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

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

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

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#+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:

The report made by the PEL is accessible [[file:documents/Spindle_report_test.pdf][here]].

* Notes
#+name: fig:setup_spindle
#+caption: Measurement setup at the PEL lab
[[file:./img/setup_spindle.png]]

| *Date*     | 2017-04-25 |
| *Location* |    PEL Lab |

The goal is to estimate all the error motions induced by the Spindle

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

** ZIP file containing the data and matlab files                     :ignore:
#+begin_src bash :exports none :results none
  if [ matlab/spindle_data_processing.m -nt data/spindle_data_processing.zip ]; then
    cp matlab/spindle_data_processing.m spindle_data_processing.m;
    zip data/spindle_data_processing \
        mat/10turns_1rpm_icepap.txt \
        mat/10turns_60rpm_IcepapFIR.txt \
        src/getAsynchronousError.m \
        spindle_data_processing.m
    rm spindle_data_processing.m;
  fi
#+end_src

#+begin_note
  All the files (data and Matlab scripts) are accessible [[file:data/spindle_data_processing.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 :exports none :results silent :noweb yes
  %% Add the getAsynchronousError to path
  addpath('./src/');
#+end_src

** Load Measurement Data
#+begin_src matlab :results none :exports code
  spindle_1rpm_table  = readtable('./mat/10turns_1rpm_icepap.txt');
  spindle_60rpm_table = readtable('./mat/10turns_60rpm_IcepapFIR.txt');
#+end_src

#+begin_src matlab :results output :exports code
  spindle_1rpm_table(1, :)
#+end_src

#+begin_src matlab :results none :exports code
  spindle_1rpm  = table2array(spindle_1rpm_table);
  spindle_60rpm = table2array(spindle_60rpm_table);
#+end_src

** Convert Signals from [deg] to [sec]
#+begin_src matlab :results none :exports code
  speed_1rpm = 360/60; % [deg/sec]
  spindle_1rpm(:, 1) = spindle_1rpm(:, 2)/speed_1rpm;  % From position [deg] to time [s]

  speed_60rpm = 360/1; % [deg/sec]
  spindle_60rpm(:, 1) = spindle_60rpm(:, 2)/speed_60rpm; % From position [deg] to time [s]
#+end_src

** Convert Signals
#+begin_src matlab :results none :exports code
  % scaling = 1/80000; % 80 mV/um
  scaling = 1e-6; % [um] to [m]

  spindle_1rpm(:, 3:end) = scaling*spindle_1rpm(:, 3:end); % [V] to [m]
  spindle_1rpm(:, 3:end) = spindle_1rpm(:, 3:end)-mean(spindle_1rpm(:, 3:end)); % Remove mean

  spindle_60rpm(:, 3:end) = scaling*spindle_60rpm(:, 3:end); % [V] to [m]
  spindle_60rpm(:, 3:end) = spindle_60rpm(:, 3:end)-mean(spindle_60rpm(:, 3:end)); % Remove mean
#+end_src

** Ts and Fs for both measurements
#+begin_src matlab :results none :exports code
  Ts_1rpm = spindle_1rpm(end, 1)/(length(spindle_1rpm(:, 1))-1);
  Fs_1rpm = 1/Ts_1rpm;

  Ts_60rpm = spindle_60rpm(end, 1)/(length(spindle_60rpm(:, 1))-1);
  Fs_60rpm = 1/Ts_60rpm;
#+end_src

** Find Noise of the ADC [$\frac{m}{\sqrt{Hz}}$]
#+begin_src matlab :results none :exports code
  data = spindle_1rpm(:, 5);
  dV_1rpm = min(abs(data(1) - data(data ~= data(1))));
  noise_1rpm = dV_1rpm/sqrt(12*Fs_1rpm/2);

  data = spindle_60rpm(:, 5);
  dV_60rpm = min(abs(data(50) - data(data ~= data(50))));
  noise_60rpm = dV_60rpm/sqrt(12*Fs_60rpm/2);
#+end_src

