# Proper Acceleration In Accelerometer ?

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1. Oct 18, 2015

### Lync

Hi guys,
I'm working on my project with accelerometer. I know that accelerometer measure specific force or proper acceleration in general by deflection of proof mass.
Proper acceleration is relative to free fall which could be analogy that proper acceleration is the relative acceleration between accelerometer(A) and a free fall object (B) .
+ Case 1: When accelerometer is on a table, it output 1G in Z direction.
proper acceleration Z = m[ F(external) + F(gravity A) - F (gravity B) ] = 1G
=> what is external force that make accelerometer output 1G, is it normal force from the table ?
+ Case 2: When accelerate in +X direction , the proof mass is deflected by fictitious force.
proper acceleration X = m[ F - F(gravity B) which I think is wrong.
The fictitious force make the proof mass deflect in -X direction, how could it output positive value?
Many thanks !!!

2. Oct 18, 2015

### Staff: Mentor

Right.
Same case as for the normal force. Non-gravitational forces always act on the case and not on the test mass. The device "knows" that and flips the sign internally.

3. Oct 18, 2015

### Lync

Can you give me some glues about non-gravitational force. Normal force and fictitious are non-gravitational force?
As I understand, since normal force is not non-gravitational force , it would not act on test mass. Gravity force is the only force that make it deflect in -Z direction in case accelerometer is on a table. That sound reasonable

If the fictitious force is also a non-gravitational, it is not acting on test mass. How can the test mass deflect ?

4. Oct 18, 2015

### Staff: Mentor

Start by looking at this system from the point of view of an observer who is not accelerating, just watching the accelerometer being accelerated past him. The test mass is trying to move in a straight line at a constant velocity because that's what's inertia says masses do. Meanwhile, the case of the accelerometer is accelerating because it's in the lab which is being accelerated by the 1G force in the +x direction you describe in your post #1 above. The test mass is connected by a spring to the case (because that's how accelerometers work) and as the case accelerates it pulls on the spring; the spring in turn applies a force to the test mass pulling it along with the accelerating case - if it didn't, the inside wall of the case would smash into the test mass. Thus, the test mass is being deflected from straight-line inertial motion by the only force acting on it, namely the very real force from the spring.

The fictitious force appears when we instead look at the system from the point of view of someone who is at rest relative to the accelerating accelerometer. Like the first observer, they see the test mass and the inside wall of the case threatening to smash together, with the collision prevented only by the tension of the spring. However, the test mass is not accelerating under the influence of that force, so they have to invent a fictitious force that appears to be pulling everything in the -x direction to cancel out the force from the spring.

5. Oct 18, 2015

### Staff: Mentor

Gravity acts on both case and test mass, it does not influence the result at all. Accelerometers cannot measure gravitational forces.
What they can measure is the force from the table, pushing the accelerometer upwards. That's completely analogous to your hand pushing it sidewards. In both cases the test mass has inertia, so relative to the case (and in this frame) it feels a fictitious force against the direction of the external force.

6. Oct 18, 2015

### A.T.

The accelerometer is measuring acceleration by real contact forces, not by fictitious forces.

7. Oct 18, 2015

### Lync

I don't understand why accelerometer cannot measure gravitational force while it is the only force that acts on the test mass, causing it to deflect. The normal force acts on the case (housing) only.

So, the ouput acceleration is the acceleration of test mass or the case ? .

Furthermore , I found a lab instruction from UC Berkeley in which they state that :
"An accelerometer measures proper acceleration of itself, which does not necessarily correlate to coordinate
acceleration; to be precise, it measures speciﬁc force. An accelerometer can be thought of as a spring and mass
system: coordinate acceleration (meaning the device physically moves in space) results in displacement of
the mass, but because the mass is freely movable, gravity also displaces it. This means that an accelerometer
measures gravity even if the device is not moving in space."

8. Oct 19, 2015

### A.T.

Will the test mass be deflected if gravitational force is the only force acting (accelerometer in free fall)?

Right, for the test mass to be deflected you need such froces that are not acting on everything, propotially to it's mass.

9. Oct 19, 2015

### Lync

yup, it's exactly what i'm wondering.
Gravitational force act all both case and I see that it the only acted on test mass.
Why one is deflected and the other is not ?

I'm undergraduate in Electrical Engineering, I'm not very familiar with these Physics knowledge :(

10. Oct 19, 2015

### A.T.

Well, have you tried to drop the accelerometer to see what it measures in free fall?

11. Oct 19, 2015

### Lync

yeah, I know that if we drop it in free fall, it will return 0G which means test mass is not deflected.
If we explain it as proper acceleration concept . The relative acceleration between it and frame of reference ( by assume a free fall of object with zero proper acceleration) is zero. The output value is 0 which is totally reasonable.

From my point of view , both cases (on table & free fall) are the same except there is normal force which acts on accelerometer while it is on the table and no other force in free fall. However, this normal force does not act on test mass. So why there is difference in output value ?

12. Oct 19, 2015

### jbriggs444

The normal force does not act on the test mass, correct. There is another force that does -- the force from the supports that bind the test mass to the case. That force is monitored by strain gauges.

In free fall, the supports for the test mass are under no strain. When the case is being deflected from a free fall trajectory by an applied force, the supports for the test mass are under a strain because they have to keep the test mass on the same trajectory as the case.

13. Oct 19, 2015

### A.T.

Because there is difference in the internal force between test mass and accelerometer case. Such internal force is required when the external forces alone cannot produce the same acceleration of both parts.

Last edited: Oct 19, 2015