# Determination of Mesh Stiffness of Gear--Analytical Approach vs. FEM Analysis
## 5 Conclusions  
This article dealt with the topic of modeling the mesh stiffness of gear teeth. Two different methods of the analytical approach to modeling the course of mesh stiffness were presented on this issue. Both of these approaches, namely AM1 and AM2, were used to determine the mesh stiffness curves for the five designed gears (Gearing 1-5), due to which it was possible to make a final comparison. To evaluate the analytical model AM1 and AM2, stiffness modeling using FEM software Abaqus CAE was mentioned. Due to the software, mesh stiffness curves of Gearing 1-5 were also obtained. Therefore, it was possible to compare and evaluate analytical and FEM curves at the end.  
From the Section 4, it can be seen that AM1 supposes the gearing mesh stiffness to be higher than in the FEM stiffness calculation. There are more factors that may cause this fact.  
1. The theoretical approach of AM1 is based on empirical relationships based on the tested samples. Although the proposed Gearings 1-4 meet the initial conditions of the analytical calculation, the theory of AM1 itself allows the possibility of up to a 20% difference in values, which is fulfilled here.
2. The FEM model considers only modeled gearing without a gear body. This can reduce the resulting FEM stiffness. However, the FEM model was created on the basis of the definition of the analytical model AM1, which also considers only gearing without a gear body to simulate the course of stiffness.  
Gearing 5 clearly does not fit the AM1. This model has been introduced intentionally to show the lack of AM1. So the shape of the tooth profile at Gearing 5 is different from the standard profile, therefore this analytical method cannot be used.  
Analytical model AM1 uses profile shift coefficients to determine the force arm. The profile shift coefficients are not intended to affect the exact shape of the fillet-foundation, therefore this analytical method is also distorting. For approximate calculations of mesh stiffness of gear with a standard profile, AM1 is a sufficient tool for determining mesh stiffness. However, for more accurate dynamic calculations, it would be more appropriate to use more precise analytical models, such as AM2 in this article.  
This model (AM2) allows to include the exact geometry of the designed gearing and does not have limiting conditions like AM1. Due to calculations based on the deformation energy of the tooth, similar results as FEM analysis can be obtained with sufficient accuracy.  
Another advantage of AM2 is that the resulting mesh stiffness curve can be used directly in the dynamic equations of gear transmission systems and unlike the AM1 method, does not affect the frequency spectrum as we avoid using the Fourier series to simulate the stiffness curve.  
Conceptualization, M.D., F.L. and J.F.; writing--original draft preparation, J.F. and M.D.; writing--review and editing, J.F.; creation of FEM model, K.P.; validation, M.D. and J.F.; supervision, J.K. and F.L.; funding acquisition, J.K. and F.L. All authors have read and agreed to the published version of the manuscript.  
This research was funded by Czech Technical University in Prague grant number SGS21/10 8/OHK2/2T/12.  
Not applicable.  
The authors declare no conflict of interest.