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Differential Calculus of Vector Functions of one Variable
Published in C. Young Eutiquio, Vector and Tensor Analysis, 2017
We observe that at each point on a curve, the two unit vectors t and n determine a plane containing that point. This plane is known as the osculating plane. It is the plane that best fits the curve at that point and thus contains the circle of curvature. For a plane curve (not a straight line), the osculating plane coincides with the plane on which the curve lies. In general, however, the osculating plane varies from point to point on the curve as shown in Fig. 2.12. Since t and n are orthogonal unit vectors, it follows that |b| = |t × n| = 1. Moreover, since b is orthogonal to both the vectors t and n, it is orthogonal to the osculating plane. We call the unit vector b the binomial vector to the curve. From the definition of cross product, it follows that the three vectors t, n, b, in that order, form a right-handed triple of orthonormal vectors varying in direction from point to point on the curve. The vectors are sometimes referred to as a moving trihedral (Fig. 2.13).
A review of road models for vehicular control
Published in Vehicle System Dynamics, 2023
When the moving trihedron is rotated so that its osculating plane is tangent to the road surface at the centre line, a third curvature variable must be introduced. In this event the moving trihedron is the Darboux frame. The Darboux frame has spatial angular velocity (in rad/m). In this coordinate system is normal to the osculating plane, is normal to the rectifying plane, while is normal to the normal plane [23,24].
Characterization and quantification of railroad spiral-joint discontinuities
Published in Mechanics Based Design of Structures and Machines, 2022
Because of the difference in the elevations of the conic surface and the osculating plane, which represents the actual motion plane that contains the velocity and acceleration vectors; distinction must be made between the track super-elevation and the super-elevation of the motion (osculating) plane which defines correctly the direction of the centrifugal force. The track (displacement-velocity) plane and the osculating (velocity-acceleration) plane share the vector tangent to the motion-trajectory curve, and therefore, they differ by a single rotation about the tangent vector The geometry of the actual motion-trajectory curve can be significantly different from the geometry of the desired-motion circular curve. In this study, whenever referring to the motion-trajectory curve, the bank angle used, refers to the super-elevation of the motion osculating plane and not the track super-elevation angle These two planes may temporarily coincide, and in this special case, the centrifugal force is in the plane of the super-elevated track; unless the motion is constrained to strictly follow a horizontal circle. The analysis presented in this paper demonstrates that the expression of the balance speed using the track super-elevation is an approximation based on the assumption that the centrifugal force remains in the horizontal plane. Nonetheless, by maintaining the track super-elevation below 6 or 7 inches, this approximation is considered a good approximation. One important observation from Fig. 2, discussed further in the appendix, is that the variation of the Curve-C curvature angle with respect to the arc length of curve is always smaller than the variation of the same angle with respect to the arc length of the circle
Space-curve Cartan matrix and exact differentiability of the curvature and torsion
Published in Mechanics Based Design of Structures and Machines, 2023
Another approach is to view curvature vector as a vector along well-defined unit normal vector and avoid using curvature vector to define normal vector. Magnitude of curvature vector along can assume zero value at zero-curvature points without having effect on definition of the normal vector For space curves, normal vector can be defined everywhere using concept of Frenet angles, which are set of Euler angles performed according to Euler sequence This sequence of rotations can be used to introduce three Frenet angles: curvature anglevertical-development-angle and bank angle (Ling and Shabana 2021; Shabana, 2021a, 2021b). In particular, bank angle defines super-elevation of curve osculating plane. In this study, another rotation sequence is used to confirm conclusions obtained using sequence used in railroad literature (Shabana, 2021b). Normal vector obtained using Frenet angles is continuous, does not flip over in neighborhood of zero-curvature points, and coincides with conventional normal vector over some curve segments and is opposite to over other curve segments when sense of curvature changes. That is, normal vector determined using Frenet angles is not always directed along curve center of curvature.