Polar numbers for Discus A

On Sunday, April 17, 2016 at 2:32:17 PM UTC+12, xcnick wrote:
On Saturday, April 16, 2016 at 4:55:01 PM UTC-7, Dan Marotta wrote:
You can make a custom polar in XCSoar and use your own numbers.

thanks, I put 51=1.22 76=2.5 97=5 and we will see.

seeyou I did at 50=1.21 65=1.8 80=3.07

but the Cambridge has only two points. Not sure what to fudge. Took me a week to find a 1995 computer with a serial port. Finally got the down audio to shut up.

Cambridge uses only two points, but one of the speeds has two constraints:

1) the L/D at that speed, AND
2) this is *best* L/D speed

Therefore, together with the 2nd speed and sink rate, you have three constraints, same as the other programs. Three constraints uniquely specifies a parabola, which is what virtually all software assumes the polar curve is (at least above min sink):

X*speed^2 + Y*speed + Z = sink

The programs will use the information you input to calculate X, Y, and Z.

For example for your XCSoar numbers 51=1.22 76=2.5 97=5

2601*X + 51*Y + 1*Z = 1.22
5776*X + 76*Y + 1*Z = 2.5
9409*X + 97*Y + 1*Z = 5

Enter these into any linear equation solver, for example at

http://wims.unice.fr/wims/wims.cgi?+...+met hod=coef

x = 0.0014749482401656, y = -0.13611842650104, z = 4.325699378881989

Put those into a spreadsheet and try with 51, 76, 97 and you'll see you get exactly 1.22, 2.5 and 5.

However with 50, 65, 80 you get 1.2071, 1.7097, 2.8759 so the numbers you've put into seeyou are not exactly consistent.

Putting your seeyou numbers into the solver:

2500*X + 50*Y + 1*Z = 1.21
4225*X + 65*Y + 1*Z = 1.8
6400*X + 80*Y + 1*Z = 3.07

x = 0.0015111111111111, y = -0.13444444444444, z = 4.154444444444445

I don't know which of those sets of inputs is best for you. If you use a SVD solver (Singular Value Decomposition) then you can input all six equations (or more!) and get a best fit solution for X, Y, and Z.

Or, we could just say that something between them is close enough. Say:

X=0.001485, Y=-0.135, Z=4.245

These give sink rates different from your inputs by +/- 0.25 fpm at 50&51 knots, +/- 6 fpm at 65&76 knots, and +/- 12 fpm at 80&97 knots.

So, lets work with:

0.001485*speed^2 - 0.135*speed + 4.245 = sink

Cambridge want different kinds of numbers.

Best L/D speed
Best L/D
speed at 2 m/s sink

The L/D is the reciprocal of the glide slope. i.e. if the L/D is 40 then the glide slope is 0.025. If the L/D is 100 then the glide slope is 0.01. etc. We'll work with glide slope for the moment.

Glide slope is sink/speed.

sink = 0.001485*speed^2 - 0.135*speed + 4.245

so

glide slope = 0.001485*speed - 0.135 + 4.245/speed

Best L/D means minimum glide slope, which happens where the 1st derivative of the slide slope is zero.

Remembering high school calculus:

0.001485 - 4.245/speed^2 = 0

= best L/D speed = sqrt(4.245/0.001485) = 53.466

Glide slope at 53.466 = 0.001485*53.466 - 0.135 + 4.245/53.466 = 0.0238

Best L/D = 1/0.0238 = 42.03

For speed at 2 m/s sink, solve:

0.001485*speed^2 - 0.135*speed + 4.245 = 3.88769 (knots)

or

0.001485*speed^2 - 0.135*speed + 0.35731 = 0

Any quadratic equation solver will give 2.73 knots and 88.18 knots. Ignore the answer that is below stall speed :-)

So for Cambridge:

Best L/D speed = 53.466 knots = 99.02 km/h
Best L/D = 42.03
speed at 2 m/s sink = 88.18 knots = 163.3 km/h.