PPRuNe Forums - View Single Post - Mathematical functions in performance diagrams
Old 24th May 2021, 12:19
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john_tullamarine
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Join Date: Apr 2001
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This is an exercise which occupied me with some regularity in years gone by.

First, might I suggest that you define your requirement –

(a) if essentially for entertainment, then go for it. The basic equations are available in any of the standard aerodynamics/performance books and flight test manuals.

Unfortunately, there are various assumptions which are built into the equations so that, when we run the initial sums for an aircraft, and then test it to determine whether the sums are fit for purpose, invariably there are bits of the envelope where things don’t quite run true to equation form. What to do ? The OEM has to fudge things a bit so that the equations (and AFM/POH stuff) match what the aircraft does to the required accuracy ... but then doesn’t release the process details for general consumption.

The end result is that trying to figure out the equations from first principles is a bit of a fool’s errand.

(b) if you are looking at commercial work, there are two competing interests –

(i) we must not be non-conservative – if something goes awry and your sums can be implicated, then you could find yourself in a difficult place at the subsequent investigation and court cases – not good. So, make sure that you don’t run non-conservatively with respect to the AFM/POH performance data.

(ii) we must not be conservative (to any degree beyond that negotiated with the customer) because that affects the customer’s profitability.

Add (i) and (ii) and we get the conclusion that we necessarily must replicate what is in the AFM/POH. To this end, scanning of graphs and zooming in to figure point values is useful. Indeed, the usual limit is the printing accuracy of the OEM AFM/POH chart data – a little easier with tabular data but data precision then can become a concern.

Now, you are not likely to find serendipity and come up with the “correct” sums. So, you are stuck with simulating the charts (tables, whatever) either with regressions or look up tables and interpolation.

If you opt for regression then, as a general rule, forget multivariate regressions – it’s not going to work. So you are faced with some set of regressions of the printed data and intermediate calculated regressions or interpolations (as your fancy might run) to get a adequately precise and accurate estimate of what the AFM/POH says for any given data point.

Then, you will usually find that the data sets (especially for heavy metal) contain various discontinuities and that just makes everything more complex.

So far as polynomials doing funny things between data points is concerned, prefer low order polynomials/high data density and run a check on each and every segment, as modeled, to detect anything out of order in the sums. (It’s not often that first order fits would be of much use).

FWIW, my approach (and it worked fine, although an absolute pita to set up) was to model each of the AFM/POH data lines, using low order polynomials segmented as I saw fit and making due allowance for discontinuities. Rather than use interpolation routines for the particular variable, I choose to establish a suitable set of data points from my equations and run a suitable minimum order polynomial through that data set from which I arrived at my answer for the required data point.

The basic test was that I couldn’t run the sums better mandraulically than I could do with my program. If I could, it was a case of back to the drawing board until the end result achieved that goal. Customers were happy.

Of course, if you can buy a program from the OEM, that probably will be the better alternative as setting up the thing from scratch is not a cheap exercise.

Australian “Old” Charts

One neat approach was followed by DCA (a predecessor regulator to CASA). When the early electronic plotters came onto the market, the chaps in the performance group came up with some fairly simple equations which were the basis of the old P-chart graphs. These, in general, were thrown out with the baby and bathwater post Yates Report some years ago - pity. Should you need to play with one of these, then the old DCA engineering reports give you the equations and you would have little trouble setting them up in the manner you so obviously desire.

Piper Chart as Attached

The PA28 chart cited is dead simple and should be a relatively quick exercise to set up.

You might get a different opinion from your insurance company, design authority and/or relevant CAA

So long as your program always produces output which is not non-conservative with respect to the AFM/POH data, no-one is going to have any interest in trying to cause you grief. Be non-conservative and things can go from bad to worse in court after the mishap, etc.

Dave highlights one significant problem in respect of some POH data ...
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