** Save all the data under spindle struct
#+begin_src matlab :results none :exports code
  spindle.rpm1.time = spindle_1rpm(:, 1);
  spindle.rpm1.deg  = spindle_1rpm(:, 2);
  spindle.rpm1.Ts   = Ts_1rpm;
  spindle.rpm1.Fs   = 1/Ts_1rpm;
  spindle.rpm1.x    = spindle_1rpm(:, 3);
  spindle.rpm1.y    = spindle_1rpm(:, 4);
  spindle.rpm1.z    = spindle_1rpm(:, 5);
  spindle.rpm1.adcn = noise_1rpm;

  spindle.rpm60.time = spindle_60rpm(:, 1);
  spindle.rpm60.deg  = spindle_60rpm(:, 2);
  spindle.rpm60.Ts   = Ts_60rpm;
  spindle.rpm60.Fs   = 1/Ts_60rpm;
  spindle.rpm60.x    = spindle_60rpm(:, 3);
  spindle.rpm60.y    = spindle_60rpm(:, 4);
  spindle.rpm60.z    = spindle_60rpm(:, 5);
  spindle.rpm60.adcn = noise_60rpm;
#+end_src

** Compute Asynchronous data
#+begin_src matlab :results none :exports code
  for direction = {'x', 'y', 'z'}
      spindle.rpm1.([direction{1}, 'async']) = getAsynchronousError(spindle.rpm1.(direction{1}), 10);
      spindle.rpm60.([direction{1}, 'async']) = getAsynchronousError(spindle.rpm60.(direction{1}), 10);
  end
#+end_src

** Save data
#+begin_src matlab
  save('./mat/spindle_data.mat', 'spindle');
#+end_src

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

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

#+begin_note
  All the files (data and Matlab scripts) are accessible [[file:data/spindle_time_domain.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 none :noweb yes
<<matlab-init>>
#+end_src

** Load the processed data
#+begin_src matlab
  load('./mat/spindle_data.mat', 'spindle');
#+end_src

** Plot X-Y-Z position with respect to Time - 1rpm
#+begin_src matlab :results none :exports code
  figure;
  hold on;
  plot(spindle.rpm1.time, spindle.rpm1.x);
  plot(spindle.rpm1.time, spindle.rpm1.y);
  plot(spindle.rpm1.time, spindle.rpm1.z);
  hold off;
  xlabel('Time [s]'); ylabel('Amplitude [m]');
  legend({'tx - 1rpm', 'ty - 1rpm', 'tz - 1rpm'});
#+end_src

#+NAME: fig:spindle_xyz_1rpm
#+HEADER: :tangle no :exports results :results raw :noweb yes
#+begin_src matlab :var filepath="figs/spindle_xyz_1rpm.pdf" :var figsize="wide-tall" :post pdf2svg(file=*this*, ext="png")
  <<plt-matlab>>
#+end_src

#+NAME: fig:spindle_xyz_1rpm
#+CAPTION: Raw time domain translation - 1rpm
#+RESULTS: fig:spindle_xyz_1rpm
[[file:figs/spindle_xyz_1rpm.png]]

** Plot X-Y-Z position with respect to Time - 60rpm
The measurements for the spindle turning at 60rpm are shown figure [[fig:spindle_xyz_60rpm]].

#+begin_src matlab :results none :exports code
  figure;
  hold on;
  plot(spindle.rpm60.time, spindle.rpm60.x);
  plot(spindle.rpm60.time, spindle.rpm60.y);
  plot(spindle.rpm60.time, spindle.rpm60.z);
  hold off;
  xlabel('Time [s]'); ylabel('Amplitude [m]');
  legend({'tx - 60rpm', 'ty - 60rpm', 'tz - 60rpm'});
#+end_src

#+NAME: fig:spindle_xyz_60rpm
#+HEADER: :tangle no :exports results :results raw :noweb yes
#+begin_src matlab :var filepath="figs/spindle_xyz_60rpm.pdf" :var figsize="wide-tall" :post pdf2svg(file=*this*, ext="png")
  <<plt-matlab>>
#+end_src

