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The Bootstrap Approach: Formulas and Graphs
Composite Bootstrap Parameters Ease Calculation
Almost everything about the airplane’s full-throttle steady-state (non-accelerating) flight performance depends on the nine BDP items plus three operational variables: weight W, bank angle f, and relative atmospheric density s. But only certain combinations of the nine BDP numbers (combinations called E, F, G, and H) actually occur in Bootstrap formulas for V-speeds or for thrust, drag, or rate or angle of climb or descent. In the V-speed formulas, in fact, only certain combinations of combinations (those to be called K, Q, R, and U) occur.
Our flight tests to determine the four harder-to-get BDP parameters were done at 5000 ft and at W = 2200 lbf. But the parameter values we got did not depend on those choices; BDP parameters only depend on the particular airplane and its flap/gear configuration. Flight tests at some other altitude or at some other weight would have given, within experimental error, the same BDP values.
But performance numbers themselves – rates of climb, V speeds, etc. – obviously do depend on gross weight and on density altitude. Again for brevity’s sake, we will consider this airplane’s behavior at one particular weight (maximum gross weight W0 = 2400 lbf) and at two particular density altitudes (MSL and 10,000 ). These choices let us compare our performance numbers – at least many of them – with the airplane’s flight manual, the POH. Looking ahead to that comparison, let us evaluate all the above composite Bootstrap parameters for those two cases. See Table 5
The composite definitions and their dependence on weight, bank angle and air density are:
|
(14) |
|
(15) |
|
(16) |
 |
(17) |
|
(18) |
 |
(19) |
|
(20) |
 |
(21) |
Variable or Composite
Case 1, MSL
Case 2, 10,000 ft
W
2400 lbf
2400 lbf
s
1.000
0.7385
F
1.000
0.7028
E
531.9
373.8
F
–0.0052368
–0.0038673
G
0.0076516
0.0056505
H
1,668,535
2,259,424
K
–0.0128884
–0.0095178
Q
–41,270.6
–39,277.5
R
–129,460,301
–237,389,461
U
218,064,595
399,861,861
Table 5. Bootstrap composite parameters for two operational situations.
Full-Throttle and Gliding V Speeds
The V speeds we are concerned with, in the Bootstrap Approach, are:
VM, maximum level flight speed;
Vm, minimum level flight speed;
Vy, speed for best rate of climb;
Vx, speed for best angle of climb (in calm air);
Vbg, speed for best (longest) glide (in calm air); and
Vmd, speed for minimum gliding descent rate.
Bootstrap formulas for these V speeds, each expressed as a true air speed in ft/sec, are:
|
(22) |
|
(23) |
|
(24) |
|
(25) |
|
(26) |
Since the three full-throttle V speeds VM, Vy, and Vx depend on only two composite parameters, Q and R, there must be a connection between them. It is:
|
(27) |
In addition, it turns out that Vx is the geometric mean between VM
and Vm. Performance specifications for most manufacturers’ airplanes
will not closely agree with Eq. (27) because the quoted values of VM are too
optimistic.
The Bootstrap Approach is not limited to predicting V speeds. There are also formulas for full-throttle power available Pa = TV, power required Pr = DV, excess power Pxs = (T–D)V, thrust T, drag D, rate of climb R/C, and flight path angle g. In the gliding case, rate of sink R/S and glide path angle can be obtained from the powered forms by setting E = F = 0 and replacing K by –G.
|
(28) |
|
(29) |
 |
(30) |
|
(31) |
|
(32) |
 |
(33) |
|
(34) |
We’ve shown all the left-hand-side variables as functions of only true air speed V.
But in all except two cases gross weight W, relative air density s, and bank angle f also
matter. It would have been more instructive, if somewhat pedantic, to have written D(V),
for instance, as D(V;W,s,f).
Where do all these formulas come from? In most cases, from
standard "power-available/power-required" analysis, which you can find in almost
any textbook treating the aircraft performance subject. The one big exception is Eq.
(31) and other equations springing from it. Eq. (31) is the Bootstrap
Approach’s "hole card," giving us a good approximation to thrust without
our having to know propeller efficiency. That Bootstrap expression for thrust is
quite easy to get. Here’s how.
|
(35) |
Next, rewrite Eq. (5) as:
|
(36) |
|
(37) |
Now, let’s look at some graphs.
But keep in mind that almost everything about an airplane’s performance depends on its weight, its altitude, its flaps/gear configuration, and whether or not its wings are level. For brevity, we’ll stick to wings level. But, as you look over the graphs, you should ask yourself such questions as: How would this graph look if it were for a more (or less) powerful airplane? What if the airplane were heavier (or lighter)? At higher (or lower) altitude? Flaps down? Banked 40 degrees?Figure 4. Thrust, drag, and their difference (excess thrust), as functions of air speed, for a Cessna 172 at sea level, flaps up, weighing 2400 pounds.
Figure 4 starts us off with graphs of thrust and drag (Eqs. (31) and (32)) and their difference, excess thrust Txs. The speed at which excess thrust is a maximum is always the speed for best climb angle, Vx. The speed at which drag is a minimum is the speed for best (longest) glide in calm air, Vbg. The thrust and drag curves cross at two places, on the right at maximum level flight speed VM and, to the left, at minimum level flight speed Vm. Vm is not marked because, in fact, you can’t get there. Stall speed VS – not a Bootstrap-calculated V speed – is higher than Vm except at high altitude.
