Category Airplane Stability and Control, Second Edition

Trailing-Edge Angle and Beveled Controls

The included angle of upper and lower surfaces at the trailing edge, or trailing- edge angle, has a major effect on control surface aerodynamic hinge moment. This was not realized by practicing stability and control engineers until well into the World War II era. For example, a large trailing-edge angle is now known to be responsible for a puzzling rudder snaking oscillation experienced in 1937 with the Douglas DC-2 airplane. Quoting from an internal Douglas Company document of July 12, 1937 (The Museum of Flying, Santa Monica, California), by L. Eugene Root:

The first DC-2s had a very undesirable characteristic in that, even in smooth air, they would develop a directional oscillation. In rough air this characteristic was worse, and air sickness was a common complaint. …It was noticed, by watching the rudder in flight, that during the hunting the rudder moved back and forth keeping time with the oscillations of the airplane.

It is common knowledge that the control surfaces were laid out along airfoil lines. Because of this fact, the rearward portion of the vertical surface, or the rudder, had curved sides. It was thought that these curved sides were causing the trouble because of separation of the air from the surface of the rudder before reaching the trailing edge. In other words, there was a region in which the rudder could move and not hit “solid” air, thus causing the movement from side to side. The curvature was increased towards the trailing edge of the rudder in such a way as to reduce the supposedly “dead” area. . . . The change that

we made to the rudder was definitely in the wrong direction, for the airplane oscillated severely…. After trying several combinations on both elevators and rudder, we finally tried a rudder with straight sides instead of those which would normally result from the use of airfoil sections for the vertical surfaces. We were relieved when the oscillations disappeared entirely upon the use of this type of rudder.

The Douglas group had stumbled on the solution to the oscillation or snaking problem, reduction of the rudder floating tendency through reduction of the trailing-edge angle. Flat­sided control surfaces have reduced trailing-edge angles compared with control surfaces that fill out the airfoil contour. We now understand the role of the control surface trailing – edge angle on hinge moments. The wing’s boundary layer is thinned on the control surface’s windward side, or the wing surface from which the control protrudes. Conversely, the wing’s boundary layer thickens on the control surface’s leeward side, where the control surface has moved away from the flow. Otherwise stated, for small downward control surface angles or positive wing angles of attack the wing’s boundary layer is thinned on the control surface bottom and thickened on the control’s upper surface.

The effect of this differential boundary layer action for down-control angles or positive wing angles of attack is to cause the flow to adhere more closely to the lower control surface side than to the upper side. In following the lower surface contour the flow curves toward the trailing edge. This curve creates local suction, just as an upward-deflected tab would do. On the other hand, the relatively thickened upper surface boundary layer causes the flow to ignore the upper surface curvature. The absence of a flow curve around the upper surface completes the analogy to the effect of an upward-deflected tab. The technical jargon for this effect is that large control surface trailing-edge angles create positive values of the derivatives Cha and Chs, the floating and restoring derivatives, respectively.

The dynamic mechanism for unstable lateral-directional oscillations with a free rudder became known on both sides of the Atlantic a little after the Douglas DC-2 experience. Unstable yaw oscillations were calculated in Britain for a rudder that floated into the wind (Bryant and Gandy, 1939). This was confirmed in two NACA studies (Jones and Cohen, 1941; Greenberg and Sternfield, 1943). The aerodynamic connection between trailing-edge angle and control surface hinge moment, including the floating tendency, completed the story (Jones and Ames, 1942).

Following the success of the flat-sided rudder in correcting yaw snaking oscillations on the Douglas DC-2, flat-sided control surfaces became standard design practice on Douglas airplanes. William H. Cook credits George S. Schairer with introducing flat-sided control surfaces at Boeing, where they were used first on the B-17E and B-29 airplanes. Trailing – edge angles of fabric-covered control surfaces vary in flight with the pressure differential across the fabric (Mathews, 1944). A Douglas C-74 transport was lost in 1946 when elevator fabric bulging between ribs increased the trailing-edge angle, causing pitch oscillations that broke off the wing tips. C-74 elevators were metal-covered after that.

Understanding of the role of the trailing-edge angle in aerodynamic hinge moments opened the way for its use as another method of control force management. Beveled control surfaces, in which the trailing-edge angle is made arbitrarily large, is such an application (Figure 5.8). Beveled control surfaces, a British invention of World War II vintage, work like balancing tabs for small control surface angles.

