The Design - A study of circular arcs.
Look at a car. Almost any car after 1929 will do, but especially a
curvy car such as the Buick Y-Job of 1938. Many times one will hear a person on the street talk about the "lines" of a certain car and how those lines are superior to all the others.
What are those "lines" of which they speak? Can they be classified? Does their classification matter? Well, I suppose that depends on who you are. If you're the consumer, you want something that looks good. If you're the machinist in charge of making the stamping dies for that car, you want something that is easy to make. If you're the designer of that car, you need to think of both the machinists and the consumers. An impossible to make car will kill your company just as quickly as a car that everyone agrees is ugly and refuses to buy.
Enter the 1920's!
In May of 1927, Alfred Sloan, Chairman of the Board of General Motors, disturbed by the ungainly appearance of the current crop of automobiles decided GM needed a central design division to handle the "artistic" side of building automobiles. Mr. Sloan was successful, hiring Harley Earl to head up the newly formed Art and Color Section of GM.
By establishing the Art and Color Section, GM effectively placed the "looks" of an automobile entirely in the hands of an artistic group, rather than leaving the finished shape entirely up to the whims of the mechanical engineers. The Art and Color Section was one of, if not the first true industrial design organization. Many other car companies and consumer products manufacturers realized they also needed an Art and Color division of their own, so they followed GM's lead.
Since "Industrial Design" was a new idea in the 20's, putting practical artists in control of the shape and functionality of everything from toasters to trains to cars, there was no such thing as an "Industrial Designer" and no set tools of the trade. The pioneers of the field at GM used anything they could get their hands on that would do the job. Somewhere early on in the history of the Art and Color Section, somebody brought a set of railroad curves to work at GM and a worldwide standard was born.
Looking at that curvy car again, you'll notice many different curves present, which make up the form of the car body. Sections of "French" or "Copenhagen" curves might be spiral, elliptic or hyperbolic in form. These curves are easy to trace with a pencil, but exceedingly difficult to machine numerically with manual equipment because they are not circular and thus can only be approximated by many circular arcs with many varying centers.
Now, look at the vast spaces, such as the top of the hood, or maybe the side of the doors. What is their profile? The GM Art and Color Section standardized their profile back in the 20's with an even older drafting tool - the Railroad Curve. Simply, the Railroad Curve, or "Sweep" as it is known in the design world is nothing more than a section of a circle, a circular arc! Railroad engineers use circular arcs to lay out railroad curves because a train following the curve is under a constant lateral acceleration which is easier on passengers and freight.
Circles and circular arcs are also easy to machine with a rotary table, because they have only one center point and only one radius. Parabolas and ellipses can only be
approximated by circular arcs, because their radius of curvature continuously accelerates.
If you've followed this far and not fallen asleep, you've discovered that circular arcs are not only a common feature of "artistically designed" commercial products, but they're also easy to machine numerically on manual machinery. So, how does one go about designing a radius attachment using circular arcs?
Looking back at the Holdridge design you'll notice that the largest swing frame is half a circle, as is the main frame of the attachment. But look the smaller swing frames for making spheres of smaller diameters are NOT circles! The radius of action of the swing frames becomes smaller, to accommodate machining of smaller spheres, but do not become shorter. At first, one might consider an ellipse, since an ellipse is a perspective view of a circle and easy enough to draw with ellipse templates. However, let's not forget I actually need to MAKE this thing. An ellipse is sheer torture to machine "Etch-A-Sketch" style numerically with manual equipment!
How about a oval? All ellipses are ovals, but not all ovals are ellipses. That is to say, mathematically speaking, there is only ONE way to draw the perspective view of a circle, an ellipse, but there are many ways to draw a "squashed circle". An egg is a "squashed circle", thus it is an oval. But an egg is not an ellipse, because it is not a perspective view of a circle. The beauty of an oval is it can be easily constructed from two circular arcs!
How does one go about calculating the radii of the arcs necessary to construct an oval of certain proportions? For that, I reference Machinery, Volume 23, August 1917, Page 1050. This is an article on machining of oval gears. It is interesting to note that the author incorrectly calls these elliptical gears, which they are not! Because.... An ellipse is the perspective view of a circle and an oval of only two arc radii merely approximates an ellipse.
Looking at the illustration we can determine how to calculate the radii and construct the oval based upon the major and minor axes of each oval. Note that the formula for the end radius converges to 0 at an aspect ratio of 3.0, limiting the formula to ovals with an aspect ratio of less than 3.0.
With the radii calculated, over at the drafting table the endpoints of each oval are sketched in. Then, the end radius of an oval is swung in:
Then, the large radius can be swung in with a trammel:
Any minor discrepancy in the transition point between the two arcs is solved with a french curve:
Finally, the construction lines are erased:
The completed oval, one of many that must be calculated and drawn by hand:
Here's the concept drawing after numerous revisions. The entire attachment is a study in circular arcs, which are easy to machine and easy on the eyes. The lightening holes actually serve a purpose. To center each part on the rotary table for various operations on the I-beam section I need locating points accessible from both sides of the part. Two of the lightening holes on each part will be bored for true location, providing the reference points needed to bring the part parallel to the machine's axis.