Roller Coaster Basics

A roller coaster has been described as ``basically an ornate means of falling''[1]. It is a thrill ride found at many amusement parks across the country and around the world. A roller coaster, as presented in this research, consists of a rigid track and a car that is constrained to move on that track. The track is typically a closed circuit, such that the it begins and ends at the same location, named the station.

Many different types of roller coasters exist today, varying in both their structural materials and in the types and orientations of the cars on the tracks. The research presented here lends itself most readily to a standard steel coaster, in which the car sits on top of the track. A wooden coaster could also be modeled with the elements presented here, except that the inversions (the loop and corkscrew elements) would not be applicable.^1.1 Newer types of coasters are the suspended and inverted coasters. Both of these are made of steel, and have the cars hanging underneath the track. The difference between the two is that suspended coasters allow the cars to sway from side to side, and do not have inversions; while the rigidity of the inverted coaster cars make inversions possible. The swaying of the suspended coaster is not modeled by the dynamics presented in this research, and the geometric elements specific to inverted coasters are not included here. It was not believed that this research could adequately handle these variations, so the focus was kept on standard steel coasters.

Some important terms that are used to describe roller coasters are:

Car

The vehicle passengers ride in as they traverse the track. A typical car has two rows seating two passengers each, although many variations exist.

Train

A group of cars connected together. Rotation occurs between the cars, allowing them to maneuver through curved sections of track. Special considerations must be given to modeling a train, as opposed to modeling a single car[4]; however, these are not discussed here.

Track

The track that will be created in this research consists only of two rails, on which the wheels of the car roll. In reality, structural elements would be needed to support the rails, but these are not dealt with here. A track will be defined by an ordered set of geometric elements, along with the parameters required to completely specify those elements.

Gage

The separation of the rails, denoted as $W$. The value of the track gage is typically about 3 feet[5].

CG height

The distance between the rails and the center of gravity of the car, $h_{cg}$. This remains a constant distance throughout the track.

Banking

The banking of the track refers to the tilt angle, $\psi$. It is how much the car is rotated about its forward vector in order to alleviate forces pushing the passengers into the sides of the car. See figure 1.1. As a car goes through a curve, the track will be banked such that the would-be lateral forces are instead directed toward the bottom of the car.

Figure 1.1: Banking Illustration.
This figure shows the tilt angle, $\psi$, of a car navigating a banked turn. The $\ensuremath \mathbf{\widehat{k}}$ vector points in the global up direction, away from the earth. The $\ensuremath \mathbf{\widehat{u}}$ vector points in the local up direction, and is car-fixed. It is what the passenger may perceive as up as the track is navigated.

The seat force^1.2 accelerations described on roller coasters are often expressed in terms of ``g's'', with 1 g being the acceleration of gravity ( $9.81 \frac{m}{s^{2}}$ or $32.2 \frac{ft}{s^{2}}$). Curtis Summers, a designer of amusement parks and roller coasters for nearly four decades[6], designed for g's on roller coasters to be no less than -0.2, and no greater than 3.5[5]. As a comparison, jet pilots may black out around 8 g's[7]. Some designers try to keep g's always positive, as they ``don't want anything that lifts passengers away from the seat''[7]. However, it is not uncommon for small negative g's to be encountered on roller coasters[8].

There are three sets of wheels on a roller coaster car, oriented differently to control the coaster in specific ways[5]. See figure 1.2. Not all sets of wheels will be in contact with the track at any one time.

Load Wheels

These are the wheels the coaster primarily rides on.

Guide Wheels

These wheels are placed on the inside of the track, and guide the car laterally along the track. There is some clearance between these wheels and the rails such that they are in contact only when the azimuth angle of the track is changing.

Upstop Wheels

These wheels are located on the underside of the track, and are used to prevent the car from disengaging from the track. They are not engaged when the seat force remains positive, and as such are added primarily as a safety measure.

Figure 1.2: Roller Coaster Wheels.
The three distinct sets of wheels are labeled here. The load wheels are the primary wheels the coaster rides on. The guide wheels help the car navigate turns, controlling its lateral motion. The upstop wheels are added as a safety measure to prevent the car from disengaging from the track.

For more introductory information about roller coasters and their dynamics, a guide has been published, intended for the general coaster enthusiast[9,10,4].

^1.1 Early attempts at looping wooden coasters were largely unsuccessful[2,3].
^1.2 Forces pushing the rider into or out of their seat. Explained in section 3.1.