LATERAL CONTROL
The lateral controls (the aileron and rudder) on a conventional airplane have three principal functions.
1. To provide trim in the presence of asymmetric thrust associated with power plant failure.
2. To provide corrections for unwanted motions associated with atmospheric turbulence or other random events.
3. To provide for turning maneuvers—that is, rotation of the velocity vector in a horizontal plane.
The first two of these purposes are served by having the controls generate aerodynamic moments about the x and z axes—rolling and yawing moments. For the third a force must be provided that has a component normal to V and in the horizontal plane. This is, of course, the component L sin ф of the lift when the airplane is banked at angle ф. Thus the lateral controls (principally the aileron) produce turns as a secondary result of controlling ф.
Ordinarily, the longterm responses to deflection of the aileron and rudder are very complicated, with all the lateral degrees of freedom being excited by each. Solution of the complete nonlinear equations of motion is the only way to appreciate these fully. Certain useful approximations of lower order are however available.
THE CONTROL EQUATIONS
Whereas the study of stability that was the subject of Chap. 6 is generally sufficiently well served by the linear model of small disturbances from a condition of steady flight, the response of an airplane to control action or configuration change can involve very large changes in some important variables, especially bank angle, pitch angle, load factor, speed, and roll rate. Consequently nonlinear effects may be present in any of the gravity, inertia, and aerodynamic terms. An accurate system model capable of dealing with these large responses must therefore begin with the more exact equations (4.7,1)—(4.7,4). These would normally be reorganized into firstorder state space form for subsequent integration by a RungeKutta or other integration scheme. Equations (4.7,3df) and (4.7,4) are already in the required form. However (4.7,1) and (4.7,2) need to be rearranged. In particular, (4.7,2a and c) need to be solved simultaneously for p and r. The functional form that results is as follows:
X
tiE = f(vE, wE, q, г в, X’) + —
m
vE = f(uE, wE, p, г, в, Ф, Y’) H– 1
m
wE = f(uE, vE, p, q, 0, </>, Z’) + —
m
P = f(p, g, r, L’, N’) + f + 4iVc
^ X
Mc
q = ftp, r, M’) + — ly
r = f{p, q,r, L’,N’) + ^ +rj, c
o = fig, Г, Ф)
Ф = fiP> g, г, в, Ф)
In the preceding equations, the subscript c denotes the control forces and moments, and the prime on force and moment symbols denotes the remainder of the aerodynamic forces and moments. The solution of these equations would require that an aerodynamic submodel be constructed for each case to calculate the forces and moments at each computing step from a knowledge of the state vector, the control vector, the current configuration, and the wind field. To follow this course in extenso would take us beyond the scope of this text, so for the most part the treatments that follow are restricted to the responses of linear invariant systems, that is, ones described by (4.9,20) with A and В constant viz
x = Ax + Be (7.1,4)
Although we are thereby restricted to relatively small departures from the steady state, these responses are nevertheless extremely useful and informative. Not only do they reveal important dynamic features, but when used in the design and analysis of automatic flight control systems that are designed to maintain small disturbances they are in fact quite appropriate.
Table 7.1 Dimensional Control Derivatives

Y 
L 
N 

8* 
Cyshpu20S 
clshpu20Sb 
C„spu20Sb 
8r 
CyahpuoS 
C, spu20Sb 
C„shpu20Sb 
In the examples that follow, we use {8e, S;,j for the longitudinal controls, elevator and throttle; and {5a, S,} for the lateral controls, aileron, and rudder. The aerodynamic forces and moments are expressed just like the stability derivatives in terms of sets of nondimensional and dimensional derivatives. The nondimensional set is the partial derivatives of the six force and moment coefficients {Cx, Cy, Cz, Ch Cm, C„} with respect to the above control variables, such as C = c)CJ<)8e or C,^ = dCJd8a, and so on. The dimensional derivatives are displayed in Table 7.1.
The powerful and welldeveloped methods of modem control theory are directly applicable to this restricted class of airplane control responses. Before proceeding to specific applications, however, we first present a review of some of the highlights of the general theory. Readers who are well versed in this material may skip directly to Sec. 7.6.
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