Derek Morrison and Gregory Neff Purdue University Calumet

Proceedings of the ASEE Illinois/Indiana Section Conference, March 13-15, 1997, Indianapolis, Indiana, page 222 - 227.

Based on Conics, mathematical descriptions of sections from a cone, we derive the general description of aircraft curves that define the shapes for the lofting the fuselage of an aircraft. This can be done by hand, or it can be done by computer. In this survey a computer program, based on conics was modified to develop the aircraft’s cross section. "Lofting" is the process of defining the external geometry of the aircraft. Conic sections include circles, ellipses, hyperbolas, and parabolas. All conic sections are described analytically by equations. Unknowns are expressed to no higher degree than the second power. All such curves may be conveniently described as second-degree curves. This survey focuses on conics as a mathematical tool to direct and assist the designer in forming the basic contours of the aircraft. The word basic here, is not meant to mislead the reader in thinking that an entire aircraft cannot be laid out with conics, but to alert the reader that modern (late 1970’s through 1990’s) aircraft may need more advanced methods, beyond the scope of this paper. However it should be pointed out that most lofting nowadays is done on computer, and to this extent, this survey focuses on how aircraft are laid out and lofted with guidance from the initial conic derivations.

Introduction

Many changes have taken effect in the evolution of aircraft design from the early years (1940’s) to the present. The aircraft industry has gone from laborious manual drafting to (CAD) computer aided design. For many years the aircraft industry designed their planes exclusively by the manual method, even up to the 1970’s, aircraft were still designed or laid out to a large existent by the manual method. Now the present design and layout of aircraft is entirely by computer, gone are the days of manually laying out the aircraft in the design process. The speed, accuracy, transferability, and mass storage capabilities, have made the computer a viable tool for design.

Some industry experts feel modern aircraft design relies too much on the Computer. According to Ben Rich¹ former Lockheed Skunk Works head, "all the airplanes in the future are going to look alike, you look at airplanes that they do on computers today -- the DC-10 looks like the L-1011-- you can lay’em on top of each other. None of the young engineers take the descriptive geometry course; they do everything by computer." It is this kind of thinking that has led to this survey. The object of this paper is to help in the understanding of conventional methods of laying out and lofting aircraft fuselages to give the designer a better understanding of the process.

Background

The name lofting comes from the shipbuilding industry. Laying out work or lofting as applied to the airplane is a direct growth from similar processes performed in building ships. It was found that work done in the mold loft served a useful function in constructions where parts with complex form had to be fitted together to build a ship. The definition of the hull shape was done in the loft over the shipyard, using enormous drawings. To provide a smooth longitudinal contour, points taken from desired cross sections were connected longitudinally on the drawing using flexible lead "ducks" (weights holding down a flexible rule -- see Figure 1).

This technique² was used for early aircraft lofting, but suffers from two disadvantages. First, it requires a lot of trial and error to achieve a smooth surface both in cross section and longitudinally.

Second and perhaps more important, this method does not provide a unique mathematical definition of the surface. To create a new cross section requires a tremendous amount of drafting effort, especially for canted cross section (i.e., a cross-sectional cut at some angle other than perpendicular to the centerline of the aircraft). In addition to the time involved, this method is prone to mismatch errors. A new method of lofting was used for the first time in the 1940's. This method, now considered traditional, is based upon a mathematical curve known as the "conic."

The advantage of the conic is the wide variety of curves that it can represent, and the ease with which they can be constructed on the drafting table.

There are other forms of lofting possible, but conic lofting has been the most widely used. Also, an understanding of conic lofting, provides the necessary foundation to learn other forms of lofting, including computer-aided lofting.

Theory

A conic is a second-degree curve whose family includes the circle, ellipse, parabola, and hyperbola³. The generalized form of the conic is given in Equation (1).

C_iX² + C_iiXY + C_iiiY² + C_ivX + C_vY + C_vi = 0 (1)

The conic is best visualized as a slanted section cut through a right circular cone (see Figure 2). The shape of the conic depends upon the angle of the cut through the cone. If the cut is flat (i.e., perpendicular to the axis of the cone), then the resulting curve will be a circle; if somewhat slanted, an ellipse; if exactly parallel to the opposite side, a parabola. A greater cut angle yields a hyperbola.

