Category AVIATORS

In addition to directional stability, the air­plane must have adequate directional control to coordinate turns, balance power effects, create sideslip, balance unsymmetrical power, etc. The principal source of directional con­trol is the rudder and the rudder must be capable of producing sufficient yawing moment for the critical conditions of flight.

The effect of rudder deflection is to produce a yawing moment coefficient according to control deflection and produce equilibrium at some angle of sideslip. For small deflections of the rudder, there is no change in stability but a change in equilibrium. Figure 4.25 shows the effect of rudder deflection on yawing moment coefficient curves with the change in equilibrium sideslip angle.

If the airplane exhibits static directional stability with rudder fixed, each angle of side­slip requires a particular deflection of the rudder to achieve equilibrium. Rudder-free directional stability will exist when the float angle of the rudder is less than the rudder deflection required for equilibrium. However at high angles of sideslip, the floating tend­ency of the rudder increases. This is illus­trated by the second chart of figure 4.25 where the line of rudder float angle shows a sharp increase at large values of sideslip. If the floating angle of the rudder catches up with the required rudder angle, the rudder pedal force will decrease to zero and rudder lock will occur. Sideslip angles beyond this point pro­duce a floating angle greater than the required rudder deflection and the rudder tends to float to the limit of deflection.

Rudder lock is accompanied by a reversal of pedal force and rudder-free instability will exist. The dorsal fin is a useful addition in this case since it will improve the directional stability at high angles of sideslip. The re­sulting increase in stability requires larger deflections of the rudder to achieve equilibrium at high sideslip and the tendency for rudder lock is reduced.

Rudder-free directional stability is appre­ciated by the pilot as the rudder pedal force to maintain a given sideslip. If the rudder pedal force gradient is too low near zero sideslip, it will be difficult to maintain zero sideslip dur­ing various maneuvers. The airplane should have a stable rudder pedal feel through the available range of sideslip.

DIRECTIONAL CONTROL REQUIRE­MENTS. The control power of the rudder must be adequate to contend with the many unsymmetrical conditions of flight. Gener­ally, there are five conditions of flight which provide the most critical requirements of di­rectional control power. The type and mission of the airplane will decide which of these conditions is most important.

ADVERSE YAW. When an airplane is rolled into a turn yawing moments are pro­duced which require rudder deflection to main­tain zero sideslip, i. e., coordinate the turn. The usual source of adverse yawing moment is illustrated in figure 4.26. When the airplane shown is subject to a roll to the left, the down­going port wing will experience a new relative wind and an increase in angle of attack. The inclination of the lift vector produces a com­ponent force forward on the downgoing wing. The upgoing starboard wing has its lift in­clined with a component force aft. The re­sulting yawing moment due to rolling motion is in a direction opposite to the roll and is hence “adverse yaw." The yaw due to roll is primarily a function of the wing lift coefficient and is greatest at high C/,.

In addition to the yaw due to rolling motion there will be a yawing moment contribution due to control surface deflection. Conventional ailerons usually contribute an adverse yaw while spoilers may contribute a favorable or “proverse” yaw. The high wing airplane with a large vertical tail may encounter an influence from inboard ailerons. Such a con­figuration may induce flow directions at the vertical tail to cause proverse yaw.

Since adverse yaw will be greatest at high CL and full deflection of the ailerons, coordi­nating steep turns at low speed may produce a critical requirement for rudder control power.

SPIN RECOVERY. In the majority of air­planes, the rudder is the principal control for spin recovery. Powerful control of sideslip at

high angles of attack is required to effect re­covery during a spin. Since the effectiveness of the vertical tail is reduced at large angles of attack, the directional control power neces­sary for spin recovery may produce a critical requirement of rudder power.

SLIPSTREAM ROTATION. A critical di­rectional control requirement may exist when the propeller powered airplane is at high power and low airspeed. As shown in figure 4.26, the single rotation propeller induces a slipstream swirl which causes a change in flow direction at the vertical tail. The rudder must furnish sufficient control power to balance this condition and achieve zero sideslip.

CROSSWIND TAKEOFF AND LANDING. Since the airplane must make a true path down the runway, a crosswind during takeoff or landing will require that the airplane be con­trolled in a sideslip. The rudder must have sufficient control power to create the required sideslip for the expected crosswinds.

ASYMMETRICAL POWER. The design of a multiengine airplane must account for the possibility of an engine failure at low airspeed. The unbalance of thrust from a condition of unsymmetrical power produces a yawing mo­ment dependent upon the thrust unbalance and the lever arm of the force. The deflection of the rudder will create a side force on the tail and contribute a yawing moment to balance the yawing moment due to the unbalance of thrust. Since the yawing moment coefficient from the unbalance of thrust will be greatest at low speed, the critical requirement will be at a low speed with the one critical engine out and the remaining engines at maximum power. Figure 4.26 compares the yawing moment coefficient for maximum rudder deflec­tion with the yawing moment coefficient for the unbalance of thrust. The intersection of the two lines determines the minimum speed for directional control, i. e., the lowest speed at which the rudder control moment can equal the moment of unbalanced thrust. It is usually specified that the minimum directional control speed be no greater than 1.2 times the stall

speed of the airplane in the lightest practical takeoff configuration. This will provide ade­quate directional control for the remaining conditions of flight.

Once defined, the minimum directional con­trol speed is not a function of weight, altitude, etc., but is simply the equivalent airspeed (or dynamic pressure), to produce a required yaw­ing moment with the maximum rudder deflec­tion. If the airplane is operated in the critical unbalance of power below the minimum con­trol speed, the airplane will yaw uncontrolla­bly into the inoperative engine. In order to regain directional control below the minimum speed certain alternatives exist: reduce power on the operating engines or sacrifice altitude for airspeed. Neither alternative is satisfac­tory if the airplane is in a marginal condition of powered flight so due respect must be given to the minimum control speed.

Due to the side force on the vertical tail, a slight bank is necessary to prevent turning flight at zero sideslip. The inoperative engine will be raised and the inclined wing lift will provide a component of force to balance the side force on the tail.

In each of the critical conditions of required directional control, high directional stability is desirable as it will reduce the displacement of the aircraft from any disturbing influence. Of course, directional control must be sufficient to attain zero sideslip. The critical control requirement for the multiengine airplane is the condition of asymmetrical power since spinning is not common to this type of airplane. The single engine propeller airplane may have either the spin recovery or the slipstream rota­tion as a critical design condition. The single engine jet airplane may have a variety of critical items but the spin recovery require­ment usually predominates.

DIRECTIONAL STABILITY

The directional stability of an airplane is essentially the “weathercock” stability and involves moments about the vertical axis and their relationship with yaw or sideslip angle. An airplane which has static directional sta­bility would tend to return to an equilibrium when subjected to some disturbance from equi­librium. Evidence of static directional sta­bility would be the development of yawing moments which tend to restore the airplane to equilibrium.

DEFINITIONS. The axis system of an air­plane will define a positive yawing moment, N, as a moment about the vertical axis which tends to rotate the nose to the right. As in other aerodynamic considerations, it is con­venient to consider yawing moments in the coefficient form so that static stability can be evaluated independent of weight, altitude, speed, etc. The yawing moment, N, is de­fined in the coefficient form by the following equation:

N = CnqSb or

where

N= yawing moment, ft.-lbs;

positive to the right q = dynamic pressure, psf T = wing area, sq. ft. b = wing span, ft.

C„ = yawing moment coefficient, positive to the right

The yawing moment coefficient, C„, is based on the wing dimensions. S’ and b as the wing is the characteristic surface of the airplane.

The yaw angle of an airplane relates the dis­placement of the airplane centerline from some reference azimuth and is assigned the short­, hand notation ф (psi). A positive yaw angle occurs when the nose of the airplane is dis­placed to the right of the azimuth direction. The definition of sideslip angle involves a sig­nificant difference. Sideslip angle relates the displacement of the airplane centerline from the relative wind rather than some reference azimuth. Sideslip angle is provided the short­hand notation /8 (beta) and is positive when the relative wind is displaced to the right of the airplane centerline. Figure 4.22 illustrates the definitions of sideslip and yaw angles.

The sideslip angle, /3, is essentially the di­rectional angle of attack of the airplane and is the primary reference in lateral stability as well as directional stability considerations. The yaw angle, ф, is a primary reference for wind tunnel tests and time history motion of an airplane. From the definitions there is no direct relationship between /3 and ф for an airplane in free flight, e. g., an airplane flown through a 360° turn has yawed 360° but side­slip may have been zero throughout the entire turn. Since the airplane has no directional sense, static directional stability of the air­plane is appreciated by response to sideslip.

The static dinctional stability of an airplane can be illustrated by a graph of yawing moment coefficient, C„} versus sideslip angle, such as shown in figure 4.22. When the airplane is subject to a positive sideslip angle, static direc­tional stability will be evident if a positive yawing moment coefficient results. Thus, when the relative wind comes from the right (+j3), a yawing moment to the right (+C.) should be created which tends to weathercock the airplane and return the nose into the wind. Static directional stability will exist when the curve of Cn versus (3 has a positive slope and the degree of stability will be a function of the slope of this curve. If the curve has zero slope, there is no tendency to return to equilibrium and neutral static directional stability exists. When the curve of Cn versus 0 has a negative slope, the yawing moments developed by side­slip tend to diverge rather than restore and static directional instability exists.

The final chart of figure 4.22 illustrates the fact that the instantaneous slope of the curve of Cn versus /3 will describe the static directional stability of the airplane. At small angles of sideslip a strong positive slope depicts strong directional stability. Large angles of sideslip produce zero slope and neutral stability. At very high sideslip the negative slope of the curve indicates directional instability. This decay of directional stability with increased sideslip is not an unusual condition. However, directional instability should not occur at the angles of sideslip of ordinary flight conditions.

Static directional stability must be in evi­dence for all the critical conditions of flight. Generally, good directional stability is a fun­damental quality directly affecting the pilots’ impression of an airplane.

CONTRIBUTION OF THE AIRPLANE COMPONENTS. The static directional sta­bility of the airplane is a result of contribution ofi each of the various airplane components. While the contribution of each component is somewhat dependent upon and related to other components, it is necessary to study each component separately.

The vertical tail is the primary source of directional stability for the airplane. As shown in figure 4.23, when the airplane is in a sideslip the vertical tail will experience a change in angle of attack. The change in lift—or side force—on the vertical tail creates a yawing moment about the center of gravity which tends to yaw the airplane into the relative wind. The magnitude of the vertical tail contribution to static directional stability then depends on the change in tail lift and the tail moment arm. Obviously, the tail moment arm is a powerful factor but essentially dic­tated by the major configuration properties of the airplane.

When the location of the vertical tail is set, the contribution of the surface to directional stability depends on its ability to produce changes in lift—or side force—with changes in sideslip. The surface area of the vertical tail is a powerful factor with the contribution of the vertical tail being a direct function of the area. When all other possibilities are ex­hausted, the required directional stability may be obtained by increases in tail area. How­ever, increased surface area has the obvious disadvantage of increased drag.

The lift curve slope of the vertical tail relates how sensitive the surface is to changes in angle of attack. While it is desirable to have a high lift curve slope for the vertical surface, a high aspect ratio surface is not necessarily practical or desirable. The stall

angle of the surface must be sufficiently great to prevent stall and subsequent loss of effec­tiveness at ordinary sideslip angles. The high Mach numbers of supersonic flight produces a decrease in lift curve slope with the consequent reduction in tail contribution to stability. In order to have sufficient directional stability at high Mach numbers, the typical supersonic configuration will exhibit relatively large vertical tail surfaces.

The flow field in which the vertical tail operates is affected by the other components of the airplane as well as power effects. The dynamic pressure at the vertical tail could depend on the slipstream of a propeller or the boundary layer of the fuselage. Also, the local flow direction at the vertical tail is in­fluenced by the wing wake, fuselage crossflow, induced flow of the horizontal tail, or the direction of slipstream from a propeller. Each of these factors must be considered as possibly affecting the contribution of the vertical tail to directional stability.

The contribution of the wing to static direc­tional stability is usually small. The swept wing provides a stable contribution depending on the amount of sweepback but the contribu­tion is relatively weak when compared with other components. –

The contribution of the fuselage and nacelles is of primary importance since these compo­nents furnish the greatest destabilizing in­fluence. The contribution of the fuselage and nacelles is similar to the longitudinal case with the exception that there is no large in­fluence of the induced flow field of the wing. The subsonic center of pressure of the fuselage will be located at or forward of the quarter- length point and, since the airplane c. g. is usually considerably aft of this point, the fuselage contribution will be destabilizing. However, at large angles of sideslip the large destabilizing contribution of the fuselage di­minishes which is some relief to the problem of maintaining directional stability at large displacements. The supersonic pressure. dis­tribution on the body provides a relatively greater aerodynamic force and, generally, a continued destabilizing influence.

