Converting Acceleration, Velocity & Displacement

From time to time I meet engineers who are interested in converting acceleration, velocity & 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.

X Y
0 1
1 2
2 3
3 4
4 5

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

v = f(t)

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,

d(t) = \int{v.dt}

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,

a(t) = dv/dt

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

Calculating Velocity Or Displacement From Acceleration Time Histories

Calculating Velocity Or Displacement From Acceleration Time Historieshttps://blog.prosig.com/2007/10/05/calculating-velocity-or-displacement-from-acceleration-time-histories/It is quite straightforward to apply “classical” integration techniques to calculate either a velocity time history from an acceleration time history or the corresponding displacement time history …
Converting Acceleration, Velocity & Displacement

Converting Acceleration, Velocity & Displacementhttps://blog.prosig.com/2010/12/16/methods-of-conversion-between-acceleration-velocity-and-displacement/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 …

 
 

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James Wren

Former Sales & Marketing Manager at Prosig
James Wren was Sales & Marketing Manager for Prosig Ltd until 2019. James graduated from Portsmouth University in 2001, with a Masters degree in Electronic Engineering. He is a Chartered Engineer and a registered Eur Ing. He has been involved with motorsport from a very early age with a special interest in data acquisition. James is a founder member of the Dalmeny Racing team.
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Matt McKean
9 years ago

When you do an FFT… you get mag, real, and imaginary… what is a practical application for using real and imaginary in interpreting phenomena or solving a noise or vibration problem?

Lutz
Lutz
9 years ago

I feel a little bit confused, when reading “acceleration [m]” or
“acceleration [m/s]” at the figures described with velocity and displacement. (i.e. fig. 8 or fig. 10) 🙂
Lutz

Prosig
Admin
9 years ago

Hello Lutz

Thanks for letting us know. Well spotted. It looks like the screenshots were messed when editing the article for the web. They have now been corrected. It might take a short while for the new images to show up due to the page and image cache.

Thanks again.

Jha, PC
9 years ago

Where is the link to article referred by you in reply to Matt? Can you not provide a download version and link (PDF) for these explanatory notes/ articles?

trackback

[…] & phase?By Adrian Lincoln – May 4, 2011Posted in: featured, signal processing In one of our recent articles a question was asked regarding the practical use of real & imaginary type plots compared with […]

Prosig
Admin
9 years ago

Hello Jha

The new article that James mentioned is now available on the blog here… https://blog.prosig.com/2011/05/04/which-should-i-use-real-imaginary-or-magnitude-phase/

Stuart
Stuart
9 years ago

I’m reading this because you’ve just sent out a newsletter with links to the article.

I think there may be some underlying assumptions in your analysis that you haven’t really explained. For example, when you say that the results in figures 6 and 9 are “clearly incorrect” I’m not sure I’d entirely agree. The results do not appear to be correct if the acceleration is, say, a measurement from the vibrating surface of an object that has no net movement – in other words your original acceleration signal erroneously contains DC and/or low frequency components. So the issue here is not so much that you have the wrong integration algorithm, but that you are using the algorithm on inherently inaccurate signals. This can happen when using piezoelectric accelerometers with inadequate bias removal and particularly for acceleration data of shock phenomena. Apparently modern MEMS sensors can be pretty good down to DC so may eliminate some of the problems that you have at source although most of us have a cupboard full of piezoelectric and/or ICP devices which are inherently AC coupled so tend to use a high-pass filter to remove low frequencies and DC and have generally found this to be perfectly successful.

A second point is that we are often interested in the frequency domain characteristics of signals, even if they start life in the time domain. In my experience it is unwise to attempt to integrate and then transform to obtain a frequency domain representation of velocity or displacement. It’s much better to go straight to the frequency domain and then peform the integration through a division by jw (or powers of this depending on whether it is single of double integration and if the spectrum has linear or power quantitites). Maybe that is at the heart of the so-called Omega Arithmetic method (I’m guessing based on the name) and you then do the transform back into the time domain, but unless the engineer actually needs the time domain data then it may be best to stay in the frequency domain once the conversion to a spectrum has taken place.

will
will
8 years ago

Dear James,

I have attempted to calculate the centre of mass displacement during human locomation over different speeds. I have used the doubly integration method of acceleration from vertical ground reaction forces and this has been unsuccessful. I am looking at multiple strides and find that I get a lot of drift in the data. I have come across the omega arithmetic method. Would this be an appropriate method to use to get the information I require? Also can you use Excel software for this method? Would you recommend other data processing software that is more appropriate for the job?

will

Nabaraj Dahal
Nabaraj Dahal
4 years ago
Reply to  will

Hi Will, Just curious to know weather you were successful in estimating the human displacement using omega arithmetic. I guess the frequency range would be between 0.3 to 10 Hz. James Wren suggested the omega arithmetic have poor performance for low frequency frequency range (< 10 Hz). Will you please share your work experience.

Narayan
Narayan
8 years ago

Hi ,

The blog is great.

When we use a digital ideal filter would not the above problem be addressed. The digital ideal filter converts time domain to frequency domain and cut’s off unwanted frequencies and reconverts using IFFT to time domain. In that scenario would not Omega arithmetic be the same.

Would this steps be fine
(1) use ideal filter remove DC from acceleration
(2) Integrate in time domain get velocity
(3) use ideal filter remove DC from velocity
(4) Integrate in time domain to get displacement

Note : The only over head I see the time spend.

Am wrong in this approach. As ideal filter do convert time to freq and converts back.