The beveled-edge control works quite well for moderate bevel angles. As applied to the North American P-51 Mustang, beveled ailerons almost doubled the available rate of roll at high airspeeds, where high control forces limit the available amount of aileron deflection.

Trailing-Edge Angle and Beveled Controls

Figure 5.8 Typical hinge moment parametervariation with bevel angle ф for a beveled control surface. (From Phillips, NACA Rept. 927, 1948)

But large bevel angles, around 30 degrees, acted too well at high Mach numbers, causing overbalance and unacceptable limit cycle oscillations (Figure 5.9). Beveled controls have survived into recent times, used for example on the ailerons of the Grumman/Gulfstream AA-5 Tiger and on some Mooney airplanes.

Nonelectronic Stability Augmentation

Really ingenious nonelectronic stability augmentation systems came out of the jet’s awkward age, as designers tried to have artificial damping without the heavy, costly, and, above all, unreliable electronics of the period. A mechanical yaw damper, invented by Roland J. White and installed on early Boeing B-52 Stratofortresses, is a good example of the genre.

Imagine a rudder tab that is free to rotate on low friction bearings. Instead of being connected to an electric actuator, or to cables leading to the cockpit, the free tab is driven by

Nonelectronic Stability Augmentation

Figure 7.1 Boeing B-52 rudder control linkages. R. J. White’s magnetically phased bobweight yaw damper operates the stability tab. (From B-52 Training Manual, 1956)

inertia forces acting on a small bobweight located ahead of the hinge line (Figure 7.1). Tab position is further modified dynamically by an eddy current damper, providing damping hinge moments proportional to tab rotational velocity.

As the airplane goes through a typical lateral or Dutch roll oscillation, the vertical tail assembly swings from side to side, accelerating the tab bobweight. Without the eddy current damper it is clear that the tab will take up deflections in phase with the lateral accelera­tion at the vertical tail. However, ideally, tab positions should be phased with respect to yawing velocity in such a way as to drive the rudder in opposition. This is the classic yaw damping action, right rudder in opposition to left yawing velocity. The function of the eddy current damper is to “tune” tab deflections to create exactly that phasing. In 1952, a similar approach was taken by M. J. Abzug and Hans C. Vetter of the A3D Skywarrior design team at Douglas Aircraft, to provide nonelectronic yaw damping for that airplane. The design method was cut and try on the analog computer, to find the proper combina­tion of bobweight mass and damper size that would phase the tab, creating effective yaw damping.

The obvious practical problem with the B-52 and A3D yaw dampers is one that is faced with any purely mechanical system, as compared with a modern electromechanical control system. In the mechanical system, the result or output depends critically on the condition of each component. If the free tab’s bearings deteriorate over time or are invaded by grit, or if the eddy current damper’s effectiveness is changed, tab phasing will be thrown off.

In the extreme case, tab action could actually add to the airplane’s lateral oscillation, instead of damping it.

In a July 1994 letter Roland White describes such a situation that actually occurred on a B-52, as follows:

A rudder tail shake on a test airplane caused the magnetic damper to lose its damping. A serious accident would have occurred if the bobweight did not jam due to a mechanical failure. After that I found when going to work the next day your friends will ask if you still work here.

A modern, electromechanical yaw damper drives the rudder in opposition to the measured rate of yaw. It does so by comparing the current rudder position with the desired value and continuing to exert torque on the rudder until that value is reached, overriding mechanical obstacles such as sticky bearings or even losses in performance of the motor that drives the rudder.

The practical shortcomings of purely mechanical yaw damping were not unknown to the Boeing and Douglas design staffs. When a chance appeared to get a yaw damper function electronically, that option was taken instead. In the case of the B-52, the spring-tab – controlled rudders were replaced by powered rudders, allowing Boeing to use the electro­mechanical yaw damper design developed successfully for the B-47.

In the Douglas case, electromechanical yaw damping was installed using components of the airplane’s well-proved Sperry A-12 automatic pilot. The Sperry Gyroscope Company’s DC-3 “dogship” proved the concept in test flights at the Sperry plant in Long Island, New York. Signals from the outer, or yaw, gimbal of the A-12’s free directional gyro were elec­tronically differentiated through a lead network and sent to the rudder servo. Differentiated yaw angle is of course yaw rate.

