Coaster Dynamics: Physics Primer Chapter 4



Coaster Dynamics
Physics Primer

Chapter 4

Model of a Roller Coaster Car

System Models and Analytical Idealizations

The objective of physics is to understand, and to develop mathematical models of the world around us. Although this may seem like a natural thing to do, the process of modeling requires a skillful combination of science and art. To begin, for example, what constitutes a "system model"?

The term "system" might seem like an intuitive concept, yet it is often very difficult to explicitly define what it is. For instance, a "system" might be a simple ball rolling down a hill... or perhaps an airplane in flight. In the case of the ball, it might be pretty easy to identity the system. But, for the airplane, what do you include in the system model? Do you include the wings... the landing gear... the jet engine? What about the exhaust gases expelled from the jet engines... or the type of paint used on the fuselage? The art in systems modeling is to be able to clearly identify and include only the important features of the system needed to make your analysis useful. Otherwise, the model may become so complex so as to render it useless.

The conceptual models we develop lead to mathematical equations which describe the system. These equations, then, comprise the mathematical model of our system.

In the case of Newtonian Mechanics, a system is a loosely defined thing which includes an object, or set of objects, and any associated objects or phenomenon which exert forces on the object(s). From the model, we can derive equations of motion which will predict the behavior of the system. However, even for relatively simple systems such as these, analytical idealizations are required to make calculations amenable.

Idealized Roller Coaster Car Model

A roller coaster is a complex dynamic system. Thus, to allow useful analysis, a number of idealizations are made in our system model.

The first simplification we make is that we model only a single roller coaster car... not an entire train of cars, as is the case in most real roller coasters. The only exception to this assumption appears in Chapter 12, for the case of analyzing the motion of a roller coaster train as it goes over the top of a hill

The next assumption we make is that the modeled roller coaster is a traditional single lift hill coaster. That is, the roller coaster car is pulled up the lift hill, and then dropped from there. No other mechanisms are employed to alter the motion of the car. The only forces acting on the car are gravity, friction, and air resistance.

A simple model of a roller coaster car is shown in Figure 4-A, which includes the car body, its wheels, the track, and the forces exerted on the car. Although highly simplified already, this model is still too detailed to analyze with reasonable complexity.

Figure 4-A. Simple Model of a Roller Coaster Car.

The following are additional assumptions and idealizations used in our model of a roller coaster car.

The fully idealized model of a roller coaster car is illustrated in Figure 4-B.

Figure 4-B. Idealized Model of a Roller Coaster Car.

Lumped Mass:   A real roller coaster car consists of a large assembly of subcomponents (seats, axles, wheels, etc.). However, it can be shown that the assembly of objects can be modeled (for the purpose of motion analysis) as a single "lumped mass" object which has the same total mass concentrated at the theoretical center of gravity of the assembly. The concept of the center of gravity will be further discussed in Chapter 12.

Single Resultant Force:   As illustrated in Figure 4-A, there are different forces acting on the roller coaster car -- gravity acting at the center of gravity, the upward force on each wheel, friction forces acting on each wheel, and air resistance acting on the front area of the car. To simplify the analysis, we neglect the different geometry of the forces, and all forces are assumed to act at the theoretical center of gravity. Furthermore, the forces can be conceptually combined into a single resultant force, which is assumed to act entirely at the center of gravity.

Neglect Rotational Dynamics:   Rotational dynamics is the study of the motion of rotating objects. In our roller coaster car model, we neglect all rotational motion -- for example the twisting of the track through a corkscrew loop (that is, the twisting that occurs along the path of the track, as opposed to the large-scale circular geometry of the track perpendicular to the primary axis of travel). In reality, the twisting motion effects the behavior of the car. However, for the speeds and geometries of a typical roller coaster track, the rotational affects are very small. Thus, we neglect them entirely in all of our analyses.

Neglect Track Banking:   In a real roller coaster, the tracks are banked. That is, the track is tilted in a curve with the wheels along the outer radius of the curve higher than the wheels along the inner radius. Banking creates additional forces on the car passengers that help keep them from being thrown out. Although banking is used in every roller coaster track (and on every freeway), the analysis of the banking forces is rather complex. Thus, for our model of a roller coaster car, we choose to neglect banking. This does not significantly change the validity our analyses.

Wheel Friction and Air Resistance:   In a real roller coaster, and in the computer calculations performed by Coaster Dynamics, friction and air resistance are vitally important factors (otherwise, the car would roll forever!). When analyzing the behavior of a roller coaster car over the entire length of the track, friction and air resistance must be included. However, if analyzing the motion during a small portion of the track (for instance, calculating the car's velocity in a loop), friction and air resistance can be judiciously ignored, without affecting any conclusions of our analyses.

Table of Contents   Return to Table of Contents


Copyright © 2001, Cyclone Software, Pleasanton, CA, USA. All rights reserved.