Hi, if you hav the modal analysis results, then based on the graph attached here, you can calculate the amount of transmissiblity im some points (frequencies) around the desired frequency ( here at each resonance frequency), so you can find the damping ratio.

If your SDoF oscillator is linear; if the dynamic input can be assumed as a white noise; if the amplitude of the input changes slowly with time; then you can use the so-called spectral moments of the dynamic response to estimate the (equivalent) viscous damping ratio.

To do this, you can calculate first the bandwidth factor:

q= sqrt(1-λ12/(λ0*λ2)) ,

where λ0, λ1 and λ2 are the spectral moments of order 0, 1 and 2, respectively; and then the viscous damping ratio as:

ζ= (π/4)*q2 , if ζ

Hi Hossein,

Thank for the answer.

The Resonant Amplification Method that you proposed is an input-output method, but my problem requires an output only technique. In words, I don't have input frequency to  investigate the resonance problem. Moreover, in this method assumed that input loading  is harmonic, however, my unknown input suppose to be non-stationary.

Dear Mr. Palmeri,

Since my system is LTI, the non-stationary content of the responses only emanates from the non stationary nature of the input (and not non-linearity in the system). So, my unknown input is narrow band non-stationary excitation which cannot be modeled as a broadband white noise.

Thanks for the answer and the good resource that you have introduced.

Dear Yousef:

What do you know about the input? Can you make any estimate about its frequency content and its amplitude?

If this is the case, then you could probably still use the spectral moments for your analysis.

As the SDoF oscillator depends on three parameters (the mass m, the undamped natural circular frequency ω0, and the equivalent viscous damping ratio ζ0, plus the excitation) you should be able to relate these three quantities to the three spectral moments λ0(t), λ1(t) and λ2(t). Just be careful, though, as the mathematical definition of λ1(t) is not straightforward.

The link below redirects you to a paper in which the time-dependent spectral moments have been successfully used to predict the peak response of linear structures subjected to a non-stationary Gaussian input (which in turn strongly depends on the amount of damping).

Good luck with your project.

Regards,

Alessandro

http://www.sciencedirect.com/science/article/pii/S004578250400413X

The SDOF system I am talking about comes from the decomposition of a MDOF system to a number of SDOF system. I mean Modal Coordinate Signals. This process is conducted via a blind modal identification method without the knowledge of input. Input can be of any frequency content and amplitude (different earthquakes).

There are methods like Random Decrement Technique and Auto Correlation Function to convert a random signal to an exponentially free decayed one. Then by simply exploiting a curve fitting technique, damping ratio can be estimated. But these methods are limited to stationary signals.

Moreover, using absolute responses for identification  makes problem more challenging. because frequency content of the input affect the frequency content of the output into great extent. So that in general frequency-based  methods do not work well in handling non stationary signals .

Anyway, I am grateful to Dr. Alessandro for providing valuable material.

by transfering your response to a RD function (Random Decrement) , then you will have a function that represent the free vibration. So you can estimate the damping ratio from the decay of that response (logarithmic decay)

Thanks Dr. Niousha for the response

In RD technique it is assumed that response signal is stationary. That is why for non stationary signals it does not yield a well behaved free decay. In same way is auto correlation function.

Yes, the method is basically developed for stationary signal, but it is shown that can be used for non-stationary one as well. As you know most ambient excitations in engineering problems are non-stationary, and the RDT applied to those excitations give good and resonable results. If you have real data you need to filter it with a bandpass filter around the frequency of intrest before appluing RDT.

Hi

I do believe Mr Ahmadian and Mr Palmiei are on to the right track.

There are different ways to do this but the simplest is indeed the Half Power bandwidth method. You can ignore the fact that the first curve happened to show Transmissibility as you can do this with output only.

For the Half Power bandwidth method, you take an FFT of your time signal, average with overlap processing & sufficient spectral resolution until it stabilizes. You find the highest peak amplitude, A1, and its frequency f1. The Half power points (assuming you show amplitude and not amplitude^2) of your FFT, then are found down the slopes from f1 at the Amplitudes A1/sqrt(2) which identify the frequencies f1L (Lower) and f1H (Higher). The structural loss factor eta = (f1H - f1L)/f1. Its inverse is called the Q factor, where Q designates Quality as this method originated as a way to identify the Quality of how well crystals were tuned for Radio reception. Here is a link http://www.allaboutcircuits.com/vol_2/chpt_6/6.html This should work find for a SDOF system but may run into difficulty if you have more modes and these overlap.