#+NAME: fig:spindle_xyz_60rpm
#+CAPTION: Raw time domain translation - 60rpm
#+RESULTS: fig:spindle_xyz_60rpm
[[file:figs/spindle_xyz_60rpm.png]]

** Plot Synchronous and Asynchronous - 1rpm
#+begin_src matlab :results none :exports code
  figure;
  hold on;
  plot(spindle.rpm1.time, spindle.rpm1.x);
  plot(spindle.rpm1.time, spindle.rpm1.xasync);
  hold off;
  xlabel('Time [s]'); ylabel('Amplitude [m]');
  legend({'tx - 1rpm - Sync', 'tx - 1rpm - Async'});
#+end_src

#+NAME: fig:spindle_1rpm_sync_async
#+HEADER: :tangle no :exports results :results raw :noweb yes
#+begin_src matlab :var filepath="figs/spindle_1rpm_sync_async.pdf" :var figsize="wide-tall" :post pdf2svg(file=*this*, ext="png")
  <<plt-matlab>>
#+end_src

#+NAME: fig:spindle_1rpm_sync_async
#+CAPTION: Comparison of the synchronous and asynchronous displacements - 1rpm
#+RESULTS: fig:spindle_1rpm_sync_async
[[file:figs/spindle_1rpm_sync_async.png]]

** Plot Synchronous and Asynchronous - 60rpm
The data is split into its Synchronous and Asynchronous part (figure [[fig:spindle_60rpm_sync_async]]). We then use the Asynchronous part for the analysis in the following sections as we suppose that we can deal with the synchronous part with feedforward control.

#+begin_src matlab :results none :exports code
  figure;
  hold on;
  plot(spindle.rpm60.time, spindle.rpm60.x);
  plot(spindle.rpm60.time, spindle.rpm60.xasync);
  hold off;
  xlabel('Time [s]'); ylabel('Amplitude [m]');
  legend({'tx - 60rpm - Sync', 'tx - 60rpm - Async'});
#+end_src

#+NAME: fig:spindle_60rpm_sync_async
#+HEADER: :tangle no :exports results :results raw :noweb yes
#+begin_src matlab :var filepath="figs/spindle_60rpm_sync_async.pdf" :var figsize="wide-tall" :post pdf2svg(file=*this*, ext="png")
  <<plt-matlab>>
#+end_src

#+NAME: fig:spindle_60rpm_sync_async
#+CAPTION: Comparison of the synchronous and asynchronous displacements - 60rpm
#+RESULTS: fig:spindle_60rpm_sync_async
[[file:figs/spindle_60rpm_sync_async.png]]

** Plot X against Y
#+begin_src matlab :results none :exports code
  figure;
  hold on;
  plot(spindle.rpm1.x, spindle.rpm1.y);
  plot(spindle.rpm60.x, spindle.rpm60.y);
  hold off;
  xlabel('X Amplitude [m]'); ylabel('Y Amplitude [m]');
  legend({'1rpm', '60rpm'});
#+end_src

#+NAME: fig:spindle_xy_1_60rpm
#+HEADER: :tangle no :exports results :results raw :noweb yes
#+begin_src matlab :var filepath="figs/spindle_xy_1_60rpm.pdf" :var figsize="wide-tall" :post pdf2svg(file=*this*, ext="png")
  <<plt-matlab>>
#+end_src

#+NAME: fig:spindle_xy_1_60rpm
#+CAPTION: Synchronous x-y displacement
#+RESULTS: fig:spindle_xy_1_60rpm
[[file:figs/spindle_xy_1_60rpm.png]]

** Plot X against Y - Asynchronous
#+begin_src matlab :results none :exports code
  figure;
  hold on;
  plot(spindle.rpm1.xasync, spindle.rpm1.yasync);
  plot(spindle.rpm60.xasync, spindle.rpm60.yasync);
  hold off;
  xlabel('X Amplitude [m]'); ylabel('Y Amplitude [m]');
  legend({'1rpm', '60rpm'});
#+end_src

