# Equivalent acceleration of an accelerometer

1. Dec 9, 2007

### ylim

Hi there,

I am reading this old journal and found this formulaes.

forces on the seismic mass, expressed in inertial reference coordinates, are given by newton's law as

sum of force in x direction = m * x''
sum of force in y direction = m * y'' = Fy = mg

the equivalent acceleration of the accelerometer are
ax = -F/m = -x''
ay = -F/m = -y'' - g

i am rather concerned with the change in sign direction. i vaguely understand the concept of inertial and non inertial ref frame. but i dont know why it will cause a diff to the axial acceleration. hope to hear from someone.

2. Dec 9, 2007

### belliott4488

I'm not sure I understand your notation (Fy is force in the y direction, and ax and ay are accelerations in the x and y directions, right? x" is also acceleration in x direction?)

In any case, however, the negative signs probably come from balancing the inertial force on the test mass, which results from the external acceleration that you're measuring, with the restoring force of the spring (assuming that's what supplies the balancing force in the accelerometer) - equal and opposite forces, you know. In other words, if the system is accelerating to the right (positive direction), the mass will swing to the left (negative direction), so it applies a force in the negative direction. If you want to think about the non-inertial frame of the accelerometer, it just means that a tiny observer in that frame would see the mass move to the left and interpret that as a force (and acceleration) to the left.

That's my guess, anyway - it's kind of hard to be sure since you haven't given much explanation.

3. Dec 9, 2007

### ylim

this will come to a qn which i have, i have an accelerometer, when i swing it to its positive direction, will it be a positive or a negative output?

so (to be more specific) to a right accelerating accelerometer (positive is right), what should be my output? positive or negative?

>> I'm not sure I understand your notation (Fy is force in the y direction, and ax and ay are accelerations in the x and y directions, right? x" is also acceleration in x direction?)
- Yes.

i guess we can see it when one is a in plane, when it is accelerating, we cannot anticipate/follow up, so we get push back to our seats? the seats gives the 0 opposite force, which is the spring in your explaination?

BUT, the weird thing is the sign of the g. which is -g? in the case of a resting accelerometer, in a inertial ref frame, it should be -1g. but in a non-inertial, it should be +1g? I notice this from a analog devices (crossbow) development board i have.

Last edited: Dec 9, 2007
4. Dec 9, 2007

### belliott4488

The output can be either - it's just a matter of convention. If the output reports the calculated acceleration on the whole system, then it will be positive (in your example). If it reports the apparent acceleration of the mass in the non-inertial internal system (which is the "raw" information it measures) and leaves you to convert that back to the system acceleration, then it would report a negative (left) acceleration.

Yes, I think you're correct about the plane example: the plane accelerates forward during take-off, but inside, in our non-inertial frame, we feel a force that accelerates us backwards into the seat. We stop when the seat cushion is compressed enough to provide an opposing force equal to the inertial force of our bodies, and if the cushion happened to calibrated to provide a nice measurement of that compression, we could determine the rate of acceleration of the plane.

As for the factor of g, that has to do with whether we're reporting the "sensed" acceleration or the "true" acceleration. This in turn depends on how the accelerometer is calibrated, i.e. under what circumstance does it read 0 vertical acceleration. It could be when it is at rest, i.e. when it senses an acceleration of -g, or it could be when it is actually falling at an acceleration of -g, so it senses no net acceleration.