# The Bootstrap Approach to Aircraft Performance(Part One – Fixed-Pitch Propeller Airplanes)

## You say your airplane's POH doesn't have some performance numbers you need? Or, because of airframe or powerplant modifications, your factory-original POH performance section is out of date? Or perhaps you need numbers for your one-of-a-kind homebuilt? Don't despair ... and don't guess! Now there's an easy way to calculate accurate light aircraft V-speeds, rates and angles of climb, thrust, drag, and much more. Aviation physicist and private pilot John T. Lowry shows you how. (This first installment deals strictly with simple fixed-pitch-prop airplanes; a follow-up article deals with constant-speed props and other complications.)

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The Bootstrap Approach is a method which lets Joe (or Jo)Pilot calculate his (or her) airplane’s performance numbers – performance for any grossweight at any density altitude – without a master’s degree in aeronautical engineering.You simply fly some sawtooth climbs and glides (less than an hour), calculate fournumbers, pick up five more numbers from your airplane’s POH, then plug them into formulas.Those formulas are taken care of automatically in the downloadable Excel spreadsheet whichaccompanies this article. The nine basic numbers you’ll need in order to use thespreadsheet make up what we’ll call the airplane’s Bootstrap Data Plate, or BDP for short.

## The Fine Print

We all learned to run before we learned to dance, and this subject is no exception. Soin this article we will restrict ourselves to steady flight performance of fixed-pitchpropeller-driven airplanes with a single normally-aspirated engine (properly leaned formaximum power) running either at full open throttle or gliding with closed throttle, atone flaps/gear configuration, with wings level in calm air. Follow-up articles will takeup the performance of constant-speed propeller aircraft, maneuvering, partial-throttleoperations, light twins, turbocharged engines, wind and updraft/downdraft effects, andtakeoff and landing operations.

Like many of us, the Bootstrap Approach has some built-in limitations. We assume thethrust line is directed along the flight path (certainly no Ospreys or other powered-liftaircraft need apply!) and that the flight path is not too steep, inclined less than 15degrees from the horizontal. The airplane’s movement is assumed ‘steady’ in the sense thatit is not being accelerated along the flight path. (Later on in the series, steady turnswill be allowed.) So the bootstrap approach spreadsheet can’t tell you what your father’sJ3 Cub would do halfway through a tail slide. (Not recommended!) But it can come up withalmost all the performance numbers most of us need.

## Bootstrap Goals

Our current aim is to come up with two kinds of performance figures. First, we’d likeour airplane’s interesting V-speeds, and performance numbers associated with those speeds,for any desired altitude and gross weight. See Table 1 for specifics and examples. Noticethat the bootstrap approach does not produce the airplane’s stall speeds, or anystructural-limitation V-speed such as maneuvering speed Va; you can get thosefrom the POH or by flying separate tests. Secondly, we’d like to know the airplane’sthrust and drag and its detailed flight path – its angle or rate of climb or descent -for any desired speed, weight, and altitude. See Table 2.

SymbolMeaningSample Value
VMMaximum level flight speed100.2 KCAS
VmMinimum level flight speedNot Applicable
VxSpeed for best angle of climb60.5 KCAS
AOCmaxBest angle of climb3.9 deg
VySpeed for best rate of climb67.5 KCAS
ROCmaxBest rate of climb491 ft/min
VbgSpeed for best glide69.0 KCAS
BGABest glide angle5.4 deg
VmdSpeed for minimum descent rate57.4 KCAS
ROSminMinimum rate of sink641 ft/min

Table 1. Bootstrap V-speeds, and corresponding performancenumbers, for a particular Cessna 172 weighing 2200 lb. at an altitude of 7000 feet.

SymbolMeaningSample Value
TFull-throttle Thrust322 lb
DDrag216 lb
ROCFull-throttle rate of climb431.9 ft/min
AOCFull-throttle angle of climb2.8 deg
ROSGliding rate of sink884.0 ft/min
AOSGlide angle5.6 deg

Table 2. Bootstrap performance numbers for a particular Cessna 172weighing 2200 lb. at altitude 7000 feet at 80 KCAS.

Since implementing the bootstrap approach is a detailed formula-based numericalprocedure, and because spreadsheets are an easy way to do repetitive lengthy calculations,at this point you should consider downloading a copy of the companion spreadsheet whichcan do the calculations for you.

