Abstract

The paper discusses difference between super-elevation of recorded motion-trajectory (MT) curves and constant super-elevation of curve plane and demonstrates that vertical curvature used in railroad literature is not necessarily associated with curve out-of-plane bending or twist. Vertical curvature can be highly nonlinear while the curve remains planar and untwisted. A computational procedure is proposed for identifying MT planar curves using computer-simulation or experimentally recorded data. Data-driven science (DDS) approach allows measuring Frenet vertical-development and super-elevation that define centrifugal inertia force. Difference between motion-independent curve-plane and motion-dependent osculating plane (OP) which contains velocity, acceleration, and inertia centrifugal forces is explained. Three Frenet angles: horizontal-curvature, vertical-development, and bank angles; are used to define planar MT geometry. Zero-torsion and zero-vertical-curvature conditions are derived for identifying planar curves and demonstrating that vertical curvature is not always associated with curve torsion. An orthogonal transformation is used to define OP sweeping angle whose derivative defines curve curvature regardless of curve-plane orientation. Difference between Frenet horizontal-curvature angle and OP sweeping angle is discussed. It is shown that super-elevation of planar-curve surface is equal to Frenet super-elevation if Frenet vertical-development angle is zero. An analytical 3D, yet planar and untwisted, curve is used to demonstrate that planar curves can have non-vanishing and nonlinear curvature, vertical-elevation, vertical curvature, and super-elevation.

摘要

本文讨论了记录的运动轨迹(MT) 曲线的超高与曲线平面的恒定超高之间的差异，并论证了铁路文献中使用的垂直曲率不一定与曲线面外弯曲或扭曲相关。垂直曲率可以是高度非线性的，而曲线保持平面且未扭曲。提出了一种计算程序，用于使用计算机模拟或实验记录的数据来识别 MT平面曲线。数据驱动科学(DDS) 方法允许测量定义离心惯性力的Frenet 垂直发展和超高. 解释了与运动无关的曲线平面与包含速度、加速度和惯性离心力的与运动相关的密切平面(OP) 之间的区别。三个 Frenet 角：水平曲率、垂直展开角和坡度角；用于定义平面 MT 几何。导出零扭转和零垂直曲率条件以识别平面曲线并证明垂直曲率并不总是与曲线扭转相关联。正交变换用于定义OP 扫描角其导数定义曲线曲率，与曲线平面方向无关。讨论了Frenet水平曲率角与OP扫掠角的区别。表明当Frenet垂直展开角为零时，平面曲面的超高等于Frenet超高。分析 3D，但平面和未扭曲的曲线用于证明平面曲线可以具有非零和非线性曲率、垂直高程、垂直曲率和超高。