Category Airplane Stability and Control, Second Edition

Invention of the Sweptback Wing

The story of the independent invention of the sweptback wing in the United States by Robert T. Jones and in Germany by Adolph Busemann has been told many times. But some accounts of the early work on the stability and control effects of wing sweep belong to this history.

The dive pullout problems of thick, straight-wing airplanes such as the Lockheed P-38 were mainly due to a large increase in static longitudinal stability at high Mach numbers. Thus, an early theoretical result (Jones, 1946) seemed too good to be true. Jones showed that the static longitudinal stability or aerodynamic center location of sharply swept delta wings is invariant with Mach number, from zero to supersonic speeds. A test of a triangular wing of aspect ratio 0.75, with leading-edge sweep of 79 degrees, confirmed the theory. The catch turned out to be that wings of that low aspect ratio are impractical for airplanes that operate out of normal airports.

More practical swept wings for airplanes have higher aspect ratios. In moderately high – aspect-ratio-swept wings there is an outboard shift in additional span loading (Figure 11.7). The outboard shift in additional span load leads to wing tip stall at low angles of attack for moderate – to high-aspect-ratio-swept wings. Outflow of the boundary layer adds to this tendency. Wing tip stall causes an unstable break in the wing pitching moment at the stall (Figure 11.8). That is, loss of lift behind the center of gravity causes the wing (and airplane) to pitch nose-up at the stall, driving the airplane deeper into stall. On the other hand, a stable pitching moment break or nose-down pitch leads to stall recovery, provided that the elevator is moved to trim at a lower angle of attack. Tip stall also leads to an undesirable wing drop and reversal or positive signs for the roll damping derivative Clp, making spins easier to enter and sustain. A condition for autorotation in spins is a positive Clp.

The situation changes for low-aspect-ratio-swept wings, where leading-edge vortex flow acts to create diving, or stabilizing, pitching moments at the stall. A striking correlation was produced showing the combinations of wing sweep and aspect ratio that produce either stable or unstable pitching moment breaks at the stall (Shortal and Maggin, 1946). Figure

11.9 shows an extended version that includes taper ratio effects (Furlong and McHugh, 1957). The stable region was shown to be broadened for sharply tapered wings.

The McDonnell-Douglas F-4 Phantom’s wing, with aspect ratio 2.0 and quarter-chord sweep of 45 degrees, is precisely on the Shortal-Maggin stability boundary, signifying a neu­tral pitching moment break at the stall. High-aspect-ratio-swept wings, typical of transport

Invention of the Sweptback Wing

Figure 11.7 The outward shift in additional span load distribution caused by using wing sweepback. Load increases at the tip at high angles of attack, leading to tip stalling. (From Furlong and McHugh, NACA Rept. 1339, 1957)


Invention of the Sweptback Wing

Figure 11.8 The effect of sweepback on the pitching moment coefficient break at the stall. The straight and 15-degree swept wings are stable beyond the stall; the 30-degree swept wing is unstable (noses up). (From Furlong and McHugh, NACA Rept. 1339, 1957)


.(Ref. 10)

Invention of the Sweptback Wing

0 20 40 60 80

Лс/4, deg

Figure 11.9 Shortal and Maggin’s celebrated empirical longitudinal stability boundary for sweptback wings, extended to include the effect of taper ratio. (From Furlong and McHugh, NACA Rept. 1339, 1957)

airplanes, are unstable at the stall without auxiliary devices. For example, the Lockheed 1011’s wing, with aspect ratio 6.95 and quarter-chord sweep of 35 degrees, is in the un­stable zone at the stall. An early attempt to evaluate in-flight the low-speed stability and control characteristics of moderate-aspect-ratio-swept wing airplanes was made by sim­ply removing the wings of a Bell P-63 Kingcobra and reattaching them to the fuselage at a sweep angle of 35 degrees. The tail length was increased at the same time by adding a constant cross-section plug to the fuselage aft of the wing trailing edge. NACA called this early research airplane the L-39. The L-39’s first flight was made by A. M. (Tex) Johnston. He was to become famous a few years later as the test pilot for Boeing’s prototype 707 jet airliner.

Wind-tunnel tests of the L-39 showed the usual increase in dihedral effect with increasing angle of attack. That is, the rolling moment coefficient in sideslip becomes quite high in the stable direction at attitudes near the stall. There was real concern on the L-39 that if sideslip angles occurred during liftoff or the landing flare, as a result of gusts or rudder use for cross-wind corrections, the rolling moment from dihedral effects would quite overpower the ailerons and the airplane would roll out of control.