#+NAME: fig:spindle_xy_1_60_rpm_async
#+HEADER: :tangle no :exports results :results raw :noweb yes
#+begin_src matlab :var filepath="figs/spindle_xy_1_60_rpm_async.pdf" :var figsize="wide-tall" :post pdf2svg(file=*this*, ext="png")
  <<plt-matlab>>
#+end_src

#+NAME: fig:spindle_xy_1_60_rpm_async
#+CAPTION: Asynchronous x-y displacement
#+RESULTS: fig:spindle_xy_1_60_rpm_async
[[file:figs/spindle_xy_1_60_rpm_async.png]]

* Model of the spindle
  :PROPERTIES:
  :header-args:matlab+: :tangle matlab/spindle_model.m
  :header-args:matlab+: :comments org :mkdirp yes
  :END:
  <<sec:spindle_model>>

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

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

** Schematic of the model
The model of the spindle used is shown figure [[fig:model_spindle]].

$f$ is the perturbation force of the spindle and $d$ is the measured displacement.

#+name: fig:model_spindle
#+caption: Model of the Spindle
#+attr_latex: :float t
[[./figs/uniaxial-model-spindle.png]]

** 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 :exports none :results silent :noweb yes
  %% Add the getAsynchronousError to path
  addpath('./src/');
#+end_src

** Parameters
#+begin_src matlab :exports code :results none
  mg = 3000; % Mass of granite [kg]
  ms = 50;   % Mass of Spindle [kg]

  kg = 1e8; % Stiffness of granite [N/m]
  ks = 5e7; % Stiffness of spindle [N/m]
#+end_src

** Compute Mass and Stiffness Matrices
#+begin_src matlab :exports code :results none
  Mm = diag([ms, mg]);
  Km = diag([ks, ks+kg]) - diag(ks, -1) - diag(ks, 1);
#+end_src

** Compute resonance frequencies
#+begin_src matlab :exports code :results none
  A = [zeros(size(Mm)) eye(size(Mm)) ; -Mm\Km zeros(size(Mm))];
  eigA = imag(eigs(A))/2/pi;
  eigA = eigA(eigA>0);
  eigA = eigA(1:2);
#+end_src

** From model_damping compute the Damping Matrix
#+begin_src matlab :exports code :results none
  modal_damping = 1e-5;

  ab = [0.5*eigA(1) 0.5/eigA(1) ; 0.5*eigA(2) 0.5/eigA(2)]\[modal_damping ; modal_damping];

  Cm = ab(1)*Mm +ab(2)*Km;
#+end_src

** Define inputs, outputs and state names
#+begin_src matlab :exports code :results none
  StateName = {...
      'xs', ... % Displacement of Spindle [m]
      'xg', ... % Displacement of Granite [m]
      'vs', ... % Velocity of Spindle [m]
      'vg', ... % Velocity of Granite [m]
              };
  StateUnit = {'m', 'm', 'm/s', 'm/s'};

  InputName = {...
      'f' ... % Spindle Force [N]
              };
  InputUnit = {'N'};

  OutputName = {...
      'd' ... % Displacement [m]
               };
  OutputUnit = {'m'};
#+end_src

** Define A, B and C matrices
#+begin_src matlab :exports code :results none
  % A Matrix
  A = [zeros(size(Mm)) eye(size(Mm)) ; ...
       -Mm\Km          -Mm\Cm];

  % B Matrix
  B_low = zeros(length(StateName), length(InputName));
  B_low(strcmp(StateName,'vs'), strcmp(InputName,'f')) =  1;
  B_low(strcmp(StateName,'vg'), strcmp(InputName,'f')) = -1;
  B = blkdiag(zeros(length(StateName)/2), pinv(Mm))*B_low;

  % C Matrix
  C = zeros(length(OutputName), length(StateName));
  C(strcmp(OutputName,'d'), strcmp(StateName,'xs')) =  1;
  C(strcmp(OutputName,'d'), strcmp(StateName,'xg')) = -1;

  % D Matrix
  D = zeros(length(OutputName), length(InputName));
#+end_src

** Generate the State Space Model
#+begin_src matlab :exports code :results none
  sys = ss(A, B, C, D);
  sys.StateName = StateName;
  sys.StateUnit = StateUnit;

  sys.InputName = InputName;
  sys.InputUnit = InputUnit;

  sys.OutputName = OutputName;
  sys.OutputUnit = OutputUnit;
#+end_src

** Bode Plot
The transfer function from a disturbance force $f$ to the measured displacement $d$ is shown figure [[fig:spindle_f_to_d]].

