When we measure vibration data, especially in conjunction with modelling systems, the measured data is often needed as acceleration, velocity, and displacement. Sometimes, different analysis groups require the measured signals in a different form. It is impractical to measure all three simultaneously, even if we could. Physically, putting three different types of transducers in the same place is impossible.
- Accelerometers are available in all types and sizes, with a very large choice. Some types will measure down to DC (0Hz), others handle high shock loading and so on.
- True velocimeters are quite rare, but they do exist. One interesting class based on a coil and magnet scheme is self-powered.
- Direct displacement measurement is not uncommon. Some use strain gauges, but many others use a capacitive effect or induced radio frequency mechanism to measure displacement directly. The capacitive and inductive types have the advantage that they are non-contacting probes and do not affect the local mass.
But in any case, it doesn’t matter because when we measure vibration, either acceleration, velocity or displacement, then it is surely simple mathematics to convert between them by judicious use of integration or differentiation, as illustrated below.
|Measured Signal Type
So now let us look at this with a classical sine wave signal and see the effects of either differentiating or integrating it. The example uses a 96Hz sinewave of unit amplitude to avoid other side effects with 32768 samples generated at 8192 samples/second. It is useful to look at these as time histories and as a function of frequency. The originally generated sinewave was processed using a DATS worksheet, as illustrated in Figure 1.
Looking at a section of the waveforms, we have a classical result, as shown in Figure 2.
In mathematical terms, if , then the differential is and the integral is where C is the so-called ‘constant of integration’. In both cases, there is a phase shift of 900, which turns the sine into a cosine. The differential is multiplied by . The integral is divided by , is also negated and has had an offset added to it, which in this case is half the resultant amplitude, resulting in the integrated signal being entirely positive. If, for example, the original signal had represented an acceleration, then the integrated signal is a velocity, and clearly, we would not expect that to be entirely positive. This integration constant is an artefact of the standard integration methods.
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For the mathematically inclined, it is the result of carrying out what is usually referred to as an indefinite integration. The solution is quite simple. After a standard time-based integration, we should automatically reduce the result to a zero mean value. That is, we ensure there is no residual DC offset. The calculation process was modified to include that action, and the result is shown in Figure 3. Note how the integrated signal is positive and negative, as expected.
It is also interesting to look at the Fourier Transforms of the three signals. These are shown in Figure 4 in modulus (amplitude) and phase form. The moduli are shown in dBs, and the phase is in degrees.
Looking first at the phase, the original sinewave has a phase shift of -900. This is entirely as expected because the FFT’s basis is a cosine. The differentiated signal has a zero phase change as it is now a pure cosine. The integrated signal has a 180-degree phase change, denoting it as 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 with double precision. This is approximately equivalent to a 50-bit accuracy ADC! The integrated signal shows a similar dynamic range, but what may appear surprising initially is that 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 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 signal and the results are shown below.
The effect on the original is essentially not noticeable. Similarly, the differentiated signal is unchanged, as would be expected. However, 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 a zero mean, often called normalising.
Note that at this juncture, we have not had to do anything to the initial signal when we are differentiating, but we have had to remove any DC offset before integration to prevent the ‘drift’ and also remove the DC offset from the integrated signal to eliminate the constant of integration. So, at this stage, one might be tempted to conclude that using a differentiating scheme might be the best way forward. However, when we add noise, the situation changes.
As a start, a small random noise signal was added the original sinewave.
The noise is not discernible to the eye on the original signal, but the differentiated signal has become very noisy. The integrated signal remains smooth. We can, however, identify the dominant frequency quite well.
If one examines the phase of the noisy signals, one can see it is now all over the place and essentially no longer has any value. Automatic phase unwrap was used, if the phase had been displayed over a 3600 range, it would have filled the phase graph area.
The dynamic range of the original signal with added noise is around 90dB, with the differentiated and integrated signals having a similar range. That is, the added noise has dominated the range.
One other aspect to notice is that the background level of the noise on the integrated signal rises at the lower frequencies. This is known as 1/f noise (one over f noise). This sets an effective lower frequency limit below which integration is no longer viable.
To emphasise the challenge of noise, the next example has a much larger noise content.
Here the noise on the original signal is evident. The differentiated signal is useless, but the integrated signal is relatively clean. To really illustrate the point, the noisy sinewave was differentiated twice. The result is shown below. All trace of the original sinewave seems to have gone or, rather, has been lost in the noise.
The conclusion is now clear. If there are no special circumstances, then experience suggests it is best to measure vibration with an accelerometer. However, care is required to remove the very low frequencies if any integration to velocity or displacement is needed.
As a final point, why should differentiation be much noisier than integration? The answer is that differentiation is a subtraction process, and at its very basic level, we take the difference between two successive values and then divide it by the time between samples. The two adjacent data points are often quite similar in size. Hence, the difference is small and will be less accurate, so we divide by what often is a small time difference, and this tends to amplify any errors. Integration, on the other hand, is an addition process. As any broadband noise tends to be successively differently-signed, then the noise cancels out.
This article, of course, does not tell the whole story, but it provides a very simple guide to best practice when we measure vibration – should we measure acceleration, velocity or displacement?
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