This dire possibility was part of the preflight briefing for the pilot Johnston. A briefer, one of this book’s authors (Abzug), remembers that Johnston showed no reaction and asked no questions about this, showing a bit more than the usual test pilot self-confidence. All turned out well. The L-39 flight tests were reasonably routine, and sweptback wings for the next generation of commercial and military jets were on their way.

14.8.3 Modern Identification Methods

The higher powered stability derivative extraction schemes that followed knob twisting engage the interest of many mathematically minded people in the stability and control community. The focus has been broadened beyond the linearized stability deriva­tives, and the subject is now usually called flight vehicle system identification. The years have seen centers of identification activity at individual laboratories, such as Calspan and the NASA Dryden Flight Research Center, and any number of university graduate students earning doctoral degrees in this area. Kenneth W. Iliff and Richard E. Maine at Dryden are leaders in the identification field in the United States.

The DLR Institute in Braunschweig is particularly active in this area, under the guidance of Dr. Peter Hamel. The state of the art up to 1995 is summarized in a paper having 183 references by Drs. Hamel and Jategaonkar (1996). This summary has been updated (Hamel and Jategaonkar, 1999). A generalized model of the vehicle system identification process is shown in Figure 14.15.

A flow chart for a widely used method known as the maximum likelihood or output error method is given in Figure 14.16. The maximum likelihood method starts with a mathematical model of the airplane, which is nothing more than the linearized equations of airplane motion in state variable form (see Chapter 18). The method produces numerical values for the constants in those equations, the airplane’s dimensional stability and control derivatives. A cost function is constructed as the difference between measured and estimated responses, summed over a time history interval. Iliff describes the workings of the maximum likelihood method as follows (see Figure 14.16):

14.8.3 Modern Identification Methods

Figure 14.15 The generalized method of vehicle system identification. (From Hamel, RTO MP-11, 1999)

14.8.3 Modern Identification Methods

Figure 14.16 Flow chart for the maximum likelihood method of airplane stability and control deriva­tive extraction from flight test data. (From Iliff, Jour, of Guidance, Sept.-Oct. 1989)

The measured response is compared with the estimated response, and the difference be­tween these responses is called the response error. The cost functions… include this response error. The minimization algorithm is used to find the coefficient values that minimize the cost function. Each iteration of this algorithm provides a new estimate of the unknown coefficients on the basis of the response error. These new estimates of the coefficients are then used to update values of the coefficients of the mathematical model, providing a new estimated response and, therefore, a new response error. The up­dating of the mathematical model continues iteratively until a convergence criterion is satisfied.

At the time of this writing, the maximum likelihood method is the most widely used of the available identification techniques. For example, in 1993, R. V Jategaonkar, W. Monnich, D. Fischenberg, and B. Krag used this method at the DLR, Braunschweig, for the Transall airplane; and M. R. Napolitano, A. C. Paris, and B. A. Seanor used it at West Virginia University for the Cessna U-206 and McDonnell Douglas F/A-18 airplanes. In an ear­lier use at the NASA Dryden Flight Research Facility, Iliff, Maine, and Mary F. Schafer used maximum likelihood estimation to get a fairly complete set of stability and control derivatives for a Cessna T-37B and a 3/8-scale drop model of the McDonnell Douglas F-15.

Identification quality is dependent on the frequency content of the control input signal in Figure 14.16. Ideal control inputs would excite the system’s modes of motion that require control, while leaving unexcited higher frequencies representing measurement artifacts, such as vibration. Koehler (1977) at DLR devised a simple input with a relatively wide, but limited, frequency content. This is the 3211 signal. The 3211 refers to alternate positive and negative pulses of relative durations 3,2, 1, and 1. The DLR 3211 signal has become a standard in vehicle systems identification.

The alternate extended Kalman filter method of stability derivative extraction has the advantage of being suited to real-time operation. That is, it can be used as an element in a

14.8.3 Modern Identification Methods

Figure 14.17 Improvement of Cma identification on the X-31A with separate surface inputs. (From Weiss, Jour, of Aircraft, 1996)

closed-loop flight control system. Many of the applications of extended Kalman filtering imply or use this feature, as in the 1983 to 1991 work at Princeton University by M. Sri- Jayantha, Dennis J. Linse, and Robert F. Stengel.

A challenging area for identification is that of the aerodynamically unstable airframe, which can be flown only with full-time stability augmentation. Data scatter can be large in these cases, using current methods. Figure 14.17 shows that exciting motion with a separate control surface from that used for closed-loop control reduces identification data scatter.

