# A Multiple and Multi-Level Substructure Method for the Dynamics of Complex Structures
## 4 Analysis of Dynamic Characteristics Based on Substructure
### Gear Box  
The object of this example is the gear box of a helicopter, which is composed of a three-level planetary gear train and a receiver. The first-level planetary gear train is to change the direction of the power input by the engine through the level gear train to reduce the speed,  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & \multicolumn{2}{c}{**Natural Frequency/Hz**} & \multicolumn{2}{c}{**Error/\%**} \\ \cline{2-6  
and to complete the output to the tail rotor. The output deceleration power is input to the tail rotor reducer through the horizontal tail drive and the inclined tail drive shaft. The tail rotor reducer realizes the power conversion of the tail wing. The second-level planetary gear train uses the output power of the bevel gear train in the first-level planetary gear train as the input power of the second-level reducer. After planetary gear reduction, the planet carrier realizes the power output. The third-level planetary gear system uses the output power of the planet carrier in the second-level planetary gear system as the output power of the third-level reducer. After the transmission of the planetary gear system, the output of the planet carrier is completed, and the decelerated power can be passed to the helicopter rotor. Figure 7 shows the exploded view of the geometric model of the whole machine. The material of the gear box is steel. Our assumption is homogeneous, isotropic, linearly elastic material. The elastic modulus is 190 GPa, the Poisson's ratio is 0.3, and the density is 7.75 g/cm\({}^{3}\). The whole structure is modeled by 'Hypermesh' with tetrahedral and hexahedral meshes. The finite element model has a total of 1,220,562 nodes and 1,111,039 elements. The receiver has 298,047 nodes and 303,696 elements. The multi-level planetary gear train has 924,789 nodes and 807,343 elements. Based on the analysis in Section 4.1, the planetary gear train teeth in the gear box is selected as the residual structure, 50% of the external nodes is selected, then the whole structure is divided into multiple and multi-level substructures which are shown in Figure 8. Free mode analysis is performed on the substructure model and the finite element model, respectively. The first five-order modal shapes and natural frequencies are selected for comparison. Table 5 shows the modal shapes of the gear box. Analyzing the natural frequency error of Table 6, the maximum error of the model appears in the third order, which is 1.4333%. The frequency errors of the other fourth orders are all less than 1%, and the calculation deviation is small. Through the comparison of the time-consuming analysis of dynamic characteristics in Table 7, the analysis of the inherent characteristics of the multi-level substructure model takes 5364 s. Compared with the finite element model, the multi-level substructure model saves 9698 s. The comparison of MAC between the full model and the substructure model is shown in Figure 9. The diagonal element of the MAC is 1, and the rest is close to 0, which in-dicates that the modal shapes are consistent. Figure 7: Explosion diagram of the gear box. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Order**} & \multicolumn{3}{c}{**Natural Frequency/Hz**} & \multirow{2}{*}{**Error/\%**} \\ \cline{2-3} \cline{5-5}  & **Finite Element Model** & & **Substructure Model** \\ \hline
1 & 47.23 & 47.60 & 0.7686 \\
2 & 48.75 & 48.80 & 0.1075 \\
3 & 94.05 & 92.70 & 1.4333 \\
4 & 102.59 & 101.99 & 0.5843 \\
5 & 121.35 & 120.23 & 0.9231 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Frequency error analysis of finite element model and substructure model. Figure 8: Substructure model of the gear box. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Time-Consuming of Finite Element Model/s**} & \multirow{2}{*}{**Time-Consuming of Substructure Model/s**} & \multirow{2}{*}{**\(\Delta t\)/s**} & \multirow{2}{*}{**The Ratio of Reduction/\%**} \\ \cline{1-1}
15062 & 5364 & 9698 & 64.39 \\ \hline \hline \end{tabular}
\end{table}