2014年美赛数学建模A题论文

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This paper presents mathematical model to illustrate how the rule of Keep Right Except To Pass influence traffic flow and safety of highway in light and heavy traffic. Firstly , we will build and analyze a mathematical model to analyze the performance of this rule among traffic flow,safety and other factors.Secondly, in order to promote better traffic flow,we will deliver an alternative of this rule .

II. The Description of the Problem

In order to indicate the origin of the overtaking problems, the following background is worth mentioning.

In countries where driving automobiles on the right is the rule (that is, USA, China and most other countries except for Great Britain, Australia, and some former British colonies), multi-lane freeways often employ a rule that requires drivers to drive in the right-most lane unless they are passing another vehicle, in which case they move one lane to the left, pass, and return to their former travel lane.

Our team has been asked to build and analyze a mathematical model to analyze the performance of this rule in light and heavy traffic. What’s more, we will examine tradecrafts between traffic flow,safety and under- or over-posted speed limits. we suggest and analyze alternatives that might promote greater traffic flow, safety, or other factors that you deem important because the previous rule is not effective in promoting better traffic flow. In countries where driving automobiles on the left is the norm, argue whether or not the solution can be carried over with a simple change of orientation, or would additional requirements be needed.

Lastly, the rule as stated above relies upon human judgment for compliance. If vehicle transportation on the same roadway was fully under the control of an intelligent system – either part of the road network or imbedded in the design of all vehicles using the roadway – to the extent change the results of the earlier analysis.

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III. Models

3.1 Symbols :

symbols Meaning radius of curvature lateral adhesion coefficient superelevation the passing distance superelevation the passing time accelerometer of passing vehicle safe distance to be back maximum safety speed natural number Rh ?h ah I h t a S vm k 3.2 Assumptions.

? Assume that the engine of all of the vehicles are equal. ? We assume that the passing line is a parabola.

? We suppose that the vehicle s which are passed drives in a constant rate. ? Assume that the driver s of all the vehicles are skillful. ? Assume that the highway is straight and in good conditions. ? Assume that the vehicle drives constant rate at first,then it

do uniformly retarded motion.

3.3 Survey of Previous Research

? Maximum safety speed : the maximum speed of the vehicle drive in a safe way . When the vehicle drive fast enough on the horizontal curve , may lead to the vehicle slip to the side of the road because of the lateral centrifugal force,so the maximum safety speed can be represent by the follow equation:

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vm?127Rh(?h?ah) （1）

Rh :radius of curvature;

?h:lateral adhesion coefficient ,defined as 0. 4;

ah :superelevation, defined as - 0. 015.

? Radius of curvature : ?y2??1?()???x? R??2?y?x232 （2）

3.4 Simple Models for Over-taking

The rule Keep-Right-Except-To-Pass is a limit of overtaking behavior.As we all

know，he goal of overtaking is to improve the speed of the vehicles,and the vehicle speed is an important reaction of traffic efficiency. So we describe the traffic efficiency with efficiency index which closely related to traffic flow and maximum safety speed .

We present a mathematical survey of some of the algorithms that have been used to solve the problem.According to the algorithms, we find it that traffic flow is negatively related to the passing time .The safety and maximum safe speed have positive correlation. In conclusion,the traffic efficiency could be precisely described by the maximum safe speed and the passing time .

In order to look for an appropriate model, we start with the simplest possible

situation.We propose the process of overtaking is even uniformly accelerated motion to work out the maximum safe speed and passing time .There is the idealized pattern graph below:

The simple figure of overtaking route:

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Just like the simple figure upon ,the parabolic curve said the full pathnames of overtaking .

I means the passing distance,

h means superelevation,

t means the passing time.

Distance equation:y?I2x x???h,0? （3） 2h2We will get the radius of curvature of the point （-h,I/2）according to equation (2)&(3)

Rh?h?1?I2?2?32I （4）

According to equation of uniformly accelerated motion:

1t1t??I?2??vB???a?()2?=vB?t?at2 （5）

42??22vB: the passing speed of vehicle B.

a :the accelerated speed of passing vehicle C.

It1t?t??vB???a????vA??s （6） S :safe distance 2222?2?Then we can get the equation of passing time :

22?(vB?vA)?(vB?vA)2?2ast? （7） a??