美赛优秀论文

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美赛论文

注：LEO 低地球轨道MEO中地球轨道GeO 同步卫星轨道risk-profit 风险利润率fixed-profit rate 固定利润率提出一个合理的商业计划，可以使我们抓住商业机会，我们建立四个模型来分析三个替代方案（水射流，激光，卫星）和组合，然后确定是否存在一个经济上有吸引力的机会，从而设计了四种模型分析空间碎片的风险、成本、利润和预测。

首先，我们建立了利润模型基于净现值（NPV）模型，并确定三个最佳组合的替代品与定性分析：1）考虑了三个备选方案的组合时，碎片的量是巨大的；2）考虑了水射流和激光的结合，认为碎片的大小不太大；3）把卫星和激光的结合当尺寸的这些碎片足够大。

其次，建立风险定性分析模型，对影响因素进行分析在每一种替代的风险，并得出一个结论，风险将逐渐下降直到达到一个稳定的数字。

在定量分析技术投入和对设备的影响投资中，我们建立了双重技术的学习曲线模型,找到成本的变化规律与时间的变化。

然后，我们开发的差分方程预测模型预测的量在未来的四年内每年发射的飞机。

结合结果我们从预测中，我们可以确定最佳的去除选择。

最后，分析了模型的灵敏度，讨论了模型的优势和我们的模型的弱点，目前的非技术性的信，指出了未来工作。

目录1，简介1.1问题的背景1.2可行方案1.3一般的假设1.4我们的思想的轮廓2，我们的模型2.1 时间---利润模型2.1.1 模型的符号2.1.2 模型建立2.1.3 结果与分析2.2 . 差分方程的预测模型2.2.1 模型建立2.2.2 结果分析2.3 双因子技术-学习曲线模型2.3.1 模型背景知识2.3.2 模型的符号2.3.3 模型建立2.3.4 结果分析2.4风险定性分析模型2.4.1 模型背景2.4.2 模型建立2.4.3 结果与分析3．在我们模型的灵敏度分析3.1 差分方程的预测模型。

3.1.1 稳定性分析3.1.2 敏感性分析3.2 双因子技术学习曲线模型3.2.1 稳定性分析3.2.2 敏感性分析4 优点和缺点查分方程预测模型优点缺点双因子技术学习曲线模型优点缺点时间---利润模型优点缺点5..结论6..未来的工作7．参考双赢模式：拯救地球，抓住机遇1..简介问题的背景空间曾经很干净整洁。

