Averaging: RMS vs Vector 1

[ebarba]

Hello all,

I'm trying to assess vibration attenuation spectrum of a flexible suspension for an automated punching machine.

My instrumental setup is made of two accelerometers, one mounted on the machine and the other one on the floor nearby the machine. The excitation comes from the machine operation. I'm acquiring the signals with a PC based digital acquisition card and I'm calculating a transfer function for the data set (both in amplitude and phase). I'm also performing a coherence analysis for the transfer function.

I have these questions regarding averaging:

1) May I use vector averaging? I tried both on my measurements and RMS averaged data differs totally from the vector averaged one, being vector averaged data more likely what I'd like to obtain. I know I can only use vector averaging on a fixed trigger situation. Since Transfer Function should have constant phase for all significative components, while all the others should be randomly changing, this would allow me to use vector averaging. Could someone confirm this?

2) Is it normal to have completely different spectra from RMS averaged data and vector averaged ones?

2) The system's desired natural frequency is 2,5 Hz. In order to get significative data for such a low range of the spectrum I'm using long sampling periods (10s), could anyone give me an idea of how many averages should I take to have significant data for the freq. range I'm interested in (0.5-100Hz)? I'm getting low coherence (68%), so I'm assuming my number of averages is low... what else could it be?

May someone send me a link to some document where can I find a mathematical definition of vector averaging? I checked on my textbooks and other documentation, even the web, but I could not find any math expression.

Thanks and regards
Ernest.

 

[SomptingGuy]

By vector averaging, I assume you mean that your crosspower spectra have their real and imaginary components treated independently? If so, this the normal way to do it.

And by RMS averaging, I assume you mean that you are averaging the RMS values and the phase values - not generally a good idea.

 

[SomptingGuy]

The answer to your question: "Is it normal to have completely different spectra from RMS averaged data and vector averaged ones" is yes. The ratio is connected to the coherence.

If your phase is wandering all over the place, the vector-averaged crosspower will have a lower amplitude than an RMS averaged crosspower, because some of the measurements will canel out other ones.

Increasing the number of averages by itself will not increase your coherence. It will give you a smoother, more reliable coherence plot, but if your underlying data are not coherent, suspect your measurement setup, your excitation level or contamination from other sources.

 

[ebarba]

Hello SomptingGuy,

What I am trying to find is the exact definitions for RMS and Vector averaging, i.e. some softwares call the same operation in different ways.

So far, what I've found on the (vague) software's help is that the RMS averaging should be a standard, non weighted average on the amplitude values and (separately) on the phase values. According to the software's help the vector average takes into account the phase when calculating the averaged values. There lies the problem: I really don't know what's going on inside the software...

I've found too many different names for the same operation and I was hoping someone could give me a math formula so I could manually check what the software is calculating.

I’ll try to upload some graphics on the web, to try to be more clear.

Thanks.

 

[SomptingGuy]

If you are looking to calculate a transfer function (a.k.a. a frequency response function), always use vector averaging. Here are the steps:

If your input is x and your output is y, for each average:

1) Take the complex fourier transform of x and y, call then Fx and Fy (both complex arrays).

2) Calculate the cross power and autopowers of that average(' means complex conjugate);

gxy = Fx*Fy'
gxx = Fx*Fx'
gyy = Fy*Fy'

3) Repeat many times to give many estimates of gxy, gxx and gyy. Take their vector averages Gxy, Gxx and Gyy. Average the real and imaginary parts independently.

4) The transfer function is given by Gxy/Gxx. You can get its magnitude and phase at each frequency from the complex values.

5) The coherence is given by (Gxy*Gxy')/(Gxx*Gyy) It has a maximum value of 1 and a minimum of 0. Note that if you only have 1 averages, coherence is exactly 1 by definition.

 

[SomptingGuy]

and...

RMS averaging has no place in transfer function measurement.

 

[ebarba]

Thank you for the information... This morning I performed the analysis successfully.

Regards