Russ Erb

Originally published April 1994

Origins

Pilot Technique

Runway Slope Correction

Wind Correction

Weight Correction

Density Altitude Correction

So Does It Work?

References

So you've finished your new mega-mongo domestically manufactured aerospace vehicle, flown off the 40 hours, and received your FAA sign-off allowing you to do something else other than simulate a 12-inches-to-the-foot scale U-Control model on a control line 25 miles long centered at Fox Field. For your first cross country, you decide to convince your significant other of the traveling utility of the project that you've seen more of than her/him the last three years (with references to an earlier Scott Horowitz article) by taking her/him to the Blue Ox at Big Bear (located within walking distance of the airport) so you can throw peanut shells on the floor. Being the model of an FAA standard pilot, while doing your flight planning you realize that the elevation of Big Bear (L35) is 6748 ft (qualitatively "high") and with the temperatures this time of year, the density altitude is even higher. The runway is 5800 ft long. Or better yet, you're heading out to Oshkosh and decide to drop in on Russ and Bruce at EAA Chapter 1000 Det 2 in Colorado Springs, where Meadowlake airport (00V) is at 6874 ft ("higher") with a 5900 ft runway. While in Colorado, you could also go up to Leadville (LXV) and pick up one of those certificates they give out for landing at the highest airport in the United States (elevation 9927 ft, runway 'only' 5300 ft long). I could go on, but La Paz, Bolivia (LPB, 13,392 ft elevation, 13,123 ft runway) is a little out of the way while traveling from California to Oshkosh.

Two questions (should) pass through your mind, namely (1) how much runway (i.e. ground roll) will I need, and (2) how much climb rate will I have after takeoff. While the second question is very important, and should definitely not be ignored (even if the FAA written exams don't emphasize it), that could be the subject of another newsletter article.

Now if you were flying the Piper Traumachicken or Cessna 152 that you took your initial flight training in, you would flip open the Pilot's Operating Handbook to the Performance section and read the charts to find your takeoff ground roll and initial climb rate. (At this point, you would probably decide to go somewhere else--personal experience.) Unfortunately, your kit/plans supplier didn't issue you takeoff data. What to do now?

Fortunately, during your 40 hours of test time, you were faithfully recording flight test data, including takeoff data (ground roll, liftoff airspeed, pressure altitude, density, and winds), according to the "Scotty 'Doc' Horowitz's Generic Homebuilt Flight Test Plan" (yet to be published due to excessive demands the government places on its astronaut's time...don't they understand priorities?). However, all of your data is for varying density altitudes, wind conditions, and aircraft weights. Besides, all of the density altitudes were in the neighborhood of 2000 to 3000 ft. How do you extrapolate that up (or down) to other altitudes and conditions?

Well, it would be nice to think "Hey, this article is being written by a professional flight test engineer, and certainly the USAF Test Pilot School gave him some easy mathematical formula (probably on a spreadsheet) to quickly and easily solve my dilemma." Yea, that would be nice. But, hey! It just isn't going to happen! Why not? Read on.

This article is a direct by-product of a project I am currently
working on. One of the objectives of this project is to collect
and analyze takeoff data for towplanes towing gliders. Unfortunately,
takeoff data is very difficult to reduce, due to the number of
factors involved. The major factors are pilot technique, runway
slope, winds, aircraft weight, and air density. Pilot technique
cannot be corrected for. Runway slope can be handled mathematically,
but the other three factors must be handled empirically. Here's
where the problem arises. The references available to me were
not clear about whether the equations given were valid just for
jet aircraft or for any aircraft. As such, I pulled out takeoff
data for the Beechcraft Sundowner and Sierra, the Piper Tomahawk
and Archer II, and the Cessna T-41C and T41D (high-powered 172s).
I foolishly thought that by analyzing these Pilot Operating Handbook
data, I could either (1) verify the numbers I had, or (2) find
the "real" answer. Unfortunately, what I got was option
(3), namely, a * different* answer from each manufacturer.
If three major light aircraft manufacturers can't agree on how
to standardize takeoff data, what hope do we have? Well, it's
not quite that bad.

Note that the order the corrections are applied is important. Basically, apply them in the order they are presented here.

The single biggest factor is pilot technique. The best way to handle pilot technique is to standardize it (i.e. do it the same way every time). This is important because there is no method to mathematically account for differing pilot technique. A possible technique could be something like

- Apply full throttle while holding the brakes.
- Release brakes.
- Accelerate to 55 knots.
- Rotate until the cowling is on the horizon.
- Wait until the aircraft lifts off.
- Climb out at 70 knots.

The first correction to be applied is for runway slope. Fortunately, this one is straight forward without any empirical constants. This formula is from Herrington, page 6-9.

where

s_{level} | takeoff distance corrected to level runway |

s_{slope} | actual takeoff distance on sloped runway |

g | acceleration of gravity (9.81 m/sec^{2}, 32.2 ft/sec^{2}) |

runway slope, measured from horizontal (+ for uphill, - for downhill) | |

V_{to-slope} | Lift-off velocity (Ground Speed) |

The wind correction is the first of the equations with an empirical constant. The equation cannot be derived rigorously because effects such as the change in thrust with airspeed are very difficult to model mathematically. As the airspeed increases, the thrust of the propeller decreases. With a wind, you are not starting at zero airspeed, so the thrust at any particular point of the takeoff is different than for the no-wind case.

