Modal Analysis - Processing of FRF

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 \(\omega_i\)

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 \(\{O\}\).

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

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).

Now that the motion of all the solid bodies are expressed in the same frame, we should be able to compare them. This can be used to determine what joints direction between two solid bodies is stiff enough that we can fix this DoF. This could help reduce the order of the model and simplify the extraction of the model parameters from the measurements.

We decide to plot the "normalized relative motion" between solid bodies \(i\) and \(j\): \[ 0 < \Delta_{ij, x} = \frac{\left| |D_{i,x}| - |D_{j,x}| \right|}{|D_{i,x}| + |D_{j,x}|} < 1 \]

Then, if \(\Delta_{ij,x} \ll 1\) in the frequency band of interest, we have that \(D_{ix} \approx D_{jx}\) and we can neglect that DOF between the two solid bodies \(i\) and \(j\).

This normalized relative motion is shown on figure 4 for all the directions and for all the adjacent pair of solid bodies.

We have computed the Frequency Response Functions Matrix FRFs_O representing the response of the 6 solid bodies in their 6 DOFs with respect to the frame \(\{O\}\).

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, supposing that the stage is a solid body.

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

From the translation \(\delta p\) and rotation \(\delta \Omega\) of a solid body and the positions \(p_i\) of the accelerometers attached to it, we can compute the response that would have been measured by the accelerometers using the following formula:

Thus, we can obtain the FRF matrix FRFs_A that gives the responses of the accelerometers to the forces applied by the hammer.

It is implemented in matlab as follow:

FRFs_A = 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];

      FRFs_A(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 FRFs with the "recovered" FRF FRFs_A 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.

From solidworks, we can export the position of the center of mass of each solid body considered.

For that, we have to suppose density of the materials. We then process the exported file.

sed -e '1,/^Center of mass/ d' mat/1_granite_bot.txt | sed 3q | sed 's/^\s*\t*[XYZ] = \([-0-9.]*\)\r/\1/' >  mat/model_solidworks_com.txt
sed -e '1,/^Center of mass/ d' mat/2_granite_top.txt | sed 3q | sed 's/^\s*\t*[XYZ] = \([-0-9.]*\)\r/\1/' >> mat/model_solidworks_com.txt
sed -e '1,/^Center of mass/ d' mat/3_ty.txt          | sed 3q | sed 's/^\s*\t*[XYZ] = \([-0-9.]*\)\r/\1/' >> mat/model_solidworks_com.txt
sed -e '1,/^Center of mass/ d' mat/4_ry.txt          | sed 3q | sed 's/^\s*\t*[XYZ] = \([-0-9.]*\)\r/\1/' >> mat/model_solidworks_com.txt
sed -e '1,/^Center of mass/ d' mat/5_rz.txt          | sed 3q | sed 's/^\s*\t*[XYZ] = \([-0-9.]*\)\r/\1/' >> mat/model_solidworks_com.txt
sed -e '1,/^Center of mass/ d' mat/6_hexa.txt        | sed 3q | sed 's/^\s*\t*[XYZ] = \([-0-9.]*\)\r/\1/' >> mat/model_solidworks_com.txt

And we import the Center of mass on Matlab.

model_com = table2array(readtable('mat/model_solidworks_com.txt', 'ReadVariableNames', false));

model_com = reshape(model_com , [3, 6]);

First, we initialize a new FRF matrix FRFs_CoM 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_CoM = 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);

    acc_pos_com = acc_pos(acc_i, :)' - model_com(:, solid_i);

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

  for exc_dir = 1:3
    FRFs_CoM((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