蜘蛛网模型.doc

参赛队号#1144第五届“认证杯”数学中国数学建模网络挑战赛承 诺 书我们仔细阅读了第五届“认证杯”数学中国数学建模网络挑战赛的竞赛规则我们完全明白，在竞赛开始后参赛队员不能以任何方式（包括、电子邮件、网上咨询等）与队外的任何人（包括指导教师）研究、讨论与赛题有关的问题我们知道，抄袭别人的成果是违反竞赛规则的, 如果引用别人的成果或其他公开的资料（包括网上查到的资料），必须按照规定的参考文献的表述方式在正文引用处和参考文献中明确列出我们郑重承诺，严格遵守竞赛规则，以保证竞赛的公正、公平性如有违反竞赛规则的行为，我们将受到严肃处理我们允许数学中国网站()公布论文，以供网友之间学习交流，数学中国网站以非商业目的的论文交流不需要提前取得我们的同意 我们的参赛队号为：1144参赛队员 (签名) ： 队员1：刘阳 队员2：吴平队员3：王臣杰 参赛队教练员 (签名)：邓昌瑞 参赛队伍组别：专科组第五届“认证杯”数学中国数学建模网络挑战赛编 号 专 用 页参赛队伍的参赛队号：（请各个参赛队提前填写好）：1144 竞赛统一编号（由竞赛组委会送至评委团前编号）：竞赛评阅编号（由竞赛评委团评阅前进行编号）：2012年第五届“认证杯”数学中国数学建模网络挑战赛 题 目 探讨蜘蛛网结构的合理性 关 键 词 捕食期望 能量守恒 整形规划 蛛网结构 摘 要自然界中绝大部分蜘蛛依靠织网捕食为生，但同一种类织网捕食的蜘蛛往往由于某种原因，其所织网的结构有所差异。

而蜘蛛网织成怎样的结构才最合理呢，对于这个问题，我们分别运用捕食期望、边界讨论、整型规划、阻尼运动等方法建立了数学模型，顺利地解决了该问题首先，蜘蛛停留在网的中心，由于蜘蛛网上每个点出现猎物的概率是相等的，运用函数方程求解出蜘蛛网上每个点的捕食期望，进而得出整个蛛网的捕食期望结构不同的蜘蛛网其捕食期望值也不同期望值越大，这种结构的蜘蛛网捕食能力越强把蜘蛛网的周长作为一个定值，可以衍生出的蜘蛛网结构有三角形，正四边形，正五边形，以此类推，当蜘蛛网半径趋于无穷大时，把此时的结构看作圆形来处理其次，选取三角形、四边形，圆形这三种蜘蛛网结构，结合极限思想分别求得其各自的捕食期望值通过对比，得出圆形结构为最合理的蜘蛛网结构根据捕丝之间存在的几何关系，运用整形规划，列出目标函数通过查得的数据，对半径丝条数与体重、捕食面积与体重，以及捕食丝间距与蜘蛛体重进行拟合，得到三个二次函数的关系式，应用软件求解，得出当蜘蛛网结构的半径条数为条，捕丝条数为条，各条捕丝之间的间距为时，该蜘蛛网结构将更加趋于合理然后对昆虫冲向蜘蛛时蛛网所做的运动进行考虑在昆虫撞向蜘蛛网的那一刻，昆虫具有一个初速度，受到阻挡作用，昆虫开始做加速度逐渐增大的减速运动，由于蛛网对昆虫具有一个黏聚力，此时蜘蛛网开始运动。

蜘蛛网具有一定的韧性，对其进行静力学分析，把昆虫的运动作为阻尼振动处理先求出蜘蛛网受到昆虫冲击时承受的回复力，对其积分得到回复力所做的功，遵循能量守恒定律，进一步求得蛛网在昆虫撞击时所能承受的回复力公式分别选取三角形，正方形，圆形的蜘蛛网结构，通过查阅资料，求得当昆虫被困在蜘蛛网上时，三角形所能承受的回复力大小为，正方形为,圆形结构为由此可以看出圆形结构才是最合理的的蜘蛛网结构但是圆形结构是理想的结构，所以蜘蛛网应织成半径丝条数为条正多边形，捕食丝边数为个，各条捕丝之间的间距为，此时的蛛网结构是最合适的最后，提出了模型的改进方向，更深一步对蜘蛛网结构的合理性进行了补充，并对模型进行了评价与推广，保证了模型的有效可行参赛密码 （由组委会填写）参赛队号 #1144 所选题目 A题 AbstractNature, most of the spider rely on netting a predator, but the same species of spider web feed on often for some reason, the difference between the structure of the web. And the structure of the spider webs made just the most reasonable how? To this problem, we are using predator expectations, boundary discussion, integer programming, damping and other sports method to establish the mathematical model, successfully solved this problem.First of all, stay in the center of the spider web, because the web each point in the probability of prey is equal, using the function equation of a spider web each point prey on expectations, and a conclusion that the entire web of the hunt expectations. The different structures of the spider webs its prey on expectations are different. The greater the expectations, the structure of the stronger ability to prey on the web. The perimeter of the web as a fixed value, can bring out web structure have triangle, are quadrilateral, pentagon is, and so on, when the radius of the web to infinite, the structure of the right now as round to deal with.Secondly, the selection of the triangle, quadrilateral, round the three web structure, combining extreme thought respectively given.according their respective hunting the expectations. By comparison, draw round structure for the most reasonable web structure. According to capture the geometric relationships between silk, using plastic planning, listed the objective function. Through the check data, to the radius of the article and weight, silk number prey on area and weight, and prey on space and the spider silk weight was fitted, get three quadratic function of the relationship, solving the application software, it is concluded that the structure of the article radius when web for article 38, hunt for article 45 article silk, catching the spacing between various silk for, the spider web structure will be more more reasonable.And then the insects to spider web do when considering the movement. In insects, crashed into the spider webs of that a moment, insects have a velocity, by blocking effect, insects begin to do is gradually increasing acceleration of the deceleration movement, because the web with a stick insect gathered force, this time the spider webs began to exercise. Web has certain toughness, and carry on the statics analysis, the movement of the insects as damping vibration processing. Seeking first by a web of shock response under insects force, the integral reply to force the work done, follow the law of conservation of energy, further seek in insects hit the web to withstand reply force formula. Were selected triangular, square, round spider webs structure, access to information, get when the insect is trapped in the spider web, triangle could take back force size of 0.183, a square is 0.3376, the circular structure for 0.958. Can see from this circular structure is the best reasonable cobwebs structure. But the circular structure is the ideal structure, so web woven wire should be the radius of the number of article 38 article is polygon, capture the size as silk 45, catching the spacing between various silk for, at this time of the web is the most appropriate st。