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stability overpowers the inertia-induced destabilhing yawing moment, the aircraft
returns to its equilibrium condition.lf not, the aircraft willexperience a divergence
in yaw or sideslip. This phenomenon is known as the roll-yaw coupling.
Now let us consider another situation where the aircraft is rolling about the
velocit)r vector as shown in Fig. 7.1c. Such a rolling motion is usually preferred
at high angles of attack to avoid the sideslip buildup. In this case, only the fuse-
lage masses representing pitch inertia will contribute to the centrifugal reaction,
which produces a destabilizing couple in pitch.lf this destabilizing inertia-induced
pitching moment is greater than the restoring moment due to static longitudinal
stability, the aircraft will experience a divergence in pitch. In other words, an air-
craft deficient in static longitudinal stability is prone to divergence in pitch due to
inertia coupling during velocity vector rolls.
The pitch inertia increases with the length of the fuselage, thereby increasing
the magnitude of the destabilizing inertia-induced pitching couple. Furthermore,
to improve performance, modern fighter aircraft compronuse static stability in
pitch, These two factors make modern combat aircraft ~ulnerable to inertia cross
coupling problems.
7-2-2 Equations of Motion of a Stead17y Rolling Ai'rcraft
We have the following force and moment equations (see Chapter 4),
Fx =m(U +qW -r V) (7.,1)
Fy = m(V + Ur ~ pW) (7.2)
Fz = m(W + pV - Uq) (7.3)
L = plx - Lz(Pq + r) +qr(lz - Iy) (7.4)
M = ql7 + rp(lr - Iz) + (p2 _ r2)/z (7.5)
N = rlz - Ixz(P - qr) + pq(ly - 1,) (7.6)
Equations (7.1-7.6) are the complete six-degree-of-freedom equations for the air-
plane motion.ln general,itis difficult to obtain analytical solutions to these equa-
tions. In the preceding chapters, we assumed that all the flight-path variables in
the disturbed motion are small. With this assumption, we obtained two sets of
linear decoupled equations, one set of three equations forlongitudinal motion and
another set of. three equations for lateral-directional motion. Here also, we will
proceed along similar lines, assuming that all the disturbance variables are small.
Note that we have a nonzero steady-state roll rate Po, and Po is not restricted to be
INERTIA COUPLING AND SPIN
631
small. However, we assume that Po is constant so that p = 0. We need to introduce
some more assumptions to simplify the problem further. We will assume that the
product of inertia Ixz is small so that it can be ignored and further assume that
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