# A theoretical study of lateral stability with an automatic pilot Page: 3 of 13

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A THEORETICAL STUDY OF LATERAL STABILITY WITH AN AUTOMATIC PILOT

Displacement and rate-of-displacement control.-

The most general type of control treated in the present

investigation is the type for which both the aileron and

the rudder deflections are varied for any deviation in

the angle of bank or in the angle of yaw or for any

change in the velocities in rolling, yawing, or side-

slipping. (See table IV.) From a study of the

characteristics of this type of control, it was found that

the total damping of the motion of an airplane, indicated

by the sum of the damping coefficients, can be increased

over that provided by the inherent stability of the

airplane only if the aileron deflection is a function of

the rolling or the yawing velocities or if the rudder

deflection is a function of the yawing or the sideslipping

velocities. No detailed characteristics of this type of

control were determined because the extensive calcula-

tions involved were considered to be unjustified owing

to the impracticability of such a complicated control.

Cross-coupled control.-Study of the preceding type

of control indicated that, by means of suitable control

operation, the total damping of the motion can be

increased over that available as a consequence of the

inherent stability of an airplane. Because of the very

large inherent damping in roll, however, it appeared

desirable to investigate the possibility of using a rela-

tively simple type of control that would distribute the

inherent damping of the airplane more uniformly among

all the components of motion.

The assumption that both the aileron and the rudder

deflections were functions of both the angle of bank and

the angle of yaw made it possible to solve the stability

equation for the special case of equal real roots, thus

assuring that the motions after a disturbance of the

airplane would contain no oscillatory components and

that all modes would be equally damped. This method

of control is called the cross-coupled control. (See

table IV.) Coupling to provide deflections of the same

control surface for two different types of deviation of the

airplane can be easily accomplished, as demonstrated by

the Smith automatic control (reference 4) for which

deflection of the rudder occurs for deviations in either

the angle of yaw or the angle of bank.

The five equal roots of the stability equation had the

value X= - 1.06 when the numerical datafor the average

airplane at C= 0.35 were used; and the control gearings,

which express the amount of control deflection applied

for a unit deviation of the airplane in bank or yaw, had

the values:

B_= 0.7311

84 Ieb_= _0.954

BS,

= -0.356

be1

b-r1.116

60(2)

The control gearings that enabled the cross-coupled

control to provide equal damping of all modes of themotion at CL= 0.35 caused instability when used at

higher lift coefficients. It is felt that, as a minimum

requirement, any fully satisfactory method of auto-

matic contol should result in stable motion of the air-

plane at all flying speeds; further study of the cross-

coupled control as a means of obtaining uniform distri-

bution of damping was therefore abandoned and

attention was directed to more simplified forms of

control.

Simple control.-In a method of control that has

been successfully used for the Sperry automatic pilot,

the aileron deflection 6, is proportional to the angle of

bank and the rudder deflection 6, is proportional to the

angle of yaw. For such a control, which is called the

simple control (see table IV), thereduced number of

variables involved made feasible more extensive

calculations than were made for the two previously dis-

cussed methods of control. The range of the control

gearings b6a/b and b66/b,4 that would provide stability

for all flight speeds was determined; and, subsequently,

the stability characteristics were studied as the control

gearings were varied within the stable range, instead of

solving for values of the control gearings that would give

certain preassigned stability characteristics to the air-

plane. It was impossible to solve for the case of equal

roots because the conditions for equal roots lead to a

greater number of equations than there are variables.

From reference 5, it can be shown that the conditions

which must be met if the motion is to be stable are: a,

b, d, and f shall be positive, and

(be--a)>0(3)

(bc-ad) (de--cf)- (be-af)2>0 (3)

where the quantities a to f are the coefficients of the

stability equation (equation (1)). The coefficients of

equation (1) are functions of the stability derivatives,

the control derivatives, the lift coefficient, the density

factor, and the control gearings. The form of the

coefficients can be obtained from equations (16) of the

appendix by assuming b d/4b and b6,/b in the equa-

tions to be zero. Thus, upon substitution of the

numerical data for the average airplane, the coefficients,

and hence the expressions of equation (3), can be

converted to functions of the control gearings b6,/bq

and b6,/b.

The vanishing of either of the expressions of equation

(3) or of any of the coefficients a, b, d, orf of equation

(1) indicates that some mode of the motion of the air-

plane will become neutrally stable. If the various ex-

pressions are set equal to zero and are solved for pairs

of values of bsa/b and bs,/bk, boundaries for neutral

stability can be defined. (See fig. 1.) Throughout

the flight range, the axis brb#=O (from the condition

.f=0) and the line representing

(bc-ad) (de-cf)-(be-af)2=0

were found to be the only conditions influencing theregion of stability; the other expressions define lines

that lie outside the region thus bounded. Because275

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Imlay, Frederick H. A theoretical study of lateral stability with an automatic pilot, report, March 4, 1940; (digital.library.unt.edu/ark:/67531/metadc66353/m1/3/: accessed November 20, 2018), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.