美国大学生数学建模竞赛优秀论文

For office use onlyT1________________ T2________________ T3________________ T4________________Team Control Number7018Problem ChosencFor office use onlyF1________________F2________________F3________________F4________________ SummaryThe article is aimed to research the potential impact of the marine garbage debris on marine ecosystem and human beings，and how we can deal with the substantial problems caused by the aggregation of marine wastes.In task one，we give a definition of the potential long-term and short-term impact of marine plastic garbage. Regard the toxin concentration effect caused by marine garbage as long-term impact and to track and monitor it. We etablish the composite indicator model on density of plastic toxin，and the content of toxin absorbed by plastic fragment in the ocean to express the impact of marine garbage on ecosystem. Take Japan sea as example to examine our model.In ask two, we designe an algorithm, using the density value of marine plastic of each year in discrete measure point given by reference，and we plot plastic density of the whole area in varies locations. Based on the changes in marine plastic density in different years， we determine generally that the center of the plastic vortex is East—West140°W—150°W, South—North30°N—40°N. According to our algorithm, we can monitor a sea area reasonably only by regular observation of part of the specified measuring pointIn task three，we classify the plastic into three types，which is surface layer plastic，deep layer plastic and interlayer between the two. Then we analysis the the degradation mechanism of plastic in each layer. Finally，we get the reason why those plastic fragments come to a similar size.In task four， we classify the source of the marine plastic into three types，the land accounting for 80%，fishing gears accounting for 10%，boating accounting for 10%，and estimate the optimization model according to the duel-target principle of emissions reduction and management. Finally, we arrive at a more reasonable optimization strategy.In task five，we first analyze the mechanism of the formation of the Pacific ocean trash vortex， and thus conclude that the marine garbage swirl will also emerge in south Pacific，south Atlantic and the India ocean. According to the Concentration of diffusion theory, we establish the differential prediction model of the future marine garbage density，and predict the density of the garbage in south Atlantic ocean. Then we get the stable density in eight measuring point .In task six, we get the results by the data of the annual national consumption ofpolypropylene plastic packaging and the data fitting method, and predict the environmental benefit generated by the prohibition of polypropylene take-away food packaging in the next decade. By means of this model and our prediction，each nation will reduce releasing 1.31 million tons of plastic garbage in next decade.Finally, we submit a report to expediction leader，summarize our work and make some feasible suggestions to the policy- makers.Task 1:Definition:●Potential short-term effects of the plastic: the hazardeffects will be shown in the short term.●Potential long-term effects of the plastic: thepotential effects, of which hazards are great, willappear after a long time.The short- and long-term effects of the plastic on the ocean environment:In our definition, the short-term and long-term effects of the plastic on the ocean environment are as follows.Short-term effects:1）The plastic is eaten by marine animals or birds.2） Animals are wrapped by plastics, such as fishing nets, which hurt or even kill them.3）Deaden the way of the passing vessels.Long-term effects:1）Enrichment of toxins through the food chain: the waste plastic in the ocean has no natural degradation in theshort-term, which will first be broken down into tinyfragments through the role of light, waves,micro-organisms, while the molecular structure has notchanged. These "plastic sands", easy to be eaten byplankton, fish and other, are Seemingly very similar tomarine life’s food,causing the enrichment and delivery of toxins.2）Accelerate the greenhouse effect: after a long-term accumulation and pollution of plastics, the waterbecame turbid, which will seriously affect the marineplants (such as phytoplankton and algae) inphotosynthesis. A large number of plankton’s deathswould also lower the ability of the ocean to absorbcarbon dioxide, intensifying the greenhouse effect tosome extent.To monitor the impact of plastic rubbish on the marine ecosystem:According to the relevant literature, we know that plastic resin pellets accumulate toxic chemicals , such as PCBs、DDE , and nonylphenols , and may serve as a transport medium and soure of toxins to marine organisms that ingest them[]2. As it is difficult for the plastic garbage in the ocean to complete degradation in the short term, the plastic resin pellets in the water will increase over time and thus absorb more toxins, resulting in the enrichment of toxins and causing serious impact on the marine ecosystem.Therefore, we track the monitoring of the concentration of PCBs, DDE, and nonylphenols containing in the plastic resin pellets in the sea water, as an indicator to compare the extent of pollution in different regions of the sea, thus reflecting the impact of plastic rubbish on ecosystem.To establish pollution index evaluation model: For purposes of comparison, we unify the concentration indexes of PCBs, DDE, and nonylphenols in a comprehensive index.Preparations:1）Data Standardization2）Determination of the index weightBecause Japan has done researches on the contents of PCBs,DDE, and nonylphenols in the plastic resin pellets, we illustrate the survey conducted in Japanese waters by the University of Tokyo between 1997 and 1998.To standardize the concentration indexes of PCBs, DDE,and nonylphenols. We assume Kasai Sesside Park, KeihinCanal, Kugenuma Beach, Shioda Beach in the survey arethe first, second, third, fourth region; PCBs, DDE, andnonylphenols are the first, second, third indicators.Then to establish the standardized model:j j jij ij V V V V V min max min --= （1,2,3,4;1,2,3i j ==）wherej V max is the maximum of the measurement of j indicator in the four regions.j V min is the minimum of the measurement of j indicatorstandardized value of j indicator in i region.According to the literature [2], Japanese observationaldata is shown in Table 1.Table 1. PCBs, DDE, and, nonylphenols Contents in Marine PolypropyleneTable 1 Using the established standardized model to standardize, we have Table 2.In Table 2，the three indicators of Shioda Beach area are all 0, because the contents of PCBs, DDE, and nonylphenols in Polypropylene Plastic Resin Pellets in this area are the least, while 0 only relatively represents the smallest. Similarly, 1 indicates that in some area the value of a indicator is the largest.To determine the index weight of PCBs, DDE, and nonylphenolsWe use Analytic Hierarchy Process (AHP) to determine the weight of the three indicators in the general pollution indicator. AHP is an effective method which transforms semi-qualitative and semi-quantitative problems into quantitative calculation. It uses ideas of analysis and synthesis in decision-making, ideally suited for multi-index comprehensive evaluation.Hierarchy are shown in figure 1.Fig.1 Hierarchy of index factorsThen we determine the weight of each concentrationindicator in the generall pollution indicator, and the process are described as follows:To analyze the role of each concentration indicator, we haveestablished a matrix P to study the relative proportion.⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡=111323123211312P P P P P P P Where mn P represents the relative importance of theconcentration indicators m B and n B . Usually we use 1，2，…，9 and their reciprocals to represent different importance. The greater the number is, the more important it is. Similarly, the relative importance of m B and n B is mn P /1（3,2,1,=n m ）.Suppose the maximum eigenvalue of P is m ax λ, then theconsistency index is1max --=n nCI λThe average consistency index is RI , then the consistencyratio isRICI CR = For the matrix P of 3≥n , if 1.0<CR the consistency isthougt to be better, of which eigenvector can be used as the weight vector.We get the comparison matrix accoding to the harmful levelsof PCBs, DDE, and nonylphenols and the requirments ofEPA on the maximum concentration of the three toxins inseawater as follows:⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡=165416131431P We get the maximum eigenvalue of P by MATLAB calculation0012.3max =λand the corresponding eigenvector of it is()2393.02975.09243.0，，=W1.0042.012.1047.0<===RI CI CR Therefore,we determine the degree of inconsistency formatrix P within the permissible range. With the eigenvectors of p as weights vector, we get thefinal weight vector by normalization ()1638.02036.06326.0'，，=W . Defining the overall target of pollution for the No i oceanis i Q , among other things the standardized value of threeindicators for the No i ocean is ()321,,i i i i V V V V = and the weightvector is 'W ,Then we form the model for the overall target of marine pollution assessment, （3,2,1=i ）By the model above, we obtained the Value of the totalpollution index for four regions in Japanese ocean in Table 3T B W Q '=In Table3, the value of the total pollution index is the hightest that means the concentration of toxins in Polypropylene Plastic Resin Pellets is the hightest, whereas the value of the total pollution index in Shioda Beach is the lowest(we point up 0 is only a relative value that’s not in the name of free of plastics pollution)Getting through the assessment method above, we can monitor the concentration of PCBs, DDE and nonylphenols in the plastic debris for the sake of reflecting the influence to ocean ecosystem.The highter the the concentration of toxins,the bigger influence of the marine organism which lead to the inrichment of food chain is more and more dramatic.Above all, the variation of toxins’ concentration simultaneously reflects the distribution and time-varying of marine litter. We can predict the future development of marine litter by regularly monitoring the content of these substances, to provide data for the sea expedition of the detection of marine