Figure 4.23 illustrates a typical buildup of the directional stability of an airplane by separating the contribution of the fuselage and tail. As shown by the graph of C„ versus 0, the contribution of the fuselage is de­stabilizing but the instability decreases at large sideslip angles. The contribution of the vertical tail alone is highly stabilizing up to the point where the surface begins to stall. The contribution of the vertical tail must be large enough so that the complete airplane (wing-fuselage-tail combination) exhibits the required degree of stability.

The dorsal fin has a powerful effect on pre­serving the directional stability at large angles of sideslip which would produce stall of the vertical tail. The addition of a dorsal fin to the airplane will allay the decay of directional stability at high sideslip in two ways. The least obvious but most important effect is a large increase in the fuselage stability at large sideslip angles. In addition, the effective aspect ratio of the vertical tail is reduced which increases the stall angle for the surface. By this twofold effect, the addition of the dorsal fin is a v useful device.

Power effects on static directional stability are similar to the power effects on static longitudinal stability. The direct effects are confined to the normal force at the propeller plane or the jet inlet and, of course, are de­stabilizing when the propeller or inlet is located ahead of the c. g. The indirect effects of power induced velocities and flow direction changes at the vertical tail are quite significant for the propeller driven airplane and can pro­duce large directional trim changes. As in the lontitudinal case, the indirect effects are negligible for the jet powered airplane.

The contribution of the direct and indirect power effects to static directional stability is greatest for the propeller powered airplane and usually slight for the jet powered airplane. In either case, the general effect of power is

EFFECT OF RUDDER FLOAT ON STATIC
DIRECTIONAL STABILITY

destabilizing and the greatest contribution will occur at high power and low dynamic pressure as during a waveoff.

As in the case of longitudinal static stability, freeing the controls will reduce the effective­ness of the tail and alter the stability. While the rudder must be balanced to reduce control pedal forces, the rudder will tend to float or streamline and reduce the contribution of the vertical tail to static directional stability. The floating tendency is greatest at large angles of sideslip where large angles of attack for the vertical tail tend to decrease aerodynamic bal­ance. Figure 4.24 illustrates the difference be­tween rudder-fixed and rudder-free static di­rectional stability.

CRITICAL CONDITIONS. The most criti­cal conditions of static directional stability are usually the combination of several separate effects. The combination which produces the most critical condition is much dependent upon the type and mission of the airplane. Tn addi­tion, there exists a coupling of lateral and di­rectional effects such that the required degree of static directional stability may be deter­mined by some of these coupled conditions.

Center of gravity position has a relatively negligible effect on static directional stability. The usual range of c. g. position on any air­plane is set by the linits of longitudinal stability and control. Within this limiting range of c. g. position, no significant changes take place in the contribution of the vertical tail, fuselage, nacelles, etc. Hence, the static directional stability is essentially unaffected by the varia­tion of c. g. position within the longitudinal limits.

When the airplane is at a high angle of attack a decrease in static directional stability can be anticipated. As shown by the second chart of figure 4.24, a high angle of attack reduces the stable slope of the curve of Cn versus 0. The decrease in static directional stability is due in great part to the reduction in the contribution of the vertical tail. At high angles of attack, the effectiveness of the vertical tail is reduced because of increase in the fuselage boundary layer at the vertical tail location. The decay of directional stability with angle of attack is most significant for the low aspect ratio air­plane with sweepback since this configuration requires such high angles of attack to achieve high lift coefficients. Such decay in directional stability can have a profound effect on the re­sponse of the airplane to adverse yaw and spin characteristics.

High Mach numbers of supersonic flight reduce the contribution of the vertical tail to direc­tional stability because of the reduction of lift curve slope with Mach number. The third chart of figure 4.24 illustrates the typical decay of directional stability with Mach number. To produce the required directional stability at high Mach numbers, a very large vertical tail area may be necessary. Ventral fins may be added as an additional contribution to direc­tional stability but landing clearance require­ments may limit their size or require the fins to be retractable.

Hence, the most critical demands of static directional stability will occur from some combination of the following effects:

(1) high angle of sideslip

(2) high power at low airspeed

0) high angle of attack

(4) high Mach number The propeller powered airplane may have such considerable power effects that the critical conditions may occur at low speed while the effect of high Mach numbers may produce the critical conditions for the typical supersonic airplane. In addition, the coupling of lateral and directional effects may require prescribed degrees of directional stability.

In order to accomplish the stability and control objectives, various configurations of control systems are necessary. Generally, the type of flight control system is decided by the size and flight speed range of the airplane.

The conventional control system consists of direct mechanical linkages from the controls to the control surfaces. For the subsonic airplane, the principal means of producing proper control forces utilize aerodynamic bal­ance and various tab, spring, and bobweight devices. Balance and tab devices are capable of reducing control forces and will allow the use of the conventional control system on large airplanes to relatively high subsonic speeds.

When the airplane with a conventional control system is operated at transonic speeds, the great changes in the character of flow can produce great aberrations in control sur­face hinge moments and the contribution of tab devices. Shock wave formation and separation of flow at transonic speeds will limit the use of the conventional control system to subsonic speeds.

The – power-boosted control system employs a mechanical actuator in parallel with the mechanical linkages of a conventional control system. The principle of operation is to pro­vide a fixed, percentage of the required control forces thus reducing control forces at high speeds. The power-boosted control system requires a hydraulic actuator with a control valve which supplies boost force in fixed proportion to control force. Thus, the pilot is given an advantage by the boost ratio to assist in deflecting the control surface, e. g., with a boost ratio of 14, the actuator provides 14 lbs. of force for each 1 lb. of stick force.

The power-boosted control system has the obvious advantage of reducing control forces at high speeds. However, at transonic speeds, the changes in control forces due to shock waves and separation still take place but to a lesser degree. The "feedback” of hinge moments is reduced but the aberrations in stick forces may still exist.

The power-operated, irreversible control system consists of mechanical actuators controlled by the pilot. The control surface is deflected by the actuator and none of the hinge moments are fed back through the controls. In such a control system, the control position decides the deflection of the control surfaces regardless of the airloads and hinge moments. Since the power-operated control system has zero feed­back, control feel must be synthesized other­wise an infinite boost would exist.

The advantages of the power-operated con­trol system are most apparent in transonic and supersonic flight. In transonic flight, none of the erratic hinge moments are fed back to the pilot. Thus, no unusual or erratic control forces will be encountered in transonic flight. Supersonic flight generally requires the use of an all-movable horizontal surface to achieve the necessary control effectiveness. Such con­trol surfaces must then be actuated and posi­tively positioned by an irreversible device.

The most important item of an artificial feel system is the stick-centering spring or bungee. The bungee develops a stick force in proportion to stick displacement and thus provides feel for airspeed and maneuvers. A bobweight may be included in the feel system to develop a steady positive maneuvering stick force gradient which is independent of airspeed for ordinary maneuvers.

The gearing between the stick position and control surface deflection is not necessarily a linear relationship. The majority of powered control systems will employ a nonlinear gear­ing such that relatively greater stick deflection per surface deflection will occur at the neutral stick position. This sort of gearing is to advantage for airplanes which operate at flight conditions of high dynamic pressure. Since the airplane at high q is very sensitive to small deflections of the control surface, the nonlinear gearing provides higher stick force stability with less sensitive control movements than the system with a linear gearing. Figure 4-21 illustrates a typical linear and nonlinear control system gearing.

The second chart of figure 4-21 illustrates the typical control system stick force variation

with control surface deflection. While it is desirable to have a strong centering of the stick near the neutral position, the amount of force required to create an initial displacement must be reasonable. If the control system “break-out” forces are too high, precise control of the airplane at high speeds is difficult. As the solid friction of the control system con­tributes to the break-out forces, proper mainte­nance of the control system is essential. Any increase in control system friction can create unusual and undesirable control forces.

The trim of the powered control system is essentially any device to produce zero control force for a given control surface deflection. One system may trim off bungee force at a given stick position while another system may trim by returning the stick to neutral position.

Flight at high supersonic Mach numbers might require a great variety of devices in the longitudinal control system. The deteriora­tion of pitch damping with Mach number may require that dynamic stability be obtained synthetically by pitch dampers in the control system. The response of the airplane to longitudinal control may be adversely affected by flight at high dynamic pressures. In such conditions of flight stick forces must be ade­quate to prevent an induced oscillation. Stick forces must relate the transients of flight as well as the steady state conditions. Such a contribution to control system forces may be provided by a pitching acceleration bobweight and a control system viscous damper.

To be satisfactory, an airplane must have adequate controllability as well as adequate stability. An airplane with high static longi­tudinal stability will exhibit great resistance to displacement from equilibrium. Hence, the most critical conditions of controllability will occur when the airplane has high sta­bility, i. e., the lower limits of controllability will set the upper limits of stability.

There are three principal conditions of flight which provide the critical requirements of longitudinal control power. Any one or combination of these conditions can de­termine the longitudinal control power and set a limit to forward c. g. position.

MANEUVERING CONTROL REQUIRE­MENT. The airplane should have sufficient longitudinal control power to attain the maxi­mum usable lift coefficient or limit load factor during maneuvers. As shown in figure 4.19, forward movement of the c. g. increases the longitudinal stability of an airplane and requires larger control deflections to produce changes in trim lift coefficient. For the example shown, the maximum effective de­flection of the elevator is not capable of trim – ing the airplane at CL/nar for c. g. positions ahead of 18 percent MAC.

This particular control requirement can be most critical for an airplane in supersonic flight. Supersonic flight is usually accom­panied by large increases in static longitu­dinal stability and a reduction in the effective­ness of control surfaces. In order to cope with these trends, powerful all-movable surfaces must be used to attain limit load factor or maximum usable CL in supersonic flight. This requirement is so important that once satis­fied, the supersonic configuration usually has sufficient longitudinal control power for all other conditions of flight.

TAKEOFF CONTROL REQUIREMENT. At takeoff, the airplane must have sufficient control power to assume the takeoff attitude prior to reaching takeoff speed. Generally, for airplanes with tricycle landing gears, it is desirable to have at least sufficient control power to attain the takeoff attitude at 80

percent of the stall speed for propeller air­planes or 90 percent of the stall speed for jet airplanes. This feat must be accomplished on a smooth runway at all normal service takeoff loading conditions.

Figure 4.19 illustrates the principal forces acting on an airplane during takeoff roll. When the airplane is in the three point attitude at some speed less than the stall speed, the wing lift will be less than the weight of the airplane. As the elevators must be capable of rotating to the takeoff attitude, the critical condition will be with zero load on the nose wheel and the net of lift and weight supported on the main gear. Rolling friction resulting from the normal force on the main gear creates an adverse nose down moment. Also, the center of gravity ahead of the main gear contributes a nose down moment and this consideration could decide the most aft loca­tion of the main landing gear during design. The wing may contribute a large nose down moment when flaps are deflected but this effect may be countered by a slight increase in downwash at the tail. To balance these nose down moments, the horizontal tail should be capable of producing sufficient nose up moment to attain the takeoff attitude at the specified speeds.

The propeller airplane at takeoff power may induce considerable slipstream velocity at the horizontal tail which can provide an increase in the efficiency of the surface. The jet airplane does not experience a similar magni­tude of this effect since the induced velocities from the jet are relatively small compared to the slipstream velocities from a propeller.

LANDING CONTROL REQUIREMENT At landing, the airplane must have sufficient control power to ensure adequate control at specified landing speeds. Adequate landing control is usually assured if the elevators are capable of holding the airplane just off the runway at 105 percent of the stall speed. Of course, the most critical requirement will exist when the c. g. is in the most forward position, flaps are fully extended, and power is set at idle. This configuration will provide the most stable condition which is most demand­ing of controllability. The full deflection of flaps usually provides the greatest wing diving moment and idle power will produce the most critical (least) dynamic pressure at the hori­zontal tail.

The landing control requirement has one particular difference from the maneuvering control requirement of free flight. As the airplane approaches the ground surface, there will be a change in the three-dimensional flow of the airplane due to ground effect. A wing in proximity to the ground plane will experience a decrease in tip vortices and downwash at a given lift coefficient. The decrease in down – wash at the tail tends to increase the static stability and produce a nosedown moment from the reduction in download on the tail. Thus, the airplane just off the runway surface will require additional control deflection to trim at a given lift coefficient and the landing con­trol requirement may be critical in the design of longitudinal control power.

