Category Airplane Stability and Control, Second Edition

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

X

ORDER OF ROTATIONS

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

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, 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).

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.

Joukowski’s theory for a monoplane wing shows that the pitching moment coeffi­cient about a point one-quarter of the chord length behind the wing leading edge is a constant at all angles of attack. This point is the wing’s aerodynamic center. The constant pitching moment coefficient at the aerodynamic center is negative, or nose-down, for positively cambered wings and positive for negatively cambered wings.

The combination of the lift coefficient, varying linearly with the angle of attack, and a constant pitching moment coefficient can be replaced with a single vector lift force. The point of action on the wing chord of the single vector force is called the wing’s center of pressure. Early longitudinal trim and stability calculations use the center of pressure concept, rather than the concept of aerodynamic center.

Dr. Charles McCutchen speculates that center of pressure methods came to the aero­nautical field through aeronautical engineers originally trained as naval architects, such as

Dr. Jerome C. Hunsaker. Ship stability against roll is determined by metacentric height, the distance between the ship’s center of gravity and the line of action of the buoyancy force in a roll. Ship buoyancy force acting through the metacenter is analogous to the wing lift force acting through the center of pressure, except that other forces are involved in airplane stability.

B. Melvill Jones (Figure 1.8) proposed the term “metacentric ratio” for what we now call static margin, the distance in wing chords between the center of gravity and the center of gravity position for neutral stability of the complete airplane (Jones, 1934). That center of gravity position is called the neutral point. According to J. C. Gibson (2000), an article published by S. B. Gates in a 1940 Aircraft Engineering magazine explained the neutral point concept in physical terms, providing a “stunning revelation” to engineers trained on center of pressure methods.

CHAPTER 2

While he was at the Douglas Company plant in El Segundo, California, Hans C. Vetter described a damping effect to be expected from jet air intakes and exhausts. He argued that the air in a jet duct travels in a radial path with respect to the center of gravity when the airplane performs rotational oscillations. Pressure forces on the structure result which are in the direction to resist angular velocities, adding “Coriolis” damping to the aerodynamic damping moments provided by the wing and tail surfaces. Jet damping moments depend on the distances from the center of gravity to the jet intake and exits and on other dimensions (Vetter, 1953).

Artificial yaw and pitch damping, used on almost all modern jet aircraft, tends to swamp out jet damping effects. Furthermore, jet damping is most significant at low airspeeds and high thrust levels, normally encountered only at low altitudes. But the airframe’s natural damping is best at low altitudes. Still, careful designers include jet damping in their calcu­lations. Vetter’s theory implies that rocket-powered aircraft also have Coriolis jet damping, but of course only for the rocket’s exhaust.

The angular momentum of propellers and the rotating parts of jet engines create inertial reaction moments when an airplane pitches or yaws. This is of interest in the analysis of inertial coupling (Chapter 8) and spins (Chapter 9).

New stability and control problems associated with geometries appropriate to tran­sonic and supersonic speeds and their approximate theoretical or empirical consequences led to the creation of handbook data for their solution in a form suitable for the use of airplane designers. Handbooks have been produced by the USAF Wright Air Development Center, the British Royal Aeronautical Society (RAeS), the Japan Society for Aeronautical and Space Sciences (JSASS), and others. The USAF version, called DATCOM, for Data Compilation (Hoak, 1976), is supplemented by a computer version intended to reduce the manual labor in using the rather bulky hard copy version of the material.

The goal of all these compilations is to show the effects of all possible design factors on aircraft forces and moments. Charts and elaborate formulas are used, as in the example of Figure 6.3, from the DATCOM. RAeS data sheets have similar function and appearance, except for a wide use of ingenious carpet plots. Dr. H. H. B. M. Thomas played a key role in the development of the RAeS data sheets.

Modified tail arrangements did not really accomplish much for the Yankee, and NASA turned its attention to modifications of the wing outer panel. In so doing, a line of research was reopened that had been followed at NACA by the versatile Fred E. Weick and Carl J. Wenzinger in the 1930s. There was also the work by R. A. Kroeger of the University of Michigan and T. W. Feistel of NASA Ames Research Center in 1975, in which the concept of a segmented or discontinuous wing leading edge was developed to control stall progression and minimize loss in roll damping at the stall. This type of wing leading edge had been seen previously on the McDonnell F-4 Phantom II.

The best NASA wing leading-edge extensions tested on general-aviation airplanes also have sharp discontinuities at their inboard ends (Figure 9.8). The sharp discontinuities evi­dently trigger vortices that slow the spread of inboard wing stalling to the outer wing panels.

For each installation, there appears to be an optimum location for the droop discontinuity, to delay spin entry and improve spin recovery. Full-span leading-edge droop was actually a detriment on the Yankee, causing easily entered flat spins where there had been none before.

The discontinuous outboard wing leading-edge droop modification provides gener­ally good spin resistance and spin recovery characteristics on a number of different

configurations. These include the modified Grumman/American AA-1B Yankee; the mod­ified Piper T-Tail Arrow; and models of the Smith Aviation Trainer, the Questair Venture, and the DeVore Aviation Corporation Trainer, an ultralight pusher (Figure 9.9).

Naval aviators have developed a distinctive landing approach procedure to make touchdown point precision a routine matter. U. S. carrier-based airplanes turn onto a short final approach path in a steep left turn, avoiding as much as possible the ship’s turbulent wake or burble. Carrier final approaches are typically less than 3/4 mile long, taking some 15 to 20 seconds to complete (Craig, Ringland, and Ashkenas, 1971).

