Technology: Ceptstrum Applications

The theory is best learned via Google, as there are many presentations at many different levels.

In practice a Cepstrum will find several dominant harmonic series in a spectrum and reduce them to a few Rahmonic peaks

which are easier to identify in an automated system. In this example there is only 1 dominant harmonic series.

The “Orders” spectrum harmonic fundamental is 2.70850 x which = 0.3692 “Turns” in the cepstrum, which agrees reasonably

well (0.3721) with the ceptsrum cursor values. The Turns accuracy is very dependent on how clear the Orders harmonics stand out.

These values are non synchronous to all shaft speeds.

Ideally, if you know all the shaft speeds in the gearbox and you automatically detect a Rahmonic series which does not coincide with

any of them, there is a good chance it has discovered a bearing fault harmonic series.

Comparison with known bearing frequencies on the various shafts will identify the bearing fault.

The scale of a Cepstrum is a theoretical thing which has no relationship to machine condition, and they are not meaningfully trendable,

but the can be a great indicator.

Machinery vibration is the result of forces from individual machine parts and it’s modal parameters.

Tooth meshing forces from a 20 tooth gear, & a “once per turn” impact from a damaged tooth are added.

The summed waveform is fft’d and multiplied by the system response fft in terms of magnitude and phase.

(Sum of Forces) (N) x Machine inertance (g/N) = g’s

The result is the measured machine vibration in g’s.

Note the general rise in the spectrum around 70 Hz due to the modal parameters (directly above).

Spectrum analysis can separate the vibration response of the 2 faults by different frequencies of occurrence

but it can’t separate the forcs from the modal parameters.

Machine vibrations can be selectively removed by filtering in the Cepstrum.

The Cepstrum has an imaginary part and a real part, the latter being the useful part for vibration analysis.

The real part can be Short Pass Liftered (high pass filtered) to remove the 1x impact harmonics and show the 20x tooth meshing frequency more clearly.

It’s not a perfect process and the peak height is lower.

The real part can be High Pass Liftered (low pass filtered) to remove the 20x tooth meshing frequency harmonic and show the 1x impact harmonics.

Understanding a Cepstrum is tricky, and the decreasing peaks at multiples of 1 turn in this example (look like harmonics) imply an event

at ½ turn, 1/3 turn, ¼ turn etc. but there were no sub harmonics on the machine. These peaks are just artifacts of the process,

which by chance help to identify a significant frequency in the original spectrum.

There is also a “Complex Cepstrum” which is used for speech analysis and echo removal. It requires both the magnitude and phase spectrum data,

and the latter requires “unwrapping”. This is a torturous task and cannot always be relied on to work properly. Fortunately vibration analysis problems don’t need it.