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A Simplified Approach for Involute Gear Tooth Flank Generation
Published in Stephen P. Radzevich, Theory of Gearing, 2018
The involute of a circle starts at a point, A, within the base circle of a radius, rb.g. The magnitude of the position vector, rm, of an arbitrary point, m, of the involute curve can be expressed in terms of the base radius, rb.g, and the central angle ε=∠(AOgM). The length of the circular arc, AB⌢, is equal to the length of the straight line segment, AB. This is because the straight line rolls with no slippage over the base circle. From △BOgM, the following equation can be composed: Rm=rb.gtanϕt
Radiometry and Photometry
Published in Vasudevan Lakshminarayanan, Hassen Ghalila, Ahmed Ammar, L. Srinivasa Varadharajan, Understanding Optics with Python, 2018
Vasudevan Lakshminarayanan, Hassen Ghalila, Ahmed Ammar, L. Srinivasa Varadharajan
We know that a circular arc of length s subtends an angle θ at the center of a circle, where θ is defined as the ratio of the arc length to the radius r of the circle. We could easily extend this two-dimensional idea to three-dimensions and define what is called a solid angle. The solid angle is defined as ) Ω=area of the surface projected onto the tangent of a sphere(radius of the sphere)2
Longitudinal-vertical dynamics of a high-speed train rescued by locomotives during braking on grades
Published in Vehicle System Dynamics, 2023
Lai Wei, Jing Zeng, Sheng Qu, Caihong Huang, Qunsheng Wang
The vertical definition of the track is made up of constant transition curve sections. Each of the sections is defined over a given track length. The unit of the track grade is per mill, i.e. one-tenth of a percent. This can be understood that the track will gain a height of 1 m over 1000 m. The maximum gradient for high-speed passenger lines in China is 20‰ with a vertical curve of 15,000 m. The vertical curve between adjacent grades can be expressed by constant, quadratic parabolic, or circular arc transition. The projected length of the circular arc to the radius of the vertical curve is given by: where, L is the section length; and are the gradients at the start and end of the transition section respectively; R is the radius of the vertical curve.
Analysis of pressure on the roof of a culvert underneath a ditch with compressible material covered by a geosynthetic layer
Published in European Journal of Environmental and Civil Engineering, 2021
Qiang Ma, Jun-Jie Zheng, Heng-lin Xiao
Zheng et al. (2009) pointed out that the curve of deflected geosynthetic could be approximately described as catenary, and when the deflection is not very large, the deflected geosynthetic curve can be idealized to be a circular arc of radius with a subtended angle 2. During the calculation, it takes the reaction of the compressible material in the load reduction ditch into account and assumes that the deformed geosynthetic curve is arc. As shown in Figure 4, the midpoint of geosynthetic is origin of horizontal x-axis and vertical y-axis, deformed geosynthetic can be expressed as (, , the mid-point deflection for the geosynthetic. It is the settlement relative to the edge of the geosynthetic, that is, the difference between the settlement of the central point of the geosynthetic and the settlement of the top surface of the filling on both sides. Thus, =s12−s11(s13), as shown in Figure 3.
Cable Surface for the Reduction of Risk Associated with Bridge Cable Ice Accretions
Published in Structural Engineering International, 2019
Lubomir Matejicka, Christos Thomas Georgakis, Andreas Schwarz, Philipp Egger
The second cable section had the innovative surface with laterally staggered concave fillets in a double-helical pattern, referred to as “the helically-staggered concave-filleted surface” or “the HSC-filleted surface” (Fig. 1b). The helically-staggered concave-filleted surface was initially developed to eliminate the rain-wind induced vibrations (RWIV) of bridge cables.24 Later, preliminary icing tests showed an improved ice-shedding performance when compared with other bridge cable surfaces.33 Based on this preliminary test series and the need to modify the fillet profile while maintaining its robustness, the HSC fillet height for the current test series was chosen to be 6.3 mm (Fig. 1c). Each fillet had a double concave cross-section with a thickness of 0.9 mm at the top descending into the surface of the pipe, where the base thickness of the fillet was 5 mm. The fillet length was defined by a circular arc with a central angle of 60°, which is equal to 1/6 of the circumference of the pipe. The spacing between the HSC fillets was 20 mm, and their lateral offset followed a helical pattern at a 60° pitch angle (Fig. 1d). Similarly to the H-filleted surface, the concave fillets of the HSC-filleted surface were made from the same grade of HDPE material as the pipe. However, in this case, the fillets were attached to the pipe through ultrasonic welding.