# Dynamic Stiffness Effect of Mechanical Components on Gear Mesh Misalignment
## 5 Numerical Study
### Numerical Results and Discussion  
In this numerical study, the applied torque and gear mesh frequency are considered as the main variables. To predict the characteristics of the gearbox according to the torque change, the applied torque was set at three different levels: 70, 105, and 140 Nm, which are defined as 50, 75, and 100% of the nominal torque of 140 Nm. In addition, the gear mesh frequency, which plays a main role in the dynamic stiffness in Equation (21), was investigated in six cases: 0, 100, 200, 300, 400, and 500 Hz. The maximum gear mesh frequency, 500 Hz, is computed from Equation (23) based on an input speed of 600 rpm, which is the maximum input speed of the gearbox. These parameters are summarized in Table 3. The first case of the dynamic stiffness at 0 Hz implies a static stiffness, from Equation (21), unlike the other cases. We first compute the static deflections of the shafts, gears, and bearings, considering the housing flexibility. The gear mesh misalignment, face load factor, transmission error, and contact pattern on the gear mesh are also predicted. Figure 8 presents the changing gear mesh misalignment, \(F_{\beta X}\), with respect to the applied torque and gear mesh frequency. The numerical results show that \(F_{\beta X}\) decreases when the applied torque and the gear mesh frequency increase. In particular, we investigated the specific gear mesh misalignments, \(F_{\beta X}\), of the components, and the results are summarized in Table 4. The numerical results imply that \(F_{\beta X}\) of the input gear is much larger than \(F_{\beta X}\) of the output gear, and that \(F_{\beta X}\) of the bearing accounts for approximately 80% of the total gear mesh misalignment. In addition, as the applied torque increased from 70 to 140 Nm, the total amount of \(F_{\beta X}\) decreases because \(F_{\beta X}\) of the bearing dramatically decreased. This result can be attributed to an increase in the reaction forces acting on the bearing with increasing torque. This leads directly to larger bearing deformation, and we can expect larger reaction moments between the rolling elements, such as the balls, rollers, and raceway. This situation can result in a relative reduction of the bearing misalignment. This tendency is clearly apparent when comparing the contact pattern of the gear tooth flanks provided in Figure 9. Figure 9a,b has similar contact patterns, and the load can be seen to be distributed unevenly along the face width. However, the contact pattern of the dynamic stiffness with \(f_{g}=500\) Hz is relatively more even than the contact pattern of the static stiffness with \(f_{g}=0\) Hz. The maximum and average normal forces in the contact patterns are summarized in Table 5. The face load factor, \(K_{HB}\), that is defined in Equation (2) also decreases when the applied torque and the gear mesh frequency increase, as shown in Figure 10. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Material** & **Young’s Modulus (E)** & **Poisson’s Ratio (\(v\))** & **Density (\(\rho\))** \\ \hline Aluminum & 68 GPa & 0.36 & 2700 kg/m\({}^{3}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Material properties of housing. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameters** & **Unit** & **Input Gear** & **Output Gear** \\ \hline Normal module & mm & 3 & 3 \\ Pressure angle & deg. & 20 & 20 \\ Center distance & mm & 71 & 71 \\ Number of teeth & - & 50 & 21 \\ Face width & mm & 13 & 15 \\ Profile shift coefficient & - & \(-\)0.0616 & 0.0616 \\ Gear quality (ISO 1328) & - & 4 & 4 \\ Material type & - & 18 CrMo4 & 18 CrMo4 \\ Young’s modulus (\(E\)) & GPa & 206 & 206 \\ Density (\(\rho\)) & kg/m\({}^{3}\) & 7830 & 7830 \\ Poisson’s ratio (\(v\)) & - & 0.3 & 0.3 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Specifications of gear pair. Figure 8: Gear mesh misalignment, \(F_{\beta X}\), in relation to torque and gear mesh frequency. Figure 9: Comparison of contact patterns when torque is 140 Nm.