# An Attempt to Establish a Mathematical Model for an Unconventional Worm Gear with Bearings
## 1 Introduction  
To fulfill the increasing demands for a high gear transmission ratio using a single stage (up to 80) with compact gear dimensions, high load-carrying capacity, and smoothness in operation, etc.--needed for lifting-and-shifting machines in the aerospace industry and modern manufacturing processes--various worm drive types are used. These space-saving gearings, depending on the external surface shape of the worm and wheel, can be cylindrical (with a cylindrical worm known as a single-enveloping worm being the most commonly used [1]); global, known as the TA or Hindley worm drive or double-enveloping worm (DEW) (in this case, either the worm is global [2] or both the worm and wheel are globoid); or special (where the worm has a conical shape, called a spiroid [1], and the other element is a face-type gear) [3]. Until now, worm drives have been the subject of many researchers' studies. In the technical literature, there are significant publications about mathematical modeling methods, various surface analyses, research regarding new types of worm gears, and manufacturing processes. A summary of the development phases of these worm gearings is presented in [4]. [PERSON] and [PERSON] [5] described a study on the theory of gearing. [PERSON], [PERSON], and [PERSON] [6] presented a detailed kinematic and functional analysis of a double-envelope worm gear. Practical design aspects are detailed [7]. [PERSON], [PERSON], and [PERSON] [8] proposed a new meshing theory for a global worm gear considering some modifications regarding the following parameters: center distance, transmitting ratio, and the height of the cutterframe. [PERSON], [PERSON], and [PERSON] [9] suggested new research on worm gear drives based on the practical manufacturing process known as the direct digital modeling method (DDMM). They presented a detailed discussion on the misalignment errors and contact deformations in the contact zone and concluded that the proposed design solution is characterized by reduced misalignment and transmission errors. [PERSON], [PERSON], and [PERSON] [10] created an algorithm for generating the helix of a double-enveloping worm gear. They generated CAD models, which were used for FEM analysis to manufacture real prototypes employing additive manufacturing processes. [PERSON] and [PERSON] [11] presented a study regarding the curvature analysis for the helicoidal surface of a modified TA worm. However, the author of this study did not investigate the meshing process. A meshing theory for the TA worm gearing is described [12]. The performance of the studied worm gear set is considerably favorable over the other types of conical worm drive. A discussion about a new type of spiroid gear set is detailed [12]. Regarding the manufacturing process, the following studies were conducted. [PERSON], [PERSON], and [PERSON] [13] developed a new method for machining double-enveloping worms (DEWs). Firstly, they carried out a virtual simulation of the cutting process, followed by an experimental test of a worm gear set. In their research, [PERSON] et al. [14] presented a land surface cutting Hob's design for increasing the machining tool's service life. [PERSON] et al. [15] proposed a five-axis tool path for worm gear flank milling. This machining process is then compared to the conventional end milling processes. The method studied offers high process efficiency and surface quality. [PERSON] et al. [16] described a method of rough turning for the toroidal worm that permits higher-efficiency manufacturing. [PERSON] et al. [17] introduced a new type of worm drive consisting of a planar internal gear and a crown worm. The components were machined and then the obtained prototype was assembled and subject to a running test, in which the efficiency and the oil temperature were checked.