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Bootstrap Approach Extensions
Maneuvering flight (wings banked, turning) was implicitly included in the various Bootstrap formulas we presented, but for lack of time and space, and to avoid sensory overload we did not pursue or exemplify that flight performance realm. From those formulas one can get so-called "steady maneuvering charts" which graphically demonstrate (for a given airplane at a given weight, configuration, and altitude) the relations between air speed, bank angle, rate of climb or descent, and either turn radius or turn rate. The charts also include a banked stall curve, a buffer curve paralleling the stall curve, and a structural load factor limit line. Reference 11 gives details and explicit formulas for constructing steady maneuvering charts.
If you bank the airplane and try to maintain level flight, you must use back stick to increase angle of attack. For level flight, the lift vector must be long enough that its vertical component balances weight. But there are limits. If you bank too far, the wing will stall. Stall speed goes up with bank angle. More surprising, for a given thrust and altitude, there is always a bank angle beyond which the airplane cannot maintain level flight at any speed. There it is at its "absolute banked ceiling." At that altitude banked values of Vx and Vy, as functions of bank angle, cross. Vx then becomes larger than Vy. In many respects, banking the airplane is tantamount to suddenly making it heavier.
There are important safety considerations, especially for underpowered trainers at high altitude, in the banked absolute ceiling concept.
There is also a Bootstrap extension to partial-throttle operations. For that, you need one additional but very simple flight test: cruise level at various air speeds (at any known gross weight and density altitude) and record engine RPM. From that information one can construct graphs of propulsive efficiency and of both propeller thrust and power coefficients. Moreover, and more to the operational point, one can answer all such questions as the following: if I take this airplane up to (say) 9000 feet, weighing 2150 pounds, flaps up, and want to put it into a 300 ft/min standard rate (3 degrees per second) descending turn, at 90 KCAS, what RPM should I throttle back to? Being able to answer that kind of question is a major Bootstrap advance. The Bootstrap partial-throttle extension also lets one get accurate cruise performance tables, with speeds down to the quite low ones for best range Vbr and for best endurance Vbe. Those safety V speeds are not and cannot be given by the standard GAMA (General Aviation Manufacturers Association) format cruise tables. The Bootstrap partial-throttle theory also allows one to take a portion of a cruise performance table, for one weight and altitude, and use scaling laws to calculate corresponding cruise performance entries at any other weight and at any other altitude.
Bootstrap Approach Advantages
Because of its relatively simple analytic (formula-based) construction, the Bootstrap Approach also lets us find values of the two "Earth-based" V speeds, Vbg and Vx, for any steady wind conditions. That way one can find, for any given headwind, tailwind, updraft, downdraft, or combinations, how much to slow down, or speed up, from the nominal calm air V speed values, to ensure optimum glide or climb. Trial-and-error calculations are required, but with a modern spread sheet program those are easily figured.
Since the Bootstrap Approach includes a good formula for propeller thrust, extension to the take off maneuver is perfectly feasible. The same is true of the landing maneuver, including the trickier portion bringing the airplane down to the runway from altitude 50 feet. Since several different forces rolling friction, braking, runway slope and contamination, ground effect are required for take off or landing analysis, we leave that subject to another time
An advantage of The Bootstrap Approach for manufacturers of small airplanes is that design changes say a different engine only require, for new performance predictions, new BDP items for that engine. The three subsystems (airframe, engine, and propeller) are relatively independent. Even after the airplanes design has been frozen, performance flight testing by "standard" methods, according to a professional performance test pilot often hired by Cessna, takes about eighty hours of flying and calculating. The Bootstrap Approach requires only two to three hours. While at least a couple of the larger kitplane manufacturers (Skystar Aircraft and RANS) do currently (1999) use the Bootstrap Approach, many more propeller aircraft manufacturers should look into doing likewise. Several "mod shops" businesses which install STOL (short take off and landing) devices such as wing cuffs, gap seals, stall fences, and larger engines and propellers use the Bootstrap Approach as a sales tool and to demonstrate to their customers that the modifications they bought will pay off with enhanced performance.
