# Mesh Stiffness and Dynamic Modeling and Analysis of Modified Straight Bevel Gears
## 5 Numerical Analysis  
In this section, considering TVMS, \(\epsilon_{s}\), and the variation in meshing position, vibration response analysis of the coupled rotor system is performed for four scenarios: unmodified, TPM, lead crown modification, and CM. The time-varying mesh stiffness is calculated as discussed in the second section, while other parameters are as Tables 7-9 shown. The natural frequencies of straight bevel gears with the TPM and standard straight bevel gears are nearly identical. The natural frequencies of standard straight bevel gears, level gears with profile modification, and bevel gears with both profile and tooth direction modifications are as shown in Tables 10-12. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Order** & **f/Hz** & **rpm/(r/min)** \\ \hline
1 & 1426 & 2312 \\
2 & 1480 & 2400 \\
3 & 1651 & 2677 \\
4 & 1727 & 2800 \\
5 & 1730 & 2805 \\
6 & 2272 & 3684 \\
7 & 2695 & 4370 \\
8 & 3511 & 5693 \\ \hline \hline \end{tabular}
\end{table} 
Table 10: Critical speed and mode description for the geared rotor system. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameters** & **Bearing 1** & **Bearing 2** & **Bearing 3** & **Bearing 4** \\ \hline \(\mathbf{K}_{b}\) & & Diag(1.6 \(\times\) 10\({}^{8}\), 1.6 \(\times\) 10\({}^{8}\), 1.6 \(\times\) 10\({}^{8}\),10\({}^{5}\), 10\({}^{5}\)) & \\ \(\mathbf{C}_{b}\) & & Diag(10\({}^{3}\), 10\({}^{3}\), 10\({}^{3}\), 10\({}^{3}\), 10\({}^{3}\)) & \\ Node position & 2 & 8 & 10 & 16 \\ \hline \hline \end{tabular}
\end{table} 
Table 9: Parameters of bearing. From Figure 16, it can be observed that for the unmodified gear pair and the gear pair with the TPM, resonance peaks occur when the meshing frequency equals the 4 th and 6 th-order natural frequencies of the gear system. Due to the influence of nonlinear stiffness excitation and geometric transmission error, superharmonic resonance peaks appear when the meshing frequency is f4/2 and f6/2. However, when the TPM is applied to the gear pair, the abrupt change in nonlinear stiffness excitation becomes smoother. Considering the modification, the superharmonic resonance peaks at f4/2 and f6/2 disappear, and the peak value of DTE significantly decreases. However, the extent of the reduction in DTE peak value varies depending on the degree of modification. Therefore, the TPM should be chosen based on the fluctuation of time-varying mesh stiffness and the vibration response. From Figure 17, it can be observed that for the unmodified gear pair and the gear pair with LCR, the reduction in the amplitude of nonlinear stiffness after LCR affects the system's natural frequencies. Resonance peaks occur when the meshing frequency equals the gear system's 3 rd and 6 th-order natural frequencies. Superharmonic resonance peaks appear at meshing frequencies f2/2 and f6/2. The amplitude of DTE significantly increases, and the resonance peaks at f6 have higher peak values (LCM3, LCM4, LCM5) compared to the unmodified resonance peaks. Figure 18 shows that resonance peaks occur when the meshing frequency equals the 3 rd and 6 th-order natural frequencies of the gear system. However, when considering a CM, the superharmonic resonance peaks at f4/2 and f6/2 disappear. The vibration response amplitude of the gear pair increases after CM, primarily due to the significant impact of LCR on the system's vibration response. LCR alters the contact performance, leading to a substantial change in nonlinear time-varying meshing stiffness and an increase in displacement amplitude in the vibration response. On the other hand, the influence of the TPM leads to a smoother transition in the nonlinear time-varying meshing stiffness, resulting in a relatively minor displacement amplitude in the vibration response. In this chapter, based on the calculation of the time-varying meshing stiffness of straight level gears in previous chapters, the vibration responses of straight level gear systems with profile modification, axial modification, and CM are studied. Firstly, a dynamic model of the gear-rotor system is established.