Polynomial – Aviation CAD TechNotes

NAA P-51D Mustang: Fuselage: Conics

In the preceding article I had some fun with polynomials and how they could be useful for determining a smooth fit spline for the development of the Mustang fuselage. As a follow up to that article I wanted to share some research relating to conics.

The Mustang P-51 was the first aircraft to be completely defined by conics. The designer Edgar Schmued worked with Roy Liming to mathematically analyze the Mustangs shapes, tangents and curves. Conics were used by NAA as far back as 1932 though many of the techniques and equations we use today however were not actually in use until 1959.

The Bézier curves for example were based on the Bernstein polynomial which had been known since 1912 but its application for graphics was not understood till much later. Bézier curves were widely publicized in 1962 by the French engineer Pierre Bézier, who used them to design automobile bodies at Renault. The study of these curves was however first developed in 1959 by mathematician Paul de Casteljau using de Casteljau’s algorithm, a numerically stable method to evaluate Bézier curves at Citroën.

So I started to wonder how did Edgar Schmued and Roy Liming actually apply conic principles and what methods did they use for the Mustang design!

The documentation I have available for the Mustang Wind Tunnel models gives us a clue at the geometric construction for the fuselage frames. The designers used smooth conic sections with key parameters controlled by longitudinal shoulder and slope control curves. The longitudinal curves defined fullness and tangency values for the conics from forward to aft of the fuselage. The P-51 designers found that this technique allowed them to accurately control sectional areas to secure the required effects for lift, drag, stability, and overall performance.

Here we see a scrap view from the wind tunnel models, showing clearly the development of the conic constrained by 2 tangent lines and a third Shoulder Point as a known point on the designed curve.

The intersections of lines extended from the Max Half breadth point and the Lower Ship Centre point illustrate a drafting technique for creating the finished curve for the lower section of a fuselage frame.

Taking this method further we can describe a curve using a series of extended lines to define any point on the curve as shown in my Cad drawing.

This is my interpretation of a technique for the drafting of a typical Mustang fuselage frame. I haven’t seen this technique applied to a full fuselage profile and whilst the design information I have suggests a similar approach by the Mustang designers I can’t verify that this was the actual technique used.

It is not possible within the scope of this article to go into the detail of this technique, but suffice to say that selecting only 3 points for the lower and upper sections contained within tangential lines provides the basis for accurately determining any other ordinate point on the particular curve. I have uploaded a short video on Youtube here: Drawing a Conic

This is actually a lot better than using the polynomial equations for frame geometry as they only give you a best fit approach based on the tabled ordinates; with limitations; whilst this construction technique will allow the flexibility of defining any point on the curve to an unprecedented degree of accuracy when created in CAD…it works!

So what else did these visionary guys do? I am really keen to further research the mathematical approach that Edgar Schmued and Roy Liming used in the other aspects of the aircraft design and uncover the methods that made the Mustang unique.

It is my hope that by sharing my research and developments that this will inspire others to also research the work of the designers from this era and hopefully in some small measure encourage support for our project “Operation Ark”.

Update: I must have spent a full day browsing through the archives to find more information that would assist with understanding the conics development and thankfully I came across this NAA lines drawing for the cowl on P-51C (NA-103).

This shows the development and tangent lines for everything including the shoulder lines and the fairing lines as well as the main profile contour lines.

Its very important to spend time verifying the information used for developing these designs to validate the research. Sometimes I could spend days just looking for small scraps of information just to verify one dimension, which happened quite a lot on the Ta-152 project!

Full profiles drawn in Autocad from comprehensive excel spreadsheet ordinate collections now available for download. See this article for details.

NAA P-51D Mustang: Fuselage Lines; Polynomials.

This evening I spent some time looking back through some old notes I had on fuselage design, particularly Conic sections and Setting-out design theory.

Checking through the archives for the Mustang P-51 we have a design set for the wind tunnel model with a line plan showing the Shoulder Points (SP) and the “point of convergence” where the upper line of the Mustang fuselage converges with the lower fuselage line and the Fuselage reference line.

