This section deals with some of the basic stability and control issues that must be addressed in order that the airplane is capable of flying at all. The section includes a general discussion on stability and control and some terminology. Basic requirements for static longitudinal stability, dynamic stability, and control effectiveness are described. Finally, methods for tail sizing and design are introduced.

The starting point for our analysis of aircraft stability and control is a fundamental result of dynamics: for rigid bodies motion consists of translations and rotations about the center of gravity (c.g.). The motion includes six degrees of freedom: forward and aft motion, vertical plunging, lateral translations, together with pitch, roll, and yaw.

Quantity | Variable | Dimensionless Coefficient | Positive Direction |
---|---|---|---|

Lift | L | CL = L/qS | 'Up' normal to freestream |

Drag | D | CD = D/qS | Downstream |

Sideforce | Y | CY = Y/qS | Right, looking forward |

Roll | l | Cl = l / qSb | Right wing down |

Pitch | M | Cm = M/qSc | Nose up |

Yaw | N | Cn = N/qSb | Nose right |

Quantity | Symbol | Positive Direction |
---|---|---|

Angle of attack | a | Nose up w.r.t. freestream |

Angle of sideslip | b | Nose left |

Pitch angle | q | Nose up |

Yaw angle | y | Nose right |

Bank angle | f | Right wing down |

Roll rate | P | Right wing down |

Pitch rate | Q | Nose up |

Yaw rate | R | Nose Right |

Aircraft velocities, forces, and moments are expressed in a body-fixed coordinate system. This has the advantage that moments of inertia and body-fixed coordinates do not change with angle of attack, but a conversion must be made from lift and drag to X force and Z force. The body axis system is the conventional one for aircraft dynamics work (x is forward, y is to the right when facing forward, and z is downward), but note that this differs from the conventions used in aerodynamics and wind tunnel testing in which x is aft and z is upward. Thus, drag acts in the negative x direction when the angle of attack is zero. The actual definition of the coordinate directions is up to the user, but generally, the fuselage reference line is used as the direction of the x axis. The rotation rates p, q, and r are measured about the x, y, and z axes respectively using the conventional right hand rule and velocity components u, v, and w are similarly oriented in these body axes.

Basic Concepts

Stability is the tendency of a system to return to its equilibrium condition after being disturbed from that point. Two types of stability or instability are important.

A static instability | A dynamic instability |
---|---|

An airplane must be a stable system with acceptable time constants. To assure
this, a careful analysis of the dynamic response and controllability is
required, but here we look only at the simplest case: static longitudinal
stability and trim. This will tell us something about the aerodynamic design
of the surfaces - the load they must carry, the effect of airfoil properties,
and the drag associated with the surfaces.

In general we want the dynamics to be acceptable, actually more than just
stable we need appropriate damping and frequency. The dynamic equations
of motion are shown below, expressed in body axes. The top six equations
are just forms of F=ma and M=I d omega / dt for each of the coordinate directions.
The bottom three equations are kinematic expressions relating angular rates
to the orientation angles Q, F, and Y, angles describing the airplane pitch,
roll, and heading angles.

In general, we must solve these nonlinear, coupled, second order differential
equations to describe the dynamics of the airplane. Many simplifying assumptions
are often justified and make the analysis more simple.

Nearly independent modes: lateral, longitudinal.

Note that this doesn't always work: large sideslip -> Cm, large alpha
-> Cn, asymmetric a/c.

1. 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*, Cm, become negative as we increase CL:

Note that:

where x is the distance from the system's center of additional lift 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 CL.

