Category AERODYNAMICS, AERONAUTICS, AND FLIGHT MECHANICS

Figure 8.1 illustrates the right-handed set of body axes that will be used in the material to follow. These axes are fixed at the center of gravity relative to the body and, hence, move with the body. The system is easily remem­bered. x is directed forward, у to the right, and z downward. X, У, and Z are the aerodynamic force components along their respective axes; u, v, and w the velocity components of the center of gravity along these axes; L, M, and N are the moments about these axes; and P, Q, and R are the angular velocities about the x-, y-, and z-axes, respectively. L, M, and N are defined as the rolling, pitching, and yawing moments, respectively. Similarly, P, Q, and R are called the roll, pitch, and yaw rates.

We will see later that for many problems concerning stability and control of an airplane, its motion in the plane of symmetry (x-z plane) can be uncoupled from the motion of the plane of symmetry. The former is referred

to as longitudinal motion and treats linear motion along the x – and z-axes, and rotation about the у-axis. Motion of the plane of symmetry, known as lateral or lateral-directional motion, deals with linear motion along the у-axis and angular rotations about the x – and z-axes.

INTRODUCTION

FAR Part 23 (Ref. 8.1) states that “the airplane must be safely controll­able and maneuverable during—(1) take off; (2) climb; (3) level flight; (4) dive; and (5) landing (power on and off) (with the wing flaps extended and retracted).” Part 23 also says that “The airplane must be longitudinally, directionally, and laterally stable.” This particular FAR, which defines the airworthiness standards for normal, utility, and acrobatic airplanes, then goes into detail to state what requirements an airplane must meet in order to satisfy the foregoing general statements on controllability and stability.

Stability of an airplane refers to its movement in returning, or the tendency to return, to a given state of equilibrium, frequently referred to as trim. More specifically, static stability refers to the tendency of an airplane under steady conditions to return to a trimmed condition when disturbed rather than any actual motion it may undergo following the disturbance. The forces and moments are examined to determine if they are in the direction to force the airplane back to its equilibrium flight conditions. If so, the airplane is statically stable.

Dynamic stability encompasses the unsteady behavior of an airplane responding to time-dependent aerodynamic forces and moments produced by the airplane’s motion. Following a disturbance of the airplane from a trimmed condition, its movement under the influence of the unsteady forces and moments is examined to determine whether or not the airplane ultimately returns to its trimmed condition. To the uninitiated, it comes as a surprise to learn that many airplanes exhibit a dynamic instability known as spiral divergence. Unless counteracted by control input, many airplanes will gradu­ally drop one wing and go into an ever tightening spiral dive. Fortunately, this spiral divergence initially is so gradual that the pilot will compensate for it without even realizing it. However, if a noninstrument-rated pilot is suddenly without a reference to the horizon, the spiral divergence instability can prove disastrous.

An airplane that is statically stable will not necessarily be dynamically stable. However, one that is statically unstable will be dynamically unstable; that is, static stability is necessary but not sufficient for dynamic stability. In this chapter we will consider static stability and control; in other words, we will consider the forces and moments acting on an airplane undergoing steady motion.

Sometimes one must give a reasonable estimate of an airplane’s charac­teristics or performance without having all the facts at hand. It is therefore a good idea to commit to memory a few principles and numbers. The “square-cube” scaling law is a good one to remember. For two geometrically similar airplanes designed to the same stress levels and using the same materials, one would expect their areas to be proportional to the characteristic

length squared and their volumes, and hence weight, proportional to the length cubed.

Socl2

Woe Iі

It follows that / a wm. One would therefore expect the wing loading of aircraft to vary as

w

~ « W113 (7.65)

Figure 7.33 was prepared with Equation 7.65 in mind. This figure in­dicates that while the square-cube law can be helpful in estimating the gross weight of an airplane, other factors must also be considered. For performance reasons, wing loadings are sometimes made purposefully higher or lower than the average. Generally, the aircraft with higher cruising speeds lie on the high side of the shaded portion in Figure 7.33. This upper boundary is given by

– 2.94 (Wm – 6) psf

= 85.5 Wm – 9.9) N/m2 (7.66)

The lower boundary is approximated by,

1.54(W1,3-6)psf

= 44.8 (W1,3 – 9.9) N/m2 (7.67)

The foregoing must be qualified somewhat. Scaling, such as this, is valid only if pertinent factors other than size remain constant, for example, the struc­tural efficiency of materials. Also, for purposes of their mission, aircraft are designed for different load factors.

In sizing an aircraft, it is also of value to note that the empty weights of aircraft average close to 50 or 60% of the design gross weights as shown in Figure 7.34. Thus, knowing the payload and fuel and having some idea of the aerodynamic “cleanliness” of the aircraft, one can undertake a preliminary estimate of its weight and performance.

