A generalized meshing analysis method and its application to toroidal surface enveloping conical worm drive

A generalized method for the meshing analysis of conical worm drive is proposed, whose mathematical model is more general and whose application scope is expanded. A universal mathematical model, which can be conveniently applied to left-handed and right-handed conical worm pairs and their tooth flanks on different sides, is established by introducing the helical spin coefficient and tooth side coefficient of the conical worm. The pressure angle at the reference point, which is a key parameter for calculating the curvature parameters and lubrication angle, is determined based on the unit normal vector of the worm helical surface and is no longer determined by the tooth profile angle in the worm shaft section. The above improvement breaks away from the limitation of the classic meshing analysis method based on the reference-point-based meshing theory and thus expands its application scope. The toroidal surface enveloping conical worm drive is taken as an instance to illustrate the proposed method and the numerical example studies are conducted. The approaches to determine the reference point, the normal unit vector, and the curvature parameters at the reference point are all demonstrated in detail. The numerical results all manifest that the method presented in the current work is correct and practicable.

Toroidal Vortices of Energy in Tightly Focused Second-Order Cylindrical Vector Beams

In this paper, we simulate the focusing of a cylindrical vector beam (CVB) of second order, using the Richards–Wolf formula. Many papers have been published on focusing CVB, but they did not report on forming of the toroidal vortices of energy (TVE) near the focus. TVE are fluxes of light energy in longitudinal planes along closed paths around some critical points at which the flux of energy is zero. In the 3D case, such longitudinal energy fluxes form a toroidal surface, and the critical points around which the energy rotates form a circle lying in the transverse plane. TVE are formed in pairs with different directions of rotation (similar to optical vortices with topological charges of different signs). We show that when light with a wavelength of 532 nm is focused by a lens with numerical aperture NA = 0.95, toroidal vortices periodically appear at a distance of about 0.45 μm (0.85 λ) from the axis (with a period along the z-axis of 0.8 μm (1.5 λ)). The vortices arise in pairs: the vortex nearest to the focal plane is twisted clockwise, and the next vortex is twisted counterclockwise. These vortices are accompanied by saddle points. At higher distances from the z-axis, this pattern of toroidal vortices is repeated, and at a distance of about 0.7 μm (1.3 λ), a region in which toroidal vortices are repeated along the z-axis is observed. When the beam is focused and limited by a narrow annular aperture, these toroidal vortices are not observed.

Substantiation of Parameters of Friction Elements of Bernoulli Grippers With a Cylindrical Nozzle

The article provides a step-by-step justification of the parameters of the friction elements of the Bernoulli gripping devices with a cylindrical nozzle. The effect of friction element parameters on the lifting force of Bernoulli grippers with the classic design of the active surface and with the rounded-off nozzle and flat and toroidal surface is considered. Influence of friction elements location radius on grip lifting force is considered. Influence of friction elements shape on grip lifting force is considered. Effect of friction coefficient between friction elements of grip and object of manipulation on minimum required lifting force in order to perform handling operation is investigated. Influence of number of friction elements on Bernoulli grip lifting force is considered. For the grip design with the rounded-off nose and flat and toroidal surface when the elliptical friction elements overlap the end gap by 73%, the lifting force will increase by 7% in torsion with the lifting force without the friction elements.

Vortex crystals on the surface of a torus

Vortex crystals are equilibrium states of point vortices whose relative configuration is unchanged throughout the evolution. They are examples of stationary point configurations subject to a logarithmic particle interaction energy, which give rise to phenomenological models of pattern formations in incompressible fluids, superconductors, superfluids and Bose–Einstein condensates. In this paper, we consider vortex crystals rotating at a constant speed in the latitudinal direction on the surface of a torus. The problem of finding vortex crystals is formulated as a linear null equation A Γ = 0 for a non-normal matrix A whose entities are derived from the locations of point vortices, and a vector Γ consisting of the strengths of point vortices and the latitudinal speed of rotation. Point configurations of vortex crystals are obtained numerically through the singular value decomposition by prescribing their locations and/or by moving them randomly so that the matrix A becomes rank deficient. Their strengths are taken from the null space corresponding to the zero singular values. The toroidal surface has a non-constant curvature and a handle structure, which are geometrically different from the plane and the spherical surface where vortex crystals have been constructed in the preceding studies. We find new vortex crystals that are associated with these toroidal geometry: (i) a polygonal arrangement of point vortices around the line of longitude; (ii) multiple latitudinal polygonal ring configurations of point vortices that are evenly arranged around the handle; and (iii) point configurations along helical curves corresponding to the fundamental group of the toroidal surface. We observe the strengths of point vortices and the behaviour of their distribution as the number of point vortices gets larger. Their linear stability is also examined. This article is part of the theme issue ‘Topological and geometrical aspects of mass and vortex dynamics’.

Influence of grinding wheel parameters on the meshing performance of toroidal surface enveloping conical worm drive

A new type of toroidal surface enveloping conical worm gearing is proposed in our recent work (Chongfei and Yaping, 2019b). According to its forming principle, the geometrical shape of the generating surface has an important influence on the geometry characteristic of the enveloping worm pair. To explore the reasonable principles for selecting the geometrical parameters of the grinding wheel, some numerical study examples are performed. In this process, the methods for the tooth crest width are developed. Simple strategies for estimating the risk of the worm tooth surface being located in the invalid area and the risk of the curvature interference on the tooth surface are proposed. The numerical result shows that increasing the radius of the toroidal-generating surface and the nominal pressure angle of the grinding wheel are beneficial to improve the engagement behavior of the conical worm pair, but the tooth crest sharpening of the conical worm may happen if they are too large. For the nominal radius of the grinding wheel, it has a negligible effect on the meshing characteristics of this worm set. In addition, the selection principle of the parameters is also suggested.

Exact solution to a Liouville equation with Stuart vortex distribution on the surface of a torus

A steady solution of the incompressible Euler equation on a toroidal surface T R , r of major radius R and minor radius r is provided. Its streamfunction is represented by an exact solution to the modified Liouville equation, ∇ T R , r 2 ψ = c e d ψ + ( 8 / d ) κ , where ∇ T R , r 2 and κ denote the Laplace–Beltrami operator and the Gauss curvature of the toroidal surface respectively, and c , d are real parameters with cd < 0. This is a generalization of the flows with smooth vorticity distributions owing to Stuart (Stuart 1967 J. Fluid Mech. 29 , 417–440. ( doi:10.1017/S0022112067000941 )) in the plane and Crowdy (Crowdy 2004 J. Fluid Mech. 498 , 381–402. ( doi:10.1017/S0022112003007043 )) on the spherical surface. The flow consists of two point vortices at the innermost and the outermost points of the toroidal surface on the same line of a longitude, and a smooth vorticity distribution centred at their antipodal position. Since the surface of a torus has non-constant curvature and a handle structure that are different geometric features from the plane and the spherical surface, we focus on how these geometric properties of the torus affect the topological flow structures along with the change of the aspect ratio α = R / r . A comparison with the Stuart vortex on the flat torus is also made.