What Are Embedded Axis Frames in Euler Equations?

[commutator]

my book says that it is actually difficult to get the true motion of a body by using these equations because it says that euler equations are written in embedded axis frame ...
what is an embedded axis frame?where is it different from normal frames that i used in before?after solving euler equations, how do i return to the normal frame??
any help will be highly appreciated.

 

[D H]

In theory, it's not all that difficult: "The polhode rolls without slipping on the herpolhode lying in the invariable plane."

Euler's equations are a simplification of the general equations of motion for a rigid body:

$$\boldsymbol I\frac{d\boldsymbol{\omega}}{dt} = \boldsymbol{\tau}_{\text{ext}} - \boldsymbol{\omega}\times(\boldsymbol I\boldsymbol{\omega})$$

Euler's equations result when the frame in which the moment of inertia tensor I is expressed makes than tensor diagonal (i.e., when the body axes are the principal axes of the rigid body).

In practice, finding those principal axes is rather difficult. One can define an orthogonal set of axes for a rigid body, but those (somewhat arbitrarily defined) axes typically are not the principal axes. The inertia tensor expressed in these body axis coordinates will have non-zero off-diagonal terms. Measuring the moments of inertia is a bit challenging; getting more than three or four places of accuracy is rather tough. Measuring the products of inertia is more than challenging. For some big, bulky objects it just cannot be done. Even when it can the accuracy is quite low. This low accuracy makes it rather difficult to accurately predict the behavior over any length of time. The motion essentially becomes chaotic.

 

[commutator]

thanks..:)
i think u are referrring to the principal axis that makes inertia tensor diagonal, but i am still in the dark about what an " embedded axis frame is" and its specialities..
am i missing something here?
thanks again.

 

[D H]

Without context, it's a bit hard to answer your question. What text? What education level (high school physics, freshman, undergrad classical mechanics)?

 

[DrDu]

I suppose "embedded axes" refers to axes fixed with respect to the body (in contrast to axes fixed with respect to the laboratory).

 

[tiny-tim]

hi commutator!

my book says that it is actually difficult to get the true motion of a body by using these equations because it says that euler equations are written in embedded axis frame ...
what is an embedded axis frame?where is it different from normal frames that i used in before?

Euler's equations are valid only in a frame of reference fixed in the body along three (perpendicular) principal axes, and therefore rotating with it.

They are valid only if the origin of the axes is at the centre of mass, or at the centre of rotation.

(And the advantage of Euler's equations over the fixed-frame equations is that the moment of inertia tensor is changing in the latter, but not in the former, so you don't have to bother with its derivative.)

See the PF Library on Euler's equations for an extended explanation.

… after solving euler equations, how do i return to the normal frame??

I don't know of any general method, but in particular examples, it's usually fairly obvious …

did you have an example in mind?​