Dear Mr Horiguchi

Most damping models are imperfect as our understanding of damping is imperfect. This implies that inverse procedures may amplify errors even when the model captures the gist of the problem. Or very crudely put, inverse easily becomes perverse.

In your elaboration above, you touch on, in my experience, the better way to deal with model fitting, i.e. to use a forward procedure where a model based on first principles is employed to obtain a best fit to the data at hand. This fit is the best you can get using such a model, i.e. nothing is gained from the inverse procedure.

A forward approach enables also the use of numerical models where you are able to capture also things such as i) uneven displacement/rotation across the damping layer, ii) variation and combination of material damping models, iii) the use of other damping models and iv) you are able to incorporate also artifacts inherent in your idealized test method.

For what it is worth, I have not tried it myself, the Bergström-Boyce model seems to be quite good for rubber type materials. There may be other models better aligned for viscoleastic materials. https://polymerfem.com/articles/material-models

Sincerely

Claes

PS - If you are interested, some ramblings of mine on the subject of damping http://qringtech.com/2014/06/22/designed-damping-types-mechanisms-application-limitation/

Dear Dr. Fredö,

Thank you for your kind answer. I have understood that the measured results contain many kinds of errors. Even for a simple free-free beam test, there may be imperfect boundary conditions at the rod. There may be some rotational vibration. The viscoelastic data used in our paper are derived from the half-power band width procedure of the measurement system. There may be some errors in viscoelastic data. Therefore as in your explanation, I have understood inverse procedures may amplify errors. I am terribly sorry for my late response in contrast to your quick answer.

Sincerely, Ryuzo Horiguchi

(PS) I visited your Gringtech website. It was full of interesting contents. I have to learn much more.

Dear Mr Horiguchi

You sum things up correctly. :)

The simplest way, perhaps, is to look at a linear spring where F = K*X. If you measure F and X to determine K, things go reasonably well as long as you do it for a single direction and with care.

However, placing the force, say, with an offset along the spring's central axis implies loading it also with a moment and picking up also rotation. In reality, you are now dealing with a stiffness matrrix. Insisting on treating this matrix as a single input-output function affect matters.

Glad you liked my web site. It is intended to serve as inspriration.

All the best

Claes

Hi ,Everybody

I agree with all the above answer.

Dear Dr. Fredö and Dr. Chiad,

At first, I wanted to know the reason why the inverse problem does not have unique solution. I still think it has mathematical or numerical problems such as the characteristics of complex functions.

But, after reading your answer, I have understood that the single input-output function treatment is not appropriate for this kind of inversion procedure. We need to perform more experiments and finite element simulation.

Thank you for your kind advice.

Sincerely,

Ryuzo Horiguchi (Kao Corporation)

Dear Mr Horiguchi

To exemplify an inverse problem of the simplest kind.

If I tell you that the answer is four - can you please tell me what the question is?

As you can see, the above inverse problem has an infinte solution space and hence, not a unique solution (as the answer might be 2+2, 3+1, 2*2, 12/3 or for that matter four days, or eight minus four chickens).

You can improve the probability that the answer to an inverse procedure is trustworthy by testing it at many occasions, several instances and so on, but you can never prove it to be true. This ties in to the core of the scientific method that you can never prove a positive only break a theory by proving a negative.

A forward approach where you use a model based on first principles can show its merits and a best fit ability. This is the approach I am suggesting.

Sincerely

Claes

Dear Dr. Fredö,

I agree that my inverse problem may have an infinite solution space.

I will try the modal based forward approach .

Thank you for your suggestion.

Sincerely,

Ryuzo Horiguchi (Kao Corporation)