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Definition of Inertia Tensor and Euler's Equations

The angular momentum [Graphics:../Images/index_gr_24.gif] of a collection of particles with momentum [Graphics:../Images/index_gr_25.gif] for each particle [Graphics:../Images/index_gr_26.gif], with a position vector [Graphics:../Images/index_gr_27.gif] relative to some origin in inertial space is

[Graphics:../Images/index_gr_28.gif]

where [Graphics:../Images/index_gr_29.gif] is given by [Graphics:../Images/index_gr_30.gif]. Expanding the summation, and using Einstein summation indices, we can define the inertia tensor [Graphics:../Images/index_gr_31.gif]that combines with the body angular momentum vector [Graphics:../Images/index_gr_32.gif] to give the net angular momentum [Graphics:../Images/index_gr_33.gif].

[Graphics:../Images/index_gr_34.gif]

[Graphics:../Images/index_gr_35.gif]

[Graphics:../Images/index_gr_36.gif]

In an arbitrary body coordinate system, for an irregular body like the Mir, [Graphics:../Images/index_gr_37.gif] is not diagonal. However, a set of coordinate axes can be found in the solid body that correspond to the eigenvectors of [Graphics:../Images/index_gr_38.gif], given by [Graphics:../Images/index_gr_39.gif]. The eigenvalues [Graphics:../Images/index_gr_40.gif] of [Graphics:../Images/index_gr_41.gif] are the three principal moments of inertia [Graphics:../Images/index_gr_42.gif]about the three orthogonal eigenvectors defined as [Graphics:../Images/index_gr_43.gif]. If the body has axes of symmetry, then the [Graphics:../Images/index_gr_44.gif]'s are in the same direction as the axes of symmetry. For arbitrary directions of [Graphics:../Images/index_gr_45.gif], the equation [Graphics:../Images/index_gr_46.gif] implies that [Graphics:../Images/index_gr_47.gif] will not be in the same direction as [Graphics:../Images/index_gr_48.gif]. The coupling of the components [Graphics:../Images/index_gr_49.gif] to each other through the principle moments of inertia [Graphics:../Images/index_gr_50.gif] is given by the Euler equations, if no external forces are acting. The general Euler equations are

[Graphics:../Images/index_gr_51.gif] = [Graphics:../Images/index_gr_52.gif]
[Graphics:../Images/index_gr_53.gif] = [Graphics:../Images/index_gr_54.gif]
[Graphics:../Images/index_gr_55.gif] = 0.

The [Graphics:../Images/index_gr_56.gif]'s are evaluated in the body axes of the rotating frame. The total angular momentum is

[Graphics:../Images/index_gr_57.gif]

for initial values for [Graphics:../Images/index_gr_58.gif] to be [Graphics:../Images/index_gr_59.gif] respectively. Obtaining solutions for the [Graphics:../Images/index_gr_60.gif]'s in the fixed body frame, these must be related to the Euler angles [Graphics:../Images/index_gr_61.gif], [Graphics:../Images/index_gr_62.gif], and [Graphics:../Images/index_gr_63.gif] of the body to inertial transformation matrices [Graphics:../Images/index_gr_64.gif].

[Graphics:../Images/index_gr_65.gif]
[Graphics:../Images/index_gr_66.gif]

These three additional equations can be solved simultaneously with the Euler equations to give solutions for the six parameters [Graphics:../Images/index_gr_67.gif], [Graphics:../Images/index_gr_68.gif], [Graphics:../Images/index_gr_69.gif], and [Graphics:../Images/index_gr_70.gif] as a function of time.


Converted by Mathematica      October 6, 1999

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