(down-elevator). These correspond to an angle of attack of 0.42 rad (maxi- mum) and 0.02 rad (minimum), respectively, as seen in Figure 4.5.
Since thrust equal to drag is being maintained at all values of elevator deflection in Figure 4.4, the variation of throttle reflects the nearly qua- dratic functionality of drag versus elevator deflection (equivalently, angle of attack). This is more explicitly seen in the plot of thrust versus angle of attack or thrust versus trim velocity in Figure 4.5, which will be useful for further analysis as well. For instance, the airplane’s maximum velocity is constrained by the upper limit of throttle setting. From Figure 4.5, this reads to be a little under 300 m/s at this particular flight condition (this number is to be taken with a pinch of salt as Mach-number-dependent effects have likely not been included in this aerodynamic model).
The balance of forces in level flight requires that (T/W) be equal to 1/(L/D); that is, the minimum thrust level corresponds to the maximum (L/D) flight. The velocity at which the maximum (L/D) flight occurs may therefore be marked in Figure 4.5 at the point of minimum thrust (η = 0.31). This value turns out to be around 120 m/s. Since maximum endurance flight implies minimum fuel consumption at every instant, and from Equation 4.10 that is seen to correspond to the lowest permis- sible throttle setting to maintain level flight, the best endurance for this airplane data and flight condition is obtained at the maximum (L/D)
×104 4.5
(a) (b)
3.5 3 2.5 Treq (N)
2 1.5
0.5 VL/Dmax
0 100 150 200
V (m/s) 0.1 0.2
Max throttle limit
Stability limit Stall limit
α (rad)0.3 0.4 0.5 250 300
1 4
×104 4.5
3.5 3 2.5 Treq (N)
2 1.5
0.5 0 1 4
FIGURE 4.5 Trim thrust for F-18 data with varying elevator 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).
condition at a velocity of 120 m/s. The EBA analysis can thus be used to analyze the aircraft performance.
The EBA analysis also provides information about the stability of each trim state. For instance, Figure 4.5 suggests that all trims on the “front side” of the power curve (as the T vs. V plot is often called) are indeed stable. (Of course, this analysis does not presently include stability to dis- turbances out of the longitudinal plane.) Interestingly, on the “back side”
of the power curve, there are multiple stable trim branches. This is in con- trast to the popular statement often seen in aircraft performance texts that trims on the “back side” of the power curve are unstable by considering a limited notion of stability in terms of thrust and drag variation with velocity alone. In fact, as marked in Figure 4.5, the lower velocity (higher α) limit of trimmed level flight can arise due to one of three reasons:
• The first reason could be the maximum throttle limit beyond which thrust can no longer balance drag and level flight cannot be main- tained; the airplane then enters a descending trim. In the present case, this limit is ruled out as the EBA analysis in Figure 4.5 marks the low-velocity η= 1 trim around α= 0.42 rad as unstable.
• The second reason could be a limit due to loss of stability, for instance, as seen in Figure 4.5 for α = 0.38 rad. The nature of the instability depends on the aerodynamic details of the airplane and the flight condition.
• The third cause may be aerodynamic stall, which, if indicated by the peak of the lift coefficient versus angle of attack curve in Figure 2.12, occurs near α= 0.35 rad. Often, one is led by textbooks to believe that after stall there is a sharp drop in lift coefficient and stable, level trimmed flight can no longer be maintained. Figures 2.12 and 4.5 show that this is not necessarily so; in this instance, the drop in lift coefficient post-stall is gradual and stable level flight is possible at stall and slightly beyond.
If the instability at α= 0.38 rad is critical, then it marks the limit of low-velocity level trimmed flight. Otherwise, the limit is due to the throt- tle setting reaching η= 1, where the trim is technically unstable but the instability may be effectively tolerable.
The other flight variables from the constrained level flight EBA analy- sis are shown in Figure 4.6. With increasing negative (up-) elevator, the
airplane trims at higher angles of attack, as expected. There are three seg- ments seen in the plots of Figure 4.6—a low-α segment with a steeper slope, and mid-α and high-α segments with a shallower slope. As can be found from Section 4.6.2 of Reference 1, the slope of the trim angle of attack versus trim elevator deflection curve is related to the airplane sta- bility in pitch. In general, a steeper slope means a lower degree of stability and a larger sensitivity of trim α to elevator deflection.
Homework Exercise: Can you correlate the slope of the various segments on α—δe plot with the appropriate aerodynamic derivatives in Chapter 2?
The gap between the stable segments in Figure 4.6 needs to be analyzed.