#+begin_src matlab :exports code :results none
  freqs = logspace(-1, 3, 1000);

  figure;
  plot(freqs, abs(squeeze(freqresp(sys('d', 'f'), freqs, 'Hz'))));
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  xlabel('Frequency [Hz]'); ylabel('Amplitude [m/N]');
#+end_src

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

#+NAME: fig:spindle_f_to_d
#+CAPTION: Bode plot of the transfer function from $f$ to $d$
#+RESULTS: fig:spindle_f_to_d
[[file:figs/spindle_f_to_d.png]]

** Save the model
#+begin_src matlab
  save('./mat/spindle_model.mat', 'sys');
#+end_src

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

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

#+begin_note
  All the files (data and Matlab scripts) are accessible [[file:data/spindle_psd.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

** Load the processed data and the model
#+begin_src matlab
  load('./mat/spindle_data.mat', 'spindle');
  load('./mat/spindle_model.mat', 'sys');
#+end_src

** Compute the PSD
#+begin_src matlab :exports code :results none
  n_av = 4; % Number of average

  [pxx_1rpm, f_1rpm] = pwelch(spindle.rpm1.xasync, hanning(ceil(length(spindle.rpm1.xasync)/n_av)), [], [], spindle.rpm1.Fs);
  [pyy_1rpm, ~]      = pwelch(spindle.rpm1.yasync, hanning(ceil(length(spindle.rpm1.yasync)/n_av)), [], [], spindle.rpm1.Fs);
  [pzz_1rpm, ~]      = pwelch(spindle.rpm1.zasync, hanning(ceil(length(spindle.rpm1.zasync)/n_av)), [], [], spindle.rpm1.Fs);

  [pxx_60rpm, f_60rpm] = pwelch(spindle.rpm60.xasync, hanning(ceil(length(spindle.rpm60.xasync)/n_av)), [], [], spindle.rpm60.Fs);
  [pyy_60rpm, ~]       = pwelch(spindle.rpm60.yasync, hanning(ceil(length(spindle.rpm60.yasync)/n_av)), [], [], spindle.rpm60.Fs);
  [pzz_60rpm, ~]       = pwelch(spindle.rpm60.zasync, hanning(ceil(length(spindle.rpm60.zasync)/n_av)), [], [], spindle.rpm60.Fs);
#+end_src

** Plot the computed PSD
The Amplitude Spectral Densities of the displacement of the spindle for the $x$, $y$ and $z$ directions are shown figure [[fig:spindle_psd_xyz_60rpm]]. They correspond to the Asynchronous part shown figure [[fig:spindle_60rpm_sync_async]].

#+begin_src matlab :exports code :results none
  figure;
  hold on;
  plot(f_1rpm, (pxx_1rpm).^.5, 'DisplayName', '$P_{xx}$ - 1rpm');
  plot(f_1rpm, (pyy_1rpm).^.5, 'DisplayName', '$P_{yy}$ - 1rpm');
  plot(f_1rpm, (pzz_1rpm).^.5, 'DisplayName', '$P_{zz}$ - 1rpm');
  % plot(f_1rpm, spindle.rpm1.adcn*ones(size(f_1rpm)), '--k', 'DisplayName', 'ADC - 1rpm');
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  xlabel('Frequency [Hz]'); ylabel('ASD [$m/\sqrt{Hz}$]');
  legend('Location', 'northeast');
  xlim([f_1rpm(2), f_1rpm(end)]);
#+end_src
#+NAME: fig:spindle_psd_xyz_1rpm
#+HEADER: :tangle no :exports results :results raw :noweb yes
#+begin_src matlab :var filepath="figs/spindle_psd_xyz_1rpm.pdf" :var figsize="wide-normal" :post pdf2svg(file=*this*, ext="png")
  <<plt-matlab>>
#+end_src

