Transforming Complex Mode Shapes into Real, Undamped Shapes

Modal test data processing methods perform curve-fitting on measured frequency response functions. These are indirect methods for extracting natural frequencies and damping ratios. The FRF methods also yield mode shapes which may be complex if the system has non-proportional damping.

Each complex mode shape can be multiplied by an independent complex scale factor to render it into an equivalent real, undamped form. These transformed modes can then be compared to analytical modes from the generalized eigenvalue problem for the mass and stiffness matrices.

A method for doing this is the least-squares method given in:

Rajeev Hiremaglur, Real-Normalization of Experimental Complex Modal Vectors with Modal Vector Contamination Download Link

Here is a Matlab code snippet for doing this for the case of a numerical simulation of a modal test with two-degrees-of-freedom. The code should work for higher systems by making the appropriate substitutions.

The imaginary parts of the complex-valued modal coefficients are taken as the dependent variable and the real parts as the independent variable for the least squares calculation.

Note that a given mode’s imaginary coefficients may remain relatively high after the rotation process if the mode is a spurious computational one or if the coherence is poor at the corresponding frequency.

% CM = complex modes in column format
% RM = real modes (approximately)
%
% The least squares curve-fit uses the form: y = mx + b
%
% A(1) is the slope term m
%

% disp(‘ Complex Modes ‘);

CM = [ 0.389-0.389i , 0.427+0.427i ;
0.59-0.591i , -0.564-0.563i ]

sz=size(CM);

num=sz(2);

RM=zeros(sz(1),sz(2));

for i=1:num

Y=imag(CM(:,i));
X(:,1)=real(CF(:,i));
X(:,2)=1;

XT=transpose(X);

A=pinv(XT*X)*(XT*Y);

theta=atan(A(1));

% Perform transformation via rotation

RM(:,i)=CM(:,i)*exp(-1i*theta);
end

% disp(‘ Approximate Real Modes ‘);
RM

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