Slant Helix Curves and Acceleration Centers in Minkowski 3-Space Ε31

In this study, some basic concepts (e.g., instant screw axis (ISA), instantaneous pole points, acceleration pole points) will be given and analyzed about an alternative one-parameter motion of a rigid-body in 3-dimensional Minkowski space Ε31 obtained by moving coordinate frame {N, C, W} along a non-null unit speed curve α = α(t), where N, C and W correspond to unit principal normal vector field, derivative vector field of unit principal normal vector field and Darboux vector field (or angular-velocity vector field) of the non-null unit speed curve α, respectively.

Generation of Noncircular Bevel Gears With Free-Form Tooth Profile and Curvilinear Tooth Lengthwise

Noncircular bevel gear is applied to intersecting axes, realizing given function of transmission ratio. Currently, researches are focused mainly on gear with involute tooth profile and straight tooth lengthwise, while that with free-form tooth profile and curvilinear tooth lengthwise are seldom touched upon. Based on screw theory and equal arc-length mapping method, this paper proposes a generally applicable generating method for noncircular bevel gear with free-form tooth profile and curvilinear tooth lengthwise, covering instant screw axis, conjugate pitch surface, as well as the generator with free-form tooth profile and curvilinear tooth lengthwise. Further, the correctness of the proposed method is verified through illustrations of computerized design.

The Synthesis of the Axodes of RCCC Linkages

As the coupler link of an RCCC linkage moves, its instant screw axis (ISA) sweeps a ruled surface on the fixed link; by the same token, the ISA describes on the coupler link itself a corresponding ruled surface. These two surfaces are the axodes of the linkage, which roll while sliding and maintaining line contact. The axodes not only help to visualize the motion undergone by the coupler link but also can be machined as spatial cams and replace the four-bar linkage, if the need arises. Reported in this paper is a procedure that allows the synthesis of the axodes of an RCCC linkage. The synthesis of this linkage, in turn, is based on dual algebra and the principle of transference, as applied to a spherical four-bar linkage with the same input–output function as the angular variables of the RCCC linkage. Examples of RCCC linkages are included. Moreover, to illustrate the generality of the synthesis procedure, it is also applied to a spherical linkage, namely, the Hooke joint, and to the Bennett linkage.

The Role of the Orthogonal Helicoid in the Generation of the Tooth Flanks of Involute-Gear Pairs With Skew Axes

Camus' concept of auxiliary surface (AS) is extended to the case of involute gears with skew axes. In the case at hand, we show that the AS is an orthogonal helicoid whose axis (a) lies in the cylindroid and (b) is normal to the instant screw axis of one gear with respect to its meshing counterpart; in general, the helicoid axis is skew with respect to the latter. According to the spatial version of Camus' Theorem, any line or surface attached to the AS, in particular any line L of AS itself, can be chosen to generate a pair of conjugate flanks with line contact. While the pair of conjugate flanks is geometrically feasible, as they always share a line of contact and the tangent plane at each point of this line, they even have the same curvature, G2-continuity, when L coincides with the instant screw axis (ISA). This means that the two surfaces penetrate each other, at the same common line. The outcome is that the surfaces are not realizable as tooth flanks. Nevertheless, this is a fundamental step toward the synthesis of the flanks of involute gears with skew axes. In fact, the above-mentioned interpenetration between the tooth flanks can be avoided by choosing a smooth surface attached to the AS, instead of a line of the AS itself, which can give, in particular, the spatial version of octoidal bevel gears, when a planar surface is chosen.

Design of Constant-Velocity Transmission Devices Using Parallel Kinematics Principle

This paper presents a new type of constant-velocity transmission devices based on parallel mechanisms with properties of equal-diameter spherical pure rolling. The method we used is essentially an extension of the planar ellipse gear to the spherical one. Both the fixed and moving axodes of a specified parallel mechanism are obtained, as traced by the spatial instant screw axis (ISA) with respect to the fixed and moving coordinate systems. Based on Poinsot’s theorem and achievements, a series of these parallel mechanisms which satisfy constant-velocity condition have been disclosed correspondingly. Their motion range and transmission performances are also explored by taking the 3-4R mechanism as an instance. As the main part of this paper, two important applications for this type of constant-velocity transmission devices are also explored. One is used as a gearless spherical gear, and the other is used as a constant-velocity universal joint (CVJ). Simulations were fulfilled on ADAMS to verify the transmission performance in terms of different applications.

The Role of the Orthogonal Helicoid in the Generation of the Tooth Flanks of Involute-Gear Pairs With Skew Axes

Camus’ concept of Auxiliary Surface (AS) is extended to the case of involute gears with skew axes. In the case at hand, we show that the AS is an orthogonal helicoid whose axis a) lies in the cylindroid and b) is normal to the instant screw axis of one gear with respect to its meshing counterpart; in general, the helicoid axis is skew with respect to the latter. According to the spatial version of Camus’ Theorem, any line attached to the AS, in particular any generator g of AS itself, can be chosen to generate a pair of conjugate flanks with line contact. While the pair of conjugate flanks is geometrically feasible, as they always share a line of contact and the tangent plane at each point of this line, there are poses where the flanks even have a common Disteli axis. Then there is a G2-contact at the striction point and the two surfaces penetrate each other. The outcome is that the surfaces are not realizable as tooth flanks. Nevertheless, this is a fundamental step towards the synthesis of the flanks of involute gears with skew axes.

The Synthesis of the Axodes of Spatial Four-Bar Linkages

The paper introduces a procedure for the motion analysis of a four-bar linkage by means of dual algebra and the Principle of Transference. This procedure allows the mapping of the motion from the Euclidean to the spherical dual space. In particular, the position analysis of a spatial four-bar linkage is formulated by referring to a spherical four-bar linkage, which moves on the dual unit sphere. Moreover, both fixed and moving axodes of the coupler link are obtained, as traced by the spatial motion of the instant screw axis (ISA) with respect to the fixed and moving frames. These ruled surfaces reproduce the spatial motion of the coupler link upon relatively rolling and sliding around and along the ISA. Finally, the proposed procedure has been implemented in MatLab, in order to analyze the motion of the different types of four-bar linkages, including the Bennett mechanism and the Hooke joint. The motion is illustrated by means of animations of the four-bar linkages and their axodes.

The Synthesis of the Pitch Surfaces of Internal and External Skew-Gears and Their Racks

The synthesis of the pitch surfaces of any pair of external and internal skew gears, using dual algebra and the principle of transference, is the subject of this paper. The spatial motion of the Euclidean space is transferred to the dual space in order to obtain a simplified dual spherical motion, thus emulating the motion of bevel gears. The relative screw motion is hence analyzed by determining the position of the instant screw axis and the angular and sliding velocities. Moreover, the hyperboloid pitch surfaces of the driving and driven gears are synthesized, along with the helicoid pitch surface of their rack. Several numerical results are reported.