The basic equation comes from Herrington, page 6-7.

where

s_{w} | takeoff distance corrected for wind |

s_{level} | takeoff distance corrected for runway slope |

V_{to} | liftoff velocity (ground speed) |

V_{w} | component of wind velocity down runway (+ for headwind, - for tailwind) |

wind exponent | exponent which gives best results. |

Analysis of the manufacturers' data gives typical values for the wind exponent. Note that some manufacturers use a higher exponent for tailwinds than for headwinds.

Headwind | Tailwind | Source |
---|---|---|

0.987 | 1.44 | Beech |

1.07 | 2.46 | Piper |

1.88 | N/A | Cessna |

1.85 | 1.85 | Herrington |

The best way to attack this seems to be to take all of your data for similar density altitudes and weights and put it in the equation, preferably in a spreadsheet, and adjust the exponent. The goal is to get all takeoff distances for varying wind speeds to reduce to a single distance. If an exponent can be found to adjust all takeoff distances for varying wind conditions to a single takeoff distance, that exponent should be in the neighborhood of those listed above.

In the simplest of takeoff distance formulas, the takeoff distance is shown to vary with the square of the aircraft weight. Again, because of factors that cannot be simply modeled, the relationship is not that simple.

The weight correction formula is derived from Herrington, page 6-14.

where

s_{wt} | takeoff distance corrected for weight |

s_{w} | takeoff distance corrected for wind |

W_{s} | standard weight |

W_{t} | actual test weight |

weight exponent | exponent which gives best results. |

Analysis of the manufacturers' data gives typical values for the wind exponent.

Exponent | Source |
---|---|

1.37 | Beech |

2.12 | Piper |

2.39 | Cessna |

2.4 | Herrington |

Note that the exponents for Piper and Cessna are close to 2, in keeping with the equation mentioned earlier. Also, it is encouraging that the Cessna exponent is very close to the exponent reported in Herrington. Again, the recommended approach is similar to that for finding the wind exponent. Find data for similar density altitudes and wind conditions with different weights, and find an exponent that reduces all takeoff distances to the same distance.

The last correction is for density altitude. You may have noticed by now that these corrections are applied in the reverse order of what you've seen in certified aircraft flight manuals. When you calculate a takeoff distance on the FAA written test, you are essentially adjusting a sea level-standard day-standard weight takeoff distance to get an actual takeoff distance for your current conditions. What we are trying to do is back up from actual takeoff distances to that sea level-standard day-standard weight takeoff distance.

Anyway, the formula is derived from Herrington, page 6-14.

where

s_{std} | takeoff distance corrected for density, final standard distance |

s_{wt} | takeoff distance corrected for weight |

_{s} | density ratio at standard altitude |

_{t} | actual density ratio |

density exponent | exponent which gives best results |

Analysis of the manufacturers' data gives typical values for the density exponent.

Exponent | Source |
---|---|

-2.34 | Beech |

-3.73 | Piper |

-2.4 | Cessna |

-2.4 | Herrington |

Again, follow the same procedure as stated above to determine the exponent. The density ratio is the ratio of the local density to the sea level density. There are two ways that you can find the density ratios, depending on what data you have. If you know your density altitude, you can use

where H_{D}is the density altitude. To calculate _{s},
use the altitude that you want to standardize your data to (usually
sea level) for H_{D}. If you don't know your density altitude,
but you do know your pressure altitude (set your altimeter to
29.92) and outside air temperature, you can use this method:

Boy, I hope so. Otherwise my project will be in trouble. Ideally,
after correcting for each of these conditions, all takeoffs would
standardize to one distance. However, because of slight differences
in pilot technique, there will still be some scatter. To effectively
apply this technique requires a * lot* of data. I hope
to report back on my results in a later newsletter. It should
be interesting to see how this exponent business works out. In
the mean time, if you want to come out to Meadowlake Airport and
collect some actual high altitude takeoff data, Penny and I have
a room open to any Chapter 1000 (or 49) member. Maybe you could
even see what will likely by the only flying example of the Orion
in the U.S., powered by a converted Chevy engine.

For the rest of the story, check out A Low Cost Method For Generating Takeoff Ground Roll Charts From Flight Test Data (Society of Flight Test Engineers (SFTE) Paper, November 1996)

- Herrington, et. al.,
*Flight Test Engineering Handbook*, AF Technical Report 6273, Air Force Flight Test Center, Edwards AFB, 1966. - Beechcraft Sundowner Flight Manual
- Beechcraft Sierra Flight Manual
- Piper Tomahawk Flight Manual
- Piper Archer II Flight Manual
- Cessna T-41C/D Flight Manual

EAA Chapter 1000 Home Page E-Mail: Web Site Director Russ Erb at erbman@pobox.com

URL: http://www.eaa1000.av.org/technicl/takeoff/takeoff.htm

Contents of The Leading Edge and these web pages are the viewpoints of the authors. No claim is made and no liability is assumed, expressed or implied as to the technical accuracy or safety of the material presented. The viewpoints expressed are not necessarily those of Chapter 1000 or the Experimental Aircraft Association.

Revised -- 22 February 1997