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The
Bootstrap Approach: Background
The Bootstrap Approach
The Bootstrap Approach is a parametric performance method. You take the airplane out and fly it, for an hour or two, doing very specific climbs, glides, and a level speed run. Those routine maneuvers must be done at known weight and altitude, but it doesn’t matter what that weight or that altitude is. That gives you the data, after you get back down, to calculate the parameters making up the "Bootstrap Data Plate" (BDP). In all, the BDP consists of nine parameters. When you want to know the airplane’s performance – say angle of climb – under specific circumstances (weight and altitude), you simply take the appropriate Bootstrap formula, substitute in the BDP parameters for that airplane, and the weight and altitude figures, and do the calculations. Out comes the airplane’s performance number. Now let’s take a look at where the Bootstrap Approach comes from.
Of the four forces acting on the airplane – thrust, drag, lift, and weight – thrust is the most difficult to measure or predict. That is why most books about aircraft performance simply assume that propeller efficiency h is some constant. Commonly cited values are h = 80% and h = 85%. Then thrust T = h P, where P is the engine power. Unfortunately, propeller efficiency is in fact not constant; it varies with air speed and RPM or, more precisely, with the dimensionless ratio of those two variables:
(1) 
where J is the "propeller advance ratio." As the propeller rotates through one circle the airplane advances a distance V/n. J is then the ratio of that advance distance to the propeller’s diameter d. Figure 1 is an example of how propeller efficiency varies with advance ratio.
Figure 1. Efficiency graph for McCauley 7557 propeller on some Cessna 172s.
The basic Bootstrap Approach, strangely enough, makes no assumption about propeller efficiency. It has an alternate way, which we now explain, for coming up with thrust. Because a section of propeller blade at distance r from the hub moves (in a sense) in two directions at once – longitudinally with velocity V and to the side with speed 2pnr – there are two distinct propeller "coefficients," one (C_{T}) having to do with thrust and the other (C_{P}) having to do with absorbed power. (In the third direction, along the length of the propeller blade, we assume the propeller is rigid enough that it doesn’t move at all.) The propeller thrust coefficient is
(2)
and the propeller power coefficient is
(3) 
Figure 2 shows, for the same propeller as in Figure 1, how these coefficients vary with advance ratio J. Propeller efficiency can be obtained, knowing the two coefficients, from
(4) 
Figure 2. All the important information about a propeller’s function can be obtained from its thrust and power coefficient functions.
The Bootstrap Approach uses a little known but close approximate relation between these
two coefficients: that the socalled "propeller polar," defined as C_{T}/J^{2}
plotted against C_{P}/J^{2}, is linear. That means that for any reasonable
propeller there are two numbers m and b so that
(5) 
The Bootstrap Approach depends upon our
finding those parameters m and b, and a few others, experimentally, by means
of flight tests. For the same propeller as above, Figure 3 shows the propeller polar and
the best fit line approximating it.
The Bootstrap Data Plate
To predict the airplane’s performance using the Bootstrap method, a socalled "Bootstrap Data Plate" or BDP, consisting of nine numbers, must first be ascertained. Table 4 is a sample BDP for a particular Cessna 172 airplane.

Value 
Units 
Aircraft 
Wing area, S 
174 
ft^{2} 
Airframe 
Wing aspect ratio, A 
7.38 
Airframe 

Rated MSL torque, M_{0} 
311.2 
ftlbf 
Engine 
Altitude dropoff parameter, C 
0.12 
Engine 

Propeller diameter, d 
6.25 
ft 
Propeller 
Parasite drag coefficient, C_{D0} 
0.037 
Airframe 