But not everything depends on all three operational variables W, s, and f. Thrust, for instance, is independent of W. But at higher altitude, lower s, the thrust or power available curve will be a lower one. How the drag curve behaves with changes in weight and altitude is harder to see; altitude affects Bootstrap composite parameter G and both altitude and weight affect H. With a spread sheet program (Lotus 1-2-3 or Quattro Pro or Excel) you can easily take the Bootstrap formulas, assume reasonable aircraft parameters, and construct graphs for all kinds of scenarios.
Figure 5 shows most of the important performance V speeds and where they come from. VM and Vm are speeds at which the Pa and Pr graphs cross. Or, alternatively, where Pxs has its zeroes. Since rate of climb R/C = Pxs/W, speed for best rate of climb Vy is where Pxs peaks. When gliding (thrust zero), Pxs = –Pr; therefore the minimum descent rate speed Vmd is at the low point of the Pr curve. Speed for best glide Vbg and speed for best climb angle Vx take a bit more analysis. Since Pxs/V = Txs, we see that the straight line tangent to the Pxs curve which has the largest slope is the one hitting the Pxs curve at Vx. Similarly for Pr/V = D and Vbg.
Figures 4 and 5 are graphs for the airplane at one weight and one altitude. Figure 6 gives a broader view, in the vertical direction, showing how the main full-throttle V speeds differ, again for the Cessna 172 weighing 2400 pounds, over the full range of accessible density altitudes.Figure 5. Power available, power required, and excess power for the Cessna 172 at 7,500 ft, flaps up, 2400 pounds.
Figure 6 shows that speed for minimum level flight Vm doesn’t come out from hiding behind the skirts of stall speed VS until the altitude is way high, above 14,000 feet. Which means, in almost all practical cases, never. There are conflicts between Figure 6 and the Cessna 172P POH. The POH has Vy not down to as low as 70 KCAS until 12,000 ft; we get it there at only 7000 ft. And the book has Vx increasing a little with altitude, about 4 KCAS over 10,000 ft, where we get that it is constant in calibrated terms. Naturally, we believe we are right. Figure 7 delves into drag. As Eq. (32) shows, total drag D is made up of two terms. DP, parasite drag, is proportional to the square of the air speed. See Eq. (16) for G to see what makes up the proportionality constant. The second term, induced drag Di (also known as "drag due to lift"), works much differently. Induced drag is inversely proportional to V2. That means induced drag is higher the slower the airplane goes through the air. See Eq. (17) for H to see what the "constant" depends on.
Figures 8 and 9 show how performance drops off with altitude for a Cessna 172 weighing 2400 pounds, flaps up. By 10,000 feet, considerably less than half the mean sea level (MSL) best rate of climb or best angle of climb remains. This is a real problem for a relatively underpowered airplane like the Cessna 172. The Civil Air Patrol, for instance, doesn’t allow its airplanes to conduct mountain search and rescue missions at altitudes where their best rate of climb is less than 300 ft/min. A fully-loaded Cessna 172 with the stock 160 HP engine could not qualify. Perhaps a special "climb" propeller would let it meet that mark.
These graphs give you a reality check, samples showing how various factors influence propeller airplane performance. The references contain many more such graphs and numerical illustrations.Figure 6. Full-throttle V speeds for Cessna 172, flaps up, 2400 pounds.
Figure 7. How drag varies with air speed for a Cessna 172, flaps up, 2400 lbf, at MSL.
Figure 8. Rate of climb graphs, at three density altitudes, for a Cessna 172 weighing 2400 pounds, flaps up. Notice that calibrated air speeds for best rate of climb decrease with altitude.
Figure 9. Angle of climb for a Cessna 172, flaps up, 2400 pounds, at three density altitudes.
So the Bootstrap formulas are
relatively simple and straightforward. But how accurate are they?
Performance
Item
Case 1, 2400 lb at MSL
Case 2, 2400 lb at 10,000 Ft
TBA
POH
TBA
POH
VM
115.3
121.0
90.8
98.8
Vy
75.8
76.0
67.5
71.0
R/C(Vy)
700.5 fpm
700.0 fpm
258.5 fpm
237.0 fpm
Vx
63.2
62.0
63.2
66.0
Vbg
72.0
66.0
72.0
66.0
g
bg–5.40 deg
–6.26 deg
–5.40 deg
–6.26 deg
Table 6. Comparison of Bootstrap (TBA) and POH performance predictions.
There is a major discrepancy where maximum level flight speed VM is concerned.
We’ve flown a number of Cessna 172s, but never one which would go 121 KTAS at
sea level. The Cessna test pilot might explain her higher speed is due to her brand
new engine, the pristine leading edge of the wing, no dents, .... Perhaps. In
glide performance, on the contrary, our airplane did better than the factory-fresh
one. There we suspect (but only suspect) the Cessna Aircraft Company corporate
attorneys came into play. For liability reasons, they certainly wouldn’t want
to claim a longer best glide than might be demonstrated by a lawyer whose
client’s engine had failed.
The only way to settle these questions of whose performance statements are more accurate is through careful, well-instrumented, and un-manipulated test flights. Our point here is that the Bootstrap Approach lets you fly the airplane for an hour or so, performing a few simple climbs and glides and a level speed run, and then lets you calculate many interesting and apparently accurate performance numbers for that actual airplane. Bugs, dents, tired engine and all.
Go to next section - Conclusion & References
The ALLSTAR network would like to thank Dr. John T. Lowry, of Flight Physics, for providing this section of material and giving ALLSTAR permission to use it. Dr. Lowry is the 1999 AIAA Flight Research Project Award winner. Though the ALLSTAR network edited the material for clarity, and maintains the copyright over the format of the material presentation, the material is wholly Dr. Lowry's and is copyrighted to him (© April 1999). Any questions about this material should be directed to Dr. Lowry.
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