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Design for Direct Shear, Torsional Shear, and Torsional Deformation
Published in Robert L. Mott, Joseph A. Untener, Applied Strength of Materials, 2016
Robert L. Mott, Joseph A. Untener
The derivation of the angle-of-twist formula depends on some basic assumptions about the behavior of a circular member when subjected to torsion. As the torque is applied, an element of length L along the outer surface of the member, which was initially straight, rotates through a small angle γ (gamma). Likewise, a radius of the member in a cross section rotates through a small angle θ. In Figure 4–18, the rotations γ and θ are both related to the arc length AB on the surface of the bar. From geometry, for small angles, the arc length is the product of the angle in radians and the distance from the center of the rotation. Therefore, the arc length AB can be expressed as either AB=γL
Berry phase of the linearly polarized light wave along an optical fiber and its electromagnetic curves via quasi adapted frame
Published in Waves in Random and Complex Media, 2022
Talat Körpinar, Rıdvan Cem Demirkol
Let be an arc-length parametrized curve in the 3D space i.e. The arc-length parametrized curve is also called a unit speed curve. A unit speed curve Ψ is said to be a Serret–Frenet curve if Serret–Frenet curves admit a Serret–Frenet orthonormal frame field along with Ψ, which satisfies following Serret–Frenet formulae Here, and τ are called the curvature and torsion functions. are called tangent, principal normal, and binormal vector fields of the Serret–Frenet curve Ψ. The vector product of Serret–Frenet vector fields is given by
Onesided offsetting and smoothing algorithm for complex 3D trimming curve of trimming insert steel for automotive panels
Published in International Journal of Computer Integrated Manufacturing, 2020
Trimming curve is an irregular 3D curve that is difficult to describe by function. The curve offset can be achieved by the method of offset point. There are three methods of curve discretization: equal chord length, equal chord height and equal arc length. The equal arc length discretization method is adopted by this algorithm, so that the discrete points uniformly distributed on the curve. The discrete arc length is set to 1 mm, because the automotive panel has a trimming deviation of ±0.5 mm, so the 1 mm discrete arc length meets the accuracy requirement of the trimming curve. Figure 9 is a discrete example of a 3D trimming curve with a 266 mm length. The discrete arc length is 1 mm and 267 discrete sample points are obtained, as shown in Figure 9 (b). The discrete sample points are then constructed into a spline curve through quadratic spline function, as shown in Figure 9 (c). The deviation analysis of the trimming curve and the rebuilt spline curve is carried out, and 1000 points were selected for comparison. The results show that the average deviation is 0.001850 mm and the maximum deviation is 0.040524 mm, which is much smaller than the deviation of the trimming curve ±0.5 mm. It is proved that the discrete arc length of 1 mm meets the accuracy requirements of the trimming curve. The discrete points set is marked .Curve s1 and discrete sampling points are projected in the trimming direction to the installation reference plane of the inserted steel, and the projection curves C2 and 2D discrete sampling points are obtained.
Discrete spectrum of interactions concentrated near conical surfaces
Published in Applicable Analysis, 2018
Thomas Ourmières-Bonafos, Konstantin Pankrashkin
Denote by the length of and set . Furthermore, let us choose an arc-length parametrization of , i.e. an injective function such that and and set . Recall that the geodesic curvature of at a point is defined through , i.e. , and the assumption (1.1) takes the form