The next three sections of this article will correspond to the spreadsheet’s threemajor sections – Parts A, B, and C.

## Part A: Obtaining Your Airplane’s Bootstrap Data Plate

You’ve seen those little aluminum data plates riveted onto the fuselage beneath thehorizontal tail, giving the airplane model, gross weight, serial number, etc. Think of theBootstrap Data Plate as something similar, but invisible. It’s a data plate which- once you decode it – tells you a lot about your airplane’s performance. But first wemust “develop” the airplane’s Bootstrap Data Plate, rendering it visible. Table3, from cell block A39:C47 of the spreadsheet, is a typical Bootstrap Data Plate.

 BDP Parameter Value Units Wing area, S 174 ft2 Wing span, B 35.83 ft Rated engine power, P0 160 hp Rated engine speed, N0 2700 RPM Propeller diameter, d 6.25 ft Parasite drag coefficient, CD0 0.037 Airplane efficiency factor, e 0.72 Slope of propeller polar, m 1.70 Intercept of propeller polar, b -0.0567

Table 3. Bootstrap Data Plate for a particular Cessna 172, flapsup.

The first five numbers in Table 3, the “easy” ones, come straight from yourPOH. The last four “harder-to-get” numbers come from – tough work, butsomebody’s got to do it – flying the airplane! See Figure 1.

Figure 1. The five “easy” BDP items come from the POH orcommon knowledge; the four “harder-to-get” items come from the sawtooth climband glide tests.

Now let’s focus on the sawtooth climbs and glides. See Figure 2. The object is to find,by trial and error, the airplane’s speed for best angle of climb Vx withcorresponding AOCmax (best or maximum angle of climb) and its speed for bestglide Vbg with corresponding BGA (best glide angle). POH values should give youfirst good guesses, but if you’re lighter than the standard weight used in the POH bothtrue Vx and true Vbg will be lower than their book counterparts.

Figure 2. Sawtooth climbs and glides (perhaps five or ten ofeach), give you sufficient test data to calculate the four “harder-to-get”Bootstrap Data Plate items.

You will need stabilized airspeed (hopefully within a knot) during each run. So thatmeans starting high on the glides, and low on the climbs, to give yourself time tostabilize at your chosen target airspeed. The measured sawteeth need not be large;altitude excursions of about 500 feet are usually sufficient. Table 4 shows the datayou’ll jot down during each climb or glide. Figure 3 shows the kinds of graphs which mightresult. As you can see, Vx here is approximately 61 KCAS and Vbg isabout 69 KCAS.

Run
No.
Clock
Time
Climb/
Glide
IAS
(knots)
CAS
(knots)
Delta-t
(seconds)
KCAS X
Delta-t
Weight
(pounds)
17:12:00C53.053.066.093502.82261
27:17:06G62.062.045.512821.92256
37:22:26C56.056.060.153368.32251
47:26:56G65.065.044.112867.22246
5

Table 4. Data collection form with entries for two climbs and twoglides. We’ve assumed the ASI has no calibration error. Gross weight can be filled inlater. The next-to-last column is calculated, and plotted against the airspeed (as inFigure 3) to help pinpoint the minimum of the climb values and the maximum of the glidevalues.

Note that when doing these climb/glide tests, each climb and glide for which you recordKCAS and delta-t must be conducted between the exact same upper and lower altitude limits.It doesn’t really matter what the altitude change is (300 feet or 800 feet) so long asit’s the same for all the data points. The climb/glide tests should also be made during arelatively short elapsed time so that the atmospheric temperature doesn’t changesignificantly between the first test and the last one.

For the best results, you should use a different (stabilized) airspeed for each climband each glide, covering a range from well under the expected Vx and Vbgto well above it. Your final test should be made at your provisional best guess at Vxand Vbg.

Either during the test flight (assuming you have an assistant along) or when you’redone recording your climb and glide test results and are safely back on the ground,calculate the seventh column of your data collection form (Table 4 above) by multiplyingthe recorded KCAS by the recorded delta-T for each test. Then plot these values of (KCAS *delta-t) against KCAS on a piece of graph paper, and try to draw a smooth curve throughall your data points. I prefer to do this inflight (using an asistant) so that I’m surewe’ve honed in on Vx and Vbg. When you’ve graphed the values, youshould come up with something that looks like Figure 4 below:

Figure 3. Graphs of (KCAS * delta-t) vs. KCAS for both the climband glide series of points. The climb minimum shows Vx = 61 KCAS and the glidemaximum shows Vbg = 69 KCAS.

