## Angular Misalignment Of Uotor Case With Respect To Spacecraft Centerline Angular Misalignment Of Principal Inertia Axes With Rs5psct To Spacecraft Centerline Offset Of Thrust Vector With Respect To Apogee Motor Case Offset Of Motor Cass With Respect To Sp

Fig. 17-6. Definition of Misalignments for a Rocket Motor Nominally Aligned With the Spin Axis of a Spinning Spacecraft

An analytic model for the spacecraft motion during the engine firing, including the above six misalignments, can be developed with the following simplifying assumptions:

1. Rigid body dynamics are applicable.

2. The engine is a solid fuel motor and the fuel burns symmetrically about the motor case centerline.

3. The total spacecraft mass, moments of inertia, and the location of the center of mass in the spacecraft are linear functions of time during the motor firing.

4. The motor firing does not distort the spacecraft; i.e.. the misalignments remain constant during the motor firing.

5. The exhaust gases carry away angular momentum equal to that of the fuel which was burned.

The last assumption is applicable to a motor which possesses a single, large, centrally mounted nozzle. A solid fuel motor of this type is generally used for large velocity changes. In this case, the exhaust gases spend so short an interval in the engine that they have no time to exchange any angular momentum with the spacecraft before being ejected and hence the spin rate of the spacecraft will not change if the alignments are correct. This is in agreement with the observed very small spin rate change during the apogee motor firing on CTS ( + 0.4 deg/s), GOES-1 (+1.8 deg/s), GOES-2 (-3.2 deg/s). and SIRIO ( + 0.4 deg/s) [Tandon and Smith (1976); Page (1975); Chen and McEnnan (1977)].

If the engine is different from the one discussed above, especially if it possesses more than one nozzle, an appropriate jet damping model should be used in place of assumption 5. The term jet damping refers to the phenomenon in which the rotation of the motor exhaust gases carries away a portion of the component of the spacecraft's angular momentum perpendicular to the nominal exhaust direction. This serves to damp the nutation induced by the motor firing. The jet damping theory is discussed by Thomson and Reiter [1965], Warner and Snyder [1968], Katz [1968], and Papis [1968]. The basic dynamics model consists of three sets of differential equations and an algebraic vector equation. These are summarized in vector form in Fig. 17-7. Nf is the portion of the torque, N. which is induced by the motor thrust, F, and Ny is the portion which models the effect of the angular momentum carried away by the exhaust gases. Ny will depend on the jet damping model used. The equation for Ny, using assumption 5, is

where all vector quantities are resolved along the spacecraft principal axes. The detailed derivation of the equations in Fig. 17-7 is given by Keat and Shear [1974],

Assumptions 1 through 4, together with an approximate jet damping model in place of assumption 5, were used to simulate the performance of the CTS spacecraft during apogee motor firing by Keat and Shear [1974], The signs of the misalignments were selected so. that their effect was cumulative (i.e., the worst case for the combined effect of all of the misalignments was simulated). The results of the simulaions indicated that for the nominal specified misalignments for the CTS spacecraft, the principal Z axis (the nominal spin axis) would wander up to 2 deg from its initial i>osition in inertial space during the motor firing and this would cause a „.I* deg -ior in the direction of the velocity change vector. The additional fuel needed to correct the effects of this directional error on the orbit would be

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