Referring back to the aerodynamic data in Figure 2.12, there is no obvious discontinuity in the plots of Cm versus α or Cm versus elevator deflection that may suggest the sudden onset of instability in Figure 4.6 in the angle of attack range of these gaps. Hence, the instability is probably not con- nected to pitch stability and the short-period dynamics, in which case it must be due to the phugoid mode. The EBA analysis may also be used to output the modal eigenvalues; hence, these can be examined to ascertain the cause of these unstable gap regions. Table 4.1 reports the longitudinal mode eigenvalues at three points (labels “1,” “2,” “3” in Figure 4.6) around one such unstable gap region. It is seen that the short-period mode is stable in all cases. The phugoid mode goes unstable at a Hopf bifurcation point
0.8
0.5
0.1 0.5
–0.1 0
0.4 0.3 0.2 0.1 0.4 0.3 0.2 α (rad)θ (rad) 0.1
q (rad/s)
δe (rad) 0.6
0.4
M
0.2 –0.1 –0.05 0
δe (rad)–0.05 0 3
12 –0.1
δe (rad)–0.05 0 δe (rad)–0.05 0 –0.1
–0.1
FIGURE 4.6 Flight variables in constrained level flight using the throttle sched- ule for F-18 data with varying elevator deflection in Figure 4.4 (full line: stable, dashed line: unstable).
(label “1”), where the eigenvalues lie on the imaginary axis. At the point labeled “2,” the phugoid mode is marginally unstable, and at the point with label “3,” it has regained stability. Thus, the small detours of the phu- goid mode eigenvalues across the imaginary axis are responsible for the limited regions of instability in the gaps between the stable segments in Figure 4.6. Note that these unstable gap regions happen to be on the “back side” of the power curve in Figure 4.5 where an unstable phugoid mode is understandable.
Homework Exercise: Traditionally, the instability on the “back side” of the power curve is attributed to loss of “speed stability,” that is, by con- sidering the dynamics in V alone due to changes in thrust and drag. On the other hand, the phugoid mode predominantly involves variation in V and γ, the flight path angle, as well. Are these two—unstable phugoid and
“speed instability”—related?
4.3.1 Level Flight Airplane Performance
As seen above, the EBA analysis can also serve as a tool to analyze the performance of the aircraft in addition to its trim and stability. Some of the performance parameters of interest in level flight are briefly discussed below with reference to the EBA results in Figure 4.5—a deeper analysis will be available in any text on aircraft performance.
1. Maximum speed: As seen previously, the maximum speed in level trim usually corresponds to the throttle limit of η= 1, subject to that trim state being stable.
2. Minimum speed: As discussed above with reference to Figure 4.5, the minimum speed is traditionally attributed to the stall limit; however, depending on the stalling behavior of the airplane and the resultant stability of the level flight trim states, it may be limited by the onset of phugoid instability (on the “back side” of the power curve) or by the maximum throttle limit.
TABLE 4.1 Longitudinal Eigenvalues for Labels Marked in Figure 4.6 Label in
Figure 4.6 Nature of Stability Phugoid Mode
Eigenvalue Short-Period Mode Eigenvalue
1 Hopf bifurcation 0.0 ± j0.1260 −0.5586 ± j0.6896
2 Unstable trim 0.0047 ± j0.1195 −0.5136 ± j0.4437
3 Stable trim −0.0206 ± j0.1908 −0.2570 ± j0.9580
3. Speed for best endurance: The best endurance or the longest duration of flight (for a given amount of fuel) is obtained when the fuel is con- sumed as slowly as possible. That is, when the SFC (Equation 4.10) is the least, which happens when the throttle setting is the minimum for level trim flight to be possible. The condition for endurance can be written formally as follows:
W dW
dt T dt dW
T dt dW
T
f f f
t
f
Wf
= = −sfc ⇒ = −sfc ⇒
∫
= −∫
sfc0
0
(4.11)
Thus, the best endurance is possible at a flight speed correspond- ing to minimum trim thrust, which happens to be the same as maxi- mum (L/D) as marked in Figure 4.5.
4. Speed for best range (or optimum cruise condition): Clearly, for a commercial airliner, it is not flying at top speed alone or consuming fuel at the lowest rate (remaining aloft for longest time) that matter.
Instead, a combination of high speed (high flight Mach number) and low fuel consumption (high aerodynamic efficiency, L/D) is desired.
Therefore, it is common to optimize a parameter such as V(L/D) (or M(L/D)) to get the ideal cruise condition. This can be seen more for- mally by writing an expression for the range below:
dWdx V
T dx V
D L dW
W x dx
V LD dW W
f
f
x
W
= − ⇒ = − ⇒ =
=
∫
−
sfc sfc
sfc
( / ).
0
1
W W W
∫
f(4.12) where the factor V(L/D) is evident. To examine this condition, we
plot the variable T/V versus velocity and angle of attack in Figure 4.7. The minima in Figure 4.7 corresponds to flight at the condition (VL/D)max that gives best range (optimal cruise condition). In this instance, the best-range velocity and angle of attack may be read off Figure 4.7 to be approximately 150 m/s and 3 deg, respectively.
Note that the best-range velocity is a compromise between the maximum velocity and the speed for minimum fuel consumption.
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