# Reduction of Coupling Interface Degrees of Freedom in Mixed-Interface Component Mode Synthesis
## 5 Numerical Examples
### Example 1
Meanwhile, the shape of sub-regions are also a key factor for precision. Though the number of the IBNs in scheme (b) is nine while that in scheme (d) is 10, the numerical precision of (b) is better. That is, the precision is better when the sub-regions divide the coupling interface evenly and the shape of sub-regions are nearly square. The validity of the MCMS-rid method is also verified by using the Modal Assurance Criterion (MAC) and the Frequency Response Functions (FRF) comparisons. The equation of MAC is,  
\[\text{MAC}=\frac{\left\|\boldsymbol{\phi}_{MCMS-rid}^{\text{T}}\boldsymbol{ \phi}_{MCMS-fid}\right\|^{2}}{(\boldsymbol{\phi}_{MCMS-rid}^{\text{T}} \boldsymbol{\phi}_{MCMS-rid})(\boldsymbol{\phi}_{MCMS-fid}^{\text{T}}\boldsymbol {\phi}_{MCMS-fid})} \tag{63}\]  
Figure 5: The first four mode shapes of the beam. (a) 1 st mode in x-direction (\(f_{1}=39.10\) Hz); (b) 1 st mode in y-direction (\(f_{2}=56.62\) Hz); (c) 2 nd mode in x-direction (\(f_{3}=104.63\) Hz); (d) 2 nd mode in y-direction (\(f_{4}=146.16\) Hz). where \(\boldsymbol{\phi}_{MCMS-fid}\) and \(\boldsymbol{\phi}_{MCMS-rid}\) are the physical partitions of the eigenvectors by the MCMS-fid method and the MCMS-rid method, respectively. The MAC between the modes by the MCMS-fid method and the MCMS-rid method with different interpolation schemes are plotted in Figure 6. It is seen that the mods of the MCMS-fid method and the MCMS-rid method are identical to each other, which gives confidence in the MCMS-rid method. The FRF comparisons, shown in Figure 7, are conducted by applying a harmonic excitation at excitation level of 1N in the Y direction in the middle of the beam. It illustrates that the MCMS-rid method with nine and 10 IBNs have better capability to capture the dynamic responses of the beam than the MCMS-rid method with four IBNs in terms of both the resonance frequency and its corresponding amplitude, confirming that both the number and distribution of the IBNs have a huge effect on the validity of the proposed method. \begin{table}
\begin{tabular}{l c c c c c c c c c} \hline \hline \multirow{2}{*}{**Scheme**} & \multicolumn{8}{c}{**Mode**} & \multirow{2}{*}{**DOFs**} \\ \cline{2-2} \cline{5-10}  & **1** & **2** & **3** & **4** & **5** & **6** & **7** & **8** \\ \hline a & 39.47 & 56.36 & 95.67 & 134.88 & 159.02 & 214.40 & 284.67 & 298.65 & 165 \\ b & 39.89 & 56.28 & 95.05 & 133.63 & 159.97 & 217.61 & 279.00 & 299.90 & 69 \\  & (1.06) & (0.14) & (0.65) & (0.93) & (0.60) & (1.50) & (1.99) & (0.42) & 69 \\ c & 42.23 & 58.22 & 88.79 & 132.56 & 165.71 & 226.48 & 293.96 & 307.52 & 39 \\  & (6.99) & (3.30) & (7.19) & (1.72) & (4.21) & (5.63) & (3.26) & (2.97) & 39 \\ d & 40.04 & 56.05 & 96.35 & 134.23 & 160.22 & 218.18 & 281.08 & 297.34 & 80 \\  & (1.44) & (0.55) & (0.71) & (0.48) & (0.75) & (1.76) & (1.26) & (0.44) & 80 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The first eight natural frequencies of the beam \(f\) (Hz) and relative error \(e\) (%). Figure 6: Modal assurance criterion between the modes by the MCMS-fid method and MCMS-rid method different interpolation schemes. (**a**) MCMS-rid:b vs. MCMS-fid; (**b**) MCMS-rid:c vs. MCMS-fid; (**c**) MCMS-rid:d vs. MCMS-fid.