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**Conclusion **

__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 V_{x} and V_{y}, as functions of bank angle,
cross. V_{x} then becomes *larger* than V_{y}. 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 V_{br} and for best endurance V_{be}.
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, V_{bg}
and V_{x}, 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 airplane’s 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!

General 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. 233–235. 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. 25–33. 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.
764–770. 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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