For example, suppose we are designing a four-place, light aircraft with fixed gear. Let us arbitrarily decide on 300 lb of fuel. The gross weight will be

474 AIRPLANE PERFORMANCE approximately

Gross weight = Weight empty + Payload + Fuel weight

W = WE + WPL + WF (7.68)

Assuming each passenger and baggage to weigh 200 lb gives

W= WE + 800 + 300

But WE — 0.55 W, so that

W = 2400 lb

From Figure 7.34 for this weight, the wing loading should be ap­proximately

WIS = 14.7 psf

This results in a wing area of S = 163 ft2. If we decide on a low-wing airplane,

Table 7.6 Summary of Approximate Relationships

Note. Output from turbojet engines, approximately proportional to density ratio, o. For piston engines, P — P0 (o–0.1)/0.9.

then an e of 0.6 is reasonable. Also, for a fixed-gear aircraft, a parasite CD of 0.037 was recommended in Chapter Four. We are now in a position to construct power-required curves for various altitudes. The next step would be to select a power plant so that performance estimates can be made. In so doing, we would make use of a typical BSFC of 0.5 lb/bhp-hr for piston engines (unless performance curves on the particular engine selected were available). The value of developing a “feeling” for reasonable values for airplane parameters should be obvious. Table 7.6 summarizes some of these.

The problem to be considered briefly in this section concerns the al­titude-velocity (or Mach number) schedule, which should be flown to mini­mize the time or fuel required to go from one speed and altitude to another speed and altitude. As pointed out in Reference 7.4, this problem can be solved by the application of variational calculus. However, the result is a formidable computer program. As an alternate method, which is approximate but close to the more exact solution, one can obtain a graphical solution by considering the energy state of the airplane.

As noted previously, Equation 7.14 is an energy relationship for the rate of climb. If we let he denote the total energy, kinetic and potential, per unit weight of the airplane, Equation 7.14 can be written in terms of this specific energy as

dhe V(T-D) dt W

where

(7.62)

dhjdt will be denoted by Ps and is called the excess specific power.

The rate of change of he with respect to fuel weight, Wf, will be denoted by fs and can be written as

r _ dhe

Is ~’dWf

dhjdt dWJdt

The time required to go from one energy level to another will be given by

(7.64)

The path to minimize At at any altitude and airspeed will be the one that gives the maximum rate of change of he for a given Ps value. Therefore, if contours of constant he and constant Ps values are plotted as a function of altitude and Mach number, the path for minimum time will be the locus of points for which the contours are parallel. Similarily, contour plots of constant fs and he values provide an altitude-Mach number schedule for minimum fuel con­sumption.

As an example, consider a hypothetical subsonic turbojet airplane with the thrust and drag given by

T=T0<r

_ pV2f 2(Wlbf 1 D~ 2 + тгре V2

In this case, one can write, for the density ratio <r

В + VB2 + 4AC

a =————– 2———-

where

PofV2 2

The curves of altitude versus airspeed presented in Figure 7.31 were prepared by evaluating a over a range of airspeeds for constant values of Ps. For the standard atmosphere, a and h are related by h =44.3 (1 – o-0235)km. Curves of constant he are also shown in Figure 7.31. The altitude-airspeed schedule for climbing from sea level to Fmax at 11.8 km is indicated by the dashed line in this figure. This line passes through points on the Ps curves where these curves would be tangent to lines of constant he. In this example, where the thrust and drag are well behaved, the result is about as one would expect.

The results are substantially different, however, for an airplane designed to operate through Mach 1, particularly if the thrust is marginal in the transonic region. Such a case is presented in Figure 7.32 (taken from Ref. 7.4). As indicated by the dashed line, in this case the optimum trajectory consists

•i

of a subsonic climb at a nearly constant Mach number to 33,000 ft followed by a descent through the transonic drag rise region to 20,000 ft and a Mach number of 1.25. A climb to 39,000 ft at increasing Mach numbers then ensues up to 39,000 ft and Mach 2.1. The remainder of the climb up to 50,000 ft is accomplished at a nearly constant Mach number, as shown.

Flight Envelope

An airplane’s flight envelope is the. region on an airspeed-altitude plot in which the airplane is capable of operating. Within this region, an airplane is limited at low speeds by stall and at high speeds by the available thrust. The stall boundary as a function of altitude is easily determined from

V2=(2WSU

yPoC^J a

The high-speed boundary is determined from power-available, power – required curves such as those presented in Figure 7.15. As an example, let us again consider the Cherokee Arrow. Figure 7.25 was prepared using a gross Weight of 11.8 kN, a of 1.6, a constant propeller efficiency of 0.85, and a gea level engine power of 149 kW. The Cherokee is capable of level flight within the region bounded by the two curves labeled “stall” and “maximum” power.

