Frenet oscillations and Frenet–Euler angles: curvature singularity and motion-trajectory analysis

Abstract

Motion-trajectory (MT) curves are used to introduce Frenet oscillations. Time-varying orientation of the motion plane that contains the absolute velocity and acceleration vectors is defined in terms of three Frenet–Euler angles; the curvature, vertical-development, and bank angles, referred to as the Frenet angles for brevity. The Frenet bank angle and the associated Frenet super-elevation of the motion plane, which measure deviation of the centrifugal inertia force from the horizontal plane, can be used to shed light on definition of the balance speed used in practice. The concept of the pre-super-elevated osculating (PSEO) plane is introduced and Rodrigues’ formula is employed to develop an orthogonal rotation matrix that provides a geometric interpretation of the PSEO plane. A new inverse-dynamics problem that utilizes experimentally or simulation recorded motion trajectories (RMT) is used to define the Frenet inertia forces and demonstrate their equivalence to the Cartesian form of the inertia forces. New expressions for the curvature vector in terms of the velocity and acceleration, limit on the magnitude of the tangential acceleration for a given forward velocity, condition required for the centrifugal force to remain horizontal, and condition of curvature singular points are derived. The Frenet bank angle can be used to prove existence of the normal vectors at the curvature singular points. It is shown that the inertia force can assume different forms, depending on the curve parameter used. The results of a simple analytical curve demonstrate Frenet oscillations and importance of distinguishing between the highway-ramp and railroad track bank angles and super-elevations, which are time-invariant, and the Frenet bank angle and super-elevation, which are motion-dependent.