Rotary derivatives and cross-derivatives, such as the rolling moment due to yawing and the yawing moment due to aileron deflection, are generally the least well-known data in the airplane equations of motion. Identification methods are at their worst for such parameters. Correlations of extracted derivatives with wind-tunnel and theoretical data generally focus on just those derivatives such as Clp and Clp whose numerical values are well known from other sources. Flight test data quality must be high for identification algorithms to work well, probably higher than needed for any other application. State noise from atmospheric turbulence and sensor noise are obvious complications.

The power of flight vehicle system identification continues to advance to the point where the derived system models can meet accuracy requirements for high-fidelity flight

14.8.3 Modern Identification Methods

Figure 14.18 Unsteady aerodynamic modeling of approach to stall, stall, and recovery on the Do 328 transport. Dots are flight test data; solid line is model output. (From Fischenberg and Jategaonkar, RTO MP-11, 1999)

simulators, such as those used in research and pilot training. Recent extensions of the theory include using the frequency domain in place of the usual time domain methods, and the application of neural networks.

Evolution of the Equations of Motion

There is a reproduction in Chapter 1 of George H. Bryan’s equations of airplane motion on moving axes, equations developed from the classical works of Newton, Euler, and Lagrange. This astonishingly modern set of differential equations dates from 1911. Yet, Bryan’s equations were of no particular use to the airplane designers of his day, assuming they even knew about them.

This chapter traces the evolution of Bryan’s equations from academic curiosities to their present status as indispensable tools for the stability and control engineer. Airplane equations of motion (Figure 18.1) are used in dynamic stability analysis, in the design of stability augmenters (and automatic pilots), and as the heart of flight simulators.

18.1 Euler and Hamilton

One of the problems faced by Bryan in developing equations of airplane motion was the choice of coordinates to represent airplane angular attitude. Bryan chose the system of successive finite rotations developed by the eighteenth-century Swiss mathematician Leonhard Euler, with a minor difference. In Bryan’s words:

In the [Eulerian] system as specified in Routh’s Rigid Dynamics and elsewhere, the axes are first rotated about the axis of z, then about the axis of y, then again about the axis of z. The objection to this specification is that if the system receives a small rotation about the axis of x, this cannot be represented by small values of the angular coordinates.

Bryan chose instead to rotate by a yaw angle Ф about the vertical axis, a pitch angle © about the lateral axis, followed by a roll angle Ф about the pitch axis – a sequence that has been followed in the field ever since. However, Bryan’s orthogonal body axes fixed in the airplane are rotated by 90 degrees about the X-axis as compared with modern practice. That is, the Y-axis is in the place of the modern Z-axis, while the Z-axis is the negative of the modern Y-axis (Figure 18.2).

Bryan’s Eulerian angles have served the stability and control community well in almost all cases. However, there were other choices that Bryan could have made that would have avoided a singularity inherent in Euler angles. The singularity shows up at pitch angles of plus or minus 90 degrees, the airplane pointing straight up or straight down. Then the equation for yaw angle rate becomes indeterminate.

The Euler angle singularity at 90 degrees is avoided by the use of either quaternions, invented by Sir W. R. Hamilton, or by direction cosines. The main disadvantage of quater­nions and direction cosines as airplane attitude coordinates is their utter lack of intuitive feel. Flight dynamics time histories calculated with quaternions or direction cosines need to be translated into Euler angles for intelligent use. Except for simulation of airplane or space-vehicle vertical launch or of fighter airplanes that might dwell at these attitudes, the Euler angle singularity at 90 degrees is not a problem.

As the term implies, there are four quaternion coordinates; there are nine direction cosine coordinates. Since, as Euler pointed out, only three angular coordinates are required

to specify rigid-body attitudes, quaternion and direction cosine coordinates have some degree of redundancy. This redundancy is put to good use in modern digital computations to minimize roundoff errors in an orthogonality check. Another advantage to quaternion as compared with Euler angle coordinates is the simple form of the quaternion rate equations, which are integrated during flight simulation. Euler angle rate equations differ from each other, are nonlinear, and contain trigonometric functions. On the other hand, quaternion rate equations are all alike and are linear in the quaternion coordinates.

The nine direction cosine airplane attitude coordinates are identical to the elements of the 3-by-3 orthogonal matrix of transformation for the components of a vector between two


Evolution of the Equations of Motion



Y, © , Ф

Figure 18.2 The Euler angle sequence in most common use as airplane attitude coordinates in flight dynamics studies. This sequence was defined by B. Melvill Jones in Durand’s Aerodynamic Theory, in 1934. (From Abzug, Douglas Rept. ES 17935, 1955)

coordinate systems. As in the quaternion case, all nine direction cosine rate equations have the advantage of being alike in form, and all are also linear. The direction cosine rate equations are sometimes called Poisson’s equations. Airplane equations of motion using quaternions are common; those using direction cosine attitude coordinates are rare.

