The Bootstrap Approach to Aircraft Performance
(Part Three — Maneuvering)

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Okay, bootstrappers ... time to cut loose and have some fun! In the first two bootstrap articles by aviation physicist John T. Lowry Part One on basics for fixed-pitch aircraft, Part Two on constant-speed-prop aircraft the airplane was kept at full throttle, or perhaps gliding, and always wings level. But eagles don't soar wings-level. (Then again, neither do buzzards.) This article takes the airplane, still at full throttle, and lets it bank and turn. Since bootstrap calculations are easy and realistic, we'll be able to calculate interesting turn performance numbers and bring up some new concepts. To do that, two downloadable Excel spreadsheets are included: banks.xls, on geometric aspects of level turns, and bootstp3.xls, details on maneuvering a fixed-pitch airplane.

AirmanshipStrictly speaking, maneuvering should include pullups and pushdowns and all sorts of aircraft antics. While those short-lived non-steady accelerations are hard to figure, turns are easily handled by the bootstrap approach. There will still be lots to talk about. First we're interested in geometric turn relations and how those influence aircraft safety and performance. It's a tale of two angles, bank angle phi(Greek small phi) and flight path angle gamma(Greek small gamma). If you compare current spreadsheet bootstp3.xls with the earlier wings-level spreadsheet, bootstp1.xls, you'll see that formulas have been modified with factors using the trigonometric cosine function cosphi. (You won't need to know trigonometry; the spreadsheet takes care of that.) We'll see the effect banking has on rates or angles of climb or descent. We'll discuss Steady Maneuvering Charts (built into bootstp3.xls) which show both level and non-level turns and introduce the "banked absolute ceiling" concept. Finally, we'll treat optimal (tightest and quickest) level turns. A full plate.

Don't be intimidated by the symbols and formulas below, because almost all the math is dealt with automatically by the two downloadable Excel 97 spreadsheets that accompany this article:

I suggest you download them now. If your spreadsheet can't accomodate Excel 97 format, the spreadsheets are also available in Symbolic Link (SYLK) format:

Level Turns

You may remember from science class that anything moving in a circle at constant speed is being subjected to a centripetal force of size: mass of object times speed squared divided by circle radius. A centripetal (center-seeking) force. For our airplane in a level turn, that centripetal force is the horizontal part of the tilted lift. (With a little luck, and perhaps a little rudder, weathervaning will keep the airplane pointed in the same direction it's moving, coordinated.) We also know that total lift must be big enough that its vertical part supports the airplane's weight. Put those three facts together and you have not just 'Mother' but Figure 1, the mother of all dense over-achieving graphs (chart TurnRadius in spreadsheet Banks.xls).

Figure 1.

Figure 1. The relation between airspeed, bank angle, and level turn radius.

For an important application, suppose you want to do turns about a point while your passenger photographs a six-point bull elk. See Figure 2.

Figure 2.

Figure 2. In making turns about a point (in calm air), with the wing lined up on the object, there is a connection between pivot height h, turn radius R, and bank angle phi. Vary any one of those and the alignment is disturbed.

If we use the geometry of Figure 2, together with the basic turn relation of Figure 1, we get the connection between pivot height and true airspeed depicted in Figure 3 (chart AboutPoint in Banks.xls). At 80 KTAS you will want to be 566 feet above the elk. At 90 KTAS, 717 feet. Otherwise it won't quite work. To make Figure 3 useful in your cockpit, you'll have to convert KIAS to KTAS. Assuming your airspeed indicator is fairly accurate, to get KTAS just add 1.5% of your KIAS reading for each 1000 feet of density altitude. Like tipping the waitress, but only one-tenth the usual. Very poor service. So if indicated airspeed is 80 KIAS, and you're at 8000 feet, add 12% to get 90 KTAS.

Figure 3.

Figure 3. Pivot AGL altitude necessary to point the wing at a stationary object on the ground.

And so, as the sun sinks behind the ridge and our camera-shy bull disappears into the aspens, we see that centripetal force rules. Or does it? The FAA believes a further force suddenly appears during turns, a mysterious balancing centrifugal force pulling the airplane horizontally outwards. They're totally wrong about that, but this is the holiday season Good Will to Men and Federal Agencies so we won't call attention to the knowledge gap in parts of Oklahoma City. By year's end we've stored up more pressing matters to gripe at them about.