A typical flight envelope for a supersonic aircraft is given in Figure 7.26. At high subsonic Mach numbers, a phenomenon known as buffet can limit flight to speeds higher than the stalling speeds. This type of buffeting is caused by an instability in the position of the shock waves near the trailing edge of the upper and lowej wing surfaces. As the stall is approached, these

Figure 7.25

waves begin to move fore and aft out of phase with each other, producing a periodic flow behind the wing that resembles a Karman vortex street. The tail, in proximity to this unsteady flow, can produce a severe shaking of the airplane.

Also shown on Figure 7.26 is a limit on the maximum dynamic pressure that can be tolerated. This boundary arises from structural considerations and involves items such as flutter, torsional divergence, and static pressure within an engine inlet diffuser.

An aerodynamic heating limit as shown in Figure 7.26 also exists for airplanes designed to operate at high Mach numbers. It is beyond the scope of this text to consider in depth the subject of aerodynamic heating. However, one can gain some appreciation for the problem by calculating the stagnation

temperature as a function of Mach Number. This can be accomplished using the relationships covered in Chapter Five with the results shown in Figure 7.27. Along the leading edge of a wing, these temperatures will be alleviated somewhat by sweep. Nevertheless, temperatures of the order of 250 °С or higher can be expected for Mach numbers exceeding 2.0.

Maneuvering Envelope (V-n Diagram)

The lift distribution on a wing is illustrated in Figure 7.28. If у represents the spanwise distance to the center of lift of one side, the bending moment at the wing root will be given approximately by

M-yf

where L is the total lift on the wing. Generally, L will be greater than the airplane’s weight, in which case the airplane is accelerating upward at a value

so that the bending moment becomes.

The term (1 + alg) is known as the load factor, n.

n = 1 + – (7.56)

g

In this example, the wing bending moment in steady flight is seen to increase by the factor n. Similarly, n is a measure generally of the increase in the loads on any member of the airplane resulting from accelerations. In I, steady, level flight, n is equal to 1. As a result of maneuvering or gusts, n can increase in magnitude to high values and can be positive or negative.

The value of n that can be achieved by maneuvering can be obtained from

But 2WlpSCLmax equals the stalling speed, Vs. Therefore

(7.57)

Since V can be appreciably greater than the stalling speed, Vs, it is not practical to design an airplane’s structure to withstand the highest possible load factors that it could produce. Instead, based on experience, airplanes are certified to withstand different limit load factors, depending on the airplane’s intended use. A limit load is one that can be supported by a structure without yielding. In addition to designing to the limit loads, FAR Parts 23 and 25 require factors of safety of 1.5 to be applied to the sizing of the structure. Since the ultimate allowable stress of aluminum alloys is approximately 50% greater than the yield stress, a factor of safety of 1.5 applied to the limit loads is approximately equivalent to designing to ultimate load factors with no factor of safety.

Civil airplanes are designed in the normal, utility, acrobatic, and transport categories. For the first three categories, FAR Part 23 states:

§ 23.337 Limit maneuvering load factors.

(a) The positive limit maneuvering load factor n may not be less than

[(1) 2.1 + J4’™ for normal category airplanes, except that n need

W t 1U, IHJU

not be more than 3.8;]

(2) 4.4 for utility category airplanes; or

(3) 6.0 for acrobatic category airplanes.

(b) The negative limit maneuvering load factor may not be less than—

(1) 0.4 times the positive load factor for the normal and utility categories; or

(2) 0.5 times the positive load factor for the acrobatic category.

(c) Maneuvering load factors lower than those specified in this section may be used if the airplane has design features that make it impossible to exceed these values in flight.

For the transport category, FAR Part 25 states:

§ 25.337 Limit maneuvering load factors.

(a) Except where limited by maximum (static) lift coefficients, the airplane is assumed to be subjected to symmetrical maneuvers resulting in the limit maneuvering load factors prescribed in this section. Pitching velocities appropriate to the corresponding pull-up and steady turn maneuvers must be taken into account.

(b) The positive limit maneuvering load factor n for any speed up to VD may not be less than 2.5.

(c) The negative limit maneuvering load factor—

(1) May not be less than -1.0 at speeds up to Vc’, and

(2) Must vary linearly with speed from the value at Vc to zero at V&

(d) Maneuvering load factors lower than those specified in this section may be used if the airplane, has design features that make it impossible to exceed these values in flight.

In these regulations, Vc is referred to as the design cruising speed. It need not exceed VH, the maximum speed in level flight at maximum con­tinuous power. Otherwise, it must not be less than VB plus 43 knots where VB is the lowest speed that can produce a load factor of 2.5. VD is the design dive speed, and for the transport category it need not be greater than VH-

Gust Load Factors

A wing suddenly penetrating a “sharp-edged” gust is pictured in Figure 7.29. The gust velocity is denoted by Udc in accordance with FAR notation. After penetrating the gust and before the wing begins to move upward, the angle-of-attack increment rdsulting from the gust, Да, equals

Да=-^ (7.58)

The increase in the wing’s lift then becomes

Д L = PV2Sa^

Before encountering the gust, in level flight,

.W = ^pV2Saa

The load factor, rt, resulting from the gust encounter therefore becomes

L

W

W + AL
W

Ude

Va

^rT777////////777T^,_______________ v_

Figure 7.29 A wing penetrating a sharp-edged gust.

