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1.4          1.2
1.0         0.8         0.6
h/b
Flg. 8.25    Effect of cross-sectional shape on static directional stability.lz,zo (Courtesy
AGARD.)
8.7   Relation Between Angle.of Attack, Sideslip, and Roll Angle
    Before we discuss the high-angle-of-attack stability and control problems, it is
necessary to understand that, at high angles of attack, the rolling motion about the
body axis reduces the effective angle of attack and builds up the sideslip as given
by the following equations:
ct = tan-l(tan u cos ~)                                         (8.1)
p = sin-l(sinc sin~)
(8.2)
where Cr iS the angle between the x-body (rolD axis and the velocity vector and
~ is the roll angle about the x-body axis (see Fig. 8.26). For # -. 0, tY - a' and
p - 0 and, for 4 - 90, ct = O and p = cr _ Thus, at extreme roll angles, all the
angle of attack gets converted to sideslip.
8.8 WingRock
      Wing rock is a limit cycle oscillation predonunantly in roll about the body axis.
The wing rock is exhibited by a variety gf aircraft configurations. Some examples
are the F-5, the F-16, the X-29, and the X-31. The principal source of wing rock
is the wing itself. The thin, low-aspect ratio, highly swept delta wings are prone
c,
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a
    I
v
u
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696            PERFORMANCE, STABILITY, DYNAMICS, AND CONTROL
u
D
D
FXg. 8.26    Delta wing at combined lugh angle of attack and roll angle.
to develop wing rock at high angles of attack. However, aircraft configurations
not ha\ring highly swept delta wings but featuring fuselages with long, slender
forebodies are also known to exbibit wing rock. This latter type of wing rock is
called forebody-induced wing rock and is known to occur even when the wing is
removed. It may also be noted that thin, sharp-edged rectangular wings of very
low aspect ratio, usually less than 0.5, also exhibit wing rock, which is generated
by the dynamic motion of the side edge vori       Because aircraft rarely have
such low-aspect ratio rectangular wmgs.   ,rfe  o~wing rock is not discussed
here.  The interested reader may refer elsewhe
    A loss of damping in roll at high angles of attack makes a configuration sus-
ceptible to wing rock but does not necessarily generate a sustained wing rock. An
example ofloss ofrolldamping at high angles of attackis presentedin Fig. 8.27 for
the F-5 and the X-29 aircraft.ltis interesting to observe that these two aircraft have
similar forebodies but radically different wing planforms, yet both lose damping
in roll at high angles of attack.
     To generate a sustained wing rock motion, some additional aerodynamic cause
is necessary. The various causes that have been identified so far arejl) nonlinear-
ities in lateral-directional aerodynamic characteristics, 2) aerodynamic hysteresis
in the variation of rolling moment with sideslip/roll angle, and 3) nonlinear vari-
ation of roll damping such that negative damt"prng (destabilizing) exists at low
values of sideslip/roll angle and positive damping (stabilizing) at higher values of
sideslip/roll angle.
       The nonlinearities in aerodynamic characteristics ofmodern fighter aircraft with
 highly swept, thin delta wings and long,pointed slender forebodies arise because of
STABILITY AND CONTROL PROBLEMS AT HIGH ANGLES OF ATTACK   697
X-29
F-5
Roll damping
-10 o  10 20 30 40  50 60
             cr, deg
nstabre
tabte
Fig. 8.27     IDustration- ofloss of damping in roll at high angles of attack.22
.6
-4
-2
  o
-. 2
-. 4
-. 6
698             PERFORMANCE, STA81LITY, DYNAMfCS, AhID CONTROL
      . {,
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U
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 rr
 E
U
Fig.8.28a   Dampingderivatrves ofa wing_bodycombinat1oDl,3
　
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