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approach zero.
   The simulation of free response to an initial disturbance in angle of yaw of
 5 deg (not shown here) shows that a disturbance involving only angle of yaw does
 not induce any sideslip, banking or roll, or the yaw rate. The disturbance in angle
of yaw remains constant and does not decay at all. This is due to the fact that
 no aerodynamic force or moment depends on the angle of yaw. Mathematically,
the associated root is zero, and the airplane is neurrally stable with respect to a
 disturbance in the angle of yaw.
       The free response to a disturbance in yaw rate of 0.2 rad/s is shown in Fig, 6.16.
 It is observed that this response is very similar to other cases discussed above.
    The free response to any arbitrary combination of intial disturbances can be
 constructed fro~ these basic responses using the principle ofsuperposit,ion because
 the given system is a linear system.
         The above free responses ofthe general aviation airplane are typicai ofa major:ity
of the airplanes. In view of this, it is usual to introduce the following lateral-
 directional approximations.
6.3.t Lateral-DirectionajtApproxrmations
    Ro/l-s-ubsidence approximation.    The motion immediately following a lat-
eral-directional disturbance is the heavily damped roll subsidence mode during
which the ainjlane motion is predominantly rolling about the body axis, and,
during this process, other variables vary very slowly so that we can assume Ap -.
A~ - r = r - 0. With this assumption, the side force and yawing moment
equations can be ignored. Furthermore, the rolling moment equation assumes the
following form:
Iii p - Ctpbi P = Cl8a A8a + Ct8t Ab'r
For free response, A8" - A8r - O so that
                                             Ixi P - Ctpbi P = 0
(6.206)
(6.207)
where
AIRPLANE RESPONSE AND CLOSED-LOOP CONTROL         573
rp+p_0
                     .  Ixl
                                                          r -- - C~pb~
The parameter r is the time constant for the rolling motion
suming p = PoeAr' and substituting in Eq. (6.208), we get
                                                              1
                                 Ar -- --.
                                    r
_ Ctpbi
         Ix I
(6.208)
 (6.209)
of the airplane. As-
(6.210)
(6.211)
Usually, for angles ofattack below staitl CiP  < 0. Therefore, Ar is real. Usually, Ar
has a large negative value. As a result, the roll subsidence mode is well damped
and is hardly felt by the pilot or passengers.
b,C,.Bttp = C_,.p Ap - (mi  - bi Cyr)r + C).8" A8a + Cy8,A8r
          (6.212)
            Izir = Cnp Ap + Cn~br Ap + Cnrbir + Cn8o A8a + Cn8,A8r           (6.213)
Substituting for A p from Eq. (6.212) in Eq. (6.213) and simplifying, we get these
two equations in state-space form
     [Atpl=[::~ ~22l[Arpl+[g:.:: g:'::]K~i:]  (6.214)
where
all =
mi - C).Bbi
a12 ~ -
mi - bi Cr.r
mi - Cypbr
(6.215)
(6.216)
574
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