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Optimization of Geometrical Engagement Parameters for Gear Honing
Published in Stephen P. Radzevich, Advances in Gear Design and Manufacture, 2019
The design of tools with such a profile is challenging, as the involute of the modification and the involute of the gear profile have no frequent real intersections. This leads to the formation of a step near the limiting point of the involute and thus to the occurrence of the interference mentioned above. Figure 2.20 shows examples of modified gear profiles with and without a step in the area of the tooth profile's limiting point. It is apparent from Figure 2.20 that the relative position of the main involute, the involute of the modification, and the connecting curve as well as their components can be definitely controlled by changing the parameters of machine engagement. However, numerical values are required for such a specific control, characterizing the tooth profile of the gear, particularly of its connecting curve. Such an integral quantity is the curvature radius of the connecting curve [8], which is established with the dependence (Equation 2.29). Figure 2.21 shows the curvature radius of the connecting curve depending on the magnitude of the actual angle of engagement αnw01. As can be seen from Figure 2.21, the curvature radius of the connecting curve changes significantly (in the examples here, it varies between 0.2 and 5.5 mm). To assess the curvature of the connecting curve, some researchers introduced various concepts—the arithmetic mean of the curvature radius, the mean integral radius, the apothem, the specific radius, and so on. This approach produced only an estimation of the connecting curve's curvature. The algorithm developed makes it possible to establish the curvature radius of the connecting curve for the tooth profile of the gear at each point depending on the parameters of machine engagement as well as to compare it with the given curvature and the form of the connecting curve. This offers the possibility for solving the inverse problem, that is, establishing the parameters of the generating contour in accordance with a given form and curvature of the connecting curve, which is realized by a multiple iterative method for the solution of a direct problem. The analysis showed that the characteristics of the model (objective functions) depend on design parameters in a complex and polyvalent way. In this connection, it is difficult and impractical to explicitly choose the objective functions desired. Hence, the objective functions in the development of technological systems should be used as results of the functioning of the model simulating the geometry of machine engagement.
A Stylized 3-D Benchmark Problem Set Based on the Pin-Fueled SmAHTR
Published in Nuclear Technology, 2020
K. Lisa Reed, Farzad Rahnema, Dingkang Zhang, Dan Ilas
In the original solid cylindrical fuel option, prismatic hexagonal graphite blocks with a 22.5-cm apothem as the outer boundary and a 16.94-cm radius opening for the cylindrical coolant channel are detailed. Each assembly has six concentric rings of pins arranged in a hexagonal lattice pattern with a pitch of 3.08 cm. The center pin and middle ring, which account for a total of 19 pins, are solely graphite pins, and the TRISO compacts make up the other 72 (fuel) pins. Each fuel pin is 1.1 cm in radius with the TRISO packed in graphite at a 50% volumetric packing fraction. The fuel pins are cladded in 0.3 cm of graphite to yield an outer radius of 1.4 cm, and the graphite pins are solely 1.4-cm-radius graphite cylindrical pins. The assembly in the original preconceptual design from the ORNL report1 is shown in Fig. 3. Within the assembly, spacers for the cylindrical pin fuel type are not specified, nor are any other support structures to suspend the pins.
Calculus of pasta, sausages, and bagels: can their surface areas be derivatives of their volumes?
Published in International Journal of Mathematical Education in Science and Technology, 2020
Surprisingly, this volume-area derivative relationship (2) has attracted the attention of mathematicians and math educators only recently. In (Tong, 1997), Tong demonstrated that although formulas (3) and (4) both fail in the case of a square with ‘the natural choice’ of r as side length, but did mention (after Miller [1978]) that the formulas do hold if r is chosen as the in-radius (apothem) of the square. Moreover, the simple use of changing variables through differentiation allowed Tong to formulate a general principle: for any Jordan region (the region bounded by a simple closed curve), a suitable linear dimension, , can be found that provides a derivative relationship between area and perimeter similar to (4).