Category Airplane Stability and Control, Second Edition

P-47 Dive Tests at Wright Field

The Republic P-47 Thunderbolt was, like the P-38 Lightning, an important fighter airplane of World War II. Like the P-38, the P-47 had a supercharged engine and could climb to altitudes of about 35,000 feet and reach high enough airspeeds in dives to have com­pressibility effects on stability and control. The P-47 experience was sufficiently different

P-47 Dive Tests at Wright Field

Figure 11.4 A Republic P-47 Thunderbolt fails to respond to 4 degrees of up-elevator in a dive at a Mach number of 0.86. (From Perkins, Jour. of Aircraft, July-Aug. 1970)

from the P-38’s to merit retelling (Perkins, 1970). Following split S entries to vertical dives at 35,000 feet, the P-47’s nose would go down beyond vertical. No recovery seemed pos­sible even with full-back stick and nose-up tab (Figure 11.4). At 15,000 feet high normal acceleration would suddenly come on, and airplanes would recover at 20,000 feet, with bent wings.

Three possible reasons for this were examined in a 1943 conference held at the NACA Langley Laboratory:

1. ice formation on the elevator hinges at 35,000 feet;

2. elevator hinge binding due to loads;

3. Mach number effects on stability and control.

As Perkins recalls, it was Theodore Theodorsen, the eminent NACA mathematician and flutter theorist who championed ice on the hinges as a possible cause for the problem. The NACA structures researcher Richard V Rhode proposed elevator hinge binding as the cause. Robert Gilruth and the forceful John Stack claimed correctly that it was all transonic aerodynamics. Quoting from the Perkins paper:

It became obvious that one simple test would resolve the major difference in the theories. When the pilot pulled on the stick, did the elevator go up or didn’t it? If it didn’t go up, then one of the first two theories would be correct; but if the elevator did go up and the airplane did not respond as it should, then the third theory would be the most likely answer.

The U. S. Air Corps at Wright Field agreed to run these tests and attempts were made to sign up a test pilot to perform the experiment. None of the contract test pilots were very anxious to do this and would have agreed only at very high fees. The problem was resolved when one of the Air Corps’s strongest and ablest test pilots, [Capt] P. [Perry] Ritchie [Figure 11.5] said he would perform the tests for nothing. He performed some thirty dive tests on an instrumented P-47 and his reward was an Air Medal.

P-47 Dive Tests at Wright Field

Figure 11.5 Capt. Perry Ritchie (1918-1944), the courageous U. S. Air Corps test pilot who made 30 test dives in the P-47 Thunderbolt. (USAF photo)

It was found at once that the elevator did go up to the predicted angle. However, while at that high airspeed the measured amount of elevator angle should have produced 20 to 30 g, the actual response was about 0.5 g, which appeared to the pilot as no response at all. This behavior was also found later by Republic Corporation test pilots. The P-47 was clearly experiencing the same Mach number phenomena as did the P-38. Compressibility burble on the inboard wing sections led to lift curve slope reductions and reductions in rate of change in downwash over the horizontal tail. This caused an increase in longitudinal static stability and a nose-down trim shift.

Rudder Lock and Dorsal Fins

Rudder lock occurs at a large angle of sideslip when reversed rudder aerodynamic hinge moments peg the rudder to its stop. The airplane will continue to fly sideslipped, rudder pedals free, until the pilot forces the rudder back to center or rolls out of the sideslip with the ailerons. Aerodynamic hinge moments can peg the rudder against its stops so securely as to defy the pilot’s efforts at centering. In that case recovery by rolling or pulling up to reduce airspeed are the only options.

Two things must happen before an airplane is a candidate for rudder lock. Directional stability must be low at large sideslip angles and rudder control power must be high. The relative size of the fuselage and vertical tail determines the general level of directional stability. Directional stability is reduced at large sideslip angles when the fin stalls. The sideslip angle or fin angle of attack (considering sidewash) at which the fin stalls de­pends on the fin aspect ratio. Unfortunately, tall, efficient, high-aspect-ratio fins stall at low fin angles of attack. As a general rule, fin stall occurs at sideslip angles of about 15 degrees.

Unlike normal wings, whose lift is proportional to angle of attack until near the stall, the lift of very low-aspect-ratio rectangular wings is proportional to the square of the angle of attack (Bollay, 1937). There is very little lift generated in the low angle of attack range. However, the angle of attack for stall is increased greatly, reaching angles as high as 45 degrees.

