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What is the difference between the modal terms "residue" and "residual"?

Experimenter
Experimenter

What is the difference between the modal terms "residue" and "residual"?  What are the used for? Are they different?

3 REPLIES

Re: What is the difference between the modal terms "residue" and "residual"?

Siemens Valued Contributor Siemens Valued Contributor
Siemens Valued Contributor

Very relevant question. Indeed the two terms, although written very similarly, refer to very different quantities of the modal model and have completely different meaning.

If you consider the representation of an FRF in the modal domain, you will have the following formulation:

FRF.png 

The last formula on the rigth is the one we commonly use in modal analysis, where the numerators contains the product between the mode shapes and the participation factor. This product, which returns the terms A in the formula in the middle, is the so-called residue (a term that comes from calculus).

 

Now obviously this equation holds as it is only if we know all the measured modes n in our system. If we build a lumped parameter model, with a finite number of masses connected by springs, then this formulation is ok. But in a real system, the number of modes is infinite and we can only observe a limited number of them in a limited frequency range. 

The FRFs we measure are an exact representation of the real system, but if we want to derive a modal model we need to take into account that we cannot include the contribution of all the modes. Even modes which are at very high frequency, outside of our band band of interest, will contribute to the response in the frequency range we are interested to, and the Residuals are what allows us to take them into account.

Basically, the Lower and Upper Residuals introduce extra terms in the equation above to compensate for the contribution of the modes respectively below  the minimum and above the maximum frequency we set for our modal analysis:

FRF_res.png

You will find different residuals formulation depending on whether your FRFs represent Acceleration, Velocity or Displacement over Force.

 

So to summarize, while the residues are a characteristic of your system, and are further decomposed into mode shapes (and participation factors), the Residuals are extra terms which ensure your modally reconstructed FRFs, in the frequency range of interest, fits as much as possible the measured one, by compensating for the inevitable truncation errors.

 

Hope this answer your question.

 

Re: What is the difference between the modal terms "residue" and "residual"?

Experimenter
Experimenter

Thanks for the explanation!  If I understand correctly, the residue is related to the mode shape (vector?). The residual is used to compensate for modal contributions outside the bandwidth of interest during a curvefit.

Re: What is the difference between the modal terms "residue" and "residual"?

Siemens Valued Contributor Siemens Valued Contributor
Siemens Valued Contributor

That is exactly what i was trying to convey. If you want to oversimplify the concepts, Residues are physical properties of the structure and relates to motion at resonance, while residuals are a smart way to compensate for inevitable truncation errors.