In practice, one never encounters a truly sharp-edged gust. Therefore Udc is multiplied by an alleviation factor less than unity, again based on experience, which lessens the acceleration due to the gust. The final result, given in FAR Part 23, for the load factor resulting from a gust is expressed as follows. ‘ 1 iJ >

where

Positive and negative values of t/de up to 50 fps must be considered at Vc at altitudes between sea level and 20,000 ft. The gust velocity may be reduced linearly from 50 fps at 20,000 ft to 25 fps at 50,000 ft. Positive and negative gusts of 25 fps at VD must be considered at altitudes between sea level and

20,0 ft. This velocity can be reduced linearly to 12.5 fps at 50,000 ft.

For FAR Part 23, the foregoing criteria for the maneuvering and gust loads results in the type of V-n diagram pictured in Figure 7.30 In certifying an airplane, one must demonstrate the structural integrity of the airplane subjected to the aerodynamic loadings that can exist throughout the V-n diagram.

Specifications of military aircraft and many larger civil aircraft indue range-payload curves. This is a graph, that for a particular mission profit presents the effect of trading off payload for fuel on the range of an airplam In determining such a curve, one must consider the operational phases th; have been treated thus far in this chapter. .

As an example in calculating a range-payload curve, consider the Cessn Citation I. Figure 7.21 presents a side-wiew sketch of this airplane. Table 7. presents a general description and some of its specifications. The rang calculation will include: [3]

The reserve fuel is calculated on the basis of holding at 25,000 ft (7620 m).

Using methods similar to those just presented and given engine per­formance curves (including installation losses), one can calculate, for a number of fixed airplane weights, information similar to that presented in Tables 7.2 to 7.5.

The maximum ramp weight for Citation I is 12,000 lb (53.4 kN). During taxi, 150 lb of fuel is assumed to be burned, so that the maximum allowable takeoff weight is 11,8501b (52.7 kN). The maximum fuel capacity at this takeoff weight allowing for the 1501b expended during taxi is 36301b (16.3 kN). This is the fuel available to fly the cruise profile. This maximum fuel together with the empty weight of 6470 lb (28.7 kN) and an oil weight of 341b leaves 17161b for payload. Thus, if 10 persons are on board (including the cre(w) at 200 lb per person, 284 lb of fuel must be removed resulting in a direct reduction in the range.

The total fuel for the range profile can be expressed as

(7.51)

where CLB, CR, HLD, and DES refer to climb, cruise, hold, and descent.

Figures 7.22 and 7.23 present in graphical form data from Tables 7.2, 7.3, and 7.5 necessary to determine the range-payload curve. In determining the range, in a manner of speaking, we begin at the ends and work toward the middle. During the hold, the average airplane weight will equal

(7.52)

In Equation 7.52 Wfms is the fuel required to descend from the holding altitude of 25,000 ft. Consider first takeoff at maximum allowable takeoff weight with full fuel. Thus,

WTO= 11,8501b Wf = 3630 lb W, DES = 83 (Table 7.4) lb

The average weight for holding thus becomes

W = 8303 + ^“-°

Referring to Figure 7.23, guess at a W of 85001b. This leads to a fuel flow rate of 540 lb/hr or, for 45 min, a-holding fuel of 405 lb. From Equation 7.53, W is then calculated to be 8708, .which is higher than the guessed value. Iterating in this manner, the holding fuel is found to equal 415 lb.

The fuel to climb to 41,000 ft from Figure 7.22 equals 8861b. To descend from 41,000 ft to sea level requires 1341b – Thus, the fuel left for cruising equals: f

W7c„ = Wf – W/HLD – VV/DES – 4V/CIB = 2195 lb

The average weight during the cruise will be

W = WTO- WfcLB-^

= 9867 lb

From Figure 7.23, the cruising speed at this weight equals 328 kt. The fuel

Source.

flow rate for maximum cruising thrust is found from Table 7.3 to equal 688 lb/hr. The total cruising fuel, fuel flow, and speed give a distance covered during the cruise of 1046 nmi. To get the range, we add to this the distance covered during the climb and descent. These are – obtained from Figure 7.22 and Table 7.4, respectively. These two distances total 283 nmi. giving a total range of 1329 nmi.

