2011年_美国大学生数学建模竞赛_论文.pdf

Getting Some Big Air
Control #10499
February 14, 2011
Abstract
In this paper we address the problem of optimizing a ramp for a snow-
boarder to travel. Our approach is two sided. We first address the “for-
ward problem” of modeling the motion of a snowboarder on a ramp as-
suming a given ramp and initial conditions for the snowboarder. We
derive a second order ODE which we solve numerically. The second as-
pect of our approach, and arguably the more interesting and useful, is the
“variational problem”, . finding the optimal ramp for a snowboarder to
travel on, given initial conditions for the snowboarder and a given jump-
ing strategy for the snowboarder. To do this, we consider the space of all
possible halfpipe curves and maximize final angular velocity by applying
an adapted version of the Euler-Lagrange equations in several variables.
We use use Taylor series approximations to get a differential equation that
is numerically tractable. The solution to this differential equation traces
out a curve which represents an approximation to a candidate solution for
a local extremum to the vertical velocity of a snowboarder on the halfpipe.
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Contents
1 The Forward Problem 3
Statement of the Problem . . . . . . . . . . . . . . . . . . . . . . 3
Outline of Solution of the Forward Problem . . . . . . . . . . . . 3
Derivation of the ODE . . . . . . . . . . . . . . . . . . . . . . . . 4
Approximating the Radius of Curvature . . . . . . . . . . . . . . 6
Deriving the ODE, continued. . . . . . . . . . . . . . . . . . . . . 6
2 The Variational Problem 7
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . 8
The Euler-Lagrange Equations . . . . . . . . . . . . . . . . . . . 8
Approximating the Euler-Lagrange Equations . . . . . . . . . . . 9
3 Results 10
References 11
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