When using vibration data, especially in conjunction with modelling systems, the measured data is often needed as an acceleration, as a velocity and as a displacement. Sometimes different analysis groups require the measured signals in a different form. Clearly, it is impractical to measure all three at once even if we could. Physically it is nigh on impossible to put three different types of transducer in the same place.
Accelerometers are available in all types and sizes and there is 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 if we measure either acceleration, velocity or displacement then it is surely simple mathematics to convert between them by a judicious use of integration or differentiation as illustrated below.
So now let us look at this with a classical sine wave signal and see the effects of either differentiating or integrating it. To avoid other side effects the example uses a 96Hz sinewave of unit amplitude with 32768 samples generated at 8192 samples/second. It is useful to look at these as time histories and as function of frequency. That is, the original generated sinewave was processed using a DATS worksheet as illustrated in Figure 1.
Looking at a section of the wave forms, 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.
For the mathematically inclined, it is the result of carrying out of what is usually referred to as an indefinite integration. The solution is quite simple. After doing a standard time based integration then we should automatically reduce the result to have 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 we would expect.
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 modulii 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 basis of the FFT is actually 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 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 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. 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 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 the best way forward. However, when we add noise the situation changes.
As a start, a small random noise signal was added the 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 any value. Automatic phase unwrap was used, if the phase had been displayed over a 3600 range it would have totally 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 very much larger noise content.
Here the noise on the original signal is evident. The differentiated signal is effectively 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 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, then we divide by what often is a small time difference and this tends to amplify any errors. Integration on the otter hand is addition. 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 good practice.
Chief Signal Processing Analyst (Retired) at Prosig
Dr Colin Mercer was formerly at the Institute of Sound and Vibration Research (ISVR), University of Southampton where he founded the Data Analysis Centre. He then went on to found Prosig in 1977. Colin retired as Chief Signal Processing Analyst at Prosig in December 2016. He is a Chartered Engineer and a Fellow of the British Computer Society.