litter and reference for government departments to make the policies for ocean governance.Task 2:In the North Pacific, the clockwise flow formed a never-ending maelstrom which rotates the plastic garbage. Over the years, the subtropical eddy current in North Pacific gathered together the garbage from the coast or the fleet, entrapped them in the whirlpool, and brought them to the center under the action of the centripetal force, forming an area of 3.43 million square kilometers (more than one-third of Europe) .As time goes by, the garbage in the whirlpool has the trend of increasing year by year in terms of breadth, density, and distribution. In order to clearly describe the variability of the increases over time and space, according to “Count Densities of Plastic Debris from Ocean Surface Samples North Pacific Gyre 1999—2008”, we analyze the data, exclude them with a great dispersion, and retain them with concentrated distribution, while the longitude values of the garbage locations in sampled regions of years serve as the x-coordinate value of a three-dimensional coordinates, latitude values as the y-coordinate value, the Plastic Count per cubic Meter of water of the position as the z-coordinate value. Further, we establish an irregular grid in the yx plane according to obtained data, and draw a grid line through all the data points. Using the inverse distance squared method with a factor, which can not only estimate the Plastic Count per cubic Meter of water of any position, but also calculate the trends of the Plastic Counts per cubic Meter of water between two original data points, we can obtain the unknown grid points approximately. When the data of all the irregular grid points are known (or approximately known, or obtained from the original data), we can draw the three-dimensional image with the Matlab software, which can fully reflect the variability of the increases in the garbage density over time and space.Preparations:First, to determine the coordinates of each year’s sampled garbage.The distribution range of garbage is about the East - West 120W-170W, South - North 18N-41N shown in the “Count Densities of Plastic Debris from Ocean Surface Samples North Pacific Gyre 1999--2008”, we divide a square in the picture into 100 grids in Figure (1) as follows:According to the position of the grid where the measuring point’s center is, we can identify the latitude and longitude for each point, which respectively serve as the x- and y- coordinate value of the three-dimensional coordinates.To determine the Plastic Count per cubic Meter of water. As the “Plastic Count per cubic Meter of water” provided by “Count Densities of P lastic Debris from Ocean Surface Samples North Pacific Gyre 1999--2008”are 5 density interval, to identify the exact values of the garbage density of one year’s different measuring points, we assume that the density is a random variable which obeys uniform distribution in each interval.Uniform distribution can be described as below:()⎪⎩⎪⎨⎧-=01a b x f ()others b a x ,∈We use the uniform function in Matlab to generatecontinuous uniformly distributed random numbers in each interval, which approximately serve as the exact values of the garbage density andz-coordinate values of the three-dimensional coordinates of the year’s measuring points.Assumptions(1)The data we get is accurate and reasonable.(2)Plastic Count per cubic Meter of waterIn the oceanarea isa continuous change.(3)Density of the plastic in the gyre is a variable by region.Density of the plastic in the gyre and its surrounding area is interdependent , However, this dependence decreases with increasing distance . For our discussion issue, Each data point influences the point of each unknown around and the point of each unknown around is influenced by a given data point. The nearer a given data point from the unknown point, the larger the role.Establishing the modelFor the method described by the previous,we serve the distributions of garbage density in the “Count Pensities of Plastic Debris from Ocean Surface Samples North Pacific Gyre 1999--2008”as coordinates ()z y,, As Table 1:x,Through analysis and comparison, We excluded a number of data which has very large dispersion and retained the data that is under the more concentrated the distribution which, can be seen on Table 2.In this way, this is conducive for us to get more accurate density distribution map.Then we have a segmentation that is according to the arrangement of the composition of X direction and Y direction from small to large by using x co-ordinate value and y co-ordinate value of known data points n, in order to form a non-equidistant Segmentation which has n nodes. For the Segmentation we get above,we only know the density of the plastic known n nodes, therefore, we must find other density of the plastic garbage of n nodes.We only do the sampling survey of garbage density of the north pacificvortex,so only understand logically each known data point has a certain extent effect on the unknown node and the close-known points of density of the plastic garbage has high-impact than distant known point.In this respect,we use the weighted average format, that means using the adverse which with distance squared to express more important effects in close known points. There're two known points Q1 and Q2 in a line ,that is to say we have already known the plastic litter density in Q1 and Q2, then speculate the plastic litter density's affects between Q1、Q2 and the point G which in the connection of Q1 and Q2. It can be shown by a weighted average algorithm22212221111121GQ GQ GQ Z GQ Z Z Q Q G +*+*=in this formula GQ expresses the distance between the pointG and Q.We know that only use a weighted average close to the unknown point can not reflect the trend of the known points, we assume that any two given point of plastic garbage between the changes in the density of plastic impact the plastic garbage density of the unknown point and reflecting the density of plastic garbage changes in linear trend. So in the weighted average formula what is in order to presume an unknown point of plastic garbage density, we introduce the trend items. And because the greater impact at close range point, and thus the density of plastic wastes trends close points stronger. For the one-dimensional case, the calculation formula G Z in the previous example modify in the following format:2212122212212122211111112121Q Q GQ GQ GQ Q Q GQ Z GQ Z GQ Z Z Q Q Q Q G ++++*+*+*=Among them, 21Q Q known as the separation distance of the known point, 21Q Q Z is the density of plastic garbage which is the plastic waste density of 1Q and 2Q for the linear trend of point G . For the two-dimensional area, point G is not on the line 21Q Q , so we make a vertical from the point G and cross the line connect the point 1Q and 2Q , and get point P , the impact of point P to 1Q and 2Q just like one-dimensional, and the one-dimensional closer of G to P , the distant of G to P become farther, the smaller of the impact, so the weighting factor should also reflect the GP in inversely proportional to a certain way, then we adopt following format:221212222122121222211111112121Q Q GQ GP GQ GQ Q Q GQ GP Z GQ Z GQ Z Z P Q Q Q Q G ++++++*+*+*=Taken together, we speculated following roles:(1) Each known point data are influence the density of plastic garbage of each unknown point in the inversely proportional to the square of the distance;(2) the change of density of plastic garbage between any two known points data, for each unknown point are affected, and the influence to each particular point of their plastic garbage diffuse the straight line along the two known particular point; (3) the change of the density of plastic garbage between any two known data points impact a specific unknown points of the density of plastic litter depends on the three distances: a. the vertical distance to a straight line which is a specific point link to a known point;b. the distance between the latest known point to a specific unknown point;c. the separation distance between two known data points.If we mark 1Q ，2Q ，…，N Q as the location of known data points,G as an unknown node, ijG P is the intersection of the connection of i Q ，j Q and the vertical line from G to i Q ，j Q()G Q Q Z j i ,,is the density trend of i Q ，j Q in the of plasticgarbage points and prescribe ()G Q Q Z j i ,,is the testing point i Q ’ s density of plastic garbage ,so there are calculation formula:()()∑∑∑∑==-==++++*=Ni N ij ji i ijGji i ijG N i Nj j i G Q Q GQ GPQ Q GQ GP G Q Q Z Z 11222222111,,Here we plug each year’s observational data in schedule 1 into our model, and draw the three-dimensional images of the spatial distribution of the marine garbage ’s density with Matlab in Figure (2) as follows:199920002002200520062007－2008(1)It’s observed and analyzed that, from 1999 to 2008, the density of plastic garbage is increasing year by year and significantly in the region of East – West 140W-150W, south - north 30N-40N. Therefore, we can make sure that this region is probably the center of the marine litter whirlpool. Gathering process should be such that the dispersed garbage floating in the ocean move with the ocean currents and gradually close to the whirlpool region. At the beginning, the area close to the vortex will have obviously increasable about plastic litter density, because of this centripetal they keeping move to the center of the vortex ,then with the time accumulates ,the garbage density in the center of the vortex become much bigger and bigger , at last it becomes the Pacific rubbish island we have seen today.It can be seen that through our algorithm, as long as the reference to be able to detect the density in an area which has a number of discrete measuring points,Through tracking these density changes ,we Will be able to value out all the waters of the density measurement through our models to determine,This will reduce the workload of the marine expedition team monitoring marine pollution significantly, and also saving costs .Task 3:The degradation mechanism of marine plasticsWe know that light, mechanical force, heat, oxygen, water, microbes, chemicals, etc. can result in the degradation of plastics . In mechanism ,Factors result in the degradation can be summarized as optical ，biological，and chemical。

