Torsion and vertical curvature of motion-trajectory curves

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.