As an example of ground effect, a typical propeller powered airplane may require as much as 15° more up elevator to trim at CL in ground effect than in free flight away from the ground plane. Because of this effect, many ai rpl an es h a ve suffici en t con trol power t о a chi eve full stall out of ground effect but do not have the ability to achieve full stall when in close proximity to the ground.

In some cases the effectiveness of the control surface is adversely affected by the use of trim tabs. If trim tabs are used to excess in trim­ming stick forces, the effectiveness of the elevator. may be reduced to hinder landing or takeoff control.

Each of the three principal conditions re­quiring adequate longitudinal control are crit­ical for high static stability. If the forward c. g. limit is exceeded, the airplane may en­counter a deficiency of controllability in any of these conditions. Thus, the forward c. g.

limit is set by the minimum permissible con­trollability while the aft c. g. limit is set by the minimum permissible stability. LONGITUDINAL DYNAMIC STABILITY.

All previous considerations of longitudinal stability have been concerned with the initial tendency of the airplane to return to equilib­rium when subjected to a disturbance. The considerations of longitudinal dynamic sta­bility are concerned with time history response of the airplane to these disturbances, i. e., the variation of displacement amplitude with time following a disturbance. From previous defi­nition, dynamic stability will exist when the amplitude of motion decreases with time and dynamic instability will exist if the amplitude increases with time.

Of course, the airplane must demonstrate positive dynamic stability for the major longi­tudinal motions. In addition, the airplane must demonstrate a certain degree of longitu­dinal stability by reducing the amplitude of motion at a certain rate. The required degree of dynamic stability is usually specified by the time necessary for the amplitude to reduce to one-half the original value—the time to damp to half-amplitude.

The airplane in free flight has six degrees of freedom: rotation in roll, pitch, and yaw and translation in the horizontal, vertical, and lateral directions. In the case of longitudinal dynamic stability, the degrees of freedom can be limited to pitch rotation, vertical and horizontal translation. Since the airplane is usually symmetrical from port to starboard, there will be no necessity for consideration of coupling between longitudinal and lateral – directional motions, Thus, the principal vari­ables in the longitudinal motion of an airplane will be:

(1) The pitch attitude of the airplane.

(2) The angle of attack (which will differ

from the pitch attitude by the inclination of

the flight path).

(3) The flight velocity.

(4) The displacement or deflection of the elevator when the stick-free condition is considered.

The longitudinal dynamic stability of an airplane generally consists of three basic modes (or manners) of oscillation. While the longi­tudinal motion of the airplane may consist of a combination of these modes, the characteristics of each mode are sufficiently distinct that each oscillatory tendency may be studied separately.

The first mode of dynamic longitudinal sta­bility consists of a very long period oscillation referred to as the phugoid. The phugoid or long period oscillation involves noticeable varia­tions in pitch attitude, altitude, and airspeed but nearly constant angle of attack. Such an oscillation of the airplane could be considered as a gradual interchange of potential and kinetic energy about some equilibrium airspeed and altitude. Figure 4.20 illustrates the char­acteristic motion of the phugoid.

The period of oscillation in the phugoid is quite large, typical values being from 20 to 100 seconds. Since the pitching rate is quite low and only negligible changes in angle of attack take place, damping of the phugoid is weak and possibly negative. However, such weak or negative damping does not necessarily have any great consequence. Since the period of oscilla­tion is so great, the pilot is easily able to counteract the oscillatory tendency by very slight and unnoticed control movements. In most cases, the necessary corrections are so slight that the pilot may be completely un­aware of the oscillatory tendency.

Due to the nature of the phugoid, it is not necessary to make any specific aerodynamic provisions to contend with the oscillation. The inherent long period of the oscillation al­lows study to be directed to more important oscillatory tendencies. Similarly, the differ­ences between the stick-fixed and stick-free phugoid are not of great importance.

The second mode of longitudinal dynamic sta­bility is a relatively short period motion that

1ST MODE OR PHUGOID

MOTION OCCURS AT ESSENTIALLY CONSTANT SPEEO

can be assumed to take place with negligible changes in velocity. The second mode consists of a pitching oscillation during which the air­plane is being restored to equilibrium by the static stability and the amplitude of oscillation decreased by pitch damping. The typical mo­tion is of relatively high frequency with a period of oscillation on the order of 6.5 to 5 seconds.

For the conventional subsonic airplane, the second mode stick-fixed is characterized by heavy damping with a time to damp to half amplitude of approximately 0.5 seconds. Usu­ally, if the airplane has static stability stick – fixed, the pitch damping contributed by the horizontal tail will assume sufficient dynamic stability for the short period oscillation. How­ever, the second mode stick-free has the possi­bility of weak damping or unstable oscilla­tions. This is the case where static stability does not automatically imply adequate dy­namic stability. The second mode stick-free is essentially a coupling of motion between the airplane short period pitching motion and ele­vator in rotation about the hinge line. Ex­treme care must be taken in the design of the control surfaces to ensure dynamic stability for this mode. The elevators must be statically balanced about the hinge line and aerodynamic balance must be within certain limits. Control system friction must be minimized as it con­tributes to the oscillatory tendency. If insta­bility were to exist in the second mode, “por­poising” of the airplane would result with possibility of structural damage. An oscilla­tion at high dynamic pressures with large changes in angle of attack could produce severe flight loads.

The second mode has relatively short periods that correspond closely with the normal pilot response lag time, e. g., 1 or 2 seconds or less. There is the possibility that an attempt to forceably damp an oscillation may actually re­inforce the oscillation and produce instability. This is particularly true in the case of powered controls where a small input energy into the control system is greatly magnified. In addi­tion, response lag of the controls may add to the problem of attempting to forceably damp the oscillation. In this case, should an oscilla­tion appear, the best rule is to release the con­trols as the airplane stick-free will demonstrate the necessary damping. Even an attempt to fix the controls when the airplane is oscillating may result in a small unstable input into the control system which can reinforce the oscilla­tion to produce failing flight loads. Because of the very short period of the oscillation, the amplitude of an unstable oscillation can reach dangerous proportions in an extremely short period of time.

The third mode occurs in the elevator free case and is usually a very short period oscillation. The motion is essentially one of the elevator flapping about the hinge line and, in most cases, the oscillation has very heavy damping. A typical flapping mode may have a period of 0.3 to 1.5 seconds and a time to damp to half­amplitude of approximately 0.1 second.

Of all the modes of longitudinal dynamic stability, the second mode or porpoising oscil­lation is of greatest importance. The por­poising oscillation has the possibility of damaging flight loads and can be adversely affected by pilot response lag. It should be remembered that when stick-free the airplane will demonstrate the necessary damping.

The problems of dynamic stability are acute under certain conditions of flight. Low static stability generally increases the period (de­creases frequency) of the short period oscil­lations and increases the time to damp to half­amplitude. High altitude—and consequently low density—reduces the aerodynamic damp­ing. Also, high Mach numbers of supersonic flight produce a decay of aerodynamic damping.

STATIC LONGITUDINAL STABILITY

GENERAL CONSIDERATIONS. An air­craft will exhibit positive static longitudinal stability if it tends to return to the trim angle of attack when displaced by a gust or control movement. The aircraft which is unstable will continue to pitch in the disturbed direction until the displacement is resisted by opposing control forces. If the aircraft is neutrally stable, it tends to remain at any displacement to which it is disturbed. It is most necessary to provide an airplane with positive static longitudinal stability. The stable airplane is safe and easy to fly since the airplane seeks and tends to maintain a trimmed condition of flight. It also follows that control deflec­tions and control “feel’’ are logical in direction
and magnitude. Neutral static longitudinal stability usually defines the lower limit of. airplane stability since it’ is the boundary between stability and instability. The air­plane with neutral static stability may be excessively responsive to controls and the aircraft has no tendency to return to trim fol­lowing a disturbance. The airplane with negative static longitudinal stability is in­herently divergent from any intended trim condition. If it is at all possible to fly the aircraft, the aircraft cannot be trimmed and illogical control forces and deflections are re­quired to provide equilibrium with a change of attitude and airspeed.

Since static longitudinal stability depends upon the relationship of angle of attack and pitching moments, it is necessary to study the pitching moment contribution of each com­ponent of the aircraft. In a manner similar to all other aerodynamic forces, the pitching

moment about the lateral axis is studied in the coefficient form.

M = pitching moment about the c. g., ft.- lbs., positive if in a nose-up directioti q~ dynamic pressure, psf i’=wing area, sq. ft.

МЛС = mean aerodynamic chord, ft.

CM—pitching moment coefficient

The pitching moment coefficients contributed by all the various components of the aircraft are summed up and plotted versus lift coeffi­cient. Study of this plot of CM versus CL will relate the static longitudinal stability of the airplane.

Graph A of figure 4.5 illustrates the variation of pitching moment coefficient, CM, with lift coefficient, CL, for an airplane with positive static longitudinal stability. Evidence of static stability is shown by the tendency to re­turn to equilibrium—or “trim"—upon dis­placement. The airplane described by graph A is in trim or equilibrium when CM = 0 and, if the airplane is disturbed to some different CL, the pitching moment change tends to return the aircraft to the. point of trim. If the airplane were disturbed to some higher CL (point У), a negative or nose-down pitching moment is de­veloped which tends to decrease angle of attack back to the trim point. If the airplane were disturbed to some lower CL (point X), a posi­tive) or nose-up pitching moment is developed which tends to increase the angle of attack back to the trim point. Thus, positive static longitudinal stability is indicated by a negative slope of CM versus CL, i. e., positive stability is evidenced by a decrease in Cm with an increase in Ch.

The degree of static longitudinal stability is indicated by the slope of the curve of pitching moment coefficient with lift coefficient. Graph

В of figure 4.5 provides comparison of the stable and unstable conditions. Positive sta­bility is indicated by the curve with negative slope. Neutral static stability would be the result if the curve had zero slope. If neutral stability exists, the airplane could be dis­turbed to «оте higher or lower lift coefficient without change in pitching moment coefficient. Such a condition would indicate that the air­plane would have no tendency to return to some original equilibrium and would not hold trim. An airplane which demonstrates a posi­tive slope of the CM versus CL curve would be unstable. If the unstable airplane were subject to any disturbance from equilibrium at the trim point, the changes in pitching moment would only magnify the disturbance. When the unstable airplane is disturbed to some higher Cl, a positive change in CM occurs which would illustrate a tendency for continued, greater displacement. When the unstable air­plane is disturbed to some lower CL, a negative change in Cm takes place which tends to create continued displacement.

Ordinarily, the static longitudinal stability of a conventional airplane configuration does not vary with lift coefficient. In other words, the slope of CM versus CL does not change with Cb – However, if the airplane has sweepback, large contribution of power effects to stability, or significant changes in downwash at the horizontal tail, noticeable changes in static stability can occur at high lift coefficients. This condition is illustrated by graph C of figure 4.5. The curve of CM versus Cl of this illustration shows a good stable slope at low values of Сь. Increasing CL effects a slight decrease in the negative slope hence a decrease in stability occurs. With continued increase in CL, the slope becomes zero and neutral stability exists. Eventually, the slope be­comes positive and the airplane becomes un­stable or “pitch-up" results. Thus, at any lift coefficient, the static stability of the air­plane is depicted by the slope of the curve of CM versus CL.

CONTRIBUTION OF THE COMPONENT SURFACES. The net pitching moment about the lateral axis is due to the contribution of each of the component surfaces acting in their appropriate flow fields. By study of the con­tribution of each component the effect of each component on the static stability may be ap­preciated. It is necessary to recall that the pitching moment coefficient is defined as:

Thus, any pitching moment coefficient—re­gardless of source—has the common denomi­nator of dynamic pressure, q, wing area, S, and wing mean aerodynamic chord, MAC. This common denominator is applied to the pitch­ing moments contributed by the fuselage and nacelles, horizontal tail, and power effects as well as pitching moments contributed by the wing.

WING. The contribution of the wing to stability depends primarily on the location of the aerodynamic center with respect to the airplane center of gravity. Generally, the aerodynamic center—or a. c.—is defined as the point on the wing mean aerodynamic chord where the wing pitching moment coefficient does not vary with lift coefficient. All changes in lift coefficient effectively take place at the wing aerodynamic center. Thus, if the wing experiences some change in lift coefficient, the pitching moment created will be a direct function of the relative location of the a. c. and

c-g-

Since stability is evidenced by the develop­ment of restoring moments, the c. g. must be forward of the a. c. for the wing to contribute to positive static longitudinal stability. As shown in figure 4.6, a change in lift aft of the c. g. produces a stable restoring moment de­pendent upon the lever arm between the a. c. and c. g. In this case, the wing contribution would be stable and the curve of CM versus CL for the wing alone would have a negative slope. If the c. g. were located at the a. c., CM would not vary with CL since all changes in lift would take place at the c. g. In this case, the wing contribution to stability would be neutral. When the c. g. is located behind the a. c. the wing contribution is unstable and the curve of Cu versus CL for the wing alone would have a positive slope.

