Stability and Control
The methods in these notes allow us to compute the overall aircraft drag.
With well-designed airfoils and wings, and a careful job of engine and fuselage
integration, L/D's near 20 may be achieved. Yet some aircraft with predicted
L/D's of 20 have actual L/D's of 0 as exemplified by any paper airplane
contest. Many aircraft have been dismal failures even though their predicted
performance is great. In fact, most spectacular failures have to do with
stability and control rather than performance.
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.
The following nomenclature is common for discussions of stability and control.
Forces and Moments
|Quantity ||Variable ||Dimensionless Coefficient ||Positive
|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 |
Angles and Rates
|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.
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
|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.
Longitudinal Static Stability
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:
where x is the distance from the system's center of additional lift to the
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.
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
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:
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
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
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
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%
Some interesting numbers for the DC-9-30: ac at 71% (rigid)
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%
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%!
1. Return to discussion of types of motion: damping and frequency
2. decoupling yields simplified solutions:
i. short period: alpha,theta goes away with high enough damping
ii. phugoid: u, h, to flee, constant alpha, slow 20-60 secs.
i. roll subsidence mode
ii. spiral mode
iii. Dutch roll
iv. effect of dihedral, v. tail, sweep
Longitudinal Control Requirements
Notes from a phone interview with Mark Page of McDonnell Douglas, 2 November 1993, 12:55pm to 1:30pm.
Notes taken by Sean Wakayama.
Control power is usually critical in sizing the tail. The MD12 is cruise trim critical. Tail sweep and thickness are selected from a plot similar to plots in Shevell's notes. The tail is sized to be buffet free or below drag divergence at dive Mach number. Drag divergence is used as a measurement of likelihood of elevator control reversal. Drag divergence is accompanied by strong shocks on the suction side of the stabilizer. Deflecting the elevator to diminish lift in this condition can improve the flow behind the shock, increasing lift instead of reducing it and causing a control reversal. Typically the tail would be designed to be below drag divergence at dive Mach number and at its mid center of gravity cruise lift coefficient, a lift coefficient of 0.2 to 0.3. On the MD12 in cruise, the tail carries almost no load at mid CG, positive load at aft CG, and negative load at forward CG. In this case the tail is probably designed to be divergence free at dive Mach number and at its worst cruise lift coefficient. Control requirements at low speed are usually critical. One requirement that determines the elevator sizing is a go around maneuver. The airplane begins in approach trim, flaps down, stabilizer set for 1g flight, no elevator. By deflecting the elevator only, the pilot should be able to get a pitch acceleration of 5¡/s^2, minimum. On new aircraft with no stretch history, the elevator would be designed to provide 10¡/s^2 pitch acceleration. 8¡/s^2 is desirable. Nosewheel liftoff may be a critical constraint, especially on advanced aircraft because of a trend toward moving the center of gravity aft relative to the aerodynamic center. In this maneuver, the aircraft is trimmed for climbout at V2 + 10 knots, which is about 1.3 Vstall. The elevator should generate enough moment to crack the nosewheel off the ground and provide 3¡/s^2 pitch acceleration. In designing the tail, one would shoot for 6¡/s^2 pitch acceleration. The approach trim constraint is often critical. This constraint involves a 1g level acceleration from approach speed, 1.3 Vstall, to maximum flaps extended speed, VFE, which is typically 1.8 Vstall. The aircraft begins in approach trim and must be reach VFE using only the elevator, not the stabilizer, to retrim. In approaching VFE, the angle of attack decreases and must be accompanied by deflecting the elevator down. For trim at 1.3 Vstall, however, the stabilizer is deflected up to generate download. At VFE, the stabilizer and elevator end up working against each other. At this condition, the tail must be 2¡ below stall. Icing affects estimation of maximum section lift. With evaporative anti icing systems the properties of the clean section can be used. For aircraft without ice protection, the tail should be oversized by as much as 30%. At VFE, it is common for the wing flap to be stalled. Because of the low angle of attack, there is no flow through the wing slat. Flow separates on the lower surface of the slat, and this disturbance impinges on the flap causing it to stall. Takeoff normally does not stall the tail. The elevator typically has a limited throw. This usually keeps the tail within 2¡ of its stall angle of attack. The DC9 tail has a maximum stabilizer deflection of about 12¡ and a maximum elevator deflection around 25¡. Pitching moments from landing gear are usually small and act opposite to one's intuition. The gear struts block the flaps and reduce their nose down pitching moment. The gear also cause a slight increase in lift. Structural sizing for fins are often set by a tail stop maneuver. Pilot applies a maximum rudder input, limited by either a pedal stop or a mechanical stop in the fin. The airplane sideslips and is carried by its inertia beyond its equilibrium sideslip angle. From the maximum equilibrium sideslip, the pilot releases the pedals causing the airplane to swing back and oscillate around zero sideslip. The maximum fin loads encountered during this maneuver are used to size the fin structure. For this reason, McDonnell Douglas uses rudder throw limiters that provide full deflection, typically ±30¡, up to 160 knots, then decrease maximum deflection inversely proportional with dynamic pressure.