What this means is that a two-part vertical tail is an efficient way to avoid loss in directional stability at large sideslip angles and rudder lock. One part is a high-aspect-ratio vertical tail, which can provide directional stiffness in the normal flight regime of low side­slip angles and give good Dutch roll damping and suppression of aileron adverse yaw. The other part is a low-aspect-ratio dorsal fin, with a reasonably sharp edge, which will carry very little lifting load in the normal flight regime. However, at a sideslip angle where the high-aspect-ratio fin component stalls, the dorsal fin can become a strong lifting surface, maintaining directional stability.

Returning to the role of the rudder, large rudder areas and control power are needed for two-engine airplanes with wing-mounted engines, for the condition of single-engine failure at low airspeeds. This is especially true for propeller-powered airplanes, since full-throttle propeller thrust is highest at low airspeeds, and wing-mounted engines tend to be further outboard than for jets, to provide propeller-fuselage clearance.

Although a four-engine rather than a two-engine airplane, rudder lock was experienced on the Boeing Model 307 Stratoliner, with its original vertical tail. This occurred during an inadvertant spin. From William H. Cook (1991):

On a demonstration flight for KLM and TWA, the KLM pilot applied rudder at low speed. The rudder locked full over in the spin, and the control forces on the rudder were too high [to center it]. Wind tunnel tests showed that a long dorsal fin would prevent the rudder locking over. A hydraulic servo on the rudder was also added.

The addition of a dorsal fin to the Stratoliner and a reduction in rudder area corrected the problem (Figure 14.13) (Schairer, 1941). George Schairer recently commented that he was unaware of the true inventor of dorsal fins, but that a member of the GALCIT 10-foot tunnel staff might have installed one during tests of one of the Douglas airlin­ers. Small dorsal fins appeared earlier than on the Stratoliner, notably on the Douglas DC-3, first produced in 1935, and on the Douglas DC-4, which had its first flight in 1938.

In spite of the small dorsal fin installed on the DC-3, that airplane is still subject to rudder lock in all configurations with power on (Figure 14.14). JohnA. Harper flew a U. S. Air Force C-47B, the military version of the DC-3, in NACA flying qualities tests in 1950. Harper later speculated that rudder lock might have contributed to some puzzling DC-3 accidents resulting from loss of power on one engine, followed by a stall and spin. In these strange

Rudder Lock and Dorsal Fins

Figure 14.13 The variation of yawing moment coefficient with sideslip angle for the Boeing Stra – toliner with original vertical tail (above) and revised vertical tail and dorsal fin (below). Rudder-free cases are shown by the dashed lines. With the original tail, adverse yawing moment due to the ailerons overcomes the low level of restoring moment at large side-slip angles, and there is rudder lock. (From G. S. Schairer, Jour oftheAeo. Sci., May 1941)

accidents, the airplane spun into the operating engine, the reverse of what one would expect. Harper argues that rudder lock and high pedal forces for recovery could have occurred if the pilot overcontrolled with the rudder to turn toward the live engine.

Rudder lock was suspected in the early Boeing 707 airplanes, which had manually operated rudders assisted by spring tabs and internal aerodynamic balance. An Air Force test of the XC-135 tanker version reported rudder lock and an American Airlines crash on Long Island may have been due to rudder lock. As a result, the 707 and KC-135 series of airplanes have powered rudders.

Rudder Lock and Dorsal Fins

Figure 14.14 Incipient rudder lock on the DC-3 airplane. The rudder force has gone to zero at a right sideslip angle of18 degrees. The rudderangle is only21 degrees left, with 9 more degrees of deflection available before reaching the rudder stop of 30 degrees. The rudder locks over at larger rudder and sideslip angles, but these are not reached in this test series because of heavy airplane buffet. (From Assadourian and Harper, NACA TN 3088, 1953)

In addition to the large rudder area requirement for the engine-out condition on multi­engine airplanes, large rudder areas are needed for spin recovery on maneuverable airplanes, to handle heavy crosswinds for airplanes intended to operate out of single-strip airports, and for gliders, to counter adverse aileron yaw. Gliders have a particular adverse yaw problem because their high-aspect-ratio wings have large negative (adverse) values of yaw­ing moment due to rolling at high lift coefficients. Pilots transitioning from light power planes to gliders, or vice versa, find vigorous rudder action in rolls is needed for coordina­tion in gliders, as compared with light planes.