Thanking you in advance

Ahmed Samir
Ahmed Samir
8 years ago

Hi,

If you have a filtered acceleration signal of frequencies 50 to 350 Hz. Then you are integrating this signal into velocity, can the velocity signal have frequencies below of above 50 and 350 Hz respectively?

Regards,
Ahmed

Sam MacLeod
Sam MacLeod
6 years ago

Hi James,
The article and discussion are very helpful. My question concerns the “filtering” done in the frequency domain. If an FFT is done on acceleration data and then converted to velocity and displacement by integration in the frequency domain, does the filtering prior to IFFT consist of removing just the DC bin, or are there also nearby bins that need to be removed due to the windowing effects?

Thank you

Ivan
Ivan
6 years ago
Reply to  James Wren

Am I right to assume that we should’t use windows (like Hanning or others) before performing Omega Arithmetic transformations?

Hannah
Hannah
5 years ago

Hi James,
Thank you for the interesting post, I am currently working on a project where I have to retrieve the displacement from the acceleration time histories, frequency range of interest in my project is 2-100 Hz. I have implemented the Omega arithmetic method in matlab and the retrieved displacement data still showed some errors. I am doubting on my skills on signal processing and matlab as I am new to this field. I am confusing with the fft and the frequency lines which I used to divide the ffft. The following is part of my script line which I confused with:

az- acceleration time histories
N=numel(az); %length of FFT
f = (0:N-1)*fs/N; %Frequency line for fft
% Calculte the fft of acceleration
AZ=fft(az,N); %Acceleration in frequency domain
V=zeros(size(AZ));
D=zeros(size(AZ));

for i=1:N
if f(i)~=0 %when frequency near 0, the peak is highes. Therefore, filter the value at this frequency
VZ(i)=AZ(i)/(2*pi*f(i)*sqrt(-1) );%Velocity frequency domain
DZ(i)=AZ(i)/(2*pi*f(i))^2; %Displacement frequency domain
else
VZ(i)=0;
DZ(i)=0;
end
end
% reverse fft
vz=ifft(VZ);%
dz=ifft(DZ);

I would truly appreciate if you could help me to see where I am wrong. I am really sorry to ask you very fundamental questions.
Thank You,
Hannah.

nasim
nasim
5 years ago
Reply to  Hannah

Dear Hannah,

I was wondering if you have found the solution with MATLAB?

Nasim

trackback

[…] and distance, a better way is to do an FFT and do the calculations in the frequency domain. https://blog.prosig.com/2010/12/16/m…tween-acceleration-velocity-and-displacement/ […]

Abraham
Abraham
5 years ago

Hi;

Heaps of thanks for sharing the information.

I have seen two different versions for converting acceleration to velocity and displacement and vice versa.

1- Some websites mentioned (Velocity = Acceleration/-i*w) where omega is the frequency in (radians/sec) = 2*pi*f with f in Hz. and (Disp=Acc/-w^2).

i is sqrt(-1)

2-While in some forum it is written (Velocity = Acceleration/i*w) and (Disp=Acc/w^2).

So which one is correct? I noticed that if the fourier transform of time displacement is taken, then the first sentence is correct. However, when the inverse of fourier transform for continuous time displacement is taken, then the second sentence is correct.

Also how about spectral densities of Omega arithmetic. In the last pages of article (http://signalprocessing.prosig.com/OmegaArithmetic.pdf), I cannot understand why negative is there!

I appreciate if some experienced mates help me out.

kate
kate
4 years ago

hello
I am working on project which determines the displacement and velocity of an elevator using an accelerometer , but the problem is , i dunno what to do after i record the acceleration signal i know i have to integrate but what program do i use and how to do it
i hope u help me 🙁

Kate
Kate
4 years ago
Reply to  James Wren

Thank you so much for your help

anonymous
anonymous
4 years ago

James, I measured using a laser doppler vibrometer and I would like to convert the set of velocity measurements to acceleration. Can you type a sample matlab code using x as a set of velocity measurements to convert to acceleration with the most accurate method.

Juan Carlos Añazco Pazos

Aceleration Unit Converter,Quick and easy tool for converting acceleration units. You can convert units like: km/s², m/s², mm/s², milla per second squared, ft/s², in/s² and others
https://play.google.com/store/apps/details?id=com.anazco.juan.acelerationunitconverter&hl=en

Leandro
Leandro
2 years ago

Hello, my name is Leandro, i’m a brazilian engineering. This post helped me very much in a project that i working. I was facing the problems, but the post answers many questions that i have.
Thank you very much. Great job.

Mehmet
Mehmet
2 years ago

Hello James,

First of all, thank you for the great article.

I have been wondering is this method suitable for vehicle level acceleration measurements. I have tested this method in a MAST table for a vehicle level acceleration values. I have measured both acceleration and displacement. After applying this method they are not correlated. Would you be so kind to comment please?

Thanks,
Mehmet

Gayathri
Gayathri
7 months ago

Hi, Am working on the project of vibration sensor. I have to calculate value in units (acceleration,frequency,velocity and displacement). In datasheet ,sensitivity is mentioned as 20mV/m/s2 at 160Hz.

I have applied the formula as mentioned below to find acceleration

Acceleration (1g) => It is about 9.8 m/s2
g = [Output reading(mV) – Zero output(mV) / Sensitivity(mV/g)]
where,
Zero output(mV) = Lets we consider the value we receiving when sensor at rest (2.480v which means 2480mV)
Sensitivity for our vibration sensor = 20

Vibration in terms m/s2 = g * 9.8

Vibration sensor is tested in motor.

My question is how to find frequency using these values? How to consider time domain?
Or otherwise can u guide me other method to find values in unit for vibration sensor?

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