This range and payload represent one point on the range-payload curve. A break in the slope of the range-payload curve will occur at this point because

of the following. For lighter payloads, the gross weight will decrease, resulting in increased ranges for the same total fuel. For heavier payloads, less fuel must be put aboard, resulting in a direct loss of range. At a cruising airspeed of around 328 kt and a fuel flow of 688 lb/hr, one can see immediately that the range will decrease by approximately 48 nmi/100 lb of additional payload over 17161b.

Consider two additional points on the range-payload curve: a payload of 2000 lb corresponding to a crew of two and eight passengers at 200 lb per person, and a minimum payload of 400 lb corresponding to the crew alone.

For the 2000-lb payload, the takeoff weight will remain the same, but the fuel weight is reduced by 2841b to 33461b after taxi. The average holding weight now becomes

W = 8587 +

By iteration, W/HLD = 421 lb

The fuel to climb remains unchanged, so that

WfcR = 1905 lb

The average weight during cruise equals 10,012 lb. From Figure 7.23, the cruising speed at this average weight will be 325 kt. This speed and a fuel rate of 688 lb/hr result in a distance of 900 nmi. The distances gained during climb and descent remain unchanged, so the total range for the 20001b payload becomes 1183 nmi.

For a minimum payload of 4001b, the takeoff gross will be reduced to 10,534 lb.

The equation for the average holding weight now reads

W = 6987 +

so VF/hid = 3441b. The fuel weight to climb to 41,000 ft at this reduced gross weight equals 5571b and the distance travelled during the climb is 110 nmi. The descent fuel and distance are assumed to remain the same. Thus the fuel for cruising equals 2595 lb. The average cruising weight becomes 8680 lb. At this weight, the cruising speed equals 339 kt with the fuel consumption rate unchanged. Therefore the cruising distance equals 1279 nmi. Added to the distances covered during climb and descent, this figure results in a total range of 1458 nmi.

The preceding three points define the range-payload curve for the Cita­tion I as presented in Figure 7.24. The calculated points are shown on the curve. Again, the break in this curve corresponds to maximum allowable takeoff weight with a full fuel load.

4

The calculation of the ground roll in landing follows along the same lines used for a takeoff ground roll, but with different parameters and initial conditions. The braking coefficient of friction varies from approximately 0.4 to 0.6 on a hard, dry surface to 0.2 on wet grass or 0.1 on snow. With spoilers the lift is essentially zero. Also, with flaps and spoilers the parasite drag coefficient may be higher.

Beginning with an initial value of VA, the equations of motion can be numerically integrated, accounting for the variation with V of any reverse thrust, drag, and possibly lift. One can also use the approximate relationship

derived previously, equation 7.6. In the case of landing, this becomes

where a is the magnitude of the deceleration evaluated at VАЫ 2.

As an example, consider the 747-100 at its maximum landing weight 2500 kN (564,000 lb). Assuming a fi of 0.4, an / of 80 m2 (260 ft2) and VA equ to 65 m/s (126 kt) gives

a = 4.05 m/s2

Therefore, the ground roll distance is estimated to be,

s = 522 m (1710 ft)

This gives a total estimated FAR landing distance, including the factor of 1.667, <

s(total) = 1670m (5480ft)

This compares favorably with the distance of 1880 m (6170 ft) quoted Reference 5.11. The difference is easily attributable to uncertainties in tl approach speed and the braking friction coefficient.

Lighter aircraft, except on an instrument approach, tend to descend at г angle steeper than 3°. With their lower wing loadings, light aircraft also touc down at much lower speeds. Hence, their landing distances are significant less than those for a jet transport. A Cherokee Arrow, for example, touchir down at approximately 65 kt can be stopped with moderate braking withi 300 m (1000 ft).

From Figure 7.20, since 0n is a small angle, the total airborne distance, sA, is given by

This assumes the flare to be a circular arc having a radius of R. If VA is the approach velocity, this velocity is assumed to remain constant throughout the flare. The acceleration toward the center of curvature, a„, will therefore be

R

However,

T ur – w

L— W = —a„

Thus,

h

g(UW-l)

If CLa denotes the lift coefficient during the steady approach then, during the flare,

The flare radius can therefore be expressed as

FAR Part 25 requires that VA exceed the stalling speed in the landing configuration by 30%. Thus,

Thus the ratio CJCLa can vary anywhere, from just above 1 to 1.69 or higher. \ typical value of this ratio for jet transports is 1.2. Using this value, but keeping in mind that it can be higher, the total airborne distance, in meters, becomes

15.2

“ T~ +

tfD

A jet transport approaches typically at a speed of 125 kt at an angle of 3°. Thus,

sA — 290 + 55

= 350 m (1155 ft.)

