nass-micro-station-measurements/disturbance-measurement/index.org

• Experimental Setup
• Signal Processing
  • Load Data
  • Time domain Data
  • ASD and Frequency domain data
  • Some plots
  • Scaling to take into account the sensibility of the geophone and the voltage amplifier
  • Computation of the ASD of the velocity
  • RMS value of the difference between the two geophones

#+TITLE:Measurement of the sample vibrations when rotating the Spindle

Experimental Setup

Signal Processing

Load Data

Measurement data_001.mat corresponds to a measurement where the spindle is not turning and data_002.mat where the spindle is turning at 1rpm. x1 is the signal coming from the geophone located on the marble and x2 is the signal from the geophone located on the sample station.

  data1 = load('mat/data_001.mat', 't', 'x1', 'x2');
  data2 = load('mat/data_002.mat', 't', 'x1', 'x2');

Time domain Data

  figure;
  hold on;
  plot(data1.t, data1.x1);
  plot(data2.t, data2.x1);
  hold off;

  figure;
  hold on;
  plot(data1.t, data1.x2);
  plot(data2.t, data2.x2)
  hold off;

ASD and Frequency domain data

  dt = data1.t(2) - data1.t(1);
  Fs = 1/dt;

  windows_psd = hanning(ceil(10*Fs));

  [pxx1m, f] = pwelch(data1.x1, windows_psd, [], [], Fs); f(1) = []; pxx1m(1) = [];
  [pxx1h, ~] = pwelch(data1.x2, windows_psd, [], [], Fs); pxx1h(1) = [];

  [pxx2m, ~] = pwelch(data2.x1, windows_psd, [], [], Fs); pxx2m(1) = [];
  [pxx2h, ~] = pwelch(data2.x2, windows_psd, [], [], Fs); pxx2h(1) = [];

Some plots

  figure;
  hold on;
  plot(f, sqrt(pxx1m));
  plot(f, sqrt(pxx2m));
  hold off;
  set(gca, 'xscale', 'log');
  set(gca, 'yscale', 'log');
  xlabel('Frequency [Hz]'); ylabel('PSD [m/s/sqrt(Hz)]')

  figure;
  hold on;
  plot(f, sqrt(pxx1h));
  plot(f, sqrt(pxx2h));
  hold off;
  set(gca, 'xscale', 'log');
  set(gca, 'yscale', 'log');
  xlabel('Frequency [Hz]'); ylabel('PSD [m/s/sqrt(Hz)]')

  figure;
  hold on;
  plot(f, sqrt(pxx2m));
  plot(f, sqrt(pxx2h));
  hold off;
  set(gca, 'xscale', 'log');
  set(gca, 'yscale', 'log');
  xlabel('Frequency [Hz]'); ylabel('PSD [m/s/sqrt(Hz)]')

  figure;
  hold on;
  plot(f, cumtrapz(f, pxx1m))
  plot(f, cumtrapz(f, pxx2m))
  set(gca, 'XScale', 'log');
  xlabel('Frequency [Hz]'); ylabel('CAS [m]')

Scaling to take into account the sensibility of the geophone and the voltage amplifier

The Geophone used are L22. Their sensibility is shown on figure fig:geophone_sensibility.

  S0 = 88; % Sensitivity [V/(m/s)]
  f0 = 2; % Cut-off frequnecy [Hz]

  S = S0*(s/2/pi/f0)/(1+s/2/pi/f0);

We also take into account the gain of the electronics which is here set to be $60dB$.

  G0_db = 60; % [dB]

  G0 = 10^(G0_db/20); % [abs]

We divide the ASD measured (in $\text{V}/\sqrt{\text{Hz}}$) by the gain of the voltage amplifier to obtain the ASD of the voltage across the geophone. We further divide the result by the sensibility of the Geophone to obtain the ASD of the velocity in $m/s/\sqrt{Hz}$.

  scaling = 1./squeeze(abs(freqresp(G0*S, f, 'Hz'))); scaling(1) = 0;

Computation of the ASD of the velocity

  figure;
  hold on;
  plot(f, sqrt(pxx1h).*scaling);
  plot(f, sqrt(pxx2h).*scaling);
  hold off;
  set(gca, 'xscale', 'log');
  set(gca, 'yscale', 'log');
  xlabel('Frequency [Hz]'); ylabel('ASD of the measured Velocity $\left[\frac{m/s}{\sqrt{Hz}}\right]$')
  xlim([0.1, 500]);

  figure;
  hold on;
  plot(f, (sqrt(pxx1).*scaling)./(2*pi*f));
  plot(f, (sqrt(pxx2).*scaling)./(2*pi*f));
  hold off;
  set(gca, 'xscale', 'log'); set(gca, 'yscale', 'log');
  xlabel('Frequency [Hz]'); ylabel('ASD of the displacement $\left[\frac{m}{\sqrt{Hz}}\right]$')
  xlim([0.1, 500]);

RMS value of the difference between the two geophones

We also compute the Power Spectral Density of the difference between the two geophones. This is done in order to estimate the relative displacement of the sample with respect to the granite.

  [pxxd1, ~] = pwelch(data1.x2-data1.x1, windows_psd, [], [], Fs); pxxd1(1) = [];
  [pxxd2, ~] = pwelch(data2.x2-data2.x1, windows_psd, [], [], Fs); pxxd2(1) = [];

  figure;
  hold on;
  plot(f, (sqrt(pxxd1).*scaling)./(2*pi*f));
  plot(f, (sqrt(pxxd2).*scaling)./(2*pi*f));
  hold off;
  set(gca, 'xscale', 'log'); set(gca, 'yscale', 'log');
  xlabel('Frequency [Hz]'); ylabel('ASD of the displacement $\left[\frac{m}{\sqrt{Hz}}\right]$')
  xlim([0.1, 500]);

  psd_d1 = ((sqrt(pxxd1).*scaling)./(2*pi*f)).^2;
  psd_d2 = ((sqrt(pxxd2).*scaling)./(2*pi*f)).^2;

  df = f(2) - f(1);

  figure;
  hold on;
  plot(f, sqrt(cumsum(df.*psd_d1, 'reverse')));
  plot(f, sqrt(cumsum(df.*psd_d2, 'reverse')));
  hold off;
  set(gca, 'xscale', 'log'); set(gca, 'yscale', 'log');
  xlabel('Frequency [Hz]'); ylabel('CAS $\left[m\right]$')
  xlim([0.1, 500]);