Since the wing is the predominating aero­dynamic surface of an airplane, any change in the wing contribution may produce a sig­nificant change in the airplane stability. This fact would be most apparent in the case of the flying wing or tailless airplane where the wing contribution determines the airplane stability. In order for the wing to achieve stability, the c. g. must be ahead of the a. c. Also, the wing must have a positive pitching moment about the aerodynamic center to achieve trim at positive lift coefficients. The first chart of figure 4.7 illustrates that the wing which is stable will trim at a negative lift coefficient if the CuAC is negative. If the stable wing has a positive С**лс it will then trim at a useful posi­tive CL. The only means available to achieve trim at a positive CL with a wing which has a negative CUac is an unstable c. g. position aft of the a. c. As a result, the tailless aircraft cannot utilize high lift devices which incur any significant changes in С**лс-

While the trim lift coefficient may be altered by a change in c. g. position, the resulting change in stability is undesirable and is unsat­isfactory as a primary means of control. The variation of trim CL by deflection of control surfaces is usually more effective and is less inviting of disaster. The early attempts at manned flight led to this conclusion.

When the aircraft is operating in subsonic flight, the a. c. of the wing remains fixed at the 25 percent chord station. When the aircraft is flown in supersonic flight, the a. c. of the wing will approach the 50 percent chord sta­tion. Such a large variation in the location of the a. c. can produce large changes in the wing contribution and greatly alter the air­plane longitudinal stability. The second chart

of figure 4.7 illustrates the change of wing contribution possible between subsonic and supersonic flight. The large increase in static stability in supersonic flight can incur high trim drag or require great control effectiveness to prevent reduction in maneuverability.

FUSELAGE AND NACELLES. In most cases, the contribution of the fuselage and nacelles is destabilizing. A symmetrical body of revolution in the flow field of a perfect fluid develops an unstable pitching moment when given an angle of attack. In fact, an increase in angle of attack produces an increase in the unstable pitching moment without the devel­opment of lift. Figure 4.8 illustrates the pres­sure distribution which creates this unstable moment on the body of revolution. In the actual case of real subsonic flow essentially the same effect is produced. An increase in angle of attack causes an increase in the unstable pitching moment but a negligible increase in lift.

An additional factor for consideration is the influence of the induced flow field of the wing. As illustrated in figure 4.8, the upwash ahead of the wing increases the destabilizing influence from the portions of the fuselage and nacelles ahead of the wing. The downwash behind the wing reduces the destabilizing influence from the portions of the fuselage and nacelles aft of the wing. Hence, the location of the fuselage and nacelles relative to the wing is important in determining the contribution to stability.

The body of revolution in supersonic flow can develop lift of a magnitude which cannot be neglected. When the body of revolution in supersonic flow is given an angle of attack, a pressure distribution typical of figure 4.8 is the result. Since the center of pressure is well forward, the body contributes a destabilizing influence. As is usual with supersonic con­figurations, the fuselage and nacelles may be quite large in comparison with the wing area and the contribution to stability may be large. Interaction between the wing and fuselage and nacelles deserves consideration in several in­stances. Body upwash and variation of local Mach number can influence the wing lift while lift carryover and downwash can effect the fu­selage and nacelles forces and moments.

HORIZONTAL TAIL. The horizontal tail usually provides the greatest stabilizing influ­ence of all the components of the airplane. To appreciate the contribution of the horizontal tail to stability, inspect figure 4-9- If the air­plane is given a change in angle of attack, a change in tail lift will occur at the aerody­namic center of the tail. An increase in lift at the horizontal tail produces a negative moment about the airplane e g. and tends to return the airplane to the trim condition. While the contribution of the horizontal tail to stability is large, the magnitude of the Contribution is dependent upon the change in tail lift and the lever arm of the surface. It is obvious that the horizontal tail will produce a stabilizing effect only when the surface is aft of the c. g. For this reason it would be inap­propriate to refer to the forward surface of a canard (tail-first) configuration as a horizontal “stabilizer.” In a logical sense, the horizontal “stabilizer” must be aft of the c. g. and— generally speaking—the farther aft, the greater the contribution to stability.

Many factors influence the change in tail lift which occurs with a change in airplane angle of attack. The area of the horizontal tail has the obvious effect that a large surface would generate a large change in lift. In a similar manner, the change in tail lift would depend on the slope of the lift curve for the horizontal tail. Thus, aspect ratio, taper, sweepback, and Mach number would deter­mine the sensitivity of the surface to changes in angle of attack. It should be appreciated that the flow at the horizontal tail is not of the same flow direction or dynamic pressure as the free stream. Due to the wing wake, fuse­lage boundary layer, and power effects, the q at the horizontal tail may be greatly different from the q of the free stream. In most ІП-

stances, the q at the tail is usually less and this reduces the efficiency of the tail.

When the airplane is given a change in angle of attack, the horizontal tail does not expe­rience the same change in angle of attack as the wing. Because of the increase in down – wash behind the wing, the horizontal tail will experience a smaller change in angle of attack, e. g., if a 10° change in wing angle of attack causes a 4° increase in downwash at the hori­zontal tail, the horizontal tail experiences only a 6° change in angle of attack. In this manner, the downwash at the horizontal tail reduces the contribution to stability. Any factor which alters the rate of change of down – wash at the horizontal tail will directly affect the tail contribution and airplane stability.

Power effects cah alter the downwash at the horizontal tail and affect the tail contribution. Also, the downwash at the tail is affected by the lift distribution on the wing and the flow condition on the fuselage. The low aspect ratio airplane requires large angles of attack to achieve high lift coefficients and this posi­tions the fuselage at high angles of attack. The change in the wing downwash can be accompanied by crossflow separation vortices on the fuselage. It is possible that the net effect obviates or destabilizes the contribu­tion of the horizontal tail and produces air­plane instability.

POWER-OFF STABILITY. When the in­trinsic stability of a configuration is of interest, power effects are neglected and the stability is considered by a buildup of the contributing components. Figure 4.10 illustrates a typical buildup of the components of a conventional airplane configuration. If the c. g. is arbi­trarily set at 30 percent MAC, the contribu­tion of the wing alone is destabilizing as indi­cated by the positive slope of CM versus CL. The combination of the wing and fuselage increases the instability. The contribution of the tail alone is highly stabilizing from the large negative slope of the curve. The contribution of the tail must be sufficiently stabilizing so that the complete configuration will exhibit positive static stability at the anticipated c. g. locations. In addition, the tail and wing incidence must be set to provide a trim lift coefficient near the design condition.

When the configuration of the airplane is fixed, a variation of c. g. position can cause large changes in the static stability. In the conventional airplane configuration, the large changes in stability with c. g. variation are primarily due to the large changes in the wing contribution. If the incidence of all surfaces remains fixed, the effect of c. g. position on static longitudinal stability is typified by the second chart of figure 4.10. As the c. g. is gradually moved aft, the airplane static sta­bility’ decreases, then becomes neutral then unstable. The c. g. position which produces zero slope and neutral static stability is re­ferred to as the “neutral point.” The neutral point may be imagined as the effective aerody­namic center of the entire airplane configura – ration, i. e., with the c. g. at this position, all changes in net lift effectively occur at this point and no change in pitching moment results. The neutral point defines the most aft c. g. position without static instability.

POWER EFFECTS. The effects of power may cause significant changes in trim lift coefficient and static longitudinal stability. Since the contribution to stability is evaluated by the change in moment coefficients, power effects will be most significant when the airplane operates at high power and low airspeeds such as the power approach or waveoff condition.

The effects of power are considered in two main categories. First, there are the direct effects resulting from the forces created by the propulsion unit. Next, there are the indirect effects of the slipstream and other associated flow which alter the forces and moments of the aerodynamic surfaces. The direct effects of power are illustrated in figure 4.11. The ver­tical location of the thrust line defines one of the direct contributions to stability. If the

thrust line is below the c. g., thrust produces a positive or noseup moment and the effect is de­stabilizing. On the other hand, if the thrust line is located above the c. g., a negative moment is created and the effect is stabilizing.

A propeller or inlet duct located ahead of the c. g. contributes a destabilizing effect. As shown in figure 4.11, a rotating propeller in­clined to the windstream causes a deflection of the airflow. The momentum change of the
slipstream creates a normal force at the plane of the propeller similar to a wing creating lift by deflecting an airstream. As this normal force will increase with an increase in airplane angle of attack, the effect will be destabilizing when the propeller is ahead of the c. g. The magnitude of the unstable contribution de­pends on the distance from the c. g. to the propeller and is largest at high power and low dynamic pressure. The normal force created

at the inlet of a jet engine contributes a similar destabilizing effect when the inlet is ahead of the c. g. As with the propeller, the magni­tude of the stability contribution is largest at high thrust and low flight speed.

The indirect effects of power are of greatest concern in the propeller powered airplane rather than the jet powered airplane. As shown in figure 4.12, the propeller powered airplane creates slipstream velocities on the various surfaces which are different from the flow field typical of power-off flight. Since the various wing, nacelle, and fuselage surfaces are partly or wholly immersed in this slip­stream, the contribution of these components to stability can be quite different from the power-off flight condition. Ordinarily, the change of fuselage and nacelle contribution with power is relatively small. The added lift on the portion of the wing immersed in the slipstream requires that the airplane oper­ate at a lower angle of attack to produce the same effective lift coefficient. Generally, this reduction in angle of attack to effect the same CL reduces the tail contribution to stability. However, the increase in dynamic pressure at the tail tends to increase the effectiveness of the tail and may be a stabilizing effect. The magnitude of this contribution due to the slipstream velocity on the tail will depend on the c. g. position and trim lift coefficient.

The deflection of the slipstream by the nor­mal force at the propeller tends to increase the down wash at the horizontal tail and reduce the contribution to stability. Essentially the same destabilizing effect is produced by the flow induced at the exhaust of the jet power – plant. Ordinarily, the induced flow at the horizontal tail of a jet airplane is slight and is destabilizing when the jet passes underneath the horizontal tail. The magnitude of the indirect power effects on stability tends to be greatest at high Ct, high power, and low flight speeds.

The combined direct and indirect power effects contribute to a general reduction of static stability at high power, high and low q. It is generally true that any airplane will experience the lowest level of static longi­tudinal stability under these conditions. Be­cause of the greater magnitude of both direct and indirect power effects, the propeller pow­ered airplane usually experiences a greater effect than the jet powered airplane.

An additional effect on stability can be from the extension of high lift devices. The high lift devices tend to increase downwash at the tail and reduce the dynamic pressure at the tail, both of which are destabilizing. However, the high lift devices may prevent an unstable contribution of the wing at high CL. While the effect of high lift devices depends on the airplane configuration, the usual effect is de­stabilizing. Hence, the airplane may experi­ence the most critical forward neutral point during the power approach or waveoff. Dur­ing these conditions of flight the static stability is usually the weakest and particular attention must be given to precise control of the air­plane. The power-on neutral point may set the most aft limit of c. g. position.

CONTROL FORCE STABILITY. The static longitudinal stability of an airplane is defined by the tendency to return to equilibrium upon displacement. In other words, the stable air­plane will resist displacement from the trim or equilibrium. The control forces of the air­plane should reflect the stability of the air­plane and provide suitable reference for precise control of the airplane.

The effect of elevator deflection on pitching moments is illustrated by the first graph of figure 4.13. If the elevators of the airplane are fixed at zero deflection, the resulting line of Cm versus CL for 0° depicts the static stability and trim lift coefficient. If the elevators are fixed at a deflection of 10° up, the airplane static stability is unchanged but the trim lift coefficient is increased. A change in elevator or stabilizer position does not alter the tail contribution to stability but the change in pitching moment will alter the lift coefficient

at which equilibrium will occur. As the ele­vator is fixed in various positions, equilibrium (or trim) will occur at various lift coefficients and the trim CL can be correlated with elevator deflection as in the second graph of figure 4.13.

When the c. g. position of the airplane is fixed, each elevator position corresponds to a particular trim lift coefficient. As the c. g. is moved aft the slope of this line decreases and the decrease in stability is evident by a given control displacement causing a greater change in trim lift coefficient. This is evidence that decreasing stability causes increased controlla­bility and, of course, increasing stability de­creases controllability. If the c. g. is moved aft until the line of trim CL versus elevator de­flection has zero slope, neutral static stability is obtained and the “stick-fixed” neutral point is determined.

