In addition to the translational motion equation, the equations for the rotational motion of a body that is moving in 3-D with an angular velocity ω relative to XYZ need to be derived.
For this purpose, it is convenient to introduce a second coordinate system xyz that translates with the rigid body and has an angular velocity Ω
relative to XYZ. Note that Ω is not necessarily equal to ω.
By definition, the sum of the moments about a fixed point O, or the center of gravity cg, is equal to the time rate of change of total angular momentum about that point:
Because xyz may now be rotating, the previously derived angular acceleration equation
needs to be used again,
dA/dt = (dA/dt)xyz + (Ω × A)
This equation can be used to relate the time derivative of the angular momentum with respect to XYZ. This gives,
Depending on the motion of the body, there are three cases for the angular velocity Ω of the xyz axes that need to be examined. |