# Determination of Mesh Stiffness of Gear--Analytical Approach vs. FEM Analysis
## 1 Introduction  
In the development of transmission systems, among other things, it is necessary to take into account the dynamic properties of gears or complete gearboxes when designing the gearbox. Properly designed gear geometry has a positive effect on the dynamic response of the system, which can be observed on the frequency spectrum of the investigated dynamic system. This geometry has a noticeable effect on the reduced emission of noise in around. What is manifested in the frequency spectrum of gears is mainly the effect of internal excitation caused by the mesh stiffness, which changes during the meshing and thus interferes with the course of torsional dynamic processes of the gear systems.  
It is therefore necessary to take into account the mesh stiffness when designing the gearbox and to incorporate its course into the dynamic equations of motion. There are various ways to visualize, simulate, and calculate the course of mesh stiffness of gearing. There are different variants of analytical calculations. One of the variants of analytical modeling is the possibility of calculating the mesh stiffness by calculating the deformation energy of individual teeth of the gear, [1, 2, 3, 4, 5]. In this method, it is possible to take into account a large number of additional calculation conditions. This is, for example, taking into account defects occurring on the involute side of the tooth, it can also be modeling cracks that are in the root of the tooth, [1, 6, 7]. These mechanical defects significantly affect mesh stiffness. Another way to improve the analytical model based on deformation energy is to include the effect of a lubricating layer between the functional sides of the teeth [8, 9], etc. Another possibility is the calculation using empirical analytical relationships, which are based on testing of real gears. This variant is elaborated, for example, in the publication of [PERSON] [10] and [PERSON] [11]. Current publications that use this option to model gear stiffness are [6, 12, 13]. With today's possibilities, the mesh stiffness can alsobe determined by means of finite element simulation [7] or dynamic simulations [14], on the basis of which we are able to determine the stiffness of the respective gear or dynamic response very precisely. The last way to determine the mesh stiffness of the gears is to perform an experimental measurement. Publications [7; 15; 16; 17] are devoted to the experimental measurement of mesh stiffness. Due to this measurement, it is possible to obtain an accurate value of the stiffness of the examined gearing. From the point of view of the accuracy of the determined stiffness, the best option for obtaining the stiffness course is experimental measurements. However, the test equipment is necessarily needed and its production is expensive, and it is not always possible to implement this measurement. Due to the expansion of FEM and computer technology, it is very favorable to use stiffness simulation with software that allows FEM analysis. Unfortunately, even FEM analysis is financially expensive and also very time-consuming. Today's analytical models are constantly being improved to be more in line and comparable with the results of the FEM analysis. As a result, they are still attractive to many research and development sites, as they are not time-consuming to calculate and do not require high financial costs.  
The main goal of this article is to present two possible analytical models that have different approaches to determining the mesh stiffness of gears. Furthermore, the analytical models are used to determine the mesh stiffness of various designed gearings. The graphical representation of mesh stiffnesses obtained by the analytical models is compared here with the courses of stiffness obtained by FEM analysis. By means of this comparison, the usability and suitability of the stiffness modeling methods used are determined and the limitations of the analytical method are emphasized.