Page 60 - Prosig Catalogue 2005

SOFTWARE PRODUCTS
VIBRATION ANALYSIS: SHOULD WE MEASURE ACCELERATION, VELOCITY OR DISPLACEMENT

Training & Support

Figure 5: Result of integrating with small DC offset

Figure 3: Adjusted to zero mean Note, that at this juncture, we have not had to do anything to the initial
Condition Monitoring how the integrated signal is positive and negative as we would expect. this stage one might be tempted to conclude that using a differentiating
signal when we are differentiating, but we have had to remove any DC
we ensure there is no residual DC offset. The calculation process was
offset before integration to prevent the ‘drift’ and also remove the DC offset
modified to include that action and the result is shown in Figure 3. Note
from the integrated signal to eliminate the constant of integration. So at
scheme might the best way forward. However, when we add noise the
It is also interesting to look at the Fourier Transforms of the three signals.
situation changes.
These are shown in Figure 4 in modulus (amplitude) and phase form. The
moduli are shown in dBs and the phase is in degrees.
As a start, a small random noise signal was added to the original sinewave.
The noise is not discernible to the eye on the original signal, but the

Software

Figure 4: Fourier transforms Figure 6: Addition of a small amount of noise
differentiated signal has become very noisy. The integrated signal remains
Looking first at the phase, the original sinewave has a phase shift of smooth. We can however identify the dominant frequency quite well.
-90°. This is entirely as expected because the basis of the FFT is actually
Hardware a pure cosine. The integrated signal has a 180 degree phase change,
a cosine. The differentiated signal has a zero phase change as it is now
denoting it is a negative cosine.
The dynamic range of the original signal is well over 300 dB which is
not surprising as it was generated in software in double precision. This
is approximately equivalent to a 50 bit accuracy ADC! The integrated
signal shows a similar dynamic range but, what may appear as surprising
initially, the differentiated signal has lost half of the dynamic range. We
will return to this point later.
Small DC offsets are not uncommon in many data acquisition systems.
Some offer AC coupling (highpass filtering) to minimise any offset. How
System Packages signal and the results are shown below.
would this affect the resultant signals? To illustrate this point a small DC
offset of 0.01 (1% of the amplitude) was added to the original sinewave
The effect on the original is essentially not noticeable. Similarly the
differentiated signal is unchanged as would be expected. But the effect on
the integrated signal is quite dramatic. The small DC offset has produced
a huge trend. We have integrated a 0.01 constant over 4 seconds, which
gives an accumulated ‘drift’ of 0.04. The underlying integrated signal is
still evident and is superimposed on this drift.
How do we avoid this? Simply reduce the input to have a zero mean,
which is often called normalizing. Figure 7: Spectra from noisy signals

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