What is the name of the fourth inertial force in a rotating 3-dimensional frame?

[Erland]

Inertial forces are not real, but apparent forces that are experienced by an observer in a non-inertial reference frame, reflecting the motion of the frame. Thus, you experience a backwards inertial force if you sit in an accelerating car, and a forward intertial force if the break is applied.

If the reference frame is rotating, well known inertial forces are the centrifugal force and the Coriolis force. We also have the Euler force, which reflects the angular acceleration of the frame.

But there is a fourth inertial force in a rotating 3-dimensional frame, which I don't know what it is called. This force reflects the change of the axis of rotation. If the frame has its origin fixed, this intertial force at a point with position vector r is F_i = - ω dn/dt × r, where ω is the angular velocity and n is the axis of rotation, a unit vector.
For example, if the centre of the Earth is considered as fixed and the Earth's axis is changing so that the north pole is moving towards Greenwich, an observer at the north pole will experience an inertial force directed east, seen from Greenwich.

Does anyone know what (if anything) this inertial force is called? Is it included in the Euler force (since it also deals with non-uniform rotation)?

Also, does the inertial force which reflects linear acceleration (such as in the car example above) have a a name?

 

[Elliot Webb]

I am not aware of a named force that will rotate an axis that is rotating itself, but there is a force that resists this - it is actually quite hard to change the position of a rotating axis, due to gyroscopic precession en.wikipedia.org/wiki/Precession. This could also be regarded as a pseudo force in the same way, as it will resist any rotation in another axis.

 

[Erland]

I am not talking about actual forces which cause anything to rotate, but about fictitious forces which are experienced by observers on bodies which rotate in some arbitrary manner. What causes this rotation is not of interest here.

 

[A.T.]

- ω dn/dt × r

Is it included in the Euler force ?

That's the definition of the Euler acceleration. See:
http://en.wikipedia.org/wiki/Euler_force#Mathematical_description

 

[Erland]

That's the definition of the Euler acceleration. See:
http://en.wikipedia.org/wiki/Euler_force#Mathematical_description

This means, then, that the Euler acceleration (yeah, I was talking about "force" before, but forgot about the mass) can be divided in two parts with quite distinct character:

The angular velocity vector ω can be written as ω = ωn, where ω is the (scalar) angular speed and n is a unit vector parallell to the axis of rotation: I call it the rotation axis vector. Let's say we choose it (one of two choices: ±n) so that ω>0.

According to Wikipedia, the Euler acceleration is a_e = - dω/dt × r. But then

a_e = - dω/dt × r = - dω/dt n × r - ω dn/dt × r.

The first term reflects the angular acceleration and is all I find treated when I read about the Euler force. The second term reflects the change of rotation axis, and this I have never seen treated in texts (OK, I did not read very many texts), and this is the term I am interested in right now.

As I wrote, the two terms have quite distinct character, and I think they should be given different names. But there seems to be no specific name of the inertial acceleration given by the second term...

 

[A.T.]

The first term reflects the angular acceleration and is all I find treated when I read about the Euler force. The second term reflects the change of rotation axis, a.

Angular acceleration can be also defined as a vector (time derivative of the angular velocity vector) . Then it includes the change of rotation axis orientation.