数学建模美赛获奖论文

Some players believe that “corking” a bat enhances the “sweet spot” effect. There are some arguments about that .Such asa corked bat has (slightly) less mass.,less mass (lower inertia) means faster swing speed and less mass means a less effective collision. These are just some people’s views, other people may have different opinions. Whethercorking is helpful in the baseball game has not been strongly confirmed yet. Experiments seem to have inconsistent results.
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2010 Mathematical Contest in Modeling (MCM) Summary Sheet
(Attach a copy of this page to each copy of your solution paper.)
Keywords:simple harmonic motion system , differential equations model , collision system

2022年美赛C题论文

2022年美赛C题论文基于相关性分析及线性规划下的交易策略的最优问题本文针对每日价格流来确定交易员是否应该购买、持有或出售其投资组合中的资产问题，主要通过建立具有不同投资项目的未来产值预测模型及其损失定量分析和敏感性评价模型，以此来对美国的黄金、现金、比特币三种投资进行组合并评价其未来的价值。

对于问题 1 中对美国的黄金和比特币的价值年变化特征进行相关分析，筛选出价值变化较高的年份以及对美国 2021 年的 10月 9 日进行具体的定量分析，这里选取美国 2016 年不同短历时黄金和比特币资料,采用频率分析、小波分析和极值分析等方法, 揭示了美国市场投资的演变规律，之后我们对于 2021 年的具体数据通过上述方法进行了具体分析。

选取价值变化趋势、中位数、成本、回报率等作为分析因素，确定各因素影响权重，从而确定现金、比特币和黄金的持有率，以达到最优的投资结果。

对于问题 2 评价模型的敏感性，根据五项指标数据，同时参照第一问求解得到的各指标对排名的的贡献度，我们综合考虑各种因素，对投资回报的指标进行赋值。

结合中国的各项数据，我们将估值水平，也就是总价值从 0 至 10 划分为十个不同的层次阶段，0-1 阶段为一等高价值，其余水平依次类推。

采用二次多项式拟合提取黄金、比特币的趋势分量，采用谐波分析法提取的周期成分，利用线性回归模型求解随机成分，最后将三者叠加，构建了各投资项目的预报模型。

模型计算结果与实测数据对比可知，应用预报模型对投资回报进行预报精度较高。

一、问题重述 1.1 背景资料市场交易员经常买卖波动性资产，其目标是使其总回报最大化。

每一次买卖通常都有一笔佣金。

其中两种资产是黄金和比特币。

1.2 需要解决的问题交易员要求开发一个模型，该模型仅使用迄今为止过去的每日价格流来确定交易员每天是否应该购买、持有或出售其投资组合中的资产。

二、问题分析2.1 问题 1 的分析通过对比美国的黄金和比特币的价值年变化特征进行分析，发现无论是黄金还是比特币，每日的资金流都处于波动之中。

2016 美国大学生数学竞赛优秀论文AB

2016年美赛A题热水澡一个人用热水通过一个水龙头来注满一个浴缸，然后坐在在浴缸中，清洗和放松。

不幸的是，浴缸不是一个带有二次加热系统和循环喷流的温泉式浴缸，而是一个简单的水容器。

过一会儿，洗澡水就会明显地变凉，所以洗澡的人需要不停地将热水从水龙头注入，以加热洗浴水。

该浴缸的设计是以这样一种方式，当浴缸里的水达到容量极限，多余的水通过溢流口泄流。

考虑空间和时间等因素，建立一个浴缸的水温模型，以确定最佳的策略，使浴缸里的人可以用这个模型来让整个浴缸保持或尽可能接近初始的温度，而不浪费太多的水。

使用你的模型来确定你的策略对浴缸的形状和体积，浴缸里的人的形状、体积、温度，以及浴缸中的人的运动等因素的依赖程度。

如果这个人一开始用了一种泡泡浴剂加入浴缸，以协助清洗，这会怎样影响你的模型的结果？除了要求的一页MCM摘要提交之外，你的报告必须包括一页的为浴缸用户准备的非技术性的说明书来阐释你的策略，同时解释为什么洗澡水的温度得到均衡地保持是如此之难。

2016年美赛B题太空垃圾在地球轨道上的小碎片的数量已引起越来越多的关注。

据估计，目前有超过500，000块的空间碎片，也被称为轨道碎片，由于被认为对空间飞行器是潜在的威胁而正在被跟踪。

2009年2月10日，俄罗斯卫星kosmos-2251和美国卫星iridium-33相撞之后，该问题受到了新闻媒体更广泛的讨论。

一些消除碎片方法已经被提出。

这些方法包括使用微型的基于太空的喷水飞机和高能量的激光来针对一些特定的碎片和设计大型卫星来清扫碎片。

碎片按照大小和质量分步，从刷了油漆的薄片到废弃的卫星都有。

碎片在轨道上的高速度飞行使得捕捉十分困难。

建立一个以时间为考量的模型，以确定最佳的方法或系列方法，为一个私营企业提供商机，以解决空间碎片问题。

你的模型应该包括定量和定性的对成本，风险，收益的估计，并考虑其他的一些重要因素。

你的模型应该能够评估某种方法，以及组合的系列方法，并能够研究各种重要的假设情况。

美赛金奖论文

1
Team # 14604
Catalogue
Abstracts ........................................................................................................................................... 1 Contents ............................................................................................................................................ 3 1. Introduction ................................................................................................................................... 3 1.1 Restatement of the Problem ................................................................................................ 3 1.2 Survey of the Previous Research......................................................................................... 3 2. Assumptions .................................................................................................................................. 4 3. Parameters ..................................................................................................................................... 4 4. Model A ----------Package model .................................................................................................. 6 4.1 Motivation ........................................................................................................................... 6 4.2 Development ....................................................................................................................... 6 4.2.1 Module 1: Introduce of model A .............................................................................. 6 4.2.2 Module 2: Solution of model A .............................................................................. 10 4.3 Conclusion ........................................................................................................................ 11 5. Model B----------Optional model ................................................................................................ 12 5.1 Motivation ......................................................................................................................... 12 5.2 Development ..................................................................................................................... 12 5.2.1 Module B: Choose oar- powered rubber rafts or motorized boats either ............... 12 5.2.2 Module 2: Choose mix of oar- powered rubber rafts and motorized boats ............ 14 5.3 Initial arrangement ............................................................................................................ 17 5.4. Deepened model B ........................................................................................................... 18 5.4.1 Choose the campsites allodium .............................................................................. 18 5.4.2 Choose the oar- powered rubber rafts or motorized boats allodium ...................... 19 5.5 An example of reasonable arrangement ............................................................................ 19 5.6 The strengths and weakness .............................................................................................. 20 6. Extensions ................................................................................................................................... 21 7. Memo .......................................................................................................................................... 25 8. References ................................................................................................................................... 26 9. Appendices .................................................................................................................................. 27 9.1 Appendix I .................................................................................................. 27 9.2 Appendix II ....................................................................................................................... 29

美赛一等奖论文-中文翻译版

目录问题回顾 (3)问题分析： (4)模型假设： (6)符号定义 (7)4.1---------- (8)4.2 有热水输入的温度变化模型 (17)4.2.1模型假设与定义 (17)4.2.2 模型的建立The establishment of the model (18)4.2.3 模型求解 (19)4.3 有人存在的温度变化模型Temperature model of human presence (21)4.3.1 模型影响因素的讨论Discussion influencing factors of the model (21)4.3.2模型的建立 (25)4.3.3 Solving model (29)5.1 优化目标的确定 (29)5.2 约束条件的确定 (31)5.3模型的求解 (32)5.4 泡泡剂的影响 (35)5.5 灵敏度的分析 (35)8 non-technical explanation of the bathtub (37)Sｕmｍary人们经常在充满热水的浴缸里得到清洁和放松。

本文针对只有一个简单的热水龙头的浴缸，建立一个多目标优化模型，通过调整水龙头流量大小和流入水的温度来使整个泡澡过程浴缸内水温维持基本恒定且不会浪费太多水。

首先分析浴缸中水温度变化的具体情况。

根据能量转移的特点将浴缸中的热量损失分为两类情况:沿浴缸四壁和底面向空气中丧失的热量根据傅里叶导热定律求出；沿水面丧失的热量根据水由液态变为气态的焓变求出。

因涉及的参数过多,将系数进行回归分析的得到一个一元二次函数。

结合两类热量建立了温度关于时间的微分方程。

加入阻滞因子考虑环境温湿度升高对水温的影响,最后得到水温度随时间的变化规律(见图＊*)。

优化模型考虑保持水龙头匀速流入热水的情况。

将过程分为浴缸未加满和浴缸加满而水从排水口溢出的两种情况，根据能量守恒定律优化上述微分方程,建立一个有热源的情况下水的温度随时间变化的分段模型，（见图*＊)接下来考虑人在浴缸中对水温的影响。

