COUPLING OF CONTROLS WITH ELASTIC DEGREES OF FREEDOM

In Sec. 5.12 we presented equations of motion for elastic modes with con­trols locked in a fixed position, and in the preceding section we have developed the control equations for a rigid airplane. Thus, coupling between controls and elastic motions has been excluded. In fact, as is clear from the existence of the aileron reversal phenomenon (Sec. 8.4), and the effect of flexibility on elevator effectiveness (Sec. 7.4), there are important couplings between the control degrees of freedom and the elastic degrees of freedom. To include these entails modifications to both the elastic equations (5.12,7) and (5.12,12) and control system equations such as (11.3,12). The details depend on

which control system is being considered—aileron, elevator, or rudder—and on its particular design features. We illustrate the process by considering the elevator surface and its coupling with z deflections of the vehicle. We treat a case of one degree of freedom by stipulating dj = 0.

The deflection of the structure from the reference position is now given by [cf. (5.12,1)]

z'(t) =^hn(xo, y0, z„)en(t) + hsMe(t)

n—0

where hd is zero except for points of the elevator, where it is hs — | and f is the distance from the elevator hinge line, as shown on Fig. 11.5. Now the displacement function represented by the last term is not in general ortho­gonal to the hn, and hence the integrals of its products with them that appear in the kinetic energy do not vanish. This leads to the appearance of an additional term on the l. h.s. of (5.12,7), viz. (an exercise for the reader)

1п(ёп + 2£иа>„ёи + mr^e) + Ins $e — ^n (11.3,23)

where InS=jhn!-dm

the integral being taken over the elevator.

Similarly, the l. h.s. of (11.3,13a) (with 6j — 0) becomes

co

= (11-3,24)

n= 0

The terms containing lni in these equations represent inertial couplings be­tween the elevator and elastic degrees of freedom. That in (11.3,23) corre­sponds to “tail wags dog,” i. e. acceleration de of the elevator generates motion in the wth elastic mode. This may be expected to be a small effect in most cases. That in (11.3,24) represents the converse, “dog wags tail,”

i. e. elastic mode accelerations en generate motion of the elevator. This contribution is very significant in relation to control-surface flutter, and is minimized by proper mass balancing of the control surface to reduce Ini for the critical elastic mode.

The remaining modifications to the equations of motion occur on the r. h.s. For the elastic modes the only addition is one aerodynamic term to #”и, i. e. AnS A. de to (5.12,12) or GnS Ade to (5.12,13). These aerodynamic contributions to elastic motion are usually important. The addition to the control equation is also an aerodynamic coupling. There He in (11.3,13c) becomes

00

He= –+ ZHnsen

n=0

In summary, the elastic and control equations are both modified by addi­tional simple inertial terms on the l. h.s and by aerodynamic terms on the r. h.s.