There are also other methods. If you have Matlab, you can look at the pmusic and the peig algorithms. These, in essence, makes a best fit of a sinus tone to you random time data as this captures the SDOF systems free vibration which arises. This approach is computationally more laborious than the half power bandwidth method.

Last, there is a whole researchfield called Operating Modal Analysis wich contains different versions. For a simple SDOF system, this is overkill, but for a multimodal system this may be the way to go.

Thanks Dr. Niousha, I think you are narrowing down my question to the right place.

Filtering as a pre-processing step to denoise a signal and to discard unwanted frequency components with lower energy, is inevitable. However, for a structural system with narrow band excitation, applying a band pass filter to convert a nonstationary signal to stationary one, seems arguable. If it were so, there would not have been any trouble with non stationary signals in any other fields.

Ambieant vibration is a general term including stationary and non stationary types. I will appreciate if I see any paper using RDT particularly considering the worst case scenario which is earthquake excitation.

Hi Dr. Fredo,

Thanks for your thorough response which is precisely detailed .

In the Half Power bandwidth method, it is assumed that dynamic loading is harmonic. So is the Resonant Amplification Method. Thus, they are not applicable here.

But about FDD method, you are right for a SDOF system it would be kind of overkill. However, it does also have limitation handling nonstationary responses, because FDD is conducted in frequency domain.

Hi Yousef

The half power bandwidth method is applicable for random loading. You can find several books written on the subject. 

The other methods I referred you to, all operate in the time domain.

All three methods apply for random, non-stationary processes.

About damping estimation.

I believe you must take to heart that damping is the most uncertain parameter we deal with in dynamics for the simple reason that its modelling is poor. We likely oversimplify the physics involved and many tend to confuse damping with amplitude reduction which may arise for a variety of reasons, none of which connected to damping, i.e. the destruction of vibration energy into heat which is a nearly irreversible process.

If you are interested about damping you might want to read some of my ramblings on the subject here http://qringtech.com/2014/06/22/designed-damping-types-mechanisms-application-limitation/

So, all in all, the half power bandwidth method is the simplest. It will not tell you everything, then again, neither will no other method I know of.

If your damping changes with time, you are dealing with a non-linear system and I believe you will find that you have opened Pandoras box. This box, of course, should be opened - but not without just cause. 

Suggestion - make a running averaged FFT, derive the structural loss factor from spectra made for different time segments and you should see damping as a function of time, as averaged over a slice of time for each value. This should allow you to assess your situation.

Sincerely

Claes

Dear Dr. Fredo,

Performing Half Power Band Width technique on the signal as a whole did not work well. However,  applying the technique on short time intervals showed promising result in comparison with other methods that I've used.

It might emanate from the fact that every non stationary signal can be considered stationary within a small portion of it. My damping ratio is constant, but time-varying damping ratio estimation can be used in health monitoring problems.

I also checked out your website. It was interesting for me.

Thanks,

Yousef

Hi Yousef

Just to eliminate. If you get largely different result between short and long duration signals, this might arise from the natural frequency moving a bit in frequency with time, i.e. that your system is not time invariant.

You should be able to see such frequency variation if you present your data as a spectrogram, i.e. time-frequency-amplitude with amplitude shown as color or, simply plot a bunch of 2D FFTs on top of each other.

Out of curiosity what kind of system is your application?

Thanks. My website was written with the intention of providing a platform for self study. You find some B&K and HP Technical notes that are old but, in my opinion, still the best written as an introduction to vibration and acoustic analysis. http://qringtech.com/learnmore/recommended-reading/

Feel free to make suggestions for improvements to my website.

Sincerely

Claes

Hi Dr. Fredo,

My system is a MDOF Spring-Mass-Damper system, which is decomposed to a number of SDOF via an Operational Modal Analysis method.

Variation in frequency bins is not significant in comparison to the damping variation in different segments. Moreover, I am using a simulated example which is an LTI system, theoretically.

One of the assumptions in Half Power Band Width method is that steady state response should be used for analysis. I guess, since a non stationary response comprises different transient stages, valid range (steady state response segment) might  be picked out meticulously.

Although,  I haven't yet  find any resource discussing damping ratio estimation of a non stationary signals using Half Power Method, I will look up for extra resources. .

