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measured roll angle or roll rate time lustory.
    The histogram of Fig. 8.35 indicates that the inner loop encloses the area in a
clockwise sense (positive) and the two outer loops enclose the area in a counter-
clockwise sense (negative) as the Ioop is traversed in the direction of increasing
time. The positive area corresponds to the addition of energy to the system, and the
negative area implies extraction of energy (dissipation) from the s;rstem. There-
fore, the positive loop for -20 deg < 4 < 20 deg is destabilizing:'and the outer
lobes for 20 deg "$"< 55 deg are stabilizing. If the net area is positive, the
amplitude of the oscillatory motion increases and leads to a divergence. On the
other hand, if the net area is negative, the oscillatory motion decays gradually
(damped oscillation). If the net area is zero, it is a case of constant amplitude or
 limit cycle motion. It may be observed that the net area of the histogram shown in
Fig. 8.35 is close to zero, confirming that the wing rock is an example oflinut cycle
oscillation.
    Now,let us study the available fiow visualization data on the delta wing model
during a wing rock cycle. The static vortex locations are shown in Fig. 8.36 for
 various roll angles corresponding to the roll angles recorded in a typical wing rock
cycle. The peak amplitude of the roll angle for this wing rock cycle is 45 deg.
rfhe numbers above the open circle symbols are the values of roll angle 4   Starting
 from 4 : O, as the model is rolled in the positive direction (right wing down and
left wing up), the right vortex moves inboard and vertically closer to the wing
surface, while the left vortex moves outboard and further away from the wing sur-
face. The picture is other way when the model rolls in the negative direction. This
alternating vortex asymmetry is a characteristic feature of the wing rock of the
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706             PERFORMANCE, STABILITY, DYNAMICS, AND CONTROL
D
M
q 130?,:lcr = -0-75
                              -/ /                 
┏━━━━━━━━━━━━━┳━━━━━━━━━━┓
┃   LEFT SIDE              ┃        RIGHT SIDE  ┃
┃   -zls                   ┃-1-.                ┃
┃"                         ┃-. .o  -105 o       ┃
┃o   30                    ┃-o82O,s 'I5 O       ┃
┃        O'  l5 (? ~5:3~ ; ┃                    ┃
┃     I[l                  ┃     itl            ┃
┗━━━━━━━━━━━━━┻━━━━━━━━━━┛
yls
o
Flg. 8.36   Static vortex locations o'ver 80-deg delta wing at various roll angles for
a = 30 deg.3o
80-deg delta wing. As said earlier, the angle of attack for the onset of wing rock is
below that for vortex asymmetry. Therefore, what actually happensis that, once the
free-to-roll delta wing model starts oscillating because ofloss of roll damping, the
vortex'asymmetry gets established and leads to a sustained,limit-cycle oscillatory
motion.
     When the model rolls to the right, the vortex on the right wing (down going)
 moves closer to the wing surface, leading to an increase in the lift, whereas that
 on the left wing moves away from the wing surface, lcading to a decrease in the
lift. This leads to a stable variation in the static rolling moment with sideslip/roll
 angle as observed in Fig. 8.33a  The closer the vortex is to the wing upper surface,
 the more the lift on that part of the wing is because the suction is higher.
        The dynamic vortex positions are presented pictorially in Fig. 8.37a.ltis evident
　
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