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The Solution of Euler's Equations

Dynamic Equations

The Euler equations are

[Graphics:../Images/index_gr_71.gif]

The total angular momentum and initial conditions are

[Graphics:../Images/index_gr_72.gif]
[Graphics:../Images/index_gr_73.gif]

The Euler angles [Graphics:../Images/index_gr_74.gif], [Graphics:../Images/index_gr_75.gif], and [Graphics:../Images/index_gr_76.gif] using the transformation matrices [Graphics:../Images/index_gr_77.gif] give the equations

[Graphics:../Images/index_gr_78.gif]

For display purposes, the initial Euler angles will be chosen so that output plots of the motion have the net angular momentum directed along the z axis. The Euler angles [Graphics:../Images/index_gr_79.gif] at [Graphics:../Images/index_gr_80.gif] are given by the transformation

[Graphics:../Images/index_gr_81.gif]
[Graphics:../Images/index_gr_82.gif]

Solve two equations for two initial angles, the angle [Graphics:../Images/index_gr_83.gif] being indeterminate.

[Graphics:../Images/index_gr_84.gif]
[Graphics:../Images/index_gr_85.gif]

The first solution in the list will be used, using a small initial [Graphics:../Images/index_gr_86.gif] with positive value, so that there is correspondence with the analytic solution found elsewhere for the case of the symmetric top when [Graphics:../Images/index_gr_87.gif].

[Graphics:../Images/index_gr_88.gif]

Initial Conditions

The task is to solve the six parameters [Graphics:../Images/index_gr_89.gif], [Graphics:../Images/index_gr_90.gif], [Graphics:../Images/index_gr_91.gif], and [Graphics:../Images/index_gr_92.gif] as a function of time, starting with initial values for each of the parameters, using the three principle moments of inertia [Graphics:../Images/index_gr_93.gif] for the Mir station. The principal axes of inertia of Mir roughly coincide with the module axes originating from the center section, known as the node. Although the Mir's moments of inertia cannot be known at any one time precisely because of the uncertain arrangement of internal stores throughout the Mir, the basic fixed hardware results in moments of inertia whose ratios can be characterized by setting [Graphics:../Images/index_gr_94.gif]. The initial values for [Graphics:../Images/index_gr_95.gif] are denoted to be [Graphics:../Images/index_gr_96.gif] respectively. The rate about the [Graphics:../Images/index_gr_97.gif] axis is denoted as n, distinguishing it from the other two axes, because it was about this axis that a spin was attempted to be established manually, using the Soyuz spacecraft. The crew attempted to begin the spin with [Graphics:../Images/index_gr_98.gif]about the A axis and [Graphics:../Images/index_gr_99.gif]about the [Graphics:../Images/index_gr_100.gif]axis as small as possible. However, because of the lack of Soyuz control moment about the [Graphics:../Images/index_gr_101.gif] axis (Soyuz is docked along this axis), [Graphics:../Images/index_gr_102.gif] could not be controlled, or reduced from the value developed during gyrodyne braking, so here [Graphics:../Images/index_gr_103.gif] is simply calculated from initial values of J and n, and [Graphics:../Images/index_gr_104.gif]. Note that we are using dimensionless quantities, and that our time parameter t should be scaled roughly by 20 to get the physical time in minutes.

tmax is the total time of integration. Define a very small real number [Graphics:../Images/index_gr_105.gif] to avoid causing difficulties in NDSolve. List the Mir physical properties as a single list, called dynamics.

[Graphics:../Images/index_gr_106.gif]

Check the consistency of our parameters, by finding the minimum value of [Graphics:../Images/index_gr_107.gif] for [Graphics:../Images/index_gr_108.gif]

[Graphics:../Images/index_gr_109.gif]
[Graphics:../Images/index_gr_110.gif]

This should always be less than the [Graphics:../Images/index_gr_111.gif] value specified in dynamics, to ensure a non-imaginary value for [Graphics:../Images/index_gr_112.gif].

[Graphics:../Images/index_gr_113.gif]
[Graphics:../Images/index_gr_114.gif]

Numerical Solutions

Using the parameters given, the Euler equations are solved simultaneously with the three equations for the Euler angles, applying our geometric constraint anglerule that [Graphics:../Images/index_gr_115.gif] be directed along the inertial z axis.

[Graphics:../Images/index_gr_116.gif]

To understand our solution from the point of view of the crew members on board Mir, plot the angular rates [Graphics:../Images/index_gr_117.gif] as observed about the three principle axes of Mir. These can be measured directly by the crew, utilizing the stars as an inertial reference. It is apparent that the motion is that of an irregular body, with most angular momentum about its middle axis of inertia, causing an unstable exchange of angular momentum between the principle axes. Although the Mir starts spinning initially about the [Graphics:../Images/index_gr_118.gif] axis, after[Graphics:../Images/index_gr_119.gif] the direction of rotation is reversed and oscillates with a period of roughly [Graphics:../Images/index_gr_120.gif]. The spin about the [Graphics:../Images/index_gr_121.gif] axis, initially small, grows periodically, but never reverses.

[Graphics:../Images/index_gr_122.gif]

[Graphics:../Images/index_gr_123.gif]

Solid curve is [Graphics:../Images/index_gr_124.gif], short dashed curve is [Graphics:../Images/index_gr_125.gif], long dashed curve is [Graphics:../Images/index_gr_126.gif]

No physical intuition is readily apparent when the Euler angles are plotted, but they determine the behavior of the physical transformation matrix [Graphics:../Images/index_gr_127.gif] used to do the animation of the Mir model. It should be noted that the roughly linear growth of [Graphics:../Images/index_gr_128.gif] with time corresponds to the continuous spin of the Mir complex about the inertial Z axis, along which the total angular momentum was constrained as an initial condition using anglerule.

[Graphics:../Images/index_gr_129.gif]

[Graphics:../Images/index_gr_130.gif]

Solid curve is [Graphics:../Images/index_gr_131.gif], short dashed curve is [Graphics:../Images/index_gr_132.gif], long dashed curve is [Graphics:../Images/index_gr_133.gif]


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