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at t - 0. Therefore,
Xo, =  [ R:e ]
EQUATIONS OF MOTION AND ESTfMATION OF.STABILITY DERIVATIVES 349
and L -  }. = 0. Then, using Eq. (4.32),
               00
            Tte= 0 : ']
                     -1 0
Furthermore,
                                    Xe = Tte(Xt _ Xoi)
or
             xt                                          + 36,250
       X;e:l = I-i  :  '] [R'6+2,50050] _ [-]  ,  '] [R:']
     Ze                 0
         = [-32gOj250]
                                             Example 4.2
   Plot the Euler angle time history for an airplane performing a velocity vector
roll with 1) LY - 30 deg, 2) a - 45 deg, and 3) ce -. 60 deg. Assume that the
angular velocity S2 about the velocity vectorin each case is 25 deg/s.
    Solution    The first step is to select a reference frame that remains fixed in
space and with respect to which the Euler angles are measured. Let xoYozo be such
a reference frame as shown -in Fig. 4.8. The Oxo axis is assumed to coincide with
the velocity vector Uo, the OYo axis points to the rightin the horizontal plane, and
 Ozo points vertically downward to form a right-hand system.
    At t - 0, the aircraftis assumed to be oriented as shown in Fig. 4.8 so that the
initial values of the Euler angles of the body axes system Oxbybzb are rjr (0) - 0,
0(0) = ct, and ~(0) = O.
zo
Fig. 4 8    Airplane in a velocit)r vector roll.
""
.:i
Ill:-:"
       cos S2eti
-.   -siri CZetl
        0
[X:E::]
350                PERFORMANCE, STABILITY, DYNAMICS, AND CONTROL
We have
p -. s-z cos a
   q -.0
 r - S2 sina
Because the angle of attack remains constant during the velocity vector roll, the
values of p, q, r will also remain constant.
    With q = 0, the Euler angle rates as given by Eqs. (4.66-4.68) are
4 = p+r tanOcos4
       0 - -r sin4
   ~ - r sec0 cos4
An integration of these equations for the above initial values was performed
using the MATLABrM (Ref. 5) routine ODE45, and the results are shown in
Fig. 4.9. We observe that the roll angle continuously increases with time.ln view
of the restrictions as shown in Eqs. (4.1-4.3), the roll angle 4 has a sawtooth
variation.
07}
0. 0
-200
Euler Angles in Velocity Vector Roll, Omega= 25 deg/s
                 I ---t~    J  
  (: ::. -/::- ;1 ;-           
┏━━━━━━━━━━━━━━┓
┃         :   . ,P---.       ┃
┗━━━━━━━━━━━━━━┛
o      .  5        10        15        20        25        30
-fhg. 4.9    Euler angle time histories in velocit}r \rector rolls.
EQUATIONS OF MOTION AND ESTIMATION OF STABILITY DERIVATIVES 351
a
Fig. 4.10    Spinning airplane of Examp,le 4.3.
Example 4.3
zo
    For a spinrung airplane as shown in Fig. 4.10, the angular velocity components
in the body axes are given by
p = s? cOs tx
　
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