This article is only a primer on the aircraft performance subject. What about designing an airplane? Knowing about performance comes first; the airplane is designed to perform a certain job. There is also the vast subject of airplane stability and control, which we did not even touch. But knowing about performance is also a prerequisite to that subject. The references suggest where you might look further. Learning the ins and outs of aircraft performance will make you a better pilot and make you a better engineer. There are few technical subjects more interesting, or more fun, than aircraft performance. Calculate thoughtfully. Fly the same way!
ReferencesGeneral Works on Aircraft Performance, Stability, and Design
1. Roskam, J., and C.-T.E. Lan, Airplane Aerodynamics and Performance, DARcorporation, Lawrence, Kan., 1997. A large book, currently the text in the aircraft performance course at the U.S. Military Academy, covering both propeller and jet airplanes.
2. Hubin, W.N., The Science of Flight, Iowa State University Press, Ames, Iowa, 1992. The author is a physicist at Kent State University and an experienced aerobatic pilot. Well illustrated. Requires only algebra and elementary physics.
3. McCormick, B.W., Aerodynamics, Aeronautics, and Flight Mechanics, Wiley, New York, 1979. A calculus-level treatment for aspiring aeronautical engineers.
4. Hurt, H.H., Aerodynamics for Naval Aviators, U.S. Navy, 1960. A good full qualitative (almost completely non-mathematical) treatment. Covers both propeller and jet airplanes.
5. Von Mises, R., Theory of Flight, Dover, New York, 1959. A somewhat old-fashioned reference (for instance somewhat confusingly separates wing drag out from the remainder of the airframe) but packed with wisdom and information. Written by the best educated person to ever clamber into a cockpit. Treats only propeller airplanes. Uses calculus and differential equations as necessary.
6. Hale, F.J., Aircraft Performance, Selection, and Design, Wiley, New York, 1984. Written by a former U.S. Air Force jet combat pilot. Uses minimal calculus. Treats propeller airplanes and the several types of jet aircraft.
7. Hiscocks, R.D., Design of Light Aircraft, published by the author (designer of the deHavilland Beaver) and distributed by Murphy Aircraft Manufacturing, Ltd., Unit #1, 8155 Aitken Road, Chilliwack, B.C., Canada V2R 4H5. Takes the reader step by step through the full range of light airplane design techniques and considerations.
8. Perkins, C.D., and R.E. Hage, Aircraft Performance Stability and Control, Wiley, New York, 1949. A classic, still in print.
Works Specific to The Bootstrap Approach
9. Lowry, J.T., "Analytic V Speeds from Linearized Propeller Polar, Journal of Aircraft, 33, No. 1 (Jan/Feb 1996), pp. 233235. The first small engineering note on the Bootstrap Approach.
10. Lowry, J.T., "The Bootstrap Approach to Predicting Airplane Flight Performance," Journal of Aviation/Aerospace Education and Research, 6, No. 1 (Fall 1995), pp. 2533. A much fuller explication, including the constant-speed extension, a since-much-improved-upon theory of partial-throttle performance, and several sample graphs.
11. Lowry, J.T., "Maneuvering Flight Performance Using the Linearized Propeller Polar," Journal of Aircraft, 34, No. 6 (Nov/Dec 1997), pp. 764770. Contains a summary of the wings-level theory and the definitions of all Bootstrap Data Plate and composite parameters. Also a detailed recipe for constructing steady maneuvering charts.
12. Lowry, J.T., "Fixed-Pitch Propeller/Piston Aircraft Operations at Partial Throttle," Journal of Propulsion and Power, 15, No. 3 (May/Jun 1999). The not-quite-here long article explaining the Bootstrap partial-throttle theory. Includes scaling rules for expanding cruise performance table entries from original values of weight and altitude to any different values.
13. Lowry, J.T., Computing Airplane Performance with The Bootstrap Approach: A Field Guide, M Press, Billings, Mont., 1995. Available from the author at 1615 Redwood Road, Apt 12A, San Marcos, TX 78666. All the necessary background, formulas (not derived), and data collection forms needed to actually perform a Bootstrap analysis on your own airplane. A disk of seventeen supporting spread sheet templates (WK1 format, but easily converted to Quattro Pro or Excel) is also available
14. Lowry, J.T., Performance of Light Aircraft, AIAA, Reston, Va., 1999. A book on all aspects of The Bootstrap Approach and on several other facets of propeller airplane flight and ground performance. Available starting in August 1999. Contact the author at the email address shown below.
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