This implies that:

With a single wing, generating a sufficient Cm at zero lift to trim with a reasonable static margin and CL is not so easy. (Most airfoils have negative values of Cmo.) Although tailless aircraft can generate sufficiently positive Cmo 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. CL or angle of attack, we can say a great deal about the airplane's characteristics.

examples

3. Cm equation, trim

The change in pitching moment with angle of attack, dCm/ dalpha , is called the pitch stiffness. The change in pitching moment with CL of the wing is given by:

Note that :

The position of the c.g. which makes Cm/CL = 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 Cmc.g. with respect to CLA, 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, CLah, 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:

CLah0 is the isolated tail lift curve slope.

etah 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: CLaw is the wing lift curve slope per radian, Lf is the fuselage length, wf is the maximum width of the fuselage, and 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 following data were taken from TR540 and Aerodynamics of the Airplane by Schlichting and Truckenbrodt:

Position of 1/4 root chord on body Kf in percent body length

.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. This is ignored here but further details may be found in the listed references.

4. contribution of different parameters a. wing (note c.g. importance - don't need a tail.) b. tail c. fuse, nacelles

5. tail efficiency terms

6. stick-fixed vs. stick free

7. tail volume definition

8. aeroelastics

9. definition of neutral point and static margin

Older aircraft, such as the DC9, were designed to be 3% stable in the worst case at aft center of gravity. Newer aircraft with active control systems, such as the MD11, may be slightly unstable, -5% static margin. For future aircraft, McDonnell Douglas is looking at aircraft that are -10% to -15% stable.

Contributions: Original DC-9 Design (At aft c.g. xcg = .4c rigid airplane):

Wing+Fuse Cma = +.021

Wing+Fuse+Pylons+Nacelles Cma = +.015

Wing+Fuse+Tail Cma = -.0143

Complete Airplane Cma = -.0157

Note that nacelles and pylons are quite stabilizing when added to the fuselage+wing, but only slightly stabilizing when added to the fuse+wing+tail. This is due to the large downwash on the tail from the nacelle/pylon combination. This airplane has a rigid sm of about .175 at aft cg

DC-9-10 revised tail due to deep stall: Cma = -.0257, sm = .285

This would mean a static margin of about .435 with the cg at 25%MAC.

At forward c.g. (14%), the static margin is then .545!

For the DC-9-30 the forward c.g. limit was even farther forward (about 6%)! The DC-9-30 has an a.c. at 71% for the rigid airplane. This corresponds to a static margin of 65% at fwd cg, 31% at aft c.g.. Not too different from the DC-9-10.

Why is this so large?

The effect of Mach number is small, moving the a.c. from 71% at M=0 to 68% at M=.75.

But aeroelastics effects are huge. At a Mach number of .75 at 30000 ft, the dynamic pressure is about 250 psf. At Mach .8 at 20,000 ft, the q is: 440 psf. At 450 psf, the horizontal tail Cla is 83% of the rigid value, excluding fuselage bending. Fuselage bending reduces the tail load to 58% of its normal value. There is also about a 4% shift in tail-off static margin.

Now for the DC-9-10, the tail provides a static margin increment of about (Cma = -.0407) sm = .452. Now if this were reduced to 55% of that, the increment would be sm = .249. Thus, the aft c.g. static margin would be: .31-.452+.249 = .107. With the 4% shift tail-off, we are left with a static margin of about 6.7% in the worst condition.

The airplane typically flys at a q of 250 psf where the aeroelastic effects produce a TO sm change of 2% and 72% of the tail force due to fuse bending. Using 70% to include tail bending, the aft c.g. static margin becomes about 15%. At more typical c.g. locations a typical cruise sm is then 30%. So, aeroelasticity is a big effect! It is even bigger for stretched DC-9's.

Note, however, that in terms of aircraft trim drag, the required tail load is relatively unaffected by fuselage bending. Thus, although the DC-9 typically flys at a reasonable 30% static margin, it has the same trim drag as a rigid airplane with a static margin of 46%!

2. decoupling yields simplified solutions:

a. longitudinal

i. short period: alpha,theta goes away with high enough damping

ii. phugoid: u, h, to flee, constant alpha, slow 20-60 secs.

b. lateral

i. roll subsidence mode

ii. spiral mode

iii. Dutch roll

iv. effect of dihedral, v. tail, sweep