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Instantaneous motion plane and zero-force axis and their relationship to Frenet geometry

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Abstract

The two vectors of applied and constraint forces form a plane, which is not directly related to the Frenet geometry. Nonetheless, the resultant of the applied and constraint forces lies in a plane defined by the Frenet geometry when the motion trajectory (MT) of the mass center of a rigid body is considered. This, however, is not the case for non-centroidal-point trajectories. For both centroidal and non-centroidal points, an instantaneous motion plane defined by the Frenet-frame osculating plane can be determined. This plane contains the absolute velocity and acceleration vectors, inertia forces including centrifugal force, and resultant of the applied and constraint forces acting at the body center of mass. It is shown that the component of the resultant of the applied and constraint forces along the Frenet bi-normal vector is zero when mass-center trajectories are considered, and therefore, the Frenet bi-normal vector is an instantaneous zero-force axis. The MT analysis is generalized to the case of non-centroidal points in which the bi-normal vector is not orthogonal to the plane formed by the two vectors of applied and constraint forces only. Complexities that arise in case of points different from the mass center are highlighted. At zero-curvature points, singularities that can lead to software crashing can be avoided by proper definition of the vector normal to the space curve. Consequently, the spatial Newton equations can always be transformed to instantaneous planar equations. Developing real-time onboard-computer MT algorithms for autonomous vehicles and positive-train control can contribute to avoiding linearization and simplifications of the equations of motion that may lead to wrong results, particularly in extreme dynamics that characterizes accidents.

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The data of the models developed in the paper are presented in the numerical example section.

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Funding

This research was supported by the National Science Foundation (Project # 1852510).

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  1. Department of Mechanical and Industrial Engineering, University of Illinois at Chicago, 842 West Taylor Street, Chicago, IL, 60607, USA

    Ahmed A. Shabana

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  1. Ahmed A. Shabana
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Correspondence to Ahmed A. Shabana.

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Shabana, A.A. Instantaneous motion plane and zero-force axis and their relationship to Frenet geometry. Nonlinear Dyn 111, 15737–15748 (2023). https://doi.org/10.1007/s11071-023-08685-3

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