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Calculate amplitude(displacement) from accelerometerby Pivskid
Tags: accelerometer 
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#1
May613, 05:12 AM

P: 8

Hi there
Given a result from an accelerometer, mounted on a vibrating machinery. I would like to be able to calculate the physical amplitude. That is the actual movement of the vibration. I'm sure it should be possible, but my attempt to manipulate the formula of acceleration has not resulted in valid results. Given is a sinusodial measurement of 20hz, 4g "peak to peak". The acceleration fomula is fairly simple: a=v/t But how do I convert it into using hz and g as input values and output the displacement (amplitude) ? 


#2
May613, 05:28 AM

P: 718

The formula ##v = at## is only true for constant acceleration. In vibrational motion, you have a changing force (e.g. changing linearly in simple harmonic motion) and so acceleration is not constant. In the general case, you need to integrate: ##v(t) = \int _{t_0} ^t a(t) dt + v(t_0)##. Since you will have a set of numerical data points, you will need to do use some technique of numerical integration. With ##t_0## chosen so that ##v(t_0)=0## (the turnaround point), you can ignore the integration constant. Since integrating over a single cycle of periodic motion is equivalent wherever in the cycle you start, you can just thrown this term out. Now, having ##v## as a function of time, you can numerically integrate it again to get displacement as a function of time.



#3
May613, 06:37 AM

P: 8

Can you outline for me more specifically how it would look like ?
I assume it is the "length" (time) from turnaroundpoint(=t0) to peak acceleration, that is necessary. That time should be =(1/hz)/4 How to get from there ? ...integrate from t0 to (1/hz)/4 ? 


#4
May613, 07:08 AM

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Calculate amplitude(displacement) from accelerometer
The usual practical way to do this is to assume the vibration is simple harmonic motion, not to integrate the acceleration numerically.
If the displacement is ##A\sin \omega t##, the acceleration is ##\omega^2 A \sin \omega t##. You need to convert "g" into units of meters/sec^2 or inches/sec^2, and ##\omega = 2 \pi f## . 


#5
May613, 08:54 AM

P: 8

If ω=2π*frequency
Then what is t ? 1/f ? 


#6
May613, 09:44 AM

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P: 6,959

t is time.
If the maximum displacement is ##A##, and the maximum acceleration is ##\omega^2 A##. 


#7
May613, 10:43 AM

P: 718




#8
May613, 11:50 AM

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Second, mesuring relatively large accelerations is cheap and simple, compared with measuring small displacements directly. For example at 50 Hz, or 3000 RPM for a rotating machine, a peak accleration of 1g corresponds to a peak displacement of about 0.01mm. 


#9
May613, 12:05 PM

P: 718

Numerical integration is easy and, if the accelerometer has sufficient time resolution, very accurate. It will give OP his answer directly without having to make any assumptions about the linearity of material. In fact, it will provide a way of checking to what degree the motion deviates from SHM, if that's something OP is interested to know. I can't for the life of me understand why someone would advocate relying on a theoretical (and imperfect) model for determining something you could actually directly determine using the available tools. It makes no sense at all. 


#10
May613, 12:20 PM

P: 718

This looks to be a nice overview of the techniques of numerical integration for definite integrals. Software packages like MATLAB and Python have built in modules for doing this as well. 


#11
May613, 01:50 PM

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The OP never said anything about "a series of acceleration measurements" either. You don't need digitallly sampled data to do this sort of vibration monitoring, in the real world. 


#12
May613, 02:24 PM

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#13
May713, 07:21 AM

P: 8

Hi again
I found the solution. http://www.spaceagecontrol.com/calcsinm.htm Very simple, and it conforms to my actual "reallife" measurements. Now I just wonder how the sin() and integration appeared on the scene. 


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