Integration of geophone to have the displacement

2018-10-15 - Marc/index.org

@@ -242,7 +242,51 @@ Measurements will be normalized by the inverse of this transfer function in orde
 #+RESULTS: fig:L4C_bode_plot
 [[file:figs/L4C_bode_plot.png]]
 
-Time domain data are just normalized using the sensibility of the sensor ($276.8 V/m/s$).
+#+begin_src matlab :results none :exports code
+  meas{1}.Track1 = (meas{1}.Track1)./276.8;
+  meas{1}.Track2 = (meas{1}.Track2)./276.8;
+  meas{1}.Track3 = (meas{1}.Track3)./276.8;
+
+  meas{2}.Track1 = (meas{2}.Track1)./276.8;
+  meas{2}.Track2 = (meas{2}.Track2)./276.8;
+  meas{2}.Track3 = (meas{2}.Track3)./276.8;
+
+  meas{3}.Track1 = (meas{3}.Track1)./276.8;
+  meas{3}.Track2 = (meas{3}.Track2)./276.8;
+  meas{3}.Track3 = (meas{3}.Track3)./276.8;
+
+  meas{4}.Track1 = (meas{4}.Track1)./276.8;
+  meas{4}.Track2 = (meas{4}.Track2)./276.8;
+  meas{4}.Track3 = (meas{4}.Track3)./276.8;
+
+  meas{5}.Track1 = (meas{5}.Track1)./276.8;
+  meas{5}.Track2 = (meas{5}.Track2)./276.8;
+  meas{5}.Track3 = (meas{5}.Track3)./276.8;
+  meas{5}.Track4 = (meas{5}.Track4)./276.8;
+#+end_src
+
+#+begin_src matlab :results none
+  meas{1}.Track1_norm = lsim(inv(L4C_G), meas{1}.Track1, t1);
+  meas{1}.Track2_norm = lsim(inv(L4C_G), meas{1}.Track2, t1);
+  meas{1}.Track3_norm = lsim(inv(L4C_G), meas{1}.Track3, t1);
+
+  meas{2}.Track1_norm = lsim(inv(L4C_G), meas{2}.Track1, t2);
+  meas{2}.Track2_norm = lsim(inv(L4C_G), meas{2}.Track2, t2);
+  meas{2}.Track3_norm = lsim(inv(L4C_G), meas{2}.Track3, t2);
+
+  meas{3}.Track1_norm = lsim(inv(L4C_G), meas{3}.Track1, t3);
+  meas{3}.Track2_norm = lsim(inv(L4C_G), meas{3}.Track2, t3);
+  meas{3}.Track3_norm = lsim(inv(L4C_G), meas{3}.Track3, t3);
+
+  meas{4}.Track1_norm = lsim(inv(L4C_G), meas{4}.Track1, t4);
+  meas{4}.Track2_norm = lsim(inv(L4C_G), meas{4}.Track2, t4);
+  meas{4}.Track3_norm = lsim(inv(L4C_G), meas{4}.Track3, t4);
+
+  meas{5}.Track1_norm = lsim(inv(L4C_G), meas{5}.Track1, t5);
+  meas{5}.Track2_norm = lsim(inv(L4C_G), meas{5}.Track2, t5);
+  meas{5}.Track3_norm = lsim(inv(L4C_G), meas{5}.Track3, t5);
+  meas{5}.Track4_norm = lsim(inv(L4C_G), meas{5}.Track4, t5);
+#+end_src
 
