Decay and Dampening

[rfulky]

How do you back the damping ratio out of a transient wave measured on an oscilloscope?

 

[GregLocock]

You have to measure the exponential decay.

I've forgotten how to do so exactly but basically draw an envelope over the peaks, this gives you the decay curve against time. It should be in any noise and vibration or dynamics text.

Cheers

Greg Locock

 

[electricpete]

Let's say have a waveform exp(-sigma*t)*sin(wd*t)

Hopefully you know how to find wd. It is the radian frequency. If you find the time between zero crossing T, then wd = 2*Pi/T

How to find sigma?

Let's say the waveform decays by ratio K (<1) between two successive peaks (separated in time by t = 2*Pi/wd)

exp(-sigma*2*Pi/wd) =
-sigma*2*Pi/wd = ln (K)
sigma = -ln(K)*wd / (2Pi).

Now you have the info to define the pole zero position at sigma +/- j*wd.

One definition of damping factor is cosin of the angle that the pole makes with the imaginary axis. That would be cos(sigma)/sqrt(sigma^2+wd^2)

There may be other definitions of damping factor. Let me know your definition and I can tell you how to express it from sigma and wd.

By the way, for lightly damped systems, you can draw the frequency response curve very easily from sigma and wd. As you know it is a bell shaped curve (for single-degree of freedom single damped mass spring system). Make your horizontal axis radian frequency. The peak of the curve is wd. The point where the curve drops to 1/sqrt(2) of it's peak is sigma away from wd. That is called the half-power bandwidth.

 

[electricpete]

There is a dangling equation:
exp(-sigma*2*Pi/wd) =

There should be a K on the right hand side of that equation

 

[electricpete]

&quot;That would be cos(sigma)/sqrt(sigma^2+wd^2)&quot; was incorrect. It is cos of angle but I don't need cos function when I express it in this form.

The damping factor is (sigma)/sqrt(sigma^2+wd^2)

For lightly damped systems it is approx sigma/wd

 

[anands78]

Let me go ahead and state the obvious.
From the decaying time graph one can get the amplitudes of the peaks. the logarithmic decrement is given by

logdec delta=ln(Xn/Xn-1)
ln=natural log
Xn,Xn-1= amplitudes of 2 successive peaks.

The llogdec and the damping ratio are related by
logdec = 2*pi*zeta/sqrt(1-zeta^2)
where xeta=damping ratio.

THese are the ones deriuved from electricpete eqns.

 

[electricpete]

Here is an attempt to simplify it:
exp(-sigma*2*Pi/wd) = K
-sigma*2*Pi/wd = ln (K) = logdec
sigma = -logdec*wd / (2Pi).

Zeta = (sigma)/sqrt(sigma^2+wd^2)
= -logdec*wd / [(2Pi) *sqrt( (logdec*wd / (2Pi))^2+wd^2)]
= -logdec*wd / [wd * sqrt(logdec^2 + (2PI)^2)]
= -logdec / sqrt(logdec^2 + (2PI)^2)

For lightly damped systems logdec^2 << 2PI^2
Zeta ~ -logdec / (2Pi)

So you can estimate damping factor zeta from logdec = ln(K) where K is ratio betweeen two sucessive peaks (K<1).

Even though sigma and wd are not required for the above calculation, sigma and wd are in my view useful parameters in their own right. They give us an estimate of the equation of the time waveform (exp(-sigma*t)*sin(wd*t)) and an estimate of the shape of the spectrum (bell shape centered on wd with half-power bandwidth of sigma).

 

[tomirvine]

I will show you how to perform this calculation if you Email the time history to me in ASCII text format.

Tom Irvine
Email: tomirvine@aol.com

http://www.vibrationdata.com

 

[electricpete]

Hi Tom - do you disagree with
Zeta ~ -logdec / (2Pi)
where &quot;logdec&quot; is ln(k)
k = ratio between two positive peaks

The error in the approximation would be <5% for k>0.2

 

[anands78]

Electricpete..The approximation has worked out quite well for me in most cases. The fact that Zeta gets squared adds to this.

 

[tomirvine]

Update...

rfulky sent me his time history data, in response to my offer. The data was interesting in that it consisted of several spectral components, which produced a slight beat frequency effect.

I have written a paper and a software program that identifies frequencies and their respective damping ratios for such data. The paper is called:

A Time Domain, Curve-Fitting Method for Accelerometer Data Analysis

I will be presenting it at the:

AIAA Conference, Session 100 DSC-10, April 10, 2003; Norfolk, Virgina

If you are interested in receiving a copy of the paper, let me know.

Tom Irvine
tomirvine@aol.com

http://www.vibrationdata.com

 

[GregLocock]

That's interesting. I think a similar technique was the basis of the MDOF curve fit in SDRC's MPLUS modal analysis suite. The modal parameters (frequency and damping) were done by some sort of successive subtraction on the impulse response. It also extracted the eigenvector at the measurement point, but must have used the FFT to do that, I guess.

I'm sure your paper would have helped me a couple of years back, now I don't get to play with FFTs at all.
Cheers

Greg Locock

 

[MikeyP]

We have developed damping identification methods based on analysis of sinusoidal decaym (known as the &quot;Resonant Decay Method&quot. This uses a combination of force appropriation and least squares regression. The benefit of this approach is that it is able to identify the degree of non-proportional damping coupling between modes. If significant NP damping is present then the concept of a damping ratio is inappplicable.

The danger of fitting any old sinusoidal response decay is that you have no idea which modes are contributing to the decay. It may well give reasonable results for a mode which is not close in frequency to any others, but this is often not the case.

If anybody wants more info the I can send you copies of papers.

M

michael.platten@man.ac.uk