## Position Plus Rate Controller

Fig. 18-2. Position-Phis-Rate Pitch Control Block Diagram. See text for explanation.

Comparing Eq. (18-9) with the second-order equation of a mass-spring-damper system (see, for example, Melsa and Shultz [19690, we define the natural frequency, <i>„, and the damping ratio, p, of our system as and rewrite Eq. (18-11) as m

UNoy s2+2puns+ We now discuss the response of this system when the input disturbance torque is a step function. Because the Laplace transform of a step function of magnitude N0 is N0/s (see Appendix F), Eq. (18-13) reduces to m=

This may be rewritten as the sum of partial fractions, to obtain for p< 1

where

Using the inverse Laplace transforms listed in Table F-l, we can obtain the time response of the pitch angle as

9(')= ^ [ 1 - (:I - p2)~ ,/2exp( - p«./)sin(u,/ + V) ] (18-16)

Figure 18-3 shows, a plot of the system response to a step function assuming that N0/K= 1. The shape of the response curve depends on the damping ratio, p, and the time scale is determined by the natural frequency, un. When p=0, the system is called undamped and undergoes a bounded sinusoidal oscillation. As p