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Fz = _m C22R cosa
(7.97)
(7.98)
(7.99)
Ignoring power effects and resolving the aerodynamic and gravity forces acting
on the airplane, we get
L sin af - D cos cr + W cos av = mC22R sin cy
-L cosa - D sin a + W sin ar = -mSl2R cos a
(7.100)
(7.101)
Multiply Eq. (7.100) by cos ct and Eq. (7.101) by sin a and add the two equations
to obX;
      D-W
With this, substitution in either Eq. (7.100) or (7.101) gives
L = IT2~2R
(7.102)
(7.103)
It so happens that we could have arrived at these simple relations directly by look-
ing at Fig. 7.12. However,in this process of deriving these results using equations
 of motion, we have obtained some understanding of the lanematics of the spinning
motion.
With
D -. 2lp U02SCD
L = ~pU02SCL
(7.104)
(7.105)
we get
INERTIA COUPLfNG AND SPiN
UO -
R=(2 )pUo2SC
From Eqs. (7.67) and (7.68), we have
                                                        CL - CR COS Ct
                                                CD = CR siri CL
655
(7.106)
(7.107)
(7.108)
(7.109)
    Suppose we could determine the angle of attack and spin rate from some other
criteria, then we could use Eqs. (7.106) and (7.107) to determine the descent
velocity and spin radius.
7,,5-2   Balance of Moments
    For balance of moments, the sum of all the moments must be zero, regardless
of the axes system chosen* Suppose we consider the airplane motion with respect
to the spin axis; then the centrifugal forces acting on all the components of the
airplane acting radially outward from the spin axis do not produce any moment
about the spin axis. Therefore, the net aerodynamic moment about the spin axis
must also be zero. This means that the resultant aerodynamic force must also pass
through the spin axis. In other words, to determine the equilibrium spin modes,
we simply have to find the combination of angle of attack, sideslip, and spin rate
at which the resultant aerodynamic force passes through the spin axis. However,
the scenario is not this simple. Usually, tre aerodynamic data does not contain
the magnitude, direction, and point of action of thejresultant aerodynamic force.
Instead, we usually have-the aerodynamic data in the form of lift, drag, side force,
pitching, rolling, and yawing moment coefficients in the stability axes system.
Using this information, we have to compute the balance of pitching, rolling, and
yawing moments about the body axes system to determine the equilibrium spin
modes.
     For steady-state spin, p = q  - r - 0. With this, moment Eqs. (7.77-7.79) take
the following form:
 / = qr(lz - /y)
M = rp(lx - Iz)
N = pq(ly - /x)
(7.110)
(7.111)
(7.112)
Here, L, M, and N denote the net external rolling, pitching, and yawing moments
acting on the airplane duning the steady-state spin. Because we have ignored the
 power effccts,the only external moments acting on the airplane are the aerodynamic
moments. The right-hand side of Eqs. (7.110-7.112) represent the moments due
to inertia cross coupling effects. Substituting for p, q, and r from Eqs. (7.91-7.93)
656            PERFORMANCE, STABfLITY, DYNAMICS, AND CONTROL
in Eqs. (7.110-7.112), take the following form:
                  Q2 .
                                    L = - 2  sin2asiriX(Iz - Iy)
                                        M = S22 sin2a cosX(lx - Iz)
                 g22 -2
                                      N = -  2  COS2 tX sin2X(ly - Ix)
　
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