Airplanes in all of these categories might be found with dorsal fins, to prevent rudder lock. For example, the Waco CG-4A and XCG-13 cargo gliders had strong rudder lock before their vertical tails were enlarged and dorsal fins were added. On the other hand, dorsal fins have been used on airplanes as a matter of style rather than for the function of augmenting static directional stability at large angles of sideslip. This can be suspected if dorsal fins are found on airplanes that have large vertical tails at a reasonable tail length, rudders of small to moderate size, and either one or more than two engines.

Pusher Propeller Problems

Although canard airplanes can have propellers in front, in the so-called tractor position, canard propeller-driven aircraft generally wind up with pusher propellers. Thus, in the context of discussing design, stability, and control problems of canards, it is appropriate to bring up some design problems of pusher propellers as well.

A tail-down landing touchdown attitude is often desired, for a minimum energy landing. In fact, many nose landing gears are not expected to take landing impact loads and are noticeably lighter and weaker than main landing gear assemblies. Pusher propellers tend to have relatively small diameters, just to provide clearance for tail-down landings. This is a constraint on propeller design, leading to lower propulsive efficiency. Alternatively, airplanes with pusher propellers tend to have relatively long, heavy main landing gear legs.

Pusher propellers generally act in the wakes of either wings or horizontal tails. While there may be no appreciable propulsive efficiency loss for such arrangements, a distinctive propeller noise generally results, which could be a problem for people on the ground. Pusher propellers have vibration problems, and their engines can have cooling problems.

Finite-Element or Panel Methods in Quasi-Static Aeroelasticity

Analyzing quasi-static aeroelastic effects requires balancing air loads against struc­tural stiffness and mass distributions. Because of the complexity of the problem, only ap­proximate methods were available for many years. The advent of finite-element or panel methods both in structural analysis and in aerodynamics made accurate quasi-static aero – elastic analysis really possible for the first time.

In the aerodynamic finite-element approach, the airplane’s surface is divided into many generally trapezoidal panels, or finite elements. Under aerodynamic and inertial loadings, the structure finds an equilibrium when boundary conditions are satisfied at control points such as the center of the 3/4-chord line of an aerodynamic panel or at the edges of a structural panel. Finite-element methods in structural analysis preceded those for aerodynamic analysis by many years.

Finite-Element or Panel Methods in Quasi-Static Aeroelasticity

Figure 19.8 Effect of dynamic pressure on the dihedral effect of the Douglas XA3D-1 airplane, at two angles of attack. The wing lift is close to zero at a (fuselage) angle of attack of -3 degrees, and there is little wing bending and change in dihedral effect. (From Rodden, AGARD Report 725, 1989)

The earliest aerodynamic finite-element method, called vortex lattice analysis, appears to have been developed independently by two people. Vortex lattice analysis is docu­mented in internal Boeing Company and Swedish Aeronautical Research Institute re­ports by P. E. Rubbert in 1962 and Sven G. Hedman in 1965, respectively, and in a few other reports of the same period. Dr. Arthur R. Dusto and his associates at the Boeing Company combined these structural and aerodynamic finite-element methods into an aero – elastic finite-element system they call FLEXSTAB (Dusto, 1974) in the period 1968 to 1974.

Finite-element methods in quasi-static aeroelasticity require generation of mass, struc­tural influence, and aerodynamic influence matrices. The mass matrix is the airframe mass assigned to each element. The structural influence coefficient matrix transforms deflec­tions at control points in an element to elastic forces and moments at the other elements. The aerodynamic influence coefficient matrix transforms angle of attack at one element to aerodynamic forces and moments acting on the other elements.

It is interesting that the advent of finite-element quasi-static aeroelastic methods coin­cided with the need for methods that account for significant chordwise structural distortions. Quasi-static aeroelastic methods based on lifting line theory were appropriate for flexible airplanes of the Boeing B-47 and Douglas DC-8 generation, subsonic airplanes with long, narrow wings. Proper quasi-static aeroelastic analysis of the lower aspect ratio, complex wing planforms of the Northrop B-2 stealth bomber and supersonic-cruise transport air­planes, requires panel methods.