Airplane efficiency factor, e 
0.72 
Airframe 

Propeller polar slope, m 
1.70 
Propeller 

Propeller polar intercept, b 
–0.0564 
Propeller 
Table 4. Bootstrap Data Plate for a particular Cessna 172.
Figure 3. For most propellers, the best fit line to its polar diagram has a goodnessoffit parameter R^{2} = 0.95 or better.
Where do these nine BDP items come from? Five come from the Pilots Operating Handbook (POH) or common knowledge. Those are:
Reference wing area S = 174 ft^{2};
Wing aspect ratio A = B^{2}/S = 7.38 (B = wing span = 35.83 ft);
Mean sea level (MSL) full–throttle rated torque M_{0} = P_{0}/2pn_{0} (P_{0} rated power, n_{0} rated propeller revolutions per second). For this Cessna 172, P_{0} = 160 HP = 88,000 ft–lbf/sec and n_{0} = RPM/60 = 2700/60 = 45 rps. Hence M_{0} = 311.2 ft–lbf. But in most of our formulas, though it makes them a little longer, we’ll retain P_{0} and n_{0;}
_{}The proportional mechanical power loss
independent of altitude, C, which can almost always be taken as 0.12. This governs
fullthrottle torque at altitude through the power dropoff factor (Greek capital
‘Phi’):
(6) 
Relative atmospheric density (Greek small 'sigma') s = r/r_{0 }where r is atmospheric density and standard density r_{0 }= 0.002377 slug/ft^{3}.The timehonored form (Gagg and Farrar, 1934) for this drop off factor is
(7) 
5. Propeller diameter d = 6.25 ft.
To simplify later calculations, it’s convenient to assume a “standard weight” for the airplane. For our sample Cessna 172 we choose W0 = 2400 lbf, maximum certified gross weight. Standard relative air density is taken to be unity.
Glide test for Drag Parameters
Of the four remaining "hardertoget" BPD items, two typify drag and two characterize thrust. The drag numbers are the usual:
6. Parasite drag coefficient, C_{D0}; and
7. Airplane efficiency factor, e.
Getting C_{D0} and e by the usual method, linear regression analysis of many glides, is overkill. Instead, simply find, by trial and error, the speed for best glide V_{bg} and its corresponding glide angle g_{bg} (Greek small ‘gamma’) at one known aircraft weight W in an atmosphere of known relative density. Let us take W = 2200 lbf and h_{r }= 5000 ft. That latter makes s = 0.86167 and F(s) = 0.84281. (For convenience of the checking reader, we carry more decimal places than makes strict sense).
Consider that we time glides from 5100 ft to 4900 ft; Dh = 200 ft. Glide angle (in calm wind) is shallowest when product V_{T}×Dt, true air speed times elapsed time, is greatest. To find
that maximizing V, one can just as well use calibrated air speed Vc.
Best glide angle is later calculated from
(8) 
The relation between true and calibrated air speeds is:
(9) 
For our sample Cessna, take V_{Cbg} = 68.9 KCAS = 116.29 ft/sec and Dt = 16.96 sec From Eq. (9), V_{Tbg} = 74.3 KTAS = 125.3
ft/sec. From Eq. (8), g_{bg} = 5.40 deg.
The two required drag parameters are obtained from:
(10) 
and
(11) 
Substituting our numbers into Eqs. (10) and (11) gives us C_{D0} = 0.0370 and e
= 0.720. Those numbers (especially C_{D0}) would have been different if we had
run the glide tests with some flaps extended.
Climb and Level Flight Tests for Thrust Parameters
Our last two BDP items are:
8. Slope of the linear propeller polar, m;
9. Intercept of the linear propeller polar, b.
Of several alternative flight test regimens for evaluating m and b, we choose: trialanderror climbs to find speed for best angle of climb, V_{x}, and subsequently b, followed by a test for maximum level flight speed, V_{M}, and then m.
V_{x} is the full–throttle partner of V_{bg}.
The latter is the most nearly positive (smallest negative) glide angle you can
achieve. Accordingly, when product V×Dt is smallest one has
found V_{x}. For our sample Cessna 172, assume V_{Cx} = 60.5 KCAS =
102.1 ft/sec. The true value is then V_{Tx} = V_{x} = 65.2 KTAS =
110.0 ft/sec. The Bootstrap formula which kinds polar intercept b is :
(12) 
Substituting our sample values into Eq. (12) gives b = –0.0564.
We conclude our flight tests with a fullspeed level run (still at 5000 ft, still at 2200 lbf) and find V_{CM} = 104.8 KCAS = 176.9 ft/sec. In the true terms needed in our formulas, V_{TM} = V_{M} = 112.9 KTAS = 190.6 ft/sec. The Bootstrap formula for polar slope m is:
(13) 
Substituting our values into Eq. (13) gives us m = 1.70. The Bootstrap Data Plate of Table 4 is complete.
Go to next sectionBoostrap Approach: Formulas and Graphs
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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Updated: January 18, 2011