## 4 Gear ACTRAN Noise Prediction
### Harmonic Response Analysis  
Gear drives are typically nonlinear systems due to the nonlinear properties introduced by the gears when they mesh, especially gear mesh shocks. It is crucial to consider factors such as the adoption of appropriate initial linearity, which play a key role in the simulation of the dynamic response of the system and can affect the accuracy and confidence of the system simulation results [31]. These nonlinear effects cause harmonic vibrations in the gear system at different frequencies. Harmonic response analysis helps to better understand and predict the impact of these nonlinear effects on acoustic performance. During the operation of a gear system, vibrations and shocks can lead to noise generation. Harmonic response analysis can reveal the vibration and noise generation mechanisms of the different frequency components of a gear system, which can help in taking appropriate control measures to counter these mechanisms. In addition, harmonic response analysis helps to determine whether resonance exists in the gear system. Resonance can lead to a dramatic increase in the amplitude of system vibration and even lead to structural damage or noise problems. By identifying potential resonance frequencies, measures can be taken to avoid resonance problems. Finally, harmonic response analysis can also be used to optimize the design of gear systems. By analyzing the vibration response at different harmonic frequencies, engineers can adjust the parameters of the gears, such as reducing the nonlinear effects of the gears or optimizing the damping, to improve the acoustic performance of the system. These measures can contribute to the design of quieter and more reliable gear systems. The model and material property settings from the previous modal analysis are copied to define the contact, delimit the mesh of the gear pair, and apply loads and constraints. The modal superposition method is shown in Figure 7:  
The modal results are inputted into the harmonic response analysis. Based on the computer's solving speed and the range of modal frequencies, the swept frequency range of the solution was set to 100-2700 Hz, and the load step was set to 50 Hz. Considering that the gear pair will produce energy consumption due to friction and other factors in the vibration state, the damping of the gear pair system is set to a constant of 0.05 in order to improve the accuracy of the harmonic response analysis results. According to the actual working condition of the gear pair, a torque of 841.5 N\(\cdot\)m is applied to the main wheel, \begin{table}
\begin{tabular}{c c c c} \hline
**Mode** & **Frequency (Hz)** & **Mode** & **Frequency (Hz)** \\ \hline
1 & 1123.2 & 6 & 2813.3 \\
2 & 2380.9 & 7 & 2889.2 \\
3 & 2439.5 & 8 & 4191.8 \\
4 & 2571.9 & 9 & 4283.9 \\
5 & 2710.3 & 10 & 4747.1 \\ \hline \end{tabular}
\end{table} 
Table 5: First 10 th-order intrinsic frequency of gear pair. Figure 7: Modeling the harmonic response analysis. and the vibration displacement frequency curve in the \(X\)-axis direction is solved by taking the surface connected to the main wheel and the shaft as the study surface, as shown in Figure 8.  
According to the results of the above harmonic response analysis, the amplitude increases first. A peak occurs between 800 Hz and 1200 Hz and begins to decrease slowly after reaching the maximum value. As a rule of thumb, when a gear pair vibrates under the impact of a dynamic load [32], the first mode contributes the most to the overall system, while its intrinsic frequency may be closer to the external excitation frequency. Therefore, when the excitation frequency matches the intrinsic frequency of the first mode, the system is more prone to resonance, which manifests itself as a pronounced peak on the displacement frequency curve. On the contrary, the intrinsic frequencies of the other modes are usually relatively far away from the excitation frequency; so, significant amplitude peaks are less likely to be observed on the displacement frequency curve.