Rocket motor gas, as it streams through the motor case and the nozzle, generally has a damping effect on spacecraft motions. The phenomenon is referred to as jet damping [9], As in aircraft, it can dampen pitch and yaw

A/ Motor Rotation

I Line ; of Sight

A/ Motor Rotation

I Line ; of Sight

Figure 6.17 Schematic of a spinning spacecraft with nutation damping wheel and horizon sensor (wheel axis parallel to xY principal axis).

motions. More significant, however, is the jet damping of spinning vehicles, because it counteracts the precessional motion by decreasing the nutation angle.

The beneficial effect of jet damping obtained during rocket motor firings is important because it mitigates the relatively large disturbance torques that are caused by unavoidable motor performance fluctuations.

Nevertheless, by itself, jet damping may not be sufficient to keep the nutation angle from growing during motor firings. There have been some upper stage vehicles with solid-propellant motors that exhibited unaccept-ably large nutation angles. To keep the angle within one or two degrees then required the addition of special thrusters [10-12],

Jet damping can be explained by the torque that is produced by the Coriolis forces resulting from the angular velocity of the spacecraft and the rocket gas velocity relative to it. The jet damping torque is the integrated moment of these Coriolis forces, starting at the combustion chamber and ending at the nozzle exit. An explanation that is both simpler and also makes it clear that the combustion chamber and the nozzle are not separately involved, is as follows.

A simple but sufficiently accurate model of jet damping is illustrated in Fig. 6.18. The model assumes that the mass-averaged gas exit velocity, uex, at the nozzle exit and relative to the spacecraft is in the direction of the nominal thrust. (This is not quite accurate because the Coriolis forces cause the gas stream at the nozzle exit to deviate slightly from this direction. The gas already enters the nozzle with some angle of attack. But because

Mean

Burn Surfac

Mean

Burn Surfac

Nozzle

Figure 6.18 Jet damping of a spinning spacecraft: O, instantaneous center of mass; 6, nutation angle; p, precession rate; uex, gas exit velocity relative to spacecraft; Mjd, jet damping moment. Shown for case of prolate spacecraft.

1 \-Rocket

\ Motor

Figure 6.18 Jet damping of a spinning spacecraft: O, instantaneous center of mass; 6, nutation angle; p, precession rate; uex, gas exit velocity relative to spacecraft; Mjd, jet damping moment. Shown for case of prolate spacecraft.

of the subsequent large acceleration in the nozzle, the effect of the angle of attack is largely canceled and the gas stream is redirected to leave the nozzle close to the nominal thrust direction. The most thorough investigation, both theoretically and experimentally, of nozzle flows with initial angle of attack has been published by Pirumov and Roslyakov [13].)

The figure refers to a spinning, inertially symmetric, prolate vehicle with a solid-propellant motor, but the results derived in the following are equally valid in the oblate case. Let x:i designate the spacecraft principal axis that coincides with the nominal thrust axis. Also, let ri be the position vector in the aft direction of x3 from the (instantaneous) center of mass, 0, to the average location of the burn surface at a specified time. Similarly, r2 is the position vector from O to the nozzle exit midpoint. The spacecraft angular velocity is designated by w, the momentum vector by L, the nutation angle by 9, the precession rate by p, and the gas mass flow rate by m. A one-dimensional, steady-state description is used. Because the damping rate is slow compared with u, it suffices to compute the jet damping moment separately at each instance, assuming that all quantities are constant at that time.

As a consequence of the precession, there is at the nozzle exit, relative to inertial space, an additional velocity component of the gas u2 = p(L/L) x r2

transverse to the thrust axis. The reactive force resulting from the momentum per unit time, mu2, is opposite to the direction of the precession and tends to decrease the nutation angle. It accounts for the major part of the jet damping effect.

A closed control surface can be introduced that contains the spacecraft and crosses the nozzle exit plane. The angular momentum per unit time (in inertial space and relative to the center of mass) of the gas that leaves the control surface at the nozzle exit is mpr2 x (L/L x r2) (6.57a)

The angular momentum loss in the control volume, per unit time, caused by the propellant diminution at the burn surface is

The torque on the spacecraft, balancing these two angular momentum changes, is the jet damping moment

The vector Mjd is seen to be perpendicular to the thrust axis and to rotate around it. The second term in the square bracket is generally much smaller than the first, since the center of mass of most spacecraft is closer to the burn surface than to the nozzle exit plane. Therefore, the averaging of the location of the burn surface, as was done in defining ri, will not generally introduce appreciable errors.

It follows that the magnitude Mjd of the jet damping moment is Mjd = mp{rl - rf) sin0 = - rf) tan0

Therefore, for small nutation angles h h

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