This worked well when the system was transferred to the A3D and flown routinely at Edwards Air Force Base. Then one day a test pilot bringing an A3D back for landing dove at the runway and pulled up into a chandelle, a natural thing to do for a high-spirited test pilot with an airplane he likes. The A3D, with yaw damper on, responded by applying bottom rudder during the nearly vertical bank, diving the ship back toward the ground.

The pilot regained control and an investigation started at once. The A-12 and yaw damper function were found to be in perfect order. The culprit turned out to be what had been called for years “gimbal error.” The A-12 directional gyro is a conventional two-gimbal free gyro, with yaw measured on the outer gimbal. The rotor, spinning in the inner gimbal, is slaved to magnetic north and the inner gimbal itself is erected to gravity by a bubble level system. The angle between the outer gimbal and the instrument’s case is true yaw or heading angle as long as the outer and inner gimbals are at right angles to each other. This holds only for zero bank angle. At the sharp bank angles of the chandelle, or in any steep turn, the yaw reading picks up errors that depend on the heading angle (Figure 7.2).

During turns, differentiation with respect to time of the erroneous yaw angle exagger­ates the ordinary gimbal errors. The A3D experience proves dramatically that one cannot in general differentiate free gyro signals to produce damping signals for stability augmen­tation, at least for airplanes that maneuver radically. After the all-mechanical and free-gyro A3D yaw damper designs were proved faulty the airplane was finally fitted with what is now the standard design, a single-degree-of-freedom yaw rate gyro driving the rudder servo.

A rather more successful nonelectronic stability augmentation system was developed at the Naval Weapons Center, China Lake, for the AIM-9 Sidewinder missile. The Sidewinder

Nonelectronic Stability Augmentation

Figure 7.2 Gimbal angles of the outer gimbal of the Sperry A-12 directional gyro, as a function of bank and yaw angles. The outer gimbal rate fluctuates strongly in turns at steep bank angles. Differentiating outer gimbal angle to obtain yaw rate caused a near-crash of a Douglas A3D-1 Sky Warrior. (From Abzug, Jour. of the Aero. Sciences, July 1956)

derives roll damping from nonelectronic, air-driven flywheels mounted at the tips of the missile’s ailerons, producing gyroscopic torques that drive the ailerons to oppose roll rate. The flywheel torques are evidently high enough to override variations in aileron bearing friction. There seems to have been no application of this all-mechanical damping system to airplanes.

Air-to-Air Missile-Armed Fighters

A price has to be paid for extreme rolling performance in terms of demands on hydraulic system size and flow rate and on structural weight required for strength and stiffness. This led to a new controversy. As in the days of P-40s versus Zeros, high roll rates were important in dogfight gun-versus-gun battles.

But what about fighters that merely fired air-to-air missiles? Sparrow I and Sidewinder air-to-air missiles both went into service in 1956. Clearly, the missiles themselves can do the end-game maneuvering, to veer left and right, climb and dive, following any feints by the airplane being attacked. Penalizing missile-armed fighters so that they could carry out dogfight tactics might be as foolish as it would have been to require Army tank crews to wear cavalry spurs.

The drive to reduce fighter airplane rolling requirements because of the advent of missile­armed fighters was led on the technical side by a former NACA stability and control engineer who had risen to a high administrative level. The then USAF Director of Requirements weighed in with a letter stating flatly that the F-103 would be the last USAF manned fighter airplane.

The need for high levels of fighter airplane rolling performance was argued back and forth at Wright Field and the Naval Air Systems Command until the issue was settled by the Vietnam War of 1964-1973. U. S. fighters went into that conflict armed with both Sparrow and Sidewinder air-to-air missiles. Nevertheless, they found themselves dogfighting with Russian-built fighters. The reason that aerial combat was carried out at dogfighting ranges was that visual target identification and missile lock-before-launch doctrines were found to be needed, to avoid missile firings at friendly targets. Ranges for positive visual identification were so small that engagements quickly became dogfights. High roll rates were once more in favor. Of course, dogfighting capability meant that guns could still be used effectively on missile-carrying fighters.

Ultralight and Human-Powered Airplanes

The category of ultralight airplanes ranges from hang gliders to light versions of general-aviation airplanes. They fill a need for experimenters and for pilots who want to fly inexpensively and with little regulation. Ultralight airplanes evolved as did the early flying machines, by much cut-and-try and flight testing. Although these designs have been useful, indications are that many commercial ultralights are deficient in stability and control.

