nass-micro-station-measurements/Spindle/index.org

Spindle Analysis

• Data Processing
  • Load Measurement Data
  • Convert Signals from [deg] to [sec]
  • Convert Signals
  • Ts and Fs for both measurements
  • Find Noise of the ADC [m/sqrt(Hz)]
  • Save all the data under spindle struct
  • Compute Asynchronous data
• Time Domain Data
  • Plot X-Y-Z position with respect to Time - 1rpm
  • Plot X-Y-Z position with respect to Time - 60rpm
  • Plot Synchronous and Asynchronous - 1rpm
  • Plot Synchronous and Asynchronous - 60rpm
  • Plot X against Y
  • Plot X against Y - Asynchronous
• Model of the spindle
  • Parameters
  • Compute Mass and Stiffness Matrices
  • Compute resonance frequencies
  • From model_damping compute the Damping Matrix
  • Define inputs, outputs and state names
  • Define A, B and C matrices
  • Generate the State Space Model
  • Bode Plot
• Power Spectral Density
  • Compute the PSD
  • Plot the computed PSD
  • Load the model of the spindle
  • Plot the PSD of the Force using the model
  • Estimated Shape of the PSD of the force
  • PSD in [N]
  • PSD in [m]
  • Compute the resulting RMS value [m]
  • Compute the resulting RMS value [m]
• Functions
  • getAsynchronousError

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Data Processing

Load Measurement Data

  spindle_1rpm_table  = readtable('./data/10turns_1rpm_icepap.txt');
  spindle_60rpm_table = readtable('./data/10turns_60rpm_IcepapFIR.txt');

  spindle_1rpm_table(1, :)

  spindle_1rpm  = table2array(spindle_1rpm_table);
  spindle_60rpm = table2array(spindle_60rpm_table);

Convert Signals from [deg] to [sec]

  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]

Convert Signals

  % 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

Ts and Fs for both measurements

  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;

Find Noise of the ADC [m/sqrt(Hz)]

  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);

Save all the data under spindle struct

  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;

Compute Asynchronous data

  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

Time Domain Data

Plot X-Y-Z position with respect to Time - 1rpm

  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'});

  <<plt-matlab>>

Raw time domain translation - 1rpm

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.

  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'});

  <<plt-matlab>>

Raw time domain translation - 60rpm

Plot Synchronous and Asynchronous - 1rpm

  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'});

  <<plt-matlab>>

Comparison of the synchronous and asynchronous displacements - 1rpm

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.

  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'});

  <<plt-matlab>>

Comparison of the synchronous and asynchronous displacements - 60rpm

Plot X against Y

  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'});

  <<plt-matlab>>

Synchronous x-y displacement

Plot X against Y - Asynchronous

  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'});

  <<plt-matlab>>

Asynchronous x-y displacement

Model of the spindle

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.

Parameters

  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]

Compute Mass and Stiffness Matrices

  Mm = diag([ms, mg]);
  Km = diag([ks, ks+kg]) - diag(ks, -1) - diag(ks, 1);

Compute resonance frequencies

  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);

From model_damping compute the Damping Matrix

  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;

Define inputs, outputs and state names

  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'};

Define A, B and C matrices

  % 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));

Generate the State Space Model

  sys = ss(A, B, C, D);
  sys.StateName = StateName;
  sys.StateUnit = StateUnit;

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

  sys.OutputName = OutputName;
  sys.OutputUnit = OutputUnit;

Bode Plot

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

  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]');

  <<plt-matlab>>

Bode plot of the transfer function from $f$ to $d$

Power Spectral Density

Compute the PSD

  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);

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.

  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)]);

  <<plt-matlab>>

Power spectral density of the Asynchronous displacement - 1rpm

  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)]);

  <<plt-matlab>>

Power spectral density of the Asynchronous displacement - 60rpm

Load the model of the spindle

  load('./mat/spindle_model.mat', 'sys');
  Tfd = abs(squeeze(freqresp(sys('d', 'f'), f_60rpm, 'Hz')));

Plot the PSD of the Force using the model

  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)]);

  <<plt-matlab>>

Power spectral density of the force - 60rpm

Estimated Shape of the PSD of the force

  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')));

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).

  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');

  <<plt-matlab>>

Power spectral density of the force - 60rpm

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.

  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');

  <<plt-matlab>>

Comparison of the power spectral density of the measured displacement and of the model

Compute the resulting RMS value [m]

  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');

  <<plt-matlab>>

Cumulative Power Spectrum - 60rpm

Compute the resulting RMS value [m]

  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');

  <<plt-matlab>>

Cumulative Power Spectrum - 1rpm

Functions

getAsynchronousError

  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