Page 49 - Prosig Catalogue 2005

SOFTWARE PRODUCTS
ALIASING, ORDERS & WAGON WHEELS

considering here then is the relationship between the rate at which we
collect data from a rotating shaft and the highest order to avoid aliasing.
The relationship depends on how we do our sampling as we could sample
at constant time steps (equi-time step sampling), or at equal angles
spaced around the shaft (equi-angular or synchronous sampling). We will
consider both of these but first let us recall the relationship for regular
equi-time step sampling and the highest frequency permissible to avoid
aliasing. This is often known as Shannon’s Theorem.
Standard Aliasing Training & Support
With regular time based sampling using uniform time steps we have a
sample rate of say S samples/second. That is digital values are taken 1/S
seconds apart. For convenience let δt be the time increment in seconds
so that δt = 1/S seconds.
With regular time domain processing we have a time and frequency
relationship. That is if we carry out a Fourier analysis of a regularly spaced
time history then we get a frequency spectrum.
Shannon’s aliasing theorem states that if we have a sample rate S then
the highest frequency we can observe without aliasing is (S/2) Hz. (S
/2) is known as the Nyquist frequency. As previously mentioned the
Figure 1 implication and results of aliasing are illustrated in an appendix below.
So if we have a time step δt then the highest frequency, f , is given by
max Condition Monitoring

This is a relationship between time steps in seconds and the highest
frequency in Hz. It is worth noting that originally frequencies were
specified in units of “cycles per second”, and that the fundamental units
of Hz are 1/second.
Highest order with time based sampling
First recall that orders are multiples of the rotation speed of the shaft.
So if a shaft is rotating at R rpm (revs/minute), then the Nth order
corresponds to a rotational rate of (N*R) rpm. So if a shaft is notating at
1000 rpm then second order is 2000 rpm but if the shaft rotation speed Software
was 1500 rpm then second order corresponds to a speed of 3000 rpm.
Orders are independent of the actual shaft speed, they are some multiple
or fraction of the current rotational speed.
The relationship between order and frequency for a given basic rotation
speed of R rpm is simply:
Figure 2
frequency of the resonance. In this case the resonant frequency is 245
Hz.
This means that this structure should probably not be used if in its
working life it will be exposed to this frequency. Figure 2 also shows that
if this structure was to be used, and only exposed to 300Hz to 400 Hz or
perhaps 0Hz to 200Hz , this particular resonant frequency would not be
excited, and therefore the structure would not vibrate abnormally. Putting this relationship into the regular time based relationship to find Hardware
the highest order to avoid aliasing gives:
Aliasing, Orders & Wagon

Wheels That is the highest order, K max , when using time based sampling at S
samples/second is given by:
These days most people collecting engineering and scientific data digitally
have heard of and know of the implications of the sample rate and the
highest observable frequency in order to avoid aliasing. For those people
who are perhaps unfamiliar with the phenomenon of aliasing then an
Appendix is included below which illustrates the phenomenon. Synchronous or Equiangular Sampling
In saying that most people are aware of the relationship concerning sample With equiangular sampling we take N points per revolution, typically by
rate and aliasing this generally means they are aware of it when dealing using a toothed wheel or similar to give exactly N points per revolution. System Packages
with constant time step sampling where digital values are measured at This is again independent of the actual shaft speed. So our sample rate
equal increments of time. There is far less familiarity with the relevant is N points/revolution.
relationship when dealing with orders, where an order is a multiple of
the rotational rate of the shaft. For example second order is a rate that With equiangular sampling we are in the “revolution” domain and the
is exactly twice the current rotational speed of the shaft. What we are corresponding domain is the order domain. That is if we carry out a
Fourier analysis of an equiangular spaced signal then we get an order

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