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Method of Complex Potential for Three-Dimensional Problems
Published in Boris Levin, Antenna Engineering, 2017
The presented consideration shows that a choice of surface for placement of the self-complementary radiation structure is not accidental. This surface is the circular cone, which in the limiting case turns into a plane. The shape of a metallic and slotted radiator, located at the circular cone, may be different – similarly to the shape of flat radiators (see Figure 8.8). The simplest shape is obtained, if the edge of a slotted antenna coincides with the cone generatrix. From (8.38) and (8.24) it follows that the magnitude m at least for infinitely long symmetrical slots of this kind located along the circular metal cone is equal to 1/2, i.e. the identical power is radiated inside and outside irrespective of the vertex angle of the cone and width of the slotted antenna. In the general case, the question of the magnitude m is open.
Adjustment of Characteristics of Self-complementary Antennas
Published in Boris Levin, Wide-Range and Multi-Frequency Antennas, 2019
The magnetic radiator, which was used in derivation of (5.8) coincides with a surface of magnetic field strength. It can be not straight but a curved one. For example, it can have the shape of a parabola. Then a metallic surface, which coincides with the surface of the magnetic field strength, will have a shape of a circular paraboloid with a vertex at the point of excitation (Fig. 5.5a). In the general case, that is a surface of revolution with a curved generatrix. On such a surface as well as on a circular cone, which is its particular case, also it is possible to place the self-complementary radiators, to which expression (5.9) is applicable. If radiators are not self-complementary, equality (5.8) is valid.
Integral Equation of Leontovich
Published in Boris Levin, Antennas, 2021
The magnetic radiator used in the analysis of the self-complementary conical structure is located on the surface along which the magnetic field lines pass. These lines may be curvilinear, for example, may have a parabolic shape. In this case, the metallic surface that coincides with the surface along which the magnetic field lines are located has the shape of a circular paraboloid. The point of excitation coincides with the top of the paraboloid (Fig. 2.13). In a general case, it is a surface of revolution with a generatrix in the shape of curve line. On such a surface, as on a circular cone, which represents this particular case, it is also possible to place complementary radiators, to which expression (2.68) is applicable.
Ballistic response of a high-strength steel
Published in Mechanics of Advanced Materials and Structures, 2023
Chun Cheng, Tianpeng Li, Guang Li, Yuanchao Wang, Yuxuan Zheng, Yingqian Fu, Yiwen Ni, Zhaoxiu Jiang
When the impact velocity was 209.13 m/s, the residual fragment was collected as shown in Figure 9. The fragment was intact as a whole, and there were only obvious signs of chiseling damage on the head. The target plates at impact velocities of 178.9 m/s and 209.13 m/s were cut along the plug (perforation) axis to obtain the profile of the plugs and target plates as shown in Figure 10. It can be seen from the figure that the plug is in the shape of a cone as a whole, the diameter of the end close to the impact surface is large, and its outer contour generatrix is perpendicular to the target plate. The reason may be that the pressure and strain rate at the moment of impact were high, and the target plate was damaged by adiabatic shear plugging. The diameter of the end far from the impact surface is small, and the included angle between the outer contour generatrix and the normal direction of the target plate is about 45°. It is likely that with the impact process, the resistance of the target plate gradually reduced the projectile velocity, the pressure and strain rate on the target plate decreased, and the target plate was damaged by shear failure due to compression. Therefore, it can be considered that the failure mode of the F1 fragment to 6 mm high-strength steel target plate is shear plugging failure, but it is not just adiabatic shear plugging, but also shear failure caused by compression. From the target plate in contact with the plug, it can be seen that the target plate around the plug has also undergone slight bending deformation.
Non-equivalent notions of the eccentricity of the conics: an exploratory study with high school teachers
Published in International Journal of Mathematical Education in Science and Technology, 2023
Antonio Rivera-Figueroa, Ernesto Bravo-Díaz
In this third approach, the conics are the curves in the three-dimensional space obtained by slicing a fixed infinite right circular double cone with a variable inclined plane that does not pass through the cone's vertex. In this definition, elements of the curves such as centre, foci, or directrix are not involved. Every plane perpendicular to the axis of the double cone will be called a horizontal plane. The name of conics (or conic sections) comes precisely from the conception of these curves in three-dimensional space. For a given double cone, these curves are classified by the inclined plane's inclination and position (Figure 3). If the inclined plane is horizontal, the curve obtained is a circle. If the inclined plane is parallel to a cone's generatrix, the plane will intersect only one nappe of the cone, and the curve is called a parabola. Suppose the inclined plane is neither horizontal nor parallel to a generatrix. Then an ellipse or a hyperbola is obtained, depending on whether the plane intersects only one nappe of the cone or intersects both nappes. The hyperbola is a curve that consists of two branches, one on each nappe. The eccentricity is not involved in these definitions of the curves but appears in the context of a proposition, established and proven by Germinal Pierre Dandelin (see, e.g. Brannan et al., 2012, p. 22; Simmons, 1996, pp. 551–552) in this proposition the concept of directrix is introduced.
Analysis of nonlinear vibrations of CNT- /fiberglass-reinforced multi-scale truncated conical shell segments
Published in Mechanics Based Design of Structures and Machines, 2022
Seyed Sajad Mirjavadi, Masoud Forsat, Mohammad Reza Barati, A. M. S Hamouda
According to differing values for fiber orientation (), Figure 9 displays the variations of nonlinear-to-linear frequency ratio of truncated conical shell segments with respect to nondimension deflection. It is supposed that Wcnt=0.2% and Vf=0.1. The first issue is that =0 is corresponding to frequency results based on fiber orientations parallel to generatrix direction. According to the findings provided in this figure, frequency ratio of truncated conical shell segments is decreasing with the growth of fiber orientation angle. This means that geometric nonlinearity effects are less prominent at larger values of fiber orientation angle within multi-scale composite.