In this article, a mathematical description of tooth flank surface of the globoidal worm and worm wheel generated by the hourglass worm hob with straight tooth axial profile is presented. The kinematic system of globoidal worm gear is shown. The equation of globoid helix and tooth axial profile of worm is derived to determine worm tooth surface. Based on the equation of meshing the contact lines are obtained. The mathematical description of globoidal worm wheel tooth flank is performed on the basis of contact lines and generating the tooth side by the extreme cutting edge of worm hob. The presented mathematical model of tooth flank of TA worm and worm wheel can be used e.g. to analyse the contact pattern of the gear.

This study describes the methodology for modelling a worm and worm wheel of a double enveloping worm gear with the use of a CAD system. An algorithm for generating a globoid helix is described. In addition, the methodology for modelling an hourglass worm thread with a straight axial tooth profile is presented. The shape of the hourglass worm tooth end with and without a trace modification is proposed. Moreover, a method for achieving a geometric modification of the tooth trace was developed. Next, the method for modelling the worm wheel teeth is described. A solid model of using a machining worm as a hob is applied. Owing to the limitations of a CAD system, which prevents the use of a direct machining simulation, an indirect modelling method is introduced. In the present study, different CAD techniques, both solid and surface, are applied. Knowledge of the correct modelling of the hourglass worm and worm wheel facilitates their generation and conducting various analyses, including a tooth contact analysis. CAD models are utilised to analyse the geometrical contact pattern in a CAD environment, to carry out FEM analysis, to manufacture real parts or to prototype models using the technique of rapid prototyping. They can be also used as master models for measurement, e.g. in optical technics.

The paper presents the method of determining geometric contact pattern by using the direct computer-aided design (CAD) method for ideal globoid worm gear in which mounting deviations are considered. The tooth contact analysis was performed for all cycle of worm rotation. Based on the results of temporary contact pattern, graphical characteristics of the contact area size depending on worm position were made. A complete analysis of the correctness of gear meshing can be obtained based on presented method. If the worm or worm wheel is incorrectly designed in terms of the geometry, the meshing simulation of CAD models can indicate the collision. Geometric contact pattern analyses were made at two pressure angles of ideal gear. The analysis of the influence of mounting deviations was done against one selected pressure angle and one gear position.