Modal Analysis - Modal Parameter Extraction

The Mode Indicator Functions are usually used on \(n\times p\) FRF matrix where \(n\) is a relatively large number of measurement DOFs and \(p\) is the number of excitation DOFs, typically 3 or 4.

In these methods, the frequency dependent FRF matrix is subjected to a singular value decomposition analysis which thus yields a small number (3 or 4) of singular values, these also being frequency dependent.

These methods are used to determine the number of modes present in a given frequency range, to identify repeated natural frequencies and to pre process the FRF data prior to modal analysis.

From the documentation of the modal software:

The MIF consist of the singular values of the Frequency response function matrix. The number of MIFs equals the number of excitations. By the powerful singular value decomposition, the real signal space is separated from the noise space. Therefore, the MIFs exhibit the modes effectively. A peak in the MIFs plot usually indicate the existence of a structural mode, and two peaks at the same frequency point means the existence of two repeated modes. Moreover, the magnitude of the MIFs implies the strength of the a mode.

We compute the Complex Mode Indicator Function. The result is shown on figure 1. The exact same curve is obtained when computed using the OROS software.

MIF = zeros(size(FRFs, 2), size(FRFs, 2), size(FRFs, 3));

for i = 1:length(freqs)
  [~,S,~] = svd(FRFs(:, :, i));
  MIF(:, :, i) = S'*S;
end

We can also compute the CMIF using the FRF matrix expressed in the same global frame. We compare the two CMIF on figure 2.

They do not indicate the same resonance frequencies, especially around 110Hz.

MIF_O = zeros(size(FRFs_O, 2), size(FRFs_O, 2), size(FRFs_O, 3));
for i = 1:length(freqs)
  [~,S,~] = svd(FRFs_O(:, :, i));
  MIF_O(:, :, i) = S'*S;
end

An alternative is the Composite Response Function \(HH(\omega)\) defined as the sum of all the measured FRF:

Instead, we choose here to use the sum of the norms of the measured FRFs:

The result is shown on figure 3.

figure;
hold on;
plot(freqs, squeeze(sum(sum(abs(FRFs)))), '-k');
plot(freqs, squeeze(sum(sum(abs(FRFs_O)))), '--k');
hold off;
xlabel('Frequency [Hz]'); ylabel('Amplitude');
xlim([1, 200]);

Modal identification can be done within the Modal software of OROS. The manual for the software is available here.

Several modal parameter extraction methods are available. We choose to use the "broad band method" as it permits to identify the modal parameters using all the FRF curves at the same times. It takes into account the fact the the properties of all the individual curves are related by being from the same structure: all FRF plots on a given structure should indicate the same values for the natural frequencies and damping factor of each mode.

Such method also have the advantage of producing a unique and consistent model as direct output.

In order to apply this method, we select the frequency range of interest and we give an estimate of how many modes are present.

Then, it shows a stabilization charts, such as the one shown on figure 4, where we have to manually select which modes to take into account in the modal model.

We can then run the modal analysis, and the software will identify the modal damping and mode shapes at the selected frequency modes.

The obtained modal parameters are:

• the resonance frequencies
• the modes shapes
• the modal damping
• the residues

They are all exported in a text file named modes.asc. Its first 20 lines as shown below.

Created by N-Modal
Estimator: bbfd
01-Jul-19 16:44:11


Mode 1
freq    = 11.41275Hz
damp    = 8.72664%
modal A =  -4.50556e+003-9.41744e+003i
modal B =  -7.00928e+005+2.62922e+005i
Mode matrix of local coordinate [DOF: Re IM]
     1X-:  -1.04114e-001   3.50664e-002
     1Y-:   2.34008e-001   5.04273e-004
     1Z+:  -1.93303e-002   5.08614e-003
     2X-:  -8.38439e-002   3.45978e-002
     2Y-:   2.42440e-001   0.00000e+000
     2Z+:  -7.40734e-003   5.17734e-003
     3Y-:   2.17655e-001   6.10802e-003
     3X+:   1.18685e-001  -3.54602e-002
     3Z+:  -2.37725e-002  -1.61649e-003

We split this big modes.asc file into sub text files using bash. The obtained files are described one table 1.

