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 Or, because of airframe or powerplant modifications, your factory-original POH performance section is out of date?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.)
The Bootstrap Approach is a method which lets Joe (or Jo) Pilot calculate his (or her) airplane's performance numbers — performance for any gross weight 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 four numbers, 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 which accompanies this article. The nine basic numbers you'll need in order to use the spreadsheet 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. So in this article we will restrict ourselves to steady flight performance of fixed-pitch propeller-driven airplanes with a single normally-aspirated engine (properly leaned for maximum power) running either at full open throttle or gliding with closed throttle, at one flaps/gear configuration, with wings level in calm air. Follow-up articles will take up the performance of constant-speed propeller aircraft, maneuvering, partial-throttle operations, light twins, turbocharged engines, wind and updraft/downdraft effects, and takeoff and landing operations.
Like many of us, the Bootstrap Approach has some built-in limitations. We assume the thrust line is directed along the flight path (certainly no Ospreys or other powered-lift aircraft need apply!) and that the flight path is not too steep, inclined less than 15 degrees from the horizontal. The airplane's movement is assumed 'steady' in the sense that it is not being accelerated along the flight path. (Later on in the series, steady turns will be allowed.) So the bootstrap approach spreadsheet can't tell you what your father's J3 Cub would do halfway through a tail slide. (Not recommended!) But it can come up with almost all the performance numbers most of us need.
Our current aim is to come up with two kinds of performance figures. First, we'd like our 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. Notice that the bootstrap approach does not produce the airplane's stall speeds, or any structural-limitation V-speed such as maneuvering speed Va; you can get those from the POH or by flying separate tests. Secondly, we'd like to know the airplane's thrust 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.
Table 1. Bootstrap V-speeds, and corresponding performance numbers, for a particular Cessna 172 weighing 2200 lb. at an altitude of 7000 feet.
Table 2. Bootstrap performance numbers for a particular Cessna 172 weighing 2200 lb. at altitude 7000 feet at 80 KCAS.
Since implementing the bootstrap approach is a detailed formula-based numerical procedure, 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 which can do the calculations for you.
|NOTE: You may want to download the spreadsheet now,
because it'll make it easier for you to follow the rest of this article. The spreadsheet
is available for download in two formats:
There are no macros or menus or anything fancy — just input, output, and formulas in between.
The next three sections of this article will correspond to the spreadsheet's three major 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 the horizontal tail, giving the airplane model, gross weight, serial number, etc. Think of the Bootstrap 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 we must "develop" the airplane's Bootstrap Data Plate, rendering it visible. Table 3, from cell block A39:C47 of the spreadsheet, is a typical Bootstrap Data Plate.
|Wing area, S||
|Wing span, B||
|Rated engine power, P0||
|Rated engine speed, N0||
|Propeller diameter, d||
|Parasite drag coefficient, CD0||
|Airplane efficiency factor, e||
|Slope of propeller polar, m||
|Intercept of propeller polar, b||
Table 3. Bootstrap Data Plate for a particular Cessna 172, flaps up.
The first five numbers in Table 3, the "easy" ones, come straight from your POH. The last four "harder-to-get" numbers come from — tough work, but somebody's got to do it — flying the airplane! See Figure 1.
Figure 1. The five "easy" BDP items come from the POH or common knowledge; the four "harder-to-get" items come from the sawtooth climb and 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 with corresponding AOCmax (best or maximum angle of climb) and its speed for best glide Vbg with corresponding BGA (best glide angle). POH values should give you first good guesses, but if you're lighter than the standard weight used in the POH both true Vx and true Vbg will be lower than their book counterparts.
Figure 2. Sawtooth climbs and glides (perhaps five or ten of each), 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 that means starting high on the glides, and low on the climbs, to give yourself time to stabilize 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 data you'll jot down during each climb or glide. Figure 3 shows the kinds of graphs which might result. As you can see, Vx here is approximately 61 KCAS and Vbg is about 69 KCAS.
Table 4. Data collection form with entries for two climbs and two glides. We've assumed the ASI has no calibration error. Gross weight can be filled in later. The next-to-last column is calculated, and plotted against the airspeed (as in Figure 3) to help pinpoint the minimum of the climb values and the maximum of the glide values.
Note that when doing these climb/glide tests, each climb and glide for which you record KCAS 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 as it's the same for all the data points. The climb/glide tests should also be made during a relatively short elapsed time so that the atmospheric temperature doesn't change significantly between the first test and the last one.
