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Converting Acceleration, Velocity & Displacement

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From time to time I meet engineers who are interested in the conversions between acceleration, velocity and displacement. Often, they have measured acceleration, but are interested in displacement or vice versa. Equally, velocity is often used to find acceleration.

This article will attempt to outline the nature of the conversion between these units and will suggest the preferred method for doing so. We will deliberately avoid some of the more complex mathematics. For the mathematically minded there are already other excellent articles on this blog that describe the mathematics involved. This article will also suggest the best method for such a conversion.

Before discussing these conversions we should consider what these measurements are.

  • Displacement is the measurement of a distance travelled. If an object has moved 10 meters then it has been displaced by 10 meters or 10 m.
  • Velocity, often incorrectly called speed, is the measurement of a certain displacement in a certain time in a specific direction. For example an object that moves 10 meters in a second is travelling at the velocity of 10 meters per second or 10 m/s.
  • Velocity can be both negative and positive as it is a vector. The magnitude of a velocity is the speed, this can only be positive.
  • Acceleration is the rate of change of velocity, it is also a vector as it implies a direction. If we had a stationary object, it would have no acceleration. The rate of change of its velocity is zero. And, it follows, that an object travelling with a constant velocity of 10 m/s is not accelerating, it is not getting any faster or any slower. Therefore the rate of change of its velocity is zero. However, if an object is changing velocity then it is accelerating. Acceleration can be both negative, which is deceleration and positive, which is acceleration.
  • If an object starts from a stationary position and accelerates up to a velocity of 10 m/s in 1 second then the acceleration is 10 meters per second per second or 10 m/s/s.
  • If the object started from a stationary position and accelerates up to a velocity of 10 m/s in 50 seconds then the acceleration is 0.2 meters per second per second or 0.2 m/s/s.
  • Units of acceleration are often shown as $m/s/s$, $m/s^2$ or $ms^{-2}$.

As can be seen from these definitions of displacement, velocity and acceleration, they are all very closely related. In fact, in mathematical terms they are directly related and simple conversions exist. The mathematical relationship involves calculus, specifically integration and differentiation.

The mathematical integral of the velocity curve against time, is the displacement. That means if you plot the velocity curve f(x) against time and measure the area under the curve you have the total displacement. This relationship is shown in Figure 1.

Figure 1: Integrating velocity to obtain displacement

The mathematical differential of the velocity curve f(x) against time, is the acceleration. That means if you plot the velocity curve against time and measure the slope of the curve a at a given point in time T you would have the acceleration at that time. This relationship is shown in Figure 2.

Figure 2: Differentiating velocity to obtain acceleration

In simple terms these curves can be described as functions, a function being a representation of a signal. For example, in Figure 3 below, a simple signal exists.

Figure 3: A simple digital signal

The (x,y) co-ordinates for this curve are as follows in Table 1.

XY
01
12
23
34
45

Table 1

The pattern is clearly repeating and simple in its nature. To express the contents of Table 1 as a function would be (x, f(x)) rather than (x, y) or more commonly denoted as y = f(x). In this case when x = 1 so then y = x + 1 and so on. Therefore the function $y = f(x)$ is actually $y = x + 1$,

So $f(x) = x + 1$

This explains the concept of a curve or signal being expressed as a function rather than sets of co-ordinates. Returning to our velocity curve, if the velocity curve against time was represented as a function, it would be

</p><p>v = f(t)</p><p>

Where $f$ is the function at a given time $(t)$ the x-axis value, that produces the y-axis value $v$.

But as a displacement curve it would be represented as the function,

</p><p>d(t) = \int{v.dt}</p><p>

Where $d(t)$ is the displacement and $dt$ is the change in displacement with time.

Hence as an acceleration curve it would be represented as the function,

</p><p>a(t) = dv/dt</p><p>

Where $a(t)$ is the acceleration and $dv$ is the change in velocity in time.

Returning from core mathematics and using DATS to visualise these conversions, we have a simple worksheet as shown in Figure 4.

Figure 4

This worksheet takes a vibration signal and performs integration using three different methods and then integrates again on the resulting signals. Thus from one acceleration signal, it is possible to convert to velocity and then to displacement.

Method 1

Integration only.

Method 2

Application of a high pass filter and then perform the integration.

Method 3

Apply the DATS Omega Arithmetic integration algorithm.

Figure 5 shows the acceleration signal  measured by an accelerometer.

Figure 6 shows the velocity signal from the "integration only" method.

Figure 7 shows the velocity signal from a high pass filter and integration method.

Figure 7

Figure 8 shows the velocity signal from the Omega Arithmetic integration method.

Figure 8

Figure 9 shows the displacement signal, created by performing the integration on a velocity signal as in Figure 6.

Figure 9

Figure 10 shows the displacement signal, created by performing a high pass filter and the integration on a velocity signal as in Figure 7.

Figure 10

Figure 11 shows the displacement signal, created by performing the Omega Arithmetic integration on a velocity signal as in Figure 8.

Figure 11

These graphs show the different results from our three methods of the conversion between acceleration, velocity and displacement, pay special attention to the magnitude of the y-axis units, these reflect the conversion in question and give some guide to the accuracy of the signals.

It can be seen that, in Figure 6 and Figure 9, the integration only method has produced results which are clearly incorrect. The displacement curve shows a movement of several metres when we know we were measuring displacements of only mm's. The shape of the curves tells us that there is a factor or constant that is affecting the shape of the curve. The data appears to go in the same direction and is constantly increasing in magnitude. This is because of the low frequency or DC content of the signal. This causes an effect which throws out the conversion process and the integration cannot account for this DC content. This error builds up as the signal is integrated and gives this growing or decreasing error effect. As can be seen from Figure 6 the actual valid part of the signal can just be seen but is very small, it appears to be super imposed on the growing error. It is clear from this that Method 1 is not the correct procedure.

As Figure 7 and Figure 8 show very similar signals, one might expect there to be no difference, but on closer inspection of the y-axis values it is possible to see the filter and integrate method has left a slight DC offset present in the signal. That is, the origin is not about zero when we know that it should be. This same pattern is repeated in Figure 9 and Figure 10. Figures 12 & 13 show the velocity and displacement from Method 2 and Method 3 superimposed.

Figure 12

Figure 13

This leaves the Omega Arithmetic method for discussion, after experimentation it has been proven to be the only valid method. The reason for this is that, with Omega Arithmetic, the integration is completed in the frequency domain and not in the time domain. The signal is converted to the frequency domain with an FFT, integrated or differentiated then using an inverse FFT converted back to the time domain.

In summary, if converting from Acceleration to Velocity to Displacement, the required conversion is integration, to go the other way differentiation is used.

Simply applying calculus to a time domain signal is not an acceptable method to perform this conversion.

Filtering in the time domain to remove any DC content, for example filtering out 5 Hz or below, then integrating does produce reasonable looking results, but they are not correct.

The correct conversion method is Omega Arithmetic.

Further Reading

If you are interesting in discovering more about this topic the following articles are also available

Acceleration, Velocity & Displacement Spectra – Omega Arithmetic
by Dr Colin Mercer

Calculating Velocity Or Displacement From Acceleration Time Histories
by Adrian Lincoln

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James Wren

Application Engineer & Sales Manager at Prosig
James Wren is an Application Engineer and the Sales Manager for Prosig Limited. James graduated from Portsmouth University in 2001, with a Masters degree in Electronic Engineering. He is a member of the Institution of Engineering and Technology. He has been involved with motorsport from a very early age with special interest in data acquisition. James is a founder member of the Dalmeny Racing team.
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