Since each value of lift coefficient corresponds to a particular value of dynamic pressure re­quired to support an airplane in level flight, trim airspeed can be correlated with elevator deflection as in the third graph of figure 4.13. If the c. g. location is ahead of the stick-fixed neutral point and control position is directly related to surface deflection, the airplane will give evidence of stick position stability. In other words, the airplane will require the stick to be moved aft to increase the angle of attack and trim at a lower airspeed and to be moved forward to decrease the angle of attack and trim at a higher airspeed. To be sure, it is desirable to have an airplane demon­strate this feature. If the airplane were to have stick position instability, the airplane would require the stick to be moved aft to trim at a higher airspeed or to be moved forward to trim at a lower airspeed.

There may be slight differences in the static longitudinal stability if the elevators are allowed to float free. If the elevators are allowed to float free as in “hands-off"’ flight, the elevators may have a tendency to “float” or streamline when the horizontal tail is given a change in angle of attack. If the horizontal tail is subject to an increase in angle of attack and the elevators tend to float up, the change in lift on the tail is less than if the elevators remain fixed and the tail contribution to stability is reduced. Thus, the “stick-free” stability of an airplane is usually less than the stick-fixed stability. A typical reduction of stability by free elevators is shown in figure 4.14(A) where the airplane, stick-free demon­strates a reduction of the slope of CM versus CL. While aerodynamic balance may be provided to reduce control forces, proper balance of the surfaces will reduce floating and prevent great differences between stick-fixed and stick-free stability. The greatest floating tendency oc­curs when the surface is at a high angle of attack hence the greatest difference between stick-fixed and stick-free stability occurs when the airplane is at high angle of attack.

If the controls are fully powered and actu­ated by an irreversible mechanism, the sur­faces are not free to float and there is no differ­ence between the stick-fixed and stick-free static stability.

The control forces in a conventional air-

і

plane are made up of two components. First, the basic stick-free stability of the airplane contributes an increment, of force which is independent of airspeed. Next, there, is an increment of force dependent on the trim tab sttting which varies with-the dynamic pres­sure or the square of equivalent airspeed. Figure 4.14(B) indicates the variation of stick force with airspeed and illustrates the effect of tab setting on stick force. In order to trim the airplane at point (1) a certain amount of up elevator is required and zero stick force is obtained with the use of the tab. To trim the airplane for higher speeds corre­sponding to points (2) and (3) less and less nose-up tab is required. Note that when the airplane is properly trimmed, a push force is required to increase airspeed and a pull force is required to decrease airspeed. In this man­ner, the airplane would indicate positive stick force stability with a stable “feel” for air-

speed. If the airplane were given a large nose down tab setting the pull force would in­crease with airspeed. This fact points out the possibility of “feel” as not being a true indi­cation of airplane static stability.

If the c. g. of the airplane were varied while maintaining trim at a constant airspeed, the effect of c. g. position on stick force stability could be appreciated. As illustrated in figure 4.14(C), moving the c. g. aft decreases the slope of the line of stick force through the trim speed. Thus, decreasing stick force stability is evident in that smaller stick forces are necessary to displace the airplane from the trim speed. When the stick force gradient (or slope) becomes zero, the c. g. is at the stick-free neutral point and neutral stability exists. If the c. g. is aft of the stick-free neutral point, stick force instability will exist, c. g. the airplane will require a push force at a lower speed or a pull force at a higher speed. It should be noted that the stick force gradient is low at low airspeeds and when the airplane is at low speeds, high power, and a c. g. position near the aft limit, the “feel” for airspeed will be weak.

Control system friction can create very un­desirable effects on control forces. Figure 4.14(D) illustrates that the control force versus airspeed is a band rather than a line. A wide friction force band can completely mask the stick force stability when the stick force stability is low. Modern flight control systems require precise maintenance to mini­mize the friction force band and preserve proper feel to the airplane.

MANEUVERING STABILITY. When an airplane is subject to a normal acceleration, the flight path is curved and the airplane is subject to a pitching velocity. Because of the pitching velocity in maneuvering flight, the longitudinal stability of the airplane is slightly greater than in steady flight condi­tions. When an airplane is subject to a pitch­ing velocity at a given lift coefficient, the air­plane develops a pitching moment resisting the pitch motion which adds to the restoring moment from the basic static stability. The principal source of this additional pitching moment is illustrated in figure 4.15.

During a pull-up the airplane is subject to an angular rotation about the lateral axis and the horizontal tail will experience a component of wind due to the pitching velocity. The vector addition of this component velocity to the flight velocity provides a change in angle of attack for the tail and the change in lift on the tail creates a pitching moment resisting the pitching motion. Since the pitching mo­ment opposes the pitching motion but is due to the pitching motion, the effect is a damping in pitch. Of course, the other components of the airplane may develop resisting moments and contribute to pitch damping but the horizontal tail is usually the largest contri­bution. The added pitching moment from pitch damping will effect a higher stability in maneuvers than is apparent in steady flight. From this consideration, the neutral point for maneuvering flight will be aft of the neutral point for unaccelerated flight and in most cases will not be a critical item. If the airplane demonstrates static stability in unaccelerated flight, it will most surely demonstrate stability in maneuvering flight.

The most direct appreciation of the ma­neuvering stability of an airplane is obtained from a plot of stick force versus load factor such as shown in figure 4.15. The airplane with positive maneuvering stability should demonstrate a steady increase in stick force with increase in load factor or “G”. The maneuvering stick force gradient—or stick force per G—must be positive but should be of the proper magnitude. The stick force gradient must not be excessively high or the airplane will be difficult and tiring to maneuver. Also, the stick force gradient must not be too low or the airplane may be overstressed in­advertently when light control forces exist. A maneuvering stick force gradient of 3 to 8 lbs. per G is satisfactory for most fighter and

Figure 4.15. Maneuvering Stability

attack airplanes. A large patrol or transport type airplane would ordinarily show a much higher maneuvering stick force gradient be­cause of the lower limit load factor.

When the airplane has high static stability, the maneuvering stability will be high and a high stick force gradient will result. A possibility exists that the forward c. g. limit could be set to prevent an excessively high maneuvering stick force gradient. As the c. g. is moved aft, the stick force gradient de­creases with decreasing maneuvering stability and the lower limit of stick force gradient may be reached.

The pitch damping of the airplane is obvi­ously related to air density. At high altitudes, the high true airspeed reduces the change in tail angle of attack for a given pitching velocity and reduces the pitch damping. Thus, a de­crease in maneuvering stick force stability can be expected with increased altitude.

TAILORING CONTROL FORCES. The control forces should reflect the stability of the airplane but, at the same time, should be of a tolerable magnitude. The design of the surfaces and control system may employ an infinite variety of techniques to provide satis­factory control forces.

Aerodynamic balance must be thought of in two different senses. First, the control surface must be balanced to reduce hinge moments due to changes in angle of attack. This is necessary to reduce the floating tendency of the surface which reduces the stick-free stability. Next, aerodynamic balance can reduce the hinge moments due to deflection of the control sur­face. Generally, it is difficult to obtain a high degree of deflection balance without incurring a large overbalance of the surface for changes in angle of attack.

Some of the types of aerodynamic balance are illustrated in figure 4.16. The-simple horn type balance employs a concentrated balance area located ahead of the hinge line. The balance area may extend completely to the leading edge (unshielded) or part’way to the leading edge (shielded). Aerodynamic balance can be achieved by the provision of a hinge line aft of the control surface leading edge. The resulting overhang of surface area ahead of the hinge line will provide a degree of balance depending on the amount of overhang. Another variation of aerodynamic balance is afl internal balance surface ahead of the hinge line which is contained within the surface. A flexible seal is usually incorporated to in­crease the effectiveness of the balance area. Even the bevelling of the trailing edge of the control surface is effective also as a balancing technique. The choice of the type of aerody­namic balance will depend on many factors such as required degree of balance, simplicity, drag, etc.

Many devices can be added to a control system to modify or tailor the stick force stability to desired levels. If a spring is added to the control system as shown in figure 4.16, it will tend to center the stick and provide a force increment depending on stick displace­ment. When the control system has a fixed gearing between stick position and surface deflection, the centering spring will provide a contribution to stick force stability according to stick position. The contribution to stick force stability will be largest at low flight speeds where relatively large control deflec­tions are required. The contribution will be smallest at high airspeed because of the smaller control deflections required. Thus, the stick centering bungee will increase the airspeed and maneuvering stick force stability but the contribution decreases at high airspeeds. A variation of this device would be a spring stiffness which would be controlled to vary with dynamic pressure, q – In this case, the contribution of the spring to stick force stability would not diminish with – speed.

A “downspring” added to a control system is a means of increasing airspeed stick force stability without a change in airplane static

TYPES OF AERODYNAMIC BALANCE

OVERHANG OR LEADING EDGE BALANCE BY OFFSET HINGE

C

d

EFFECT OF STICK CENTERING SPRING

stability. As shown in figure 4.17, a down­spring consists of a long preloaded spring at­tached to the control system which tends to rotate the elevators down. The effect of the downspring is to contribute an increment of pull force independent of control deflection or airspeed. When rhe downspring is added to the control system of an airplane and the air­plane is retrimmed for the original speed, the airspeed stick force gradient is increased and there is a stronger feel for airspeed. The down­spring would provide an “ersatz” improve­ment to an airplane deficient in airspeed stick force stability. Since the force increment from the downspring is unaffected by stick position or normal acceleration, the maneuvering stick force stability would be unchanged.

The bobweight is an effective device for im­proving stick force stability. As shown in figure 4.17, the bobweight consists of an eccen­tric mass attached to the control system which—in unaccelerated {light—contributes an increment of pull force identical to the downspring. In fact, a bobweight added to the control system of an airplane produces an effect identical to the downspring. The bob – weight will increase the airspeed stick force gradient and increase the feel for airspeed.

A bobweight will have an effect on the maneuvering stick force gradient since the bob – weight mass is subjected to the same accelera­tion as the airplane. Thus, the bobweight will provide an increment of stick force in direct proportion to the maneuvering acceleration of the airplane. Because of the linear contribu­tion of the bobweight, the bobweight can be applied to increase the maneuvering stick force stability if the basic airplane has too low a value or develops a decreasing gradient at high lift coefficients.

The example of the bobweight is useful to point out the effect of the control system dis­tributed masses. All carrier aircraft must have the control system mass balanced to prevent undesirable control forces from the longi­tudinal accelerations during catapult launching.

Various control surface tab devices can be utilized to modify control forces. Since the de­flection of a tab is so powerful in creating hinge moments on a control surface, the possible application of tab devices is almost without limit. The basic trim tab arrangement is shown in figure 4.18 where a variable linkage connects the tab and the control surface. Ex­tension or contraction of this linkage will de­flect the tab relative to the control surface and create a certain change in hinge moment coef­ficient. The use of the trim tab will allow the pilot to reduce the hinge moment to zero and trim the control forces to zero for a given flight condition. Of course, the trim tab should have adequate effectiveness so that control forces can be trimmed out throughout the flight speed range.

The lagging tab arrangement shown in figure

4.18 employs a linkage between the fixed sur­face and the tab surface. The geometry is such that upward deflection of the control surface displaces the tab down relative to the control surface. Such relative displacement of the tab will aid in deflection of the control surface and thus reduce the hinge moments due to deflection. An obvious advantage of this device is the reduction of deflection hinge moments without a change in aerodynamic balance.

The leading tab arrangement shown in figure

4.18 also employs a linkage between the fixed surface and the tab surface. However, the geometry of the linkage is such that upward deflection of the control surface displaces the tab up relative to the control surface. This relationship serves to increase the control sur­face hinge moments due to deflection of the surface.

The servo tab shown in figure 4.18 utilizes a horn which has no direct connection to the control surface and is free to pivot about the hinge axis. However, a linkage connects this free horn to the tab surface. Thus, the control system simply deflects the tab and the resulting hinge moments deflect the control surface.

SERVO TAB

,HORN FREE TO PIVOT ON HINGE AXIS

^SPRING

-HORN FIXED TO SURFACE

Since the only control forces are those of the tab, this device makes possible the deflection of large surfaces with relatively small control

forces.

A variation of the basic servo tab layout is the spring tab arrangement of figure 4.18. When the control horn is connected to the control surface by springs, the function of the tab is to provide a given portion of the required control forces. The spring tab arrangement can then function as a boost to reduce control forces. The servo tab and spring tab are usually applied to large or high speed subsonic airplanes to provide tolerable stick forces.