The Wind Tunnel drawings are a quarter scale but are quite accurate.

Here we can see the “point of convergence” actually defined on the the wind tunnel drawing at the scaled sta 92. Technically station 92 does not exist as it is outwith the fabric of the WT aircraft, but for convenience I have defined it!

So with this in mind I decided to undertake an experiment to calculate the “point of convergence” with the fuselage ref line according to the manufactured ordinates.

For this exercise I used the upper line of the fuselage, shown here as X,Y values starting from Station 113 and created a line chart.

I applied an third order polynomial equation to the line chart with a scientific value to 5 decimal places to increase the accuracy.

I recalculated the values of the Y ordinate to check that the formula produced an accurate result; shown in red. As you can see the resulting values are very close to the original Y values.

The last X value is the projected value I want to calculate to achieve a “close to zero” Y coordinate thus by definition being the calculated “point of convergence”. This value is 9518mm (374.725 inches) which compares quite well with the Wind Tunnel drawings showing this to be 92*4=368 inches.

Should I recreate this exercise but instead use a fifth or sixth order polynomial equation I am quite sure the resulting value for the point of convergence would be closer yet to the scaled up wind tunnel value.

Normally for this type of exercise I would work with tangent lines and the start points of the upper and lower fuselage lines from predefined Shoulder Points.

This was a bit of fun just to demonstrate how we can use the power of spreadsheets and mathematical equations to assist with developing our Cad designs.

Update: I decided to play about with this a bit more and had a look at the fuselage lines for the Bf109. I don’t have the design “point of convergence” for comparison but decided to do it anyway to find the convergence between the Lower and Upper fuselage lines.

These points are measured from a ground datum at 800mm below the fuselage reference line.

The stations/frames are from 2 – 8 inclusive. As you can see the calculated values verify the existing ordinate dimensions with the projected “point of convergence” calculated at 4832mm from station/frame 2.

These are the fuselage lines on the vertical plane which in theory should share the same convergence point for the fuselage lines on the horizontal plane (technically plan of max width)…an exercise for some other time!

What is even more interesting is that a line equation can be used to generate a spline in both the Inventor & Solidworks cad products… as a check to verify the cad work this is enormously useful!

Another example of application would be for the frames or station profiles.

In this example I have applied a polynomial equation to a set of ordinates for the top section of station 300 for the P-51 Mustang.

This needs a full profile as an arc to achieve an accurate result, which I’ve applied as a sixth order polynomial…you cant get much more accurate than this with Excel!

Ideally we would wish to extend this arc to the max width ordinate, which would add another negative ordinate (below the base line) to the graph…for some unknown reason Excel finds it difficult to compute an acceptable polynomial with 2 sets of negative values, so I would have to transpose the ordinates accordingly.

The Mustang ordinates induce a minuscule negative curvature on the top rear fuselage frames when you create a CAD profile just using the ordinate values from the NAA drawings. Its not detrimental in anyway but it is rather annoying…so to obviate these issues I could utilize a polynomial solution to adjust the ordinates to get a positive curvature. The adjustment is micro millimeters, but hey that’s the way that CAD works.

Another Update: Out of curiosity I recalculated; to a higher degree of accuracy; the upper fuselage line for the P-51 and contrasted that with a similar calculation for the lower line of the fuselage.

The calculated point of convergence of both lines based on a 4th order polynomial to 5 decimal places is at 9375mm and slightly above the fuselage reference line at +18mm. Factoring in error based on the original ordinates being accurate to 1/16th inch and possible error as a consequence of a higher order polynomial I think this is a reasonable result. Its interesting to note the variation with the results we got before.

This is certainly closer to the expected values based on the wind tunnel data. The squiggly line by the way on the lower part of the fuselage is the plotted max half breadths; which is rather interesting!

Confirmation; have received confirmation that the intended point of convergence for the upper and lower fuselage lines is at Sta 368, which is at 9347.2mm…this is great!!

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