Problem Definition

Most aircraft designs or concepts, start with profile drawing. This drawing serves as a way to enclose the internal components of the aircraft i.e., engine, people, landing gear, fuel, etc. The profile drawing is made to determine the side view shape of the aircraft. A good drawing will show the overall aerodynamic concept. The inboard profile view (detailed drawing of all internal components) is probably the most important and most useful view of the aircraft. A fuselage shape is drawn around the internal fuselage components and the height and depth of the fuselage shape are thus defined. A view from the top called the plan view is made. This view shows the maximum width of the fuselage. Two maximum width lines are normally used and a flat surface is defined between these lines. See Figure 3 for terminology.

Method

In his book Aircraft Design: A Conceptual Approach, Daniel P. Raymer⁴ explains that a conic curve is constructed from the desired start and end points ("A" and "B") and the desired tangent angles at those points. These tangent angles intersect at point "C." The shape of the conic between the points A and B is defined by some shoulder point "S." The points labeled "E" in Figure 2 are the type of shoulder points characteristic of various conics. Figure 4 illustrates the rapid graphical layout of a conic curve.

The first illustration in Fig. 4 shows the given points A, B, C, and S. In the second illustration, lines have been drawn from A and B passing through S.

The remaining illustrations show the generation of one point on the conic. In the third illustration a line is drawn from point C at an arbitrary angle. Note the circled points where this line intersects the A-S and B-S lines.

Lines are now drawn from A and B through the circled points found in the last step. The intersection of these lines is a point "P" which is on the desired conic curve. To generate additional points, the last two steps are repeated. Another line is drawn from point C at a different arbitrary angle. The lines from A and B are drawn, and their intersection is found. When enough points have been generated, a French or flexible curve is used to draw the conic. With a little practice a good designer can construct an accurate conic in less than a minute. Figure 5, illustrates a conic generated in this manner.

In creating a smooth lofted fuselage using conics, one must also ensure that points A, B, C and S in each of the individual cross sections can be connected longitudinally by a smooth line. Figure 6 shows the upper half of a simple fuselage, in which the A, B, C, and S points in the three cross sections are connected by smooth longitudinal lines. These lines are called longitudinal control lines, because they control the shape of the conic cross sections.

From point A, each point is defined by two measurements, one from side view and one from top view. New cross sections can be drawn from these points using the conic layout procedure illustrated in Figure 4.

In the next figures, are some examples of common applications of conic lofting. In Figure 7 a fighter fuselage is laid out using five control stations. The stations start with zero, which is the nose point. This is the starting point for all longitudinal control lines. The stations continue through 500 inches. Each conic has its own A, B, C, and S points. Then the longitudinal control lines come back together smoothly at the end of the fuselage.

Results

In this study a method similar to that described above was attempted. By starting with commercial computer program written to generate fuselage cross sections based on conic techniques, an aircraft shape was generated. The cross sections were defined in the program by the Stallion aircraft’s designers by "tweaking" a variable "k" (making small changes and seeing the result), which controls the curvature of the cross section, similar to how the S shoulder point controls the cross section’s curvature. This program was written in BASIC and wrote 2 D coordinates to a data file for points on the cross section generated for any station value. In AutoCAD, cross sections were joined by smooth curves called splines to generate the general aircraft shape. One advantage over hand layout of the fuselage is that this program allows you to enter station distances which we used to define cross sections 3-dimensionally, rather than stacking and arranging 2-dimensional geometry after the geometry was defined in the CAD system. The methodology continues by connecting the cross section with a smooth fair line, just like the old hand lofting process.

In the process of designing and laying out the aircraft, it imperative that there is a logical and systematic approaching to the development of the shapes that define the fuselage. As mentioned in the previous sections, the profile view is drawn first, which acts as guide for developing the plan view and cross sections. Next there must be some controlled way of defining the cross sections. Here the conics technique and computer use comes in. This is simplistic compared to what aircraft manufacturers are using, but is a good starting point since a sizable variety of aircraft can be designed with only slight deviations from this procedure. In this survey an approach was taken using conics through the use of an existing source code commercial program in QBASIC with the "k" values already inserted for one aircraft shape⁵,⁶. If desired, one could change angles, heights, widths, distances, and tweak arbitrary k values until a new cross sectional shape is achieved.

