2009美国建模四篇论文合集

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美国建模，2009年的几篇论文。

Contents

1. Introduction...................................................................................................................................................3

1.1 Why does toll way collects toll? ………………………………………….…………………….3

1.2 Toll modes………………………………………………………………………………………………3

1.3 Toll collection methods……………………………………………………………………….……3

1.4 Annoyance in toll plazas………………………………………….………….…………………….3

1.5 The origin of the toll way problem……………………...……………………………………...3

1.6 Queuing theory…………………………………………………………………………………...…...4

2. The Description of Problem….............................................................................................................5

2.1 How do we approximate the whole course of paying toll? ............................................5

2.2 How do we define the optimal configuration? ........................…………….………….5

2.2.1 From the perspective of motorist…………………………………….………………….5

2.2.2 From the perspective of the toll plaza…………………………………………………6

2.2.3 Compromise…………………………………...……………………………………...………..6

2.3 Overall optimization and local optimization……………………………..……………….…6

2.4 The differences in weights and sizes of vehicles………………………………………..…7

2.5 What if there is no data available? ..............................................................................................7

3. Models……………...........................................................................................................................................7

3.1 Basic Model.............................................................................................................................................7

3.1.1 Symbols and Definitions………………………………..…...……………………………...7

3.1.2 Assumptions……………………………………………………………….……..……………..8

3.1.3 The Foundation of Model………………………………………………………………….9

3.1.4 Solution and Result……………………………………………………………….………...11

3.1.5 Analysis of the Result……………………………………………….………………………..……….….11

3.1.6 Strength and Weakness………………………………………………….…………….…..13

3.2 Improved Model.................................................................................................14

3.2.1 Extra Symbols……………………………………..………………………...…………………......................14

3.2.2 Additional Assumptions………………………………………………...…..…………………………..14

3.2.3 The Foundation of Model………………………………..…………………………………………….14

3.2.4 Solution and Result………………………………………….……………………………..……………...15

3.2.5 Analysis of the Result…………………………………………….……………………..…………….….18

3.2.6 Strength and Weakness……………………………………………….……………….…..19

4. Conclusions..................................................................................................................................................19

4.1 Conclusions of the problem……………………………………..……………..19

4.2 Methods used in our models…………………………………...……………………..…………19

4.3 Application of our models…………………………………………………..………………..….19

5. Future Work..............................................................................................................................................19

5.1 Another model………………………………………………………………………………………19

5.2 Another layout of toll plaza………………………………………………………..……………23

5.3 The newly- adopted charging methods………………………………………..……………23

6.
References...................................................................................................................................................23

7.
Appendix......................................................................................................................................................23

Programs and codes………………………………………………………………..……………………24

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The Sweet Spot: A Wave Model of Baseball Bats
本帖来自: 数学中国 作者: miaoshi 日期: 2010-10-3 23:52 您是本帖第643个浏览者

The Sweet Spot: A Wave Model of Baseball Bats

Rajib Quabili

Peter Diao

Yang Mou

Princeton University

Princeton,NJ

Advisor:Robert Calderbank

Abstract


; |# X3 y- [4 N; Q! k  ^We determine the sweet spot on a baseball bat. We capture the essential physics of the ball-bat impact by taking the ball to be a lossy spring and the bat to be an Euler-Bernoulli beam. To impart some intuition about the model, we begin by presenting a rigid-body model. Next, we use our full model to reconcile various correct an incorrect claims about the sweet spot found in the literature. Finally, we discuss the sweet spot and the performances of corked an aluminum bat, with a particular emphasis on hoop modes.

Introduction


& e/ @$ W/ ?+ V& w( s5 y' |# qAlthough a hitter might expect a model of the bat-baseball collision to yield insight into how the bat breaks, how the bat imparts spin on the ball, how best to swing the bat, and so on, we model only the sweet spot.

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There are at least
: V- g( e1 A/ u4 s! @  S8 v/ ]two notions of where the sweet spot should be-an inpact location on the bat that either

. minimizes the discomfort to the hands, or

. maximizes the outgoing velocity of the ball.

We focus exclusively on the second definition.


+ [+ y4 g9 _" |5 @The velocity of the ball leaving the bat is determined by

. the initial velocity and rotation of the ball,

. the relative position and orientation of the bat and ball, and

. the force over time that the hitter’s hands applies on the handle.

We assume that the ball is not rotating and that its velocity at impact is perpendicular to the length of the bat. We assume that everything occurs in a single plane, and we will argue that the hands’ interaction is negligible. In the frame of reference of the center of mass of the bat, the initial conditions are completely specified by

. the angular velocity of the bat,

. the velocity of the ball, and

. the position of impact along the bat.

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The location of the sweet spot depends not on just the bat alone but also on the pitch and on the swing.


: d, j: H% E' Q( H+ b% lThe **st model for the physics involved has the sweet spot at the center of percussion[Brody1986], the impact location that minimizes discomfort to the hand. The model assumes the ball to be a rigid body for which there are conjugate points: An
9 y; p0 `9 W4 d% Y8 w  x0 y; Y* nimpact at one will exactly balance the angular recoil and linear recoil at the other. By gripping at one and impacting at the other (the center of percussion), the hands experience minimal shock and the ball exits with high velocity. The center of percussion depends heavily on the moment of inertia and the lacation of the hands. We cannot accept this model because it both erroneously equates the two definitions of sweet spot and furthermore assumes incorrectly that the bat is a rigid body.

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