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Cnr = -k,7u V2av(T2br + cr)
Cn8r ~ -kr1u V2aur2
(3.316)
(3.317)
Cnr = qAl~b                        (3.318)
                       a cn                                                                      (3.319)
C"8r = a8r
We assume that r7y -  7r   :he method to evaluate r7r was presentedin Section 3.3.
    The rudder deflection to generate a sideslip t3 is given by
(cnp)rxP + Cn8r8r = 0
(3.320)
                                                                          (C tp)fixP
            (3.321)
           8r=- ,8
 From this relation, we observe that the higher thelevel of static directional stability,
the higher the rudder deflection to generate a given sideslip will be. Typically, a
value of q.iBr of --O.OOl is considered satisfactory
      The rudder effectiveness, Jike that of any other aerodynamic control surface,is
 nearly constant at low and moderate sideslip but falls off rapidly at high values of
 sideslip because of flow separation and stall.
STATIC STABILITY AND CONTROL
281
   Rudder requirements.   An airplane having an adequate level of static di-
rectional stability and symmetric power generally tends to maintain zero sideslip
condition and, as such, the defiection of the rudder may not be usually warranted.
However, under some critical conditions, it is possible that the static directional
sta'oility alone may not be sufficient to maintain zero sideslip, and the operation of
the rudder becomes absolutely essential. The rudder should be designed to provide
sufficient control authority under such circumstances as discussed in the following
sections.
   Crosswind takeoff and landing.   During the ground run, if the aircraft en-
counters a crosswind, the resultant 'velocit)t vector falls out of the airplane's plane
of symmetry, producing sideslip. An aircraft with positive directional stability
(Cnp  >  O) will tend to realign itself with the direction of the resultant wind so that
the sideslip is eliminated. Takeoff with this kind of aircra:ft orientation (Fig. 3.85a)
during the ground run can pose safety problems. To prevent this, the rudder should
be capable of generating a yawing moment to counter that due to directional stabi-
lity so that the aircraft sideslips but is properly oriented with respect to the runway
(Fig. 3.85b). However, the takeoff or landing performance will be below normal
because the sideslipping aircraft experiences a higher drag.
     Adverse yaw.     In alevelcoordinatedmrn,the aircraftis moving approximately
in a circular path in a horizontal plane as shown in Fig. 3.86. The angular velocity
Q about the vertical axis passing through the center of the turn is equal to V/R.
During the turn, the outer wing is moving with a higher velocity compared to
the inner wing. As a result, the outer wuy; experiences relatively higher drag,
and this imbalance in drag (AD) induces an yawing moment that tends to turn
the nose of the aircraft away from the center of the turn. This phenomenon is
known as adverse yaw. Because of this adverse yaw, a rolling motion may also
be induced because of the dynamic derivative CLr (which is usually positive) as
we will study in Chapters 4 and 6. This rolling motion caused by Clr iS called
adverse roll because it tends to bank the aircraft away from the direction in which
the aircraft is turning. Generally, the magnitude of the adverse yawing moment
is small. However, for rapid turns at lugh angles of attack, it can pose problems,
particularly if the adverse roll overpowers the proverse roll because of ailerons
　
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