After touchdown, an approximately 2-sec delay is allowed while the pilot changes from the landing to the braking configuration. During this period the airplane continues to roll at the speed VA. Actually, practice has shown that the speed decreases during the flare by approximately 5 kt typically. Denoting this portion by a subscript “tran” for transition,

Stran=2(FA)

= 130 m (427 ft)

Endurance refers to the time that elapses in remaining aloft. Here one is concerned with the time spent in the air, not the distance covered. A pilot in a holding pattern awaiting clearance for an instrument landing is concerned with endurance. The maximum endurance will be obtained at theairspeed that requires the minimum fuel flow rate. In the case of a turbojet on turbofan engine, the product of TSFC and the drag is minimized. v—^

ii Assuming a constant TSFC for the jet airplane leads to Equations 7.36 and 7.37, which relate to the maximum range for a propeller-driven airplane. Thus, for a given weight of fuel, Wf, the maximum endurance, te, of a turbojet will be

1 ЖГ7гМТ/2

‘ 2(TSFC) W/ L(//S)J C7 41)

Notice that the endurance is independent of altitude. This follows from

the fact that the minimum drag does not vary with altitude. To obtain this endurance, the airplane is flown at the airspeed given by Equation 7.37.

Assuming a constant SFC, a propeller-driven aircraft will have its maxi­mum endurance when flown at the airspeed for minimum required power. This speed has been given previously as Equation 7.22. At the minimum power, the induced power is three times the parasite power. Hence,

Pmin — 2pfV3

= _2_ r4(W/b)2~l3/4 fm Vcr L 3ne J Po1’2

The endurance time for a propeller-driven airplane then becomes

t = ^a Г 37ГЄ Г Po n A1

te 2(SFC) |4(w/b)2J 1th (7 43)

(SFC) in this equation has units consistent with the other terms; that is, weight per power second where the weight is in newtons or pounds with the power in newton meters per second or foot-pounds per second. For example, if

BSFC = 0.5 lb/bhp-hr

then

SFC = 2.53 x КГ7 lb/(ft-lb/sec)/sec

Actually SFC used in this basic manner has the units of 1/length. In the Engli^system this becomes ft-1 and in the SI system it is m-‘.

Notice that the endurance of a propeller-driven airplane decreases with altitude. This follows from the fact that the minimum power increases with altitude.

Some of the foregoing equations for range and endurance contain the weight which, of course, varies with time as fuel is burned. Usually, for determining the optimum airspeed or the endurance time, it is sufficiently accurate to assume an average weight equal to the initial weight minus half of the fuel weight. Otherwise, numerical and graphical procedures must be used to determine range and endurance.

DESCENT

^ The relationships previously developed for a steady climb apply as well to descent. If the available thrust is less than the drag, Equation 7.15 results in a negative R/C. In magnitude this equals the rate of descent, R/D. The angle of descent, in radians, во, is given by,

(7.44)

Civil aircraft rarely descend at angles greater than 10°. The glide slope for an ILS (instrument landing system) approach is only 3°. Steeper slopes for noise abatement purposes are being considered, but only up to 6°.

The minimum eD value in the event of an engine failure is of interest. From Equation 7.44 we see that this angle is given by

0Dmin = emin rad (7.45)

Thus, the best glide angle is obtained at the CL giving the lowest drag-to-lift ratio. This angle is independent of gross weight. However, the greater the weight, the higher the optimum airspeed will be. The minimum e and corresponding airspeed have been given previously as Equations 7.36 and 7.37. Of course, in the event of an engine failure, one must account for the increase in / caused by the stopped or windmilling propeller, or by the stopped turbojet.

LANDING

The landing phase of an airplane’s operation consists of three segments; the approach, the flare, and the ground roll. FAR Part 25 specifies the total landing distance to include that required to clear a 50-ft (15.2-m) obstacle. A sketch of the landing flight path for this type of approach is shown in Figure

7.20. The ground roll is not shown, since it is simply a continuous deceleration along the runway. FAR Part 25 specifies the following, taken verbatim.

§ 25.125 Landing.

(a) The horizontal distance necessary to land and to come to a complete stop (or to a speed of approximately 3 knots for water landings) from a point 50 feet above the landing surface must be determined (for standard temperatures, at each weight, altitude, and wind within the operational limits established by the applicant for the airplane) as follows:

(1) The airplane must be in the landing configuration.

(2) A steady gliding approach, with a calibrated airspeed of not less than 1.3 Vs, must be maintained down to the 50 foot height.

(3) Changes in configuration, power or thrust, and speed, must be made in accordance with the established procedures for service operation.

4 (4) The landing must be made without excessive vertical acceleration,

tendency to bounce nose over, ground loop, porpoise, or water loop. (5) The landings may not require exceptional piloting skill or alertness.

(b) For landplanes and amphibians, the landing distance on land must be determined on a level, smooth, dry, hard-surfaced runway. In addition—

(1) The pressures on the wheel braking systems may not exceed those specified by the brake manufacturer;

(2) The brakes may not be used so as to cause excessive wear of brakes or tires; and

(3) Means other than wheel brakes may be used if that means—

(i) Is safe and reliable;

(ii) Is used so that consistent results can be expected in service; and /(iii) Is such that exceptional skill is not required to control the

airplane.

