Vs - Vmp - Vmd

[Alex Whittingham]

I'm trying to find what these speeds are for a range of aircraft from pistons to jets so that we can test a theory that suggests VMP should generally be about 0.75VMD.

So far we've drawn a blank, this information seems to be quite difficult to find. I would be most grateful if anyone can provide these three speeds for any one configuration for any aircraft! Thanks in anticipation.

[Stan Evil]

It's a straight mathematical relationship, irrespective of jet/piston. Differentiate the drag equation (split into zero-lift and lift induced drag) with respect to velocity and set to zero to get the minumum. Then do the same for power (drag x velocity). Solve the 2 equations simultaneously and the speed ratio pops out. You can use the same method to derive range speed for a jet aircraft.

My excuse for not doing it here is that I can't get all of the symbols I need!

QED

[Alex Whittingham]

Thanks Stan Evil. We've done the maths and are looking for empirical data to check the theory.

[hawk37]

Alex, I did some tests on a falcon 50 simulator a while back and came up with e=.80 and Cdo =.225
I then computed for 39000 lbs weight and sea level that Pmin was at 142 ktas, and Thrust min was at 187 ktas.
And thus Pr min divided by Th min = .759

As a check on my figures, the flight manual gives Vfs as 1.5 V stall.
V stall at 39000 lbs is 129 ktas, giving V fs of 194 ktas
This may seem reasonable, although V fs is for one engine out, giving increased rudder drag and windmilling drag.

The falcon 900 ex manual lists V fs as 1.43 V stall.

If you come up with any data/conclusions/equations, I'd love to hear. If posting here isn't appropriate, I can send you my email.
Good luck with your project

Hawk

[Alex Whittingham]

Thanks for that, that's very useful.

[Old Smokey]

Hawk 37,

Good work, but accept my caution that most simulators offer a good approximation of general performance characteristics, but beware of extracting any empiric performance data from them.

I have had the pleasure of co-ordinating and fine tuning the performance aspects of 3 simulators for aircraft in airline service for fidelity acceptance with the relevant authority, and in each case, almost all of the performance data needed correction, some of it relatively minor, some of it major - an example is 6500 feet difference between 1 Eng Inop absolute ceiling in the simulator and the real aircraft. This latter correction was, in the end, verified by taking the actual aircraft out, shutting one down, and observing the results. (The AFM was exactly right).

Much of the problem arises from the simulator being developed at the same time as the aircraft before any verification by flight testing. In fairness to the Sim manufacturers, this is necessary so that crews may be trained prior to taking on the real aircraft.

The worst case that I saw was a simulator which had stall characteristics that my Grand-Mother could handle, full back-stick testing in the real aircraft showed it to be a wildcat.


BEWARE!

[djpil]

Anytime that the drag polar can be represented by a straight line of CD vs Cl^2 (even if its just within that CL range under consideration) you will find that relationship between Vmp & Vmd (0.76 by my calcs).
You'll find some test data at the NASA server. (http://ntrs.nasa.gov/)
A good place to start, for small airplanes, is with the Cessna 172 info here (http://www.allstar.fiu.edu/aero/BA-Background.htm) or the info here (http://www.cafefoundation.org/services.htm)

The question I have - why do you want to know?

[Alex Whittingham]

Thanks for that, too. I understand the maths that fixes the relastionship between VMP, VMD and Best range speed still air once you make certain assumptions about the shape of the drag curve. I'm more interested in the relationship between VS and VMP, which depends in turn on the relationship between VS and VMD, so that I can explain the subject, with examples if possible, to ATPL students. It is always comforting, as well, to find real life data supporting the theory!

[djpil]

Alex, I've been caught out before with my assumptions on terminology: By VS, do you mean stall speed?

Lift coefficient for VMD is sqrt(Cdo * pi * Ar / k)
Lift coefficient for VS is CLmax

[hawk37]

Old Smokey, thanks for your insights.

Using my “extracted” data, I’ve generated power required curves, for different altitudes, using graphs of thrust available versus mach from the engine manufacturer. I then took the excess power, and computed best rate of climb. The results were quite close to reality. However, the speeds are substantially less than the flight manual in mach and cas. I’m considering making some adjustments to e and Cdo, however I can only imagine this will skew out some other data, such as climb rate, absolute ceiling, max speed, endurance, range etc.
You’re probably wondering why I’m trying to re-create the wheel.

Hawk

[Old Smokey]

hawk 37,

No, not really wondering why you're trying to re-create the wheel, it sounds as though you have an octagonal wheel and trying to convert it back to a round one !

There are a great many areas where the software creators for simulators use generally correct, generic, and simplified algorythms for simulators. Don't forget that the primary function of a simulator is to reasonably reproduce control characteristics for pilot handling qualities, often at the expense of loss of performance fidelity.

As an example, you quoted that you've tweaked your simulator for realistic best rate of climb, but the "speeds are substantially less than the flight manual in mach and cas". This one is familiar, the original programmer has done it perfectly for EAS (which we all would really like to have), but not accounted for compressibility which directly affests the CAS and Mach No., both of which you've found to be in error (in line with what you would expect if EAS were presented for CAS).

I'm no fortune teller, but can reliably predict that your prediction of correctly adjusting e / Cdo etc to fit one operational situation will affect other areas.

Lots of luck, find the happy medium.

Regards,

Smokey

[hawk37]

Alex. I too get the Vmp/Vmd ratio to be about .76
Actually, my math gives (3)^.25 as the "exact" answer.

I'm thinking there can’t be any generic relationship of Vmd to V stall, except that a wing designed for higher speeds will have a greater Vmd/Vs ratio

Examples:

1. Slow speed, C172 Vcal mph = 57 flaps up, power off, max weight
Emergency procedures give 75 mph for engine fail after takeoff, 73.5 cas, likely max L/D speed.

Vmd/Vs = 73.5/57 = 1.29

2. Westwind, Mmo .765

Vmd/Vs = 1.43 at low altitudes

3. Falcon 50, Mmo .86

Vmd/Vs somewhere between 1.5 and 1.7

4. concorde Mmo 2.2 (?)
Vmd/Vs unknown, but I’d expect greater than any subsonic aircraft.

thoughts?
Hawk