depending on the nature of the dihedral effect 
of the airplane. 
It should be noted that initial inclination of 
the inertia axis above the aerodynamic axis 
will cause the inertia couple to provide adverse 
yaw with rolling motion. If the inertia axis 
were initially inclined below the aerodynamic 
axis (as may happen at high 4 or negative load 
factors), the roll induced inertia couple would 
provide proverse yaw. Thus, roll coupling 
may present a problem at both positive and 
negative inclination of the inertia axis depend- 
ing on the exact aerodynamic and inertia 
characteristics of the configuration. 
As a result of the aerodynamic and inertia 
coupling, rolling motion can induce a great 
variety of longitudinal, directional, and lateral 
forces and moments. The actual motion of 
the airplane is a result of a complex combina- 
tion of the aerodynamic and inertia coupling. 
Actually, all airplanes exhibit aerodynamic 
and inertia coupling but of varying degrees. 
The roll coupling causes no problem when the 
moments resulting from the inertia couple are 
easily counteracted by the aerodynamic re- 
storing moments. The very short span, high 
speed modern aircraft has the capability for 
the high roll rates which cause large magni- 
tudes of the inertia couple. The low aspect 
ratio planform and flight at high Mach number 
allow large inclination of the inertia axis with 
respect to the aerodynamic axis and also add 
to the magnitude of the inertia couple. In 
addition, the aerodynamic restoring moments 
deteriorate as a result of high Mach number 
and angle of attack and can create the most 
serious roll coupling conditions. 
Since the roll coupling induces pitching and 
yawing motion, the longitudinal and direc- 
tional stability is important in determining the 
overall characteristics of the coupled motion. 
A stable airplane, when disturbed in pitch and 
yaw, will return to equilibrium after a series 
of oscillations. For each flight condition, the 
airplane will have a coupled pitch-yaw fte- 
quency between the uncoupled and separate 
NAVWEPS 00-8OT-80 
STABILITY AND CONTROL 
pitch frequency and yaw frequency. Gen- 
erally, the greater the static longitudinal and 
directional stability, the higher will be the 
coupled pitch-yaw frequency. When the air- 
plane is subject to roiling motion, the inertia 
couple disturbs the airplane in pitch and yaw 
with each roll revolution and provides a dis- 
turbing forcing function.’ If the airplane is 
rolled at a rate equal to the coupled pitch-yaw 
frequency, the oscillatory motion will either 
diverge or stabilize at some maximum ampli- 
tude depending on the airplane characteristics. 
The longitudinal stability of the typical high 
speed configuration is much greater than the 
directional stability and results in a pitch fre- 
quency higher than the yaw frequency. In- 
creasing the directional stability by increasing 
the vertical tail area, addition of ventral hns, 
or use of stabilization systems will increase the 
coupled pitch-yaw frequency and raise the roll 
rate at which a possible divergent condition 
could exist. Increasing directional stability 
by the addition of ventral fins rather than by 
addition to the vertical tail has an advantage 
of not contributing to the positive dihedral 
effect at low or negative angles of attack. 
High dihedral effect makes higher roll rates 
more easily attainable in roll motion where 
proverse yaw occurs. 
Since the uncoupled yawing frequency is 
lower than the pitching frequency, a divergent 
condition would lirst reach critical proportions 
in yaw, closely followed by pitch. Of course, 
whether the airplane motion becomes divergent 
directionally or longitudinally is of academic 
interest only. 
There is one additional type of coupling 
problem that is referred to as “autorotative 
rolling.” A rolling airplane which has a high 
positive dihedral effect may reach a large pro- 
verse sideslip as a result of the inertia couple and