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Paolo De Leva
12-17-2000, 04:28 AM
Dear subscribers,

Ton van den Bogert's messages always help me to get to the point. His
contributions to BIOMCH-L are unvaluable, and I often received important
discussions with him, and I'll never forget how much I learned from his
interesting comments. So, I hope Ton won't be offended reading what follows,
but I believe in his latest message he drew a misleading conclusion.

First, I'll underline here an appearently simple concept: D'Alembert's
equation is based on the acceleration measured in the inertial frame.

So, unfortunately, you can't just fust forget the inertial (Newtonian)
frame when you use D'Alembert's principle. It might seem that Ton van den
Bogert, using "accelerometers rigidly attached to the non-inertial frame" as
described in his latest message, could deny my previous statement:

> Example 2: Some years ago I was involved in a study on inverse dynamic
> analysis of downhill skiing... 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...
>[omissis]... 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.

Notice that Ton clearly wrote that he needed and obtained the orientation
of an inertial frame (the earth is quasi-inertial, but we can neglect in
this case the effects of its relatively slow rotation).
Notice, also, that the accelerations measured by the accelerometers are
observed from an inertial frame. This might not seem obvious,
but it is absolutely true, in my opinion. The inertial frame used by the
accelerometers is, of course, tilted relative to the usual horizontal and
vertical axes of the frame attached to the earth, and its
orientation changes with time. However, since the accelerometer senses true
accelerations and not imaginary ones, in a particular instant when you
sample the accelerometer readings, that frame should be considered
motionless, i.e.
"frozen" as in a snap-shot. And
being motionless it must be seen as an inertial frame. Otherwise, an
accelerometer would always give null acceleration. In fact, the
acceleration you would observe from a non-inertial reference system rotating
and translating with the accelerometer (and the segment to which it is
attached) would be obviously zero.

Here's how I conclude:

The accelerometer uses, at each particular instant, a different
motionless reference frame, which happens to have, at that instant, the same
orientation of the non-inertial reference frame attached to the
accelerometer. Thus, by all means the accelerometer uses inertial frames..

Here's how Ton concludes:

> So, inverse dynamics can theoretically be done in a
> non-inertial frame

Here Ton seemed to say he didn't need the inertial "Newtonian" frame
(the earth) at all,
although he just stated above he did. (What did you mean, Ton?). This
conclusion might be misleading for those who will read it too quickly. And I
think it is crucial, in this particular
discussion, not to be mislead in that direction.

Of course, I am not saying that non-inertial frames are useless. I am
just saying inertial frames are necessary.

Eventually, I want to warn you that there is a parallel discussion on
"More on inertia?".

Together with Anatol Feldman, I wish you happy Holidays.....
...without doubts about the need of inertial frames.

With my kindest regards,

Paolo de LEVA

Ist. Universitario di Scienze motorie
Biomechanics Laboratory
P. Lauro De Bosis, 6
00194 ROME - ITALY

----- Original Message -----
From: Ton van den Bogert
To:
Sent: Tuesday, December 12, 2000 10:01 PM
Subject: Re: Centrifugal Force

> 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
>
> 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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