That's it for turn geometry. But the burden of added lift in a turn has important safety implications, raising the stall speed and bringing the airplane closer to its structural load factor limit. There are also performance implications. Safety first.

In order to stay level when you bank, you use back-stick to urge more lift from your wings, raising the angle of attack. Since the wing always stalls at the same angle of attack, at the same maximum lift coefficient, there's a set relation (for the same airplane at the same weight and same configuration) between stall speed with wings level and stall speed banked in a coordinated level turn. Figure 4 (chart BankedStall in Banks.xls) shows how stall speed changes with bank angle for a Cessna 172 which stalls wings level, flaps up, C.G. forward and weighing 2400 pounds at 51 KCAS.

Figure 4.

Figure 4. Stall speed for a Cessna 172 (flaps up, 2400 pounds) which stalls, wings level, at 51 KCAS.

The structural damage limiting load factor, 3.8 for Normal category airplanes, is a numerical way of saying, "Don't come back on the control stick or column too hard or you might pull the wings off." Or at least bend them. You've probably seen so-called "V-n" or "V-g" diagrams such as Figure 5. Load factor n is the ratio of lift to gross weight.

Figure 5. V-n diagram.

Figure 5. V-n diagram for a Cessna 172, flaps up, weighing 2400 pounds.

The structural damage load limit 3.8 can be brought home in terms of the largest level turn bank angle the airplane can safely take on. Turns out that angle is 74.74 ; that's why we made the largest angle in Figure 4 about 75 degrees. Maneuvering speed Va in Figure 5 is the speed where the structural damage load limit meets the banked stall curve. For our standard Cessna 172, 2400 pounds, flaps up, that's about 99 KCAS. The idea behind maneuvering speed is that when you encounter turbulence you should slow below Va; that way a sharp upgust will stall the wing rather than break or bend it. Of course you won't want to be down close to the rocks and trees under those circumstances. So much for safety. Now what about performance?

Banking affects aircraft performance almost always for the worse by increasing induced drag, "drag due to lift." You remember that little devil "induced," the part of drag which is higher at lower speeds. See Figure 6. Since turns require more lift, turns automatically also increase induced drag. Banking may also add a little control surface drag, but we'll ignore that.

Figure 6.

Figure 6. Parasite drag DP increases as the square of airspeed V. Induced drag Di, on the contrary, goes as 1/V2. These are representative drag figures for a Cessna 172 in the clean configuration.

Banking at angle phidivides the original induced drag of wings-level flight by the square of the cosine of phi. Since cosphiis always less than or equal to unity, banking always increases induced drag. With wings level, the induced drag multiplier is 1; nada new. Banked fifteen degrees, the factor is 1.07; negligible. Thirty degrees, 1.33; doable. Forty-five degrees, 2; oh-oh. Sixty degrees, 4 times as much induced drag; bad news! To see the practical effects banking has on performance, see Figures 7, 8, and 9.

You might recall that wings-level bootstrap approach drag was expressed as D = GV2 + H/V2 , where G and H are composite bootstrap parameters which depend on the airplane's nine Bootstrap Data Plate numbers and on operational variables gross weight and density altitude. The induced drag composite bootstrap parameter H goes as the square of gross weight W, but that's really because it goes as the square of lift. Being banked at angle phioften (but not always) has the same effect as multiplying gross weight by load factor n = . When we move to the maneuvering extension of the bootstrap approach, the old wings-level value of H is called H(0) (the zero standing for zero bank) and our new banked value of H is the older one times the square of load factor n. There's now a new operational variable, bank angle phi.

Figure 7.

Figure 7. Banking the airplane adds to induced drag, reduces excess power, and thereby reduces rate of climb at any given airspeed. Even at MSL, this Cessna 172 can't bank more than 56.5 degrees and still fly level. Banked only 30 degrees or 45 degrees, if airspeed isn't too high it can still make climbing turns.

The bottom line is that besides needing operational inputs weight W, density altitude , and air speed V, for turns the bootstrap approach also needs bank angle phias an operational input. It's that simple.

Figure 8.

Figure 8. Increased drag also lowers excess thrust T- D and thereby reduces climb angle at any airspeed. When banked 56.5 degrees, sea level is this airplane's "banked absolute ceiling."

In summary each of stall speed, lift, and induced drag increase, in various ways, with increasing bank angle. See Figure 9.

Figure 9.