#+NAME: fig:spindle_psd_xyz_1rpm
#+CAPTION: Power spectral density of the Asynchronous displacement - 1rpm
#+RESULTS: fig:spindle_psd_xyz_1rpm
[[file:figs/spindle_psd_xyz_1rpm.png]]

#+begin_src matlab :exports code :results none
  figure;
  hold on;
  plot(f_60rpm, (pxx_60rpm).^.5, 'DisplayName', '$P_{xx}$ - 60rpm');
  plot(f_60rpm, (pyy_60rpm).^.5, 'DisplayName', '$P_{yy}$ - 60rpm');
  plot(f_60rpm, (pzz_60rpm).^.5, 'DisplayName', '$P_{zz}$ - 60rpm');
  % plot(f_60rpm, spindle.rpm60.adcn*ones(size(f_60rpm)), '--k', 'DisplayName', 'ADC - 60rpm');
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  xlabel('Frequency [Hz]'); ylabel('ASD [$m/\sqrt{Hz}$]');
  legend('Location', 'northeast');
  xlim([f_60rpm(2), f_60rpm(end)]);
#+end_src
#+NAME: fig:spindle_psd_xyz_60rpm
#+HEADER: :tangle no :exports results :results raw :noweb yes
#+begin_src matlab :var filepath="figs/spindle_psd_xyz_60rpm.pdf" :var figsize="wide-normal" :post pdf2svg(file=*this*, ext="png")
  <<plt-matlab>>
#+end_src

#+NAME: fig:spindle_psd_xyz_60rpm
#+CAPTION: Power spectral density of the Asynchronous displacement - 60rpm
#+RESULTS: fig:spindle_psd_xyz_60rpm
[[file:figs/spindle_psd_xyz_60rpm.png]]

** Compute the response of the model
#+begin_src matlab :exports code :results none
  Tfd = abs(squeeze(freqresp(sys('d', 'f'), f_60rpm, 'Hz')));
#+end_src

** Plot the PSD of the Force using the model
#+begin_src matlab :exports code :results none
  figure;
  plot(f_60rpm, (pxx_60rpm.^.5)./Tfd, 'DisplayName', '$P_{xx}$');
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  xlabel('Frequency [Hz]'); ylabel('ASD [$N/\sqrt{Hz}$]');
  xlim([f_60rpm(2), f_60rpm(end)]);
#+end_src
#+NAME: fig:spindle_psd_f_60rpm
#+HEADER: :tangle no :exports results :results raw :noweb yes
#+begin_src matlab :var filepath="figs/spindle_psd_f_60rpm.pdf" :var figsize="wide-normal" :post pdf2svg(file=*this*, ext="png")
  <<plt-matlab>>
#+end_src

#+NAME: fig:spindle_psd_f_60rpm
#+CAPTION: Power spectral density of the force - 60rpm
#+RESULTS: fig:spindle_psd_f_60rpm
[[file:figs/spindle_psd_f_60rpm.png]]

** Estimated Shape of the PSD of the force
#+begin_src matlab :exports code :results none
  s = tf('s');

  Wd_simple = 1e-8/(1+s/2/pi/0.5)/(1+s/2/pi/100);
  Wf_simple = Wd_simple/tf(sys('d', 'f'));
  TWf_simple = abs(squeeze(freqresp(Wf_simple, f_60rpm, 'Hz')));

  % Wf = 0.48902*(s+327.9)*(s^2 + 109.6*s + 1.687e04)/((s^2 + 30.59*s + 8541)*(s^2 + 29.11*s + 3.268e04));
  % Wf = 0.15788*(s+418.6)*(s+1697)^2*(s^2 + 124.3*s + 2.529e04)*(s^2 + 681.3*s + 9.018e05)/((s^2 + 23.03*s + 8916)*(s^2 + 33.85*s + 6.559e04)*(s^2 + 71.43*s + 4.283e05)*(s^2 + 40.64*s + 1.789e06));

