## 2 Structural Description and Numerical Modeling
### Static Analysis  
When studying the vibration characteristics of the fan frame, the focus is on the vibration characteristics of the frame structure. The mass and rotational inertia of the motor and the centrifugal impeller produce a prestressing influence in the structural modal calculations, which cannot be ignored. So, the mass point is used to attach the mass characteristics to the frame [22; 23] and is coupled at the frame connection position. The motor plate, connecting tube, and connection block are connected by welding, and welding points are used in the finite element model to simulate structural welding. The mounting plate, connection block, and current collector are connected by rigid bolts [24; 25]. The fan frame is only subjected to gravity and provides fixed constraints on the installation holes of the base plate. The numerical model is defined as shown in Figure 2, where the \(x\)-axis is vertical, the \(y\)-axis is axial, and the \(z\)-axis is lateral.  
According to Equation (2), the moment of inertia of the mass point can be calculated,  
\[A=\begin{bmatrix}I_{xx}&I_{xy}&I_{xz}\\ I_{yx}&I_{yy}&I_{yz}\\ I_{zx}&I_{zy}&I_{zz}\end{bmatrix} \tag{2}\]  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Material Properties** & **SS400** & **Q195** \\ \hline Density/kg\(\cdot\)m\({}^{-3}\) & 7860 & 7690 \\ Young’s modulus/Pa & 2.12 \(\times\) 10\({}^{11}\) & 2.12 \(\times\) 10\({}^{11}\) \\ Poisson’s ratio & 0.288 & 0.286 \\ Yield strength/MPa & 235 & 195 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Material attributes.  
Figure 1: Three-dimensional model.  
In Equation (2), \(A\) is the inertia tensor; \(I_{xx}\), \(I_{yy}\), and \(I_{zz}\) are the moments of inertia. The non-diagonal values in the equation are changed to 0, and the diagonal matrix is calculated.  
Static analysis of the structure can yield equivalent stress and total deformation results through post-processing. The fourth strength theory calculates the equivalent stress, and the equivalent stress \(\sigma\) must be less than the allowable stress \([\sigma]\) of the material, where \(\sigma_{\mathrm{s}}\) is the yield strength, and \(\gamma_{\mathrm{s}}\) is the safety factor:  
\[\sigma=\sqrt{\frac{1}{2}\big{[}\big{(}\sigma_{1}-\sigma_{2}\big{)}^{2}+\big{(} \sigma_{2}-\sigma_{3}\big{)}^{2}+\big{(}\sigma_{3}-\sigma_{1}\big{)}^{2}\big{]} }\leq[\sigma]=\frac{\sigma_{\mathrm{s}}}{\gamma_{\mathrm{s}}} \tag{3}\]  
Tetrahedral mesh partitioning is performed in Workbench finite element calculations, and mesh independence is verified. In order to verify that the finite element model does not exhibit stress singularities, mesh independence verification was performed on the maximum equivalent stress, and the results are shown in Table 2.  
According to Table 2, the results meet the accuracy requirements when the mesh number is 1,141,436.  
The stress cloud map and deformation cloud map are shown in Figure 3. From Figure 3, it can be seen that the maximum equivalent stress is 48.5 MPa, and the maximum deformation is 0.75 mm. According to Equation (3), the safety factor is 4.8, and the strength meets the requirements.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Number of** & **503,564** & **786,524** & **945,823** & **1,141,436** & **1,356,854** \\ \hline Maximum & & & & & \\ equivalent & 69.33 & 60.63 & 53.78 & 48.44 & 49.23 \\ stress/MPa & & & & & \\ Maximum & 1.02 & 0.96 & 0.86 & 0.75 & 0.75 \\ deformation/mm & & & & & \\ \hline \hline \end{tabular}
\end{table}
Table 2: Mesh independence verification.  
Figure 2: (a) Solder joint setting; (b) mass point of structural assembly connection settings; (c) fixing and loading.