NASTRAN is a widely used finite-element structural analysis computer program. The MacNeal-Schwendler Corporation’s proprietary version, called MSC/NASTRAN, adds

Finite-Element or Panel Methods in Quasi-Static Aeroelasticity

In this equation:

u = displacement vectors or column matrices К — structural stiffness matrices M = structural mass matrices P = aerodynamic force matrices D = a rigid body mode matrix

Figure 19.9 One form of the NASTRAN quasistatic aeroelastic matrix equations. Additional manipulations are needed to arrive at the unrestrained aeroelastic stability and control derivatives. (From Rodden and Johnson, eds., MSC/NASTRAN Aeroelastic Analysis User’s Guide, 1994)

aerodynamic finite-element models to the existing structural models with splining or inter­polation techniques to connect the two. This version can perform quasi-static aeroelastic analysis (Figure 19.9). This accomplishment is credited to a number of people, including Drs. Richard H. MacNeal and William P Rodden, and E. Dean Bellinger, Robert L. Harder, and Donald M. McLean.

Pilot Equalization with the Crossover Model

All airplane transfer functions, such as the pitch response to elevator and the roll response to aileron, have first – or second-order denominator functions, arising from mass or inertia. To satisfy the crossover model the pilot must supply a canceling numerator function over the same frequency range. This amounts to lead or anticipation, agreeing with common sense as to what is required for the error elimination in compensatory operation.

The amount of lead or compensation required by the pilot is a direct measure of workload. The pilot lead is reflected in the positive slope of the pilot model amplitude ratio in the Bode diagram, in the vicinity of the crossover frequency. A large positive slope corresponds to excessive lead, high workload, and poor pilot rating. A numerical connection can be made between pilot rating by the Cooper-Harper scale, discussed in Chapter 3, and required lead equalization (Figure 21.3).

Contrasting Design Philosophies

Comparison of the 1917 British (Royal Aircraft Factory) S.(scouting) E.(experimental)-5 and the Fokker D-VII shows an interesting contrast between the design philosophies of the Royal Aircraft Factory designers, who had been exposed to primitive airplane stability theory, and Anthony H. G. Fokker and his co-worker, Reinhold Platz, neither of whom had any formal technical training. Platz had been trained in the art of acetylene gas welding, which he applied to the construction of steel tube airplane fuselages, while Fokker was an experienced craftsman, pilot, and small boat sailor with an instinct for aerodynamics.

The strong dihedral (5 degrees) of the S. E.-5 wings (Figure 1.4) is evidence of an attempt to give the airplane inherent spiral stability. On spirally stable airplanes, if the pilot establishes a banked turn, the rudder and elevator have to be held in a deflected position to continue the turn. If the pilot centers the rudder bar and control stick, a correctly rigged airplane will automatically, but slowly, regain wings-level flight.

Contrasting Design Philosophies

Figure 1.4 The British paid attention to inherent spiral stability during World War I days, building 5 degrees of dihedral into the S. E.-5. (From Jane’s All the World’s Aircraft, 1919. Jane’s used a German source for these drawings since the S. E.-5 was still classified in Britain in 1919.)

The S. E.-5’s control surfaces had no aerodynamic balance and were difficult to move at diving speeds. Thin wing sections were used. The designers also had embraced a whim for numerology; the wings had 250 square feet of area and 5-foot chords; they were set at 5 degrees with respect to the thrust line, and so on.

Modern flight tests of World War I fighters (using the Shuttleworth Collection) give the S. E.-5A high ratings. Ronald Beaumont says this airplane was

perhaps the best handling fighter on either side, with excellent pitch and yaw control and inherent stability on both axes, and with light and responsive ailerons up to the quite high speed of 130 mph.

The Fokker D VII (Figure 1.5) had wooden-frame cantilever wings, almost without dihedral, with a thick airfoil section, an early result of Prandtl/Lanchester circulation theory. David Lednicer reported (2001) that the D VII wing airfoil was close to the Gottingen 418. The D VII had a steel-tube-welded fuselage and tail assembly. Horn balances (called elephant ears) were provided to lessen the pilot effort to deflect the ailerons, elevators, and rudder.

