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Fig. 16-3. Graphs of the Jacobian Elliptic Functions sn(jr|m), cn(jc|m), and dn(x|fn) for m—0.7. The quarter-period, K, is equal to 2.07536 for this m.

Fig. 16-3. Graphs of the Jacobian Elliptic Functions sn(jr|m), cn(jc|m), and dn(x|fn) for m—0.7. The quarter-period, K, is equal to 2.07536 for this m.

of the body rate components along the three axes are m'

WFV)

The body nutation rate is co = +

where the upper sign applies for 7, > I2 > I3 and the lower for 7, < 72 < 73.

The values of m given by Eq. (16-71) are always between 0 and 1. Plots of the three elliptic functions are shown in Fig. 16-3, and useful equations involving them are collected in Table 16-1. Because dn is always positive, we choose the sense of the P3 axis such that <03 is always positive, as in the axial symmetry case. For m<m 1, the first two terms in a power series expansion in m of the Jacobian elliptic

Table 16-1. Identities for Jacobian Elliptic Functions. In Eqs. (9), (10), and (11), the dependence on the parameter, m, has been omitted for notational convenience.

(1)

sn (<fr|ml - en (*|ml dn »lm|

(81

dn2Wm) + ro»>2(«|ni)-1

0 0

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