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

________________________________

See also:  Peter Avitable article

 
– Tom Irvine
 
 

Material Damping

hyster2

Internal friction and heat generation cause a deviation from perfect elasticity, resulting in the hysteresis loop in the stress-strain curve.

The damping capacity  ψ is:

ψ =  ΔW/W

where W is the initial strain energy and ΔW is the energy lost during one cycle of oscillation.

Some damping capacities of common materials are:
Sdamping

Ldamping

There is some tendency for damping capacity to increase with at higher stresses. Damping also depends on temperature, strain rate and other parameters.

References:

Maringer, Damping Capacities of Materials

Luxfer, Magnesium Fixtures

See also:    Damping References

–  Tom Irvine

Shock & Vibration Test Fixture Design

fixxxx

fitxx

Introduction

The best vibration fixtures are simple, rigid, lightweight with the minimum number of features.  Shaker and power amplifier systems are limited in terms of voltage, current and force pounds.  Lighter fixtures are need to achieve higher acceleration levels.

Materials

Aluminum alloy is a good choice for small to medium fixtures because it is cheaper than other materials such as magnesium.  Aluminum 6061-T6 & 7075-T6 are suitable alloys.

Magnesium is a better choice than aluminum in terms of weight to stiffness ratio and can provide higher damping.

Magnesium Zirconium K1A alloy in cast form has excellent hysteresis damping.

Magnesium Alloy AZ31B-Tool Plate also has very good damping.  It is a free machining material with good weldability properties.

Fixtures should have high damping in order to limit their resonant response.

Higher Damping also provides a transmissibility ratio relatively closer to unity gain across the entire frequency spectrum.

Joints

Some fixtures require joints. Bolted joints have more damping than welded ones. But welded joints have more rigidity if they are continuous.

Welded joints can be used with caution. They may have material distortion and high residual stress from thermal expansion and contraction during the welding process. After welding and before machining, the fixture should be left overnight in an oven at 250 °C to reduce warping and residual stress.

Welded joints may also have porosity, inclusions and microcracks. Some amount of microcracking is inevitable but needs to be managed via inspection and fracture calculations.

The quality of the weld is paramount. The weld joints should be continuous and smooth, with two or even three passes if possible.

References

Welded Joint Concerns for Shock, Vibration & Fatigue

Bruel & Kjaer Fixture Guidelines

See also:  Material Damping

– Tom Irvine