Load Analysis of a Vertical Corkscrew Roller Coaster Track.
Problem Statement The situation analyzed here in involves investigation of the loads experienced by a roller coaster passenger as her or she travels through a vertical helix. Complete and accurate determination of these values is absolutely critical to ensure the safety of the occupant. The challenge here is to design a "thrilling ride" without reducing the track to rubble or the occupant to a pile of goo. To achieve this, the maximum safe acceleration, based on physical limitations of the human body, should not be reached. Given the maximum velocity of the car along the track and the helix geometry (height, radius, etc.), pitch of the helix can be altered to safely maximize the acceleration of the occupant. The specific case observed here is that of a vertical helix track with constant pitch.
Why does this interest me?
This problem is interesting because it exemplifies how engineering applies to a broad range of activities including some, such as this one, that most people would never suspect. The real attraction here stems from the child within. After all, amusement parks are all about amusement, and every amusement park has a roller coaster. Most people enjoy a thrilling roller coaster ride, Six Flag’s annual attendance is a testament to this. Peaking interest even further is the hidden complexity of what appears to be a very simple system of ramps, turns and ….corkscrews. Finally, there is some satisfaction in unraveling the boring engineer paradigm by engineering excitement!
Analysis Hold on. This ride can get bumpy! Before this situation can be analyzed, several assumptions must be made. This is necessary to counteract some of the inherent complexity of the task, yet still produces a relatively reliable result. First is the assumption that the car and passenger can be modeled by a particle moving along the track. Second is the assumption that there is no friction between the car and the track. This goes hand in hand with the first assumption. Some other ground rules include ; The velocity above is not uncommon in today’s roller coaster. The maximum acceleration was chosen to minimize the hazards of a high G environment; unconsciesness and death. The human threshold is around +7.5 G’s, for a trained pilot in a "G – suit." If any further developments should arise, the analysis may be easily altered to reflect the necessary change.A helix of constant radius, r Constant velocity along the track, Vc = 55 mile/hour Acceleration is not to exceed 4 G’s
The analysis consists primarily of the search for an appropriate helix pitch with respect to the required velocity, acceleration and helix diameter. In this case, pitch can be described as the change in elevation per full revolution of the car about the z – axis and algebraically represented by the equation;To further simplify analysis, the velocity and acceleration will be represented using cylindrical coordinates. The velocity and acceleration equations are therefore; However, for this problem, the radius is constant. This means that the first and second derivatives of the radius are zero. Additionally, our ground rules state that the velocity is constant, so the second derivatives of theta and z are zero. The above equations reduce to; Notice that the car is only accelerated in the radial direction. Next, we know that the maximum acceleration cannot exceed 4 G’s. Therefore, ac must, at most, equal 4. This yields; This quantity can now be substituted back into the reduced velocity equation to determine the first derivative of z. Finally, the pitch can be calculated using the first derivatives of theta and z. Conclusion
The presentation here serves two purposes; it allows us to examine and safely design a section of roller coaster track, and it provides some insight into how interesting engineering can be. A complete description of the loads experienced by a roller coaster passenger is presented within a proposed design approach. This allows some insight into the practical usage of these equations, and provides some insight into the complexity involved in roller coaster design. Additionally, applications of this procedure are not as limited as one might think. What has been described here is simply particle motion along a simple helix. As a result, this procedure may be applied to countless applications with admittedly far reaching effects.
References and Related Links An Introduction to Dynamics, McGill and King
