Formula for fictitious moments

[taalf]

Hi All,

Everyone knows so called "fictitious" forces, also known as "inertial" forces. They are forces felt by some mass point placed in a non-inertial frame. For example: a ball in a moving car or in a carousel.

Maybe most intuitive fictitious forces are centrifugal forces, but there are also Euler forces and Coriolis forces. Formulas for such forces are well known (bold means "vector"):

Euler force: F_E = -m ⋅ dΩ/dt × OP
Centrifugal force: F_Ce = -m ⋅ Ω × (Ω × OP)
Coriolis force: F_Co = -m ⋅ 2Ω × V

with:
m the mass of the point,
Ω the rotation vector of the non inertial frame,
OP the position of the point in the non inertial frame,
V the velocity of the point in the non inertial frame.

Know, consider the object is not a mass point, but some solid with a given inertia tensor:
___|Ixx Ixy Ixz|
I = |Iyx Iyy Iyz|
___|Izx Izy Izz|

This solid, put in a non inertial frame, should not only feel fictitious forces, but also fictitious moments.

The question is: how to formulate them?

 

[vanhees71]

That's a really interesting question. I guess it's quite complicated. That's why I right away would use the action principle and write down the Lagrangian using center-mass coordinates and Euler angles between body-fixed and space-fixed Cartesian bases but with the space-fixed frame as a non-inertial (rotating) reference frame.

A special case of this is the theoretical treatment of the gyrocompass, which you can find in the Wikipedia (although I found this more complicated than necessary):

https://en.wikipedia.org/wiki/Gyrocompass

My own attempt to explain it, you can find in my lecture notes on mechanics, which are, however, in German (p. 129ff)

https://th.physik.uni-frankfurt.de/~hees/publ/theo1-l3.pdf