For many airplanes, the typical value of best-range angle of attack is 4 ± 1 deg (as is the case in Figure 4.7 as well).
Homework Exercise: Assuming a parabolic drag polar, obtain analytical relations for the best-endurance speed and the best-range speed. Evaluate these speeds and match them against the values from the EBA analysis in the plots.
5. Speed for minimum power: Just as the case of speed for minimum thrust in Figure 4.5, one can also find the speed at which the airplane consumes the minimum power, where P = TV. However, this is more a figure of merit for propeller-powered airplanes and not so useful for jet-powered ones. Nevertheless, such a plot can be produced from the EBA analysis as shown in Figure 4.8. In this case, we plot the variable T V. versus velocity and angle of attack. The velocity and angle of attack for minimum power flight may be read off Figure 4.8 to be approximately 100 m/s and 5.5 deg, respectively.
4.4 CLIMBING/DESCENDING FLIGHT TRIM
be used to either accelerate the airplane (gain kinetic energy) or to increase its altitude (gain potential energy). Presently, we shall look at steady climb, that is, with no acceleration. Then the difference between thrust and drag is entirely used to gain altitude and clearly this gain in altitude will be maximum when the thrust is a maximum. Therefore, analysis of the best climb flights should be carried out with maximum throttle, that is, η = 1.
However, it still remains to determine the ideal velocity at which to carry out a climb maneuver so that either the climb angle is maximized (steepest climb) or the climb rate is maximized (fastest climb). To calcu- late this, an SBA analysis may be carried out with fixed throttle η = 1 and elevator deflection varied as the parameter. Results from such an analysis are shown in Figure 4.9. The trim angle of attack and the correspond- ing steady-state climb/descent angle are plotted along with the stability of each trim state. More positive elevator deflections create a nose-down moment that push the trim angle of attack to lower values, hence drag is lower and the thrust–drag difference puts the airplane into a climbing trim (γ > 0). On the other hand, the larger negative (up-) elevator deflec- tions push the airplane nose up into higher angle of attack trims where the higher drag overwhelms the thrust, which is limited by the throttle at η = 1. The airplane then steadily descends (γ < 0) as it would, for instance, during the landing approach. As in the case of level flight (Figure 4.6),
16
(a) ×106 (b) ×106
14 12 10
Treq × V (N m/s) Treq × V (N m/s)
8 6 4 2 0
16 14 12 10 8 6 4 2 100 150 200 0
V (m/s) 250 300 0.1 0.2α (rad)0.3 0.4 0.5 FIGURE 4.8 Trim power (thrust times velocity) for F-18 data with varying eleva- tor deflection to satisfy level flight constraint plotted as a function of (a) trim velocity and (b) trim angle of attack (full line: stable, dashed line: unstable).
there are multiple stable trim branches with brief interludes of instability, largely due to the phugoid mode as seen previously. With increasing α, the final transition to instability at the point labeled “PB” is due to loss of short-period damping leading to pitch bucking—same as in Figure 4.2.
To deduce the climb figures of merit referred to earlier, it is preferable to plot the climb angle γ versus the velocity (or Mach number) for steep- est climb and the climb rate V sin γ in case of the fastest climb. These are shown versus the Mach number in Figure 4.10.
From the upper panel of Figure 4.10, the steepest climb angle is approx- imately 20 deg and this occurs at a Mach number of around 0.35. These numbers are quite representative of aircraft of this class. The velocity at this Mach number is pretty close to the V(L D/ )max value of 120 m/s in Figure 4.5, which is fairly obvious since that is the point of minimum drag and with thrust set to maximum at η = 1, the difference between thrust and drag is the most when drag is the least.
The lower panel of Figure 4.10 yields the best climb rate, which works out to nearly 60 m/s (3.6 km/min) when multiplied by the speed of sound.
0.8 PB
0.6 0.4 α (rad) 0.2
0 –0.2 –0.5
–0.5
–0.4 –0.3 –0.2 δe (rad)
δe (rad)
–0.1 0 0.1
0.5
0
–0.5
γ (rad)
–0.5 –0.4 –0.3 –0.2 –0.1 0 0.1
PB
FIGURE 4.9 Analysis of climbing flights by SBA, throttle η = 1, and elevator deflection varied as the parameter (full line: stable, dashed line: unstable).
The corresponding Mach number for the fastest climb is around 0.62, which corresponds to a velocity of 210 m/s. Note that this is the sustained (steady) climb rate. At this Mach number, the climb angle would be around 17 deg, marginally lower than the steepest climb angle of 20 deg.
While the steepest climb angle may be of interest in clearing an obstacle after take-off, it is usually the fastest climb (maximum climb rate) that is more useful as a performance figure of merit.
In Figure 4.10, the SBA analysis reveals that the fastest climb trim is stable, whereas the steepest climb trim occurs at a point of phugoid insta- bility. This is not an unusual occurrence and some airplanes are known to have a mildly unstable phugoid dynamics during climb following take-off.
Pilots can usually handle this instability manually. (See Exercise 2 for an interesting case study.)