The spring loaded tab of figure 4.18 consists of a free tab preloaded with a spring which furnishes a constant moment about the tab hinge line. When the airplane is at zero air­speed, the tab is rotated up to the limit of deflection. As airspeed is increased, the aero­dynamic hinge moment on the tab will finally equal the spring torque and the tab will begin to streamline. The effect of this arrangement is to provide a constant hinge moment to the control system and contribute a constant push force requirement at speeds above the preload speed. Thus, the spring loaded tab can im­prove the stick force gradient in a manner similar to the downspring. Generally, the spring loaded tab may be more desirable because of greater effectiveness and the lack of undesirable control forces during ground operation.

The various tab devices have almost un­limited possibilities for tailoring control forces. However, these devices must receive proper care and maintenance in order to function properly. In addition, much care must be taken to ensure that no slop or play exists in the joints and fittings, otherwise destructive flutter may occur.

In order to visualize the forces and moments on the aircraft, it is necessary to establish a set of mutually perpendicular reference axes originating at the center of gravity. Figure 4.4 illustrates a conventional right hand axis system. The longitudinal or X axis is located in a plane of symmetry and is given a positive direction pointing into the wind. A moment about this axis is a rolling moment, L, and the positive direction for a positive rolling moment utilizes the right hand rule. The vertical or Z axis also is in a plane of symmetry and is estab­lished positive downward. A moment about the vertical axis is a yawing moment, N, and a positive yawing moment would yaw the air­craft to the right (right hand rule). The lateral or Y axis is perpendicular to the plane of symmetry and is given a positive direction out the right side of the aircraft. A moment about the lateral axis is a pitching moment, Л1, and a positive pitching moment is in the nose – up direction.

An aircraft is said to be trimmed if all moments in pitch, roll, and yaw are equal to zero. The establishment of equilibrium at various conditions of flight is the function of the controls and may be accomplished by pilot effort, trim tabs, or bias of a surface actuator.

The term “controllability" refers to the ability of the aircraft to respond to control surface displacement and achieve the desired condition of flight. Adequate controllability must be available to perform takeoff and landing and accomplish the various maneuvers in flight. An important contradiction exists between stability and controllability since adequate controllability does not necessarily exist with adequate stability. In fact, a high degree of stability tends to reduce the controlla­bility of the aircraft. The general relation­ship between static stability and controlla­bility is illustrated by figure 4-3-

Figure 4-3 illustrates various degrees of static stability by a ball placed on various surfaces. Positive static stability is shown by the ball in a trough; if the ball is displaced from equilibrium at the bottom of the trough, there is an initial tendency to return to equilib­rium. If it is desired to “control" the ball

and maintain it in the displaced position, a force must be supplied in the direction of displacement to balance the inherent tendency to return to equilibrium. This same stable tendency in an aircraft resists displacement from trim by pilot effort on the controls or atmospheric disturbances.

The effect of increased stability on con­trollability is illustrated by the ball in a steeper trough. A greater force is required to “control” the ball to the same lateral dis­placement when the stability is increased. In this manner, a large degree of stability tends to make the aircraft less controllable. It is necessary to achieve the proper balance be­tween stability and controllability during the design of an aircraft because the upper limits of stability are set by the lower limits of controlla­bility.

The effect of reduced stability on controlla­bility is illustrated by the ball on a flat surface. When neutral static stability exists, the ball may be displaced from equilibrium and there is no stable tendency to return. A new point of equilibrium is obtained and no force is required to maintain the displacement. As the static stability approaches zero, controlla­bility increases to infinity and the only resist­ance to displacement is a resistance to the motion of displacement—damping. For this reason, the lower limits of stability may be set by the upper limits of controllability. If the stability of the aircraft is too low, control deflections may create exaggerated displace­ments of the aircraft.

The effect of static instability, on controlla­bility is illustrated by the ball on a hill. If the ball is displaced from equilibrium at the top of the hill, the initial tendency is for the ball to continue in the displaced direction. In order to “control” the ball to some lateral displacement, a force must be applied opposite to the direction of displacement. This effect would be appreciated during flight of an un­stable aircraft by an unstable “feel” of the air­craft. If the controls were deflected to in­crease the angle of attack, the aircraft would be trimmed at the higher angle of attack by a push force to keep the aircraft from con­tinuing in the displacement direction. Such control force reversal would evidence the air­plane instability; the pilot would be supply­ing the stability by his attempt to maintain the equilibrium. An unstable aircraft can be flown if the instability is slight with a low rate of divergence. Quick reactions coupled with effective controls can allow the pilot to cope with some degree of static instability. Since such flight would require constant at­tention by the pilot, slight instability can be tolerated only in airships, helicopters, and certain minor motions of the airplane. How­ever, the airplane in high speed flight will react rapidly to any disturbances and any in­stability would create unsafe conditions. Thus, it is necessary to provide some positive static stability to the major aircraft degrees of freedom.

An aircraft must have satisfactory handling qualities in addition to adequate performance. The aircraft must have adequate stability to maintain a uniform flight condition and recover from the various disturbing influences. It is necessary to provide sufficient stability to minimize the workload of the pilot. Also, the aircraft must have proper response to the controls so that it may achieve the inherent performance. There are certain conditions of
flight which provide the most critical require­ments of stability and control and these condi­tions must be understood and respected to accomplish safe and efficient operation of the aircraft.

DEFINITIONS

STATIC STABILITY

An aircraft is in a state of equilibrium when the sum of all forces and all moments is equal

POSITIVE STATIC STABILITY

to zero. When an aircraft is in equilibrium, there are no accelerations and the aircraft continues in a steady condition of flight. If the equilibrium is disturbed by a gust or deflec­tion of the controls, the aircraft will experi­ence acceleration due to unbalance of moment or force.

The static stability of a system is defined by the initial tendency to return to equilibrium conditions following some disturbance from equilibrium. If an object is disturbed from equilibrium and has the tendency to return to equilibrium, positive static stability exists. If the object has a tendency to continue in the direction of disturbance, negative static stability or static instability exists. An intermediate condition could occur where an object dis­placed from equilibrium remains in equilibrium in the displaced position. If the object subject to a disturbance has neither the tendency to return nor the tendency to continue in the dis­placement direction, neutral static stability ex­ists. These three categories of static stability are illustrated in figure 4.1. The bail in a trough illustrates the condition of positive static stability. If the ball is displaced from equilibrium at the bottom of the trough, the initial tendency of the ball is to return to the equilibrium condition. The ball may roll back and forth through the point of equilib­rium but displacement to either side creates the initial tendency to return. The ball on a hill illustrates the condition of static insta­bility. Displacement from equilibrium at the hilltop brings about the tendency for greater displacement. The ball on a flat, level surface illustrates the condition of neutral static sta­bility. The ball encounters a new equilibrium at any point of displacement and has neither stable nor unstable tendencies.

The term "static” is applied to this form of stability since the resulting motion is not considered. Only the tendency to return to equilibrium conditions is considered in static stability. The static longitudinal stability of an aircraft is appreciated by displacing the aircraft from some trimmed angle of attack. If the aerodynamic pitching moments created by this displacement tend to return the air­craft to the equilibrium angle of attack the aircraft has positive static longitudinal stability.

DYNAMIC STABILITY

While static stability is concerned with the tendency of a displaced body to return to equilibrium, dynamic stability is defined by the resulting motion with time. If an object is disturbed from equilibrium, the time history of the resulting motion indicates the dynamic stability of the system. In general, the system will demonstrate positive dynamic stability if the amplitude of motion decreases with time. The various conditions of possible dynamic behavior are illustrated by the time history diagrams of figure 4.2.

The nonoscillatory modes shown in figure

4.2 depict the time histories possible without cyclic motion. If the system is given an initial disturbance and the motion simply subsides without oscillation, the mode is termed "sub­sidence" or "deadbeat return.” Such a motion indicates positive static stability by the tend­ency to return to equilibrium and positive dy­namic stability since the amplitude decreases with time. Chart В illustrates the mode of "divergence" by a noncyclic increase of ampli­tude with time. The initial tendency to con­tinue in the displacement direction is evidence of static instability and the increasing ampli­tude is proof of dynamic instability. Chart C illustrates the mode of pure neutral stability. If the original disturbance creates a displace­ment which remains constant thereafter, the lack of tendency for motion and the constant amplitude indicate neutral static and neutral dynamic stability.

The oscillatory modes of figure 4.2 depict the time histories possible with cyclic motion. One feature common to each of these modes is that positive static stability is demonstrated in the cyclic motion by tendency to return to

equilibrium conditions. However, the dy­namic behavior may be stable, neutral, or un­stable. Chart D illustrates the mode of a damped oscillation where the amplitude de­creases with time. The reduction of amplitude with time indicates there is resistance to mo­tion and that energy is being dissipated. The dissipation of energy—or “damping”—is nec­essary to provide positive dynamic stability. If there is no damping in the system, the mode of chart E is the result, an undamped oscilla­tion. Without damping, the oscillation con­tinues with no reduction of amplitude with time. While such an oscillation indicates posi­tive static stability, neutral dynamic stability exists. Positive damping is necessary to elimi­nate the continued oscillation. As an example, an automobile with worn shock absorbers (or “dampers”) lacks sufficient dynamic stability and the continued oscillatory motion is neither pleasant nor conducive to safe operation. In the same sense, the aircraft must have sufficient damping to, rapidly dissipate any oscillatory motion which would affect the operation of the aircraft. When natural aerodynamic damp­ing cannot be obtained, a synthetic damping must be furnished to provide the necessary positive dynamic stability.

Chart F of figure 4.2 illustrates the mode of a divergent oscillation. This motion is stat­ically stable since it tends to return to the equilibrium position. However, each subse­quent return to equilibrium is with increasing, velocity such that amplitude continues to increase with time. Thus, dynamic insta­bility exists. The divergent oscillation occurs when energy is supplied to the motion rather than dissipated by positive damping. The most outstanding illustration of the divergent oscillation occurs with the short period pitch­ing oscillation of an aircraft. If a pilot un­knowingly supplies control functions which arc near the natural frequency of the airplane in pitch, energy is added to the system, nega­tive damping exists, and the “pilot induced oscillation” results.

In any system, the existence of static sta­bility does not necessarily guarantee the existence of dynamic stability. However, the existence of dynamic stability implies the existence of static stability.

Any aircraft must demonstrate the required degrees of static and dynamic stability. If the aircraft were allowed to have static in­stability with a rapid rate of divergence, the aircraft would be very difficult—if not impos­sible—to fly. The degree of difficulty would compare closely with learning to ride a uni­cycle. In addition, positive dynamic stability is mandatory in certain areas to preclude objectionable continued oscillations of the aircraft.

When air flows over any aerodynamic surface certain reductions in velocity occur with cor­responding increases in temperature. The greatest reduction in velocity and increase in temperature will occur at the various stagna­tion points on the aircraft. Of course, similar changes occur at other points on the aircraft but these temperatures can be related to the ram temperature rise at the stagnation point. While subsonic flight does not produce temper­atures of any real concern, supersonic flight can produce temperatures high enough to be of major importance to the airframe and power- plant structure. The graph of figure 3-21 il – | lustrates the variation of ram temperature rise with airspeed in the standard atmosphere. The ram temperature rise is independent of altitude and is a function of true airspeed. Actual temperatures would be the sum of the temperature rise and the ambient air temper­ature. Thus, low altitude flight at high Mach numbers will produce the highest temperatures.

In addition to the effect on the crew member environment, aerodynamic heating creates special problems for the airplane structure and the powerplant. The effect of tempera­ture on the short time strength of three typical structural materials is shown in figure 3.21.

Higher temperatures produce definite reduc­tions in the strength of aluminum alloy and require the use of titanium alloys, stainless steels, etc., at very high temperatures. Con­tinued exposure at elevated temperatures effects further reductions of strength and magnifies the problems of “creep” failure and structural stiffness.

The turbojet engine is adversely affected by high compressor inlet air temperatures. Since the thrust output of the turbojet is some func­tion of the fuel flow, high compressor inlet air temperatures reduce the fuel flow that can be used within turbine operating temperature limits. The reduction in performance of the turbojet engines with high compressor inlet air temperatures requires that the inlet design produce the highest practical efficiency and minimize the temperature rise of the air delivered to the compressor face.

High flight speeds and compressible flow dictate airplane configurations which are much different from the ordinary subsonic airplane. To achieve safe and efficient operation, the pilot of the modern, high speed aircraft must under­stand and appreciate the advantages and dis­advantages of the configuration. A knowledge of high speed aerodynamics will contribute greatly to this understanding.