When Fokker flew the first version he realized he had created a dangerous airplane. Before the German Air Ministry officials could get a good look at it, he rebuilt it secretly in the hangar, moving the wings aft to make it less unstable, lengthening the fuselage, and modifying the vertical tail to incorporate a fixed fin. As a result of the D VII’s long tail moment arm; blunt-nosed, cambered airfoil sections; and mechanically limited up elevator

Contrasting Design Philosophies

Figure 1.5 The Fokker D-VII, built without wing dihedral, showing no concern for spiral stability. This machine had horn aerodynamic balances at the tips of all control surfaces, to reduce control forces. (From Progress in Airplane Design Since 1903, NASA Publication L-9866, 1974.)

deflection, stability and control at low speeds and climb rate were quite good. In its final form it pleased everyone so much that it was mentioned in the Treaty of Versailles as a military airplane that had to be surrendered to the Allied authorities, the only one so designated.

Jet and Rocket Effects on Stability and Control

Except for the V/STOL case, jet and rocket effects on airplane stability and control tend to be small compared with those for high-powered propeller airplanes. This is because of the absence of slipstream and direct propeller effects. Yet, they are not negligible. By the time the first jet aircraft were being tested, the necessary theory was in place. Two new factors needed to be accounted for-jet intake normal force and airstream deviation due to inflow into the jet or rocket exhaust.

4.8.1 Jet Intake Normal Force

The NACA engineer Dr. Herbert Ribner, who was also an important contributor to the body of knowledge on propeller forces in yaw, or propeller normal forces, provided the analogous theory for jet air intakes, or an algorithm for jet normal force (Ribner, 1946). Ribner’s jet intake normal force formulation is based on the mass flow of air into each jet intake and the angle turned by this air to get into the duct. This method neatly avoids having to estimate or measure pressure distributions inside and outside of the intake ducts, although the resultant normal forces must be generated by those pressures.

One interesting refinement is to take into account the upflow before each jet intake caused by the wing’s lifting system. In sideslip, the corresponding correction is for any sidewash ahead of the intake. Fortunately, wing upflow angles are readily available in chart form for any wing planform.

Herbert Ribner was active until 2001, spending winters at the Langley Research Center as a NASA Distinguished Research Associate and the rest of the year at the University of Toronto.

Jet and Rocket Effects on Stability and Control

Figure 4.6 Calculated isoclines of the surrounding flow into a jet wake. Destabilizing downwash for tails mounted above the jet wake is indicated by the positive deviation angles. (From Squire and Trouncer, British R & M 1974, 1944)

Estimation from Drawings

6.2.1 Early Methods

The elements of stability and control prediction from drawings started to be avail­able as early as aerodynamic theory itself. That is, aside from elements such as propellers and jet intakes and exhausts, airplane configurations are combinations of lifting surfaces and bodies. However, it took some time before the lift and moment of lifting surfaces and bodies were codified into a form useful for preliminary stability and control design. Simple correlations of lift and moment with geometrical characteristics such as wing aspect and taper ratios and the longitudinal distributions of body volume were needed.

6.2.2 Wing and Tail Methods

For stability and control calculations at the design stage, the variations of lift coefficient with angle of attack, or lift curve slope, are needed for airplane wings and tail surfaces. Wing and tail lift curve slopes are to first-order functions of aspect ratio and sweepback angle, and to a lesser extent of Mach number, section trailing-edge angle, and taper ratio. The primary aspect ratio effect is given by Ludwig Prandtl’s lifting line theory and can be found as charts of lift curve slope versus aspect ratio in early stability and control research reports. The sweepback effect was added by DeYoung and Harper (1948).

However, classical lifting line theory for wings and tails fails for large sweep angles and low aspect ratios, even at low Mach numbers. A 1925 theory of supersonic airfoils in two-dimensional flow due to Ackeret existed, and also in the 1920s Prandtl and Glauert showed how subsonic airfoil theory could be corrected for subsonic Mach number effects. Both the Ackeret theory and Prandtl-Glauert subsonic Mach number correction theory fail at Mach 1. R. T Jones (1946) developed a very low aspect-ratio wing theory, valid for all Mach numbers, which applies to highly swept wings, that is, wings whose leading edges are well inside the Mach cone formed at the vertex.

6.2.3 Bodies

A fundamental source for the effects of bodies on longitudinal and directional stability is the momentum or apparent mass analysis of Max M. Munk (1923). This models the flow around nonlifting bodies such as fuselages, nacelles, and external fuel tanks in terms of the growing or diminishing momentum imparted to segments of the air that the body passes through. Pitching and yawing moments as functions of angle of attack and sideslip are found by this method.

