Team 9582: A half-blood half-pipe, a perfect performance - Gong, E.; Wang, X.; Li, R. (UMAPJournal, 109, 2011)
Our basic model can be divided into two parts. The first part is to find a half-pipe shape which can maximize "Vertical Air"; in the second part, we adapt the shape to maximize the possible total angle of rotation. In the extended model, we analyze the snowboarder's subjective influence on "Vertical Air" and the total degree of rotation. Finally, we discuss the feasibility and the trade-off of building a practical course. The major assumption is that the resistance includes the friction of snow and air drag. The former is proportional to the normal force and the latter is proportional to the square of velocity. Plus, the Snowboarder is perpendicular to the tangential surface of the course at his locus during his movement on the half-pipe. The variables of our model include: coordinate variables x, y, z, normal force IM, resistance f, snowboarder's mass m, mechanical energy lost due to friction of the snow and air drag E|0St, vertical air Hf, and the initial angular momentum of the fly L0. In the first part of our basic model, we obtain a differential equation of E|0St based on force analysis and Energy Conservation. Then we derive the representation of E|0St by solving the equation, which is a functional. Thus we calculate the representation of "Vertical Air" by analyzing the projectile motion. In the second part of the basic model, we derive the representation of the initial angular momentum before the fly, and discuss the factors influencing it. In our extended model, we calculate the "Vertical Air" when taking the snowboarder's subjective influence into account. We uso numerical mothod to solve the model and compare analytical results wlth reality and validate our method to be correct and robust. We analyzed the cffects of fjctors such as wldth, hoight and gradient angle of half-pipe on "vertical Air" by controlling other factors, and found that wider, steeper course with proper depth and the best path chosen by a skilled Snowboarder are preferred to obtain locally larger "vertical air". Using Gene Algorithm, we globally optimlzed the course shape to provide largest "vertical air" and total degree of rotation. There is a trade-off between maximum "vertical air" and maximum rotation. Implementing a hybrid scoring system as the objective function, we optimize the course shape to a "half blood" shape that would provide the eclectically best snowboard performance. Furthermore, we take practical problems into consideration and provide answers to more questions such as what roles the flat bottom and "vert" play in reality. We finely model the entire complicated proceeding and give numerical solution of optimal course, and establish an objective evaluating function to analyze local and global optical course. Diverse subjective influence of Snowboarders is also considered. Yet, the model hasn't provided analytic solution of optimal course.