Aircraft configurations developed for high speed flight will have significant differences in shape and planform when compared with air­craft designed for low speed flight. One of the outstanding differences will be in the selection of airfoil profiles for transonic or supersonic flight.

AIRFOIL SECTIONS. It should be ob­vious that airfoils for high speed subsonic flight should have high critical Mach num­bers since critical Mach number defines the lower limit for shock wave formation and subsequent force divergence. An additional complication to airfoil selection in this speed range is that the airfoil should have a high maximum lift coefficient and sufficient thickness to allow application of high lift devices. Otherwise an excessive wing area would be required to provide maneuverability and reasonable takeoff and landing speeds.

Revised January 19Л5

F6U MODEL AT VARIOUS MACH NUMBERS в «0® $’0°

Figure 3.11. Schlieren Photographs of Supersonic Flight (sheet 1 of 2)

However, if high speed flight is the primary consideration, the airfoil must be chosen to have the highest practical critical Mach number.

Critical Mach number has been defined as the flight Mach number which produces first evidence of local sonic flow. Thus, the air­foil shape and lift coefficient—which determine the pressure and velocity distribution—will have a profound effect on critical Mach number. Conventional, low speed airfoil shapes have relatively poor compressibility characteristics because of the high local velocities near the leading edge. These high local velocities are inevitable if both the maximum thickness and camber are well forward on the chord. An improvement of the compressibility character­istics can be obtained by moving the points of maximum camber and thickness aft on the chord. This would distribute the pressure and velocity more evenly along the chord and produce a lower peak velocity for the same lift coefficient. Fortunately, the airfoil shape to provide extensive laminar flow and low profile drag in low speed, subsonic flight will provide a pressure distribution which is favor­able for high speed flight. Figure 312 illustrates the pressure distributions and variation of critical Mach number with lift coefficient for a conventional low speed airfoil and a high speed section.

In order to obtain a high critical Mach number from an airfoil at some low lift coefficient the section must have:

(a) Low thickness ratio. The point of maximum thickness should be aft to smooth the pressure distribution.

(b) Low camber. The mean camber line should be shaped to help minimize the local velocity peaks.

In addition, the higher the required lift coefficient the lower the critical Mach number and more camber is required of the airfoil. If supersonic flight is a possibility the thick­ness ratio and leading edge radius must be small to decrease wave drag.

Figure 3.13 shows the flow patterns for two basic supersonic airfoil sections and pro­vides the approximate equations for lift, drag, and lift curve slope. Since the wave drag is the only factor of difference between the two airfoil sections, notice the configuration fac­tors which affect the wave drag. For the same thickness ratio, the circular arc airfoil would have a larger wedge angle formed between the upper and lower surfaces at the leading edge. At the same flight Mach num­ber the larger angle at the leading edge would form the stronger shock wave at the nose and cause a greater pressure change on the circular arc airfoil. This same principle applies when investigating the effect of airfoil thickness. Notice that the wave drag coefficients for both airfoils vary as the SQUARE of the thickness ratio, e. g., if the thickness ratio were doubled, the wave drag coefficient would be four times as great. If the thickness were increased, the airflow at the leading edge will experience a greater change in direction and a stronger shock wave will be formed. This powerful variation of wave drag with thick­ness ratio necessitates the use of very thin air­foils with sharp leading edges for supersonic flight. An additional consideration is that thin airfoil sections favor the use of low aspect ratios and high taper to obtain lightweight structures and preserve stiffness and rigidity.

The parameter — appears in the denominator of each of the equations for the aerodynamic coefficients and indicates a de­crease in each of these coefficients with an increase in Mach number. Essentially, this means that any aerodynamic surface becomes less sensitive to changes in angle of attack at higher Mach numbers. The decrease in lift curve slope with Mach number has tremendous implications in the stability and control of high speed aircraft. The vertical tail becomes less sensitive to angles of sideslip and the directional stability of the aircraft will deteri­orate with Mach number. The horizontal tail of the airplane experiences the same

general effect and contributes less damping to longitudinal pitching oscillations. These ef­fects can become so significant at high Mach numbers that the aircraft might require com­plete synthetic stabilization.

PLANFORM EFFECTS. The development of surfaces for high speed involves considera­tion of many items in addition to the airfoil sections. Taper, aspect ratio, and sweepback can produce major effects on the aerodynamic characteristics of a surface in high speed flight. Sweepback produces an unusual effect on the high speed characteristics of a surface and has basis in a very fundamental concept of aero­dynamics. A grossly simplified method of visualizing the effect of sweepback is shown in figure 3.14. The swept wing shown has the streamwise velocity broken down to a com­ponent of velocity perpendicular to the leading edge and a component parallel to the leading edge. The component of speed perpendicular to the leading edge is less than the free stream speed (by the cosine of the sweep angle) and it is this velocity component which determines the magnitude of the pressure distribution.

The component of speed parallel to the lead­ing edge could be visualized as moving across constant sections and, in doing so, does not contribute to the pressure distribution on the swept wing. Hence, sweep of a surface pro­duces a beneficial effect in high speed flight since higher flight speeds may be obtained be­fore components of speed perpendicular to the leading edge produce critical conditions on the wing. This is one of the most important ad­vantage of sweep since there is an increase in critical Mach number, force divergence Mach number, and the Mach number at which the drag rise will peak. In other words, sweep will delay the onset of compressibility effects.

Generally, the effect of wing sweep will apply to either sweep back or sweep forward. While the swept forward wing has been used I in rare instances, the aeroelastic instability of such a wing creates such a problem that sweep back is more practical for ordinary applica­tions.

In addition to the delay of the onset of com­pressibility effects, sweepback will reduce the magnitude of the changes in force coefficients due to compressibility. Since the component of velocity perpendicular to the leading edge is less than the free stream velocity, the magni­tude of all pressure forces on the wing will be reduced (approximately by the square of the cosine of the sweep angle). Since compressi­bility force divergence occurs due to changes in pressure distribution, the use of sweepback will "soften” the force divergence. This effect is illustrated by the graph of figure 3.14 which shows the typical variation of drag coefficient with Mach number for various sweepback angles. The straight wing shown begins drag rise at M=0.70, reaches a peak near M=1.0, and begins a continual drop past _M= 1.0. Note that the use of sweepback then delays the drag rise to some higher Mach number and reduces the magnitude of the drag rise.

In view of the preceding discussion, sweep – back will have the following principal ad­vantages :

(1) Sweepback will delay the onset of all compressibility effects. Critical Mach num­ber and force divergence Mach number will increase since the velocity component affect­ing the pressure distribution is less than the free stream velocity. Also, the peak of drag rise is delayed to some higher supersonic speed—approximately the speed which pro­duces sonic flow perpendicular to the leading edge. Various sweeps applied to wings of moderate aspect ratio will produce these approximate effects in transonic flight:

EFFECT OF SWEEPBACK ON LOW SPEED LIFT CURVE

SWEPT WING AT SWEPT WING IN A

ZERO SIDESLIP SIDESLIP TO THE RIGHT

SWEPT WING SWEPT WING IN A

IN LEVEL FLIGHT SIDESLIP TOWARD

THE DOWN WING

Figure 3.15. Aerodynamic Effects Due to Sweepback

(2) Sweepback will reduce the magnitude of change in the aerodynamic force coeffi­cients due to compressibility. Any change in drag, lift, or moment coefficients will be reduced by the use of sweepback. Various sweep angles applied to wings of moderate aspect ratio will produce these approximate effects in transonic flight.

These advantages of drag reduction and preser­vation of the transonic maximum lift coefficient are illustrated in figure 314.

Thus, the use of sweepback on a transonic aircraft will reduce and delay the drag rise and preserve the maneuverability of the aircraft in transonic flight. It should be noted that a small amount of sweepback produces very little benefit. If sweepback is to be used at all, at least 30° to 35° must be used to produce any significant benefit. Also note from figure 3-14 that the amount of sweepback required to delay drag rise in supersonic flight is very large, e. g., more than 60° necessary at M = 2.0. By comparison of the drag curves at high Mach numbers it will be appreciated that extremely high (and possibly impractical) sweepback is necessary to delay drag rise and that the lowest drag is obtained with zero sweepback. There­fore, the planform of a wing designed to operate continuously at high Mach numbers will tend to be very thin, low aspect ratio, and unswept. An immediate conclusion is that sweepback is a device of greatest application in the regime of transonic flight.

A few of the less significant advantages of sweepback are as follows:

(1) The wing lift curve slope is reduced for a given aspect ratio. This is illustrated by the lift curve comparison of figure 3-15 for the straight and swept wing. Any reduction of lift curve slope implies the wing is less sensitive to changes in angle of attack. This is a beneficial effect only when the effect of gusts and turbulence is con­sidered. Since the swept wing has the lower lift curve slope it will be less sensitive to gusts and experience less “bump” due to gust for a given aspect ratio and wing loading. This is a consideration particular to the aircraft whose structural design shows

‘ a predominating effect of the gust load spectrum, e. g., transport, cargo, and patrol types.

(2) “Divergence” of a surface is an aero – elastic problem which can occur at high dynamic pressures. Combined bending and twisting deflections interact with aerody­namic forces to produce sudden failure of the surface at high speeds. Sweep forward will aggravate this situation by "leading” the wing into the windstream and tends to lower the divergence speed. On the other hand, sweepback tends to stabilize the surface by “trailing” and tends to raise the divergence speed. By this tendency, sweep- back may be beneficial in preventing di­vergence within the anticipated speed range.

(3) Sweepback contributes slightly to the static directional—or weathercock—stability of an aircraft. This effect may be appre­ciated by inspection of figure 3-15 which shows the swept wing in a yaw or sideslip. The wing into the wind has less sweep and a slight increase in drag; the wing away from the wind has more sweep and less drag. The net effect of these force changes is to produce a yawing moment tending to return the nose into the relative wind. This directional stability contribution is usually small and of importance in tailless aircraft only.

(4) Sweepback contributes to lateral sta­bility in the same sense as dihedral. When the swept wing aircraft is placed in a side­slip, the wing into the wind experiences an increase in lift since the sweep is less and the wing away from the wind produces less lift since the sweep is greater. As shown in figure 3.15, the swept wing aircraft in a sideslip experiences lift changes and a sub­sequent rolling moment which tends to right the aircraft. This lateral stability contribution depends on the sweepback and the lift coefficient of the wing. A highly swept wing operating at high lift coefficient usually experiences such an excess of this lateral stability contribution that adequate controllability may be a significant problem. As shown, the swept wing has certain im­portant advantages. However, the use of sweepback produces certain inevitable disad­vantages which are important from the stand­point of both airplane design and flight oper­ations. The most important of these disad­vantages are as follows:

0) When sweepback is combined with taper there is an extremely powerful tendency for the wing to stall tip first. This pattern of stall is very undesirable since there would be little stall warning, a serious reduction in lateral control effectiveness, and the for­ward shift of the Center of pressure would contribute to a nose up moment (“pitch up” or “stick force lightening”). Taper has its own effect of producing higher local lift coefficients toward the tip and one of the effects of sweepback is very similar. All outboard wing sections are affected by the upwash of the preceding inboard sections and the lift distribution resulting from sweep – back alone is similar to that of high taper.

An additional effect is the tendency to develop a strong spanwise flow of the bound­ary layer toward the tip when the wing is at high lift coefficients. This spanwise flow produces a relatively low energy boundary layer near the tip which can be easily sep­

arated. The combined effect of taper and sweep present a considerable problem of tip stall and this is illustrated by the flow pat­terns of figure 3-16. Design for high speed performance may dictate high sweepback, while structural efficiency may demand a highly tapered planform. When such is the case, the wing may require extensive aero­dynamic tailoring to provide a suitable stall pattern and a lift distribution at cruise condi­tion which reduces drag due to lift. Wash­out of the tip, variation of section camber throughout span, flow fences, slats, leading edge extension, etc., are typical devices used to modify the stall pattern and minimize drag due to lift at cruise condition,

(2) As shown by the lift curve of figure 3.13 the use of sweepback will reduce the lift curve slope and the subsonic maximum lift coefficient. It is important to note this case is definitely subsonic since sweepback may be used to improve the transonic ma­neuvering capability. Various sweep angles applied to wings of moderate aspect ratio produce these approximate effects on the subsonic lift characteristics:

Parent reduction of subsonic maximum lift

coefficient and lift

The reduction of the low speed maximum lift coefficient (which is in addition to that lost due to tip stall) has very important implications in design. If wing loading is not reduced, stall speeds increase and sub­sonic maneuverability decreases. On the other hand, if wing loading is reduced, the increase in wing surface area may reduce the anticipated benefit of sweepback in the transonic flight regime. Since the require­ments of performance predominate, certain increases of stall speeds, takeoff speeds,

STRAIGHT WING OF SAME AREA, ASPECT RATIO, AND

STRUCTURAL TAPER

and landing speeds usually will be accepted. While the reduction of lift curve slope may be an advantage for gust considerations, the reduced sensitivity to changes in angle of attack has certain undesirable effects in subsonic flight. The reduced wing lift curve slope tends to increase maximum lift angles of attack and complicate the problem of landing gear design and cockpit visi­bility. Also, the lower lift curve slope would reduce the contribution to stability of a given tail surface area.

