Introduction

Sometimes, it is necessary to transition from one airframe diameter to another somewhere in the middle of the body of a rocket, or near or even at the aft end of the rocket. There are hundreds of examples of transitions, like a Nike-Apache or a D-Region Tomahawk. Often times, you need a structure underneath the transition to support other airframe elements, fins, or motor mounts, etc. But the hardest thing to design is the transition unless you have the right tools.

Figuring out a transition is actually fairly easy with a little bit of math. With a look at the following drawing, there are three things that you need to know about your transition before you can do that math. You need the small diameter (or radius), the large diameter, and the length of the transition. From these, you can do all of the calculations that you need to do in order to come up with a flat pattern for cutting your transition from a flat sheet of material, be it wood, paper, or almost any other material.

Let's Do The Math

Understand that the desired shape is actually a truncated cone. First, let's rearrange some of these letters, and add the others that we need to solve this problem. We have:

• a - Diameter of the base of the cone.
• b - Diameter of the truncated top of the cone.
• c - Height of the truncated cone.

Now, imagine we extend the shape of this truncated cone to a point so we can add the rest of the dimensions that we will need to do the rest of the math.

• d - Distance between a and b along the surface of the cone.
• e - Distance between b and the point of the cone.
• f - Height of the extension of the cone from b to the point.

We've identified all of the dimensions that we need, so let's get to work!

Let's use an example as we work through this, so that we can understand what's going on. In this example, we'll set a to 4", b to 3.5", and c to 5". First, we need to find the radius of both a and b.

A cone is the formed revolution of a right-triangle around one of it's right-axes. In order to calculate how to flatten this pattern, we need to actually calculate for two triangles. One triangle is formed by the radius of a, the line c+f, and the line e+d. The other tirangle is formed by the radius of b, the line f, and the line e. We need to find the length of c and c+f, so we first need to find the length of f.

Now solve for e and e + d using the pythagorean theorum and find d:

Now, in order for this transition to be laid flat, we must draw two concentric circles with one radius for the base, and one radius for the truncation. Starting with a vertical line from the center of the cicles to the bottom quadrant, and another at the appropriate angle, the included arcs will match the circumference of a and b, a length d apart. We have the radius e for the inner circle, which will form the circumference for the top of the truncated cone, and the radius e+d which will form the circumference for the base of the cone.

We need to figure out the angle g to complete the flat pattern. First, let's determine the circumference for the base i and the truncated tip of the cone h.

Now, to determine the included angle g that will give us the appropriate circumference for our base and cap. To calculate the angle, we'll convert the circumference to an arc-length matching the radius we drew, and then convert radians from that into an angle.

And finally, convert radians to an angle

Wow! I don't know how you feel about math, but that's a lot of work.

Submitted by Tim Scott

And now, the easy way

This Excel spreadsheet is a calculator for paper transition sections.

Transition Calculator (Excel Spreadsheet)

Submitted by Jim Malone