数学建模-Snowboard-Course

美赛2011年题目

Problem A: Snowboard Course

Determine the shape of a snowboard course (currently known as a “half-pipe”) to maximize the production of “vertical air” by a skilled snowboarder.

设计一个滑雪道的形状(当前已知是半个管道)来最大化一个熟练地滑雪运动员的vertical air。

“Vertical air” is the maximum vertical distance above the edge of the half-pipe.

vertical air指的是在半圆管道之上最大的垂直距离。

Tailor the shape to optimize other possible requirement, such as maximum twist in the air.

设计一个合适的形状去尽可能优化所需条件，如最大化在空中的扭曲动作。

What tradeoffs may be required to develop a “practical” course?

开发这样一个实际的滑雪道需要怎样的权衡？

A Half-Blood Half-Pipe, A perfect Performance

一个半Blood半管道，一个完美的表现

Abstract

摘要

Our basic model has two parts: to find a half-pipe shape that can maximize vertical air, and to adapt the shape to maximize the possible total angle of rotation.

我们的基础模型有两部分: 寻找一个half-pipe形状，这种形状能够最大化运动员的vertical air，并且运动员能够适应这个形状能在空中产生最大程度的翻滚角。

In an extended model, we analyze the snowboarder’s effect on vertical air and on rotation.

在扩展模型中，我们分析了滑雪运动员在vertical air 和旋转角度的效果。

Finally, we discuss the feasibility and the tradeoffs of building a practical course.

最后我们进行了可行性分析并且权衡了构建一个实际的滑雪道。

The major assumption is that resistance includes the friction of snow plus air drag, with the former proportional to the normal force. We find air drag negligible.

主要的假设是雪和空气的摩擦阻力

我们发现空气阻力可以忽略不计。

We first obtain and solve a differential equation for energy lost to friction and drag based on force analysis and energy conservation.

我们首先基于受力分析与能量守恒解决并得到了几个不同的能量损失方程。

We calculate vertical air by analyzing projectile motion. We then calculate the angular momentum before the flight and discuss factors influencing it.

我们通过抛体运动计算了vertical air，然后又计算了飞起前的角动量及其影响因素。

In an extended model, we take the snowboarder’s influence into account.

在扩展模型中，我们把运动员的因素也考虑进去了。

We compare analytical and numerical results with reality, using default parameters; we validate that out method is correct and robust.

我们使用默认参数比较了解析解及数值解与实际的出入; 我们验证了我们的方法是正确且稳定的。

We analyze the effects on vertical air of width, height, and gradient angle of the half-pipe. We find that a wider, steeper course with proper depth and the path of a skilled snowboarder are best for vertical air.

我们分析了half-pipe的宽度，高度及梯度对vertical air的影响。我们发现了越宽，越陡峭的滑雪道，在适当的深度和运动员的滑雪路线下可以使得vertical air最大。

Using a genetic algorithm, we globally optimize the course shape to provide either the greatest vertical air or maximal potential rotation; there is a tradeoff.

使用遗传算法，我们全局地优化了滑雪道的形状，解决了最大vertical air或最大的旋转势能; 注意这两个优化目标只能两者选一个，故需要根据实际情况折中考虑。

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.

实现了混合计分系统作为目标函数，我们优化了滑雪道的形状为一个"half-blood"，将提供一个折中的最好的滑雪成绩。

Background

背景介绍

A half-pipe is the venue for extreme sports such as snowboarding and skateboarding. It usually consists of two concave ramps (including a transition and a vert), topped by copings and decks, facing each other across a transition as shown in Figure 1.

half-pipe

是一种极限运动场所，就像滑雪运动与滑板运动那样。它通常由两个凹面的斜坡所组成(包含在平移段和垂直段之间)。。。。如图一所示

Figure 1. End-on schematic view of a half-pipe. (Source: Wikimedia Commons; created by Dennis Dowling.)