Figure 9. Just because induced drag rises so quickly at large bank angles, don't neglect the smaller increase in stall speed. Doubling your stall speed can bite.

Through induced drag, banking also has an effect on the airplane's other V speeds. For one example look at Figure 10. VM, maximum level flight speed, is the right hand place where power available Pa and power required Pr cross. There are three Pr curves in Figure 10, for bank angles 0, 20, and 40 degrees. You can see that top level flight speed is reduced a small amount if one banks 20 degrees, a larger but still only nominal amount if one banks 40 degrees. The effect is small because induced drag itself is small at high speed.

Figure 10.

Figure 10. Effect of banking on the high and low speeds for level flight. The low speed is only attainable at quite high altitudes; down low, stall speed intervenes.

As you bank and pull back to stay level, drag will increase and the airplane will slow down. Steep turns usually require more throttle. Deceleration from induced drag can be an effective if somewhat dangerous way to quickly slow down to approach speed if you've charged into the landing pattern in something of a hot lather: crank over to a sizeable bank angle and pull back on the stick. But watch out! If for instance you're in a Cessna 172 at 100 KTAS, wings level, then bank 70 and pull back hard enough to stay level, you will stall in about 2.1 seconds! And you'll likely be too low to recover.

Induced drag is only part of the whole drag picture, but at low speeds, which is where you'd be flying small circles around large elk, induced drag overshadows parasite drag. Induced drag is greater than parasite drag at any speed below speed for best glide Vbg (about 72 KCAS for a fully loaded Cessna 172). It's something we have to live with.

Can we always fly level at any chosen bank angle? No, of course not. Figures 7 and 8 showed that the little Cessna at maximum gross weight couldn't stay level, even at sea level, if it banked more than 56.5 . Induced drag will get you if you don't watch out! The airplane's steady-state flight path angle (that excepts zooms and similar temporary excursions which trade speed for altitude) depends on its excess of thrust over drag, T- D. We gave the formula in an earlier article, but it's important enough to repeat:

If thrust is greater than drag, flight path angle is positive and you climb. If drag is greater than thrust, is negative and you descend. You're not stalled, just descending. Let's look at some details.

Full-Throttle Descending Turns

In level turns, we got factors involving the bank angle. Moving on to climbing or descending turns we get additional factors involving the flight path angle, . But only in extreme cases, jet fighters and such, are the flight path angles large enough to seriously disturb the picture. There's an additional turn radius wrinkle in going from a circle (level turn) to a helix or spiral (climbing or descending turn). You can see this effect by thinking of a slinky first tightly compressed, then considerably stretched open. As you stretch the spring, the radius of the phantom cylinder on which the spring is wound gets slightly smaller and the local radius of curvature of the spring at any point gets larger. Or, consider the helical stripe on a barber's pole. Because the helix of the stripe is so stretched out, its radius of curvature is a lot larger than the radius of the pole itself.

You'll seldom need flight path angle correction factors in a non-aerobatic general aviation setting. The factors are usually very close to unity and so can be ignored. An exception is forensic aircraft accident calculations one might present as an expert witness. Then we roll out the full enchilada, following the American legal system's Guiding Principle: Even the slightest benefit, to our side, is worth any amount of the other guy's money.

Time for a reality break, an airplane which either because it's up so high or because it's banked too much is not able to make a level turn. Take our Cessna trainer at 12,000 feet. Because bootstrap formulas are relatively simple, it's not hard to get a graph like Figure 11, a so-called "Steady Maneuvering Chart," which gives us a global view of maneuvering performance for this airplane at this weight and altitude.

Figure 11.

Figure 11. Turn radius Steady Maneuvering Chart for a Cessna 172, 2400 pounds, flaps up, full throttle, at 12,000 feet density altitude. If the airplane is flying level and banks more than 32 degrees, it will descend. For instance if it banked 40 degrees, at 85 KCAS, it would descend at 250 ft/min. The tightest level turn is at 58 KCAS banked 28 degrees; the minimum level turn radius is then about 810 feet.

Notice the banked stall curve in Figure 11, compared to Figure 4, has been flipped over. Now airspeed is the horizontal axis and bank angle the vertical axis. In Figure 11 we also added a five knot buffer zone to warn the pilot away from the stall curve. Airspeed indicators aren't perfect and, especially up in the mountains, there's always that occasional wayward gust.

