/s, of(:, 2)-of(:, 1), of(:, 3)), 'DisplayName', 'OFF');
  hold off;
  xlabel('Time [s]'); ylabel('Displacement [m]');
  xlim([0, 100]);
  legend('Location', 'northeast');
#+end_src

#+HEADER: :tangle no :exports results :results none :noweb yes
#+begin_src matlab :var filepath="figs/rz_relative_motion.pdf" :var figsize="wide-normal" :post pdf2svg(file=*this*, ext="png")
  <<plt-matlab>>
#+end_src

#+NAME: fig:rz_relative_motion
#+CAPTION: Relative displacement between the Hexapod and the marble ([[./figs/rz_relative_motion.png][png]], [[./figs/rz_relative_motion.pdf][pdf]])
[[file:figs/rz_relative_motion.png]]

We compute the PSD of the relative velocity between the sample and the marble.
#+begin_src matlab
  [pxof_r, f] = pwelch(of(:, 2)-of(:, 1), win, [], [], Fs);
  [pxsr_r, ~] = pwelch(sr(:, 2)-sr(:, 1), win, [], [], Fs);
  [pxsp_r, ~] = pwelch(sp(:, 2)-sp(:, 1), win, [], [], Fs);
#+end_src

The Power Spectral Density of the Granite Velocity, Sample velocity and relative velocity are compare in Fig. [[fig:rz_psd_sample_granite_relative_comp]].
#+begin_src matlab :exports results
  figure;
  hold on;
  plot(f, sqrt(pxof_m)./(2*pi*f), 'DisplayName', 'Granite');
  plot(f, sqrt(pxof_s)./(2*pi*f), 'DisplayName', 'Sample');
  plot(f, sqrt(pxof_r)./(2*pi*f), 'DisplayName', 'Diff');
  hold off;
  set(gca, 'xscale', 'log'); set(gca, 'yscale', 'log');
  xlabel('Frequency [Hz]'); ylabel('ASD of the relative velocity $\left[\frac{m/s}{\sqrt{Hz}}\right]$')
  legend('Location', 'southwest');
  xlim([2, 500]);
#+end_src

#+HEADER: :tangle no :exports results :results none :noweb yes
#+begin_src matlab :var filepath="figs/rz_psd_sample_granite_relative_comp.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
  <<plt-matlab>>
#+end_src

#+NAME: fig:rz_psd_sample_granite_relative_comp
#+CAPTION: Comparison of the PSD of the velocity of the Sample, Granite and relative velocity ([[./figs/rz_psd_sample_granite_relative_comp.png][png]], [[./figs/rz_psd_sample_granite_relative_comp.pdf][pdf]])
[[file:figs/rz_psd_sample_granite_relative_comp.png]]

Then, we display the PSD of the relative velocity for all three cases in Fig. [[fig:sr_sp_psd_relative_compare]].
#+begin_src matlab :results none
  figure;
  hold on;
  plot(f, sqrt(pxsp_r), 'DisplayName', 'Spindle - 6rpm');
  plot(f, sqrt(pxsr_r), 'DisplayName', 'Slip-Ring - 6rpm');
  plot(f, sqrt(pxof_r), 'DisplayName', 'OFF');
  hold off;
  set(gca, 'xscale', 'log');
  set(gca, 'yscale', 'log');
  xlabel('Frequency [Hz]'); ylabel('ASD of the relative velocity $\left[\frac{m/s}{\sqrt{Hz}}\right]$')
  legend('Location', 'southwest');
  xlim([2, 500]);
#+end_src

#+NAME: fig:sr_sp_psd_relative_compare
#+HEADER: :tangle no :exports results :results value raw replace :noweb yes
#+begin_src matlab :var filepath="figs/sr_sp_psd_relative_compare.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
  <<plt-matlab>>
#+end_src

#+NAME: fig:sr_sp_psd_relative_compare
#+CAPTION: Comparison of the ASD of the relative velocity
#+RESULTS: fig:sr_sp_psd_relative_compare
[[file:figs/sr_sp_psd_relative_compare.png]]

The Cumulative Power Spectrum of the relative velocity is shown in Fig. [[fig:dist_rz_cps]] and in Fig. [[fig:dist_rz_cps_reverse]] (integrated in reverse direction).
#+begin_src matlab :results none :exports results
  figure;
  hold on;
  plot(f, cumtrapz(f,pxsp_r), 'DisplayName', 'Spindle - 6rpm');
  plot(f, cumtrapz(f,pxsr_r), 'DisplayName', 'Slip-Ring - 6rpm');
  plot(f, cumtrapz(f,pxof_r), 'DisplayName', 'OFF');
  hold off;
  set(gca, 'xscale', 'log');
  set(gca, 'yscale', 'log');
  xlabel('Frequency [Hz]'); ylabel('CPS of the relative velocity $\left[\frac{(m/s)^2}{Hz}\right]$')
  legend('Location', 'southwest');
  xlim([2, 500]);
#+end_src

