## Time

Fig. 7-30. General Thrust Profile for Gas Jet Control Systems electrically before propellant is fed in. As with pressure dependence of the force, the time dependence is sometimes available directly from the calibration data in a suitable form; otherwise, the initial thrust and the firing time required to reach full thrust can be used to create a simple piecewise linear model of the increase in force.

The response of the spacecraft to control torques is proportional to the time integral of the torque. When the thruster firing time is long compared with turn-on and turn-off delays, rise and fall times, and warm-up times, response will depend only on the peak force and the time. If the response of the spacecraft is a rate change about an axis about which the rate is directly measured, a detailed model may be unnecessary even for short thruster firing times, if commands can be sent until the measured rate equals the desired rate. As an example of a case in which neither of the above simplifications ordinarily applies, consider precession of a spinning spacecraft, in which the thruster is fired for a series of short intervals, each a fraction of a spin period. (See Section 1.3.) The direction of the applied torque changes with time, so that an average direction and magnitude must be computed, and rise and fall times can be expected to be significant.

The geometry for the computation of the average torque is shown in Fig. 7-31. Here, L is the angular momentum vector; r is the radius vector from the center of mass to the thruster; F(/) is the thrust, assumed to lie in the L/r plane; and N = rxF is the resultant torque. N2 and N4, corresponding to the forces F(/j) and F(/4), are shown relative to Nc, which is the direction of the average torque. The

centroid, or time (or equivalent angle) at which the instantaneous torque is parallel to the average torque, is computed by requiring that the integral of the torque component, Nx, perpendicular to Nc vanish:

0= J'N±(t)dt~rsin&J'F(t)sino(t-,te)dt (7-147a)

where u is the spin rate, 0 is the angle between r and F, and the integral is computed over one spin period. The effective torque or impulse, lc, is then calculated as the time integral of the torque component, N^, parallel to Nr:

The required integrals can be computed numerically from the thrust profile. If a trapezoidal approximation, as shown in Fig. 7-32, is sufficiently accurate, the integrals can be performed analytically, with the results that tan utc=b/a and

(Time points are labeled to correspond to Fig. 7-30.) The trapezoid model is commonly used for modeling gas-jet thrust profiles at Goddard Space Flight Center.