# Order Analysis Signal Processing

I'm currently working on vibration detection machine used to detect unbalance from a rotating shaft, rotating from 300-2500 Hz. I have both the tacho signal ( 1 PPR ) and the vibration from transducer.

Now I'm really confused. After the transducer there is sweep filter that uses the tacho signal as center frequency so at 1 kHz the output sine wave is 1 kHz.

How do i extract useful data from this signal ?

I've read for the past week on the internet about order analysis and different methods but im so confused about how to implement it, how fast should the ADC sample rate be for the order tracking?

Should I do Fourier transform over the sine wave?

How to transfer the signal to angular domain?

Everything is just confusing. A little help on order analysis, or how to extract information from rotating shaft would be very appreciated!

Most signal processing starts with time-sampling a signal: you have two signals, the vibration signal, $v$, and the tachometer signal, $t$. The first thing you need to be sure of is that these are sampled synchronously in time. Do you know if that's the case?

The reason order analysis is used in the analysis of rotating machinery is that the various frequencies present in the vibration signal often depend on the rotational speed of the main shaft --- which can vary with time.

Order analysis gets around this by resampling the vibration signal synchronously with rotation angle of the main (reference) shaft.

So what you need to do is to take the time-sampled signals you have and produce a single rotation-angle-sampled signal.

To generate the rotation angle signal, you need to take the tachometer signal, $t[n]$, and find the instantaneous phase of it. The way I'd start this is to find the Hilbert transform of $t$ and form the analytic signal: $$a[n] = t[n] + j H\left\{ t[n] \right\}$$ where $H$ is the Hilbert transform and then the instantaneous phase is: $$\theta[n] = {\rm atan2}\left( {\tt Im}\{a[n]\} , {\tt Re}\{a[n]\} \right)$$

From there you should be able to resample both $a$ and $v$ so that you get each evenly sampled in phase.

Once $v$ is evenly sampled in phase, you can then just use an FFT to do order analysis.

yes they are sampled synchronously in time, does the hilbert transform still need to be applied ?

If they were not sampled synchronously, then it would be almost impossible to perform order analysis at all.

What is the purpose of the sweep filter ?

The aim of the sweep filter is to turn the tacho signal into as pure a tone as possible. If it's not a tone, then the Hilbert transform trick will not work correctly for finding the instantaneous phase.

do i still need to apply FFT? Or does FFT only works on the broadband signal?

Once the vibration signal is resampled, it will be synchronous with the main shaft. To get the orders (frequencies, in effect), you need to have some sort of frequency plot. The FFT is the easiest.

The effect of this analysis is illustrated in this Matlab application note. I believe the matlab function rpmordermap effectively computes the short-time Fourier transform (but for "orders" rather than "frequencies").

• Thank you Peter for your clear answer i feel more familiar with the concept now thanks to you. In my application the tacho signal starts ADC sampling so yes they are sampled synchronously in time, does the hilbert transform still need to be applied ? Another question that confuses me a lot. What is the purpose of the sweep filter ? ( the passband filter that changes frequency according to the tacho signal frequency ) , do i still need to apply FFT? Or does FFT only works on the broadband signal? Can you please explain if you are familiar. Thanks a bunch ! Feb 22, 2016 at 21:08
• I'm so thankful @Peter K. for your edit, I want to ask some more questions but I'm afraid I'm bothering you. I still don't understand how the sweep filter affects the tacho signal, in the application it affects the vibration signal. It only lets certain frequencies to pass, so basically the vibration becomes simple sine wave with frequency changing as RPM increases. Feb 23, 2016 at 21:02