Part 3: Downside-up

Well, it's been a while since the last update. I spent a lot of time working on a fancy formula to generate a perfect ride path which, while very interesting to myself, doesn't really have a good bearing on the class. With that in mind, I've put the ridepath generator on hold and moved back to the coaster at hand. When we last left off I had built a formula to generate airtime hills. That's done and finished, so we can move on to the third element in our ride, the vertical loop. Loops are one of the easiest inversions, and one of the most popular elements. They provide a great wow factor and are relatively simple to work with. First, some history. Loops appeared as early as the 1800s. These loops, however, weren't exactly fun to ride on. For starters coasters didn't have wheel assemblies as we see them today, and were therefore held on the track solely by the normal force experienced by traveling along the path. Therefore, if the train were to go slower than expected (due to some mechanical failure or some weather condition) it would fall off of the track. This, of course, is very bad. An additional problem with early loops was the shape of the loop itself. Here's one of the early loops:

For simplicity, it's a nearly circular shape. Let's do some analysis to see why this doesn't really work. Starting at 20.83 m/s (the speed we left off at last time), we want a loop that generates 4 vertical g forces at the bottom to match the pull-up from the airtime hill. Calculation of the radius of loop required to generate this force is pretty trivial. We'll also see what the situation at the top is.

Well, we've got a big problem. To have rideable forces at the bottom of our circular loop, the coaster stalls before it can complete the loop. If we keep positive forces at the top of the loop, we see the force at the bottom raises to unsafe levels. The differences between N1 and N2 are too high in every case! So, what do we do now? Well, we wait until 1976 when Werner Stengel and Anton Schwarzkopf, both Germans, made the jump to modern design. Both men are largely credited with making modern loops work, and also pioneered the heart-lining concept which makes complex modern maneuvers possible as explained in part 1. Stengel still works in the coaster field today, completing almost every facet of design from ride path generation to structural analysis for the majority of rides built today (seriously, almost every new ride comes through his company). Anyway, the first real modern vertical loop was Revolution at Six Flags Magic Mountain. The only real change was using a radius of curvature that decreased as the cars turned through the loop (in a clothoid or euler spiral configuration). This allowed the force at the top of the loop to be much closer to that of the bottom, and for a much better overall experience. Here's the loop that started modern inversions:

Modern loops can have all sorts of shapes and sensations. There were loops by Arrow Dynamics, whose sharp teardrop shape created heavy positive gforce throughout. Here's an example of this, again at Six Flags Magic Mountain. Note the extreme change in shape from the circular loop.

Other types of loop are more passive, and give riders a small sense of floating over the crest. This can be desirable when a designer wants to slow down the pacing, and gives riders more time to notice that they are, in fact, very upside-down and very high in the air. Here's an example at Cedar Point. See how it curves less aggressively over the crest.

To suit the style of our ride, I want to make a loop with heavy positives at the bottom leading to about 1 g at the top. It's hard to show the dynamics of this problem due to the constantly changing radius, so I'll just highlight the method I used and the final result. To make our loop I utilized an alternate version of my airtime generator from the last update. I set a switch based on the value of the ascent angle. For the first and last quarters of the loop (0 < Theta < pi/2 and 3*pi/2 < Theta < 2*pi) I set the vertical g force at 4. For the top half of the loop, I set the gforce to vary as a sin function from 4 to 1 and back (Gforce=1+3*abs[sin[theta]]). Then I forced a constant offset to the side so the track does not cross through itself. This isn't really how loops are designed in the field, but I found it a fun use of my formulas and the results were very solid. This formula gave an intense, fast loop but still allowed for a little time to breathe. Here's a picture of the final loop, with a little preview of the next part of the layout.

That's it for today, tomorrow I'll get Part 4 (corkscrews and rolls) and Part 5(The little things) done tomorrow. I'll also have some video of the ride sometime over the weekend. To conclude, Here's a pic showing how silly a circular loop looks on a modern ride.