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the horizontal plane. Once again, we will assume that the response of the airplane
is sufficiently slow so that yaw rate r and roll rate p can be ignored.
  The nomenclature used to describe the motion involving the six degrees of
freedom is presented in Fig. 3.66. The longitudinal motion of the aircraft involves
forward velocity u, vertical velocity w, pitch angle 0, and pitch rate q. Sometimes
the longitudinal variables u, w, O, and q  are called symmetric degrees of freedom.
The directional motion involves sideslip velocity vl'yaw angle /t, and yaw rate r,
and thelateral motioninvolves bank angle ~ and rollrate  p.However, thelateral and
the directional degrees of freedom are always coupled because a sideslip induces
both rolling and yawing motions. Similarly, a yawing motion induces both rolling
motion and sideslip. In this section, we will study the directional stability of the
aircraft with respect to a disturbance in sideslip, and we will study lateral stability
in the next section.
   Static directional stability is a measure of the aircraft's ability to realign it-
self along the direction of the resultant wind so that the disturbance in sideslip
is effectively eliminated. A disturbance in sideslip could be caused by horizontal
gust, wind turbulence, or momentary (small) rudder deflection. Therefore, on en-
countering a disturbance in the horizontal plane, the aircraft orientation in space
 changes butits heading remains the same as before with respect to the Earth. These
 concepts are illustrated in Fig. 3.67. The aircraft is in a steady (undisturbed) level
 flight in Fig. 3.67a, encounters a horizontal gust Vw blowing from starboard side
 resulting in a sideslip as shown in Fig. 3.67b, and realigns itself with the resultant
ra
Q
Fig. 3.66    Axis system and nomenclature used in static stability analysis.
STATIC STABILITY AND CONTROL
         IJ   LV = ~tg     
                           
┏━━━━━┳━━━━━━┓
┃          ┃            ┃
┃          ┣━━━━━━┫
┃          ┃/~-- -,  J- ┃
┃C::~      ┃            ┃
┣━━━━━╋━━━━━━┫
┃          ┃            ┃
┗━━━━━┻━━━━━━┛
             
  l,    L    
┏━━┳━━┓
┃/   ┃    ┃
┃    ┃C-J ┃
┃    ┣━━┫
┃    ┃    ┃
┗━━┻━━┛
Fig. 3.67    Aircraft orientation in horizontal plane.
259
velocity VR in Fig. 3.67c. Now, the sideslip is zero but the aircraft orientation in
space has changed. Notice that in Fig. 3.67c the aircraft is still moving with the
same velocity  V with respect to the Earth as before. If the disturbance vanishes,
the aircraft orientation will also be restored.
    The angles of sideslip and yaw (Fig. 3.68) are two important parameters in the
study of directional stability. Both of these angles are measured in a horizontal
plane. The angle of sideslip is an aerodynamic angle defined as the angle between
the velocity vector and the airplane's plane of symmetry as shown in Fig. 3.68b.
The angle of sideslip is usually denoted by p and is given by
sin p
   v
=V
(3.237)
where v is the sideslip velocity and V is the fiight velocity.
   The usual sign convention is to assume B positive if the airplane sideslips to
starboard (right wing leading into sideslip) as shown in Fig. 3.68b and p negative
if the airplane sideslips towards port side (left wing leading into sideslip as shown
in Fig. 3.68c).
     The angle of yaw, usually denoted by ~, is a kinematic angle and is a measure
of the change in the heading or orientation of the aircraft relative t,o the Earth. It is
the angle between the airplane's plane of symmetry and a reference plane fixed in
space as shown in Fig. 3.68b. Usually, this reference plane is assumed to coincide
with the airplane's plane of symmetry when the airplane is in steady level flight
　
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