Modal Analysis - Processing of FRF

We load the measured FRF and Coherence matrices. We also load the geometric parameters of the station: solid bodies considered and the position of the accelerometers.

load('./mat/frf_coh_matrices.mat', 'FRFs', 'COHs', 'freqs');
load('mat/geometry.mat', 'solids', 'solid_names', 'acc_pos');

Let's consider the schematic shown on figure 1 where we are measuring the motion of a (supposed) solid body at 4 distinct points in x-y-z.

The goal here is to link these \(4 \times 3 = 12\) measurements to the 6 DOFs of the solid body expressed in the frame \(\{O\}\).

From the figure 1, we can write:

With

\(\vec{v}\) and \(\Omega\) represent to velocity and rotation of the solid expressed in the frame \(\{O\}\).

We can rearrange the equations in a matrix form:

and then we obtain the velocity and rotation of the solid in the wanted frame \(\{O\}\):

This inversion is equivalent to resolving a mean square problem.

The question we wish here to answer is how to choose the reference frame \(\{O\}\) in which the DOFs of the solid bodies are defined.

The goal is to compare the motion of each solid body to see which relative DOFs between solid bodies can be neglected, that is to say, which joint between solid bodies can be regarded as perfect (and this in all the frequency range of interest). Ideally, we would like to have the same number of degrees of freedom than the number of identified modes.

The possibles choices are:

• One frame for each solid body which is located at its center of mass
• One common frame, for instance located at the point of interest (\(270mm\) above the Hexapod)
• Base located at the joint position: this is where we want to see the motion and estimate stiffness

As the easiest choice is to choose a common frame, we start with that solution.

First, we initialize a new FRF matrix FRFs_O which is an \(n \times p \times q\) with:

• \(n\) is the number of DOFs of the considered 6 solid-bodies: \(6 \times 6 = 36\)
• \(p\) is the number of excitation inputs: \(3\)
• \(q\) is the number of frequency points \(\omega_i\)

FRFs_O = zeros(length(solid_names)*6, 3, 801);

Then, as we know the positions of the accelerometers on each solid body, and we have the response of those accelerometers, we can use the equations derived in the previous section to determine the response of each solid body expressed in the frame \(\{O\}\).

for solid_i = 1:length(solid_names)
  solids_i = solids.(solid_names{solid_i});

  A = zeros(3*length(solids_i), 6);
  for i = 1:length(solids_i)
    acc_i = solids_i(i);

    A(3*(i-1)+1:3*i, 1:3) = eye(3);
    A(3*(i-1)+1:3*i, 4:6) = [ 0                  acc_pos(acc_i, 3) -acc_pos(acc_i, 2) ;
                             -acc_pos(acc_i, 3)  0                  acc_pos(acc_i, 1) ;
                              acc_pos(acc_i, 2) -acc_pos(acc_i, 1)  0];
  end

  for exc_dir = 1:3
    FRFs_O((solid_i-1)*6+1:solid_i*6, exc_dir, :) = A\squeeze(FRFs((solids_i(1)-1)*3+1:solids_i(end)*3, exc_dir, :));
  end
end

First, we can compare the motions of the 6 solid bodies in one direction (figure 2)

We can also compare all the DOFs of one solid body (figure 3).

We have some of elements of the full FRF matrix: \[ \frac{D_{1x}}{F_x},\ \frac{D_{1y}}{F_x},\ \frac{D_{1z}}{F_x},\ \frac{D_{2x}}{F_x},\ \dots \]

\[ \frac{D_{1x}}{D_{2x}} = \frac{\frac{D_{1x}}{F_x}}{\frac{D_{2x}}{F_x}} \] Then, if \(\left| \frac{D_{1x}}{D_{2x}} \right| \approx 1\) in all the frequency band of interest, we can block the \(x\) motion between the solids 1 and 2.

\[ \frac{D_{2x} - D_{1x}}{D_{1x} + D_{2x}} = \frac{\frac{D_{2x}}{F_x} - \frac{D_{1x}}{F_x}}{\frac{D_{1x}}{F_x} + \frac{D_{2x}}{F_x}} \]

Then if \(\left| \frac{D_{2x} - D_{1x}}{D_{1x} + D_{2x}} \right| \ll 1\) in all the frequency band of interest, we can block the \(x\) motion between the solids 1 and 2.

