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Three-Dimensional Graphical Depiction of Mir Rotation

A Graphical Model of the Mir


A Mir model is constructed, using polygons, and then rotated repeatedly to create an animation of the Mir rotation in inertial space. However, it is necessary to modify the behavior of the function RotationMatrix3D, in the package Graphics`Shapes`, so that the Euler transformation matrices defined above are used to perform the rotations as a function of [Graphics:../Images/index_gr_135.gif], [Graphics:../Images/index_gr_136.gif], and [Graphics:../Images/index_gr_137.gif] rather than the matrix built into the package, which uses a different Euler sequence. To carry out the transformation from the body axes to the inertial axes, the transformation [Graphics:../Images/index_gr_138.gif] is used.


The Mir graphical model is limited in its fidelity, but includes a depiction of the solar arrays, critical in their arrangement toward the sun.



The three principle axes [Graphics:../Images/index_gr_143.gif] are shown, and the Soyuz spacecraft is represented as a slender cone on the [Graphics:../Images/index_gr_144.gif] axis. The center of mass of the Mir is roughly located at the central sphere, representing the node. It should be noted that the moment arm of the Soyuz reaction control jets at the end of the arm from the center of mass is small compared to the dimensions of the Mir. Effectively, only rotations about the [Graphics:../Images/index_gr_145.gif] and [Graphics:../Images/index_gr_146.gif] axes are achievable without excessive usage of Soyuz fuel. The orientation of the Mir is roughly that desired at the start of a manual rotation, if the sun is located on the +z axis. The critical solar arrays are mounted perpendicular to the base block of Mir, which are the two cylinders along the [Graphics:../Images/index_gr_147.gif] axis. The arrays are able to slowly rotate about their long axes to be as perpendicular as possible to the line of sight toward the sun.

Animated 3D Rotation of Mir

The Mir model is animated using the calculated interpolation functions for [Graphics:../Images/index_gr_148.gif], [Graphics:../Images/index_gr_149.gif], and [Graphics:../Images/index_gr_150.gif].



It is apparent that although the Mir, with an initial spin about the C axis, is able to present the plane of the solar arrays roughly perpendicular to the sun on the z axis, the condition does not hold for long. After roughly [Graphics:../Images/index_gr_184.gif]9, or about 180 minutes, the [Graphics:../Images/index_gr_185.gif] axis is roughly along the z axis, toward the sun, rotating about the [Graphics:../Images/index_gr_186.gif] axis so that the solar arrays cannot properly face the sun. However, this condition is also fairly short-lived, and after another 180 minutes the Mir is effectively oriented opposite to the starting condition, but with the [Graphics:../Images/index_gr_187.gif] axis again roughly perpendicular to the sun, as can be the solar arrays if they have rotated their active surface toward the sun.

Analysis of the Euler equations and their solutions shows that in general, pure rotations about the principle axis whose moment is the middle value of the three moments [Graphics:../Images/index_gr_188.gif] (in this case C) are unstable and always result in the behavior shown above. Pure rotations initiated about axes with the lowest or highest moments of inertia are stable. This suggests that an attempt to spin about the [Graphics:../Images/index_gr_189.gif] axis using the Soyuz would result in a stable spin. This is true for Mir, and can be demonstrated using the method above, but unfortunately the solar arrays in that case cannot be oriented continuously with their planes perpendicular to the spin axis, and therefore toward the sun, due to the Mir geometry.

Although clearly not an optimum situation, the rotation described about the [Graphics:../Images/index_gr_190.gif] axis was the only realizable option available. Fortunately, it nonetheless provided enough electrical energy for the most essential Mir life-support systems to be powered, including communication with the ground. This permitted a stable situation in which repairs could be made, and Mir attitude control could eventually be recovered.

Converted by Mathematica      October 6, 1999 [Prev Page][Next Page]