# Stability and Control

## Introduction

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.

## Definitions

The following nomenclature is common for discussions of stability and control.

Forces and Moments
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

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.

## 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 instabilityA 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:

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.

## Aeroelastics

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% 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%!

## Dynamic Stability

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

## 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 4. correlations 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.