数学建模美赛一等奖优秀专业论文

For office use onlyT1________________ T2________________ T3________________ T4________________ Team Control Number52888Problem ChosenAFor office use onlyF1________________F2________________F3________________F4________________Mathematical Contest in Modeling (MCM/ICM) Summary SheetSummaryIt’s pleasant t o go home to take a bath with the evenly maintained temperature of hot water throughout the bathtub. This beautiful idea, however, can not be always realized by the constantly falling water temperature. Therefore, people should continually add hot water to keep the temperature even and as close as possible to the initial temperature without wasting too much water. This paper proposes a partial differential equation of the heat conduction of the bath water temperature, and an object programming model. Based on the Analytic Hierarchy Process (AHP) and Technique for Order Preference by Similarity to Ideal Solution (TOPSIS), this paper illustrates the best strategy the person in the bathtub can adopt to satisfy his desires. First, a spatiotemporal partial differential equation model of the heat conduction of the temperature of the bath water is built. According to the priority, an object programming model is established, which takes the deviation of temperature throughout the bathtub, the deviation of temperature with the initial condition, water consumption, and the times of switching faucet as the four objectives. To ensure the top priority objective—homogenization of temperature, the discretization method of the Partial Differential Equation model (PDE) and the analytical analysis are conducted. The simulation and analytical results all imply that the top priority strategy is: The proper motions of the person making the temperature well-distributed throughout the bathtub. Therefore, the Partial Differential Equation model (PDE) can be simplified to the ordinary differential equation model.Second, the weights for the remaining three objectives are determined based on the tolerance of temperature and the hobby of the person by applying Analytic Hierarchy Process (AHP) and Technique for Order Preference by Similarity to Ideal Solution (TOPSIS). Therefore, the evaluation model of the synthesis score of the strategy is proposed to determine the best one the person in the bathtub can adopt. For example, keeping the temperature as close as the initial condition results in the fewer number of switching faucet while attention to water consumption gives rise to the more number. Third, the paper conducts the analysis of the diverse parameters in the model to determine the best strategy, respectively, by controlling the other parameters constantly, and adjusting the parameters of the volume, shape of the bathtub and the shape, volume, temperature and the motions and other parameters of the person in turns. All results indicate that the differential model and the evaluation model developed in this paper depends upon the parameters therein. When considering the usage of a bubble bath additive, it is equal to be the obstruction between water and air. Our results show that this strategy can reduce the dropping rate of the temperatureeffectively, and require fewer number of switching.The surface area and heat transfer coefficient can be increased because of the motions of the person in the bathtub. Therefore, the deterministic model can be improved as a stochastic one. With the above evaluation model, this paper present the stochastic optimization model to determine the best strategy. Taking the disparity from the initial temperature as the suboptimum objectives, the result of the model reveals that it is very difficult to keep the temperature constant even wasting plentiful hot water in reality.Finally, the paper performs sensitivity analysis of parameters. The result shows that the shape and the volume of the tub, different hobbies of people will influence the strategies significantly. Meanwhile, combine with the conclusion of the paper, we provide a one-page non-technical explanation for users of the bathtub.Fall in love with your bathtubAbstractIt’s pleasant t o go home to take a bath with the evenly maintained temperature of hot water throughout the bathtub. This beautiful idea, however, can not be always realized by the constantly falling water temperature. Therefore, people should continually add hot water to keep the temperature even and as close as possible to the initial temperature without wasting too much water. This paper proposes a partial differential equation of the heat conduction of the bath water temperature, and an object programming model. Based on the Analytic Hierarchy Process (AHP) and Technique for Order Preference by Similarity to Ideal Solution (TOPSIS), this paper illustrates the best strategy the person in the bathtub can adopt to satisfy his desires. First, a spatiotemporal partial differential equation model of the heat conduction of the temperature of the bath water is built. According to the priority, an object programming model is established, which takes the deviation of temperature throughout the bathtub, the deviation of temperature with the initial condition, water consumption, and the times of switching faucet as the four objectives. To ensure the top priority objective—homogenization of temperature, the discretization method of the Partial Differential Equation model (PDE) and the analytical analysis are conducted. The simulation and analytical results all imply that the top priority strategy is: The proper motions of the person making the temperature well-distributed throughout the bathtub. Therefore, the Partial Differential Equation model (PDE) can be simplified to the ordinary differential equation model.Second, the weights for the remaining three objectives are determined based on the tolerance of temperature and the hobby of the person by applying Analytic Hierarchy Process (AHP) and Technique for Order Preference by Similarity to Ideal Solution (TOPSIS). Therefore, the evaluation model of the synthesis score of the strategy is proposed to determine the best one the person in the bathtub can adopt. For example, keeping the temperature as close as the initial condition results in the fewer number of switching faucet while attention to water consumption gives rise to the more number. Third, the paper conducts the analysis of the diverse parameters in the model to determine the best strategy, respectively, by controlling the other parameters constantly, and adjusting the parameters of the volume, shape of the bathtub and the shape, volume, temperature and the motions and other parameters of the person in turns. All results indicate that the differential model and the evaluation model developed in this paper depends upon the parameters therein. When considering the usage of a bubble bath additive, it is equal to be the obstruction between water and air. Our results show that this strategy can reduce the dropping rate of the temperature effectively, and require fewer number of switching.The surface area and heat transfer coefficient can be increased because of the motions of the person in the bathtub. Therefore, the deterministic model can be improved as a stochastic one. With the above evaluation model, this paper present the stochastic optimization model to determine the best strategy. Taking the disparity from the initial temperature as the suboptimum objectives, the result of the model reveals that it is very difficult to keep the temperature constant even wasting plentiful hotwater in reality.Finally, the paper performs sensitivity analysis of parameters. The result shows that the shape and the volume of the tub, different hobbies of people will influence the strategies significantly. Meanwhile, combine with the conclusion of the paper, we provide a one-page non-technical explanation for users of the bathtub.Keywords:Heat conduction equation; Partial Differential Equation model (PDE Model); Objective programming; Strategy; Analytical Hierarchy Process (AHP) Problem StatementA person fills a bathtub with hot water and settles into the bathtub to clean and relax. However, the bathtub is not a spa-style tub with a secondary hearing system, as time goes by, the temperature of water will drop. In that conditions,we need to solve several problems:(1) Develop a spatiotemporal model of the temperature of the bathtub water to determine the best strategy to keep the temperature even throughout the bathtub and as close as possible to the initial temperature without wasting too much water;(2) Determine the extent to which your strategy depends on the shape and volume of the tub, the shape/volume/temperature of the person in the bathtub, and the motions made by the person in the bathtub.(3)The influence of using b ubble to model’s results.(4)Give a one-page non-technical explanation for users that describes your strategyGeneral Assumptions1.Considering the safety factors as far as possible to save water, the upper temperature limit is set to 45 ℃;2.Considering the pleasant of taking a bath, the lower temperature limit is set to 33℃;3.The initial temperature of the bathtub is 40℃.Table 1Model Inputs and SymbolsSymbols Definition UnitT Initial temperature of the Bath water ℃℃T∞Outer circumstance temperatureT Water temperature of the bathtub at the every moment ℃t Time hx X coordinates of an arbitrary point my Y coordinates of an arbitrary point mz Z coordinates of an arbitrary point mαTotal heat transfer coefficient of the system 2（）⋅/W m K1SThe surrounding-surface area of the bathtub 2m 2S The above-surface area of water2m 1H Bathtub’s thermal conductivity/W m K ⋅（） D The thickness of the bathtub wallm 2H Convection coefficient of water2/W m K ⋅（） a Length of the bathtubm b Width of the bathtubm h Height of the bathtubm V The volume of the bathtub water3m c Specific heat capacity of water/()J kg ⋅℃ ρ Density of water3/kg m ()v t Flooding rate of hot water3/m s r TThe temperature of hot water ℃Temperature ModelBasic ModelA spatio-temporal temperature model of the bathtub water is proposed in this paper. It is a four dimensional partial differential equation with the generation and loss of heat. Therefore the model can be described as the Thermal Equation.The three-dimension coordinate system is established on a corner of the bottom of the bathtub as the original point. The length of the tub is set as the positive direction along the x axis, the width is set as the positive direction along the y axis, while the height is set as the positive direction along the z axis, as shown in figure 1.Figure 1. The three-dimension coordinate systemTemperature variation of each point in space includes three aspects: one is the natural heat dissipation of each point in space; the second is the addition of exogenous thermal energy; and the third is the loss of thermal energy . In this way , we build the Partial Differential Equation model as follows:22212222(,,,)(,,,)()f x y z t f x y z t T T T T t x y z c Vαρ-∂∂∂∂=+++∂∂∂∂ (1) Where● t refers to time;● T is the temperature of any point in the space;● 1f is the addition of exogenous thermal energy;● 2f is the loss of thermal energy.According to the requirements of the subject, as well as the preferences of people, the article proposes these following optimization objective functions. A precedence level exists among these objectives, while keeping the temperature even throughout the bathtub must be ensured.Objective 1(.1O ): keep the temperature even throughout the bathtub;22100min (,,,)(,,,)t t V V F t T x y z t dxdydz dt t T x y z t dxdydz dt ⎡⎤⎡⎤⎛⎫=-⎢⎥ ⎪⎢⎥⎢⎥⎣⎦⎝⎭⎣⎦⎰⎰⎰⎰⎰⎰⎰⎰ (2) Objective 2(.2O ): keep the temperature as close as possible to the initial temperature;[]2200min (,,,)tV F T x y z t T dxdydz dt ⎛⎫=- ⎪⎝⎭⎰⎰⎰⎰ (3) Objective 3(.3O ): do not waste too much water;()30min tF v t dt =⋅⎰ (4) Objective 4(.4O ): fewer times of switching.4min F n = (5)Since the .1O is the most crucial, we should give priority to this objective. Therefore, the highest priority strategy is given here, which is homogenization of temperature.Strategy 0 – Homogenization of T emperatureThe following three reasons are provided to prove the importance of this strategy. Reason 1-SimulationIn this case, we use grid algorithm to make discretization of the formula (1), and simulate the distribution of water temperature.(1) Without manual intervention, the distribution of water temperature as shown infigure 2. And the variance of the temperature is 0.4962. 00.20.40.60.8100.51 1.5200.5Length WidthH e i g h t 4242.54343.54444.54545.5Distribution of temperature at the length=1Distribution of temperatureat the width=1Hot water Cool waterFigure 2. Temperature profiles in three-dimension space without manual intervention(2) Adding manual intervention, the distribution of water temperature as shown infigure 3. And the variance of the temperature is 0.005. 