Three-dimensional high-order continuous curve alignment design of horizontal and vertical overlapping region of high-speed railways
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摘要: 针对平竖重合曲线段存在几何连续性衰减并引起列车运动状态突变的现象, 以三维曲线的Frenet标架为基础, 结合曲率、挠率建立三维车体运动状态模型, 得到了曲率、挠率与车体加速度、急动度的关系, 并通过该模型从三维角度分析了三维曲线的几何连续性等级对车体运动的影响; 考虑几何连续性对曲率、挠率的要求, 提出以曲线曲率、挠率变化最小为目标的线形选择方法, 利用三维欧拉曲线创建平竖重合段高阶连续曲线。研究结果表明: 传统平竖重合段曲线连接点处几何连续性存在衰减, 仅为1阶几何连续, 曲率、挠率对列车加速度和急动度起主导作用, 几何连续性的衰减是竖向急动度突增的主要原因; 二维设计曲线在起点处的竖向急动度为1.206~1.264 m·s-3, 超过乘客舒适性运动学阈值0.240 m·s-3, 难以实现二维线形的高阶几何连续; 提出的曲线设计方法对连接点处的曲率和挠率都有明确定义, 容易在连接点处实现高阶几何连续, 且不存在几何连续性衰减, 曲线的曲率、挠率变化最小, 可有效降低线形参数变化给车体运动带来的不良影响; 所建曲线的加速度与急动度在全程均连续且满足运动学阈值, 实现了2阶几何连续, 最大竖向急动度为0.149 m·s-3, 为阈值的62.0%, 为二维设计的11.7%~12.3%, 有效地改善了行车稳定性与乘客舒适性; 所建曲线路径与二维设计相比变化小, 在2%~3%坡度差时, 水平、竖向坐标差分别为0.907~2.305、1.085~2.498m;所建曲线的设计参数同时也是车体运动状态的计算参数, 从而可根据列车运行条件直接优化线路的设计。Abstract: For the phenomenon that the horizontal and vertical overlapping region of curves is of geometric continuity degradation and then leads to sudden change of train motion state, based on the Frenet frame of three-dimensional curve, a three-dimensional carbody motion state model was established by combining curvature and torsion.The relationship among curvature, torsion, acceleration and jerk of carbody was obtained, and then the influence of geometric continuity order of three-dimensional curve on carbody motion was analyzed from three-dimensional point ofview by this model.Considering the geometric continuity requirement for curvature and torsion, an alignment selection method minimizing the changes of curvature and torsion in curve was proposed.A three-dimensional Euler curve was used to establish a high-order continuous curve in horizontal and vertical overlapping region.Research result shows that there is a geometric continuity degradation at the connection point of the traditional horizontal and vertical overlapping region of curves, and it is first order geometric continuity only.The curvature and torsion play dominant roles in the acceleration and jerk of trains. Geometric continuity degradation is the main reason for the sudden increase in vertical jerk.The vertical jerk of twodimensional design curve at the connection point is 1.206-1.264 m·s-3, exceeding the passenger comfort threshold value (0.240 m·s-3), so it is difficult to realize the high-order geometric continuity of two-dimensional alignment.The proposed curve design method has clear definitions on curvature and torsion at connection points, so it can easily achieve high-order geometric continuity at connection points without geometric continuity degradation.The changes of curvature and torsion in this curve are minimized.The proposed curve can effectively reduce the adverse effects on carbody motion caused by changes of curve parameters.The acceleration and jerk are continuous and meet the threshold values over the entire curve, realizing the second order geometric continuity.The maximum vertical jerk is 0.149 m·s-3, 62.0% of threshold value, and 11.7%-12.3% of that of two-dimensional design.The proposed curve effectively improves the running stability of train and passenger comfort.The difference between paths of the proposed curve and the two-dimensional design is small.When the difference in gradient is 2%-3%, the horizontal and vertical coordinate differences are 0.907-2.305 and 1.085-2.498 m, respectively.the design parameters of the proposed curve are also used as calculation parameters of the carbody motion state, so this curve can be directly applied to the optimized design according to train running conditions.
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图 4 竖向急动度与平竖曲线半径的关系
Figure 4. Relationships between vertical jerk and horizontal and vertical curves radii
图 9 迭代过程产生的系列欧拉变坡曲线
Figure 9. Series of Euler curves of gradient variation created by iterative process
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