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 depends on the angle X,we observe that a wing tilt one way or the otheris necessary
to achieve the rolling-moment balance.
                                                                                                                             F
INERTIA COUPLING AND SPIN
o
Fig. 7.19   Schematicillustration ofgyroscopic precision.
Precision
659
      A physical explanation using the lumped wing or fuselage masses is difficult in
this case. The gyroscopic analogy comes in handy.lf the gyroscope in Fig, 7.18b
with moment ofinertia Iy and rotating about the y-axis with an angular velocity
q is disturbed with yaw rate r, it will roll to the right. Similarly, the gyroscope in
Fig. 7.18c with moment of inertia Iz aligned along the z-body axis and rotating
with an angular velocity r disturbed by a pitching velocity q will roll to the left.
The net rolling motion will be the difference between the two induced motions.
Usually, /z  > Iy so that the inertia-induced rolling moment is negative or antispin.
Ba/ance of yawing moments.   We have
                         Nt = (/x - Iy)pq
=  ( g;  ) .OS2 Ct sin 2X (Iy - /x)
(7.129)
(7.130)
      As in the case of balance of rolling moments, we observe that the wing tilt plays
an important role in the balance of yawing moments.
    The concept of lumped fuselage and wing masses can be used to understand
the inertia yawing moment as shown in Fig. 7,20 for an aircraft in positive or
right spin. In Fig, 7.20a, the wing tilt is negative (right wing above the horizontal,
Oy < O, x  > O), and we observe that the fuselage masses Mt and M2 representing
inertia in pitch Iy produce a prospin yawing moment and the wing masses M3
and M4 representing inertia in roll Ix produce an antispin yawing moment. The
net inertia-induced yawing moment is the difference of these two contributions.
IfIy > Ix, then the fuselage masses dominate and the induced yawing moment
                                                                                              /
is prospin. On the other hand,iflx > Iy, the wing masses will dominate and the
inertia-yawing couple will be antispin.
    If the wing tilt is the other way, i.e., Oy  > 0 or x  < O as shown in Fig. 7.20b,
then the nature ofeach contribution changes. The wing masses M3 and M4 produce
    : - ~.
    ,
  :
  ,
t
of aerodynamic rolling moment is the dihedral effect Cip. However, beyond the
stalling angle, the magnitude of Cip is quite small. Therefore, for airplanes with
unswept wings, a major contribution to rolling moment comes from the wings,
which produce autorotative or prospin rolling moments for ce > cestaii  Therefore,
for such airplanes the inertia-rolling moment must be antispin to achieve the re-
quired balance of rolling moments. The inert.ia-rolling moment is given by
                                   L, = (Iy - Iz)qr                                            (7.127)
PERFORMANCE, STABILITY, DYNAMICS, AND CONTROL
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Airplme
a) Negative wing tilt, O  < O,V > O
　
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