A theoretical study of lateral stability with an automatic pilot Page: 2 of 13
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REPORT NO. 693-NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS
CONTROL DERIVATIVES AND VALUES OF V AND r
FOR THE AVERAGE AIRPLANE
C ' . Y* ; I, (, V
L- r a a r, (fps) (see)
0.35 -0.0347 2.10 -0.106 0.474 150 0.815
1.0 -.0347 2.09 -.300 .475 88.5 1.38
1.8 -.0347 2.11 -.298 .473 66.0 1.85
In addition to the data contained in tables I and II,
the following physical characteristics were assumed
for the average airplane:
Wing span, b.... ----------------------------- feet.. 32. 0
Wing area, S--------------------------square feet-. 171
Wing loading, WIS---------pounds per square foot._ 9. 36
Ratio of wing span to radius of gyration about X, b/kx-_ 6. 47
Ratio of wing span to radius of gyration about Z, b/kz-- 5. 47
Relative-density factor, p(=m/pSb) --------- 3. 82
Inherent-stability characteristics.-In studies of sta-
bility, only the inherent stability of an airplane is
usually determined, that is, the stability with the con-
trol surfaces fixed in their neutral positions. When
the question of automatic control is considered, the
matter of inherent stability is still of interest inas-
much as the character of the motion with the controls
fixed forms a useful basis for comparing the effective-
ness of various control methods.
As is indicated in the appendix, the lateral-stability
characteristics can be determined from the roots of a
stability equation of the form
When the control surfaces are fixed, the coefficients
a to f are functions only of the stability derivatives,
the lift coefficient, and the density factor p. (See
equation (12) of the appendix.) The inherent-
stability roots for the average airplane with controls
fixed are given in table III.
INBERENT-STABILITY ROOTS FOR THE AVERAGE
CL X.2 X )4 X'
0.35 -0.409=11.99i -4. 49 -0.00661 0
1.O -. 5412.311 --4.7 .0736 0
1.8 -1. 03 2.36 -4.81 .0909 0
The pair of conjugate complex roots X5.2 in table III
represents an oscillatory component of the motion
usually called the lateral, or Dutch roll, oscillation.
The airplane used in the calculations has characteristics
generally considered satisfactory for this mode. The
real root X3 shows the damping of rolling motion. The
pilot is ordinarily unaware of this rolling component of
the motion following a disturbance of the airplane be-
cause the mode is so highly damped. The root X4 indi-
cates the degree of spiral stability present. At cruising
speeds the average airplane has almost neutral spiral
stability; and, at lower speeds, it becomes definitely
unstable spirally. For most airplanes, stability of this
mode is either very poor or lacking. The root )s defines
the stability in azimuth, that is, the tendency to follow
a given compass course. The uncontrolled airplane
always has neutral stability in this mode, as indicated
by the zero value for the root \5.
STABILITY WITH CONTROL
When automatic control is introduced, the XI 2 lateral
oscillation is retained but the spiral-stability root X4
may be real or combined with X3 or with X to form
either of two distinct types of oscillation, depending on
the control assumptions. For control conditions such
that the X3.4 or the X45 oscillation is present, it should
be noted that the oscillation will exist in addition to
the Xl.2 oscillation. For some flight conditions, the
X1.2 oscillation and the added X3.4 or X45. oscillation may
have very similar characteristics as regards period and
damping, so that distinguishing between them in flight
The criterion used in judging the desirability of any
of the methods of automatic control subsequently dis-
cussed was the extent to which it improved the stability
characteristics of the average airplane. The factors
governing control deflection in the types of control
considered are given in table IV.
DESCRIPTION OF AUTOMATIC CONTROLS
Type of control Aileron deflection propor- Rudder deflection propor-
tional to- tonal to-
Displacement and Displacement in bank; d Displacement in bank; dis-
rate-of-displace- placement in azimuth; placement in azirimuth;
ment. sideslipping velocity; roll. sidesllppng velocty; roll-
ing velocity; yawing e- ing velocity; yawing ve-
Cross-coupled...... Displacement in bank: dis- Displacement in bank; dis-
placement in azimuth. placement in azimuth.
Simple-----------............. Displacement in bank... Displacement In azimuth.
Attention was chiefly confined to the simple control
for which the effect on the stability characteristics was
determined of varying the amount of control deflection
resulting from a unit change in the quantity governing
deflection. The effect of lag in control application was
Questions of mechanical difficulty in obtaining the
various methods of control have been given but little
consideration. Only systems dependent for their
operation on displacements or rates of displacement
have been treated because past experience with auto-
matic control has demonstrated that such displace-
ments and rates of displacement can be detected with
relatively simple mechanisms. (See reference 3.)
Although the control deflections might also be made to
depend on the accelerations in roll, yaw, and sideslip,
no such methods of control were considered in the
present study because preliminary investigation in-
dicated that they would be of little assistance in improv-
ing the character of the motion of the airplane.
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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/m1/2/: accessed July 16, 2018), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.