Parameter Estimation Methods for Spins

The use of rotary balances of ever-increasing complexity for measuring aero­dynamic forces and moments in spins is avoided if aerodynamic forces and moments can be inferred directly from free-spinning model or airplane tests. Two promising approaches to this application of parameter estimation have been reported.

The first approach (Fremaux, 1995) extends the Gates-Bryant equilibrium or steady-state spin analysis to include the nonequilibrium angular acceleration terms p, q, and r and the spin acceleration term f2. Calculated aerodynamic moments by this method vary with time if the spin is oscillatory. The calculated moments oscillate about the Gates-Bryant values, which also can be measured independently on a rotary balance. This method requires the investigator to record rapid angular motions in a spin, feasible now with the advent of modern data-acquisition techniques.

A second parameter estimation approach for spins (Jaramillo and Nagati, 1995) appears to have been inspired by the finite-element methods used in structural analysis. A set of control points are established. Aerodynamic force coefficients at these points are correlated with local angles of attack and sideslip during spinning motions. These aerodynamic force (and moment) coefficientsare in effect influence coefficients. The influence coefficientsare found by minimizing cost functions based on the errors between measured vehicle accelerations and those calculated using forces and moments derived from the influence coefficients. Once the dimensionless influence coefficients are found, the method appears to have predictive capabilities. An improved version (Lee and Nagati, 1999) of the original method reduces the number of unknown parameters to be solved for by using static wind-tunnel test data.

Supersonic Altitude Stability

A somewhat strange lack of stability cropped up when airplanes began to operate at supersonic speeds above about Mach 2 at quite high altitudes. This showed up as an inability to control altitude and airspeed precisely in flights of the North American XB-70 Valkyrie and the Lockheed SR-71A. The Concorde SST with a maximum speed around Mach 2 is believed to have difficulties of this sort, as well. According to Glenn B. Gilyard and John W. Smith (1978), on the SR-71A:

Decreased aircraft stability, low static pressures, and the presence of atmospheric distur­bances are all factors that contribute to this degraded control. The combination of high altitude and high speed also contributes to an unfavorable balance between kinetic and po­tential energy, thereby requiring large altitude changes to correct for small Mach number errors when flying a Mach hold mode using the elevator control.

In simulating SR-71A supersonic altitude and airspeed control problems NASA found it necessary to add to the normal equations of aircraft motion inlet geometry effects on airplane motion, inlet operating characteristics up to the unstart boundary, and the afterburning equations for the two engines. While with these additions simulation presents no unusual difficulties, attempts to find suitable theoretical models are another matter.

The applicable body of theory begins with Lanchester’s 1897 analysis of phugoid mo­tion. Lanchester’s model, and the Bryan and Williams analyses that followed, neglected atmospheric density changes as an airplane’s height changes during a longitudinal oscilla­tion. F. N. Scheubel added density gradient to the mathematical model in 1942. In 1950, Stefan Neumark added the effects of thrust and sound speed variations with altitude to the equations.

While the classical Bryan-Williams model leads to a fourth-degree characteristic equa­tion, with both short – and long-period longitudinal oscillations, density gradient increases the characteristic equation degree to 5. A new aperiodic height mode appears, typically a very slow divergence (Figure 11.16). The height mode was first identified, or rather predicted, by Neumark. The supersonic altitude stability problems thus far encountered probably involve both the phugoid or long-period mode and the height mode.

Thrust effects are significant on both the phugoid and height modes (Stengel, 1970; Sachs, 1990). Aside from the effects of possible thrust offsets from the airplane’s center of gravity, the throttle-fixed variation in thrust with airspeed affects both the phugoid and height modes. Both modes are stabilized when thrust decreases with increasing airspeed, and vice versa.

Lanchester’s original analysis (Durand, 1934) assumes an airplane whose lift is always at right angles to the flight path and numerically proportional to the square of the airspeed. These simple assumptions and small-angle approximations lead to Lanchester’s phugoid motion, an undamped oscillation of period ^2n V/g, where V is the flight velocity and g is the acceleration of gravity.