Steady Maneuvering Charts are somewhat confusing, at first, because there are so many curves: in Figure 11, one load factor limit line, two stall and buffered stall curves, three curves for various rates of climb or descent, and five curves for various turn radii. But once you get used to them those charts give you valuable information.

The horizontal Load Factor Limit line at the top of Figure 11 corresponds to the normal category +3.8 load limit, what you get if you bank level to 74.74 . Looking over Figure 11 one might think the tightest turn would come from banking that maximum structurally safe amount at maneuvering speed Va = 99 KCAS, off the chart to the upper right. It looks like turn radius would then be about 350 feet. But there's a problem with that attractive scenario. A quick calculation shows that the rate of descent under those conditions would be 3940 ft/min. Not advised! (Using the small flight path angle approximation one gets 4390 ft/min, so this is one place where the flight path angle is large enough to make a difference.) To keep on the safe side, below we'll only consider level turns as candidates for "optimal." And we'll make sure we can indeed stay level under the prescribed atmospheric and loading conditions. Trying a steep banked turn, when you're high and heavy, especially in an underpowered trainer, is a siren song you don't want to listen to. Far down in the charts.

Figure 11 also demonstrates that the Cessna 172 "banked absolute ceiling," for bank angle 32 , is 12,000 feet density altitude. Since we're at an altitude, and can then easily bank to any angle, it is perhaps better to call this "absolute ceiling bank." As usual with absolute ceiling performance, the ceiling can only be attained with the airplane at a particular airspeed. In this case, about 69 KCAS.

Your airplane can't get to its wings-level absolute ceiling unless it's dropped off there by some larger higher-flying entity. On its own, an airplane requires an infinite amount of both time and fuel to get to absolute ceiling. But banked absolute ceilings are different. You can easily get to the above banked absolute ceiling (32 at 12,000 feet). Simply fly, wings level, up to say 13,000 feet. Then bank 32 . The airplane will spiral down and stabilize at 12,000 feet. You're there!

This maneuver also gives us a way to "experimentally" determine an airplane's wings-level absolute ceiling. There is a simple bootstrap formula relating banked and unbanked absolute ceilings. You say you aren't surprised at that? Yep, there's a bootstrap formula for danged near anything. See Section E of Bootstp3.xls. Using some of the numbers we got from Figure 11 pretending we got them experimentally the spreadsheet suggests wings-level absolute ceiling for this maximum gross weight 160 horsepower Cessna 172 is 16,144 feet.

While it's somewhat more work, the bootstrap approach can also calculate Steady Maneuvering Charts for constant-speed aircraft. Figure 12 is such a chart for a Cessna 206 at 10,000 feet, flaps up, weighing 3400 pounds.

Figure 12.

Figure 12. Steady Maneuvering Charts for constant-speed propeller-driven aircraft tend to be flatter (one might say "calmer") than for fixed-pitch airplanes. Especially at higher altitudes, it's nice to have that extra power and control.

Optimal Level Turns

When it comes to flying tightest (minimum radius) and quickest (maximum rate) turns, you need to ask yourself two dare we call them "pivotal"? questions:

    1. How close to banked stall speed am I willing to go?
    2. How big a descent rate can I live with?

For reasons given above, we'll settle for level turns to find concrete answers to those optimal (tightest and quickest) turn problems. We think we know, from Figure 11, how to achieve a tightest turn for our Cessna 172 at 12,000 feet. When we confirm with formulas in Section F of Bootstp3.xls we do indeed find that speed required for a minimum radius (811 feet) level turn is 117.9 ft/sec = 69.8 KTAS = 58.1 KCAS. And the correct bank angle for that tightest turn is 28.0 . An alternative version of the Steady Maneuvering Chart shows curves of constant turn rate, instead of radius, but we'll stick to radius. Fighter pilots worry about turn rate a lot, wanting to get that first shot off; we focus on turn radius. Turn rate is normally less important for civilian aircraft. Section G of Bootstp3.xls calculates that the airplane's quickest turn (8.7 deg/sec at this weight, altitude, and flaps setting) is at 63.2 KCAS banked 31.1 degrees.

It is usually necessary to check tightest or quickest level turn calculations to make sure one can actually get to the suggested turn speed and bank angle without stalling. You'll notice Sections F and G of Bootstp3.xls do that. Here we didn't have to check explicitly on our tightest turn because we had the Steady Maneuvering Chart (for that airplane at that altitude, weight, and configuration) which laid bare the entire maneuvering situation " ... like a patient etherized upon a table."