Human-powered airplanes are extreme ultralights, designed not for practicality but to push the engineering and human limits of aviation. Early efforts at human-powered flight were discouraging because of the poor performance and extreme fragility of the machines that were constructed before the first successful one, the Gossamer Condor.

13.1 Apparent Mass Effects

For very light airplanes, not much heavier than the air in which they fly, apparent mass effects must be considered. These effects were first noticed in 1836 by George Green, who found that pendulum masses in a fluid medium were apparently greater than in a vacuum. The apparent mass effect can be described as follows (Gracey, 1941):

The apparent increase in mass can be attributed to the additional energy required to establish the field of flow about the moving body. Inasmuch as the motion of the body may be defined by considering its mass as equal to the actual mass of the body plus a fictitious mass, the effect of the inertia forces of the fluid may be represented as an apparent additional mass; this additional mass, in turn, may be considered as the product of an imaginary volume and the density of the fluid. The effect of the surrounding fluid has accordingly been called the additional mass effect. The magnitude of this effect depends on the density of the fluid and the size and shape of the body normal to the direction of motion.

The primary motivation for Gracey’s work was to be able to correct airplane and wind – tunnel model moments of inertia measured by suspending the airplanes or models and swinging them aslarge pendulums. To the extent that the NACA wasinvolved with equations of motion for the airships of those days, this would have given Gracey yet another motivation to study apparent mass.

The 1941, the NACA apparent mass tests were made by swinging covered frameworks of various shapes as compound pendulums. The test specimens were swung both in air and in a vacuum tank. It is interesting that Gracey started out with balsa wood shapes, but found that their weights varied with air pressure and humidity. Gracey’s training in this exacting experimental work must have helped him to appear later on as NASA’s expert in airspeed and altitude measurement methods.

Interest in apparent mass effects returned with the advent of the plastic and fiber materials that could be used to build very light airplanes, such as the human-powered Gossamer Condor and the high-altitude, long-duration pilotless airplanes Pathfinder and Helios, all built by Aero Vironment, Inc., of Monrovia, California. Apparent mass effects are important as well for lighter-than-air and for underwater vehicles. Mathematical models of these craft for dynamic stability analysis include apparent mass terms, as a matter of course. In the series expansions for aerodynamic forces and moments originated by G. H. Bryan, apparent mass terms appear as derivatives with respect to linear and angular accelerations.

Lacking the vacuum swinging apparatus of Green and Gracey, one can approximate apparent mass terms in the equations of airplane motion by adding cylindrical air masses to the lifting surfaces, with diameter equal to the surface chord for motions normal to the chord and equal to the surface thickness for motions in the chord plane. This approximation yields the following astonishing results for the Gossamer Condor. The apparent masses in lateral and vertical motions are 21 and 170 percent of the actual airplane mass. The apparent moments of inertia in pitch and roll are 140 and 440 percent of the actual moments of inertia.

In addition to measurements on swinging models and the approximations mentioned above, panel computer codes can be used for apparent mass estimation. David A. Lednicer reports that the VSAERO code is used routinely for apparent mass calculations on under­water vehicles.

The Rotation-Only Breakthrough

The rotation-only concept for variable sweep was pioneered by Dr. Barnes Wallis at Vickers-Armstrongs, Weybridge, around 1954. Starting in 1959, brilliant work by a NASA Langley Laboratory team, including Dr. Wallis, made variably swept wings a practical design option. Team members William J. Alford, Jr., Edward C. Polhamus, and Wallis found a practical way to eliminate the translation, or fore-and-aft motion of the wing inboard ends, drastically simplifying the variable-sweep rotation/translation mechanism to rotation alone.

The clue was to pivot the wings well out from the airplane centerline and to bring the wing trailing edges when fully swept parallel and close to the horizontal tail leading edges. In the Alford-Polhamus-Wallis design, the wing pivots are on the outboard ends of a glove, a diamond-shaped, highly swept inboard fixed-wing section. Wing spanwise loads are carried primarily on the outboard or unswept panels when the wings are in the forward position. The wing’s spanwise load shifts relatively inboard, to the glove, when the wings are in the aft position. This relative load shift is exactly what one wants in order to minimize movement of the total wing aerodynamic center when the wings go through its sweeping routine (Loftin, 1985). Alford and Polhamus jointly hold the U. S. patent on this design.

