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248             PERFORMANCE, STABILITY, DYNAMICS, AND CONTROL
   Using Fig. 3.37b, we obtain Cl8/(CL3)theory = 0.817 and, from Fig. 3.37a,
(Cl8)theory = 3.77 so that CLb - 3.08]rad. We have cra = -C18/ao. Then, with
ao -. 5.8365, we get a8 - -0.5277. Using cj{ld = 0.226 and A - 4.0 from
Fig. 3.51a, we get ACh8, = 0.0156(CL8 B2Ka cos  Ac/4 cos  AHL).
   We have T71 - 0.25 and T,o - 0.65. With these values and from Fig. 3.51b.
we obtain (K6),,, = 1.30 and (K8),ro = 2.25. Then, using Eq. (3.130), we obtain
K8 = 0.4688. Further substitution grves AChB.e - 0.01036/rad.
        Finally, substitutingin Eq. (3.126), we find Cha.e -. -0.3457/rad or -0.0060/deg.
3.4   Stability in Maneuvering Flights
        The class of flight paths when the load factor exceeds unity is called maneuvers.
The load factor n is defined as the ratio oflift to weight.
   L
n = W                              (3.171)
       Therefore, during a maneuver, the lift exceeds the weight and the airplane struc-
turc is subjected to higher stresses than those encountered in steady level flight
with aload factor ofunity.ln general, during a maneuver, the airplane experiences
accelerations, which makes it necessary for us to consider inertia forces in the
analysis. We will study such motions later in the text. However, by introducing
some simplifications and ignoring inertia forces, we can extend th~ methods of
static stability and controlto study simple maneuvers like the prdl-up from a dive in
a vertical plane and the coordinated turn in a horizontal plane. Even though not ex-
act, this approach gives us an idea about thelevel of stability, control requirements,
and the stick force gradients that can be expected in such maneuvers.
    We introduce the following assumptions. 1) During the maneuver, a change in
the forward speed is small and ignored.ln other words, we assume that the airplane
is moving at a uniform speed along its maneuver path. 2) The airplane is disturbed
only in angle of attack and load factor,- and these disturbances are small.
   If a maneuver is performed at transonic or low supcrsonic speeds, the above
assumptions may not be justified because even small changes in forward speed
can give rise to large variations in aerodynamic forces and moments.
    While studying the static longitudinal stability, we considered onfy one distur-
bance, which was the angle of attack. Here, wc have an additional disturbance in
the form of load factor. In the following analysis, we will develop a theory for
predicting stability, control requirements, and stick force gradients for a pull-up
from a dive in a vertical plane and the steady tum in a horizontal plane.
3.4.t   Pull-Up in a Vertical Plane
     Consider an aircraft to be initially flying in a steady level fiight at A as shown
in Fig. 3.62. At A, L -  W and the load factor n is un~ty. Let the aircraft climb to
point B and enter into a drve (C), and let the pilot effect a pull-out such that at the
bottom of the pull-out (D), the aircraft is at the same altitude as it was at A, and
the fiight path BCDE is approximately semicircular. Thus, the aircraft at D is at
the same altitude and forward speed as in A but is operating at a different angle
of attack and load factor.ln other words, the aircraft at D is disturbed in angle of
attack and load factor compared to the steady level fl:ight at A.
STATIC STABILITY AND CONTROL
/
E
D
             --- 4----
  '~2.
   ,
 -
-
Fig 3.62   Airplanein a pull-out maneuver.
249
   An important point here is to observe that, during the pull-up maneuver, the
aircraft experiences a steady rate of rotation in vertical plane, which is equivalent
to a pitch rate about the y-body axis. We will now show that, on account of this
pitch rate, the aircraft will experience higher levels of static stability compared to
that in steady level flight and this apparent increase in stability demands additional
elevator deflection.
    Consider the equilibrium of forces in vertical direction at D. Let L' denote lift
at D. Then,
so that
and
L' : n W
 . = W+ WR~J
n=l+~g
(3.172)
(3.173)
　
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