# Determination of Mesh Stiffness of Gear--Analytical Approach vs. FEM Analysis
## 2 Analytical Models of Mesh Stiffness
### Analytical Model in Terms of Deformation Energy  
Across various sources in the form of scientific articles, the method of deformation energy is considered to be the ideal analytical approach to modeling the stiffness of gears, as an internal excitation element of dynamical systems. The following findings are summarized from the authors' scientific contributions: [PERSON] and [PERSON] [1]; [PERSON], [PERSON] and [PERSON] [2]; [PERSON], [PERSON] and [PERSON] [3]; [PERSON], [PERSON] and [PERSON] [5]; [PERSON] et al. [4] and last but not least [PERSON], [PERSON], [PERSON], [PERSON] and [PERSON] [6]. The second analytical model (AM2) for obtaining the course of mesh stiffness of gear is a model based on the mentioned publications. In this case, it is not an empirical model, which was created on the basis of experimental testing of gear samples, but is directly based on the theory of elasticity. The individual teeth are considered as cantilever beams placed in the dendum circle with a variable cross-section, where a force acts in the direction of the line of action. The deformation energy accumulated in the tooth is calculated individually for each tooth coming into an engagement. The potential energies that contribute to the calculation of gear stiffness are: bending energy \(U_{b}\), shear energy \(U_{s}\), and axial compressive energy \(U_{a}\). They can be determined using Equation (5). \[U_{b} =\frac{F^{2}}{2K_{b}}, \tag{5}\] \[U_{s} =\frac{F^{2}}{2K_{s}},\] \[U_{a} =\frac{F^{2}}{2K_{a}}\]  
\(K_{b}\), \(K_{s}\) and \(K_{a}\) denote the bending, shear and axial compressive stiffness. \(F\) is the force in direction of line of action. By means of beam theory, the calculation of the potential energy for a gear tooth can be defined using these Equation (6)  
\[U_{b} =\int_{0}^{d}\frac{M^{2}}{2 EI_{x}}dx, \tag{6}\] \[U_{s} =\int_{0}^{d}\frac{1.2F_{a}^{2}}{2 GA_{x}}dx,\] \[U_{a} =\int_{0}^{d}\frac{F_{a}^{2}}{2 EA_{x}}dx,\]where \(E\) is Young modulus, \(G\) denotes shear modulus and \(A_{x},I_{x}\) represent the area of cross-section and moment area of inertia where the distance between the section and the acting point of the applied force is \(x\) which is from interval \(<\)0; \(d\)\(>\) which shows Figure 2[1]. \(F_{b}\),\(F_{a}\) and \(M\) can be calculated from the following Equation (7). \[ \begin{split} F_{b}&=F\cos\alpha_{1},\\ F_{a}&=F\sin\alpha_{1},\\ M&=F_{b}\cdot x-F_{a}\cdot h\end{split}  \tag{7}\]  
The required stiffnesses (8) can be calculated from Equations (5)-(7). \[ \begin{split}\frac{1}{K_{b}}&=\int_{0}^{d}\frac{(x \cos\alpha_{1}-h\sin\alpha_{1})^{2}}{EI_{x}}dx,\\ \frac{1}{K_{s}}&=\int_{0}^{d}\frac{1.2\cos^{2}\alpha _{1}}{GA_{x}}dx,\\ \frac{1}{K_{a}}&=\int_{0}^{d}\frac{\sin^{2}\alpha_{1 }}{EA_{x}}dx\end{split}  \tag{8}\]  
The contact stiffness is also reflected in the overall mesh stiffness of the gearing. This stiffness is referred to a Hertzian contact stiffness \(K_{h}\). This stiffness is calculated as follows:  
\[\frac{1}{K_{h}}=\frac{4(1-\
u^{2})}{\pi EW}, \tag{9}\]  
where \(\
u\) is Poisson's ratio, \(W\) is face width. The last thing that affects the mesh stiffness of gearing is called fillet-foundation stiffness \(K_{f}\). It is an expression of stiffness that takes into account the fillet radius, which is given by Equation (10). \[\frac{1}{K_{f}}=\frac{\delta_{f}}{F} \tag{10}\]  
Figure 2: Geometrical parameters for mesh stiffness calculation [1]. There is \(\delta_{f}\) in Equation (10). This expresses the deformation of the tooth at its fillet radius and a special Equation (11) using empirical constants is used to calculate it. \[\delta_{f}=\frac{Fcos^{2}\alpha_{m}}{WE}\bigg{\{}L^{*}\Big{(}\frac{u_{f}}{S_{f}} \Big{)}^{2}+M^{*}\Big{(}\frac{u_{f}}{S_{f}}\Big{)}+P^{*}(1+Q^{*}tan^{2}\alpha_{ m})\bigg{\}} \tag{11}\]  
The coefficients \(L^{*}\), \(M^{*}\), \(P^{*}\), \(Q^{*}\) are given by a general polynomial function (Equation (12)), where \(X^{*}\) represents the individual coefficients.