In designing an airplane we would compute eigenvalues and vectors (modes
and frequencies) and time histories, etc. But we don't need to do that at
the beginning when we don't know the moments of inertia or unsteady aero
terms very accurately. So we start with static stability.

If we displace the wing or airplane from its equilibrium flight condition
to a higher angle of attack and higher lift coefficient:

we would like it to return to the lower lift coefficient.

This requires that the pitching moment about the rotation point, C_{m},
become negative as we increase C_{L}:

Note that:

where x is the distance from the system's aerodyanmic center to the c.g..

So,

If x were 0, the system would be neutrally stable. x/c represents the margin
of static stability and is thus called the static margin. Typical values
for stable airplanes range from 5% to 40%. The airplane may therefore be
made as stable as desired by moving the c.g. forward (by putting lead in
the nose) or moving the wing back. One needs no tail for stability then,
only the right position of the c.g..

Although this configuration is stable, it will tend to nose down whenever
any lift is produced. In addition to stability we require that the airplane
be trimmed (in moment equilibrium) at the desired C_{L}.

This implies that:

With a single wing, generating a sufficient C_{m} at zero lift to
trim with a reasonable static margin and C_{L} is not so easy. (Most
airfoils have negative values of C_{mo}.) Although tailless aircraft
can generate sufficiently positive C_{mo} to trim, the more conventional
solution is to add an additional lifting surface such as an aft-tail or
canard. The following sections deal with some of the considerations in the
design of each of these configurations.

If we are given a plot of pitching moment vs. C_{L} or angle
of attack, we can say a great deal about the airplane's characteristics.

For some aircraft, the actual variation of C_{m} with alpha is
more complex. This is especially true at and beyond the stalling angle of
attack. The figure below shows the pitching characteristics of an early
design version of what became the DC-9. Note the contributions from the
various components and the highly nonlinear post-stall characteristics.

The analysis of longitudinal stability and trim begins with expressions
for the pitching moment about the airplane c.g..

Where:

x_{c.g.} = distance from wing aerodynamic center back to the c.g.
= x_{w}

c = reference chord

C_{Lw} = wing lift coefficient

l_{h} = distance from c.g. back to tail a.c. = x_{t}

S_{h} = horizontal tail reference area

S_{w} = wing reference area

C_{Lh} = tail lift coefficient

C_{macw} = wing pitching moment coefficient about wing
a.c. = C_{mow}

C_{mc.g.body} = pitching moment about c.g. of body, nacelles,
and other components

The change in pitching moment with angle of attack, C_{ma},
is called the pitch stiffness. The change in pitching moment with C_{L}
of the wing is given by:

Note that: when

The position of the c.g. which makes dCm/dCL = 0 is called the neutral
point. The distance from the neutral point to the actual c.g. position is
then:

This distance (in units of the reference chord) is called the static margin. We can see from the previous equation that:

(A note to interested readers: This is approximate because the static
margin is really the derivative of C_{mc.g.} with respect
to C_{LA}, the lift coefficient of the entire airplane.
Try doing this correctly. The algebra is just a bit more difficult but you
will find expressions similar to those above. In most cases, the answers
are very nearly the same.)

We consider the expression for static margin in more detail:

The tail lift curve slope, C_{Lah},
is affected by the presence of the wing and the fuselage. In particular,
the wing and fuselage produce downwash on the tail and the fuselage boundary
layer and contraction reduce the local velocity of flow over the tail. Thus
we write:

where: C_{Lah0} is the
isolated tail lift curve slope.

The isolated wing and tail lift curve slopes may be determined from experiments,
simple codes such as the wing analysis program in these notes, or even from
analytical expressions such as the DATCOM formula:

where the oft-used constant h accounts for the
difference between the theoretical section lift curve slope of 2p
and the actual value. A typical value is 0.97.

In the expression for pitching moment, h_{h}
is called the tail efficiency and accounts for reduced velocity at the tail
due to the fuselage. It may be assumed to be 0.9 for low tails and 1.0 for
T-Tails.

The value of the downwash at the tail is affected by fuselage geometry,
flap angle wing planform, and tail position. It is best determined by measurement
in a wind tunnel, but lacking that, lifting surface computer programs do
an acceptable job. For advanced design purposes it is often possible to
approximate the downwash at the tail by the downwash far behind an elliptically-loaded
wing:

We have now most of the pieces required to predict the airplane stability. The last, and important, factor is the fuselage contribution. The fuselage produces a pitching moment about the c.g. which depends on the angle of attack. It is influenced by the fuselage shape and interference of the wing on the local flow. Additionally, the fuselage affects the flow over the wing. Thus, the destabilizing effect of the fuselage depends on: Lf, the fuselage length, wf, the fuselage width, the wing sweep, aspect ratio, and location on the fuselage.

Gilruth (NACA TR711) developed an empirically-based method for estimating
the effect of the fuselage:

where:

C_{Law} is the wing lift curve
slope per radian

Lf is the fuselage length

wf is the maximum width of the fuselage

Kf is an empirical factor discussed in NACA TR711 and developed from an
extensive test of wing-fuselage combinations in NACA TR540.

Kf is found to depend strongly on the position of the quarter chord of the wing root on the fuselage. In this form of the equation, the wing lift curve slope is expressed in rad-1 and Kf is given below. (Note that this is not the same as the method described in Perkins and Hage.) The data shown below were taken from TR540 and Aerodynamics of the Airplane by Schlichting and Truckenbrodt:

Position of 1/4 root chord |
Kf |

.1 |
.115 |

.2 |
.172 |

.3 |
.344 |

.4 |
.487 |

.5 |
.688 |

.6 |
.888 |

.7 |
1.146 |

Finally, nacelles and pylons produce a change in static margin. On their
own nacelles and pylons produce a small destabilizing moment when mounted
on the wing and a small stabilizing moment when mounted on the aft fuselage.

With these methods for estimating the various terms in the expression
for pitching moment, we can satisfy the stability and trim conditions. Trim
can be achieved by setting the incidence of the tail surface (which adjusts
its CL) to make Cm = 0:

Stability can simultaneously be assured by appropriate location of the c.g.:

Thus, given a stability constraint and a trim requirement, we can determine
where the c.g. must be located and can adjust the tail lift to trim. We
then know the lifts on each interfering surface and can compute the combined
drag of the system.