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0
p = 4 sin a
p - psina
D
p  = p sin ot - Ycoi a
(8.9)
(8.10)
(8.11)
a) Roll about body axis              b) Combined roll and yaw about body axis
Fig. 8.65    Rolling motion at high angles ofattack.
STABILI-FY AND CONTROL PROBLEMS AT HIGH ANGLES OF ATTACK   737
Thus, at low angles of attack, a pure rolling motion about the x-body axis does
not generate any significant sideslip excursions. However, at high angles of attack,
body axis roll produces significantly large and undesirable sideslip excursions. To
avoid this adverse sideslip buildup, the aircraft must roll about the velocity vector
as shown in Fig. 8.65b. To understand the concept of velocity vector roll, consider
an aircraft that simultaneously rolls and yaws about the body axes with rates p and
r, respectively. For this case, the rate of sideslip buildup is given by
[3 : P sin ty - r cos a
(8.12)
To avoid sideslip buildup, we must have ,8 = O. Then, the above equation gives
r - p tanLy
(8.13)
In other words, to produce a body axis roll rate of p while suppressing the buildup..
of adverse sideslip, the aircraft must simultaneously yaw about the z-body axis at
a rate, r  -  p tan a. This coordinated rolling and yawing about the body. axes is
equivalent to the velocity vector roll at a rate SZ, which is given by
s2 - pcosa +r sinty
    p
:= --
      cos a
(8.14)
(8.15)
Quite often, S2 is also called the stability axis roll rate.
   From Eq. (8.13), we observe that the proportion of body axes yaw rate in a
velocit)r vector rollincreases as the angle of attack increases. This places a severe
requirement on yaw control during velocity vector rolls at high angles of attack at
a time when the conventional aerodynamic controls like rudder become ineffective
because of shielding by the forebody and wing wakes. With current fighter aircraft,
the rudder capability degrades rapidly as the angle of attack approaches the stall
angle due to the rudder becoming immersed in the low-energy stalled wake shed
from the wing and fuselage.A typical example of the level ofavailable and required
yaw control is shown in Fig. 8.66, which depicts that the required yaw control far
exceeds the available amount even below the stall angle. TXis deficiency in yaw
control inherently limits the high-alpha roll rate capability, hence the maneuver
effectiveness of the aircraft.
    A combined rolling and yawing motion induces a pitching acceleration, which
is given by
q
For velocity vector roll, r = P tan
q-
   pr(lz -
=  )'
ct so that
P2 tan a(lz
~y
lx)
Ix)
(8.16)
(8.17)
For a typical combat aircraft of current generation, Iz >  / so that the pitch accel-
eration or the inertia-induced pitching moment is positive (noseup). If sufficient
                                                               .
.N
  .
~
p
 N
.,
:cD
  N
 c
 o
 -
 ~
 h
 V
-
 N
 U
 U
<
z
 U
--
.N
o-
738             PERFORMANCE, STABILITY, DYNAMICS, AND CONTROL
Yaw
control
Angle of attack
Fig. 8.66    Required and available yaw control at high angles of attack.47
nosedown control authority is not available to counter this noseup pitching mo-
ment  the aircraft may experience departure from controlled flight (divergence in
 pitch as discussed in Chapter 7) and getinto a deep stall Orim point while executing
a velocitjr vector roll.
          The deep stalllul  is one of the most significant tlight dynanuc problems associated
with nonlinear variation of pitching-moment coefficient with angle of attack as
illustrated in Fig. 8.67. We have two stable trim points: one is the conventional
 trim point at low angles of attack below the stall: ~nd the other is a poststall trim
 point, which is called the deep-stall trim point. This kind of variation ofthe pitching
　
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