where the total angular momentum is L= / « + h, and we have assumed that u»>max(|(^|,|£|,|^|) and have neglected all second-order terms including those involving hx and h, in Eq. (18-53b). Equation (18-53) is central to the remainder of this section. We first describe its general characteristics and underlying approximations and then apply it to several representative spacecraft.

The pitch equation is decoupled from the roll and yaw equations which are coupled through the bias momentum, hy, and the orbit rate term (Ix - Jy + Jt)u0. Control torques, including dampers, generally increase the coupling between the roll and yaw equations but leave the pitch equation uncoupled.* For gravity-gradient stability, I, > I, and the gravity-gradient force provides a restoring torque proportional to pitch with frequency V3®^ wo where oy*=(Ix- Iz)/Iy. The effect of orbital eccentricity on the pitch behavior of gravity-gradient stabilized satellites may be seen with the aid of Fig. 18-13. The rate of change of angular momentum about the pitch axis, ignoring environmental torques and with the pitch wheel speed constant, is

where v is the true anomaly. For an orbit with small eccentricity, e, we have (see Eq. (3-11»,

* Off-diagonal terms in the moment-of-inertia tensor lead to coupling and are usually treated as disturbance torques.

where M = u0t+M0 is the mean anomaly. Substitution of Eq. (18-55) into Eq. (18-54) gives

y with the solution tp=> 3^3ysin3/+C,cas(\^(18-57)

where C, and <j>j are integration constants associated with the complementary solution.

The particular solution to the differential Eq. (18-57), L~2esinM/(3o -1), results in a sinusoidal steady-state error, which for GEOS-3 (e=0.0054, o=0.984), had an amplitude of 0J deg. For spacecraft with oyœ 1 /3, there is a pitch resonance and therefore this configuration is avoided.

The coupled roll/yaw expressions from Eq. (18-53) in the absence of roll and yaw wheels, control, and environmental torques (other than gravity-gradient) are lx I+Aiol (/,-/, %-<oe (/,-/,+1, %=0 AÏ+«<?(/, - )iy+»o(I* ~ )i=0 (18-58)

With the notation ax=(ly-/,)/Ix and o2'=(Iy-Ix)/Iz, these may be rewritten in Laplace transform notation (see Appendix F) as

s2+4o,;ox -<o0(l-ax)s e(i) «„(1 -ojs s*+»¡o, Jlefc) with the characteristic equation i4+«2(3ax-H + oxo>2+4«X®,'=0 (18-60)

For roll/yaw stability, the roots to Eq. (18-60) must have no positive real part (see Section 18.1) and hence j2 ^ - (3ax +1 + qxq.) ± [ (3a, +1 + axa,)2 - 16«xa£ ]1/2 61' 2

must be real and negative.* Therefore, a necessary and sufficient condition for stability is

where

Figure 18-14 illustrates the regions of gravity-gradient stability defined by the

*lf s, is a root of Eq. (18-60), then is ateo. Thus,.lor both », and -i, to have no positive teal part, must be pm imaginary and is teal and negative.

ai ai

Fig. 18-14. Gravity-Gradient Stability for Various Moments of Inertia. Three-axis stability corresponds to the unshaded region (adapted from Kaplan [1976]). The letters 5, G, and R denote the configurations of-three gravity-gradient stabilized spacecraft—SEASAT-A, GEOS-3, and RAE-2. (See Appendix 1.)

Fig. 18-14. Gravity-Gradient Stability for Various Moments of Inertia. Three-axis stability corresponds to the unshaded region (adapted from Kaplan [1976]). The letters 5, G, and R denote the configurations of-three gravity-gradient stabilized spacecraft—SEASAT-A, GEOS-3, and RAE-2. (See Appendix 1.)

above inequalities. (As described previously, Ix > lz or lxox > lza: is required for pitch stability.)

The remainder of this section concerns the analysis of Eq. (18-53) for four representative spacecraft configurations and control systems, as described in Table 18-1. Although all four spacecraft are of the momentum bias design as discussed in Section 18.2, they are physically very different. The moments of inertia of the asymmetric HCMM spacecraft are three orders of magnitude smaller than those of the symmetric SEASAT-A spacecraft. All four spacecraft orbits are near-circular, but the CTS orbit is equatorial at synchronous altitude, whereas the others are at 500 to 850 km in polar orbits. GEOS-3, shown in Fig. 19-17, uses an extendable 6.5-m boom with a 45-kg end mass to achieve a large, gravity-gradient restoring torque which, combined with a damper, provides passive 1-deg pitch control. For

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