Modal Shape from Experimental FRF 3

[Xtina]

Hi all, I am performing an experimental modal analysis of a beam and I am trying to get the mode shapes from the FRF measured. I have read in many articles and books that the mode shapes (unscaled) may be obtained from the peak amplitudes of the imaginary part of the FRF, as in lightly damped structures the imaginary part reaches maximum values while the real part gets zero values at the resonant frequency. My problem comes when I plot the real and the imaginary parts of the FRF, it seems that they are interchanged, so I can observe peaks in the real part that provide the correct modal shapes and values of zero in the real part for the resonant frequencies. I have also found a thesis that states that for structures with small damping, the undamped modal shapes match with the real part of the FRF. I find the information is contradictory, can someone help me? and what is the reason why any of the parts (real/imaginary) give the modal shapes?

 

[Xtina]

I correct:

"so I can observe peaks in the real part that provide the correct modal shapes and values of zero in the *imaginary* part for the resonant frequencies"

 

[electricpete]

what is the reason why any of the parts (real/imaginary) give the modal shapes?

A lightly damped system approximates an undamped system. In the case of undamped system, the DISPLACEMENT is in-phase below resonance and 180-out-of-phase above resonance. Therefore the FRF derived from displacement should be primarily real for lightly-damped systems.

If you are looking at FRF derived from VELOCITY (derivative of displacement), these phase relationships would be shifted by 90, and the FRF would be primarily imaginary for lightly-dampedsystem.

If you are looking at FRF derived from ACCELERATION (derivative of velocity), these phase relationships would be shifted by another 90, and the FRF would be primarily real again for lightly damped system system (opposite the polarity of the FRF dervied from displacement).


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(2B)+(2B)' ?

 

[electricpete]

I have read in many articles and books that the mode shapes (unscaled) may be obtained from the peak amplitudes of the imaginary part of the FRF,

I’m not sure how you are going to infer mode shape from an FRF, since FRF which contains info on response vs frequency. (mode shape is response vs spatial variable). Maybe I misunderstood your context.

However, to add to previous discussion. Let’s talk about FRF from displacement for lightly damped system. We said it was primarily real. That simply means the magnitude of FRF is closely approximated by the real part over most of the range.

Another thing to mention: the imaginary part (of FRF from displacement) will be close to zero everywhere except resonance. Don’t know if that’s what you were referring to.

I’ll shut up now.


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(2B)+(2B)' ?

 

[Xtina]

Thanks electricpete for your explanations. The way I am using to obtain the mode shapes is explained very simply and fast in this pdf:

http://macl.caeds.eng.uml.edu/umlspace/aug99.pdf

It says that the imaginary part of the FRF is directly related to the residues, and the residues to the modal shape, so that is the reason why one can know the shape, but not in the adequate scale.
As it is shown in the pdf, the plot of the imaginary part of the FRF presents some positive and negative peaks, and their amplitude and sign stablish the shape.
The problem is that the plot of the imaginary part of my FRF does not correspond with the graphics shown in the article. Instead of that, the real part present this shape. I tried to get the mode shapes from the real part and the results seem correct (the first, second and third bending mode shapes). So, it would be correct?
And referring to your comment, I know that the imaginary part (of FRF from displacement) should be close to zero everywhere except resonance, but that is not what happens to mine, it reaches maximum values at resonance (it behaves as the real part!!).
Apart from that, my objective is to get the modal shapes of the beam, do you know any method to calculate them?
Thanks!!!

 

[electricpete]

Yes, I can see what they’re doing. They are considering FRF from displacement.

At resonance: real part of FRF is 0 (as it transitions between plus and minus), imaginary part is non-zero. Therefore at resonance, the magnitude and sign of imaginary part completely characterize the FRF at resonance (since real part is zero). Combining this info (magnitude and sign of imaginary part of displacement FRF at a specific resonant frequency) from several spatial positions allows to construct mode shape at that frequency.

The problem is that the plot of the imaginary part of my FRF does not correspond with the graphics shown in the article. Instead of that, the real part present this shape.

Is your FRF derived from displacement ? (if from velocity, I could imagine it would act the way you describe).

Perhaps you can post an example.


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(2B)+(2B)' ?

 

[Xtina]

No, I measure accelerations and obtain the accelerance, but then I get the receptance (as if I had taken displacements) with the expression:
H(w)=Ha(w)/(-w^2)
where H(w) is the receptance, Ha(w) the accelerance, and w the frequencies.
In both cases, accelerance or receptance, the real part should be zero at resonance. In case of velocity, the mobility has the contrary behaviour, and it shows a maximum in the real part at resonance...I can't see what is happening.
So, what do you think about the thesis I mentioned before? That one that states that for structures with small damping, the undamped modal shapes match with the real part of the FRF... they are describing what I obtain, but I don't see the sense...

 

[electricpete]

So, in re-reading the entire thread, I see you already well understood the expected behavior. It is as described in the linked pdf (we can get the modeshapes from the imaginary part).