 * Measurement 1 - Effect of Ty stage
 The configuration for this measurement is shown table [[tab:conf_meas1]].
@@ -267,9 +311,9 @@ We then plot the measurements in time domain (figure [[fig:meas1]]).
 #+begin_src matlab :exports none :results silent
   figure;
   hold on;
-  plot(t1(ceil(300/dt):ceil(340/dt)), (1/276.8).*meas{1}.Track1(ceil(300/dt):ceil(340/dt)));
+  plot(t1(ceil(300/dt):ceil(340/dt)), meas{1}.Track1(ceil(300/dt):ceil(340/dt)));
-  plot(t1(ceil(300/dt):ceil(340/dt)), (1/276.8).*meas{1}.Track2(ceil(300/dt):ceil(340/dt)));
+  plot(t1(ceil(300/dt):ceil(340/dt)), meas{1}.Track2(ceil(300/dt):ceil(340/dt)));
-  plot(t1(ceil(300/dt):ceil(340/dt)), (1/276.8).*meas{1}.Track3(ceil(300/dt):ceil(340/dt)));
+  plot(t1(ceil(300/dt):ceil(340/dt)), meas{1}.Track3(ceil(300/dt):ceil(340/dt)));
   hold off;
   xlabel('Time [s]'); ylabel('Velocity [m/s]');
   legend({meas{1}.Track1_Name, meas{1}.Track2_Name, meas{1}.Track3_Name}, 'Location', 'northeast')
@@ -287,17 +331,14 @@ We then plot the measurements in time domain (figure [[fig:meas1]]).
 [[file:figs/meas1.png]]
 
 To understand what is going on, instead of looking at the velocity, we can look at the displacement by integrating the data. The displacement is computed by integrating the velocity using =cumtrapz= function.
-#+begin_src matlab :exports code :results silent
-  tdisp = t1(ceil(300/dt):ceil(340/dt));
-  xdisp = cumtrapz(tdisp, (1/276.8).*meas{1}.Track3(ceil(300/dt):ceil(340/dt)));
-#+end_src
 
 Then we plot the position with respect to time (figure [[fig:meas1_disp]]).
 #+begin_src matlab :exports none :results silent
   figure;
   hold on;
-  plot(tdisp, xdisp);
+  plot(t1, cumtrapz(t1, meas{1}.Track3));
   hold off;
+  xlim([300, 340]);
   xlabel('Time [s]'); ylabel('Displacement [m]');
 #+end_src
 
@@ -428,9 +469,9 @@ We plot the time domain (figure [[fig:meas2]]) and we don't observe anything spe
 #+begin_src matlab :exports results :results silent
   figure;
   hold on;
-  plot(t2(ceil(300/dt):ceil(350/dt)), (1/276.8).*meas{2}.Track1(ceil(300/dt):ceil(350/dt)));
+  plot(t2(ceil(300/dt):ceil(350/dt)), meas{2}.Track1(ceil(300/dt):ceil(350/dt)));
-  plot(t2(ceil(300/dt):ceil(350/dt)), (1/276.8).*meas{2}.Track3(ceil(300/dt):ceil(350/dt)));
+  plot(t2(ceil(300/dt):ceil(350/dt)), meas{2}.Track3(ceil(300/dt):ceil(350/dt)));
-  plot(t2(ceil(300/dt):ceil(350/dt)), (1/276.8).*meas{2}.Track2(ceil(300/dt):ceil(350/dt)));
+  plot(t2(ceil(300/dt):ceil(350/dt)), meas{2}.Track2(ceil(300/dt):ceil(350/dt)));
   hold off;
   xlabel('Time [s]'); ylabel('Velocity [m/s]');
   legend({meas{2}.Track1_Name, meas{2}.Track2_Name, meas{2}.Track3_Name}, 'Location', 'northeast')
@@ -448,6 +489,29 @@ We plot the time domain (figure [[fig:meas2]]) and we don't observe anything spe
 #+RESULTS: fig:meas2
 [[file:figs/meas2.png]]
 
