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.
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!