Application of Frenet Geometry and Recorded Motion Trajectories to Railroad Vehicle System Dynamics

Geometry description plays a central role in many engineering applications and directly influences the quality of the computer simulation results. This thesis is focused on developing and applying a procedure for quantifying the forces of railroad vehicle dynamics during curve negotiation with the goal of identifying the root-causes of derailments based on Frenet geometry and recorded-motion trajectory analysis (RMT). One of the goals of this thesis is to demonstrate the noncommutativity of finite rotations when Euler angles are used as geometry field variables to define curve geometric properties, curvature and torsion, when different rotation sequences are considered. Therefore, a systematic approach for developing the relationships between Euler angles and the curvature and torsion of a space curve will be developed. Adopting the sequence used in practice, the Euler-angle numerical representation of the railroad track geometry can be developed in terms of the horizontal curvature, vertical development, and track bank angles. These equations will help explain the relationships between the track input parameters and introduce a new definition, Frenet bank angle, to better understand the difference from the track bank angle, used to describe track geometry in railroad vehicle algorithms. It is demonstrated in this thesis that Frenet bank angle defines the actual direction of the centrifugal force and defines the actual balance speed, which is not a priori known. This thesis will contribute to developing a data-driven science (DDS) solution framework for characterization and interpretation of railroad vehicle oscillations and inertia forces. An approach based on the Frenet geometry and RMT will be introduced in this thesis to examine forward-kinematics effect on the motion geometry and inertia forces and to obtain a general expression for the ratio between the amplitudes of the centrifugal and gravity forces. The new concepts of instantaneous motion plane (IMP) and instantaneous zero-force axis (IZFA) are used to convert three-dimensional Cartesian forces into two-dimensional Frenet forces, based on the Frenet-Newton equations. To this end, a new spatial L/V formulation based on nonlinear dynamics and the assumptions of Nadal’s limit will be introduced, which employs non-generalized coordinates. The single-degree-of-freedom wheel-slide model developed accounts for curving behavior can be used to develop an inverse problem to study different wheel slide patterns. It is shown that the spatial L/V limit can approach four if the direction of the friction force is properly accounted for, highlighting the need for accurate measurement of the components of the relative velocity at the wheel/rail contact point. The application of the RMT and Frenet force analyses can have a significant impact on enhancing positive train control algorithms (PTC).