## 4 Gear ACTRAN Noise Prediction
### Modal Analysis  
Modal analysis is usually used for linear systems, but it is equally applicable in nonlinear systems. Modal analysis provides an effective approximation in the study of nonlinear systems, enabling an understanding of the vibration modes and dynamic properties of the system. Modal analysis can still provide useful information about the response of the system. In this paper, harmonic response analysis is carried out using the modal superposition method because it is effective in demonstrating the response of a nonlinear system under different excitations. Although nonlinear effects may change the vibration characteristics of a system, the modal superposition method enables a better understanding of the vibration response of nonlinear systems under different operating conditions by superimposing the linear responses of different modes. Therefore, the application of modal analysis, taking into account the nonlinear characteristics of the system, enables a more comprehensive exploration and understanding of the dynamic characteristics and vibration response of the traction gear system, which, in turn, enables an effective prediction and analysis of its radiated noise.  
The results of the modal analysis can provide the transmission gear pair In a certain frequency range of each order of the intrinsic frequency and its vibration characteristics. Based on these results, it is possible to predict the actual vibration response of the gear pair when it is subjected to various excitations in this frequency range, which provides a reliable basis for the analysis of the vibration characteristics of the gear transmission system, as well as its optimized design.  
The methodology for modal analysis of a gear pair typically involves the use of the ANSYS Workbench module, and the following are the specific steps for implementing this process:  
1. Define material properties. Material properties such as Poisson's ratio, density, and modulus of elasticity corresponding to Table 2 were assigned to the large and small gear models, respectively.
2. Divide the grid. The finer the mesh, the higher the accuracy of the solution, but the corresponding amount of work will also increase. Therefore, it is necessary to ensure the accuracy of the results under the premise of a reasonable definition of the range of values of the mesh edge length.
3. Define boundary conditions. The boundary constraints of the gear train are set such that they conform to the real state as much as possible. If the constraints are not set reasonably, the solution results will deviate significantly; therefore, the boundary conditions should be set in accordance with the actual installation conditions of the transmission gear pair.  
Solving and post-processing. In modal analysis of gears, it is common to focus only on the first few orders of modes. Lower-order modes usually correspond to lower intrinsic frequencies and may be similar to the operating frequencies of gear systems. By identifying these lower-order modes, resonance phenomena can be avoided. Resonance can lead to a dramatic increase in amplitude, which may result in structural damage or a significant increase in noise levels. Therefore, by analyzing the lower-order modes, measures can be taken to avoid resonance with the operating frequency. The Selection of lower-order modes for gear modal analysis is critical to improve design, reduce vibration and noise, increase safety and durability, and optimize acoustic performance. Low-order modes provide critical information on the vibration characteristics of gear systems, providing the basis for effective problem solving and performance improvement. Here, the first ten modes of the gear pair are set to be solved, the system performs a finite element solution for the first ten modes, and the results are shown in Table 5. Figure 6 shows the first-order vibration model, and the rest is omitted.  
Figure 6: First-order intrinsic vibration modes of gear pair.