An additional benefit of the Alford-Polhamus-Wallis arrangement arises from downwash changes with wing sweep. Bringing the wing trailing edge close to the horizontal tail drastically increases the downwash rate of change with angle of attack, reducing the tail’s stabilizing effect. That is, the tail’s increasing up-load with increasing angles of attack is reduced. In effect, the wing acts as a huge turning vane, aligning with itself the airflow into the horizontal tail. Reduced stability from the horizontal tail is just what is needed when the wing is swept back by rotation alone.

Another way of thinking of the Alford-Polhamus-Wallis arrangement is to consider the horizontal tail as an extension of the wing when the latter is fully swept back. Surface area at the rear of a lifting surface carries a smaller airload than does the same amount of area as an independent lifting surface. The lower airloads on the horizontal tail result in reduced static longitudinal stability, again just what is needed.

Additional Special Forms of the Equations of Motion

Trajectory or point-mass equations of airplane motion, lacking the torque or mo­ment equations, have been found useful for flight performance studies. In these applications, angles of attack and sideslip are assumed functions of time or are found in simple closed loops, instead of being the result of attitude adjustments influenced by control surface an­gles. Trajectory equations of motion have only 6 nonlinear state equations, as compared with 12 for the complete rigid-body equations. The savings in computer time are unimpor­tant with modern digital computers, but there is a conceptual advantage for performance studies in needing to specify only lift, drag, and thrust parameters.

Another special form of the equations of airplane motion puts the origin of body axes at an arbitrary location, not necessarily the center of gravity. The first use of such equations seems to have been for fully submerged marine vehicles, such as torpedoes and submarines. With the center of body axes at the center of buoyancy, there are no buoyancy moment changes due to changes in attitude (Strumpf, 1979). An equivalent set for airplanes came later (Abzug and Rodden, 1993).

Apparent mass and buoyancy terms in the equations of airplane motion are discussed in Chapter 13, “Ultralight and Human-Powered Airplanes.” The various special forms of the equations of airplane motion for representing aeroelastic effects are discussed in the next chapter, “The Elastic Airplane.”

Equationsof motion for an airplane with an internal moving load that isthen dropped were developed by Bernstein (1998). The motivation is the parachute extraction and dropping of loads from military transport airplanes. A control strategy using feedback from disturbance variables to the elevator was able to minimize perturbations in airplane path and airspeed during the extraction and dropping process.

The Advent of Digital Stability Augmentation

Airplane digital fly-by-wire flight control systems, which make possible digital stability augmentation, go back to the 1970s. Priority is difficult to establish, since many organizations were doing this work at about the same time. One early application was at the NASA Dryden Flight Research Center, using digital flight hardware from the Apollo program. Although overdesigned in many ways for the airplane application, it made possible an early demonstration of the possibilities of airplane digital augmentation.

That program used a Vought F-8C airplane (Jarvis, 1975). The first step was to fly single-channel digital flight control systems on the F-8C, with backup analog controls in case of failure. The next step was a big one from the standpoint of system complexity: the development of a triplex digital system, using redundancy management and data bus concepts. The subsequent routine use in modern airplanes of redundant, fail-operational digital flight control and stability augmentation is at least partially the result of this early NASA effort.

Another early application was the quadruplex redundant digital fly-by-wire system flown in the BAe FBW Jaguar. Design commenced in the late 1970s, and it flew between 1981 and 1984 in configurations ranging from normal to highly unstable. The BAe FBW Jaguar technology led to the EAP (Experimental Aircraft Programme) and ultimately to the Eurofighter.

Two Famous Airplanes

NACA measured the flying qualities of the Supermarine Spitfire VA fighter in 1942 and the Douglas DC-3 transport in 1953, both at the Langley Laboratory. These airplanes had been built in large numbers, had served magnificently in World War II, and had inspired great affection among their pilots. Yet neither of these famous airplanes had the specified

Two Famous Airplanes

Figure 3.15 The venerable Douglas DC-3 exhibits static longitudinal instability in the normal rated power, clean configuration, at an aft center of gravity position of 25.5 percent MAC. (From Assadourian and Harper, NACA TN 3088, 1953)

level of the most basic stability of them all, static longitudinal stability, as measured by the elevator angles required for steady flight at various airspeeds. This form of stability is often called stick-fixed stability.