This method is half plug and run, half computer aided drafting. The drafting part still must follow good design logic in laying out smooth longitudinal lines that will ultimately define the aircraft’s shape. What was attempted in this survey was to integrate computer modeling with the output of the commercial cross section program. The authors of the program only used numeric and plotted data to design and market the small 4-place airplane called the "Stallion". They did not integrate CAD into the process. In fact, in order to produce CAD renderings and photos for publicity, they hired consultants who took "certain liberties with the design⁷." These AutoCAD experts used a hand drawn set of lofts for the fuselage, a dimensioned top view, side view, airfoil data for the wings and horizontal stabilizer. They took over 140 hours and admitted that the drawings for the shape produced may vary slightly from the actual plane.

We wrote a QBASIC program to make DXF CAD files out of cross section data from the commercial program. The DXF files representing the cross sections were brought into AutoCAD using the DXFIN command. A complete fuselage was designed using the commercial computer program, the program written to translate the exact shape of the cross sections into DXF format and AutoCAD as shown in Figure 8. It was discovered that smooth longitudinal lines can be created and the curvature controlled with the correct CAD techniques^8,9,10,. The CAD model output may be compared to graphical BASIC output from the commercial program which is shown in Figure 9.

With practice one should be able to produce 3-D surfaced models that can be used for engineering analysis and visualization as well as molds, dies and fixtures for manufacturing accurate fuselages. A surfaced 3-D model obtained from Figure 8 is shown in Figure 10.

Conclusions

There have been great advances in aircraft design since the early days of aviation. But there is still a lack of educational material on how to actually draft and mathematically represent the aircraft’s shape in the design process. The bulk of the information seems to be in the possession of the industries that manufacture these machines and is proprietary. This is not to say that there are no good books or materials out there, but they are hard to find. What confounds this dilemma even more is the total dependence on computer CAD programs for modern aircraft layout. CAD is a very powerful tool for aircraft design and layout¹¹ only when the underlying geometry and techniques are understood. It is hoped that this basic introduction to lofting with conics will prompt potential aircraft design engineers to explore more avenues in developing their design technique.

References

¹Tierney, John, "The Real Stuff," Science85, Volume 6, No. 7, pp 24-35, September, 1985. [Return]

²Raymer, Daniel P., Aircraft Design: A Conceptual Approach, page 126, American Institute of Aeronautics and Astronautics Education Series, AIAA, Washington DC, 1992. [Return]

³Liming, A. Roy, Practical Analytic Geometry with Applications to Aircraft, The Macmillan Company, 1944. [Return]

⁴Raymer, Daniel P., Aircraft Design: A Conceptual Approach, page 127. [Return]

⁵Hollmann, Eric and Martin Hollmann, Modern Aircraft Drafting: How to make aircraft drawings using a plotter and personal computer, Aircraft Designs, INC., 25380 Boots Road, Monterey, CA, 1992. [Return]

⁶Hollmann, Martin, Advanced Aircraft Design: Designing Aircraft using a 386-PC or a Macintosh, Aircraft Designs, Inc., 1993. [Return]

⁷Banks, Doug and Kelly Kernan in Hollmann, Eric and Martin Hollmann Modern Aircraft Drafting, pp 87-94. [Return]

⁸Head, George, Charles Pietra, Kenneth Segal, The AutoCAD 3D Book, Ventana Press, 1989. [Return]

⁹Rogers, David and Adams Alan, Computer Graphics, McGraw-Hill Book Company, 1976. [Return]

¹⁰Foley, James, Andries van Dam, Steven Feiner, John Hughes, Computer Graphics: Principles and Practice Addison-Wesley Publishing Company, 1990. [Return]

¹¹Conversations with Gordan Gibson, Lockheed Tactical Aircraft Systems. [Return]