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A three dimensional Cartesian coordinate system, with origin O and axis lines X, Y and Z, oriented as shown by the arrows. The tick marks on the axes are one length unit apart. The black dot shows the point with coordinates x = 2, y = 3, and z = 4, or (2, 3, 4).
if one of the axis coincides with the axis of symmetry, the tensor of inertia has a simple diagonal form. For an axisymmetric body, the moments of inertia about the two axis in the plane will be equal.
We fix the x, y, z axis to the body and instantaneously align them with x, y, z. Referring to the figure, we see the components of ω,—- ω1, ω2 and ω3—- and the components of the angular moment vector H, which in general is not aligned with the angular velocity vector.
In ballistics and flight dynamics, axes conventions are standardized ways of establishing the location and orientation of coordinate axes for use as a frame of reference. Mobile objects are normally tracked from an external frame considered fixed.
To change the reference frame of the inertia properties, defined by x, y, and z, to a different axis system, denoted by x’, y’, and z’, recall the parallel axis theorem: (1) Ipxx = Icx’x’ + m(y2 + z2) Ipxy = Icx’y’ − mxy.
Rotating about the Z axis. With this in mind, let's think about how we can express the 2D rotations we already know. First, we're going to need to add a z z value to our coordinate vector to give us: \begin {bmatrix}x \\ y \\ z\end {bmatrix} x y z. Now our existing 2D rotation is equivalent to rotating our x x and y y coordinates around the Z axis.
We first rotate from an initial X,Y,Z system into an x ,y ,z system through a rotation φ about the Z,z axis. The angle φ is called the angle of precession.