Lateral Control Requirements Notes from a phone interview with Mark Page of McDonnell Douglas, 4 November 1993, 10:35am to 10:50am. Notes taken by Sean Wakayama. For older and current aircraft up through the MD12, stability requirements such as Dutch roll were an issue in sizing the vertical tail. In these aircraft, despite the presence of active control systems, the design philosophy was that the aircraft should be flyable with all electrons dead. An alternate philosophy is to examine how much reliance is placed on the control system and estimate the number of failures expected based on statistical data on failure rates. Control systems would then be designed with sufficient redundancy to achieve two orders of magnitude more reliability than some desired level. The alternate philosophy that trusts active control will be used for future advanced aircraft design work; it will have to be used in any HSCT design. Some basic control will still be available even without active control in that pitch trim and rudder will still be mechanically activated. In the future, vertical tails will not be sized for Dutch roll, so long as the control system has sufficient authority to stabilize the airplane. There is a limit to the instability that can be tolerated; the control system cannot be saturated. For this purpose, the rudder should be designed to return aircraft from a 10¡ sideslip disturbance at any altitude. For reliability, rudders will be split into upper and lower halves, with independent signals and actuators plus redundant processors. The critical control sizing constraint is VMCG, minimum controlled ground speed. In this condition, flight is straight and unaccelerated laterally. Nose gear reaction is zero. Aerodynamic moments must balance engine thrust with one engine out and creating windmilling drag, and the other engine at max thrust plus a thrust bump for a "hot" engine. If the moment balance is done about the aircraft center of gravity, main gear reactions caused by rudder sideforce must be considered. If the main gear reactions were ignored, rudder force would be underestimated by 15% to 20%. Alternately, the moment balance can be done about the main gear center, which lies in line with the gear and halfway between them. Engine thrust imbalance should be controllable with full rudder deflection. VMCG is relatively independent of flap setting or aircraft weight because it is primarily a matter of balancing engine thrust imbalance with the rudder. Flaps may affect rudder performance sometimes because of aerodynamic interaction. Aircraft weight does not enter the moment balance because, when moments are taken about the main gear, there are no ground moment reactions and there are no inertial forces because there is no lateral acceleration. The engine thrust imbalance is constant because full thrust is always used for takeoff, regardless of aircraft weight. To determine a required VMCG speed, one would examine an aircraft in its lightest commercial weight. This would be the weight with a minimum passenger load to break even on a particular range, say a 30% passenger load. At low takeoff weights, more flaps will be used as a result of optimizing flap deflection for best lift to drag in second segment climb. The light weight and large flap deflection should reduce speeds for second segment climb and rotation. In establishing the balanced field length for this condition, VMCG should be set at the speed where second segment climb or rotation becomes critical. For the DC9 or DC10 this speed is about 110 knots. For heavier aircraft such as the MD12, VMCG is higher, 120 knots. VMCA, minimum control airspeed, is usually not critical because dynamic pressure is higher, making the rudder more effective, the thrust imbalance is smaller, because of thrust lapse, plus the airplane is allowed to sideslip to trim. The VMCG condition is at zero sideslip; rudders are double hinged to enable large lift coefficients to be achieved on the fin at this condition. While VMCG is critical for 2 engine airplanes, on 4 engine airplanes VMCL2 may be critical. In this landing condition, 2 engines are out on same side of the airplane while the other two are at max takeoff thrust. The rudder is more effective since this is done at airspeed approach speed, 1.3 Vstall. One airborne condition that might size the rudder is a crosswind landing decrab. This condition is at 1.3 Vstall with a 35 knot crosswind. The rudder is used to control an aerodynamic sideslip of 13¡ to 15¡. This is critical on the C-17. Increasing the vertical tail area does not help here because it increases the resistance to sideslip. If this condition is critical the proportion of rudder to vertical tail area should be adjusted.
