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**The Bootstrap Approach:
Formulas and Graphs**

__Composite Bootstrap Parameters Ease Calculation__

Almost everything about the airplane’s full-throttle steady-state
(non-accelerating) flight performance depends on the nine BDP items plus three operational
variables: weight *W*, bank angle f, and relative
atmospheric density s. But only certain combinations of
the nine BDP numbers (combinations called E, F, G, and H) actually occur in Bootstrap
formulas for V-speeds or for thrust, drag, or rate or angle of climb or descent. In
the V-speed formulas, in fact, only certain combinations of combinations (those to be
called K, Q, R, and U) occur.

Our flight tests to determine the four harder-to-get BDP parameters were done at 5000
ft and at *W* = 2200 lbf. But the parameter values we got did not depend on those
choices; BDP parameters only depend on the particular airplane and its flap/gear
configuration. Flight tests at some other altitude or at some other weight would
have given, within experimental error, the same BDP values.

But performance numbers themselves – rates of climb, V speeds, etc. –
obviously do depend on gross weight and on density altitude. Again for
brevity’s sake, we will consider this airplane’s behavior at one particular
weight (maximum gross weight W_{0} = 2400 lbf) and at two particular density
altitudes (MSL and 10,000 ). These choices let us compare our performance numbers
– at least many of them – with the airplane’s flight manual, the POH.
Looking ahead to that comparison, let us evaluate all the above composite Bootstrap
parameters for those two cases. See Table 5

The composite definitions and their dependence on weight, bank angle and air density are:

(14) |

(15) |

(16) |

(17) |

(18) |

(19) |

(20) |

(21) |

Variable or Composite

Case 1, MSL

Case 2, 10,000 ftW

2400 lbf

2400 lbf

s

1.000

0.7385

F

1.000

0.7028

E

531.9

373.8

F

–0.0052368

–0.0038673

G

0.0076516

0.0056505

H

1,668,535

2,259,424

K

–0.0128884

–0.0095178

Q

–41,270.6

–39,277.5

R

–129,460,301

–237,389,461

U

218,064,595

399,861,861

Table 5. Bootstrap composite parameters for two operational situations.

__Full-Throttle and Gliding V Speeds__

The V speeds we are concerned with, in the Bootstrap Approach, are:

V_{M}, maximum level flight speed;

V_{m}, minimum level flight speed;

V_{y}, speed for best rate of climb;

V_{x}, speed for best angle of climb (in calm air);

V_{bg}, speed for best (longest) glide (in calm air); and

V_{md}, speed for minimum gliding descent rate.

Bootstrap formulas for these V speeds, each expressed as a true air speed in ft/sec, are:

(22) |

(23) |

(24) |

(25) |

(26) |

Since the three full-throttle V speeds V_{M}, V_{y}, and V_{x}
depend on only two composite parameters, *Q* and *R*, there must be a connection
between them. It is:

(27) |

In addition, it turns out that V_{x} is the geometric mean between V_{M}
and V_{m}. Performance specifications for most manufacturers’ airplanes
will not closely agree with Eq. (27) because the quoted values of V_{M} are too
optimistic.

__Additional Flight Performance Quantities__

The Bootstrap Approach is not limited to predicting V speeds. There are also
formulas for full-throttle power available P_{a} = *TV*, power required P_{r}
= *DV*, excess power P_{xs} = (*T*–*D*)*V*, thrust *T*,
drag *D*, rate of climb R/C, and flight path angle g.
In the gliding case, rate of sink R/S and glide path angle can be obtained from the
powered forms by setting E = *F* = 0 and replacing *K* by –*G*.

(28) |

(29) |

(30) |

(31) |

(32) |

(33) |

(34) |

We’ve shown all the left-hand-side variables as functions of only true air speed *V*.
But in all except two cases gross weight *W*, relative air density *s*, and bank angle *f* also
matter. It would have been more instructive, if somewhat pedantic, to have written *D*(*V*),
for instance, as *D*(*V*;*W*,*s*,*f*).

Where do all these formulas come from? In most cases, from
standard "power-available/power-required" analysis, which you can find in almost
any textbook treating the aircraft performance subject. The one big exception is Eq.
(31) and other equations springing from it. Eq. (31) is the Bootstrap
Approach’s "hole card," giving us a good approximation to thrust without
our having to know propeller efficiency. That Bootstrap expression for thrust is
quite easy to get. Here’s how.

