轨道交通弯道设计中缓和曲线的精细计算方法  

An Intensive Computing Method for Easement Curve in Orbit Curving Road Design

作　　者：张合沛 周振建 褚长海 万建军 赵永成 宋士仓 ZHANG He-pei;ZHOU Zhen-jian;CHU Chang-hai;WAN Jian-jun;ZHAO Yong-cheng;SONG Shi-cang(State Key Laboratory of Shield Machine and Boring Technology,Zhengzhou 450001,China;School of Mathematics and Statistics Zhengzhou University,Zhengzhou 450001,China)

机构地区：[1]盾构与掘进国家重点实验室,河南郑州450001 [2]郑州大学数学与统计学院,河南郑州450001

出　　处：《数学的实践与认识》2021年第21期222-228,共7页Mathematics in Practice and Theory

基　　金：国家自然科学基金(61873245);基于事件触发的非线性网络控制系统分析与设计,国家重点研发计划(2018YFB1701404);中铁隧道局集团科技创新计划“基于深度强化学习的盾构姿态实时智能控制技术研究”(合同编号:隧研合2018-40)。

摘　　要：回旋线是一类重要的缓和曲线,是公路、铁路和隧道建设中弯道设计的一种重要规划曲线.因其轨迹不能用解析表达式表示,通常用近似计算方案来代替.传统的方法是将轨迹上的点坐标用关于弧长的Taylor展式表示,缺点是随着弧长的增加,舍入误差比较大,轨迹坐标随曲线的延伸越来越不精确.提出了将描述轨迹曲线坐标位置用关于夹角的奇异积分来表示,通过变量替换转化为正常Riemann积分,然后用复化Simpson公式的数值积分方法去计算轨迹曲线坐标.由Simpson公式的误差估计定理理论分析得出该方法关于剖分尺度h具有关于位置的一致四阶逼近精度.以该数值积分方法作为标准,将传统Taylor公式方法与之比较,数值实验进一步说明新方法可靠,看出传统方法误差确实是误差随曲线延伸而加大,验证了理论分析结果.Circle round curve is a class of easement curves,which plays an important part in railway,subway and tunnel digging.Since this sort of curves can not expressed in analytic way,the numerical methods are often performed in obtaining their trace.The usual method is based Taylor formula,in respect to variable arc length value,by replacing integrand with a polynomial,that is a crude method as the error may become larger and larger as the curve gradually extends its way.We suggest an effective scheme by firstly applying variable angle a nonlinear transformation which converts singular integral to a normal Riemann integral and then being numerically calculated with composite Simpson rule.The approximate estimation depending on interval partition scale h can approach O(h^(4))uniformly with respect to location.Compared with traditional Taylor method and the new numerical integral method,what the numerical experiments show this new method is more reliable.

关 键 词：缓和曲线 回旋线 弯道设计 数值积分 精细计算 

分 类 号：U212.332[交通运输工程—道路与铁道工程]