+#+begin_src matlab :exports none :results silent
+  figure;
+  hold on;
+  plot(t2, cumtrapz(t2, meas{2}.Track1));
+  plot(t2, cumtrapz(t2, meas{2}.Track2));
+  plot(t2, cumtrapz(t2, meas{2}.Track3));
+  hold off;
+  xlim([300, 350]);
+  xlabel('Time [s]'); ylabel('Displacement [m]');
+  legend({meas{2}.Track1_Name, meas{2}.Track2_Name, meas{2}.Track3_Name}, 'Location', 'northeast')
+#+end_src
+
+#+NAME: fig:meas2_disp
+#+HEADER: :tangle no :exports results :results raw :noweb yes
+#+begin_src matlab :var filepath="figs/meas2_disp.pdf" :var figsize="wide-normal" :post pdf2svg(file=*this*, ext="png")
+  <<plt-matlab>>
+#+end_src
+
+#+LABEL: fig:meas2_disp
+#+CAPTION: Time domain - measurement 2
+#+RESULTS: fig:meas2_disp
+[[file:figs/meas2_disp.png]]
+
 We compute the PSD of each track and we plot them (figures [[fig:meas2_ry_z_psd]], [[fig:meas2_ry_tilt_psd]] and [[fig:meas2_ty_y_psd]] ).
 #+begin_src matlab :exports code :results silent
   [pxx211, f21] = pwelch(meas{2}.Track1(1:ceil(326/dt)),   psd_window, [], [], Fs);
@@ -557,9 +621,9 @@ The time domain result is shown figure [[fig:meas3]].
 #+begin_src matlab :exports results :results silent
   figure;
   hold on;
-  plot(t3(ceil(380/dt):ceil(420/dt)), (1/276.8).*meas{3}.Track1(ceil(380/dt):ceil(420/dt)));
+  plot(t3(ceil(380/dt):ceil(420/dt)), meas{3}.Track1(ceil(380/dt):ceil(420/dt)));
-  plot(t3(ceil(380/dt):ceil(420/dt)), (1/276.8).*meas{3}.Track2(ceil(380/dt):ceil(420/dt)));
+  plot(t3(ceil(380/dt):ceil(420/dt)), meas{3}.Track2(ceil(380/dt):ceil(420/dt)));
-  plot(t3(ceil(380/dt):ceil(420/dt)), (1/276.8).*meas{3}.Track3(ceil(380/dt):ceil(420/dt)));
+  plot(t3(ceil(380/dt):ceil(420/dt)), meas{3}.Track3(ceil(380/dt):ceil(420/dt)));
   hold off;
   xlabel('Time [s]'); ylabel('Velocity [m/s]');
   legend({meas{3}.Track1_Name, meas{3}.Track2_Name, meas{3}.Track3_Name}, 'Location', 'northeast')
@@ -576,6 +640,29 @@ The time domain result is shown figure [[fig:meas3]].
 #+RESULTS: fig:meas3
 [[file:figs/meas3.png]]
 
+#+begin_src matlab :exports none :results silent
+  figure;
+  hold on;
+  plot(t3, cumtrapz(t3, meas{3}.Track1));
+  plot(t3, cumtrapz(t3, meas{3}.Track2));
+  plot(t3, cumtrapz(t3, meas{3}.Track3));
+  hold off;
+  xlim([350, 450]);
+  xlabel('Time [s]'); ylabel('Displacement [m]');
+  legend({meas{3}.Track1_Name, meas{3}.Track2_Name, meas{3}.Track3_Name}, 'Location', 'northeast')
+#+end_src
+
+#+NAME: fig:meas3_disp
+#+HEADER: :tangle no :exports results :results raw :noweb yes
+#+begin_src matlab :var filepath="figs/meas3_disp.pdf" :var figsize="wide-noral" :post pdf2svg(file=*this*, ext="png")
+  <<plt-matlab>>
+#+end_src
+
+#+LABEL: fig:meas3_disp
+#+CAPTION: Time domain - measurement 3
+#+RESULTS: fig:meas3_disp
+[[file:figs/meas3_disp.png]]
+
 We then compute the PSD of each track before and after turning off the hexapod and plot the results in the figures [[fig:meas3_hexa_z_psd]], [[fig:meas3_ry_z_psd]] and [[fig:meas3_ty_y_psd]].
 #+begin_src matlab :exports code :results silent
   [pxx311, f31] = pwelch(meas{3}.Track1(1:ceil(400/dt)),   psd_window, [], [], Fs);
@@ -685,10 +772,11 @@ The slip ring is turned on at 300s, then the spindle is turned on at 620s (table
 #+begin_src matlab :exports results :results silent
   figure;
   hold on;
-  plot(t4, (1/276.8).*meas{4}.Track1);
+  plot(t4, meas{4}.Track1);
-  plot(t4, (1/276.8).*meas{4}.Track2);
+  plot(t4, meas{4}.Track2);
-  plot(t4, (1/276.8).*meas{4}.Track3);
+  plot(t4, meas{4}.Track3);
   hold off;
+  xlim([t4(1), t4(end)]);
   xlabel('Time [s]'); ylabel('Velocity [m/s]');
   legend({meas{4}.Track1_Name, meas{4}.Track2_Name, meas{4}.Track3_Name}, 'Location', 'southwest')
 #+end_src
@@ -704,6 +792,41 @@ The slip ring is turned on at 300s, then the spindle is turned on at 620s (table
 #+RESULTS: fig:meas4
 [[file:figs/meas4.png]]
 