__The Bootstrap Formula for Full-Throttle Thrust__

First, solve Eq. (2) for thrust:

(35) |

Next, rewrite Eq. (5) as:

(36) |

Then substitute Eq. (36) into Eq. (35), using the definition of *C _{P}*
(Eq. (3)) and the definition of

(37) |

One additional "physical" fact is needed. For an
internal combustion engine at given altitude, throttle position determines torque output,
irrespective of load and resulting RPM, to a good approximation. So "full
throttle" means "full torque," or at least as "full" as
ambient density altitude allows under the direction of Eqs. (6) and (7). Finally use
Eqs. (14) and (15), the definitions of Bootstrap composite parameters *E* and *F*,
to get Eq. (31).

Now, let’s look at some graphs.

__Bootstrap Performance Graphs__

There’s no better way to learn the ins and outs of airplane performance than by looking carefully at various performance graphs. The graphs drive home the definitions of the various V speeds, the airplane’s "optimal" speeds.

Figure 4. Thrust, drag, and their difference (excess thrust), as functions of air speed, for a Cessna 172 at sea level, flaps up, weighing 2400 pounds.

But keep in mind that almost everything about an airplane’s performance depends on its weight, its altitude, its flaps/gear configuration, and whether or not its wings are level. For brevity, we’ll stick to wings level. But, as you look over the graphs, you should ask yourself such questions as: How would this graph look if it were for a more (or less) powerful airplane? What if the airplane were heavier (or lighter)? At higher (or lower) altitude? Flaps down? Banked 40 degrees?

Figure 4 starts us off with graphs of thrust and drag (Eqs. (31) and (32)) and their
difference, excess thrust T_{xs}. The speed at which excess thrust is a maximum is
always the speed for best climb angle, V_{x}. The speed at which drag is a
minimum is the speed for best (longest) glide in calm air, V_{bg}. The
thrust and drag curves cross at two places, on the right at maximum level flight speed V_{M}
and, to the left, at minimum level flight speed V_{m}. V_{m} is not
marked because, in fact, you can’t get there. Stall speed V_{S} – *not*
a Bootstrap-calculated V speed – is higher than V_{m} except at high
altitude.

But not *everything* depends on all three operational variables *W*, *s*, and f. Thrust, for
instance, is independent of *W*. But at higher altitude, lower *s*, the thrust or power available curve will be a lower one.
How the drag curve behaves with changes in weight and altitude is harder to see;
altitude affects Bootstrap composite parameter *G* and both altitude and weight
affect *H*. With a spread sheet program (Lotus 1-2-3 or Quattro Pro or Excel)
you can easily take the Bootstrap formulas, assume reasonable aircraft parameters, and
construct graphs for all kinds of scenarios.

Figure 5 shows most of the important performance V speeds and where they come from.
V_{M} and V_{m} are speeds at which the P_{a} and P_{r}
graphs cross. Or, alternatively, where P_{xs} has its zeroes. Since
rate of climb R/C = P_{xs}/W, speed for best rate of climb V_{y} is where
P_{xs} peaks. When gliding (thrust zero), P_{xs} = –P_{r};
therefore the minimum descent rate speed V_{md} is at the low point of the P_{r}
curve. Speed for best glide V_{bg} and speed for best climb angle V_{x}
take a bit more analysis. Since P_{xs}/V = T_{xs}, we see that the
straight line tangent to the P_{xs} curve which has the largest slope is the one
hitting the P_{xs} curve at V_{x}. Similarly for P_{r}/V = *D*
and V_{bg}.

Figure 5. Power available, power required, and excess power for the Cessna 172 at 7,500 ft, flaps up, 2400 pounds.

Figures 4 and 5 are graphs for the airplane at one weight and one altitude. Figure 6 gives a broader view, in the vertical direction, showing how the main full-throttle V speeds differ, again for the Cessna 172 weighing 2400 pounds, over the full range of accessible density altitudes.