00.5100.51 1.5200.5 Length WidthH e i g h t 44.744.7544.844.8544.944.9545Distribution of temperatureat the length=1Distribution of temperature at the width=1Hot water Cool waterFigure 3. Temperature profiles in three-dimension space with manual interventionComparing figure 2 with figure 3, it is significant that the temperature of water will be homogeneous if we add some manual intervention. Therefore, we can assumed that222222()0T T T x y zα∂∂∂++≠∂∂∂ in formula (1). Reason 2-EstimationIf the temperature of any point in the space is different, then222222()0T T T x y zα∂∂∂++≠∂∂∂ Thus, we find two points 1111(,,,)x y z t and 2222(,,,)x y z t with:11112222(,,,)(,,,)T x y z t T x y z t ≠Therefore, the objective function 1F could be estimated as follows:[]2200200001111(,,,)(,,,)(,,,)(,,,)0t t V V t T x y z t dxdydz dt t T x y z t dxdydz dt T x y z t T x y z t ⎡⎤⎡⎤⎛⎫-⎢⎥ ⎪⎢⎥⎢⎥⎣⎦⎝⎭⎣⎦≥->⎰⎰⎰⎰⎰⎰⎰⎰ (6) The formula (6) implies that some motion should be taken to make sure that the temperature can be homogeneous quickly in general and 10F =. So we can assumed that: 222222()0T T T x y zα∂∂∂++≠∂∂∂. Reason 3-Analytical analysisIt is supposed that the temperature varies only on x axis but not on the y-z plane. Then a simplified model is proposed as follows:()()()()()()()2sin 000,0,,00,000t xx x T a T A x l t l T t T l t t T x x l π⎧=+≤≤≤⎪⎪⎪==≤⎨⎪⎪=≤≤⎪⎩ (7)Then we use two ways, Fourier transformation and Laplace transformation, in solving one-dimensional heat equation [Qiming Jin 2012]. Accordingly, we get the solution:()()2222/22,1sin a t l Al x T x t e a l πππ-=- (8) Where ()0,2x ∈, 0t >, ()01|x T f t ==(assumed as a constant), 00|t T T ==.Without general assumptions, we choose three specific value of t , and gain a picture containing distribution change of temperature in one-dimension space at different time.00.20.40.60.811.2 1.4 1.6 1.8200.511.522.533.54Length T e m p e r a t u r e time=3time=5time=8Figure 4. Distribution change of temperature in one-dimension space at different timeT able 2.V ariance of temperature at different timet3 5 8 variance0.4640 0.8821 1.3541It is noticeable in Figure 4 that temperature varies sharply in one-dimensional space. Furthermore, it seems that temperature will vary more sharply in three-dimension space. Thus it is so difficult to keep temperature throughout the bathtub that we have to take some strategies.Based on the above discussion, we simplify the four dimensional partial differential equation to an ordinary differential equation. Thus, we take the first strategy that make some motion to meet the requirement of homogenization of temperature, that is 10F =.ResultsTherefore, in order to meet the objective function, water temperature at any point in the bathtub needs to be same as far as possible. We can resort to some strategies to make the temperature of bathtub water homogenized, which is (,,)x y z ∀∈∀. That is,()(),,,T x y z t T t =Given these conditions, we improve the basic model as temperature does not change with space.112213312()()()()/()p r H S dT H S T T H S T T c v T T c V V dt D μρρ∞⎡⎤=++-+-+--⎢⎥⎣⎦(9) Where● 1μis the intensity of people’s movement ;● 3H is convection between water and people;● 3S is contact area between water and people;● p T is body surface temperature;● 1V is the volume of the bathtub;● 2V is the volume of people.Where the μ refers to the intensity of people ’s movement. It is a constant. However , it is a random variable in reality, which will be taken into consideration in the following.Model T estingWe use the oval-shaped bathtub to test our model. According to the actual situation, we give initial values as follows:0.19λ=,0.03D =,20.54H =,25T ∞=,040T =00.20.40.60.8125303540Time T e m p e r a t u r eFigure 5. Basic modelThe Figure 5 shows that the temperature decreases monotonously with time. And some signs of a slowing down in the rate of decrease are evident in the picture. Reaching about two hours, the water temperature does not change basically and be closely to the room temperature. Obviously , it is in line with the actual situation, indicating the rationality of this model.ConclusionOur model is robust under reasonable conditions, as can be seen from the testing above. In order to keep the temperature even throughout the bathtub, we should take some strategies like stirring constantly while adding hot water to the tub. Most important of all, this is the necessary premise of the following question.Strategy 1 – Fully adapted to the hot water in the tubInfluence of body surface temperatureWe select a set of parameters to simulate two kinds of situation separately.The first situation is that do not involve the factor of human1122()()/H S dT H S T T cV dt D ρ∞⎡⎤=+-⎢⎥⎣⎦(10) The second situation is that involves the factor of human112213312()()()/()p H S dT H S T T H S T T c V V dt D μρ∞⎡⎤=++-+--⎢⎥⎣⎦(11) According to the actual situation, we give specific values as follows, and draw agraph of temperature of two functions.33p T =,040T =204060801001201401601803838.53939.540TimeT e m p e r a t u r eWith body Without bodyFigure 6a. Influence of body surface temperature50010001500200025003000350025303540TimeT e m p e r a t u r eWith body Without bodyCoincident pointFigure 6b. Influence of body surface temperatureThe figure 6 shows the difference between two kinds of situation in the early time (before the coincident point ), while the figure 7 implies that the influence of body surface temperature reduces as time goes by . Combing with the degree of comfort ofbath and the factor of health, we propose the second optimization strategy: Fully adapted to the hot water after getting into the bathtub.Strategy 2 –Adding water intermittentlyInfluence of adding methods of waterThere are two kinds of adding methods of water. One is the continuous; the other is the intermittent. We can use both different methods to add hot water.1122112()()()/()r H S dT H S T T c v T T c V V dt D μρρ∞⎡⎤=++-+--⎢⎥⎣⎦(12) Where r T is the temperature of the hot water.To meet .3O , we calculated the minimum water consumption by changing the flow rate of hot water. And we compared the minimum water consumptions of the continuous with the intermittent to determine which method is better.A . Adding water continuouslyAccording to the actual situation, we give specific values as follows and draw a picture of the change of temperature.040T =, 37d T =, 45r T =5001000150020002500300035003737.53838.53939.54040.5TimeT e m p e r a t u r eadd hot waterFigure 7. Adding water continuouslyIn most cases, people are used to have a bath in an hour. Thus we consumed that deadline of the bath: 3600final t =. Then we can find the best strategy in Figure 5 which is listed in Table 2.T able 3Strategy of adding water continuouslystart t final tt ∆ vr T varianceWater flow 4 min 1 hour56 min537.410m s -⨯45℃31.8410⨯0.2455 3mB . Adding water intermittentlyMaintain the values of 0T ,d T ,r T ,v , we change the form of adding water, and get another graph.5001000150020002500300035003737.53838.53939.540TimeT e m p e r a t u r et1=283(turn on)t3=2107(turn on)t2=1828(turn off)Figure 8. Adding water intermittentlyT able 4.Strategy of adding water intermittently()1t on ()2t off 3()t on vr T varianceWater flow 5 min 30 min35min537.410m s -⨯45℃33.610⨯0.2248 3mConclusionDifferent methods of adding water can influence the variance, water flow and the times of switching. Therefore, we give heights to evaluate comprehensively the methods of adding hot water on the basis of different hobbies of people. Then we build the following model:()()()2213600210213i i n t t i F T t T dtF v t dtF n -=⎧=-⎪⎪⎪=⎨⎪⎪=⎪⎩⎰∑⎰ (13) ()112233min F w F w F w F =++ (14)12123min ..510mini i t s t t t +>⎧⎨≤-≤⎩Evaluation on StrategiesFor example: Given a set of parameters, we choose different values of v and d T , and gain the results as follows.Method 1- AHPStep 1:Establish hierarchy modelFigure 9. Establish hierarchy modelStep 2: Structure judgment matrix153113511133A ⎡⎤⎢⎥⎢⎥=⎢⎥⎢⎥⎢⎥⎣⎦Step 3: Assign weight1w 2w3w 0.650.220.13Method 2-TopsisStep1 ：Create an evaluation matrix consisting of m alternatives and n criteria, with the intersection of each alternative and criteria given as ij x we therefore have a matrixStep2：The matrix ij m n x ⨯（）is then normalised to form the matrix ij m n R r ⨯=（）, using thenormalisation method21r ,1,2,,;1,2,ijij mij i x i n j m x====∑…………，Step3：Calculate the weighted normalised decision matrix()(),1,2,,ij j ij m n m nT t w r i m ⨯⨯===⋅⋅⋅where 1,1,2,,nj j jj w W Wj n ===⋅⋅⋅∑so that11njj w==∑, and j w is the original weight given to the indicator,1,2,,j v j n =⋅⋅⋅.Step 4： Determine the worst alternative ()w A and the best alternative ()b A()(){}{}()(){}{}max 1,2,,,min 1,2,,1,2,,n ,min 1,2,,,max 1,2,,1,2,,n ,w ij ij wjbijij bjA t i m j J t i m j J t j A t i m j J t i m j J tj -+-+==∈=∈====∈=∈==where, {}1,2,,J j n j +==⋅⋅⋅ associated with the criteria having a positive impact, and {}1,2,,J j n j -==⋅⋅⋅associated with the criteria having a negative impact. Step 5： Calculate the L2-distance between the target alternative i and the worst condition w A()21,1,2,,m niw ij wj j d tt i ==-=⋅⋅⋅∑and the distance between the alternative i and the best condition b A()21,1,2,,m nib ij bj j d t t i ==-=⋅⋅⋅∑where iw d and ib d are L2-norm distances from the target alternative i to the worst and best conditions, respectively .Step 6 ：Calculate the similarity to the worst condition Step 7 ： Rank the alternatives according to ()1,2,,iw s i m =⋅⋅⋅ Step 8 : Assign weight1w2w 3w 0.55 0.170.23ConclusionAHP gives height subjectively while TOPSIS gives height objectively. And the heights are decided by the hobbies of people. However, different people has different hobbies, we choose AHP to solve the following situations.Impact of parametersDifferent customers have their own hobbies. Some customers prefer enjoying in the bath, so the .2O is more important . While other customers prefer saving water, the .3O is more important. Therefore, we can solve the problem on basis of APH . 1. Customers who prefer enjoying: 20.83w =,30.17w =According to the actual situation, we give initial values as follows:13S =,11V =,2 1.4631S =,20.05V =,33p T =,110μ=Ensure other parameters unchanged, then change the values of these parameters including 1S ,1V ,2S ,2V ,d T ,1μ. So we can obtain the optimal strategies under different conditions in Table 4.T able 5.Optimal strategies under different conditions2.Customers who prefer saving: 20.17w =,30.83w =Just as the former, we give the initial values of these parameters including1S ,1V ,2S ,2V ,d T ,1μ, then change these values in turn with other parameters unchanged. So we can obtain the optimal strategies as well in these conditions.T able 6.Optimal strategies under different conditionsInfluence of bubbleUsing the bubble bath additives is equivalent to forming a barrier between the bath water and air, thereby slowing the falling velocity of water temperature. According to the reality, we give the values of some parameters and gain the results as follows:5001000150020002500300035003334353637383940TimeT e m p e r a t u r eWithour bubbleWith bubbleFigure 10. Influence of bubbleT able 7.Strategies (influence of bubble)Situation Dropping rate of temperature (the larger the number, the slower)Disparity to theinitial temperatureWater flow Times of switchingWithout bubble 802 1.4419 0.1477 4 With bubble 34499.85530.01122The Figure 10 and the Table 7 indicates that adding bubble can slow down the dropping rate of temperature effectively . It can decrease the disparity to the initial temperature and times of switching, as well as the water flow.Improved ModelIn reality , human ’s motivation in the bathtub is flexible, which means that the parameter 1μis a changeable measure. Therefore, the parameter can be regarded as a random variable, written as ()[]110,50t random μ=. Meanwhile, the surface of water will come into being ripples when people moves in the tub, which will influence the parameters like 1S and 2S . So, combining with reality , we give the range of values as follows:()[]()[]111222,1.1,1.1S t random S S S t random S S ⎧=⎪⎨=⎪⎩Combined with the above model, the improved model is given here:()[]()[]()[]11221121111222()()()/()10,50,1.1,1.1a H S dT H S T T c v T T c V V dt D t random S t random S S S t random S S μρρμ∞⎧⎡⎤=++-+--⎪⎢⎥⎣⎦⎨⎪===⎩(15)Given the values, we can get simulation diagram:050010001500200025003000350039.954040.0540.140.15TimeT e m p e r a t u r eFigure 11. Improved modelThe figure shows that the variance is small while the water flow is large, especially the variance do not equals to zero. This indicates that keeping the temperature of water is difficult though we regard .2O as the secondary objective.Sensitivity AnalysisSome parameters have a fixed value throughout our work. By varying their values, we can see their impacts.Impact of the shape of the tub0.70.80.91 1.1 1.2 1.3 1.433.23.43.63.84Superficial areaT h e t i m e sFigure 12a. Times of switching0.70.80.91 1.11.21.31.43890390039103920393039403950Superficial areaV a r i a n c eFigure 12b. V ariance of temperature0.70.80.91 1.1 1.2 1.3 1.40.190.1950.20.2050.21Superficial areaW a t e r f l o wFigure 12c. Water flowBy varying the value of some parameters, we can get the relationships between the shape of tub and the times of switching, variance of temperature, and water flow et. It is significant that the three indexes will change as the shape of the tub changes. Therefore the shape of the tub makes an obvious effect on the strategies. It is a sensitive parameter.Impact of the volume of the tub0.70.80.91 1.1 1.2 1.3 1.4 1.533.544.55VolumeT h e t i m e sFigure 13a. Times of switching。

