View Full Version : Re: Centrifugal Force

Ton Van Den Bogert
12-18-2000, 06:05 AM
Dear subscribers,

I thank Paolo de Leva for his challenges. Out of such discussions always
comes more understanding. First of all, I want to emphasize that there are
practical issues, not just the philosophical ones. Equations can be used for
two purposes: educational and technical. For education, it is important that
equations help us understand. In technical problems, equations provide
relationships between things we can measure and the unknowns that we really
are interested in but can't measure. From that point of view, if nothing
in an equation can be measured, it is not a useful equation. This is what
I see as the greatest problem in insisting on using only inertial reference

Feldman's comments are also important. The human sensory system mostly does
not have an inertial reference (vision can be inertial referenced) and yet
we can use it control complex movements. Obviously the required sensory
information is available.

Paolo de Leva wrote:

> 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:
> > 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).

I apologize for not being more clear, but my point was that you do *not* need
that orientation, and also that you really *can* forget about the inertial
frame, as long as you know those extra terms in the equations of motion.

One of those terms in the equations of motion written for a non-inertial
reference frame is a term due to gravity and acceleration of the origin of the
reference frame. But the beauty of this is that these two effects always
are combined into one term that can be measured with accelerometers.
Intuitively, this should make sense: body segments "feel" the same forces
that the mass inside an accelerometer feels. And it does not matter
if that "feeling" comes from gravity or from an accelerating reference
frame. And Einstein says you can't distinguish between those anyway.

To show this mathematically, let's start with the familiar equation of motion
for a particle in an inertial reference frame:

(1) F + m*g = m*A

where F = (Fx,Fy,Fz)' represents the sum of all forces except gravity, g =
(0, 0, -9.81) m/s2 and A = (Ax,Ay,Az)'. The symbol ' indicates transpose, so
these are column vectors. Now let Fm and Am be the same variables but
measured in a moving reference frame. For simplicity we assume that
angular velocity and acceleration can be neglected (for the full equations
see my article in J Biomech 29:949-954, 1996). Let R be a rotation matrix
describing the orientation of the moving reference frame relative to the
inertial reference frame, and Ao be the acceleration of the origin of the
moving reference frame measured in the inertial reference frame.

The relationship between F and Fm is simply a rotation of the reference
frame: F = R*Fm. The relationship between A and Am can be derived by
twice differentiating the rigid body transformation for coordinates of
a point P: P = T + R*Pm, where T is the translation and R is the rotation
of the reference frame. In the absence of angular velocity and angular
acceleration, the result is: A = Ao + Ro*Am. Substituting these into
equation (1) gives:

R*Fm + m*g = m*(Ao + R.Am)

Pre-multiplying by the inverse of R gives:

(2) Fm + inv(R)*m*(g - Ao) = m*Am

Now this looks exactly like an equation of motion for a particle
again, but there are two differences. First, gravity has been removed.
Second, there is an extra "force" on the left-hand side which
depends on orientation R and acceleration Ao of the moving reference
frame. But, and this is important, we never need to know R and Ao!
We only need to know the combination inv(R)*m*(g-Ao). And this
quantity can be measured with an accelerometer. So for example, if
your reference frame were accelerating towards the center of the earth
with an acceleration of 2g, you would "feel" exactly the same as if the
reference frame had simply been turned upside down. In both cases you
would be pulled towards the ceiling by a 1g force field. And you never
need to know what is really happening, as long as you measure the force
field. Until you start considering that in one of the two scenarios
you will hit the ground sooner or later :-)

> 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

It is important to realize that accelerometers are sensitive to acceleration
and to gravity. It is simply a mass attached to a little force transducer. And
the XYZ components of the signal are measured along axes fixed in the moving
reference frame. Yes, it is the acceleration relative to the inertial frame,
but it is always combined with gravity and rotated to the moving frame. An
accelerometer measures inv(R)*(g-Ao) and this is all you need to know. Again
I am not considering terms related to angular velocity and angular acceleration,
which you can find elsewhere. The principle stays the same.

> 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.

What I meant is that you do not need the Earth or any other inertial
frame, as long as you use accelerometers to determine the force field
due to the reference frame being non-inertial. This requires four triaxial
accelerometers in the general case.

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

And here we disagree. In fact, I would say non-inertial frames are needed
because usually we can only collect data in a non-inertial frame, and we can't
transform the measured variables to a noninertial frame. The Earth is a good
example. It is a non-inertial reference frame and yet we measure forces and
accelerations with our video cameras and force plates in a coordinate system
attached to the Earth. It just happens that the pseudo-force terms are too small
to have an influence on human movement, so we can get away with ignoring them
in our equations. But it is important to know that you don't *have* to ignore
them! That immediately opens up some interesting possibilities. For instance,
you can do inverse dynamic analysis on a ship, no matter how wild its motion is,
provided that you attach four accelerometers to the ship and include that
information in the analysis. People who measure and predict weather patterns
have no choice. They always collect movement data relative to the Earth, but
they *need* to add the pseudo-forces to the equations because they can be as
large as the other forces. Yes, they could write the equations for some
inertial frame, and these equations would be very simple, but they could never
collect the data to actually use those equations.

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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