Below we plot the normalized relative motion between each stage: \[ 0 < \frac{\left| D_{ix} - D_{jx} \right|}{|D_{ix}| + |D_{jx}|} < 1 \]

DOFs = {'$T_x$', '$T_y$', '$T_z$', '$\theta_x$', '$\theta_y$', '$\theta_z$'}

dirs_i = 1:6;
exc_dir = 1;

figure;

for i = 2:6
  subaxis(3, 2, i);
  hold on;
  for dir_i = dirs_i
    H = (squeeze(FRFs_O((i-1)*6+dir_i, exc_dir, :))-squeeze(FRFs_O((i-2)*6+dir_i, exc_dir, :)))./(abs(squeeze(FRFs_O((i-1)*6+dir_i, exc_dir, :)))+abs(squeeze(FRFs_O((i-2)*6+dir_i, exc_dir, :))));
    plot(freqs, abs(H));
  end
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
  xlim([1, 200]); ylim([0, 1]);
  % xlabel('Frequency [Hz]'); ylabel('Relative Motion');
  title(sprintf('Normalized motion %s - %s', solid_names{i-1}, solid_names{i}));
  if i > 4
    xlabel('Frequency [Hz]');
  else
    set(gca, 'XTickLabel',[]);
  end
end

for i = 1:length(dirs_i)
  legend_names{i} = DOFs{dirs_i(i)};
end
lgd = legend(legend_names);

hL = subplot(3, 2, 1);
poshL = get(hL,'position');

set(lgd,'position', poshL);
axis(hL, 'off');

We wish here to compare the FRF in order to verify if there is any mistake.

dir_names = {'X', 'Y', 'Z', '$\theta_X$', '$\theta_Y$', '$\theta_Z$'};

solid_i = 6;
acc_dir_O = 1;
acc_dir = 1;
exc_dir = 1;

figure;

ax1 = subplot(2, 1, 1);
hold on;
for i = solids.(solid_names{solid_i})
  plot(freqs, abs(squeeze(FRFs(acc_dir+3*(i-1), exc_dir, :))));
end
plot(freqs, abs(squeeze(FRFs_O((solid_i-1)*6+acc_dir_O, exc_dir, :))), '-k');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
set(gca, 'XTickLabel',[]);
ylabel('Amplitude');
title(sprintf('%s motion measured by the Acc. vs %s motion computed in the common frame - %s', dir_names{acc_dir}, dir_names{acc_dir_O}, solid_names{solid_i}));

ax2 = subplot(2, 1, 2);
hold on;
for i = solids.(solid_names{solid_i})
  plot(freqs, mod(180+180/pi*phase(squeeze(FRFs(acc_dir+3*(i-1), exc_dir, :))), 360)-180);
end
plot(freqs, mod(180+180/pi*phase(squeeze(FRFs_O((solid_i-1)*6+acc_dir_O, exc_dir, :))), 360)-180, '-k');
hold off;
ylim([-180, 180]); yticks(-180:90:180);
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
set(gca, 'xscale', 'log');

linkaxes([ax1,ax2],'x');
xlim([1, 200]);

We have computed the Frequency Response Functions Matrix FRFs_O representing the response of the 6 solid bodies in their 6 DOFs.

From the response of one body in its 6 DOFs, we should be able to compute the FRF of each of its accelerometer fixed to it during the measurement.

We can then compare the result with the original measurements. This will help us to determine if:

• the previous inversion used is correct
• the solid body assumption is correct in the frequency band of interest

FRF_recovered = zeros(size(FRFs));

% For each excitation direction
for exc_dir = 1:3
  % For each solid
  for solid_i = 1:length(solid_names)
    v0 = squeeze(FRFs_O((solid_i-1)*6+1:(solid_i-1)*6+3, exc_dir, :));
    W0 = squeeze(FRFs_O((solid_i-1)*6+4:(solid_i-1)*6+6, exc_dir, :));

    % For each accelerometer attached to the current solid
    for acc_i = solids.(solid_names{solid_i})
      % We get the position of the accelerometer expressed in frame O
      pos = acc_pos(acc_i, :)';
      posX = [0 pos(3) -pos(2); -pos(3) 0 pos(1) ; pos(2) -pos(1) 0];
             [0 acc_pos(i, 3) -acc_pos(i, 2) ; -acc_pos(i, 3) 0 acc_pos(i, 1) ; acc_pos(i, 2) -acc_pos(i, 1) 0]

      FRF_recovered(3*(acc_i-1)+1:3*(acc_i-1)+3, exc_dir, :) = v0 + posX*W0;
    end
  end
end

We then compare the original FRF measured for each accelerometer with the recovered FRF from the global FRF matrix in the common frame.

The FRF for the 4 accelerometers on the Hexapod are compared on figure 5. All the FRF are matching very well in all the frequency range displayed.

The FRF for accelerometers located on the translation stage are compared on figure 6. The FRF are matching well until 100Hz.