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system rotates with the Earth and has a constant angular velocity S2e with respect
to the OxiYtz, system. Here, we assume that xiy,zt serves as an inertial frame of
components in the Oxt ybZb system.
cra
lb
Fig. 4.16    Inertial, narr,gational, and body axes systems.
EQUATIONS OF MOTION AND ESTIMATION OF STABILITY DERIVATfVES 363
Then,
Rr - Re+RO+rb  ,                '  (4.236)
(ddR  ). =  dd (Re + Ro + r+~,)
                    = (dR )t+(ddR ).+(dd ).
Using the moving axis theorem shown in Eq. (4.226), we have
                          (ddR ).= (dc~R )e + S2e x Re
                    .        = (2e x Re
(4.237)
(4.23 8)
(4.239)
(4.240)
where ?2e iS the angular 'velocity of the Earth-fixed OxeYeze system with respect
to the Oxiy,zi system but having components in the OxeYeze system as given by
                                                S?,e = le S-Ze COS A, - keSZe siri )L
where S2e  =  l~el, the magnitude of l2e. Furthermore,
                               (ddR )t = (ddR )e + (2e X Ro
                                        = Vo + g3e x Ro
(4.241)
(4.242)
(4.243)
Here,  V o  = (dRo/dt)e is the velocity ofthe body with respect to the inertial systcm
having components in the Oxeyeze system.
(dd ). = (dd )b+(-t3rb X 7b                  (4.244)
                                      = t3{.,b x rb                              (4.245)
The first term on the right-hand side of Eq. (4.244) vanishes because we have
assumed that the body is rigid. Then,
                   (ddR ): = ~e X --e + -,~+ g2e X Ro+O}:.b x rb         (4.246)
      (ddR  ). =   dd (S2e X Re)le + g2e X f2. x f/e + (ddV, )e + ?2e X veA
                                       ~l
+ ?2e x V~ + g2e X 62e x Ro + [dd (o3l.b X 7,)]b + (3J.f, X  (CObt, X rt)
          (4.247)
364             PERFORMANCE, STABILITY, DYNAMICS, AND CONTROL
We note that the first term on the right-hand side is zero because both S2e and Re
are constant vectors in the OxeYeze system. Then,
(dcfR ). = ~e X (~e X Re) + (ao)e+2Qe x VZ
+ (2e X (l2c x Ro) + (dd   )b X 7b + c~Obb'X tob.,l, X 7b    (4.248)
= g2~ X Qe x (Re + Ro) + (ao)e+2t2e x V~+ (d     )h .X 7b
+ cOlb X 03lb x rb                                                                (4.249)
where (ao)e  = (d V o/dt)e. Furthermore, we have assumed that the body is rigid so
that (drb/dt)b - 0.
   In Eq. (4.249), various terms have the following interpretation:
    S2e x S2e x (Re + Ro): centripetal acceleration at the origin of the body-fixed
　
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