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Ton Van Den Bogert
12-19-2000, 06:18 AM
Paolo de Leva wrote:

> My point was clear and simple (in my honest opinion, of course). Briefly: accelerometers do
> use inertial frames.

We still disagree on this. In my opinion, accelerometers do not measure acceleration,
they measure force. And they do not show how much of that force was produced by
acceleration and how much by gravity. And my point is that we do not need to know,
for the purpose of doing dynamics calculations in a non-inertial reference frame.

> Isn't it obvious that an accelerometer has zero acceleration in its own local frame?"...

The force vector that is measured by an accelerometer indeed can be written as a
function of its acceleration in the global inertial reference frame, and gravity:

F = m.a - m.g

F, a, and g are all 3x1 column matrices expressed in the global reference frame.
However, an accelerometer rotates with the frame it is attached on, so the XYZ
components of that force vector are determined along the XYZ axis of the local frame.
So the three signals we get are Fm = (Fmx, Fmy, Fmz), which can be written as:

(1) Fm = inv(R)*F = inv(R)*m*(a-g)

This is proportional to the pseudo-force that we needed to add to the equation of motion
for a particle in the moving reference frame. So this shows that the accelerometer
gives sufficient information to complete the equation of motion. This does not include
the effects of a rotating frame, since R is assumed to be constant, but terms
related to angular velocity and angular acceleration can be added too.

Now read the following carefully:

What may be confusing here is that we have a 3-D acceleration relative to the
global frame, but quantify that acceleration vector by its XYZ components in the
local frame. This is a technical necessity: the sensors are attached to the
moving frame, even though the force vector may be the result of an acceleration
relative to the global frame.

To prevent further confusion, let me make it clear that the symbol Am that I used in
my previous posting refers to the acceleration of a particle relative to the local frame,
expressed as XYZ components relative to that local frame. For an accelerometer, Am is
zero, because it does not move in the local frame. Equation (1) above can be derived
as a special case of that general equation of motion by setting Am = 0.

So this should resolve that paradox: an accelerometer has zero acceleration in its
local frame, but its signal is not zero. Its signal is a combination of *global*
acceleration and gravity, expressed as XYZ components along the *local* axes.
This is only a rotation of the reference frame, the magnitude of the vector
is not affected.

> 1) Ton knows accelerometers better than his wife :-).

I hope not...

> I ask you, Ton, and all readers: should we conclude that the force field in a non-inertial frame is different from zero? (I remind you Necip Berne's statement about the need for equilibrium, i.e. net force = 0).
>
> MY ANSWER: Probably not.

My answer: yes. The force field is not zero and it needs to be added to the equations
of motion in a non-inertial reference frame. Only then will the movement obey the
equation sum(Fm) = m*Am. The force field becomes one of the forces on the left hand side.

> Well, what's the meaning of Ton's statements then? Did he maintain that accelerometers can measure apparent-imaginary-fictitious (i.e. inertial) forces?
>
> MY ANSWER: No, I don't think so. Ton perfectly knows that an accelerometer is "simply a mass attached to a little force transducer" (or some equivalent device embedded in an integrated circuit).

My answer: yes. An accelerometer measures exactly the force that is required to
keep a mass from moving in the local frame. And that is exactly the same force
(after dividing by the accelerometer mass, and multiplying by the particle mass)
that must be added to the equation of motion for a particle that moves around
in the local frame.

> I wonder how should we describe the net external force and the weight of the small
> mass. Ton, should we call them imaginary or real forces?

You can call them what you want, it is not important. However, when explaining this
is might be helpful to say that some of the force is "real" (gravity) and some of it
is caused by accelerating or rotating the frame to which the accelerometer is
attached.

> But only one wrote Einstein's equivalence principle correctly: Dr. Kris Kirtley.
>
> "..cannot be distinguished...by any 'interior' experiment."

Thanks for clarifying this. This is indeed an important part of the principle.

> Referring to Ton's example about weather forecasting, please let's not forget we
> need to know the angular velocity of the earth and the radius of rotation of the air
> particles to compute the value of the Coriolis and centrifugal forces thei are acted
> upon. Ton, I have a latter question for you: were these data measured in a "fixed" -
> inertial reference frame or in a non-inertial reference frame?

Good question. These data were probably derived by observing the motion of the
stars relative to the earth, then assuming that the stars do not move so this
must reflect rotation of the earth. So an inertial frame (the stars) was used.
However, if we had cloudy skies and could see no stars, we could have measured
these forces with a (very sensitive) accelerometer.

Ton van den Bogert

--

A.J. (Ton) van den Bogert, PhD
Department of Biomedical Engineering
Cleveland Clinic Foundation
9500 Euclid Avenue (ND-20)
Cleveland, OH 44195, USA
Phone/Fax: (216) 444-5566/9198

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