# Reduction of Coupling Interface Degrees of Freedom in Mixed-Interface Component Mode Synthesis
## 5 Numerical Examples
### Example 1  
A beam (length = 6 m, width = 0.6 m and height = 0.4 m) with two fixed ends is taken as example 1. The three-dimensional elastic FE model with 725 nodes and 2175 DOFs is shown in Figure 3. The material properties of the beam are: \(\rho\) = 2500 kg/m\({}^{3}\), \(E\) = 30 GPa and \(\
u\) = 0.3. We divide the beam into two components (numbered \"1\" and \"2\"), and constrain the coupling interface of the component 1 while set free that of the component 2. As shown in Figure 4a, the coupling interface is a rectangular with 25 nodes and 75 DOFs. In the following modal analysis based on MCMS method, the number of truncated normal modes is 10 for component 1 and 5 for component 2, respectively. Figure 3: Finite element model of the beam with two ends fixed. We firstly compare eigenvalue features of the MCMS-rid method with other three existing MCMS methods, including the B-H method, B-H method with normal mode correction (BHC) and the revised MCMS method with full coupling interface DOFs (MCMS-fid) [43]. For the MCMS-rid method, the coupling interface is divided into four sub-regions (Figure 4b) and the corner nodes of each sub-region are chosen as IBNs. Thus, the number of the coupling interface DOFs are reduced from 75 to 27. The first eight natural frequencies calculated by each method are listed in Table 1, where \"FEM\" indicates the exact numerical solutions of the whole beam. The relative errors of the natural frequencies for MCMS methods are also done by Equation (61), and the \"Method\" in the equation refers to \"B-H, BHC, MCMS-fid and MCMS-rid\". The numbers of the retained DOFs involved in docking and mode synthesis for each method are also shown in Table 1 (in parenthesis). \[e_{i}=\frac{\left|f_{\text{Method}}^{i}-f_{\text{FEM}}^{i}\right|}{f_{\text{ FEM}}^{i}}\times 100\%,\quad(i=1,2,...,8) \tag{61}\]  
From Table 1 we can see that, the accuracy of the B-H method is worst of all. It is because the B-H method ignores the influence of higher-order of the master substructure on the modal synthesis. The accuracy of the BHC method has been improved by considering the effect of normal modes correction on the dynamic characteristics of the system. Though the MCMS-rid method is less accurate than the MCMS-fid method, it improves the computational efficiency enormously by using only 69 DOFs. Also, the relative errors for the MCMS-rid method are less than 10%, which ensures the accuracy and reasonableness of the modal analysis. Figure 5 gives the first four mode shapes of the beam by using different methods. Figure 4: The coupling interface and interpolating schemes. (**a**) the coupling interface; (**b**) scheme with 9 interpolation basic nodes; (**c**) scheme with 4 interpolation basic nodes; (**d**) scheme with 10 interpolation basic nodes. The effect of the number and distribution of the IBNs on the eigenvalue performance of the MCMS-rid method is studied as well. As shown in Figure 4, four interpolation schemes are considered for mode synthesis. For scheme (a), all the coupling interface nodes are involved in docking of substructures, i.e., MCMS-fid. For case (c), the four corner nodes of the coupling interface are assigned as the IBNs (only one sub-region). In case (b) and (d), the coupling interface is divided into four sub-regions with different shape, and all the corner nodes of the sub-regions are assigned as the IBNs. Scheme (a) is taken as exact solutions, and the relative errors for other schemes are calculated by Equation (62), where \"Scheme\" in the equation indicates \"(b), (c) and (d)\". \[e_{i}=\frac{\left|f_{\text{Scheme}}^{i}-f_{\text{(a)}}^{i}\right|}{f_{\text{(a )}}^{i}}\times 100\%,\quad(i=1,2,...,8) \tag{62}\]  
The first eight natural frequencies of the beam are listed in Table 2. The results indicates that both the number and distribution of the interpolation basic nodes have great influence on the accuracy of the MCMS-rid method. The accuracy is improved as the number of the IBNs chosen for mode synthesis are increasing.