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approach can be extended to some class of maneuvers like pull-up from a dive
in vertical plane or a steady turn in horizontal plane. This background will also
be helpful to understand the dynanuc motion of the airplane. In this chapter, we
will study the basic principles of static stability and control and discuss methods
that are suitable for preliminary estimation of the static aerodynamic and stability
characteristics of typical airplane configurations.
3.2   Concept of Equilibrium and Stability
     Consider a ball resting on three different types of surfaces as shown in Fig. 3.1.
If the ball is disturbed, it will return to its original equilibrium state in Fig. 3.la,
move away in Fig. 3.lb, and assume a new equilibrium state in Fig. 3.lc. In other
words, the forces and moments acting on the ball in its disturbed state are such
that they cause it to move towards its original equilibrium state in Fig. 3.la and
away from it in Figs. 3.lb and 3.lc. The ball never attains any equih~rium state
in Fig. 3.lb but attains a new equilibrium state in Fig. 3.lc. Thus, the equilibrium
states of the ball can be classified as stable, unstable, and neutrally stable.
   Let us see how these simple concepts can be applied to an airplane in steady
fiight as shown in Fig. 3.2. For simplicity, let us restrict ourselves to angles of
attack below the stall. To begin with, let us consider steady flights in vertical
(longitudinal) plane. The steady flights in horizontal (lateral-directional) plane
will be considered later. For a steady level flight in vertical plane, L - W and
T - D. Now if this airplane is disturbed in angle of attack, the response analogous
to that of the ball could be one of three types. The disturbance could be in the form
of a vertical gust, air turbulence, or a rapid movement of the elevator. In Fig. 3.2a,
the induced nosedown pitching moment M is stabilizing because it tends to restore
the airplane to its original angI6e of attack. In Fig. 3.2b, the induced noseup pitching
moment is dcstabilizing because it tends to increase the angle of attack and stall
the zurplane. In Fig. 3.2c, the induced pitching moment is zero, and the airplane
will assume a new angle of attack depending on the magnitude of disturbance like
the ball in Fig. 3.lc. Thus, the criterion for longitudinal or pitch stability can be
166           PERFORMANCE, STABILITY, DYNAMICS, AND CONTROL
a) Stable
b) Unstable
c) Neutrally stable
Fig.3.1   Various forms ofequilibrirun.
Gust
a) Stable equilibrium
b) Unstable equilibrium
Gust
c) Neutrally stable equilibrium
Fig.3.2   Various forms ofequilibriumin pitch.
STATIC STABILITY AND CONTROL
expressed mathematically as
or in coefficient form
dM
da
dCm
da
<O
<0
167
(3.1)
(3.2)
where Cm = M/qSc, q is the dynamic pressure, S is the reference area, and
c is the reference length. Usually, wing area is used as reference area, and the
mean aerodynanuc chord (mac) of the wing is used as reference length. Thus, an
airplane with dCmlda < O is statically stable; that with dCm/dcr > O is statically
unstable. IfdCmlda  = 0, the airplane is neutrally stable in pitch. In control system
terminology, this type of system stability is known as open-loop stability.
  The equilibrium condition in pitch is usually called trim condition. For pitch
trim, the net pitching moment about the center of gravity is zero. For an airplane to
be flyable,it must be capable of trimming at all values of angles ofattack within the
permissible range of angles of attack. Typical variations of the pitching moment
with angle of ar,t,ack are shown in Fig. 3.3. To establish a stable pitch trim the
+
     C mo
Ctn
     o
┏━━━━━━━┓
┃      Cmo >O  ┃
┃ c"oc<o       ┃
┃\- ,,,        ┃
┣━━━━━━━┫
┃\             ┃
┗━━━━━━━┛
a) Stable equilibrium
+
Z
+
Cm
   o
Cmo
┏━━━━━━━━━┓
┃     Cmo <O       ┃
┃      Cmtr >O     ┃
┣━━━━━━━━━┫
┃,,i- /         tr ┃
┃/-//              ┃
　
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