+#+begin_src matlab :exports none :results silent
+  figure;
+  subaxis(1, 2, 1);
+  hold on;
+  plot(t4, cumtrapz(t4, meas{4}.Track1));
+  plot(t4, cumtrapz(t4, meas{4}.Track2));
+  plot(t4, cumtrapz(t4, meas{4}.Track3));
+  hold off;
+  xlim([250, 350]);
+  xlabel('Time [s]'); ylabel('Displacement [m]');
+  legend({meas{4}.Track1_Name, meas{4}.Track2_Name, meas{4}.Track3_Name}, 'Location', 'northwest')
+
+  subaxis(1, 2, 2);
+  hold on;
+  plot(t4, cumtrapz(t4, meas{4}.Track1));
+  plot(t4, cumtrapz(t4, meas{4}.Track2));
+  plot(t4, cumtrapz(t4, meas{4}.Track3));
+  hold off;
+  xlim([600, 650]);
+  xlabel('Time [s]'); ylabel('Displacement [m]');
+#+end_src
+
+If we integrate this signal, we obtain Figure [[fig:meas4_int]].
+
+#+NAME: fig:meas4_int
+#+HEADER: :tangle no :exports results :results raw :noweb yes
+#+begin_src matlab :var filepath="figs/meas4_int.pdf" :var figsize="full-normal" :post pdf2svg(file=*this*, ext="png")
+  <<plt-matlab>>
+#+end_src
+
+#+LABEL: fig:meas4_int
+#+CAPTION: Time domain - measurement 4
+#+RESULTS: fig:meas4_int
+[[file:figs/meas4_int.png]]
+
 The PSD of each track are computed using the code below.
 #+begin_src matlab :exports none :results silent
   [pxx411, f41] = pwelch(meas{4}.Track1(1:ceil(280/dt)),            psd_window, [], [], Fs);
@@ -717,6 +840,22 @@ The PSD of each track are computed using the code below.
   [pxx431, ~] = pwelch(meas{4}.Track3(1:ceil(280/dt)),            psd_window, [], [], Fs);
   [pxx432, ~] = pwelch(meas{4}.Track3(ceil(280/dt):ceil(600/dt)), psd_window, [], [], Fs);
   [pxx433, ~] = pwelch(meas{4}.Track3(ceil(640/dt):end),          psd_window, [], [], Fs);
+
+  f41 = f41(2:end);
+  f42 = f42(2:end);
+  f43 = f43(2:end);
+
+  pxx411 = pxx411(2:end);
+  pxx412 = pxx412(2:end);
+  pxx413 = pxx413(2:end);
+
+  pxx421 = pxx421(2:end);
+  pxx422 = pxx422(2:end);
+  pxx423 = pxx423(2:end);
+
+  pxx431 = pxx431(2:end);
+  pxx432 = pxx432(2:end);
+  pxx433 = pxx433(2:end);
 #+end_src
 
 #+begin_src matlab :exports none :results silent
@@ -745,6 +884,51 @@ The PSD of each track are computed using the code below.
 #+RESULTS: fig:meas4_hexa_z_psd
 [[file:figs/meas4_hexa_z_psd.png]]
 