  Wf = (s+1697)^2*(s^2 + 114.5*s + 2.278e04)*(s^2 + 205.1*s + 1.627e05)*(s^2 + 285.8*s + 8.624e05)*(s+100)/((s+0.5)*3012*(s^2 + 23.03*s + 8916)*(s^2 + 17.07*s + 4.798e04)*(s^2 + 41.17*s + 4.347e05)*(s^2 + 78.99*s + 1.789e06));

  TWf = abs(squeeze(freqresp(Wf, f_60rpm, 'Hz')));
#+end_src

** PSD in [N]
Above $200Hz$, the Amplitude Spectral Density seems dominated by noise coming from the electronics (charge amplifier, ADC, ...). So we don't know what is the frequency content of the force above that frequency. However, we assume that $P_{xx}$ is decreasing with $1/f$ as it seems so be the case below $100Hz$ (figure [[fig:spindle_psd_xyz_60rpm]]).

We then fit the PSD of the displacement with a transfer function (figure [[fig:spindle_psd_d_comp_60rpm]]).

#+begin_src matlab :exports code :results none
  figure;
  hold on;
  plot(f_60rpm, (pxx_60rpm.^.5)./Tfd, 'DisplayName', '$\sqrt{P_{xx}}/|T_{d/f}|$');
  plot(f_60rpm, TWf, 'DisplayName', 'Wf');
  plot(f_60rpm, TWf_simple, '-k', 'DisplayName', 'Wfs');
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  xlabel('Frequency [Hz]'); ylabel('ASD [$N/\sqrt{Hz}$]');
  xlim([f_60rpm(2), f_60rpm(end)]);
  legend('Location', 'northeast');
#+end_src
#+NAME: fig:spindle_psd_f_comp_60rpm
#+HEADER: :tangle no :exports results :results raw :noweb yes
#+begin_src matlab :var filepath="figs/spindle_psd_f_comp_60rpm.pdf" :var figsize="wide-tall" :post pdf2svg(file=*this*, ext="png")
  <<plt-matlab>>
#+end_src

#+NAME: fig:spindle_psd_f_comp_60rpm
#+CAPTION: Power spectral density of the force - 60rpm
#+RESULTS: fig:spindle_psd_f_comp_60rpm
[[file:figs/spindle_psd_f_comp_60rpm.png]]

** PSD in [m]
To obtain the PSD of the force $f$ that induce such displacement, we use the following formula:
\[ \sqrt{PSD(d)} = |T_{d/f}| \sqrt{PSD(f)} \]

And so we have:
\[ \sqrt{PSD(f)} = |T_{d/f}|^{-1} \sqrt{PSD(d)} \]

The obtain Power Spectral Density of the force is displayed figure [[fig:spindle_psd_f_comp_60rpm]].

#+begin_src matlab :exports code :results none
  figure;
  hold on;
  plot(f_60rpm, pxx_60rpm.^.5, 'DisplayName', '$\sqrt{P_{xx}}$');
  plot(f_60rpm, TWf.*Tfd, 'DisplayName', '$|W_f|*|T_{d/f}|$');
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  xlabel('Frequency [Hz]'); ylabel('ASD [$m/\sqrt{Hz}$]');
  xlim([f_60rpm(2), f_60rpm(end)]);
  legend('Location', 'northeast');
#+end_src

#+NAME: fig:spindle_psd_d_comp_60rpm
#+HEADER: :tangle no :exports results :results raw :noweb yes
#+begin_src matlab :var filepath="figs/spindle_psd_d_comp_60rpm.pdf" :var figsize="wide-tall" :post pdf2svg(file=*this*, ext="png")
  <<plt-matlab>>
#+end_src

#+NAME: fig:spindle_psd_d_comp_60rpm
#+CAPTION: Comparison of the power spectral density of the measured displacement and of the model
#+RESULTS: fig:spindle_psd_d_comp_60rpm
[[file:figs/spindle_psd_d_comp_60rpm.png]]