The Euler parameter form of quaternions uses direction cosines to define an axis of rotation with respect to axes fixed in inertial space. A rotation of airplane body axes about that axis brings body axes to their proper attitude at any instant (Figure 18.3). This goes back to one of Euler’s theorems, which states that a body can be brought to an arbitrary attitude by a single rotation about some axis. There is no intuitive feel for the actual attitude corresponding to a set of Euler parameters because the four parameters are themselves trigonometric functions of the direction cosines and the rotation angle about the axis.

The first published report bringing quaternions to the attention of airplane flight simu­lation engineers was by A. C. Robinson (1957). Robinson’s contribution was followed in 1960 by D. T. Greenwood, who showed the advantages of quaternions in error checking nu­merical computations during a simulation. A detailed historical survey of all three attitude coordinate systems is given by Phillips, Hailey, and Gebert (2001). The flight simulation

Evolution of the Equations of Motion

Figure 18.3 The Euler parameter form of quaternions used in some flight simulations to calculate airplane attitude. The upper group of equations defines the Euler parameters in terms of an axis of rotation of XYZ to a new attitude. {x}body are vector components on the rotated axes; {x} earth are the same components on the original axes. Transformations between Euler parameters and Euler angles are given in the lower two sets of equations.

community appears to be divided on the choice between Euler angles and quaternions. In some cases, both are used in different flight simulators within a single organization. However, it is interesting that so many modern digital computations of airplane stability and control continue to use Euler angle coordinates in the 1911 Bryan manner.

Normal Mode Analysis

Normal mode analysis, as applied to aeroelastic stability and control problems, is actually a form of the small oscillation theory about given states of motion. This goes back to the British teacher of applied mechanics E. J. Routh, in the nineteenth century. A body is supposed to be released from a set of initial restraints and allowed to vibrate freely. It will do so in a set of free vibrations about mean axes, whose linear and angular positions remain unchanged. The free vibrations occur at discrete frequencies (eigenfrequencies), in particular mode shapes (eigenvectors).

Of course, the airplane does not vibrate freely, but under the influence of aerodynamic forces and moments. These forces and moments are added to the vibration equations through a calculation of the work done during vibratory displacements. Likewise, the changes in aerodynamic forces and moments due to distortions must have an effect on the motion of mean axes, or what we would call the rigid-body motions.

According to Etkin’s criterion, if the separations in frequency are not large between the vibratory eigenfrequencies and the rigid-body motions such as the short-period longitudinal or Dutch roll oscillations, then normal mode equations should be added to the usual rigid – body equations. Each normal mode would add two states to the usual airframe state matrix (Figure 19.10). A useful example of adding flexible modes to a rigid-body simulation is provided by Schmidt and Raney (2001). Milne’s mean axes are used.

Normal mode aeroelastic controls-coupled analyses were made in recent times for the longitudinal motions of both the Northrop B-2 stealth bomber and the Grumman X-29A research airplane. In both cases, the system state matrix that combines rigid-body, nor­mal mode, low-order unsteady aerodynamic and pitch control system (including actuator dynamic) states was of order about 100 (Britt, 2000).

Gibson Approach

In his 1999 thesis at TU Delft, John C. Gibson proposes a different categorization of PIO from that of McRuer (Sec. 6). In one category are PIOs that arise from conventional low – order response dynamics. The pilot can back out of these by reducing gain or abandoning the task. In this category the lag in angular acceleration following a control input is insignificant, giving the pilot an intimate linkage to the aircraft response.

In the second category are PIOs arising from high-order dynamics in which the pilot is locked in and is unable to back out. High-order dynamics such as excessive linear control law lags or actuator rate and/or acceleration limiting create large lags in acceleration response, disconnecting the pilot from the response.

In the first category, solutions can be developed assuming only the simplest of pilot models. The basic idea is that fly-by-wire technology can be used to shape the response so that the control laws provide the McRuer crossover model for the airplane-pilot combina­tion, with the pilot required only to provide simple gains. Of course, other factors such as sensitivity, attitude and flight path dynamics, and mode transitions must be considered.

The second category, involving high-order dynamics, requires detailed examination of the evidence to define the limit of high-order effects that can be tolerated. Stop-to-stop stick inputs at critical frequencies must be evaluated.