图1. half-pipe的端面示意图。（资料来源：Wikimedia Commons;由Dennis Dowling创建）

Half-pipe snowboarding has been a part of the Winter Olympics since 2002; the riders take two runs, performing tricks such as straight airs, grabs, spins, flips, and inverted rotations.

half-pipe滑雪运动是2002年冬奥会的一部分; 参赛选手需要运动来回两圈，期间表演一些技巧，如straight airs, grabs, spins, flips, and inverted rotations。

这些技巧相关介绍参见网址tutorial-5-cool-snowboard-tricks-for-beginners

We find no analysis of the “best” shape for a half-pipe. However, ususally it is 100-150m long, 17-19.5m wide, and 5.4-6.5m from floor to crown, with slope angle 16-18.5°[Postins n.d.].

In addition, the the Fédération Inter-nationale de Ski . (FIS) recommends that the width, height, transition, and the Half-pipe bottom flat snowboarding w be 15 m, 3.5 m, 学 is currently 5 m, and 5 judged m, respectively using subjective [2003, 36].
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Terminology and Definitions

术语及定义

Cycle: The start of a cycle is when the snowboarder reaches the edges of the half-pipe after a flight, and the end of a cycle is the next start.

周期: 一个周期的以滑雪运动员在运动到half-pipe的边缘为开始，在运动到下一个周期的开始为结束。

Flight: the part of the movement when the snowboarder is airborne.

飞行: 滑雪者在空中飞行的那部分运动。

Flight distance$(S_f)$: displacement along the $z$ direction during the flight.

飞行距离: 飞行期间在z方向上的移动距离

Flight time$(t_f)$: duration of the flight.

飞行时间: 飞行花费的时间

Cycle distance($S_c$): 一个周期内在$z$方向上的移动距离

Assumptions

模型假设

• The cross section of the half-pipe is a convex curve that is smooth (has
  second-order derivative) everywhere except the endpoints.

• half-pipe的横截面是一个除末端外处处光滑(至少二阶可导)的凹曲线。

• The snowboarder crouches during the performance until standing up to
  gain speed right at the edge of the half-pipe before the flight.

• 滑雪运动员在表演开始时是蹲下的，直到起飞前在half-pipe的边缘处获得了正确的速度后站立。

• We neglect the rotational kinetic energy of the snowboarder before considering the twist performance.

• 我们考虑在具备一定扭曲性表演之前暂时忽略运动员的转动动能

• The friction of the snow is proportional to the normal force of the snow exerted on the snowboarder but has nothing to do with velocity (that is, the angle between the direction of the snowboard and the snowboarder’s
  velocity is constant).

• 雪的摩擦力与雪施加在滑雪板上的法向力成正比，但与速度无关（那是因为滑雪板运动与滑雪运动员的速度之间的角度是恒定的）

• Air drag is proportional to the square of speed.

• 空气阻力与速度的二次方成正比

• The snowboarder’s body is perpendicular to the tangential surface of the
  half-pipe during movement on the half-pipe.

• 滑雪运动员在half-pipe上运动时，身体垂直于half-pipe表面的切面

• The force exerted on the board can be considered as acting at its center.

• 施加在滑雪板上的力可以考虑看做施加在其中心

• We neglect the influence of natural factors such as uneven sunshine
  (which may result from an east or west orientation), altitude, etc.

• 我们忽略自然因素的影响，如不均匀的阳光（）

Basic Model

基础模型

Model Overview

模型概述

A cycle can be divided into two parts: movement on the half-pipe, and the airborne performance.

一个周期可以划分为两个部分: 在half-pipe上的运动与在空中的表演。

For the first, we focus on the conversion and conservation of energy.

首先，我们专注于能量转化与守恒。

The loss of mechanical energy $E_{lost}$ due to the resistance of snow and air is the key.

机械能主要损失是由于雪面和空气的阻力导致的。

We derive a differential equation for it. We cannot neglect the snowboarder’s increasing the mechanical ennergy by stretching the body (standing up) and doing work against the centrifugal force.

我们得到了一个不同的等式。我们不能忽略滑雪运动员通过拉伸身体(站立)而增加的机械能和对抗离心力所做的功。

To derive an expression for vertical air, we apply Newton’s Second Law.

为了得到vertical air的表达式，我们应用了牛顿第二定律。
If we neglect air drag during the flight (we later show that it is indeed neg-ligible), we can calculate vertical air, duration of the flight, flight distance,gravitational potential decrease, etc.

