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The x,y,z axis are body fixed axis, rotating with the body; the solutions for ω x(t),ω y (t) and ω z give the components of ω following these moving axis. If angular velocity transducers were mounted on the body to measure the components of ω, ω x(t),ω y(t) and ω z from the solution to the Euler equations would be obtained,
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.
It can be proved that if we orient the orthogonal axis system XYZ to a specific orientation, denoted by the directions 1, 2 & 3, the shear stress from all faces will vanish and there will be only normal stresses.
Body-Fixed Axis. We formulate the governing equations of motion in an axis system fixed to the body, paying the price for keeping track of the motion of the body in order to have the inertia tensor remain independent of time in our reference frame.
To solve a system of equations by elimination, write the system of equations in standard form: ax + by = c, and multiply one or both of the equations by a constant so that the coefficients of one of the variables are opposite.
As an example of the theorem, consider the situation depicted in Fig. 13.3, where a cylin-drically symmetric mass distribution is rotated about is symmetry axis, and about an axis tangent to its side. The component Izz of the inertia tensor is easily computed when the origin lies along the symmetry axis:
Body axis system: {Eb} Origin is fixed to the aircraft c.g. Xb lies in plane of symmetry and points towards the nose. Yb is perpendicular to the plane of symmetry and is directed to the right wing. Zb is perpendicular to Xb and Yb. Yaw angle (body axis)
