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Ton Van Den Bogert
12-12-2000, 07:01 AM
Dear Biomch-L subscribers,

This is one of those academic discussions which make no real difference
in the end, but they are fun and stimulating. So here is my two cents
worth...

"Dr. Chris Kirtley" wrote:

> Incidentally, I've always wondered why there are no centrifugal forces
> included David Winter's 2D inverse dynamics analysis (in Biomechanics
> and motor control of human movement and elsewhere). Did David leave
> these out because they are negligible in gait, or for soem other reason?

I think the centrifugal force is not included in these equations because
the equations of motion were written for motion measured in an inertial
reference frame. Centrifugal force only appears in equations of motion
written for movement in a rotating coordinate system.

Knowing that Stalin did not allow non-inertial reference frames (thanks
to Arnold Mitnitski for this interesting piece of information), I can't
resist offering a few examples where using non-inertial frames seems to be
a good way to do the calculations.

Example 1: Weather forecasting is done by solving large finite element
models on a mesh that is attached to the earth. And since the earth is
not an inertial reference frame, Coriolis forces and centrifugal forces
(the latter are probably insignificant) must be added to the equations.
This does not make the calculations more difficult; these "pseudo-forces"
are very well known. One advantage of this is that it makes things
easier to understand, for instance why the Coriolis force makes hurricanes
spin counterclockwise in the northern hemisphere. But the main reason is
convenience in the computational work. Even though it is true that the
solution would be the same when the equations are solved in an inertial
frame, one can imagine the difficulties when weather forecasting would be
done on a mesh attached to the sun, or the galaxy, or the center of mass
of the universe...

Example 2: Some years ago I was involved in a study on inverse dynamic
analysis of downhill skiing. Because of the large volume needed for
movement analysis, we considered using a system to measure only the motion
of the body segments relative to the boot, definitely a non-inertial frame.
When transforming the equations of motion to this reference frame, "pseudo-force"
terms appear that include the state of acceleration (linear acceleration,
angular acceleration, and angular velocity) and orientation of the reference
frame relative to the earth. It also appeared that these terms could be
determined from a number of accelerometers rigidly attached to the
non-inertial frame. So, inverse dynamics can theoretically be done in a
non-inertial frame with a completely body-mounted instrumentation system.
In this case, transforming all motion data to an inertial reference frame
is not even possible, because we don't know the motion of the non-inertial
frame. We only know its state of acceleration. We did the project somewhat
differently in the end, but at least I know that it is theoretically possible
and that it requires equations of motion to be written for the non-inertial
frame. And those equations include pseudo-force terms. I don't think this
type of analysis can be done in an inertial reference frame.

In both cases, I guess the reason for using a non-inertial frame is the
difficulty of collecting movement data relative to an inertial frame. It's
fine to write the equations in an inertial frame, but what if you don't have
the data that is needed to do something with the equations?

Finally, I fully agree with Chris Kirtley mentioning Einstein's principle of
general relativity. According to that principle, there is no way of knowing
whether a force that we measure (e.g. gravity) is "real", or "just a pseudo-force"
which is a consequence of doing measurements in a non-inertial reference frame.
General relativity treats gravity as a pseudo-force just like the centrifugal
force. Even Stalin would agree that gravity belongs in a free body diagram,
but in fact gravity is no more "real" than a centrifugal force.

Ton van den Bogert

P.S. For an explanation of the effect of Coriolis forces on the weather,
and some critical comments on draining sinks, see
http://www.ems.psu.edu/~fraser/Bad/BadCoriolis.html

For an introduction to general relativity, see
http://www.svsu.edu/~slaven/gr/index.html

--

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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