+We plot the PSD of the displacement.
+#+begin_src matlab :exports none :results silent
+  figure;
+  hold on;
+  plot(f41, sqrt(pxx411)./abs(squeeze(freqresp(L4C_G, f41, 'Hz')))./(2*pi*f41));
+  plot(f42, sqrt(pxx412)./abs(squeeze(freqresp(L4C_G, f42, 'Hz')))./(2*pi*f42));
+  plot(f43, sqrt(pxx413)./abs(squeeze(freqresp(L4C_G, f43, 'Hz')))./(2*pi*f43));
+  xlim([fmin, fmax]);
+  xticks([1, 10, 100]);
+  set(gca, 'xscale', 'log'); set(gca, 'yscale', 'log');
+  xlabel('Frequency [Hz]'); ylabel('PSD [$m/\sqrt{Hz}$]');
+  title(sprintf('%s', meas{4}.Track1_Name));
+  legend({'0-280', '320-600', '640-end'}, 'Location', 'southwest');
+  hold off;
+#+end_src
+
+#+NAME: fig:meas4_hexa_z_psd_int
+#+HEADER: :tangle no :exports results :results raw :noweb yes
+#+begin_src matlab :var filepath="figs/meas4_hexa_z_psd_int.pdf" :var figsize="wide-normal" :post pdf2svg(file=*this*, ext="png")
+  <<plt-matlab>>
+#+end_src
+
+#+LABEL: fig:meas4_hexa_z_psd_int
+#+CAPTION: PSD_INT of the Z velocity of the Hexapod - measurement 4
+#+RESULTS: fig:meas4_hexa_z_psd_int
+[[file:figs/meas4_hexa_z_psd_int.png]]
+
+And we compute the Cumulative amplitude spectrum.
+#+begin_src matlab :exports none :results silent
+  figure;
+  hold on;
+  plot(f41, sqrt(cumsum(pxx431./abs(squeeze(freqresp(L4C_G, f41, 'Hz'))).^2./(2*pi*f41).*(f41 - [0; f41(1:end-1)]))));
+  plot(f42, sqrt(cumsum(pxx432./abs(squeeze(freqresp(L4C_G, f42, 'Hz'))).^2./(2*pi*f42).*(f42 - [0; f42(1:end-1)]))));
+  plot(f43, sqrt(cumsum(pxx433./abs(squeeze(freqresp(L4C_G, f43, 'Hz'))).^2./(2*pi*f43).*(f43 - [0; f43(1:end-1)]))));
+  xlim([fmin, fmax]);
+  xticks([1, 10, 100]);
+  set(gca, 'xscale', 'log'); set(gca, 'yscale', 'log');
+  xlabel('Frequency [Hz]'); ylabel('CAS [$m$ rms]');
+  title(sprintf('%s', meas{4}.Track1_Name));
+  legend({'0-280', '320-600', '640-end'}, 'Location', 'southwest');
+  hold off;
+#+end_src
+
+
+
 #+begin_src matlab :exports none :results silent
   figure;
   hold on;
@@ -818,10 +1002,10 @@ The time domain signals are shown on figure [[fig:meas5]].
 #+begin_src matlab :exports results :results silent
   figure;
   hold on;
-  plot(t5, (1/276.8).*meas{5}.Track1);
+  plot(t5, meas{5}.Track1);
-  plot(t5, (1/276.8).*meas{5}.Track2);
+  plot(t5, meas{5}.Track2);
-  plot(t5, (1/276.8).*meas{5}.Track3);
+  plot(t5, meas{5}.Track3);
-  plot(t5, (1/276.8).*meas{5}.Track4);
+  plot(t5, meas{5}.Track4);
   hold off;
   xlabel('Time [s]'); ylabel('Velocity [m/s]');
   legend({meas{5}.Track1_Name, meas{5}.Track2_Name, meas{5}.Track3_Name, meas{5}.Track4_Name}, 'Location', 'northeast')