(c) For seaplanes and amphibians, the landing distance on water must be determined on smooth water.

(d) For skiplanes, the landing distance on snow must be determined on smooth, dry, snow.

(e) The landing distance data must include correction factors for not more than 50 percent of the nominal wind components along the landing path opposite to the direction of landing, and not less than 150 percent of the nominal wind components along the landing path in the direction of landing.

(f) If any device is used that depends on the operation of any engine, and if the landing distance would be noticeably increased when a landing is made with that engine inoperative, the landing distance must be determined with

that engine inoperative unless the use of compensating means will result in a landing distance not more than that with each engine operating.

The total distance thus calculated must be increased by a factor of 1.667.

FAR Part 23 is somewhat simpler in defining the landing for airplanes certified in the normal, utility, or acrobatic categories. It states the following,

§ 23.75 Landing.

(a) For airplanes of more than 6,000 pounds maximum weight (except skiplanes for which landplane landing data have been determined under this paragraph and furnished in the Airplane Flight Manual), the horizon­tal distance required to land and come to a complete stop (or to a speed of approximately three miles per hour for seaplanes and amphibians) from a point 50 feet above the landing surface must be determined as follows:

(1) A steady gliding approach with a calibrated airspeed of at least 1.5 VSt must be maintained down to the 50 foot height.

(2) The landing may not require exceptional piloting skill or exceptionally favorable conditions.

(3) The landing must be made without excessive vertical acceleration or tendency to bounce, nose over, ground loop, porpoise, or water loop.

(b) Airplanes of 6,000 pounds or less maximum weight must be able to be landed safely and come to a stop without exceptional piloting skill and without excessive vertical acceleration or tendency to bounce, nose over, ground loop, porpoise, or water loop.

The range of an aircraft is the disfance that the aircraft can fly. Range is generally defined subject to other requirements. In the case of military aircraft, one usually works to a mission profile that may specify a climb segment, a cruise segment, a loiture, an enemy engagement, a descent to unload cargo, a climb, a return cruise, a hold, and a descent. In the case of civil aircraft, the range is usually taken to mean the maximum distance that the airplane can fly on a given amount of fuel with allowance to fly to an alternate airport in case of bad weather.

‘ Let us put aside the range profile for the present and consider only the actual distance that an airplane can fly at cruising altitude and airspeed on a given amount of fuel. For a propeller-driven airplane, the rate at which fuel is

Wf = (BSFQ(bhp) lb/hr W, = (SFC)(kW) N/s

where SFC is in units of newtons per kilowatt per second.

Using the SI notation, the total fuel weight consumed over a given time will be

W,= [ (SFC)(kW) dt

Jo

This can be written as

Wf = J‘(SFC^kW)

Since the shaft power equals the thrust power divided by the propeller efficiency,

1 f (SFC)D,

ioooj.——* where D is the drag. The constant represents the fact that lkW equals 1000 mN/s.

Given the velocity and weight, Equation 7.31 can be integrated numeric­ally. One of the difficulties in evaluating Equation 7.31 rests with the weight, which! is continually decreasing as fuel is burnt off.

A closed-form solution can be obtained for Equation 7.31 by assuming that the SFC and tj are constant and that the airplane is flown at a constant CL. With these assumptions, the fuel flow rate, with respect to distance, becomes

dW, _ (SFC)e ds 1000 tj

where e is the drag-to-lift ratio, which is a function of CL, and W is the airplane weight. dWflds is the negative of dW/ds. Thus,

dW (SFC)e

w 1000 tj dS

W, (SFC)e W 1000 tj

where Wі is the initial weight of the airplane. If WF is the total fuel weight, the distance, or range, R, that the airplane can fly on this fuel is finally, in meters,

WE denotes “weight empty,” meaning “empty of fuel.” Normally, weight empty refers to the airplane weight without any fuel or payload. This equation, which holds only for propeller-driven aircraft, is a classical one known as the Breguet range equation.

In the case of a turbojet-propelled airplane, the fuel flow becomes

Wf = (TSFC)D

so that

dW (TSFC)e ds W V

In order to integrate this relationship, we must assume that the airplane operates at a constant e/V and that TSFC is constant. When this is done, the modified Breguet range equation for jet-propelled aircraft is obtained.

R (TSFC)e ln 0 + wO (7’33)

Thus, for maximum range, e should be minimized for propeller-driven

airplanes and e/V should be minimized for turbojets. In the case of turbojets, this can lead to the airplane cruising slightly into the drag rise region that results from transonic flow.

