Part 1: Banked Curves

Hi 274 students! Upon looking at the homework assigned for this weekend, I could not help but apply the second problem to my dearest hobby: rollercoaster simulation. Rollercoasters, as you may well know, are mechanical devices designed to go very fast and move you from where you are now to that exact spot a few minutes in the future. They serve absolutely no purpose whatsoever and have no practical applications. However they are a perfect mix of engineering and art, and I have spent countless hours creating and tweaking ride paths to push the envelope of ridiculousness. The program I use is called “Nolimits,” and it is the most popular simulation software short of the rollercoaster tycoon series (which we will not be talking about today). A demo is available on its website. It allows manipulation of the ride path through placement of nodes (using a 3 point bezier system (not important)), and has a simulator mode where g-force is calculated in real-time and everything is rendered all shiny and pretty. Today, I will make for you a small section of a rollercoaster attempting to apply our coursework along the way. This will be in three parts, ‘banked curves’, ‘airtime hills’, and ‘everything else’.

Let’s begin! Today I’ll be using a track and train type from Intamin, the good Swiss people responsible for such rides as Millennium Force and Top Thrill Dragster. I’ll use a simple 3 car train. Before we begin our curve, we need a station and something to make us go. Lift hills and launches are both options to get up to speed, but for control purposes we’ll have a flat launch limited to 80km/hr (about 50mi/hr and 22.22 m/s). The launch is set to have an acceleration of 1.6 gforces (1.6*9.81=15.696m/s^2), a pretty strong push. The length of launch track required can be calculated from our coursework, but to save time I’ll skip the simple calculation and move onto the main part of this post. Here we have our humble beginnings:

Before we move on, I'll give a brief discussion of the options available in launches and lift hills. First, there's the traditional chain lift. This engages a catch on the bottom of the train and imparts potential energy using an electric motor. The clicking noise heard when climbing a lift hill is due to a safety feature known as the anti-rollback dog. This device locks into a step arrangement mounted on teh lift hill and, in case of chain failure, will hold the entire train in place. Here's a better description of the process if you are interested. There's also a more modern version of the lifthill, using a cable and a catch-car which connects to the train, pulling it to the top with a strong motor mounted on the ground. Here's a view of the crest of Millennium Force's cable lift hill and you can see the catch car (the long gray thing in the middle of the track).

This system allows for a faster and steeper lift, which is also quiet (the silence is because the anti-rollback devices are electromagnetically disengaged by the passing train and automatically close after it passes, a really cool technology).
Launch systems are also a solid option. Many options exist, including electromagnets, pneumatics, hydraulics, flywheels. Electromagnetic propulsion involves using strong electrical impulses to attract magnetic fins attached to the trains. This type of system is fairly popular as it offers very precise control of speed and quite powerful acceleration. Pneumatic and hydraulic systems try to compress a gas using either mechanical work or hydraulic pressure (respectively), then release this pressure by spinning a drum attached to a launch cable (and subsequently the train). These systems are typically less reliable, but achieve the most powerful results (128 miles per hour plus). The final type of launch involves a strong engine spinning a wheel which is quickly engaged through a clutch system to pull the car by use of the same cable system as before. This system is the cheapest and is used on most of the slower launches.

Now that we're up to speed, let's begin our ride. First, we'll add our curve. Flat curves, although simple, can be extremely powerful elements if used properly. A high-speed, high-force flat curve just above the ground can be a tremendously exciting element, and one of the cheapest. First up, you may ask why a rollercoaster would have banked curves in the first place. Most chiefly, this is done because riders absorb vertical g-forces much more kindly than lateral ones. It feels much more natural to be compressed than to be forced to the side. Although most wooden coasters thrive on such sensations, their steel brethren push, as much as possible, to eliminate these forces. Additionally lateral forces put strain on weird places in the wheel assemblies, another bad point. So, as in problem 3-III, we seek to completely eliminate lateral g-forces from a flat turn. We’ll look at the banked track halfway through the curve, where the bank will be constant (at the sides of the curve there will be transitions from and to 0 degrees of bank). The radius of the curve we want will be 13.5 meters, small enough for some real force. The calculations become very easy as speed is known and we’ll ignore thrust and friction. What angle should we bank the turn so riders are squashed into their seats, not tossed painfully to the side? It’s pretty simple:

So we need a 75 degree bank. Well, this shan’t be too hard. Using an element generator I can make a flat curve of 13.5 meters. I’ll bank it to 75 degrees as calculated and simulate the result.

You can see for yourself that, although the car has slowed to 79 km/hr, we have eliminated almost all lateral g-force. Additionally, we have +3.6 gs vertically, a forceful but certainly safe number. Let’s see for a second what happens when we lower the banking to a measly 50 degrees.

Yikes, 1.4 lateral gs! Such a force would probably injure riders, as the restraints are not set up to handle that type of force. Well, this is why we do our calculations ahead of time. You may be wondering about the lateral g-force at the entrance to the turn. Right now, there is a dangerous spike as there is no transition between straight and unbanked to curved and banked. To help smooth this out, modern coasters are designed using what is called a ‘heartline.’ This refers to looking at the center of the rider’s path (through the ‘heart’ of the passengers), and rotating the ride path around this point. Here is a visualization (credit) :

To achieve this, I need to add a small section of straight track to serve as the transition. Here’s a view of the track from above with the heartline highlighted in yellow to further illustrate the point:

See how the heartline remains straight while the track banks in preparation for the turn? This creates a smooth, painless transition that is the basis for most of the complex elements in coasters today. Let’s see that in the simulator:

It looks weird, but works very well. All modern coasters begin turns this way. Here’s a picture of a real-world example of this, a transition on Hershey Park’s Storm Runner.

Alright that’s it for part one. Next time we’ll go vertical. If you enjoyed this or couldn’t understand anything, please leave some feedback so I can make part 2 better. Also you can vote for the color of the ride if you wish. Please make all comments on the 274 course blog.