Note that Equation 2.4-44 only includes g. A first integral can be obtainted as where the subscript o denotes initial conditions. By using the identity cos 2g =1-2 sin^g, equation (2.4-45) can be put in the form sn(X|m) is a Jacobi elliptic function of argument X and parameter m and F (X/m) is the incomplete Jacobi elliptic integral of the first kind of argument X and parameter m. The proper solution of Equation 2.4-50 is for the value of the sin-1 to be in the same quadrant as the argument of sn. The actual solution in general is a complicated function of the initial conditions and ratio of moments of inertia. However, several observations are possible. First, the motion consists of periodic (oscillatory) and secular (rotation) terms. Second, it is nonlinearly dependent on the initial conditions. Third, by proper choice of the initial conditions, the secular term can be driven to zero so that only the oscillatory terms remain. Figure 2.4-13 illustrates what happens physically when the secular term is eliminated. Starting at the top, the vehicle is rotating but has no gravity gradient torques. As it moves counterclockwise, a gravity gradient torque develops which resists the rotation. Eventually the rotation stops, then reverses itself. 90° later, the satellite again faces the sun, has no gravity gradient torques, but is rotating in the opposite direction. The process reverses itself every 90°.
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