Vibration : Measure Acceleration, Velocity or Displacement?

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.

Measured Signal Type







double differentiate












double integrate


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.

Figure 1: Worksheet calculating frequency spectra

Looking at a section of the wave forms, we have a classical result as shown in Figure 2.

Figure 2: Original, differentiated and integrated time histories

In mathematical terms if y(t) = Asin(2{\pi}ft) then the differential is \{2{\pi}Afcos(2{\pi}ft)\} and the integral is \{-(A/2{\pi}f) cos(2{\pi}ft)\}+C 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 2{\pi}f. The integral is divided by 2{\pi}f, 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.

Figure 3: Adjusted to zero mean

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.

Figure 4: Fourier transforms

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.

Figure 5: Result of integrating with small DC offset

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.

Figure 6: Addition of a small amount of noise

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.

Figure 7: Spectra from noisy signals

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.

Figure 8: Time series with more noise added


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.

Figure 9: Noisy signal differentiated twice

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.

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Dr Colin Mercer

Chief Signal Processing Analyst 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.

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16 thoughts on “Vibration : Measure Acceleration, Velocity or Displacement?

  1. Boyd Hardonk

    You mention in your article “Vibration Analysis – Should We Measure Acceleration, Velocity or Displacement?: Acceleration + Integrate ==> Displacement

    To my opinion this should be: Acceleration + Double Integrate ==> Displacement

    Kind regards,

    Boyd Hardonk

    1. Chris Mason

      Dear Boyd

      Thank you for pointing that out. It seems a typo crept in when transferring the article to the blog. It has now been corrected.

      Thank you again

  2. Yeugeny Chirkov

    It is known the better way to difrerentiate the noisy signal -to approximate the signal before.
    Best Regards,

  3. Pingback: Velocity, Displacement & Acceleration: Science vs. Engineering « Sharad Sinha

  4. jim

    Informative article.But would be much helpful if a basic(lame man’s)explanation could be given..
    does speed of a drive/machine (RPM) has something to do with selecting a right sensor??

  5. Colin

    This is really a very basic article trying to just show some guidance if you want Acceleration, Velocity and Displacement. The conclusion is that it is better to use an accelerometer transducer. The article does not attempt to be a layman’s guide on all the aspects in choosing a transducer, that is actually a tricky task because there are so many criteria and constraints. There might be environmental problems such as a very low or high temperature.
    In the situation you mention then yes the speed of the machine on which you are taking measurements does matter. If the machine runs at a speed of up to say 6000 RPM this is a frequency of 100Hz. But how many orders (harmonics) of the basic speed do we need to measure? If we need to go to say the 15th order then that is a frequency of 1500Hz. So we want to make sure the accelerometer has a linear response bandwidth of say twice that value. Then we also need to ensure the acquisition system has a good analogue low pass anti-alias filter (it must have an analogue filter somewhere in the acquisition chain). We need to be able to sample at a rate typically 2.5 times the anti alias filter cut off frequency to allow for the finite cut off rate of the filter.
    So there are multiple questions and scenarios. There are other Prosig articles which cover other aspects.


  6. Shivneel Goundar

    i am using vibration sensor which gives voltage as output.
    how can i calculate to get acceleration from these voltage values?

    1. Colin

      all transducers have a voltage to physical units conversion factor for example an accelerometer could be 50mV/g. Beware MEMS type transducers as they may not be sufficiently stable. If there is no calibration certificate than I would not trust the device. What transducer are you using?


  7. Lady

    If using an accelerometer to measure vibration during CNC milling, what kind of output signal would i expect for no chatter vibration?

  8. Colin

    Chatter or No Chatter? Generally speaking chatter in a milling environment is a resonance phenomenon. There are two types of chatter: tool chatter or work-piece chatter. Tool chatter is when the cutting tool vibrates resonantly. And obviously work-piece chatter is when the work- piece is resonating.

    When there is no chatter the signal would basically look random. When chattering there are large harmonic components. A word of caution however is to beware when milling work-pieces that have non uniform stiffness as these can generate cyclic forces.

    In both chattering cases the vibration output will suddenly rise as milling speed increases and most often look like a large sine wave. Typically this is at a frequency corresponding to the rotational speed of the milling machine but it could be a harmonic.

    So a simple, non intelligent, monitor would be on signal (acceleration) amplitude, probably RC smoothed with a short time constant to avoid transients.

    In both cases it is probably best to monitor in the frequency domain. This will then show as a large ‘spike’ at the resonant frequency or possibly as a set of ‘spikes’ related to the harmonics of the resonant frequency, typically at 1st, 3rd , 5th and so on.

    If chattering occurs then change the rotating speed, or if not possible reduce the milling force.


  9. Muhammad

    Dear Mr. Colin, Thanks for the valuable information shared. I have few queries which are stated below:
    We have large sea water pumps in our plant of 60000 cu.m/hr of rated flow. These are tubular casing vertical single stage pumps having 300 RPM. The motor bearings have BN 330505 Velocity sensors installed. The output is displayed in peak on DCS after BN 1900 monitor processes the signal. Now the problem is that Alarm and Trip limits for vibration are set at 5.8 & 8.9 mm/s RMS and we see frequent alarms due to high peak values of vibration signal transmitted by the probes. Can you throw some light on how I can receive the output in RMS from these probes? Do I have to modify the existing BN 1900 or change the probes?

  10. prashanth

    i am doing research on monitoring of ground vibration due to blasting in mines. and i am getting few doubts.please clarify me
    1.which accelerometer is suitable to measure ground vibration
    2.the blast induced ground vibration frequency range in between 10-20 Hz only can i convert accelerometer data into velocity

  11. Reginaldo Vieira

    Hello, how are you.
    I have a doubt about gs acceleration in vibration analysis.
    What is the function of the gas acceleration in the analysis of vibration?
    What acceleration gs measures in the analysis of vibration?
    Would I like a more easy to understand explanation.

    They say the acceleration gs measures energy, others say that friction me, I need a better explanation on this subject.

    Thanks for listening.


    Reginaldo Vieira
    Sertaozinho, SP Brazil

  12. colin mercer

    I do not understand what you mean by ‘gs acceleration’? Do you mean acceleration measured in units of g’s
    For reference an acceleration of 1g is 9.806 65 meters per second per second (m/s2).
    If one has an acceleration it is not a measure of energy. Assuming you mean energy in a moving object then that is the kinetic enrgy which is related to velocity.

    fo a steady velocity ov v m/s then kinetic energy is mv^2/2 where m is the mass.

    Weight, W, and mass are related by 1g so m= W/g so kinetic energy becoms Wv^2/(2g)


  13. Dave D

    You talk about a low bound frequency when double integrating the acceleration PSD to displacement. I am working on a project where we are taking acceleration readings and comparing to displacement acceptance criteria. The comparisons are made using RMS values. Our excitation is random broadband turbulence coupled with structural responses. The low frequency information is dominating the RMS value. Do you have guidance on how to select an appropriate lower bound for integration to displacement?

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