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/, J2 - cos'A + ( - /j,

+ (kp + 2u0/t„)cos2X

+ hu0 + A„u<>sin2A/2

The treatment of the variable u0i = X as a constant in Eq. (18-88) should be noted: The nonlinear differential equation may be solved without resort to this approximation by the technique of multiple time scales [Alfriend, 1975] which is based on the two widely differing periods that Characterize the HCMM dynamics, i.e, the nutation period (20 seconds) and the orbital period (100 minutes). Here, we are concerned with a qualitative description of the HCMM dynamics as a function of the mean anomaly or subsatellite latitude.

Although many approximations were employed in obtaining Eq. (18-88) and simulations using detailed hardware and environmental models (particularly for the magnetic field) are required to evaluate control system performance, most of the characteristics of the HCMM control system are contained in the relatively simple model described by that equation. For a given latitude and control gains, the zeros of the characteristic equation det(M(s))=0

may be computed. In general, there are four roots to the fourth-order Eq. (18-89). In the absence of control torques, these roots are pure imaginary, ± iu, and ± /(J2, where is the nutation frequency and u2f>su0 is the orbital frequency.

With nutation control but no precession control (Ar„<0, kp=> 0), the roots are complex conjugate pairs with negative real parts and the system is damped and stable. The damping time constant associated with the nutation, t„, is shown as a function of latitude in Fig. 18-17(a); at the Equator (A=0) tnas-0.6/k„.

The Routh-Hurwitz criteria (see Section 18.1) may be applied to Eq. (18-89) to obtain the necessary conditions for stable precession control as

[^¡{ly ~h) + K - sin2A(2*„«0 + kp) ] [<o2 (Iy-/,) + K + i^sin 2X ]

' : l ' - +R<o0sin22X(2fcn«0+*;,)>0 (18-90a)

-4knsm2\[{lf-lx )u>î + hu>0+ **„«>! 2A]

- *„cos2A[4(/, - )«2 + - (2 k„u>0 + kp) sin 2A ]

- (2knu0+ kp)cos2X [(/,-/,-/>„+/■ + fc„sin 2X ] >0

SOUTH LATITUDE NORTH LATITUDE

(a) NUTATION DAMPING TIME CONSTANT

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