# Determination of Mesh Stiffness of Gear--Analytical Approach vs. FEM Analysis
## 4 Results and Discussion
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & \multicolumn{3}{c}{**Single-Pair Mesh Stiffness**} & \multicolumn{3}{c}{**Double-Pair Mesh Stiffness**} \\ \hline  & **AM2** & **FEM** & **Difference** & **AM2** & **FEM** & **Difference** \\ \hline
**Gearing 1** & 3.155 & 3.059 & 3.1\% & 5.323 & 4.772 & 11.5\% \\ \hline
**Gearing 2** & 3.123 & 2.984 & 4.6\% & 5.291 & 4.661 & 13.5\% \\ \hline
**Gearing 3** & 3.091 & 2.969 & 4.1\% & 5.258 & 4.608 & 14.1\% \\ \hline
**Gearing 4** & 3.143 & 3.009 & 4.5\% & 5.311 & 4.719 & 12.5\% \\ \hline
**Gearing 5** & 0.142 & 0.064 & 54.9\% & 0.315 & 0.349 & 9.8\% \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Mean values of mesh stiffness (\(\times 10^{8}\) N/m)—comparison of AM2 and FEM. Figure 8: Comparison of analytical (AM2) and FEM calculation of mesh stiffness (Gearing 1–4). Figure 7: Comparison of analytical (AM1) and FEM calculation of mesh stiffness (Gearing 1–4). The Figure 8 shows a different situation. The results of the AM2 method are significantly closer to the FEM method. This can be seen from the Table 5. The theoretical model (AM2) represents individual gears stiffer compared to the FEM method too, but values of stiffnesses of single pair of teeth are significantly closer to the results of the FEM analysis. The theoretical (AM2) and FEM stiffnesses of single pair of teeth differ by 4%. Values of stiffnesses of double pairs of teeth also approached the FEM model more than AM1. The stiffness of double pairs of teeth differs by 13%. Results of the AM2 can be attributed to the fact that this method better describes the actual shape of the tooth. When using AM2, there is also no problem of non-compliance with the initial conditions, as in the model AM1 in the calculation of the mesh stiffness of the Gearing 4. The comparison in the Figure 8 shows that the stiffest gearing is the Gearing 1. The Gearing 5 was intentionally non-standardly designed in order to see an extreme case, which is not suitable for analytical calculations in terms of the geometry of the tooth root. Teeth do not have a standard fillet radius to be clearly recognizable, which AM1 cannot describe. Furthermore, a large modulus and a small face width were chosen to make the teeth pliable and to better demonstrate the difference in the resulting values. This results in a huge difference in the values of the theoretical(AM1) and FEM stiffness of the gears during the contact of single pair of teeth and double pairs of teeth. For stiffness of single pair of teeth, the theoretical and FEM calculations differ by up to 91.5%. For the stiffness of double pairs of teeth, this difference is 65.5%. The influence of the tooth profile, which is not standard in the case of Gearing 5, is considerably manifested, here. The large undercut of teeth and small fillet radius significantly reduce tooth stiffness. When using the AM2 method, the results are also not ideal as in the case of the Gearing 1-4, but they are still significantly better than results of AM1. For stiffness of single pair of teeth, the theoretical and FEM calculations differ by up to 54.9%. For the stiffness of double pairs of teeth, this difference is 9.8%.AM2 has made it possible to include the geometry of a specific Gearing 5. However, non-standard tooth geometry is still better to avoid and to design standard gearing. Despite the fact that the values of single and double-pair mesh stiffnesses are significantly closer to the results of the FEM simulation,  
Figure 9: Comparison of analytical and FEM calculation of mesh stiffness (Gearing 5). the shape of the stiffness curve is still very different and it can be seen how this unusual shape manifests itself during the FEM calculation.