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is not available and 2) the equations of motion are coupled and nonlinear. An
example of determining the steady-state spin modes using the static and dynamic
wind-tunnel test data may be found elsewhere.16
    In this chapter, we will discuss various factors that influence the balance be-
tween inertia and aerodynamic moments and also discuss various methods of spin
recovery. Finally, we will discuss some of the recent methods for improving the
spin resistance of airplanes.
7.5   Equations of Motion for Steady-State Spin
   The equations governing the spinning motion are the complete six-degree-of-
freedom equations [Eqs. (7.1-7.6)]. For simplicity, we ignore t~e product ofinertia
t.erm Ixz, With this assumption, Eqs. (7.1-7.6) assume the following form:
Fx =m(U +qW -rV)
Fy=m(V+Ur-pW)
Fz = m(W + pV - Uq)
 L = plx + qr(lz - Iy)
M = qly +rp(lx - Iz)
N = r Iz + pq(ly - Ix)
(7.74)
(7.75)
(7.76)
(7.77)
(7.78)
(7.79)
      In a steady-state spin, the spin axis is nearly vertical, and the center of gravity of
the airplane moves downward in a helical path around the spin axis with a constant
velocity. Let Uo denote the velocity of descent. Resolving the velocit)r vector Uo
650          PERFORMANCE, STABILI-fY, DYNAMICS, AND CONTROL
Fig. 7.12    Forces acting on an airplane in steady-state spin.
in the body axes system, we have
U  - Uo cosa
W - Uo sina;
(7.80)
(7.81)
where a is the angle of attack. Because the descent velocity is in a vertical direc-
tion, ce is the angle between the chordline and the vertical as shown in Fig. 7.12.
Note that the spin axis is also vertical.
      In a steady-state spin, when viewed from the top,the airplane's center of gravity
appears to be moving in a circular path with a constant angular velocity. Let S2
denote this constant angular velociojr. Let us assume that the airplane is spuuung
to its right, i.e., it is rotating in a clockwise direction when viewed from the top.
Because of this angular velocity, the airplane will have a component of velocity
along the y-body axis. At the center of gravity, the velocity component along the
y-body axis is given by
V - -QR
(7.82)
where R is the radius ofthe helix or the spin radius.As said earlier, the spin radius is
usually about one-halfofthe wing span for steep spins and still smaller for flat spins.
     Because in a steady-statc spin U,  V , and W  arc constants
U - V - W -0                      (7.83)
In spin, the angle of attack is well above the stalling angle.At such angles of attack,
the resultant aerodynamic force is approximately normal to the wing chordline.
Note that the lift acts in the horizontal plane and drag is directed in the vertical
plane opposite to the gravity as shown in Fig. 7.12.
Spin
INERTIA COUPLING AND SPIN
Fig. 7.13    Angular velocity components in spin.
651
     The angular velocity vector S2 can be resolved along the x- and z-body axes (see
Fig. 7.13) as
p = g? cos (y
r - S-2 sin a
If the wings are in the horizontal plane,
         q-0
(7.84)
(7.85)
(7.86)
     Because of the helical motion, the spinning airplane experiences a sideslip. For
example, in a positive spin (spin to the right), the sideslip is towards the lefi or port
wing, which is also the leading wing. In right spin, the right or starboard wing is
the trailing wing. As shown in Fig. 7.14, the sideslip angle is related to the helix
angle by the following relation:
where the helix angle y is given by
tr - -Y
(7,87)
y = t  .' ('2UR)                          (7.88)
Now consider a more general case where the wings are tilted out of the horizontal
plane. Let Oy denote the wing tilt. We assume that Oy is positive when the right
wing is tilted down and the left wing is raised with respect to the horizontal plane
as shown in Fig. 7.15. The sideslip is now given by
f3 = Oy - y
(7.89)
652             PERFORMANCE, STABILITY, DYNAMICS, AND CONTROL
  Velocity
thg. 7.14    Schematic iLlustration of sideslip in spin.
 q-b
　
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