Tail Design 1. Tail penalties: wetted area, CLmax, structural weight, induced drag
2. scissors curve
3. control requirement method see handout for detail
5. other parameters
a. AR 4, 1.2-1.8
b. taper .4-.6
c. sweep +5¡
d. tail configuration T, V, H, + , Y, ^
e. canards a very short discussion
Tail should have lower thickness and/or higher sweep than the wing (about 5¡ usually) so that we do not get strong shocks on the tail in normal cruise.
Typical aspect ratios are about 4. Maybe 5 for T-Tails.
ARv is about 1.2 to 1.8 based on exposed area - lower values for T-Tails.
Taper ratios are about .4 to .6 except for T-Tail verticals.
It is now possible to compute the required tail size for a given stability level as a function of c.g. position. The procedure is as follows:
For c.g. positions ranging from the leading edge of the M.A.C. to about 60% of the M.A.C. compute and plot the required tail volume coefficient, for the desired level of static stability. The minimum static margin would typically be about .10 but it must be increased because bending of the wing and the fuselage at high speeds reduces the rigid airplane stability. ( Assume sm = -.10 for swept wing transport aircraft. sm due to aeroelasticity can usually be neglected in preliminary design of general aviation aircraft.) In addition, the desired static margin may be increased by about .10 for T-tail airplanes to improve high angle of attack stability.
In order to compute the required tail volume, you will need to find the distance from the c.g. to the wing a.c.. The position of the wing a.c. may be computed using the program Wing that was used in a previous assignment. The lift curve slope of the isolated tail and wing may also be computed using this program.
The second requirement for the horizontal tail is that it provide sufficient control power. It must not only be possible to trim the airplane in cruise but also in more critical conditions. Typical critical conditions include:
Rotation and nosewheel lift-off on take-off at forward c.g., trim without tail stall at maximum flap extension speed, and trim at forward c.g. with landing flaps at CLmax.
For this exercise we will consider only the problem of take-off rotation. We assume that the tail incidence and elevator angle settings are such that the horizontal tail can achieve a certain maximum lift coefficient CLHmax (in the downward direction). The force required from the tail to rotate the airplane depends on the wing and body pitching moments to some extent but largely on the weight moment about the rear wheels.
At aft c.g. the force is smallest, but a certain amount is required since the c.g. must lie in front of the rear wheels to prevent the airplane from tipping over on its tail. Actually, the requirement is not so much to avoid tipping backward but rather providing sufficient weight on the nosewheel to permit acceptable traction for steering. This is satisfied with about 8% of the weight on the forward wheels. With this load on the forward wheels, the moment about the rear wheels due to the forward position of the c.g. is at least: |M| = .08 lg W where lg is the distance from the main gear to the nose gear.
The pitching moment coefficient at take-off is then:
We will ignore the aerodynamic term for now, although a detailed study would include this. For rotation, then, the load on the tail must be:
The minimum tail volume required can then be calculated with the assumed CLHmax. (For airplanes with variable incidence stabilizers and elevators CLHmax = 1.0 will be an acceptable estimate.)
At forward c.g. positions, a larger tail is required since the moment about the rear wheels is:
M = Maft-c.g. + W c.g.
(Note that c.g. is the c.g. range. It is not the static margin, discussed earlier.)
The required tail volume may be determined from this analysis at the forward c.g. position (c.g. is determined after a detailed weight and balance calculation but may be estimated from the figure in the AA241 notes.) It may be interesting to compare your results with the statistical method in the 241 notes. Also note that we have previously estimated the main gear position at 50% of the MAC. If we desire 8% of the load on the nose gear at aft c.g. this means that the main gear must be located .08 lg behind the aft c.g.