# An Attempt to Establish a Mathematical Model for an Unconventional Worm Gear with Bearings
## 2 Materials and Methods  
The \"classic\" globoid worm gear (Figure 1) consists of a globoid worm and a similar globoid worm wheel [4]. The globoid worm is one in which the helix coils around a toroidal reference surface. Kinematic and functional analyzes of this type of gear are carried out in works [1; 10; 13; 16; 23; 24; 25]. Constructive design aspects are presented in [4]. The materials used include hardened steels (generally case-hardening steels) for the worm and an anti-friction material (tin bronze or aluminum, nodular graphite cast iron or gray) for the worm wheel. The advantages of this gear include its quiet operation, high degree of coverage, and high bearing capacity (smaller gauge at the same power to be transmitted). The main disadvantage is that of frictional losses (due to the sliding friction between the worm turn and the worm gear teeth). As the friction is intense, there is a significant release of heat and, implicitly, the transmission efficiency will be reduced. Another disadvantage is that of positioning accuracy because there is quite a large interplay (denoted \(j\) in Figure 1) between the elements in contact. To reduce the effects of these disadvantages, several variants of \"unconventional\" gears have been proposed in which the sliding friction is replaced by rolling friction (the rolling friction coefficient is about ten times lower than the sliding friction coefficient). Also, the play in the gear can be precisely controlled; even pre-tensioning can be achieved. Such a construction is the worm gear with bearings (Figure 2), which is the subject of study in this paper. In this gear, the shape of the worm does not change, but the driven wheel will have radial bearings instead of teeth. The contact between the worm coil and the bearings is a linear contact (Hertz contact theory). The bearings will rotate around their own axis during operation. By changing the pitch of the spiral of the auger, the contact between its flanks and the bearings will be in different directions (Figure 2), which leads to the cancellation of the play between the elements of the gear components. Various studies on specific aspects of this gear can be found in works [2; 3; 4; 5; 6; 18; 19; 26; 7; 27; 28; 29; 30; 31; 32; 33; 34; 35; 36; 37; 38; 39; 40; 41]. In these articles, theoretical and experimental research, 3D modeling, finite element analysis, and research on concrete physical models are presented. Figure 1: Globoid worm gear. The mathematical model of this type of gear, however, is treated in a cursory manner. Rather, the defining equations seem to be obtained after the model generation and not the other way around, as would normally be the case. Geometry of the Unconventional Worm Gear with Bearings, Kinematic Analysis, and Mathematical Model Formulation  
The main geometric elements of the gear are shown in Figure 3. In the axial section, the flanks of the spiral of the worm are rectilinear and tangent to the profiling circle (with the center in \(O_{2}\) and diameter \(d\), equal to the outer diameter of the bearing), which is concentric with the worm wheel. The inclination of the helix profile will be equal to the angular pitch \(\theta\) between the rollers of the worm wheel. Consider point \(M\) positioned in the center of the roller (bearing) and points \(A\), \(B\), \(D\) and \(E\) at its ends. The displacement of the lines \(AB\) and \(DE\) in the \(O_{1 xyz}\) system (related to the worm gear) results in the generation of the flanks of the spiral (Figure 3). The equations of point \(M\) related to the worm wheel (system \(O_{1 xyz}\)) are:  
\[\left\{ \begin{matrix}x=\frac{d_{2}}{2}\cdot\sin\varphi_{2}\\ y=\left(a-\frac{d_{2}}{2}\right)+\frac{d_{2}}{2}\cdot\left(1-\cos\varphi_{2} \right)\\ z=0\end{matrix} \right.\left\{ \begin{matrix}x=\frac{d_{2}}{2}\cdot\sin\varphi_ {2}\\ y=a-\frac{d_{2}}{2}\cdot\cos\varphi_{2}\\ z=0\end{matrix} \right. \tag{1}\]  
Relative to the driven wheel, the trajectory of point \(M\) is an arc of a circle.