Extra-Material-Free-Precession-of Earth

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Unformatted text preview: 11/6/13 1:39 AM Next: Free precession of Earth Up: Rigid body rotation Previous: Euler's equations Euler angles We have seen how we can solve Euler's equations to determine the properties of a rotating body in the corotating body frame. Let us now investigate how we can determine the same properties in the inertial fixed frame. The fixed frame and the body frame share the same origin. Hence, we can transform from one to the other by means of an appropriate rotation of our coordinate axes. In general, if we restrict ourselves to rotations about one of the Cartesian axes, three successive rotations are required to transform the fixed frame into the body frame. There are, in fact, many different ways to combined three successive rotations in order to achieve this goal. In the following, we shall describe the most widely used method, which is due to Euler. We start in the fixed frame, which has coordinates , , , and unit vectors is counterclockwise (if we look down the axis) through an angle coordinates , , , and unit vectors , , about the , , . Our first rotation -axis. The new frame has . According to Section A.6, the transformation of coordinates can be represented as follows: (496) The angular velocity vector associated with has the magnitude , and is directed along (i.e., along the axis of rotation). Hence, we can write (497) Clearly, is the precession rate about the -axis,...
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