# Determination of Mesh Stiffness of Gear--Analytical Approach vs. FEM Analysis
## 3 FEM Model  
Firstly, according to the design calculation of gears, five models of gearing with a real teeth shape were created (Gearing 1, 2, 3, 4, 5), because of the calculation, simulation, and subsequent comparison of the engagement of gears and pinions. The basic parameters of these designed gears are listed in the Table 3. Gearing 1, 2, 3, and 4 are based on one designed gearing. The individual variants differ from each other only by the selected different profile shift coefficients \(x_{1}\) and \(x_{2}\). Real tooth profiles of gears 1-4 are shown in the Figure 4.  
Gearing 5 is a gearing that is not typical. Due to the atypical shape of the file-foundation, the tooth profile can't be produced by a standard gear hob. These teeth are not usable in practice, but in this article, they will be used to compare the results and define the validity of the theoretical approach. The tooth profile of the gear 5 is shown in Figure 5.  
Furthermore, the model for the subsequent calculation was simplified by not modeling the whole wheels, but only their sections, which contain four teeth for the pinion and the  
Figure 4: Gear teeth profiles of Gearing 1–Gearing 4.  
Figure 5: Gear tooth profile of Gearing 5.  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline  & \(m_{n}\) [mm] & \(z_{1}[-]\) & \(z_{2}[-]\) & \(x_{1}[-]\) & \(x_{2}[-]\) & \(b\) [mm] & \(\mathbf{\alpha_{n}}[^{\circ}]\) & \(\mathbf{\beta}[^{\circ}]\) \\ \hline
**Gearing 1** & 1.5 & 29 & 30 & 0.2696 & 0.2606 & 23 & 20 & 0 \\ \hline
**Gearing 2** & 1.5 & 29 & 30 & 0.35 & 0.1802 & 23 & 20 & 0 \\ \hline
**Gearing 3** & 1.5 & 29 & 30 & 0.4 & 0.1302 & 23 & 20 & 0 \\ \hline
**Gearing 4** & 1.5 & 29 & 30 & 0.1823 & 0.3479 & 23 & 20 & 0 \\ \hline
**Gearing 5** & 16 & 13 & 13 & 0 & 0 & 6 & 20 & 0 \\ \hline \hline \end{tabular}
\end{table}
Table 3: The basic parameters of designed gearings.  
wheel, which come into the common engagement. This is shown in the Figure 6. Single-pair and double-pair contact of the teeth was realized on these four pairs of teeth, by defining the contacts on the flanks of the teeth and the subsequent rolling of the pinion and the gear. The individual gearbox members (pinion and wheel) were converted from solid models to surfaces. These surfaces were further subdivided into smaller sections, mainly because of creating a better mesh of surface elements in the Abaqus CAE software in which the calculation was performed.  
For the calculation, boundary conditions are determined, which take into account the placement of the pinion and gear (only rotation about its own axis of rotation is allowed) and also contain predefined rotations of pinion and gear, which are used to demarcate tooth backlash and subsequently for the common rotation. In three computational steps, the boundary conditions and the torque load (100 Nm) of the wheel are gradually realized.  
The very fine mesh of pinions (111,968 elements) and gears (83,952 elements) is formed by elements of the CPS8R type, i.e., quadrangular elements suitable for planar stress.  
Based on the FEM analysis, the deformation rotation of the pinion and the gear was determined. Then it was converted to the stiffness of spare compression springs inserted between perfectly rigid teeth. This obtained the stiffness of the gears in the N/m units.  
The resulting stiffness \(k_{c}\) obtained from the FEM calculation was determined from the Equation (15)  
\[k_{c}=\frac{k_{p}\cdot k_{g}}{k_{p}+k_{g}}, \tag{15}\]  
where \(k_{p}\) is the stiffness of the pinion teeth and \(k_{g}\) is the stiffness of the gear teeth. The stiffnesses \(k_{p}\) and \(k_{g}\) depend on the load and the deformation rotation of pinion \(\delta_{p}\) and gear \(\delta_{g}\), which are the result of the FEM analysis.