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.

New method for determining singularities on enveloped surface and its application to study curvature interference theory of involute worm drive

In this paper, a more computationally convenient singularity condition of the enveloped surface is proposed using the theory of linear algebra. Its preconditions are only the tangential vector of the enveloping surface, the relative velocity vector, and the total differential of the meshing function. It avoids calculating the curvature parameters of the enveloping surface. It is proved that the singularity conditions of enveloped surface from different references are equivalent to each other and the relational equations among them are obtained. The curvature interference theory for the involute worm drive is established using the proposed singularity condition. The equation for the singularity trajectory is obtained. The calculation method for the singularity trajectory is proposed and its numerical result is obtained. The influence of the design parameters on the singularity trajectory is studied using the proposed curvature interference theory. The study results show that the risk of curvature interference is high when the transmission ratio is too small, especially in the case of the single-threaded worm and large modulus. The proposed singularity condition can also be applied to study the curvature interference mechanism in other types of the worm drive and to study the undercutting mechanism when machining the worm drive.

Worm addendum thickness and gear curvature interference for enveloping cylindrical worm drive with arc-toothed worm

The purpose of this paper is to provide the calculation methods on worm addendum thickness and curvature interference limit line, and find the feasible value range of the technological crossing angle to avoiding addendum sharpening and curvature interference for enveloping cylindrical worm drive with arc-toothed worm. In accordance with the features of the proposed worm, the mathematical models of cutting and working are established. Based on this, the tooth profile geometry of the worm in its axial section and the worm addendum thickness are obtained by geometric analysis and calculation, and then, the feasible value range of the technological crossing angle is given. In virtue of vector rotation and elimination method, the nonlinear equation with one variable for solving the interference limit line is determined. In the process of solving nonlinear equation, the method of geometric construction is used to judge the existence of solutions and provide an initial value for the subsequent iterative calculation. The numerical example results show that with the increases of the technological crossing angle, the interference limit line is close to the boundary line of the conjugate region of the worm pair, and the hazard of curvature interference evident increases. Generally, a smaller value of the technological crossing angle within its available value range can completely avoid the occurrence of the curvature interference.

An Analysis of Manufacturing Precision of Involute Worms Using a Kinematical Model

The manufacturing precision of involute worms constitutes a major requirement. On the one hand, the worm constitutes the input element of the worm drive; secondly, the involute helical surface is the basic surface of an involute worm-hob. This paper presents an analytic comparison between the involute surfaces obtained using theoretical equations, kinematic simulation of the cutting and the surface charged with errors. The surface error is considered the distance along the normal direction to the theoretical surface, measured between this and the surface charged with simulated manufacturing errors. The main sources of errors are considered the center-error of the edge plane, the edge profile error and deviation of the axial feed direction from the axis of the worm. The theoretical results allow us to conclude that the influence of the edge profile error is the largest. It is followed by the parallelism error between the feed direction and the axis of the worm, and finally, the center error of the tool edge.

Meshing theory of axial arc tooth profile cylindrical worm drive

In this paper, the axial arc tooth profile cylindrical worm drive is proposed, whose worm is cut by turning tool. Due to the simple processing equipment, short manufacturing time and low cost, this kind of worm can replace the cylindrical worm ground by grinding wheel under some conditions. The meshing theory of the worm drive is founded comprehensively. Moreover, a movable orthogonal frame is established on the non-orthogonal parametric curves net helical surface. Based on the founded meshing theory, the simulating study on the meshing quality of the worm drive is performed systematically. The numerical outcome shows that the meshing quality of this worm drive is quite favorable, and the condition of forming lubricating oil film is excellent. The working orthogonal clearance of the turning tool is decreased with the increase of the worm thread number, which must be a positive value in the process of worm cutting. This explains that the number of the worm thread is ≤4.