Referring to Equation 7.18, e will be a minimum when

/С ^4

Vop, = (^j (propeller-driven airplane) (7.34)

e/V will have a minimum at

ip l/4

-jjf) (turbojet) (7.35)

For propeller-driven airplanes this leads to a minimum e value of

e ■ = 2(ЩШ
mm rreA)

r 4 (wis)2y14

opt

For turbojet-propelled airplanes, e/V has a minimum value of

The optimum V for the above is equal to that given by Equation 7.37 multiplied by 31/4.

Some interesting observations can be made based on Equations 7.36, 7.37, and 7.38. For either propeller or turbojet airplanes, the indicated air-

speed for maximum range is a constant independent of altitude. However, for the same wing loading, elfective aspect ratio, and parasite drag coefficient, the optimum cruising speed for the turbojet airplane is higher than that for the propeller-driven case by a factor of 1.316. The optimum range for a propeller – driven airplane is independent of density ratio and hence altitude. However, with the indicated airspeed being constant, the trip time will be shorter at a higher altitude.

The optimum range for a turbojet is seen to increase with altitude being inversely proportional to the square root of the density ratio. This fact, together with the increase in true airspeed with altitude, results in appreciably higher cruising speeds for jet transports when compared with a propeller – driven airplane. As an example in the use of Equations 7.38 and 7.37 (multiplied by 1.316), consider once again the 747-100 at a gross weight of ^2700 kN. In this case,

Y = 5284 N/m2

fls = 0.0182 Ae – 4.9

Vopt = 167/Vo – m/s

Dividing this velocity by the speed of sound, the optimum Mach number as a function of altitude shown in Figure 7.17 can be obtained. Above a Mach number of approximately 0.8, this curve cannot be expected to hold, since drag divergence will occur.

The second curve shown in Figure 7.17 presents the optimum range divided by the sea level value of this quantity. This curve is calculated on the basis of Equation 7.38 and the TSFC values for the JT9D-7A engine presented in Figure 6.38 as a function of altitude and Mach number. This curve is reasonably valid up to an altitude of 7500 m. Above this, because of Mach number limitations, the range ratio will level off. However, despite the Mach number limitations, the gains to be realized in the range by flying at the higher altitudes фе appreciable, of the order of 30% or more.

In the case of propeller-driven airplanes, the optimum cruising velocity given by Equation 7.37 does not reflect practice. To see why, consider the Cherokee Arrow. In this case, at a gross weight of 26501b, the optimum velocity is calculated to equal 87.8 kt. This velocity is appreciably slower than the speeds at which the airplane is capable of flying. It is generally true of a piston engine airplane that the installed power needed to provide adequate climb performance is capable of providing an airspeed appreciably higher than the speed for optimum range. Therefore, ranges of such aircraft are quoted at some percentage of rated power, usually 65 or 75%.

The cruising speed at some specified percentage of the rated power can be found from the power curves, such as those presented in Figure 7.15 for the Cherokee Arrow. For example, 75% of the rated power corresponds to approximately 81% of the available power shown in Figure 7.15. This increase^ results from the rating of 200 bhp at 2700 rp. m as compared to only 185 bhp output at 2500 rpm for which the figurefwas prepared. A line that is 81% of the available power crosses the power-required curve at a speed of 223 fps or 132 kt. This speed is therefore estimated to be the cruising speed at 75% of lirated power at this particular rpm.

The penalty in the range incurred by cruising at other than the optimum speed can be found approximately from Equations 7.18 and 7.34. The ratio of the drag at any speed to the minimum drag can be expressed as a function of the ratio of the speed to the optimum speed. The result is

This relationship is presented graphically in Figure 7.18. In the preceding example of the Cherokee, this figure shows a loss of approximately 25% in the range by cruising at 75% power instead of the optimum. Of course, the time required (to get to your destination is 33% less by cruising at 75% power.

The effect of wind on range is pronounced. To take an extreme, suppose you were cruising at the optimum airspeed for no wind into a headwind of equal magnitude. Your ground speed would be zero. Obviously, your airspeed is no longer optimum, and it would behoove you to increase your airspeed. Thus, without going through any derivations, we conclude that the optimum airspeed increases with headwind.

Correcting Equation 7.32 for headwind is left to you. If V* denotes the headwind, this equation becomes (now expressed in the English system),

The effect of headwind on the optimum cruising airspeed can be obtained by minimizing e/(l – VJ V). Without going into the details, this leads to the following polynomial

v

+ —= 0

v.

* opt

Here V is the optimum cruising velocity for a given headwind and Fopt is the value of V for a Vw of zero.

Vw

v;pt

т^= 1 + AVIVopt

‘ Opt

A VIVopt is presented as a function of VJVapt in Figure 7.19. This figure shows, for example, that if one has a headwind equal to 50% of the optimum velocity for no wind, he or she should cruise a. t an airspeed 20% higher than the optimum, no-wind velocity.