2013国赛数模C题第二问扭曲.doc

我们计算扭曲的方法为：计算一层的中心到某一角与其上面一层的中心到对应的该一角的向量之间的夹角作为角度，类似的两层之间可以得到八个夹角，将该八个角加起来即为总的扭曲度。计算式为：若第一层中点为第二层中心为，第一层对的某一角的对应点为得到向量第二层对应的对的某一角的对应点为得到向量，将向量、的、平移到同一点有、，则、的扭转角即为、对应的扭转角，设为，有()内为向量乘以顺时针为正，当sin很小的时候可近似等于)，总的扭转角即为求出剩下七个点（B、C、D、E、F、G、H）对应的、，八个角相加即可。Matlab示意图为：Our method to calculate the the angle of twist the way we the consider the angle which is the sum of eight angles as the criterion to evaluate the torsion resistance and each little angle is the angle between the two vectors ,one vector is from the center of one floor to one specific corner and the other vector is from the center of the floor above to the corresponding corner .The formula is that we set the center of one floor as ,the center of floor above as , is the vector from to a specific corner , is from to ,is the angle between the two vectors.,”means the vector cross product ,the value is positive when clockwise,and sin,when is very small and close to 0;we can calculate the total the same way .The total angle is =+;Matlab Figure：Computed data：angle23456720110.010209690.0108571430.0092636760.0107143850.0292378230.01189328320090.0102122750.0108538030.0092740110.0107189230.029216820.01190441919960.0181923140.0191502650.0164939590.018795420.019751780.00984409519860.0182005920.0191407270.0164940550.0188041030.0197471360.009835995angle891011121320110.021159140.023479128-0.001557151-0.002180168-0.002436809-0.00264278420090.0211552180.023448085-0.001515942-0.002195775-0.002427393-0.0026225919960.0106943230.012327270.013334137-0.001687675-0.0017705760.00356468519860.0107047240.0123219160.013337863-0.001705758-0.0017182860.000248296 我们认为倾斜方向就是13层每层的中心拟合后的斜线与水平面的夹角。通过对中心拟合的计算和观察我们发现这四个年份的倾斜应该分为三段倾斜直线来拟合（这样基本可以保证误差最小），第一段为1-5层，第二段为6-10层，第三段为11-14（包括塔尖），即最终得到3个倾斜的斜率。We consider the angle between the fitting curve of the centers of 13 floors and the horizontal line as the angle of the tilt .After calculating those values,we find that those fitting curves in those years can be divided in to 3 parts of Sloping straight lines,the first part is from 1st floor to 5st floor,the second floor is from 6st floor to 10st floor,the third floor is from