Supersonic Altitude Stability

Figure 11.16 Effect of altitude on the phugoid and height modes of a hypothetical SST, cruising at a Mach number of 3.0. (From Stengel, Jour of Aircraft, Sept.-Oct. 1970)

However, the linear increase of period with airspeed predicted by Lanchester does not occur at high airspeeds. The reason is that the density gradient effect that Scheubel wrote about in 1942 becomes very important at high airspeeds. The phugoid period is shortened compared with the Lanchester case. In effect, as the airplane noses down, picking up speed and giving up potential energy for kinetic energy, higher density at the lower altitude increases lift, bending the path upward again. Higher density at lower altitudes acts as an extra spring, shortening the period.

A simplified model developed in 1965 by Lockheed’s John R. McMaster predicts drastic reductions in the phugoid period relative to the classic Lanchester values at high airspeeds. McMaster’s predicted period at a Mach number of 3.0 is about 150 seconds, compared with the Lanchester value of 401 seconds. Calculations by Stengel in 1970 for an SST configuration at a Mach number of 3.0 give a phugoid period of about 160 seconds, close to the McMaster value. A later simplified model (Etkin, 1972) shows a reduction from the Lanchester values, but not quite so large.

A formulation by Regan (1993) also corrects the Lanchester approximation for density gradient. Regan’s approximation may be derived from the small-perturbation longitudinal equations of motion of Figure 18.4 by adding a height degree of freedom and the height derivative dZ/dh, where Z is the Z-axis aerodynamic force and h is altitude perturbation from equilibrium flight. The Regan approximation for phugoid period is

Подпись: 400
Supersonic Altitude Stability
Supersonic Altitude Stability
Supersonic Altitude Stability
Подпись: Lanchester
Подпись: altitude= 800.

Supersonic Altitude StabilityMach number

Figure 11.17 Variation of approximate phugoid period with altitude and Mach number, including density gradient with height effects.

Period = 2ж(V0/g)/(2 – (1/p)(dp/dh)(Vo/g))1/2,

where V0 = equilibrium flight speed,

p = equilibrium air density,

g = acceleration of gravity at the equilibrium altitude,

dp/dh = density gradient with altitude at the equilibrium altitude.

This relationship is used to show the general trend of phugoid period with Mach number and altitude in Figure 11.17. Omitted are possible thrust effects. Figure 11.17 shows that density gradient causes the phugoid period to reach asymptotic values as airspeed increases indefinitely, which is at odds with the classical Lanchester approximation. Neglect of density gradient incorrectly doubles the approximated phugoid period at an altitude of 200,000 feet and Mach number = 2.

Figure 11.17 predicts a phugoid period of 154 seconds at Mach 3 at altitudes below 400,000 feet. This is close to McMaster’s predicted value for Mach number 3 and also to Stengel’sresultsin Figure 11.16. Eigenvalue calculationsfor the NASA GHAME hypersonic vehicle give remarkably close results to the Regan approximation for the phugoid period at Mach numbers above 2.

Flight test data that would support one simplified phugoid model or another seem not to be available, for good reason. Typically, airplanes are flown at those very high Mach numbers and altitudes with either or both altitude and Mach hold control loops active, as a direct result of the altitude stability problem. Loop closures modify the basic phugoid

Supersonic Altitude Stability

Figure 11.18 YF-12A altitude hold performance with an optimized autopilot, at Mach 3, altitude 77,500 feet. Altitude oscillation about the desired value is held to within plus or minus 25 feet. (From Gilyard and Smith, NASA CP 2054, 1978)

motion to the point where its period and damping would be difficult to detect, even if tests could be devised to measure periods as long as 160 seconds.

NASA flight tests of the YF-12A are encouraging in that properly designed and com­pensated altitude and Mach hold systems, working through the pitch and thrust controls, respectively, seem to be able to hold reasonably stable cruise conditions at Mach 3 (Figure 11.18). One physical limitation is instability in the atmosphere itself, notably tem­perature shears that change the indicated Mach number even though the true airspeed and altitude have not changed. Gilyard and Smith noted that baseline YF-12A altitude hold mode operation varied from day to day. “Occasionally altitude could be held reasonably constant; at other times, it diverged in an unacceptable manner.”

A related phenomenon was found in flight testing the XB-70 around a Mach number of 3. Indicated altitude changes of 1,000 feet were seen in 2 or 3 seconds, quite evidently the result of atmospheric temperature gradients, since the airplane could not have possibly changed altitude so quickly. To avoid having altitude and Mach hold systems chase after atmospheric instabilities, it may be necessary to smooth atmospheric data with inertial data or position measurements derived from satellites.