If you compare bootstrap recipes in Bootstp3.xls for tightest and quickest turns against formulas in textbooks, you'll likely find they don't agree. Most books consider the stall possibility but ignore the high descent rate. The bootstrap approach takes both into account and gives accurate answers.

Appendix A: Using Spreadsheet Banks.xls

Banks.xls is quite simple. There is one spreadsheet (Sheet 1) containing numbers behind four graphs:

  1. Numbers for text Figure 1, graph TurnRadius, start at cell A3.
  2. Numbers for text Figure 3, graph AboutPoint, start at cell A32.
  3. Numbers for text Figure 4, graph BankedStall, start at cell A59.
  4. Numbers for text Figure 9, graph PerformanceFactors, start at cell A82.

Make a backup copy of the sheet and then feel free to modify it (different speed ranges, different turn radii) any way you'd like.

Appendix B: Using Spreadsheet Bootstp3.xls

Bootstp3.xls is quite complicated. There is a spreadsheet, 12K_2400, so named because the current density altitude value (cell B49) is 12,000 feet and the current gross weigh value (cell B48) is 2400 pounds. There is also a graph, 12KMC172, so named because it's the steady Maneuvering Chart, text Figure 11, for a Cessna 172 in the above situation.

As is usual for our AVweb bootstrap spreadsheets, operator input cells are painted light green. Here are the major sections on the spreadsheet and something about their contents:

  1. Section A starts at cell A10 and contains the airplane's Bootstrap Data Plate plus its standard weight and maximum lift coefficient (needed for the stall curves) in this configuration (in our sample case, flaps up).
  2. Section B starts at cell A36 and contains base values of the main bootstrap composite parameters. No operator inputs.
  3. Section C starts at cell A44 and consists of three sub-sections:
    1. Section C1 starts at cell A46 and is very important to the operator because it contains (B48:B50) input cells for the three operational variables: (1) gross weight, (2) density altitude, and (3) bank angle.
    2. Section C2 starts at cell A53 and just contains some intermediate calculations of the full range of composite bootstrap parameter values under the assumed operational conditions.
    3. Section C3 starts at cell A69 and holds only banked absolute ceiling speed calculations. No operator input.
  4. Section D starts at cell I6 and has many columns supporting the Cessna 172 Steady Maneuvering Chart. The light green operator input areas are:
    1. I17:I52, KCAS values for the horizontal axis. The current graph 12KMC172 only runs up to 90 KCAS and only uses I17:I25. For your airplane (assuming you have done the flight tests and obtained its Bootstrap Data Plate) you may want to use a considerably different range of airspeeds. This is where you put those airspeeds. Don't forget to then appropriately modify the Source Data ranges used in the graph.
    2. L14, the stall buffer size in KTAS. Currently 5 KTAS (not the 5 KCAS mentioned in Figure 11, a mistake).
    3. M13, the damage load factor limit. Currently 3.8.
    4. N12:T12, turn radius values, currently between 200 and 1500 feet. Not all of these are currently being used in the graph.
    5. U12:W12, rate of climb values, currently +100, 0, and -250 ft/min. For your airplane, you may want more of these or different values.
  5. Section E starts at cell I58. It uses your the airplane's Bootstrap Data Plate, and operational inputs for an experimentally determined banked absolute ceiling situation to calculate wings-level maximum gross weight absolute ceiling and speed.
  6. Section F starts at cell I67. It calculates (for given weight, altitude, and configuration) the bank angle and air speed needed to achieve the airplane's minimum radius (tightest) level turn.
  7. Section G starts at cell I77. It is much like Section F but focuses on turn rate instead of turn radius.

Again, make a backup copy of the sheet and then feel free to modify it for your airplane. I didn't include a spreadsheet supporting Steady Maneuvering Charts for constant-speed aircraft because those are considerably more complicated.

I hope you enjoy using these spreadsheets. For more details on how the bootstrap approach works and how to figure other aspects of your airplane's performance (including takeoff and landing) go to my Web site http://www.mcn.net/~jlowry and click on the book Performance of Light Aircraft available either from AIAA or from Amazon.com. And don't neglect to share your experiences using bootstp2.xls with your fellow AVweb pilots in the feedback section (buttons below). If you run into a problem or have a question, either post it to feedback, which I'll check periodically, or e-mail me directly: jlowry@mcn.net.