The Spitfire shows neutral stick-fixed stability under all flight conditions. The DC-3 is stable only in power-off glides or with cruise power. With normal rated power or in a power approach condition at aft loadings, increasing amounts of down elevator are needed as the airspeed is reduced, along with push column forces (Figure 3.15). For both airplanes there are other less striking deviations from NACA and military stability and control specifications. What should be made from all of this?

The Spitfire and DC-3 cases should not furnish an excuse to dismiss flying qualities requirements. It is reasonable to assume that if the Spitfire and DC-3 were longitudinally stable under all flight conditions, both of these fine airplanes would have been even better. In fact, the Spitfire Mark 22, developed at the end of the war, had a 27 percent increase in tail areas and flew “magnificently,” according to one account. The bottom line is that

nobody has ever found it feasible to run definitive, statistically valid experiments on the value of good flying qualities in terms of reducing losses in accidents or success in military missions. Instead, we rely on common sense. That is, it is highly plausible that good handling qualities in landing approach conditions will reduce training and operational accidents and that precise, light, effective controls will improve air-to-air combat effectiveness. That plausibility is essentially what energizes the drive for good flying qualities, in spite of apparent inconsistencies, such as for the Spitfire and DC-3.

Mechanical Control System Design Details

Connections between a pilot and the airplane’s control surfaces are in a rapid state of evolution, from mechanical cables or push rods, to electrical wires, and possibly to fiber optics. Push rod mechanical systems have fallen somewhat into disuse; flexible, braided, stainless steel wire cable systems are now almost universal. In an unpublished Boeing Company paper, William H. Cook reviews the mature technology of cable systems:

The multi-strand 7×19 flexible steel cables usually have diameters from 1/8 to 3/16 inch.

They are not easily damaged by being stepped on or deflected out of position. They are

usually sized to reduce stretch, and are much over-strength for a 200-pound pilot force.

The swaged end connections, using a pin or bolt and cotter pin, are easily checked. The

turnbuckles which set tension are safety-wired, and are easily checked. A Northwest Airlines early Electra crashed due to a turnbuckle in the aileron system that was not secured with safety wire wrap.

Since the cable between the cockpit and the control is tensioned, the simplest inspection is to pull it sideways anywhere along its length to check both the tension and the end connections. In a big airplane with several body sections this is good assurance. To avoid connections at each body section joint, the cable can be made in one piece and strung out after joining the sections. The avoidance of fittings required to join cable lengths also avoids the possibility of fittings jamming at bulkheads. Since the cable is rugged, it can be installed in a fairly open manner…. Deterioration of the cables from fatigue, as can happen in running over pulleys, or from corrosion, can be checked by sliding a hand over its length. If a strand of the 7×19 cable is broken, it will “draw blood.”

A recurrent problem in all mechanical flight control systems is possible rigging in reverse. This can happen on a new airplane or upon re-rigging an old airplane after disassembly. Mod­ern high-performance sailplanes are generally stored in covered trailers and are assembled only before flying. Sailplane pilots have a keen appreciation of the dangers of rigging errors, including reversals. Preflight checks require the ground crew to resist pilot effort by holding control surfaces and to call out the sense of surface motions, up or down, right or left.

A few crossed cable control accidents have occurred on first flights. The aileron cables were crossed for the first flight of Boeing XB-29 No. 2, but the pilot aborted the takeoff in time. Crossed electrical connections or gyros installed in incorrect orientations are a more subtle type of error, but careful preflight procedures can catch them, too.

Later Developments

An interesting inertial coupling development that came after the great rush of interest in the 1950s was the finding that moderate amounts of sideslip could add to the problem (Stengel, 1975). Perturbation motions about combined sideslip and rolling equi­librium solutions are less stable than perturbations about pure rolling motions.

Also important to the inertial coupling problem are some developments in related fields of airplane dynamics. Chapter 9, on “Spinning and Recovery,” notes the advent of the advanced bifurcation analysis method for study of stall-spin divergence, steady spinning, and wing rock. Bifurcation analysis is also able to predict jump phenomena in rolls or two equilibrium states for the same control surface angles (Schy and Hannah, 1977).

The 1977 study by Schy and Hannah was extended a year later to include nonlinear variations of the stability derivatives with angle of attack (Young, Schy, and Johnson, 1978). The authors correctly observed that the main utility of the bifurcation analysis method as applied to inertial coupling in rolls is to predict the flight conditions and control surface angles for which jumps may occur. These combinations should be examined in detail in complete time history solutions.