(3) The use of sweepback will reduce the effectiveness of trailing edge control surfaces and high lift devices. A typical example of this effect is the application of a single slotted flap over the inboard 60 percent span to both a straight wing and a wing with 35° sweepback. The flap applied to the straight wing produces an increase in maximum lift coefficient of approxi­mately 50 percent. The same type flap applied to the swept wing produces an increase in maximum lift coefficient of approximately 20 percent. To produce some reasonable maximum lift coefficient on a swept wing may require unsweeping the flap hinge line, application of leading edge high lift devices such as slots or slats, and possibly boundary layer control.

(4) As described previously, sweepback contributes to lateral stability by producing stable rolling moments with sideslip. The lateral stability contribution of sweepback varies with the amount of wing sweepback and wing lift coefficient—large sweepback and high lift coefficients producing large contribution to lateral stability. While sta­bility is desirable, any excess of stability will reduce controllability. For the majority of airplane configurations, high lateral sta­bility is neither necessary nor desirable, but adequate control in roll is absolutely neces­sary for good flying qualities. An excess of lateral stability from sweepback can aggra­vate “Dutch roll’’ problems and produce

marginal control during crosswind takeoff and landing where the aircraft must move in a controlled sideslip. Therefore, it is not unusual to find swept wing aircraft with negative dihedral and lateral control de­vices designed principally to meet cross wind takeoff and landing requirements,

(5) The structural complexity and aero – elastic problems created by sweepback are of great importance. First, there is the effect shown in figure 3-17 that swept wing has a greater structural span than a straight wing of the same area and aspect ratio. This effect increases wing structural weight since greater bending and shear material must be distributed in the wing to produce the same design strength. An additional problem is created near the wing root and “carry – through” structure due to the large twisting loads and the tendency of the bending stress distribution to concentrate toward the trail­ing edge. Also shown in figure 3-17 is the influence of wing deflection on the spanwise lift distribution. Wing bending produces tip rotation which tends to unload the tip and move the center of pressure forward. Thus, the same effect which tends to allay divergence can make an undesirable contri­bution to longitudinal stability.

EFFECT OF ASPECT RATIO AND TIP SHAPE. In addition to wing sweep, plan – form properties such as aspect ratio, and tip shape, can produce significant effects on the aerodynamic characteristics at high speeds. There is no particular effect of aspect ratio on critical Mach number at high or medium aspect ratios. The aspect ratio must be less than four or five to produce any apparent change in critical Mach number. This effect is shown for a typical 9 percent thick sym­metrical airfoil in the graph of figure 3.18. Note that very low aspect ratios are required to cause a significant increase in critical Mach number. Very low aspect ratios create the extremes of three dimensional flow and sub­sequent increase in free stream speed to create

local sonic flow. Actually, the extremely low aspect ratios required to produce high critical Mach number are not too practical. Generally, the advantage of low aspect ratio must be combined with sweepback and high speed airfoil sections.

The thin rectangular wing in supersonic flow illustrates several important facts. As shown in figure 318, Mach cones form at the tips of the rectangular wing and affect the pressure distribution on the area within the cone. The vortex develops within the tip cone due to the pressure differential and the resulting average pressure on the area within. thc-Cone is approximately one-half the pressure between the cones. Three-dimensional flow on the wing is then confined to the area within the tip cones, while the area between the cones experiences pure two-dimensional flow.

It is important to realize that the three­dimensional flow on the rectangular wing in supersonic flight differs greatly from that of subsonic flight. A wing of finite aspect ratio in subsonic flight experiences a three-dimen­sional flow which includes the tip vortices, downwash behind the wing, upwash ahead of the wing, and local induced velocities along the span. Recall that the local induced veloc­ities along the span of the wing would incline the section lift aft relative to the free stream and result in “induced drag.” Such a flow condition cannot be directly correlated with the wing in supersonic flow. ‘ The flow pattern for the rectangular wing of figure 3.18 dem­onstrates that the three-dimensional flow is confined to the tip, and pure two-dimensional flow exists on the wing area between the tip cones. If the wing tips were to be “raked” outside the tip cones, the entire wing flow would correspond to the two-dimensional (or section) conditions.

Therefore, for the wing in supersonic flow, no upwash exists ahead of the wing, three­dimensional effects are confined to the tip cones, and no local induced velocities occur along the span between the tip cones. The

supersonic drag due to lift is a function of the section and angle of attack while the subsonic induced drag is a function of lift coefficient and aspect ratio. This comparison makes it obvious that supersonic flight does not demand the use of high aspect ratio planforms typical of low speed aircraft. In fact, low aspect ratios and high taper are favorable from the standpoint of structural considerations if very thin sections are used to minimize wave drag.

If sweepback is applied to the supersonic wing, the pressure distribution will be affected by the location of the Mach cone with respect to the leading edge. Figure 3-19 illustrates the pressure distribution for the delta wing plan – form in supersonic flight with the leading edge behind or ahead of the Mach cone. When the leading edge is behind the Mach cone the com­ponents of velocity perpendicular to the leading edge are still subsonic even though the free stream flow is supersonic and the resulting pressure distribution will greatly resemble the subsonic pressure distribution for such a plan – form. Tailoring the leading edge shape and camber can minimize the components of the high leading edge suction pressure which are inclined in the drag direction and the drag due to lift can be reduced. If the leading edge is ahead of the Mach cone, the flow over this area will correspond to the two-dimensional supersonic flow and produce constant pressure for that portion of the surface between the leading edge and the Mach cone.

CONTROL SURFACES. The design of con­trol surfaces for transonic and supersonic flight involves many important considerations. This fact is illustrated by the typical transonic and supersonic flow patterns of figure 3.19. Trail­ing edge control surfaces can be affected ad­versely by the shock waves formed in flight above critical Mach number. If the airflow is separated by the shock wave the resulting buffet of the control surface can be very objec­tionable. In addition to the buffet of the sur­face, the change in the pressure distribution due to separation and the shock wave location can

create very large changes in control surface hinge moments. Such large changes in hinge moments create very undesirable control forces and present the need for an "irreversible" con­trol system. An irreversible control system, would employ powerful hydraulic or electric actuators to move the surfaces upon control by the pilot and the airloads developed on the surface could not feed back to the pilot, Of course, suitable control forces would be syn­thesized by bungees, “q” springs, bobweights, etc.

Transonic and supersonic flight can cause a noticeable reduction in the effectiveness of trailing edge control surfaces. The deflection of a trailing edge control surface at low sub­sonic speeds alters the pressure distribution on the fixed portion as well as the movable portion of the surface. This is true to the extent that a 1-degree deflection of a 40 percent chord eleva­tor produces a lift change very nearly the equivalent of a 1-degree change in stabilizer setting. However, if supersonic flow exists on the surface, a deflection of the trailing edge control surface cannot influence the pressure distribution in the supersonic area ahead of the movable control surface. This is especially true in high supersonic flight where supersonic flow exists over the entire chord and the change in pressure distribution is limited to the area of the control surface. The reduction in effective­ness of the trailing edge control surface at tran­sonic and supersonic speeds necessitates the use of an all movable surface. Application of the all movable control surface to the horizontal tail is most usual since the increase in longi­tudinal stability in supersonic flight requires a high degree of control effectiveness to achieve required controllability for supersonic maneu­vering.

SUPERSONIC ENGINE INLETS. Air which enters the compressor section of a jet engine or the combustion chamber of a ramjet usually must be slowed to subsonic velocity. This process must be accomplished with the least possible waste of energy. At flight speeds

just above the speed of sound only slight modi­fications to ordinary subsonic inlet design pro­duce satisfactory performance. However, at supersonic flight speeds, the inlet design must slow the air with the weakest possible series or combination of shock waves to minimize en­ergy losses and temperature rise. Figure 3.20 illustrates some of the various forms of super­sonic inlets or "diffusers."

One of the least complicated types of inlet is the simple normal shock type diffuser. This type of inlet employs a single normal shock wave at the inlet with a subsequent internal subsonic compression. At low supersonic Mach j numbers the strength of the normal shock wave is not too great and this type of inlet is quite practical. At higher supersonic Mach num­bers, the single normal shock wave is very strong and causes a great reduction in the total pressure recovered by the inlet. In addition, it is necessary to consider that the wasted J energy of the airstream will appear as an addi­tional undesirable rise in temperature of the captured inlet airflow.

If the supersonic airstream can be captured, the shock wave formations will be swallowed and a gradual contraction will reduce the speed to just above sonic. Subsequent diverging flow | section can then produce the normal shock wave which slows the airstream to subsonic. Further expansion continues to slow the air to lower subsonic speeds. This is the convergent- divergent type inlet shown in figure 3.20. If the initial contraction is too extreme for the inlet Mach number, the shock wave formation will not be swallowed and will move out in front of the inlet. The external location of the normal shock wave will produce subsonic flow immediately at the inlet. Since the airstream is suddenly slowed to subsonic through the strong normal shock a greater loss of airstream energy will occur.

Another form of diffuser employs an external oblique shock wave which slows the super­sonic airstream before the normal shock occurs. Ideally, the supersonic airstream could be

slowed, gradually through a series of very weak oblique shock waves to a speed just above sonic velocity. Then the subsequent normal shock to subsonic could be quite weak. Such a combination of the weakest possible waves would result in the least waste of energy and the highest pressure recovery. The ef­ficiency of various types of diffusers is shown in figure 3-20 and illustrates this principle.

An obvious complication of the supersonic inlet is that the optimum shape is variable with inlet flow direction and Mach number. In other words, to derive highest efficiency and stability of operation, the geometry of the inlet would be different at each Mach number and angle of attack of flight. A typical super­sonic military aircraft may experience large variations in angle of attack, sideslip angle, and flight Mach number during normal oper­ation. These large variations in inlet flow conditions create certain important design considerations.

(1) The inlet should provide the highest practical efficiency. The ratio of recovered total pressure to airstream total pressure is an appropriate measure of this efficiency.

(2) The inlet should match the demands of the powerplant for airflow. The airflow captured by the inlet should match that necessary for engine operation.

(3) Operation of the inlet at flight condi­tions other than the design condition should not cause a noticeable loss of efficiency or excess drag. The operation of the inlet should be stable and not allow “buzz” conditions (an oscillation of shock location possible during off-design operation).

In order to develop a good, stable inlet design, the performance at the design condition may be compromised. A large variation of inlet flow conditions may require special geometric features for the inlet surfaces or a completely variable geometry inlet design.

SUPERSONIC CONFIGURATIONS. When all the various components of the supersonic airplane are developed, the most likely general configuration properties will be..as follows:

(1) The wing will be of low aspect ratio, have noticeable taper, and have sweepback depending on the design speed range. The wing sections will be of low thickness ratio and require sharp leading edges.

(2) The fuselage and nacelles will be of high fineness ratio (long and slender). The supersonic pressure distribution may create significant lift and drag and require con­sideration of the stability contribution of these surfaces.

(3) The tail surfaces will be similar to the wing—low aspect ratio, tapered, swept and of thin section with sharp leading edge. The controls will be fully powered and ir­reversible with all movable surfaces the most likely configuration.

(4) In order to reduce interference drag in transonic and supersonic flight, the gross cross section of the aircraft may be “area ruled” to approach that of some optimum high speed shape.

One of the most important qualities of high speed configurations will be the low speed flight characteristics. The low aspect ratio swept wing planform has the characteristic of high induced drag at low flight speeds. Steep turns, excessively low airspeeds, and steep, power-off approaches can then produce extremely high rates of descent during landing. Sweepback and low aspect ratio can cause severe deterioration of handling qualities at speeds below those recommended for takeoff and landing. On the other hand, thin, swept wings at high wing loading will have rela­tively high landing speeds. Any excess of this basically high airspeed can create an im­possible requirement of brakes, tires, and arrest ing gear. These characteristics require that the pilot account for the variation of optimum speeds with weight changes and adhere to the procedures and techniques outlined in the flight handbook.