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I am attempting to validate an accelerometer. I have a triaxial accelerometer and placed it on a wheel which goes at various frequencies from 1 to 3 Hz. Am I right in thinking that I would expect the acceleration in z to be 1g?

Then in the x and y plane I would expect a read out of acceleration in $X(t) = -2\pi fr\cos{(2\pi f t)}$ and acceleration in $Y(t) = -2\pi fr\sin{(2\pi f t)}$ where $r$ is the radius of the wheel and $f$ is frequency.

How can I compare these with my readout from the sensor, and what is the best way to report accuracy? I'm confused as surely I will need to quite accurately time align the two traces.

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  • $\begingroup$ Does your wheel fall? $\endgroup$
    – Bernhard
    Commented Jun 16, 2014 at 14:13
  • $\begingroup$ No its actually a mechanical wheel that sits on a table and I have placed my accelerometer at different positions on it. I have measured the radius to the nearest mm $\endgroup$
    – branny12000
    Commented Jun 16, 2014 at 14:16
  • $\begingroup$ Then why do expect an acceleration of $g$ in the $z$-direction? $\endgroup$
    – Bernhard
    Commented Jun 16, 2014 at 14:22
  • $\begingroup$ I thought for example if the accelerometer was placed on a flat surface it wil always show 1g in the z direction $\endgroup$
    – branny12000
    Commented Jun 16, 2014 at 14:31
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    $\begingroup$ Except that it does show 1g when sitting flat. The accelerometer cannot distinguish between acceleration and sitting still in a gravitational field. My accelerometer certainly shows ~1g in one direction when sitting still. $\endgroup$
    – BowlOfRed
    Commented Aug 27, 2014 at 5:06

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Then in the x and y plane I would expect a read out of acceleration in X(t)=−2πfrcos(2πft) and acceleration in Y(t)=−2πfrsin(2πft) where r is the radius of the wheel and f is frequency.

The planes on your accelerometer are aligned with respect to the unit, not with respect to your room. When you attach it to the wheel, you're going to fix it in place. So two of the axes will rotate with it. You would put it on the wheel and align one axis (say x) radially. Then the entire centripetal acceleration will appear on that axis. You won't have to worry about any time component. Your y axis should only show accelerations as you spin up and spin down the wheel.

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An accelerometer measures the force on a small test mass and will therefore read 1g total acceleration when at rest on the earth's surface. depending on how well the axes are aligned with the housing and the housing with the vertical, it is quite possible that your device will not have the full value on one axis but that you need to take the vector sum $\sqrt{x^2+y^2+z^2}$.

When you have a trace of the response while it's rotating, the trace should have a different magnitude (and a new apparent direction of force) but no sinusoidal variation unless the wheel "wobbles" or unless the accelerometer rotates with respect to the wheel. In the rotating frame of reference of the wheel, the accelerometer is stationary and the apparent acceleration is the (vector) sum of the centrifugal acceleration (observed in rotating frame of reference) and gravitational acceleration.

You said you had obtained traces. Maybe you can show what they looked like?

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  • $\begingroup$ Another way of describing the reason for no sinusoidal variation is that the accelerometer is measuring acceleration relative to axes along its own body, not axes in some external inertial frame--so if a particular axis is aligned with the radial direction at one moment, it should continue to be as the wheel rotates, so if the rotation is constant that axis just gives a measure of the constant centripetal acceleration. $\endgroup$
    – Hypnosifl
    Commented Jan 11, 2015 at 22:44
  • $\begingroup$ @Hypnosifl - thanks for the comment. That was what I was trying to say; maybe when it is said in two different ways it becomes more obvious to the OP. $\endgroup$
    – Floris
    Commented Jan 11, 2015 at 22:46
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As someone in the comments pointed out, it won't show any acceleration in the "z" direction if it's just sitting there. What people mean when they talk about gravitational acceleration is when the object is freely falling.

Your other method with the wheel might be sound, but I'd suggest a better method: since your device definitely doesn't know any "absolute" directions (that is, it's always measured the acceleration in certain directions, no matter how the device is oriented), figure out the directions relative to the device itself, and then drop it off some height, with the device oriented in that direction, so you can measure only that one (for example, have a weight, and then a "tail", like a lawn dart, and put the device on one of the fins of the tail. It will always point in the same direction). This will be a lot simpler than a rotating setup, and allow you to isolate them.

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  • $\begingroup$ To clarify (maybe :-) ), $g$ is a force . Acceleration is a change in velocity which only happens when there's a net force. your accelerometer's net force in the $\hat{z}$ direction is zero due to the normal force of the surface it's sitting on. $\endgroup$
    – Carl Witthoft
    Commented Jun 16, 2014 at 14:48
  • $\begingroup$ Ah I see! Thank you guys for helping me. Well I have considered just a simple drop experiment but the test has already been done by another person and I just must analyse the result $\endgroup$
    – branny12000
    Commented Jun 16, 2014 at 15:11
  • $\begingroup$ Hey guys as I have already acquired a trace from this method any ideas of how to work out error? And align the traces so they can be compared? $\endgroup$
    – branny12000
    Commented Jun 17, 2014 at 0:05
  • $\begingroup$ @Carl Witthoft - But accelerometers don't measure acceleration relative to a Newtonian inertial frame, nor do they measure net Newtonian "force"--they are real-world devices and hence they are better understood using general relativity which is more accurate than Newtonian gravity. In general relativity, the only frame-invariant notion of acceleration is proper acceleration, and since accelerometers don't make use of any particular reference frame, naturally that's what they measure. $\endgroup$
    – Hypnosifl
    Commented Jan 11, 2015 at 14:40
  • $\begingroup$ (continued) According to general relativity, gravity is not really a "force" at all but rather curvature of spacetime, and general relativity's equivalence principle says that being in freefall in a gravitational field is locally equivalent to moving inertially (zero proper acceleration) in non-curved spacetime, and likewise sitting at a constant radius in a gravitational field is locally equivalent to accelerating in non-curved spacetime (so that's when an accelerometer should measure a nonzero proper acceleration). $\endgroup$
    – Hypnosifl
    Commented Jan 11, 2015 at 14:43

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