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Effect of Nutation on Observed Sun Angles and Spin Rate for a Symmetric Spacecraft With JjlT-1.238, Nutation Angle of S Deg, and Average Spin Rate of 40 Deg/Sec too 110 120

TIME ISECl

Effect of Nutation on Observed Sun Angles and Spin Rate for a Symmetric Spacecraft With JjlT-1.238, Nutation Angle of S Deg, and Average Spin Rate of 40 Deg/Sec whose slit plane contains the x and z axes and having a nutation angle, 0, of 5 deg and an average measured spin period of 9 sec. We have just shown that the Sun angle oscillates at the inertial nutation rate <o, (=<£), which for this example correspond^ to a period of 7.24 sec. It can be shown in a derivation similar to that in Section 16.3.1 (but one which solves in terms of the variation of Sun crossing times) that the deviation in crossing time oscillates at the body nutation rate up (=<£), or for this example, a period of 37.8 sec. Section 16.3.2 shows that for a symmetric spacecraft whose attitude sensor measures the coelevation of the inertial vector only once per spin period, the measured coelevation angles vary at a rate which is roughly approximate to the body nutation rate. This is shown by the dashed line in Fig. 16-5.

Section 16.3.1 derives a technique for determining the dynamic motion of a dual-spin symmetric spacecraft from body measurements of the coelevation of Inertial vectors. This technique involves solving for the Euler angle rates and initial values and is suitable for a time development of the rotational dynamic motion. Section 16.3.2 derives techniques for monitoring spacecraft nutation with a digital Sun sensor. It presents approximations for the amplitude and the phase of L in the body system for a symmetric spacecraft and extends these approximations to an asymmetric spacecraft Techniques in this section are particularly suitable for determining the amplitude and phasing of torques for active nutation damping (see Section 18.4).

163.1 Dynamic Motion of a Symmetric Dual-Spin Spacecraft

In this case, we assume a dual-spin spacecraft having a momentum wheel with known moments of inertia (Iwheei) and known constant spin rate (uj relative to the body; further, we assume that the wheel rotational axis is aligned with the body z axis. The components of the total angular momentum, L, in the body frame are then

Recall from Section 16.1 that the body moments of inertia (Ix,Iy,It) are assumed measured with the momentum wheel "caged." Substitution of Eq. (16-98) into Eq. (16-12) (and noting that here x,y,z, are used in place of u,v,w in Section 16.1) gives

Lx = Ixux = L sin 0 sin^ Ly = Iyuy = L sin 0 cosip - A", + hl = L cos0

where z^td wheel

For the axially symmetric case, using Ix = iy = IT, Eq. (16-99) becomes

* IjCosO

For small 9, cosflssl, and Eq. (16-11c) reduces to where u is approximately the average measured spin rate (ignoring nutation*). Thus, Eq. (16-100) becomes

Because the right sides of Eq. (16-101) are known, the problem of determining the time development of <?>, 9, and ^ reduces to determining 9 and the initial values of <j> and $ from sensor data.

Using small angle approximations for 9, the 3-1-3 transformation from the inertial to the body frame given in Eq. (12-20) reduces to cossin(4>+9 sin if* - sin(<¡> + if>) cos(«J>+ $) 9 cos ^ 0sin<£ -9cos<f> 1

where 9 is in radians. Thus, for each observation of a vector, SB, in the body frame whose inertial position, S,, is known (e.g., a star, the Sun, the magnetic field vector, the nadir vector), we have

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