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 could come near the observed motion of the Fairey Delta aircraft. At the same time,
the very same approach gave perfectly good results when the roll rate was small.
    These two examples demonstrate vividly the essential features of the problem
 now known as inertia coupling or roll coupling.It has been widely recognized that
inertia coupling will occur on aircraft that have long slender fuselages and short
aspect ratio wings and have most of their mass concentrated in the fuselage. Such
 aircraft have a low value of moment of inertia about the longitudinal (x) axis and
fairly large values of moments of inertia about y- and z-axes. Furthermore, it is
 also understood that a loss of either longitudinal (pitch) stability or the directional
stability compounds this problem further. This type of mass/inertia distribution
and stability characteristics are typical of modern high-speed fighter aircraft.
       A fundamental analysis ofinertia coupling was presented by Philipsin 1948.1 He
examined the stability of a steadily rolling aircraft  He assumed that all the distur-
 bance variables except the steady roll rate were small. He also ignored the damping
in pitch and yaw. Based on this analysis, he demonstrated that a steadily rolling
 airplane deficient in pitch stability experiences a drvergence in pitch, whereas one
deficient in directional stability experiences a divergence in yaw or sideslip. Even
though this simple analysis did not actually consider the real rolling maneuvers
 like that of the Heinkel 162 or Fairey Delta aircraft,it helped the flight dynamicist
identify the root cause of the problem. Since then, the problem ofinertia coupling
has received considerable attention from many authors who have performed more
rigorous analyses of this problem.Interested readers may refer to the literature for
more information on this subject.2-6
          In this section,  we will first present a briefphysical explanation of the problem of
 inertia coupling. Then, we will develop the theory of stability of a steadily rolling
aircraft and der:ive Philip's criteria for divergence in pitch or yaw.
72- I    Yawand Pitch Divergence in Rolling Maneuvers
     To understand the basic physical principles of ineffla coupling, let us assume
that the mass of the airplaneis concentratedin four distinctlumps at the extremities
of the fuselage and wings as shown in Fig* 7.1. The two masses Mi and M2 at
either end of the fuselage represent the inertia in pitch Iy, and the two masses M3
and M4 at the wingtips represent the roll inertia Ix. All the four masses acting
together contribute to the inertia in yaw Iz.
  Now let us consider what happens when the aircraft starts rolling about its
longitudinal (x) axis as shown in Fig. 7.1a. The fuselage masses Mi and M2
representing the inert.ia in pitch do not develop any centrifugal reaction, whereas
the wing masses representing the rollinertia de velop equal and opposite centrifugal
reactions that will try to tear the wings apart.. Because they cancel each other, these
forces do not cause any inertia coupling but concern the structuralengineer who has
to provide enough strength and rigidity to the airframe to deal with this situation.
   Now let us'assume that, for some reason, the rolling aircraft is disturbed in
yaw from an equilibrium condition. For this case, all four masses are subjected
to centrifugal reactions as shown in Fig. 7.lb. T.he fuselage masses Mi and M2
INERTIA COUPLING AND SPIN
:/L--
~:-:-~:,.- ------:-

a)
b)
                   c)
Fig. 7.1    Schematic illustration ofinertia coupling effects.
~
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630            PERFORMANCE, STABILITY, DYNAMICS, AND CONTROL
representing the inertia in pitch will attempt to further increase the yaw, and the
wing masses M3 and M4 representing the inertia in roll will try to reduce yaw
and restore the aircraft to its original equilibrium condition. Whether the net result
is stabilizing or destabilhing depends on the relative magnitudes of these two
centrifugal reactions. If the inertia in pitch exceeds the inertia in roll, which is
usually the case for most of the modem combat aircraft, then the destabilizing
couple will dominate. Whether the aircraft will experience a divergence in yaw
because of this destabilhing inertia-induced yawing couple depends on the level
of static directional stability. If the restoring yawing moment due to directional
　
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