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┗━━━━━━━━━┛
b) Unstable equilibriium
┏━━━━━━━━━┓
┃Crn o =  CmcC= o  ┃
┣━━━━━━━━━┫
┃               tx ┃
┗━━━━━━━━━┛
c) Neutrally stable equilibrium
Fig. 3.3    Concept of stability and trim.
168           PERFORMANCE, STABILITY, DYNAMICS, AND CONTROL
necessary and sufficient conditions are
(3.3)
(3.4)
If Gm,  < O but Cmo  < 0, the airplane cannot be trimmed. In other words, the
 airplane is not flyable even though it has static stability, On the contrary,if Cm r  >  O
 and Cmo  <  0, th~ airplane is fiyable, butitis statically unstable,i.e., the equilibrium
condition is unstable like the ball in Fig. 3,1b. However, with proper feedback
control, such an'ayplane can be'made closed-loop stable as we will study later
in Chapter 6. In this chapter, we will study the open-loop stability of the airplane
with following assumptions:
        1) The airplane has a verticalplane ofsymmetry,i.e.,it has a symmetnic geometry
and mass distribution with respect to this plane.
      2) Defiection of longitudinal controls like elevators do not generate side force,
rolling, or yawing moments. Similarly, deflection ofthe lateral-directional controls
like ailerons or rudders do not produce lift or pitching moments.
   3) Aerodynamic forces and moments vary linearly with aerodynamic/control
variables.
   4) Total forces and moments acting on the airplane are equal to the sum of
forces/moments on individual components.
      With these assumptions,longitudinal and lateral-directional motions of the air-
plane can be decoupled and studied separately. We will evaluate the contributions
of fuselage, wing, and tail surfaces to static longitudinal and lateral-directional
stabilities. The analysis presented here, in general, applies to aicplane type config-
urations at all speed ranges. However, the methods for evaluation of aerodynamic
coefficients are presented only for subsonic (O < M < 0.8) and supcrsonic speeds
(1.2 < M  < 5). At transonic speeds (0.8 < M < 1.2), satisfactory methods suit-
able for prelinunary estimation of aerodynamic characteristics are not available.
In view of this, one may have to carefully interpolate subsonic and supersonic
aerodynamic data to obtain some crude estimates of aerodynamic coe:fficients in
the transonic Mach number range. The estimation of aerodjynamic coefficients at
hypersonic speeds (M  > 5) is not discussed here.
  We will be referring quite often to Datcom,l short for Data Compendium.
Datcorn is a collection of empirical design methods for estimating stability and
control derivatives for subsonic, supersoruc, and hypersonic speed regimes.
3.3  Static LongitudinaIStability
    The forces and moments acting on wing, fuselage, and horizontal tail surfaces
in a steady, unaccelerated level fiight are shown schematically in Fig. 3.4. We will
be considering only such class of flights unless otherwise stated. As said before,
we assume that the net aerodynamic force or moment coefficient is equal to the
sum of the individual contributions from fuselage, wing, and tail surfaces. In the
 following subsections, we will discuss the methods that are suitable for preliminary
estimation of these contributions.
STATIC STABILITY AND CONTROL
Lw
                      Aerodynamic Center
Fig. 3.4    Forces and moments acting on an airplane inlevel flighL
169
  The aerodynamics of streamlined bodies, typical of airplane fuselages, was
studied by Munk2 in the early 1920s. He ignored fluid viscosity and assumed ideal
fiuid flow. According to this theory, the pressure distribution over a streamlined
body at an angle of attack yields a zero net force accompanied by a pure couple
that is of a de~abilizing nature as shown in Fig. 3.5. In other words: bPoth lift avnd
drag are equal to zero, but the pitching moment is nonzero. Mathematically, this is
equivalent to a zero lift (normal force) acting at an infinite distance from the body so
that the product (pitching moment) is finite. Generally, the fuselage contribution to
static longitudinal stability is quite significant and is of a destabilizing nature. The
destabilizing pitching moment varies linearly with angle of attack cr as given by
　
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