## 5 Finite Element Modelling and Simulation  
Direct-coupled transient thermal-structural analysis was used to assess the heat flux distribution of the worm-gear pair. The modelling and solving of the finite element model were conducted using ANSYS 19.2 software. The rate of frictional dissipation in the contact elements was evaluated using the default ANSYS equation [28]:  
\[q=FHTG\cdot\tau\cdot v, \tag{1}\]  
where:  
\(FHTG\)--the dissipation factor, which takes into account the part of friction energy that is converted into heat;  
\(\tau\)--the equivalent stress that depends on the contact pressure and the friction coefficient;  
\(v\)--the relative sliding velocity. All frictional dissipated energy was considered to be converted into heat, and the distribution of friction-generated heat between the worm and the worm gear was considered equal. The numerical analysis was carried out via three main steps according to the operating conditions listed in Table 3. The model's boundary conditions are defined in Figure 6. The first step was the introduction of the axial force load to establish proper contact between the worm and the wheel. The second step was to increase the speed of the worm drive to the operating speed of 1500 min\({}^{-1}\). The third step was used to assess the heat flux distribution in the operating conditions. The analysis time of the third step that corresponded to the actual operating conditions was 0.4 s, which enabled the gear pair to rotate for several  
Figure 4: Contact pattern at the worm gear with the contact shifted to the outlet side. Figure 5: Contact pattern at the worm gear with the contact shifted to the inlet side. mesh cycles and to achieve consistent results, showing dynamic changes in temperature and contact stress. The central rotation points of the pinion and gear were fixed to the inner side of the worm and wheel. The geometry of the worm and wheel was sliced to obtain simple bodies for optimising the mesh size and quality. The final mesh contained 24,703 elements with 27,926 nodes. The mesh of the worm-gear tooth is shown in Figure 7, and the contact surfaces of the worm and worm gear are shown in Figure 8. The mesh was generated with higher-order elements (SOLID226 [29]) in order to accurately capture the geometry of the worm and wheel. A mesh sensitivity test was performed, and mesh sizing of 1 mm on tooth flanks was adopted to ensure that contact pressure results did not differ by more than 5%. \begin{table}
\begin{tabular}{l c c} \hline \hline
**Parameter** & **Worm** & **Worm Gear** \\ \hline Power [kW] & & 4.5 \\ Speed of worm [min\({}^{-1}\)] & & 1500 \\ Torque at worm gear [Nm] & & 587.28 \\ Mean tooth friction number & & 0.0258 \\ Heat transfer coefficient [(W/m2/K)] & & 24,440 \\ Young’s modulus [Mpa] & 206,000 & 88,300 \\ Poisson’s ratio & 0.300 & 0.350 \\ Specific heat [J/kgK] & 434 & 384 \\ Thermal conductivity [W/(m\(\cdot\)K)] & 60.5 & 60.4 \\ Coefficient of thermal expansion [C\({}^{-1}\)] & \(1.2\times 10^{-5}\) & \(1.85\times 10^{-5}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Operating, material, and thermal data of the worm-gear pair used. Figure 6: FEM setup for the directly coupled transient thermal–structural analysis. The friction coefficient value necessary for the definition of contact pairs was determined based on ISO/TS 14521. The friction coefficient, i.e., mean tooth friction number, as defined in the standard was calculated as:  
\[\mu_{zm}=\mu_{0T}\cdot Y_{S}\cdot Y_{G}\cdot Y_{W}\cdot Y_{R}=0.0245, \tag{2}\]  
In Equation (2), the size factor \(Y_{S}\) takes into account the influence of the centre distance via the following formula:  
\[Y_{S}=(100/a)^{0.5}=1, \tag{3}\]  
The geometry factor \(Y_{G}\) considers the gear geometry's influence on the lubricating gap thickness. It depends on the characteristic value for mean lubrication gap width \(h^{*}=0.0692\) that is calculated according to ISO 14521.