functions given by Eqs. (12), (13), and (14) of Table 16-1 provide an analytic approximation for the rotational motion for near-axial symmetry or small nutation. These equations also show that for m = 0, the Jacobian elliptic functions are trigonometric functions. This limit arises in the axial symmetry case, /, = /2, and also in the case of no nutation, L2 = 2I3Ek. In the axial symmetry limit, Eqs. (16-69) through (16-73) become equal to Eqs. (16-59) and (16-63). The m = 1 limit is attained when L2=2/2Ek. In this limit, the Jacobian elliptic functions can be expressed as hyperbolic functions, which is in agreement with the exponential behavior for rotation about an axis of intermediate moment of inertia, for which L2«2/2£a.
Equations (16-69) through (16-73) can be verified by substitution into the Euler equations of motion, with the use of Equations (1), (2), (3), (7), and (8) of Table 16-1. We can derive equations in terms of the initial body rate vector, «„, for the asymmetric case by using the addition laws for Jacobian elliptic functions, Eqs. (9), (10), and (11) of Table 16-1, and Eqs. (16-69) through (16-73). This gives
u01cn apt + (i»to02(o03/(o3m)sn upl dn u>pt i-(/"Ww3m)2sn2'y
Was this article helpful?