# A Multiple and Multi-Level Substructure Method for the Dynamics of Complex Structures
## 4 Analysis of Dynamic Characteristics Based on Substructure  
### Assembly Structure with Three Panels  
The research object of this example is the assembly structure with three panels. A complete FE model of the assembly structure with three panels was modeled by the FE analysis software 'Nastran' and 'Hypermesh' based on the size diagram of the assembly structure with three panels, as shown in Figure 2. The assembly structure with three panels is a top and bottom axisymmetric structure, composed of three plates of the same material and the same geometric shape [31]. The finite element model is composed of shell elements and beam elements. The periphery of the slab is surrounded by beam elements, and the slab is connected by beam elements. A total of 12,397 nodes and 12,784 elements were established by using 'Hypermesh'. Among them, the shell element has 12,397 nodes and 12,000 elements, and the beam element has 780 nodes and 784 elements. All the nodes are hinged. The beam element is steel and the cross-sectional area of the beam is A\({}_{1}\) = 4 \(\times\) 10\({}^{-4}\) m\({}^{2}\). The shell element is an aluminum alloy and the thickness is t = 4 \(\times\) 10\({}^{-3}\) m. Our assumption is a homogeneous, isotropic, linearly elastic material. The material and its parameters of the finite element model are as they are shown in Table 1.  
There are two substructure classifications for the assembly structure with three panels. The specific classification is shown in Figure 3. For the analysis of the substructure, we selected the first 20 order models for free mode analysis (it mainly studies the improvement of the efficiency of modal calculation, so it is only carried out at a normal temperature). The first 20 natural frequencies are as they are shown in Table 2. Analyzing the natural frequency error of Table 2, we can see that the maximum error occurs in the seventeenth and eighteenth steps, respectively, which are 0.18462% and 0.43966%, indicating that different \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Type of Element** & **Material** & **Elastic Modulus/GPa** & **Density/g-cm\({}^{-3}\)** & **Poisson’s Ratio** \\ \hline Shell element \"50\" & aluminum & 70 & 2.7 & 0.34 \\ Shell element \"60\" & aluminum & 70 & 2.7 & 0.34 \\ Shell element \"70\" & aluminum & 70 & 2.7 & 0.34 \\ Beam element & steel & 200 & 7.85 & 0.32 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Material parameters of the assembly structure with three panels. Figure 2: Finite element model of the assembly structure with three panels. classification methods do not affect the calculation accuracy. The modal assurance criteria (MAC) is used to verify the modal. The comparison of MAC between the full model and the substructure model is shown in Figure 4. The diagonal element of the MAC is 1, and the rest is close to 0, which indicates that the modal shapes are consistent. \begin{table}
\begin{tabular}{c c c c c c} \hline  & \multicolumn{3}{c}{**Natural Frequency/Hz**} & \multicolumn{3}{c}{**Error/\%**} \\ \cline{2-6} \cline{2-6} \multicolumn{1}{c}{**Order**} & \multirow{2}{*}{**Full Model**} & \multirow{2}{*}{**Substructure III**} & \multirow{2}{*}{**Substructure IV**} & \multirow{2}{*}{**Comparison of Full Model and**} & \multirow{2}{*}{**Comparison of Full Model and**} \\  & & & & & **Substructure III** & **Substructure IV** \\ \hline  
\hline \end{tabular}
\end{table} 
Table 2: Natural frequency error analysis of full model and multi-level substructure model. Figure 3: Different classification of substructure for the assembly structure with three panels. Further considering the influence of the selection of external nodes in the substructure model on the analysis results, 25%, 50%, and 75% of external nodes are selected for calculation and free mode analysis. The natural frequency error curve is shown in Figure 5. The maximum absolute value errors of the corresponding frequencies are 0.255%, 0.077%, and 0.026%, respectively.