Thanks,

Yousef

Hi Yousef

You get a stationary response assumption in there also for the time period you select when doing OMA, i.e. expect damping values to vary depending on the time period you select if/when your damping is non-stationary. That said, OMA is the more general approach as it can resolve nearby modes.The Half Power approach works only for well separated modes.

FYI - It may be useful to know that most modal methods work up to a modal overlap of 0.3. https://math.la.asu.edu/~milner/PDF/61.pdf

Have fun

Claes

Dear Dr Claes,

My problem with half power is that even for sinusoidal input, when I divide the window length by 2, damping ratio is multiplied by 2. you can replace 2 by 3 or 4 or any integer number (for windows of length 1sec or less).

I cannot figure out why is that this sensitivity?

My damping is nearly constant in all windows. And the modes are well separated.

Moreover, it is to note that half-power is restricted for mass proportionally damped systems. It is not a general method while applying to MDOF structures.

I am looking for a new method. By now I have implemented different methods, but no one of them have yielded a reliable result.

Anyway thanks for your contributions to this discussion...

Hi Yousef.

Half power is one of the oldest and simplest methods around. Therefore, younger brothers/sisters will be more evolved at the cost of being more complex. I am not advocating half power over, say, OMA. In fact, I use OMA myself. That said, I do check results using half power to see that things are in the right ballpark.

Why?

Because you are faced with solving an inverse problem. See here. https://www.researchgate.net/post/What_are_the_disadvantages_limitations_of_inverse_finite_element_method_for_soft_tissue_experiments

You have your measured or simulated responses. These are 'truths' that you wish to fit an answer to. Unfortunately, the possible solution space is infinite. Therefore, OMA,, or for that matter any method, truly, can provide beautiful as well as awful answers. The more complex the method, the greater the capacity to provide strange results. This is where half power is good for ball park estimates. It is so crude that it is hard to fool.

For what it is worth, your problem with half power sounds like more of a programing problem (maybe in how you do your FFT) than damping method assessment issue to me, as the half power approach uses ratios and therefore, should be rather 'insensitive' to spectral resolution.

The structural loss factor eta, originates from force-displacement hysteresis. A linear lossless system would show a Force = K*X, where K is spring stiffness, while a lossy system shows Force = K*(1 + j*eta)*X. 

The half power bandwidth definition arises from the so called Q-factor, where Q = 1/eta and Q denotes Quality - for radio crystals. The Q factor is defined for a single resonance and hence, works well only for mdof systems when resonances are well separated in frequency.

Using Google, I find this work. I have not read it myself but the tutors behind this work are among the best in the field. I expect you will find lots of good stuff in there. http://www-g.eng.cam.ac.uk/dv_library/Theses/sondiponthesis.pdf

About mass proportional damping. The only mass proportional damping I know of is Rayleigh damping which is a pure mathematical artefact, i.e. lacking a physical phenomenon. The only physical mass-acceleration proportional damping mechanism I can think of would be throwing mass away from a system, e.g. as is done with the water that is sprayed onto the skin of a rocket during take off. I have seen damper devices that are stated to provide acceleration proportional damping. In any case, mass proportional damping is unusual. 

About your problems. When having problems, I find it always is good to step back and execute a problem that is so simple that no excuses are accepted. So, why not run a linear sdof case. You can try both structural as well as viscous damping as well as solve the sdof equation in the time and spectral domain.

Once you master the linear sdof system, step up to linear mdof systems. Once these are mastered, go back and make your sdof system non-linear, time variant or what ever is your preference.

Start from weakly non-linear and make the non-linearity/time variance stronger to see what happens. Once you know what you are doing, step up to mdof systems.

That said, never in your work, assume a linear model ( y = a*x + b) to be a good fit for a non-linear case (say, y = a*x3 + b*x2 + c*x +d) .

Hope this helps

Claes

Thanks for your insightful comments.

I will give them a try.

Best regards,

Dear Yousef Hedayati,

in order to evaluate the modal frequency and the related equivalent viscous damping factor, I suggest you the Short Time Impulse Response Function (STIRF) described in the following paper. For any question do not esitate to contact me.

Regards,

Rocco Ditommaso

Article Automatic evaluation of the fundamental frequency variations...

How can we estimate the damping factor of a 2nd order SDOF system from circle fitted method