Figure 6 shows that speed for minimum level flight V_{m} doesn’t come out
from hiding behind the skirts of stall speed V_{S} until the altitude is way high,
above 14,000 feet. Which means, in almost all practical cases, never. There
are conflicts between Figure 6 and the Cessna 172P POH. The POH has V_{y} not down
to as low as 70 KCAS until 12,000 ft; we get it there at only 7000 ft. And the book
has V_{x} increasing a little with altitude, about 4 KCAS over 10,000 ft, where we
get that it is constant in calibrated terms. Naturally, we believe *we* are
right. Figure 7 delves into drag. As Eq. (32) shows, total drag *D* is made up of two
terms. D_{P}, parasite drag, is proportional to the square of the air speed.
See Eq. (16) for *G* to see what makes up the proportionality constant. The
second term, induced drag D_{i} (also known as "drag due to lift"),
works much differently. Induced drag is *inversely* proportional to V^{2}.
That means induced drag is higher the *slower* the airplane goes through the
air. See Eq. (17) for *H* to see what the "constant" depends on.

Figures 8 and 9 show how performance drops off with altitude for a Cessna 172 weighing 2400 pounds, flaps up. By 10,000 feet, considerably less than half the mean sea level (MSL) best rate of climb or best angle of climb remains. This is a real problem for a relatively underpowered airplane like the Cessna 172. The Civil Air Patrol, for instance, doesn’t allow its airplanes to conduct mountain search and rescue missions at altitudes where their best rate of climb is less than 300 ft/min. A fully-loaded Cessna 172 with the stock 160 HP engine could not qualify. Perhaps a special "climb" propeller would let it meet that mark.

Figure 6. Full-throttle V speeds for Cessna 172, flaps up, 2400 pounds.

These graphs give you a reality check, samples showing how various factors influence propeller airplane performance. The references contain many more such graphs and numerical illustrations.

Figure 7. How drag varies with air speed for a Cessna 172, flaps up, 2400 lbf, at MSL..

Figure 8. Rate of climb graphs, at three density altitudes, for a Cessna 172 weighing 2400 pounds, flaps up. Notice that calibrated air speeds for best rate of climb decrease with altitude.

Figure 9. Angle of climb for a Cessna 172, flaps up, 2400 pounds, at three density altitudes.

So the Bootstrap formulas are
relatively simple and straightforward. But how accurate are they?

__Comparison of Bootstrap and POH Performance Figures__

That depends on whom you ask. Certainly the airplane’s POH
won’t be *far* wrong, so let’s take a look at it. Sifting through the
Cessna 172P Pilots Operating Handbook (for the airplane *without* speed fairings),
and making occasional use of the air speed indicator calibration curve given there, one
can come up with the two columns headed ‘POH’ in Table 6. We’ve translated
all the speeds in that table into KCAS.

Performance

Item

Case 1, 2400 lb at MSL

Case 2, 2400 lb at 10,000 Ft

TBA

POH

TBA

POHV

_{M}115.3

121.0

90.8

98.8

V

_{y}75.8

76.0

67.5

71.0

R/C(V

_{y})700.5 fpm

700.0 fpm

258.5 fpm

237.0 fpm

V

_{x}63.2

62.0

63.2

66.0

V

_{bg}72.0

66.0

72.0

66.0

g

_{bg}–5.40 deg

–6.26 deg

–5.40 deg

–6.26 deg

Table 6. Comparison of Bootstrap (TBA) and POH performance predictions.

There is a major discrepancy where maximum level flight speed V_{M} is concerned.
We’ve flown a number of Cessna 172s, but never one which would go 121 KTAS at
sea level. The Cessna test pilot might explain her higher speed is due to her brand
new engine, the pristine leading edge of the wing, no dents, .... Perhaps. In
glide performance, on the contrary, our airplane did *better* than the factory-fresh
one. There we suspect (but only suspect) the Cessna Aircraft Company corporate
attorneys came into play. For liability reasons, they certainly wouldn’t want
to claim a *longer* best glide than might be demonstrated by a lawyer whose
client’s engine had failed.

The only way to settle these questions of whose performance statements are more accurate is through careful, well-instrumented, and un-manipulated test flights. Our point here is that the Bootstrap Approach lets you fly the airplane for an hour or so, performing a few simple climbs and glides and a level speed run, and then lets you calculate many interesting and apparently accurate performance numbers for that actual airplane. Bugs, dents, tired engine and all.

Go to next section - Conclusion & References

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**