# Determination of Mesh Stiffness of Gear--Analytical Approach vs. FEM Analysis
## 4 Results and Discussion  
For the proposed Gearing 1-5, the course of gear stiffness was determined according to the AM1 and AM2 mentioned in Sections 2.1 and 2.2. The AM1 was chosen because it is limited by conditions of use that limit the range of suitable gearing for calculating mesh stiffness of gear and it is advisable to test this method. Theoretical courses of mesh stiffness (by AM1 and AM2) are graphically compared with the courses of mesh stiffness, which are the result of FEM analysis. A standard tooth profile was chosen for Gearing 1-4, which meets the initial conditions of the AM1, but at the same time, these gearings differ from each other in the choice of profile shift coefficients for the gears and pinions. Profile shift coefficients of Gearing 1-3 meet the initial conditions of the theoretical calculation of the AM1. Gearing 4 does not maintain the condition of profile shift coefficients \(x_{1}\geq x_{2}\). For the AM2 these notes are not relevant, because this method not demand to adhere to the initial conditions of the  
Figure 6: Abaqus CAE: the four-tooth engagement model. method AM1. The AM2 was chosen because it seems to be a universal tool for calculating of mesh stiffness. Gearing 5 completely deviates from the assumptions of calculation according to the AM1. Conditions concerning profile shift coefficient are not observed here, neither the standard profile of teeth. The total results of mesh stiffness obtained by theoretical calculation using AM1 and the results of mesh stiffness obtained by the FEM method are summarized in the summary table of results Table 4.  
The resulting stiffness courses of gears are also shown in the Figures 7-9. The total results of mesh stiffness obtained by theoretical calculation using AM2 and the results of mesh stiffness obtained by the FEM method are summarized in the summary table of results Table 5.  
As can be seen from the Figures 7 and 9 and summary Table 4 the results of the FEM analysis of Gearing 1-4 differ from the AM1. The theoretical model (AM1) represents individual gears stiffer compared to the FEM method. The theoretical (AM1) and FEM stiffnesses of single pair of teeth differ by 26%. The stiffness of double pairs of teeth differs by 17%. Therefore the applicability of the AM1 can be accepted, with consideration of its application. If a 26% difference in values is accepted, this theoretical model can be considered satisfactory. It should be noted that if initial conditions are not observed as with Gearing 4, the FEM stiffness course will slightly deviate from the stiffness course defined by the AM1. According to the theoretical calculation, of all gearing models 1-4, Gearing 4 should show the greatest stiffness. However, Gearing 1 shows to be the stiffest according to the FEM simulation. Therefore, the influence of the initial conditions of the theoretical calculation is manifested, here. \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  & **AM1** & **FEM** & **Difference** & **AM1** & **FEM** & **Difference** \\ \hline
**Gearing 1** & 4.085 & 3.059 & 25.1\% & 5.646 & 4.772 & 15.5\% \\ \hline
**Gearing 2** & 4.073 & 2.984 & 26.7\% & 5.625 & 4.661 & 17.1\% \\ \hline
**Gearing 3** & 4.062 & 2.969 & 26.9\% & 5.605 & 4.608 & 17.8\% \\ \hline
**Gearing 4** & 4.091 & 3.009 & 26.4\% & 5.653 & 4.719 & 16.5\% \\ \hline
**Gearing 5** & 0.759 & 0.064 & 91.5\% & 1.011 & 0.349 & 65.5\% \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Mean values of mesh stiffness (\(\times 10^{8}\) N/m)—comparison of AM1 and FEM.