Airborne Distance

Becoming airborne at the speed FLof, an airplane continues to accelerate to the speed V2 over the obstacle height. An airplane that is accelerating both normal to and along its flight path is pictured in Figure 7.9. As shown, V(deldt) is the acceleration normal to the flight path and dVIdt is the acceleration along the flight path. The equations of motion normal to and along the flight path can be written as

The rate of climb, dh/dt, and denoted by Vc in Chapter One, is found * from

dhldt =V sine (7.12)

Rate of climb is also denoted in the literature by R/C. Solving for sin в from Equation 7.10, Equation 7.12 can be written as

dh_ Г(Г-£>) dV/dtl

dt V[ W t g J

This can also be expressed as

(T-d>V-W£+£(Wv!) <714)

n

TV is the available power and DV is the required power. Thus (T-D)V is the excess power that, as Equation 7.14 shows, can be used either to climb or to accelerate. Actually, Equation 7.14 is an energy relationship that states that the excess power equals the sum of the time rates of change of the potential energy and the kinetic energy.

Let us now apply these relationships to the calculation of the horizontal

distance required during the takeoff flare to attain a specified height. The actual flight path that is followed during the flare, or transition, segment of the takeoff depends on pilot technique. Referring to Equations 7.10 and 7.11, V and в are the independent variables, while g, W and T are known, the latter as a function of V. L and D are functions of V and the airplane’s angle of attack a. By controlling a, and hence CL, the pilot can fly a desired trajectory (i. e., the pilot can accelerate or climb or do some of each). During the takeoff, however, in attempting to clear an obstacle, FAR Part 25 limits the operating CL to approximately CimJ.2 at FLof and to C^J 1.44 at V2. Therefore, in calculating the flare distance, it will be assumed that CL varies linearly with V between these limits and is constant for speeds above V2. Thus, if V2 is £ ’ і before the specified obstacle height is reached, the stall margin on CL

і tained.

turning to the example of the 747 at 733,000-lb takeoff weight, let us assume a with partially deflected flaps of 1.8. In addition, let FLOf be 10% above the stall speed. Thus, at sea level,

Flop — 83.7 m/s CLlof= 1.49 V2 = 91.2 m/s = 1.25

Assuming an / of 18.6 m2 (200 ft2) and calculating the induced drag obtained using CL, the effective aspect ratio, and Figure 7.4, the drag can be determined at V’lqf to be

where q = 4289 Pa and CD; = 0.0203. Thus,

D = 225.8 kN

The rated takeoff thrust at this speed for the four JT9D-7A engines equals 654.7 kN. From Equation 7.10 for an initial в of zero,

Initially at Vlof, F and W are exactly equal, so that dd/dt is zero from Equation 7.11. For illustrative purposes, however, let us take a time increment of 1 sec and assume the preceding acceleration to be constant over that time. At the end of 1 sec, V then becomes 85.0 m/s and CL is reduced slightly to 1.45. Thus, at this time,

L = 3280 kN (737,400 lb)

dd/dt from Equation 7.11 then becomes 0.000693 rad/sec. Averaged over the
1 sec, this value of d8/dt gives a climb angle of approximately 0.02° 1 sec after lift-off.

The rate of climb, dhldt is obtained from Equation 7.12. Over the 1 sec, the average rate of climb will equal

dhldt (average) = 1(85) sin (0.02)

= 0.0148 m/s

Using this average rate of climb results in an altitude gain of 0.0148 m over the 1 sec.

The complete numerical solution of this example is shown in Figure 7.10. A time increment of 0.1 sec was used in the numerical integration. It can be see that a distance of 3140 ft (957 m) and 10.3 sec are required to attain an altitude of 35 ft (10.7 m). For the first half of the flare following lift-off, the airplane is primarily accelerating with little of the excess power going into climb. After approximately 7 sec, the acceleration decreases and the rate of climb increases rapidly.

The flare distance to clear 35 ft together with the ground roll distance of 6350 ft shown in Figure 7.6 to reach Vlof gives a total calculated takeoff distance of 9490 ft. Reference 5.11 quotes a value of 9450 ft at the same gross weight. This agreement may be fortuituous in view of the uncertainty on / with the gear and flaps down.

Figure 7.10 Calculated takeoff flare performance for a Boeing 747-100.