What you quoted from a thesis (we can get them from the real part) sounds wrong, unless they are talking about velocity.

Why your data doesn’t match the expected results as shown in the linked pdf: hard to figure. Maybe an example plot would provide some small clue. There are a lot of knowledgeable people on the forum (many moreso than me).


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(2B)+(2B)' ?

 

[Xtina]

Yes, it sounds wrong but I drew my modal shapes from the real parts (as they do in the thesis) and it seems to be correct, I obtained the typical bending shapes of a simply supported beam... I found the first and second bending shapes at the first and second frequencies and the third bending shape at the 5th frequency (I guess that the modes between them correspond to torsion).
Do you know any method to calculate the mode shapes?

 

[GregLocock]

In a traditional EMA if you have the normal setup the imaginary peaks give the amplitude and sign of the mode shape. The real part should be zero at quadrature, by definition.

Integrating to displacement from acceleration does not alter that.

If you post one or more of your FRFs as a bode plot or preferably raw data I'll have a look at it and tell you whether you are dealing with bad data.

With very lightly damped systems it can be difficult to get enough resolution at the peaks, this is one reason why your coherence drops at the peaks.

It can be very instructive to look at the Nyquist (or Argand) domain of the data, and if you have 3 or more lines in the bandwidth of the mode a circle fit is a nice way of doing curve fits, but for a lightly damped system it'll give the same answer, since the diameter of the circle is the imaginary value at quadrature.

http://cp.literature.agilent.com/litweb/pdf/5954-7957E.pdf

is a good intro, p30 and there onwards.



Cheers

Greg Locock


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[Xtina]

Thanks Greg for your help. You can find attached the plot of my FRF and also an excel with the accelerations measured in case you can process them. I have to say that my acceleration data have been obtained from a transient analysis in ANSYS, in which I impacted the simply supported concrete beam with a punctual load in a point and then let it vibrate. That is the way I have thought to calibrate my model with the experimental data. Do you think it is correct? Or instead of that I should ignore damping and just run a modal analysis?
As a said, I want to calulate the mode shapes, and I found some formulae (pdf attached). Are they correct? What other methods do you know to calulate them?

 

[Xtina]

Here is the pdf of the modal shape formulae.

 

[Xtina]

And here the FRF plot.

¡ Thanks for your attention !

 

[GregLocock]

Um. Oh. I assumed we were talking about real test data. Still, no harm done.

At this point we are deeply into bizarro world since the linear analysis of your FEA model will give you the mode shapes directly.

So what you are doing is synthesising time histories based on known mode shapes and assumed damping and scaling, for each mode, and then trying to re-extract the mode shape you first thought of.

Is that really a sensible approach?





Cheers

Greg Locock


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[Xtina]

The experimental data behaves in the same way as the numerical model, that is, it also provides FRFs which real parts are maximum at resonance instead of zero. So my experimental data also contradicts the theory...

My objective is to create a model that behaves as the real beam, and the way of calibrating it is looking at the natural frequencies and the mode shapes (with modal assurance criteria MAC)of both.

I am trying to simulate the real setup with the model: the impact hammer is the punctual impact force and the accelerations are obtained with the transient analysis instead of the accelerometers. Then the postprocess is the same for both: I obtain the FRF with the FFT of the input and output with Matlab and then represent them and extract the parameters that allow me to compare them. I am doing this to represent the real process and to consider damping, since the modal analysis does not take it into account. Maybe it would be better just to run the modal analysis assuming very little damping, I don't know. But the problem of obtaining the experimental mode shapes would continue...

 

[GregLocock]

OK, I'll analyse your FEA time hstory today.

I'm afraid there must be something wrong with your experimental setup if you are getting zero phase at resonance, if you are plotting displacement/force or acceleration/force.



Cheers

Greg Locock


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[GregLocock]

Here's better plots of the fft of that data.

The first peak and the response below it look very odd.

The second peak is at quadrature, ie is behaving correctly.

Try reducing the damping of all the modes, particularly the low frequency ones.




Cheers

Greg Locock


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[GregLocock]

I suspect you have a lot of low frequency damping which is masking the usually m k behaviour. try running it with a dmaping of 0.1 % or so for each mode.

Cheers

Greg Locock


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[GregLocock]

Incidentally I think synthesising time histories to try out your analysis technique is a good move.You should be aware that there are MANY approaches to estimating modal parameters, peak picking and circle fitting are rarely used in the automotive industry these days. They both work well with lightly damped systems, but as soon as you see close coupled modes you run into problems. The acid test is always how well you can recreate the original FRF from your modal parameter list and mode shapes.

Cheers

Greg Locock


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[hacksaw]

if not mistaken the inferred mode shape can only be accomplished in the case where the modes are well separated and then only for a single variable system such as for bending modes only.

If you examine the change in phase at the second resonance it is clear that you are dealing with overlapping modes, very likely bending and torsional.