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              C32 -.- -            zC  ]
  ~33
     =-[ 04323-. )*00578+0.8665*(-0.:
                                                            0.9666              0 2496 ]
                    - 0.2496
 Substituting in Eqs. (4.125) and (4.126), we get
                              0.8999 * (-0.4323) + 0.8665C21 + 0.2496C31  - 0
                              0.8999 * 0.0578 + (-0.2496)Czi + 0.9666C31  -. o
Solving, we get, C21 - 0.4323 and C31 - 0.0578. Now to verif)r our calcu-
lations, we have to check whether the remaining three redundancy relations in
EQUATIONS OF MOTION AND ESTIMATION OF STABILITY DERIVATIVES 359
o)
a
o
;.
r:L
cn
q)
D
d
6
-C
-
Eulcr Rate Eqruktion
DCM
Quatcmions
Fig. 4.14    Euler angles (SZ = 30 deg/s, a = 60 deg).
Eqs. (4.122-4.124) are satisfied. Substituting, we find
                                          Cizl + C~2 + C321 = 1.00004
                                            C1?2 + C~2 + C~2 = 1.000006
                                            C~3 + Cln + C~3 = 0.999996
 Hence, the calculated values of missing elements are satisfactory.
4.3  Equations of Motion
   The equations governing the aircraft motion are based on Newton's laws of
motion. We have the force and moment equations in the form
      (4.223)
F = m'(ddV, ).
M = (ddH)
(4.224)
Eakr Rrite Eqr=tion
DCM
Qurkrmon
360               PERFORMANCE, STABILITY, DYNAMICS, AND CONTROL
where  V  is the velocity and H 'is the angular momentum. The suffix / implies that
the acceleration dV/dt and the rate of change of angular momentum dHldt are
supposed to be measured with respect to an inertial frame of reference. We have
H  -.  It.o where I is the moment ofinertia of the body and co is the angular velocity
of the body with respect to an inertial system. Then Eq. (4.224) takes the form
M -.
GI),, +i (dd ),
(4.225)
If a space-fixed inertial system of reference is used to compute the angular mo-
mentum, the moment ofinertia I will continuously vary with time as the aircraft
translates and rotates in space. As a result,it will be extremely difficult to solve the
equations of motion because, at each time step, we have to evaluate the moment
o~:nertia I and its time derivative (dl/dt),. For the aircraft, I is a dyad with nine
components, three principal moments of inertia lxx, Iyy, Izz  and six products of
inertia Ixy, Iyx, Iyz, Izy, Ixz, Izx.
   One way of o'verconung this difficulty of computing time-varying moments
and products of inertia is to use a moving or body axes system, which is fixed
to the aircraft all the time and moves with it. Then the moments and products of
inerLia calculated with respect to this axes system will be constant except for such
variations as fuel consumption or control surface detlections that can be easily
accounted for.
        Theintroduction ofa moving axes system avoids the problem ofcomputing time-
varying moments and products ofinertia but creates another problem because the
　
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