sed '/^\s*[0-9]*[XYZ][+-]:/!d' mat/modes.asc > mat/mode_shapes.txt
sed '/freq/!d'                 mat/modes.asc | sed 's/.* = \(.*\)Hz/\1/' > mat/mode_freqs.txt
sed '/damp/!d'                 mat/modes.asc | sed 's/.* = \(.*\)\%/\1/' > mat/mode_damps.txt
sed '/modal A/!d'              mat/modes.asc | sed 's/.* =\s\+\([-0-9.e]\++[0-9]\+\)\([-+0-9.e]\+\)i/\1 \2/' > mat/mode_modal_a.txt
sed '/modal B/!d'              mat/modes.asc | sed 's/.* =\s\+\([-0-9.e]\++[0-9]\+\)\([-+0-9.e]\+\)i/\1 \2/' > mat/mode_modal_b.txt

Then we import the obtained .txt files on Matlab using readtable function.

shapes  = readtable('mat/mode_shapes.txt',              'ReadVariableNames', false);  % [Sign / Real / Imag]
freqs   = table2array(readtable('mat/mode_freqs.txt',   'ReadVariableNames', false)); % in [Hz]
damps   = table2array(readtable('mat/mode_damps.txt',   'ReadVariableNames', false)); % in [%]
modal_a = table2array(readtable('mat/mode_modal_a.txt', 'ReadVariableNames', false)); % [Real / Imag]
modal_b = table2array(readtable('mat/mode_modal_b.txt', 'ReadVariableNames', false)); % [Real / Imag]

modal_a = complex(modal_a(:, 1), modal_a(:, 2));
modal_b = complex(modal_b(:, 1), modal_b(:, 2));

We guess the number of modes identified from the length of the imported data.

acc_n = 23; % Number of accelerometers
dir_n = 3; % Number of directions
dirs = 'XYZ';

mod_n = size(shapes,1)/acc_n/dir_n; % Number of modes

As the mode shapes are split into 3 parts (direction plus sign, real part and imaginary part), we aggregate them into one array of complex numbers.

T_sign = table2array(shapes(:, 1));
T_real = table2array(shapes(:, 2));
T_imag = table2array(shapes(:, 3));

modes = zeros(mod_n, acc_n, dir_n);

for mod_i = 1:mod_n
  for acc_i = 1:acc_n
    % Get the correct section of the signs
    T = T_sign(acc_n*dir_n*(mod_i-1)+1:acc_n*dir_n*mod_i);
    for dir_i = 1:dir_n
      % Get the line corresponding to the sensor
      i = find(contains(T, sprintf('%i%s',acc_i, dirs(dir_i))), 1, 'first')+acc_n*dir_n*(mod_i-1);
      modes(mod_i, acc_i, dir_i) = str2num([T_sign{i}(end-1), '1'])*complex(T_real(i),T_imag(i));
    end
  end
end

The obtained mode frequencies and damping are shown below.

We arrange the obtained modal parameters into matrices: \[ \Omega = \begin{bmatrix} \omega_1^2 & & 0 \\ & \ddots & \\ 0 & & \omega_n^2 \end{bmatrix}; \quad \Psi = \begin{bmatrix} & & \\ \{\psi_1\} & \dots & \{\psi_n\} \\ & & \end{bmatrix} \] with \(n\) the number of identified modes and: \[ \{\psi_1\} = \begin{Bmatrix} \psi_{1_x} & \psi_{2_x} & \dots & \psi_{23_x} & \psi_{1_y} & \dots & \psi_{1_z} & \dots & \psi_{23_z} \end{Bmatrix}^T \]

eigen_value_M = diag(freqs*2*pi);
eigen_vector_M = reshape(mode_shapes, [mod_n, 6*length(solid_names)])';

We normalize each column of the eigen vector matrix.

Then, each eigenvector as a norm of 1.

eigen_vector_M = eigen_vector_M./vecnorm(eigen_vector_M);

From the figure above, 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 a mean square problem.

We arrange the obtained modal parameters into matrices: \[ \Omega = \begin{bmatrix} \omega_1^2 & & 0 \\ & \ddots & \\ 0 & & \omega_n^2 \end{bmatrix}; \quad \Psi = \begin{bmatrix} & & \\ \{\psi_1\} & \dots & \{\psi_n\} \\ & & \end{bmatrix} \] with \[ \{\psi_1\} = \begin{Bmatrix} \psi_{1_x} & \psi_{2_x} & \dots & \psi_{6_x} & \psi_{1_y} & \dots & \psi_{1\Omega_x} & \dots & \psi_{6\Omega_z} \end{Bmatrix}^T \]

eigen_value_M = diag(freqs*2*pi);
eigen_vector_M = reshape(mode_shapes_O, [mod_n, 6*length(solid_names)])';

We normalize each column of the eigen vector matrix.

Then, each eigenvector as a norm of 1.

eigen_vector_M = eigen_vector_M./vecnorm(eigen_vector_M);