In manually flown approaches the pilot relies on an optical projection device for a vertical reference, rather than the view of the ship’s landing area. The optical device, mounted on the carrier’s deck, is gimbaled to project a stable glide slope. The pilot sees a projection of a solid circle and short horizontal datum bar. Below the glide slope the ball appears below the bar, and vice versa. Radar-controlled automatic carrier landings have also been developed, using an SPN-42 tracking radar mounted on the ship. Aircraft attitude changes are sent as commands to the airplane’s pitch attitude autopilot, to correct perceived height errors relative to the ideal glide path.

In order to control airspeed closely, the final approach is made at a constant angle of attack. Precise control of angle of attack as a means of controlling airspeed is considered so important that a special throttle control system – the Approach Power Compensation System, or APCS – was developed for that purpose. There were many experiments with different feedbacks; the final APCS design uses angle of attack and normal acceleration feedbacks to the throttle and some pilot stick feedforward (Figure 12.1).

Unusual aerodynamic arrangements, such as canards, floating main wings (Regis, 1995), fore and aft horizontal tails, tailless configurations, and vee-tails represent particular hazards in the designs of personal airplanes because of the limited opportunity for thorough aerodynamic testing. Designers expect to work out stability and control problems in pro­totype flight testing, but flight testing can leave many areas unexplored, as compared with systematic wind-tunnel testing.

Possible tail stalling on the vee-tailed Beech Model 35 in sideslips at forward center of gravity positions, flaps down, leading to abrupt noseovers is mentioned in Chapter 14. The special stability and control problems of canard airplanes are mentioned in Chapter 17. Canard applications to military airplanes have a better chance of success than those for personal airplanes. Military canard applications take place under the protection of large wind-tunnel test programs and electronic stability augmentation, protections not available to designers of personal airplanes.

Bryan used quite conventional cartesian coordinates in the derivation of the equa­tions of airplane motion on moving axes, in 1911. Cartesian coordinates were used as well by subsequent investigators, such as B. Melvill Jones (1934), Charles Zimmerman (1937), and Courtland Perkins (1949). The first author who applied vector methods to the derivation of these equations appears to have been Louis M. Milne-Thomson, in his book Theoretical Aerodynamics (1958).

The most notable thing about the Milne-Thomson vector derivation is the way in which a fundamental moving axis rate equation is developed. This is the relationship between vector

rates of change referred to inertial axes, required for application of Newton’s law of motion, and vector rates of change as seen on moving axes. A simple vector cross-product connects the two rates of change. Milne-Thomson’s vector equations of airplane motion on moving axes is of course far more compact than the cartesian form. In Theoretical Aerodynamics, Milne-Thomson did extend the vector derivation to the small-perturbation case.

Dyadics are generalized vectors, having nine instead of three components. Rigid-body moments of inertia and angular momenta have particularly simple dyadic forms. Dyadic versions of the torque or rotational equations of airplane motion are readily found, but there is no particular advantage to the dyadic form of the ordinary equations of motion. An advantage does occur for the semirigid case where the relative angular velocities of linked rigid bodies are computed (Abzug, 1980).

On the other hand, matrix forms of the equations of airplane motion on moving axes now have a significant role in flight dynamics. This is the result of the marvelous matrix manipulation capability of modern digital computers. The linearized equations of airplane motion are put into matrix form by first expressing the equations in state-variable form. In the state-variable formulation, a first-order differential equation is written for each degree of freedom of the system.

The matrix form is {x} = [A ] {x} + [B] [u }, where, for the airplane,

{x} is a N-by-1 state vector formed of the perturbation airplane motion states, such as u, v, and w.

{x} is the time derivative of {x}.

[A ] is a N-by-N system matrix formed of stability derivatives, such as dX/du, mass, and dimensional properties.

[B ] is an N-by-M control matrix formed of control derivatives such as дX/dS.

{u} is a M-by-1 control vector formed of perturbation control surface angles.

For the perturbation longitudinal equations a typical state vector {x} is the 5-by-1 vector {u а в q h}. A typical control vector {u} is the 1-by-1 vector, hence scalar, {Sh}, the stabilizer angle. For the perturbation lateral equations {x} is typically the 6-by-1 vector {в ф p ф ry}. The control vector is usually the 2-by-1 vector {Sa Sr}.

Modern matrix flight control analysis and synthesis methods generally augment the airplane state vector with control system states and manipulate matrices [A] and [B] in closed-loop operations. All of the classical Bryan, Gates, Zimmerman, and Perkins anal­yses for the unaugmented airframe can be carried out with standard computerized matrix manipulations. A prime example is the method of finding transient solutions by forming transition equations from one interval to the next. Transition matrices are computed using large numbers of successive matrix multiplications.

Matrix methods are fundamental to a number of commercially available flight dynamics computer programs. Systems Technology, Inc., of Hawthorne, California, offers the “Linear System Modeling Program,” which does every possible form of linearized stability anal­ysis, including time-vector analysis. The Design, Analysis and Research Corporation of Lawrence, Kansas, produces the “Advanced Aircraft Analysis” program, which does stabil­ity and control preliminary design, trim, and flight dynamics. Large general-purpose matrix manipulation computer programs such as “MATLAB” from MathWorks and “Mathcad” from MatSoft are also commercially available to the stability and control engineer.

The remarks about the dyadic form of the equations of airplane motion apply as well to tensor forms. That is, there is no special advantage in expressing the ordinary equations of rigid-body airplane motion in tensors, as compared with cartesians or vectors. Zipfel (2000) uses a tensor form of the rigid-body equations of airplane motion on moving axes.