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The Design of Snowboard HalfpipeAbstract: Based on the snowboard movement theory, the flight height depends on the out- velocity. We take the technical parameters of four sites and five excellent snowboarders for statistical analysis. As results show that the size of halfpipe (length, width and depth, halfpipe slope) influence the in- velocity and out- velocity. Help ramp, the angle between the snowboard’s direction and speed affect velocity ’s loss.For the halfpipe, we established the differential equation model, based on weight, friction, air density, resistance coefficient, the area of resistance, and other factors and the law of energy conservation. the model’s results show that the snowboarders’ energy lose from four aspects(1) the angle between the direction of snowboard and the speed, which formed because of the existing halfpipe(2) The friction between snowboard and the surface(3) the air barrier(4) the collision with the wall for getting vertical speed before sliping out of halfpipe.Therefore, we put forward an improving model called L-halfpipe，so as to eliminate or reduce the angle between the snowboard and the speed .Smaller radius can also reduce the energy absorption by the wall.At last, we put forward some conception to optimize the design of the halfpipe in the perspective of safety and producing torsion.Key words:snowboard; halfpipe; differential equation model;L-halfpipeContents1. Introduction (3)简介1.1the origin of the snowboard course problems (3)滑雪课程的起源问题。

1.2 the background (3)背景2. The Description of Problem (3)问题的描述2.1Practical halfpipe’s requirements (3)实用halfpipe的需求2.1.1 the maximum vertical and the largest body twist (3)最大垂直和最大的身体扭曲2.1.2 Speed analysis (3)速度分析2.2 Halfpipe’s own conditions (4)Halfpipe自身的条件2.2.1 Friction (4)摩擦2.2.2 the size of halfpipe (4)halfpipe的大小3. Model (4)模型3.1 Definitions and Symbols (4)定义和符号3.2 Assumptions (5)假设3.3 the simple analysis of gravity and friction when sliding in the halfpipe (5)简单的分析重力和摩擦力的halfpipe时滑动3.4 in-velocity of factors (6)速度的因素3.4.1 the snowboarder’ angle when in and the speed loss (6)滑雪在角和速度上的损失3.5 out-velocity of factors (8)初速度的因素3.5.1 Help ramp (8)帮助坡道3.5.2 the force point and the plate angle when out (9)力的点和板角3.5.3 the snowboarder’ angle when out and the speed loss (9)滑雪在角和速度上的损失3.5.4 H alfpipe’s Radius (11)Halfpipe的半径3.6 the in-velocity comparison with the out- velocity (14)速度与速率的对比3.7Snowboarder’s position impact on the speed (14)滑雪的位置影响速度3.8 the entire movement of the energychange in the halfpipe (15)在halfpipe中整个运动的能量变化3.9 the balance of speed after considering the air resistance (18)后速度的平衡考虑空气阻力3.10 L-halfpipe (19)左halfpipe3.11 Solution and Result (20)解决方案和结果4. Conclusions (21)总结4.1 Conclusions of the problem (21)结论的问题5. Future Work (21)工作展望5.1 other models (21)其他模型5.1.1 H alfpipe’s location outdoor (22)Halfpipe位置的户外5.1.2 H alfpipe’s material (22)Halfpipe的材料6. References (22)参考文献1. IntroductionIn order to indicate the origin of the snowboard course problems, the following background is worth mentioning.1.1 The origin of the snowboard course problemsIn the past, a significant amount of half pipe anxiety was due to the learning curve of a new sport, and educating resorts and pipe construction person nelson how to prepare the best shapes with basic resort equipment. This mode of operation is changing with the advent of new snowboard specific technology both in machine and hand tools. As technology has made half pipes better, the standards have also been proved. Most half pipe riders have a vision of what an ideal pipe should look like, but shifting that vision into reality seems to be a quantum leap.1.2 The backgroundThe problem lies in the fact that too many people who control the decision making process view of the half pipe as a fixed and static feature, and that once built, a pipe is left to the forces of nature. A severe change of opinions needed, as the half pipe needs to be thought of as an elastic form (almost lifelike) that changes daily and which needs continual maintenance. Another huge factor in developing consistent half pipes is a set of standards. Over the years, the NASBA, OP, USASA, USSA, ISF, and FIS have given differing pipe dimensions to resorts. All this help from various organizations has left pipe building more of an art than a science. Both the ISF and the FIS are now promoting similar versions of half pipe dimensions. So we need to redesign the shape of a snowboard course to maximize the production of vertical air by a skilled snowboarder.2. The Description of the Problem2.1 Practical halfpipe’s requirements2.1.1 the maximum vertical and the largest body twistSnowboarders’ greatest height, the number of rotations (the larges t body twist) and the beautiful action will affect the athlete's score. the longer the spare time left, the more rotations to do for snowboarders. The basic physics principle at work here is the conservation of angular momentum. The angular momentum of the snowboarder is determined at takeoff, and cannot be changed once the snowboarder is airborne. So to make turns in the air the snowboarder must give himself initial rotation upon takeoff. In order to reach the maximum height, the maximum out-velocity would be required.so we analyzed the in- velocity and the out-velocity, and the shape of space (length, width, depth, field gradient) affect the in- velocity and the out-velocity obviously.But the height can not be too high, because too high speed would be a big threat to the safety of snowboarders. Therefore, in order to control the maximum speed, we need to redesign the halfpipe.2.1.2 Speed analysisWhether to reach the maximum vertical height or to produce the largest body twist speed is is a reflection of practical indicators to the halfpipe design.The composition of the factors in the action.Including the fly height, difficulty, diversity, qualitycompletion of the action, Site use and landing conditions and so on because the height have an limit effect on difficulty, diversity, quality of action completement, so the fly height is the core elements of many factors.To conclude,no height,no no flight time and no flight time,no difficult action.As the free fall shows:V=.The height snowboarders can reach have a veryyclose relationship with the speed.2.2 Half pipe’s own conditions2.2.1 FrictionFriction, including friction between the board and the snow as well as air friction.The dynamic friction coefficient between Snow and the board changes from 0.03 to 0.2.Take 0.2 for example, the maximum friction coefficient and the full effect of body weight to calculate the vertical friction0.2Wf=, that the acceleration less due to friction is generated to accelerate the role of body weight 0.2 times, much smaller than resulting in the acceleration of gravity effect. Air friction 2f C Av, in our model, we do not consider the influence of air friction.0.5a a d2.2.2 the size of halfpipeUnder certain circumstances,as the length, depth, tilt angle increases, the speed will be. In view of snowboard safety, speed can not be infinite, which has some of the value of the constraints.3. Models3.1 Definitions and SymbolsFlat：the bottom ground of U grooveTransitions：the transition zone between Horizontal and vertical groove bottom wall Verticals：the vertical parts of the walls between the Lip and the Transitions Platform：the level platform on the snow wall surfaceEntry Ramp：the slippery position of U-shaped slotm：Athlete's qualityg：Gravity accelerationV：Athletes’ speed when first enter u-shaped slot1V：Athletes’ speed when last sliding out u-shaped slottl：under side rectangular width of U-shaped slot1l：the length of U-shaped slot2R：the deep of U-shaped slotn：Athletes emptied timesβ：Angle between Athletes’ speed and slot edge horizontal when first enteru-shaped slotu ： the frictional factor between Skateboarding and snowf A ：how much work friction do when Athletes vertically into a u-shaped slot in arc d C ：Air resistance coefficienta ρ：Air densityν：Athletes’ speed relative air movementA ：Corresponding to the projective area of v3.2 Assumptions1.Assuming frictional factor is a constant when athletes are in taxiing process2.Assuming no melting snow when athletes are in taxiing process)3.Assuming the maximizing friction is gravity, frictional factor as the biggest 0.2, when compared friction work and gravity work4.Assuming the loss of speed is 2 meters per second because of the Angle between the speed and direction of existence with blade when athletes come into (out) the slots every time3.3 the simple analysis of gravity and friction when sliding in the halfpipeIf the athlete slip into the half pipe with a certain speed. Athletes in motion of constantly falling in vertical direction Increasing gravitational potential energy. The process in motion need to overcome the frictional resistance acting between the skate and snow acting must also overcome the air resistance acting. We use all ski areas in China to analyze the data[1] as follows in Table1:Table 2 National snowboard half pipe skiing skill to the situation Championship Series17 ° slope of more than 100 meters along the length of glide in the groove The competition is in the17 ° slope and along the length of more than 100 meters slide in the grooves and do all kinds of flip, twist, grasp the difficulty of board action, the action is completed in a certain vertical height of drop. The standards of international competition venues, can be obtained by calculating the U-groove vertical drop 150*sin17h =.Those athletes complete the maneuver in the vertical direction to produce the height of 40 meters gap. A gap of more than 40 meters in the vertical direction athletes can have a very substantial increase in the rate. A gap of more than 40 meters in the vertical direction athletes can have a very substantial increase in the rate. In terms of free fall calculations 22104020y V hg =≈⨯⨯≈m / s, However the snow and the board’s dynamic friction coefficient between 0.03 to 0.2, the maximum friction coefficient and the full body weight to calculate the friction force acting perpendicular 0.2W of t =.That the speed less is due to the friction resistance, it is weight generated to accelerate the role of body weight 0.2 times, far less than the acceleration of gravity produces results. Therefore, venue’s height of fall is an important way for athletes obtained the vertical velocity. Athletes can complete the vertical velocity and level velocity conversion with a reasonable technology, So that Athletes most likely to get to the maximum vacate height at the last vacate.3.4 in-velocity of factors[1]3.4.1 the snowboarder’ angle when in and the speed lossPlayers control the skis taxiing around the edge of the board into the slot ，both the before and the after of snowboard have the effect of braking, so in order to reduce the loss of speed, so that ，the speed of the body center of gravity in the same direction with the board's longitudinal axis as far as possible ，to reduce the braking effect when the snowboard have instant contact with the snow, and homeopathic slide, taking fulladvantage of wall height difference obtained acceleration. It can be seen the speed of full contact is less than the speed of front panel from Table 3, indicating that the human body has a loss of speed when completely into the slot, Since the existence of wall resistance, the speed loss is normal. However, if the speed of body center of gravity has the same direction with the blade, the speed of the losses will be reduced. As can be seen from Table 3, the athlete’ gravity speed direction has an angle with direction of blade center, the minimum is 1.2, and the maximum is 5.4, the speed ofdirection and the direction with the blade did not reach exactly the same. Decrease the maximum rate reached 27.5%, a minimum rate of 6.8%.Figure 1 the angle between the rate of speed loss and direction with the blade when into the slotIt can be seen that the speed loss rate and direction with the blade angle has not exactly the same trend from Figure 1, there may be several reasons as follows:(1)players is not very skilled when sliding into the slot, the ability of controlling board is not strong(2) It may require different sliding speed for the different air movement in the next time, resulting in players want to control taxi speed on purpose (3)the center of gravity is too forward, the gravity torque is too large, have Side effect, So the technology will have a major impact in speed.3.5 out-velocity of factors [1]3.5.1 help rampAthletes for the first time into the slot before sliding into the slot with help, Athletes should be actively obtained the speed of access to controlled, If the snowboarder into the slot before , after slide a certain distance at the edge of the slot, Obtain a certain speed. and before leaping into the slot and in a certain height 0E , you'll get some initial energy reserves 000E E E 动势(0E Representative athlete ofthe initial energy, 0动E representing athletes initial kinetic energy,0势E .Representing athletes Initial potential) With the completion of the action into the groove, getting smaller and smaller potential energy athletes to complete, in the case of gravity does positive work, the potential energy of the players is correspondingincrease, that the athletes will get the vertical speed by energy