What equipment does it take to collect and process this flight data? Besides theairplane, paper, graph paper, and a pencil, you will need:

• Calibrated airspeed indicator (and the calibration curve needed to get from KIAS to KCAS);
• Calibrated altimeter (set to 29.92 in. Hg, so that it reads pressure altitude);
• Stop watch (and a calibrated thumb to operate it);
• Calculator (to get the seventh column in Table 4;
• Ordinary watch (to later figure fuel burn and hence gross weight); and
• OAT thermometer (to later help the spreadsheet correct pressure altitude to density altitude).

Part A of the downloadable spreadsheet will hold the experiment’s identifiers, the five”easy” BDP numbers, and will also process your collected flight test data toproduce the four “harder-to-get” BDP items. All the numbers you need to input gointo green cells in the spreadsheet. A millisecond after finishing your input you willhave the corresponding Bootstrap Data Plate output. Phase I of the Bootstrap process – aone-time job – is now complete.

That last, about the one-time job, is important. To some, even to a fewengineers who’ve heard about the bootstrap approach, the process seems circular. Theythink “Get Vx to later find out what Vx is? That’s a bigadvance? What’s wrong with this picture?” What they fail to realize is that (stillfocusing on Vx) it’s more like: Get Vx once, for one knownweight and altitude, to be able to get Vx (and much more) later, for anyweight and for any altitude. It’s somewhat like your pilot training. You first takelessons, say for forty hours, to then later be able to fly around the countryside for 400hours. Or for 4000 hours.

You’ve got your Bootstrap Data Plate. The training wheels are off.

## Part B: Calculating Your Airplane’s V-Speeds and V-Speed Performance

Think for a minute about your flight controls. Since we’ve restricted ourselves in thisfirst article to fully open or fully closed throttle, you have no choices there. Wingsstay level, and flight stays coordinated, so there’s nothing creative to do with eitherailerons or rudder. Flaps stay fixed, probably up. All you have left in your control bagis the ability to change airspeed, using the elevator control, and the freedom to move toa different altitude. Only airspeed V and altitude h are at your disposal. And those onlywithin limits. In a sense you can also choose the airplane’s gross weight, W, but you’dbest do that on the ground, while fueling. Passengers strongly protest being ejected fromthe airplane in midflight. So in our current scenario the “operating” variables,those somewhat under the pilot’s control, consist of only airspeed V, density altitude h,and gross weight W.

In the spreadsheet’s V-speed section, Part B, speeds are outputs. Here you’re down toonly the two operating variables h and W. Further, since a set of altitudes h has alreadybeen provisionally chosen for you (see Table 5, though you are perfectly free tosubstitute your own altitude choices), the only free choice left is gross weight W. Youmay want printouts of Part B for three or four different gross weights. Stash them away inyour airplane’s glove box or in your map case.

 Gross Weight, W: 2000 lbs. h ft VM KCAS Vm KCAS Vx KCAS AOCmax deg Vy KCAS ROCmax ft/min Vbg KCAS BGA deg Vmd KCAS ROSmin ft/min 0 117.0 28.5 57.7 8.3 74.2 962 65.8 5.4 50.0 550 2000 112.6 29.6 57.7 7.3 72.3 858 65.8 5.4 50.0 566 4000 108.2 30.8 57.7 6.3 70.4 758 65.8 5.4 50.0 584 6000 103.7 32.1 57.7 5.4 68.7 659 65.8 5.4 50.0 602 8000 99.2 33.6 57.7 4.5 66.9 563 65.8 5.4 50.0 620 10000 94.6 35.2 57.7 3.7 65.3 469 65.8 5.4 50.0 640 12000 89.8 37.1 57.7 2.9 63.7 377 65.8 5.4 50.0 661 14000 84.8 39.3 57.7 2.2 62.2 286 65.8 5.4 50.0 682 16000 79.3 42.0 57.7 1.5 60.7 196 65.8 5.4 50.0 705 18000 73.0 45.6 57.7 0.8 59.3 108 65.8 5.4 50.0 728 20000 63.9 52.1 57.7 0.1 58.0 20 65.8 5.4 50.0 754

Table 5. Part B of the accompanying spreadsheet Bootstp1.xls for aCessna 172 weighing 2000 pounds.

