# Determination of Mesh Stiffness of Gear--Analytical Approach vs. FEM Analysis
## 2 Analytical Models of Mesh Stiffness
### Analytical Model by ISO Standard  
The analytical calculation model (AM1), which is described in this paragraph, is based on the publications of the Czech authors [PERSON] [10] and [PERSON] [11]. This model is still widely used in research of gearing dynamics. Other authors who mention this model in their publications are [PERSON], [PERSON], [PERSON], [PERSON] and [PERSON] [6], [PERSON] and [PERSON] [12] and, last but not least, [PERSON] [13]. This is a model that is not universally applicable and is very approximate. It was derived from the testing of many gear samples and can be described as a computational model based on empirical relationships. It is useable when it is not absolutely necessary to create perfectly accurate dynamic models of transmission systems and we are only interested in approximate values of the stiffness of single pair of teeth and double pair of teeth in engagement. The stiffness of single pair in contact was determined by Equation (2):  
\[\frac{1}{c^{\prime}}=C_{1}+\frac{C_{2}}{z_{1}}+\frac{C_{3}}{z_{2}}-C_{4}\cdot x_{ 1}-C_{5}\cdot\frac{x_{1}}{z_{1}}-C_{6}\cdot x_{2}-C_{7}\cdot\frac{x_{2}}{z_{2}}+ C_{8}\cdot x_{1}^{2}+C_{9}\cdot x_{2}^{2}, \tag{2}\]  
where \(z_{1}\), \(z_{2}\) denote the number of teeth of pinion and gear, \(x_{1}\), \(x_{2}\) are profile shift coefficients of the pinion and gear. The individual empirical constants \(C_{1}\)-\(C_{9}\) are listed in the Table 1.  
From the point of view of the simplicity of the stiffness calculation of double pair of teeth, the calculation relation for determining the stiffness of double pairs of teeth is convenient as well. In publications [10; 11], this stiffness is called the engagement stiffness and is again the approximate value of the stiffness of double pairs of teeth in engagement. This engagement stiffness implies from the Equation (3):  
\[c_{\gamma}=c^{\prime}\cdot(0.75\cdot e+0.25), \tag{3}\]  
\(c_{\gamma}\) denotes the stiffness of double pair of teeth or engagement stiffness, \(c^{\prime}\) is the stiffness of single pair of teeth in engagement given by the computational relation--Equation (2) and \(e\) is the designation of the engagement factor or contact ratio of gear. Using Equations (2) and (3), there are obtained two approximate values that correspond to the stiffness of single and double pairs of teeth in the engagement. The values related to the face width are given as a result of its calculation. These values have unit N/mm\({}^{2}\). To obtain real values that correspond to the time-varying mesh stiffness with the unit N/m, we must multiply the stiffness \(c^{\prime}\) and \(c_{\gamma}\) by the actual face width in millimeters and then convert N/mm to N/m. Figure 1 shows the theoretical course of mesh stiffness of gearing during an engagement. From the calculation of relations Equations (2) and (3), the constant values of the stiffness of single and double pairs of teeth are obtained. In order to obtain a similar course of mesh stiffness as in Figure 1, it is necessary to use the prescription of the Fourier series, which allows plotting the course of mesh stiffness with periodic alternation of the stiffness of single and double pairs of teeth in engagement. The Fourier series has the form (4) [18]:  
\[k(\tau)=c^{\prime}+(c_{\gamma}-c^{\prime})\cdot(e-m)+\sum_{n=1}^{\infty}\frac{ 2\cdot(c_{\gamma}-c^{\prime})}{\pi n}\cdot\sin\pi n\cdot(e-m)]\cdot\cos n\tau, \tag{4}\]  
where \(e\) is contact ratio of gearing, \(m\) is equal to 1 for the case of calculating the course of stiffness of the gearing with staight teeth, \(\tau\) is dimensionless time. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \(C_{1}\) & \(C_{2}\) & \(C_{3}\) & \(C_{4}\) & \(C_{5}\) & \(C_{6}\) & \(C_{7}\) & \(C_{8}\) & \(C_{9}\) \\ \hline
0.04723 & 0.15551 & 0.25791 & 0.00635 & 0.11654 & 0.00193 & 0.24188 & 0.00529 & 0.00182 \\ \hline \end{tabular}
\end{table} 
Table 1: Constants for calculating the stiffness of single pair of teeth [10; 11]. Figure 1: The course of the total stiffness of spur gearing [10].