transfer. After get some of the vertical velocity into the tank, the athletes have a certain amount of kinetic energy reserves; athletes using the kinetic energy reserves, transformation to the potential when out the half pipe, it can achieve the purpose of improving flight altitude; flight altitude do reserve for potential of the next action into the half pipe for the next action to provide time and space to ensure the successful completion However, athletes in the kinetic and potential energy conversion, to achieve the speed must be controllable. If the speed is not contro llable, it will affect the athlete’s performance, Otherwise it will lead to serious accidents. From Table 4, it can be seen that the athletes Lei Pan rear positive blade rate of 540movement into the tank thelargest; is s.14, the minimum Shi wan Cheng's anti-blade rate of om93720front foot movement, is s11. The actions are successful action, but also a national athlete,.m06so you can give a preliminary conclusion: the speed of athletes in the following speed control 15 meters per second.3.5.2 the force point and the plate angle when outIn the trench wall of the moment, because of losing the support of the front skis, then, the stress point should be to leave the center of board, and gradually transition back to the board, so that the stress point is always forcing plate wall, front foot homeopathic slide, back foot should be gradually forced pedal. When reaction force in sufficient, maintain parabolic path smooth, increasing the speed, and maintain a reasonable angle of the slot. At the same time of achieving the goal of increasing height highly effective, also get into the appropriate slot speed and angle of twist. Reaching movements while floating high, reducing the level of speed and the effect of resistance into the half pipe, reasonably come into the groove; do energy reserves for the next the action.3.5.3 the snowboarder’ angle when out and the speed lossCheng（20）Sun ZhiFeng front o72010.20 8.24 4.0 19.2Huang Shi Ying Anti-fronto72013.73 12.09 3.9 11.9ZenXiao Hua front o72011.65 11．55 0.3 0.9Liu JiaYu behind o54011.209.82 4.1 12.3Pan Lei behind o54012.00 9.11 3.0 24.0 Table 4 is part of the e lite athlete’s slotting board kinematic parameters. By comparing the data in Table 4, we can find what the speed of completely clear out the slot is less than the speed of the front panel instantaneous slip out the slot. It can be seen that five players’ speed and the direction of blade angle have positively correlated with the loss rate in Figure 2, indicating that the greater of angle between speed and direction with the blade, the greater of loss speed, so you need to control the sliding board direction, letting the long axis have the same direction with the speed of human body.Figure 2 the angle between the rate of speed loss and direction with the snowboardwhen out of the halfpipe3.5.4 H alfpipe’s RadiusAppropriate reduced orbit radius can increase the speed when athletes slip out half pipe, and favor the athletes to make various actions in the air. Sides rail identifiable by two 1/4arcs, we can deduces the formulaof tfti o i oi f I dt M w I w ∑⎰+=)(Then taking orbit design into consideration, the optimal speedup method is to reduce the rail depth (by our hypothesis know depth and arc radius is equal), namely decreases of r , and so can reduce of I , effectively increase f w . But, taking the athlete's safety into consideration, the athletes' speed may not excessive, namely orbit radius cannot be too small. General provisions half pipe orbit radius scope for 3-4.5m, guarantee the slot speed are not more than 15, also ensures the athlete's safety.The basic snowboarding physics behind this phenomenon can be understood by applying the principle of angular impulse and momentum.The schematic of the physics of snowboarding in this analysis is given below.Figure 3 the analysis of forceWhere:i w is the initial angular velocity of the body (consisting of snowboarder plusboard), at position (1)f w is the final angular velocity of the body, at position (2), which is the point at which the snowboarder exits the half-pipei V is the initial velocity of the center of mass G of the body, at position (1)f V is the final velocity of the center of mass G of the body, at position (2)i r is the initial distance from the center of rotation o to the body's center of massG, at position (1)f r is the final distance from the center of rotation o to the body's center of mass G , at position (2)g is the acceleration due to gravityN is the normal force acting on the snowboard, as shownF is the friction force acting on the snowboard, as shownIt is assumed that the half-pipe is a perfect circle with center at o. The physics of snowboarding in this analysis can be treated as a two-dimensional problem. Now, apply the equation for angular impulse and momentum to the system (consisting of snowboarder plus board):tfoi i o of fti I w M dt I w +=∑⎰Where:oi I is the initial moment of inertia of the body (consisting of snowboarder plus board) about an axis passing through point o and pointing out of the page, at position(1)of I is the final moment of inertia of the body (consisting of snowboarder plus board) about an axis passing through point o and pointing out of the page, at position(2)o M ∑ is the sum of the moments about point o. These moments are integrated between an initial time i t (at position 1) and a final time f t (at position 2)Here we are assuming that the body can be treated as rigid at positions (1) and (2), even though the snowboarder does in fact change his moment of inertia between thesetwo positions. But as it turns out, when using this equation we only need to know the initial and final values of the moment of inertia of the body.The line of action of the normal force N passes through point o, so it does not exert a moment on the body about point o. The friction force F is small so it can be neglected in terms of its moment contribution. This leaves only the gravitational force which exerts a moment on the body about point o. (Note that the gravitational force acts through the center of mass of the body, consisting of snowboarder plus board). In the above equation isolate f w . Thus,tfoi i o tif of I w M dtw I +=∑⎰Now,22oi Gi i of Gf f I I mr I I mr =+=+Where:Gi I is the initial moment of inertia of the body about an axis passing through point G and pointing out of the page, at position (1)Gf I is the final moment of inertia of the body about an axis passing through point G and pointing out of the page, at position (2)m is the mass of the bodyIn the above equation for f w , if we decrease of I the angular velocity f w will increase beyond the value it would be if we did not decrease of I . In practice this can be accomplished by sufficiently reducing the distance from the center of mass of the body G to the point o. In other words, make f r small enough and f w will increase. Note also that the terms Gf I and o M ∑ may also change somewhat. But the dominant effect will be that of reducing f r .At positions (1) and (2), the velocity of the center of mass G is given byi i if f fV w r V w r ==These two velocities are parallel to the half-pipe since the body is rigid at positions (1) and (2).r small appropriate, Looking at the above equations for velocity, if we makesfw. This in turn will result in his velocity the snowboarder will significantly increasefV) being greater than otherwise.exiting the pipe (f3.6 The in-velocity comparison with the out- velocity [1]It can be seen that the speed of athletes when athletes slip out half pipe is less than the speed of athletes when athletes slip out half pipe from Figure 4. The biggest difference between the two is the Shi wan Cheng, the smallest difference between the two is that Zen Xiao Ye. The average speed is 11.69m s when slip into half pipe, the average down is1.94m s,the speed decline will lead to altitude declining when slip out half pipe, having effect on the speed of slipping into half pipe next time, which restricts movements of athletes and sports techniques to improve the difficulty level of play, but also make the action quality greatly reduced, so the players should pay attention to the completion of a continuous action of the hair lower limb muscle strength.Figure 4 the chart of comparison about speed change when into (out of)half pipe 3.7 Snowb oarder’s position impact on the speedPumping on a half-pipe is used by snowboarders to increase their vertical take-off speed when they exit the pipe. This enables them to reach greater height and performmore aerial tricks, while airborne. The principle is exactly the same as for skateboarders pumping on a half-pipe.The snowboarder is able to increase his speed on the half-pipe with his feet remaining firmly on the board. This begs the question, what is the physics of snowboarding taking place that enables the snowboarder to increase his speed on the half-pipe?To increase his speed, the snowboarder crouches down in the straight part of the half-pipe. Then when he enters the curved portion of the half-pipe he lifts his body and arms up, which results in him exiting the pipe at greater speed than he would otherwise.r small Looking at the above equations for velocity, if the snowboarder makesfw. This in enough (by lifting his body and arms up), he will significantly increasefV being greater than if he did not lift turn will result in his velocity exiting the pipe (fhis body and arms up.By continually pumping his body (by crouching down and lifting his body and arms up in the curved portion of the half-pipe), the snowboarder is able to continually increase his velocity, eventually allowing sufficient height to be reached (upon exiting the half-pipe) to perform a variety of mid-air tricks.A more intuitive (non-mathematical) explanation of the physics of snowboarding taking place here is that pumping adds energy to the system in the same way that a child pumping on a swing adds energy, and results in him swinging higher. Therefore, the physics of snowboarding related to pumping on a half-pipe is similar to pumping on a swing.As a snowboarder lifts his arms and body up he feels resistance due to the force of centripetal acceleration which tends to push his body away from the center of rotation o. This resistance is proof that work is being done, and therefore energy is being added to the system.3.8 the entire movement of the energy change in the halfpipeHow the energy change during the Athletes’ entire movement in the half pipe.Figure 5 3-D half pipeFigure 6 halfpipe’s cr oss-sectionFrom Figure 6, we can know both sides of the curved part is the 1 / 4 cylinder in the side, the middle is rectangle.As shown, we assume that the depth of half pipe is R, the middle part length is 1l , the width of half pipe is 12l R , the half pipe ’s length is 2l , half pipe ’s inclination angle is α.When the athletes straight down into the tank by the vertical speed, we analysis the friction’s work in this process.When the athletes straight down into the tank where has friction, the friction’ work can be applied to functional principle, considering the given state can find out friction ’s work, But this does not consider the specific forms of friction force. By the analysis of analytical solution, we can describe its distribution characteristics.[2] α1l R 2l βAs shown in Fig 7, Objects satisfied Newton equations, the tangent of the form and normal directions form is (considered f uN =),dt dv m uN mg =-θcos ，————————————————（1）R v m mg N 2sin =-θ，————————————————（2）Figure 7 objects in circular orbit forcePray for (2) a derivative timedt dv v R m dt d mg dt dN 2cos =-θθ， dtd R v θ= and (1) David into the type dtd uN mg dt d mg dt dN θθθθ)2cos 2(cos -=-， θθcos 32mg uN d dN =+—————————————————— (3) Solving (3) type is the key to solve the solution of friction()()[]()[]⎰⎰⎰⎰+-=+-=c d u mg u c d ud mg ud N θθθθθθθθ*2exp *cos 3)2exp(*2exp *cos 32exp ——————————（4） Among them;()()()()()θθθθθθθθθθθθθθθd u u u u u u d u u u u d u *2exp *cos 412exp 4sin 2exp 2cos 2exp 2sin )2exp(2cos 2exp *cos 22⎰⎰⎰-+=+=namely ())sin cos 2(14)2exp(*2exp *cos 2θθθθθθ++=⎰u u u d u ， So:)2exp()sin cos 2(4132θθθu C u umg N -+++=， fBecause of: 0,0==N θ，will,24132U mg uC +-=. So:))2exp(2sin cos 2(4132θθθu u u umg N --++=———————————（5） And:))2exp(*2sin cos 2(4132θθθu u u umg uN f --++== ())cos sin 2(4132sin cos 241322202θθθθθθθθθu u f e u u umgR d ue u u umgR uNRd uNds A --+-+-=-++-=-=-=⎰⎰⎰3.9 the balance of speed after considering the air resistance [3]If in the process of straight downhill snow is flat and snowboard does not leave the ground can be approximately described by plane hinged to the relationship between ski and snow we watch skis and skiers as a whole force people ski and snowboard in the force of both concentration a nd reduced to a couple Torques’s’. At this point slide in the snow is equivalent to a single degree of freedom motion system as Figure 8When the system is in static equilibrium with⎩⎨⎧=-=+-0cos 0)(sin ααmg f f f mg sr a Which 20.5a a d f C Av ρ=,r s f uf =.Joint Solution available ：cos r f umg α=Can be seen, friction and gravity components is balance at the balance.Figure8 single degree of freedom motion systemInto the above equation can be obtained:0)cos 5.0(sin 2=+-αραumg Av C mg d aAC u mg v d a ραα)cos (sin 2-= suppose 12sin (1)u u β-=+，。