# Understanding use of accelerometers to measure angular (rotational?) accelerataion

Imagine this: A rod is attached at one end to a rotating shaft. So, as the shaft rotates from 0 to 90 degrees back-and-forth (in the xy plane), so does the rod. My goal is to measure the instantaneous acceleration of the rod while its moving. I have a few 3-axis accelerometers, but I feel like I'm really confused. I guess I'm not really understanding how accelerometers can work best in this scenario because it's not really a "linear" acceleration. I think its a rotational or angular acceleration (are those the same?). Also, even though the rod is only moving in the xy plane, since it rotating, doesn't that mean that it's rotating about the z-axis? Although I'm assuming I only need data in the x and y dimensions.

I guess here are my more specific questions: Should I use more than one accelerometer here? If so, where would be the best locations to mount them? How should I mathematically combine these data to give me an overall acceleration (perhaps just sum the squares, and then take the square root)? And lastly, let's say the rod ramps up from 0 degrees, stays at a constant velocity, and then ramps down. How should I expect the data to look like in this simple case?

I'm hoping someone can shed some light on the use of accelerometers in this particular scenario. It got to the point where doing more research online has only confused me more, so I'd really appreciate any help I can get here.

Let me guess the location of the accelerometer.

My goal is to measure the instantaneous acceleration of the rod while its moving.

I will assume it's on the extremity of the rod. There would be 2 forces acting on your accelerometer:

• gravity
• centrifugal force

I think that you want to measure the 2nd one, of course.

Now, I can see 2 configurations depending on how you fix the accelerometer on your rod.

# 1. If it has the possibility to rotates (what I called floating) Then you will have to deal with data on both x and y axis

# 2. It is fixed on the rod Then you'll have only one axis data.

let's say the rod ramps up from 0 degrees, stays at a constant velocity, and then ramps down. How should I expect the data to look like in this simple case?

I'll try to predict, I'm not 100% sure. So, in case you set-up the fixed accelerometer, you would have a ramp-up (linear, or whatever depending on the transient) to a constant value, then stays at this constant value until or ramp-down or stops.

• Hmm I'm not sure I'm picturing what you're saying. How could you have a setup where the accelerometer isn't fixed on the rod? Right now the accelerometer is definitely fixed on the rod in my mind. Also, is there any advantage to using more than 1 here? If I should only use one, would it be better to mount it near the center of rotation, or at the rod's extreme end like you said? Thanks by the way. Mar 19 '15 at 14:25
• By not fixed I mean it's on a kind of bearing.. I'd no idea of your exact configuration so I tried to imagine. I think you don't need more than 1 accelerometer. If you put it near the rod's extreme it would have more intensive force applied to it, so the signal magnitude would be higher, thus increasing the signal-to-noise ratio, which is good. Mar 19 '15 at 14:32
• OK. I think that helps a lot. I'm gonna try collecting some data and see if I can make sense of it. Anything else I should be looking out for you think? I'm definitely still getting the hang of all this. Mar 19 '15 at 15:01
• Also, how do I account for the distance from the center of rotation in data? Like in terms of the math/physics? Mar 19 '15 at 15:02
• The force that is applied to your accelerometer is proportional to the distance from the center multiplied by the radial speed (in rad.s-1). By the way, don't forget that reward by voting up is welcome on this site ;-) Mar 19 '15 at 15:22

You certainly can do this. Kionics app note AN019 Using two tri-axis accelerometers for rotational measurements' outlines exactly what you are talking about. This method essentially exploits, from the rigid body planar kinematics of a rotating disk, that the linear acceleration $a = r\dot\theta$. ($\dot\theta$ being the angular acceleration and the linear acceleration $a$ is what the accelerometer measures).

This is all based around rigid body motion. See the Wikipedia entry on rigid body motion, particularly the point Acceleration of two points fixed on a rigid body.' From this you can get both the angular acceleration and angular velocity. However, the angular velocity is only available in quadratic form, so you have to find the correct answer somehow. Some people integrate the angular acceleration and use this result to find the sign of the angular velocity. You will have trouble with sensor noise and this will limit the performance of your sensor. You might try extending the length of your rod ($r$) or finding lowew noise accelerometers or even filtering.

As an aside, this method has been extended into - wait for it- the third dimension! The aim of this is to build IMUs that do not rely on gyros. Check out accelerometer only IMU's and have your mind blown.