Definition of Inertia Tensor and Euler's Equations
The angular momentum of a collection of particles with momentum for each particle , with a position vector relative to some origin in inertial space is
where is given by . Expanding the summation, and using Einstein summation indices, we can define the inertia tensor that combines with the body angular momentum vector to give the net angular momentum .
In an arbitrary body coordinate system, for an irregular body like the Mir, is not diagonal. However, a set of coordinate axes can be found in the solid body that correspond to the eigenvectors of , given by . The eigenvalues of are the three principal moments of inertia about the three orthogonal eigenvectors defined as . If the body has axes of symmetry, then the 's are in the same direction as the axes of symmetry. For arbitrary directions of , the equation implies that will not be in the same direction as . The coupling of the components to each other through the principle moments of inertia is given by the Euler equations, if no external forces are acting. The general Euler equations are
The 's are evaluated in the body axes of the rotating frame. The total angular momentum is
for initial values for to be respectively. Obtaining solutions for the 's in the fixed body frame, these must be related to the Euler angles , , and of the body to inertial transformation matrices .
These three additional equations can be solved simultaneously with the Euler equations to give solutions for the six parameters , , , and as a function of time.
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