如果我们忽视飞行中的空气阻力(我们稍后将展示它的确可以忽略)，我们能够计算出vertical air，飞行时间，飞行距离，降低的重力势能等等。

Next, we discuss the airborne rotation of the snowboarder. Since the shape of the half-pipe directly influences the initial angular momentum of the snowboarder, and the angular momentum cannot change during the flight, the relationship between the half-pipe shape and the initial angular momentum is the key to our discussion.

下一步，我们讨论了运动员在空中的旋转，由于half-pipe的形状直接影响了滑雪运动员的初始角动量，并且在飞行期间角动量无法改变，因此helf-pipe形状与初始角动量之间的关系是我们讨论的关键。

After deriving an expression for the initial angular momentum, we can find the optimal shape of the half-pipe.

在得到了初始角动量的表达式之后，我们能够找到一个优化后的half-pipe的形状。

The Model

Vertical Air

Step 1. Force Analysis: The top part of Figure 2 shows the definition of the coordinate variables: x is the free variable, while y and z are functions of x. The relationship between y and x depends on the shape of the half-pipe, while the relationship between z and x depends on the path chosen by the snowboarder.

第一步: 受力分析：上面的图2展示了坐标变量的定义: $x$是一个自由变量，y和z是x的函数。y与x的关系取决于half-pipe的形状，z和x的关系取决于滑雪运动员所选的路线

• $s$: length of the path

  • 路径长度

• $E_0$: Initial mechanical energy at the beginning of a cycle

  • 最开始的一个周期的初始机械能

• $E_{leave}$: Kinetic energy right before the flight

  • 准备起飞时候的动能

• $E_{reach}$: Kinetic energy at the end of the flight

  • 准备结束飞行时的动能

• $E_{lost}$: Mechanical energy lost due to friction of the snow and air drag

  • 由于雪面摩擦和空气阻力而导致的机械能的损失量

• $N$: Normal force of the snow exerted on the snowboarder

  • 雪面施加在滑雪者上的法向力

• $W_{human}$: Work done by the snowboarder at the edge of the half-pipe when he or she stands up.

  • 当运动员站起来时，滑雪板在half-pipe的边缘的做功

• $W_G$: Decrease in gravitational potential during the flight

  • 在飞行时的重力势能减少量

• $f$: Friction of the snow plus air drag

  • 雪和空气的摩擦力之和

• $m$: Mass of the snowboarder

  • 运动员的质量

• $\alpha$: Friction coefficient between the snow and snowboard

  • 雪与滑雪板的动摩擦因数

• $\beta$: Drag coefficient of air

  • 空气摩擦因数

• $\theta$: Angle between z-axis and the horizontal plane

  • z轴和水平面的夹角

• $\Delta h$: Rise of the mass point of the snowboarder when he or she stands up from a crouching position

  • 质点的升高量

• $\rho$: Radius of curvalure at a point on the cross section of the half-pipe

  • 曲率半径

• $x_t, y_t, z_t, y'_t, z'_t, \dot{y}_t,\dot{z}_t,y''_t$: Values right before the flight

  • 准备起飞时的各个量

• $H_f$: Vertical air

to the direction of $N$ needs to be considered:

考虑在$N$的方向上有，

$\rho = \frac{(1+{y'}^{3/2})}{y''}$

$N=\frac{\dot{x}^2+\dot{y}^2}{\rho}m+ \frac{mgcos \theta}{\sqrt{1+{y'}^2}}= (y'' \dot{x}^2+g cos\theta) \frac{m}{\sqrt{1+{y'}^2}}$

Path length unit can be represented as

考虑路径的长度微元记作

$ds = \sqrt{1+\dot{y}^2+\dot{z}^2}dx$

Step 2. Energy Conservation: According to the Energy Conservation Principle, we have

第二步. 能量守恒: 根据能量守恒定律，我们有

$\frac{1}{2}m(1+{y'^2}+{z'^2})\dot{x}^2 =E_0-E_{lost}-mg(ycos\theta-zsin\theta)$

Then we have

变形后我们得到，

$\dot{x}^2 =\frac{2}{m(1+{y'^2}+{z'^2})}[E_0-E_{lost}-mg(ycos\theta-zsin\theta)]$

第三部. 机械能损失$E_{lost}$

\begin{align} E_{lost}&=\int_{-x_0}^{x} {f\cdot ds}\\ &=\int_{-x_0}^{x} []\\ \end{align}