For the best results, you should use a different (stabilized) airspeed for each climb and each glide, covering a range from well under the expected Vx and Vbg to well above it. Your final test should be made at your provisional best guess at Vx and Vbg.
Either during the test flight (assuming you have an assistant along) or when you're done 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 multiplying the 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 through all your data points. I prefer to do this inflight (using an asistant) so that I'm sure we've honed in on Vx and Vbg. When you've graphed the values, you should come up with something that looks like Figure 4 below:
Figure 3. Graphs of (KCAS * delta-t) vs. KCAS for both the climb and glide series of points. The climb minimum shows Vx = 61 KCAS and the glide maximum shows Vbg = 69 KCAS.
What equipment does it take to collect and process this flight data? Besides the airplane, 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 to produce the four "harder-to-get" BDP items. All the numbers you need to input go into green cells in the spreadsheet. A millisecond after finishing your input you will have the corresponding Bootstrap Data Plate output. Phase I of the Bootstrap process — a one-time job — is now complete.
That last, about the one-time job, is important. To some, even to a few engineers who've heard about the bootstrap approach, the process seems circular. They think "Get Vx to later find out what Vx is? That's a big advance? What's wrong with this picture?" What they fail to realize is that (still focusing on Vx) it's more like: Get Vx once, for one known weight and altitude, to be able to get Vx (and much more) later, for any weight and for any altitude. It's somewhat like your pilot training. You first take lessons, say for forty hours, to then later be able to fly around the countryside for 400 hours. 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 this first article to fully open or fully closed throttle, you have no choices there. Wings stay level, and flight stays coordinated, so there's nothing creative to do with either ailerons or rudder. Flaps stay fixed, probably up. All you have left in your control bag is the ability to change airspeed, using the elevator control, and the freedom to move to a different altitude. Only airspeed V and altitude h are at your disposal. And those only within limits. In a sense you can also choose the airplane's gross weight, W, but you'd best do that on the ground, while fueling. Passengers strongly protest being ejected from the 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 to only the two operating variables h and W. Further, since a set of altitudes h has already been provisionally chosen for you (see Table 5, though you are perfectly free to substitute your own altitude choices), the only free choice left is gross weight W. You may want printouts of Part B for three or four different gross weights. Stash them away in your airplane's glove box or in your map case.
Table 5. Part B of the accompanying spreadsheet Bootstp1.xls for a Cessna 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 h and a gross weight W. Then thrust and drag and rate of climb, etc. — all the data items of Table 2 — are given for a range of airspeeds V. Again, feel free to tailor those suggested speeds to your faster or slower airplane. If you want readily-available printed output for say three altitudes and three gross weights, you'll end up with nine different Part C performance sheets.
Table 6. Part C of the accompanying spreadsheet Bootstp1.xls for a Cessna 172 weighing 2200 pounds at 5000 feet.
AVweb readers may prefer to turn this kind of tabular material (their own versions of Tables 5 and 6) into graphs. Figures 4, 5, and 6 below are examples. Using various spreadsheet techniques (or by simply recalculating and jotting down changed results) one can also put together graphs which do not come directly from the spreadsheet cells. For example see Figure 7 for how maximum rate of climb depends on altitude for three different gross weights.
Figure 4. Major powered V-speeds vs. altitude for a Cessna 172 weighing 2400 pounds.
Figure 5. ROC vs. V for a Cessna 172 weighing 2400 pounds at three altitudes.
Figure 6. Thrust and drag vs. airspeed for the Cessna 172, flaps up, at MSL weighing 2400 pounds.
Figure 7. Maximum rate of climb vs. altitude for the Cessna 172 at three gross weights.
Since no one can (or should!) do these calculations in the cockpit, not even after they've had time to absorb what they mean and how to do them, you'll want to sit down with a cup of coffee and think through which graphs or charts you really want. That includes selecting density altitudes at which you normally (and perhaps abnormally!) fly, at which weights, and what the most appropriate ranges of featured airspeeds should be. Figure out the least amount of most important information before you construct the final output spreadsheets and graphs. Less is more, so start small. You can always add another graph or table 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 AVweb readers. In the meantime, those of you wanting more background on the bootstrap approach can 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 firstname.lastname@example.org.
Lots of bootstrap formulas but almost no derivations. There is a companion disk of
- 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
- 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 right here on AVweb, through the article feedback facility (see buttons below). And to stay tuned for future articles.