# Reduction of Coupling Interface Degrees of Freedom in Mixed-Interface Component Mode Synthesis
## 3 Normal Mode Correction for the Master Substructure
### Mode Synthesis in the Revised MCMS Method [40]  
Mode synthesis process for the revised MCMS method is much more complicate than the CCMS method and FCMS method as two different kinds of interface (i.e., constrained-interface and the free-interface) should be coupled. In this case, the coordinate transformation modes contain truncated mode \(\mathbf{\Phi}\), including \(\mathbf{\Phi}_{m}\) for the constrained-interface substructure and \(\mathbf{\Phi}_{k}\) for the free-interface substructure, constrained mode \(\mathbf{\Phi}_{c}\) and residual attachment mode \(\mathbf{\Psi}_{d}\) simultaneously. The displacement-coordination and force-equilibrium between adjacent substructure can be written as  
\[\left\{\mathbf{u}_{f}^{B}\right\}=\left\{\mathbf{u}_{c}^{B}\right\} \tag{21}\]  
\[\left\{\mathbf{g}_{f}\right\}+\left\{\mathbf{g}_{c}\right\}=\mathbf{0} \tag{22}\]  
When describing \(\mathbf{u}_{f}^{B}\) by the generalized coordinates, the displacement-coordination in Equation (21) can be stated as follows:  
\[\left\{\mathbf{u}_{c}^{B}\right\}=\left[\mathbf{\Phi}_{k}^{B}\right]\left\{\mathbf{q}_{k} \right\}+\left[\mathbf{\Psi}_{d}^{B}\right]\left\{\mathbf{g}_{f}\right\} \tag{23}\]  
where, \(\mathbf{\Phi}_{k}^{B}\) and \(\mathbf{\Psi}_{d}^{B}\) are the truncated modal matrix and the remaining adherent mode relating to all the interface DOFs for the master substructure. Multiplying the both sides of Equation (23) by \(\left[\mathbf{\Psi}_{d}^{B}\right]^{-1}\), then \(\mathbf{g}_{f}\) can be written as  
\[\left\{\mathbf{g}_{f}\right\}=\left[\mathbf{\Psi}_{d}^{B}\right]^{-1}\left\{\mathbf{u}_{c }^{B}\right\}-\left[\mathbf{\Psi}_{d}^{B}\right]^{-1}\left[\mathbf{\Phi}_{k}^{B} \right]\left\{\mathbf{q}_{k}\right\} \tag{24}\]  
For the slave substructure, the following equation of motion is used to obtain the pseudo static matrix \(\mathbf{\Phi}_{c}\),  
\[ \begin{bmatrix}\mathbf{K}_{c}^{II}&\mathbf{K}_{c}^{IB}\\ \mathbf{K}_{c}^{BI}&\mathbf{K}_{c}^{BB}\end{bmatrix} \left\{ \begin{array}{l}\mathbf{u}_{c} ^{I}\\ \mathbf{u}_{c}^{B}\end{array} \right\}=\left\{ \begin{array}{l}\mathbf{0}\\ \mathbf{g}_{c}\end{array} \right\} \tag{25}\]  
The interface force \(\mathbf{g}_{c}\) can be obtained from the second equation in Equation (25),  
\[\left\{\mathbf{g}_{c}\right\}=\left[\mathbf{K}_{c}^{BB}+\mathbf{K}_{c}^{BI}\cdot\mathbf{\Phi}_ {c}\right]\left\{\mathbf{u}_{c}^{B}\right\}+\left[\mathbf{K}_{c}^{BI}\right]\left[\bm {\Phi}_{m}\right]\left\{\mathbf{q}_{m}\right\} \tag{26}\]  
Substituting Equations (24) and (26) into Equation (22) leads to  
\[\left\{\mathbf{u}_{c}^{B}\right\}=\left[\mathbf{P}_{1}\quad\mathbf{P}_{2}\right]\left\{ \begin{array}{l}\mathbf{q}_{k}\\ \mathbf{q}_{m}\end{array} \right\} \tag{27}\]  
where,  
\[\mathbf{P}_{1}=\left(\mathbf{I}+\mathbf{\Psi}_{d}^{B}\cdot\mathbf{K}_{c}^{BB}+\mathbf{\Psi}_{d}^{ B}\cdot\mathbf{K}_{c}^{BI}\cdot\mathbf{\Phi}_{c}\right)^{-1}\cdot\mathbf{\Phi}_{k}^{B} \tag{28}\]  
\[\mathbf{P}_{2}=-\left(\left[\mathbf{\Psi}_{d}^{B}\right]^{-1}+\mathbf{K}_{c}^{BB}+\mathbf{K}_{ c}^{BI}\cdot\mathbf{\Phi}_{c}\right)^{-1}\cdot\mathbf{K}_{c}^{BI}\cdot\mathbf{\Phi}_{m} \tag{29}\]  
Substituting Equation (27) into Equation (24), then \(\mathbf{g}_{f}\) becomes  
\[\left\{\mathbf{g}_{f}\right\}=\left[\mathbf{Q}_{1}\quad\mathbf{Q}_{2}\right]\left\{ \begin{array}{l}\mathbf{q}_{k}\\ \mathbf{q}_{m}\end{array} \right\} \tag{30}\]  
where,  
\[\mathbf{Q}_{1}=\left[\mathbf{\Psi}_{d}^{B}\right]^{-1}\cdot\left(\mathbf{P}_{1}-\mathbf{\Phi} _{k}^{B}\right) \tag{31}\]  
\[\mathbf{Q}_{2}=\left[\mathbf{\Psi}_{d}^{B}\right]^{-1}\cdot\mathbf{P}_{2} \tag{32}\]  
Thus, the second coordinate transformation for the two substructures are \[\left\{ \begin{array}{l}\mathbf{q}_{k}\\ \mathbf{g}_{f}\end{array} \right\}= \begin{bmatrix}\mathbf{I}_{k}&\mathbf{0}\\ \mathbf{Q}_{1}&\mathbf{Q}_{2}\end{bmatrix} \left\{ \begin{array}{l}\mathbf{q}_{k}\\ \mathbf{q}_{m}\end{array} \right\}= \begin{bmatrix}\mathbf{T}_{f}^{2}\end{bmatrix}  \left\{ \begin{array}{l}\mathbf{q}_{k}\\ \mathbf{q}_{m}\end{array} \right\} \tag{33}\] \[\left\{