** Compute the resulting RMS value [m]
#+begin_src matlab :exports code :results none
  figure;
  hold on;
  plot(f_60rpm, 1e9*cumtrapz(f_60rpm, (pxx_60rpm)).^.5, '--', 'DisplayName', 'Exp. Data');
  plot(f_60rpm, 1e9*cumtrapz(f_60rpm, ((TWf.*Tfd).^2)).^.5, '--', 'DisplayName', 'Estimated');
  hold off;
  set(gca, 'XScale', 'log');
  xlabel('Frequency [Hz]'); ylabel('CPS [$nm$ rms]');
  xlim([f_60rpm(2), f_60rpm(end)]);
  legend('Location', 'southeast');
#+end_src

#+NAME: fig:spindle_cps_d_comp_60rpm
#+HEADER: :tangle no :exports results :results raw :noweb yes
#+begin_src matlab :var filepath="figs/spindle_cps_d_comp_60rpm.pdf" :var figsize="wide-normal" :post pdf2svg(file=*this*, ext="png")
  <<plt-matlab>>
#+end_src

#+NAME: fig:spindle_cps_d_comp_60rpm
#+CAPTION: Cumulative Power Spectrum - 60rpm
#+RESULTS: fig:spindle_cps_d_comp_60rpm
[[file:figs/spindle_cps_d_comp_60rpm.png]]

** Compute the resulting RMS value [m]
#+begin_src matlab :exports code :results none
  figure;
  hold on;
  plot(f_1rpm, 1e9*cumtrapz(f_1rpm, (pxx_1rpm)), '--', 'DisplayName', 'Exp. Data');
  plot(f_1rpm, 1e9*(f_1rpm(end)-f_1rpm(1))/(length(f_1rpm)-1)*cumsum(pxx_1rpm), '--', 'DisplayName', 'Exp. Data');
  hold off;
  set(gca, 'XScale', 'log');
  xlabel('Frequency [Hz]'); ylabel('CPS [$nm$ rms]');
  xlim([f_1rpm(2), f_1rpm(end)]);
  legend('Location', 'southeast');
#+end_src

#+NAME: fig:spindle_cps_d_comp_1rpm
#+HEADER: :tangle no :exports results :results raw :noweb yes
#+begin_src matlab :var filepath="figs/spindle_cps_d_comp_1rpm.pdf" :var figsize="wide-normal" :post pdf2svg(file=*this*, ext="png")
  <<plt-matlab>>
#+end_src

#+NAME: fig:spindle_cps_d_comp_1rpm
#+CAPTION: Cumulative Power Spectrum - 1rpm
#+RESULTS: fig:spindle_cps_d_comp_1rpm
[[file:figs/spindle_cps_d_comp_1rpm.png]]

* Functions
** getAsynchronousError
  :PROPERTIES:
  :header-args:matlab+: :tangle src/getAsynchronousError.m
  :header-args:matlab+: :comments org :mkdirp yes
  :END:
  <<sec:getAsynchronousError>>

This Matlab function is accessible [[file:src/getAsynchronousError.m][here]].

#+begin_src matlab :eval no :results none
  function [Wxdec] = getAsynchronousError(data, NbTurn)
  %%
      L = length(data);
      res_per_rev = L/NbTurn;

      P = 0:(res_per_rev*NbTurn-1);
      Pos = P' * 360/res_per_rev;

      % Temperature correction
      x1 = myfit2(Pos, data);

      % Convert data to frequency domain and scale accordingly
      X2 = 2/(res_per_rev*NbTurn)*fft(x1);
      f2 = (0:L-1)./NbTurn; %upr -> once per revolution

      %%
      X2dec = zeros(size(X2));
      % Get only the non integer data
      X2dec(mod(f2(:), 1) ~= 0) = X2(mod(f2(:), 1) ~= 0);

      Wxdec = real((res_per_rev*NbTurn)/2 * ifft(X2dec));

      %%
      function Y =  myfit2(x,y)
          A = [x ones(size(x))]\y;
          a = A(1); b = A(2);
          Y = y - (a*x + b);
      end
  end
#+end_src