#+HEADER: :tangle no :exports results :results none :noweb yes
#+begin_src matlab :var filepath="figs/dist_rz_cps.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
  <<plt-matlab>>
#+end_src

#+NAME: fig:dist_rz_cps
#+CAPTION: Cumulative Power Spectrum of the relative velocity ([[./figs/dist_rz_cps.png][png]], [[./figs/dist_rz_cps.pdf][pdf]])
[[file:figs/dist_rz_cps.png]]


#+begin_src matlab :exports results
  figure;
  hold on;
  plot(f, flip(-cumtrapz(flip(f), flip(pxsp_r))), 'DisplayName', 'Spindle - 6rpm');
  plot(f, flip(-cumtrapz(flip(f), flip(pxsr_r))), 'DisplayName', 'Slip-Ring - 6rpm');
  plot(f, flip(-cumtrapz(flip(f), flip(pxof_r))), 'DisplayName', 'OFF');
  hold off;
  set(gca, 'xscale', 'log'); set(gca, 'yscale', 'log');
  xlabel('Frequency [Hz]'); ylabel('CPS of the relative velocity $\left[\frac{(m/s)^2}{Hz}\right]$')
  legend('Location', 'southwest');
  xlim([2, 500]);
#+end_src

#+HEADER: :tangle no :exports results :results none :noweb yes
#+begin_src matlab :var filepath="figs/dist_rz_cps_reverse.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
  <<plt-matlab>>
#+end_src

#+NAME: fig:dist_rz_cps_reverse
#+CAPTION: Cumulative Power Spectrum of the relative velocity (integrated from high to low frequencies) ([[./figs/dist_rz_cps_reverse.png][png]], [[./figs/dist_rz_cps_reverse.pdf][pdf]])
[[file:figs/dist_rz_cps_reverse.png]]


Finally, the Cumulative Amplitude Spectrum of the relative position between the hexapod and the marble is shown in Fig. [[fig:dist_rz_cas]].
#+begin_src matlab :results none
  figure;
  hold on;
  plot(f, sqrt(flip(-cumtrapz(flip(f), flip(pxsp_r./(2*pi*f).^2)))), 'DisplayName', 'Spindle - 6rpm');
  plot(f, sqrt(flip(-cumtrapz(flip(f), flip(pxsr_r./(2*pi*f).^2)))), 'DisplayName', 'Slip-Ring - 6rpm');
  plot(f, sqrt(flip(-cumtrapz(flip(f), flip(pxof_r./(2*pi*f).^2)))), 'DisplayName', 'OFF');
  hold off;
  set(gca, 'xscale', 'log'); set(gca, 'yscale', 'log');
  xlabel('Frequency [Hz]'); ylabel('CAS of the relative displacement $\left[\frac{m}{\sqrt{Hz}}\right]$')
  legend('Location', 'southwest');
  xlim([2, 500]);
#+end_src

#+HEADER: :tangle no :exports results :results none :noweb yes
#+begin_src matlab :var filepath="figs/dist_rz_cas.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
  <<plt-matlab>>
#+end_src

#+NAME: fig:dist_rz_cas
#+CAPTION: Cumulative Amplitude Spectrum of the relative motion Hexapod/Granite ([[./figs/dist_rz_cas.png][png]], [[./figs/dist_rz_cas.pdf][pdf]])
[[file:figs/dist_rz_cas.png]]

** Save
The Power Spectral Density of the relative velocity and of the hexapod velocity is saved for further analysis.
#+begin_src matlab
  save('mat/pxsp_r.mat', 'f', 'pxsp_r', 'pxsp_s');
#+end_src

** Conclusion
#+begin_important
The relative motion below 20Hz is dominated by another effect than the rotation of the Spindle (probably ground motion).

The Slip-Ring rotation induce almost no relative motion of the hexapod with respect to the granite (only a little above 400Hz).

The Spindle rotation induces relative motion of the hexapod with respect to the granite above 20Hz.
#+end_important

#+begin_important
There is a huge peak at 24Hz on the sample vibration but not on the granite vibration
- The peak is really sharp, could this be due to magnetic effect?
- Should redo the measurement with piezo accelerometers.
#+end_important