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level oflateral stability or dihedral effect compared to the static directional stability.
If not, it is possible that the spiral mode becomes unstable. However, it should be
remembered that this inference is based on spiral approximation and not on the
complete fifth-order lateral-directional system.
576           PERFORMANCE, STABILITY, DYNAMICS, AND CONTROL
  The physical motion of an aircraft experiencing a spiral divergence can be
approximately described as follows.
   On encountering a gust that raises one wing relative to the other, the aircraft
banks, the nose drops, and the aircraft develops sideslip in the direction ofthe lower
wing. Let us assume that the left wing is raised, the right wing is dropped, and the
aircraft sideslips to the nght. Because the aircraft has a low level oflateral stability
or only a small stable value of Cip, it cannot come to the wing level condition
immediately. Instead, because ofthe strong directional stability (Cnp  >  O), it yaws
to the right to orient the nose in the direction of the relative wind. In this process
it develops a positive yaw rate and, because of this, it further rolls to the right due
to Clr because Clr iS usually positive. This rolling motion increases the sideslip
further. Because the lateral stability Cip is insufficient, the condition worsens, with
the bank angle continuously increasing and the nose dropping further. The net
result is that the aircraft is losing altitude, gaining air speed, and banking more and
more to the right with an ever increasing turn rate. The aircraft is essentially in a
tightening spiral motion. What distinguishes the spiral motion from a spin is that
the angle of attack is below the stall and control s~rfaces are still effective.
       To avoid the spiral divergence, we have to improve spiral stability of the aircraft
while keeping a check on the level of directional stability. This can be done by
 increasing the dihedral effect (for example increase wing dihedral) and simultane-
ously keep the static directional stabilit)r level to a minimum.
6.3.2 Accuracyof Lateral-DirectionaIApproximations
       Generally, the accuracy ofthese approximations depends on type of the airplane.
It may so happen that for one aircraft these approximations may be satisfactory
but for the other they can be considerably in error.
   To have an idea of the accuracy of lateral-directional approximations, let us
refer to Table 6.1, which gives various values ofthe lateral-directional roots based
on the complete fifth-order system, roll subsidence, Dutch roll, and spiral mode
approximations for the genera7 aviation airplane.
    We observe that the roll subsidence and Dutch-roll approximations are satisfac-
tory but the spiral approximation is in poor agreement with the complete fifth-order
lateral-directional system.
  The computed free responses based on the complete fifth-order system and
Iateral-directional approxirnations for the general aviation airj?lane are shown in
Figs. 6.18-6.20. It may be observed that the motion corresponding to spiral root
differs considerably.
     The unit-step responses based on complete fifth-order system and lateral-direc-
tional approximations are shown in Figs. 6.21-6.25.ltis observed that the transient
    rfable 6.1    Lateral-directional roots of the general aviation airplane
 ----~-- -_-_-
Mode         Approximation   Completesystem Comment
 Roll  subsidence, ,\r                   -8.4481                               -8.4804                 Excellent
Dutch roll, Adr              -0.5102 +  j2.1164     -0.4897 :1: 2.3468        Good
Spiralmode       -0.1446      -0.0087     Poor
  ---_-_---_-_-- -_ N
AIRPLANE RESPONSE AND CLOSED-LOOP CONTROL         577
:  --- Roll SubsidcnceApproximarion
-------  Complete Fifth.Order System
Fig. 6.18    Free response Oateral-directional) of the general aviation airplane.
　
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