## Part C: Calculating Thrust, Drag, and Rate or Angle of Climb/Descent

In this “any speed” section, Part C, you must choose a (density) altitude hand a gross weight W. Then thrust and drag and rate of climb, etc. – all the data itemsof Table 2 – are given for a range of airspeeds V. Again, feel free to tailor thosesuggested speeds to your faster or slower airplane. If you want readily-available printedoutput for say three altitudes and three gross weights, you’ll end up with nine differentPart C performance sheets.

 Gross Weight, W: 2200 lb Density Altitude, h: 5000 ft. Airspeed, V KCAS Thrust, T lb Drag, D lb Full-Throttle ROC ft/min Full-Throttle AOC degrees Gliding ROS ft/min Gliding AOS degrees 40 424 343 162 2.1 680 9.0 50 411 251 396 4.2 623 6.6 60 395 215 534 4.7 640 5.6 70 375 207 583 4.4 719 5.4 80 353 216 541 3.6 858 5.6 90 327 237 402 2.3 1058 6.2 100 299 267 158 0.8 1322 7.0 110 267 304 -200 -1.0 1657 7.9 120 233 347 -682 -3.0 2067 9.1 130 195 397 -1298 -5.3 2557 10.4

Table 6. Part C of the accompanying spreadsheet Bootstp1.xls for aCessna 172 weighing 2200 pounds at 5000 feet.

AVweb readers may prefer to turn this kind of tabular material (their ownversions of Tables 5 and 6) into graphs. Figures 4, 5, and 6 below are examples. Usingvarious spreadsheet techniques (or by simply recalculating and jotting down changedresults) one can also put together graphs which do not come directly from the spreadsheetcells. For example see Figure 7 for how maximum rate of climb depends on altitude forthree different gross weights.

Figure 4. Major powered V-speeds vs. altitude for a Cessna 172weighing 2400 pounds.

Figure 5. ROC vs. V for a Cessna 172 weighing 2400 pounds at threealtitudes.

Figure 6. Thrust and drag vs. airspeed for the Cessna 172, flapsup, at MSL weighing 2400 pounds.

Figure 7. Maximum rate of climb vs. altitude for the Cessna 172 atthree gross weights.

Since no one can (or should!) do these calculations in the cockpit, not even afterthey’ve had time to absorb what they mean and how to do them, you’ll want to sit down witha cup of coffee and think through which graphs or charts you really want. That includesselecting density altitudes at which you normally (and perhaps abnormally!) fly, at whichweights, and what the most appropriate ranges of featured airspeeds should be. Figure outthe least amount of most important information before you construct the final outputspreadsheets and graphs. Less is more, so start small. You can always add another graph ortable later on.

## Next Bootstrap Steps

In the next article of this series,we’ll shift gears to the constant-speed propeller aircraft, the type flown by most AVwebreaders. In the meantime, those of you wanting more background on the bootstrap approachcan find it in books and articles, at selected Web sites, or in feedback comments here on AVweb.The following references were all written by the author:

• Performance of Light Aircraft, AIAA 1999, available for purchase from AIAA at www.aiaa.org, from Amazon at www.amazon.com, or from Barnes and Noble at www.bn.com. The mathematics level is mostly that of early college with an occasional (but optional) bout of heavy lifting.
• Computing Airplane Performance with The Bootstrap Approach: A Field Guide, M Press 1995, available from the author at jlowry@mcn.net. Lots of bootstrap formulas but almost no derivations. There is a companion disk of spreadsheet templates.
• Aircraft Performance at High Density Altitude, USAF 1999, is a short book written for Civil Air Patrol mountain search and rescue teams. Graphics instead of algebra. I’m uncertain whether the Air Force version will be available other than to CAP members.
• Several professional engineering bootstrap articles have appeared in such publications as Journal of Aircraft, Journal of Aviation/Aerospace Education and Research, and Journal of Propulsion and Power. Contact the author, by email, for specific citations.
• The author’s Web site, www.mcn.net/~jlowry will have rotating articles on aircraft performance (some bootstrap, some not), and has a link to four bootstrap articles on the AllStar Network (supported by NASA and hosted by Florida International University).

The best and simplest way to get your bootstrap questions answered is to ask them righthere on AVweb, through the article feedback facility (see buttons below). And tostay tuned for future articles.