2011年MCM美国大学生数学建模大赛题目

2011年MCM美国大学生数学建模大赛题目

PROBLEM A: Snowboard Course

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

"Vertical air" is the maximum vertical distance above the edge of the halfpipe.

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

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

请设计一个单板滑雪场（现为“半管”或“U型池”）的形状，以便能使熟练的单板滑雪选手最大限度地产生垂直腾空。

“垂直腾空“是超出“半管”边缘以上的最大的垂直距离。

定制形状时要优化其他可能的要求，如：在空中产生最大的身体扭曲。

在制定一个“实用”的场地时哪些权衡因素可能需要？

PROBLEM B: Repeater Coordination

The VHF radio spectrum involves line-of-sight transmission and reception. This limitation can be overcome by “repeaters,” which pick up weak signals, amplify them, and retransmit them on a different frequency. Thus, using a repeater, low-power users (such as mobile stations) can communicate with one another in situations where direct user-to-user contact would not be possible. However, repeaters can interfere with one another unless they are far enough apart or transmit on sufficiently separated frequencies.

In addition to geographical separation, the “continuous tone-coded squelch system” (CTCSS), sometimes nicknamed “private line” (PL), technology can be used to mitigate interference problems. This system associates to each repeater a separate subaudible tone that is transmitted by all users who wish to communicate through that repeater. The repeater responds only to received signals with its specific PL tone. With this system, two nearby repeaters can share the same frequency pair (for receive and transmit); so more repeaters (and hence more users) can be accommodated in a particular area.

For a circular flat area of radius 40 miles radius, determine the minimum number of repeaters necessary to accommodate 1,000 simultaneous users. Assume that the spectrum available is 145 to 148 MHz, the transmitter

2011年MCM美国大学生数学建模大赛题目

frequency in a repeater is either 600 kHz above or 600 kHz below the receiver frequency, and there are 54 different PL tones available.

How does your solution change if there are 10,000 users?

Discuss the case where there might be defects in line-of-sight propagation caused by mountainous areas.

甚高频无线电频谱包含信号的发送和接受。这种限制可以被中继站所克服。中继站可以捕捉到微弱的信号，然后把它放大，再用不同的频率重新发送。这样，低功耗的用户，例如移动电话用户，在不能直接与其他用户联系的地方可以通过中继站来保持联系。然而,中继站之间会互相影响,除非彼此之间有足够远的距离或通过充分分离的频率来传送。

除了地理的分离、“连续编码音调控制系统”(CTCSS),有时被称为“私人专线”(PL)、通过这项技术可以减轻干扰问题。该系统连接每个中继站，靠的是所有通过同一个中继站连接的用户发送的独立的亚音频音调来连接。中继站只回应接收到的具有特殊PL的语调的信号。通过这个系统,两个附近的中继站可以共享相同的频率对(包括接收和发送);对于更多的中继站(并且更多的用户)可以提供在一个特定的区域。

在一个半径40英里的圆形区域，请你设计一个方案，用最少量的中继站来容纳1000同时在线用户。假设频谱范围是145到148兆赫,在中继站中的发射机的频率要么是600千赫以上，要么低于接收机频率600千赫、并且这里有54个不同的PL可用。

如果这里有10,000个用户，如何改变你的解决方案。

在由于山区引起信号传播的阻碍的地区，讨论这样的情形。

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