# Reduction of Coupling Interface Degrees of Freedom in Mixed-Interface Component Mode Synthesis
## 2 A Brief Review of the Traditional MCMS  
This section provides a brief review of the traditional MCMS proposed by [PERSON] [PERSON] (also called the B-H method for short) [13]. In a global domain, the linear equation of motion (EOM) of a N-DOFs structure is written as  
\[\left[\mathbf{M}\right]\left\{\ddot{\mathbf{u}}\right\}+\left[\mathbf{C}\right]\left\{\dot {\mathbf{u}}\right\}+\left[\mathbf{K}\right]\left\{\mathbf{u}\right\}=\left[\mathbf{f}\right] \tag{1}\]  
where \(\mathbf{M}\), \(\mathbf{C}\) and \(\mathbf{K}\) are the mass matrix, the damping matrix and the stiffness matrix of the global structure, respectively; \(\ddot{\mathbf{u}}\), \(\dot{\mathbf{u}}\) and \(\mathbf{u}\) are the acceleration vector, the velocity vector and the displacement vector, respectively; \(\mathbf{f}\) is the external force vector. In the B-H method, the original system is divided into two groups: master substructures and slave substructures. All the substructures are non-overlapping so that each node belongs to exactly one substructure except for the nodes on the coupling interface. In Figure 1, only one master substructure and one slave substructure are shown as demonstration. Different interconnection conditions are imposed on each part, where the interconnection of the master substructure is constrained while that of the slave substructure is free. Then, Equation (1) can be rewritten in block matrices as  
\[ \begin{bmatrix}\mathbf{M}_{f}^{I}&\mathbf{0}&\mathbf{0}\\ \mathbf{0}&\mathbf{M}^{B}&\mathbf{0}\\ \mathbf{0}&\mathbf{0}&\mathbf{M}_{c}^{I}\end{bmatrix} \left\{ \begin{array}{c}\dot{\mathbf{u}} _{f}^{I}\\ \ddot{\mathbf{u}}_{f}^{B}\\ \ddot{\mathbf{u}}_{c}^{I}\end{array} \right\}+ \begin{bmatrix}\mathbf{C}_{f}^{I}&\mathbf{0} &\mathbf{0}\\ \mathbf{0}&\mathbf{C}^{B}&\mathbf{0}\\ \mathbf{0}&\mathbf{0}&\mathbf{C}^{I}_{c}\end{bmatrix} \left\{ \begin{array}{c}\dot{\mathbf{u} }_{f}^{I}\\ \dot{\mathbf{u}}_{g}^{B}\\ \mathbf{0}&\mathbf{0}&\mathbf{K}_{c}^{I}\end{array} \right\}+ \begin{bmatrix}\mathbf{K}_{f}^{I} &\mathbf{0}&\mathbf{0}\\ \mathbf{0}&\mathbf{K}^{B}&\mathbf{0}\\ \mathbf{0}&\mathbf{0}&\mathbf{K}_{c}^{I}\end{bmatrix} \left\{ \begin{array}{c}\mathbf{u}_{f}^ {I}\\ \mathbf{u}_{g}^{B}\\ \mathbf{u}_{c}^{I}\end{array} \right\}=\left\{ \begin{array}{c}\mathbf{f}_{f}^{I}\\ \mathbf{f}^{B}\\ \mathbf{f}_{c}^{I}\end{array} \right\} \tag{2}\]  
in which, subscripts \(c\) and \(f\) indicate the master substructure with constrained interconnections and the slave substructure with free interconnections, respectively; superscripts \(I\) and \(B\) indicate the internal DOFs and the interface DOFs. The linear equations of motion of each structure are,  
\[ \begin{bmatrix}\mathbf{M}_{f}^{I}&\mathbf{0}\\ \mathbf{0}&\mathbf{M}_{f}^{B}\end{bmatrix} \left\{ \begin{array}{c}\dot{\mathbf{u}}_{f}^{ I}\\ \ddot{\mathbf{u}}_{f}^{B}\end{array} \right\}+ \begin{bmatrix}\mathbf{C}_{f}^{I}&\mathbf{0}\\ \mathbf{0}&\mathbf{C}_{f}^{B}\end{bmatrix} \left\{ \begin{array}{c}\mathbf{u}_{f}^{I}\\ \dot{\mathbf{u}}_{f}^{B}\end{array} \right\}+ \begin{bmatrix}\mathbf{K}_{f}^{I}&\mathbf{0} \\ \mathbf{0}&\mathbf{K}_{f}^{B}\end{bmatrix} \left\{ \begin{array}{c}\mathbf{u}_{f}^{I}\\ \mathbf{u}_{f}^{B}\end{array} \right\}=\left\{ \begin{array}{c}\mathbf{f}_{f}^{I}\\ \mathbf{f}_{f}^{B}\end{array} \right\}+\left\{ \begin{array}{c}\mathbf{0}\\ \mathbf{g}_{f}\end{array} \right\} \tag{3}\]  
\[ \begin{bmatrix}\mathbf{M}_{c}^{I}&\mathbf{0}\\ \mathbf{0}&\mathbf{M}_{c}^{B}\end{bmatrix} \left\{ \begin{array}{c}\ddot{\mathbf{u}}_{c}^{ I}\\ \ddot{\mathbf{u}}_{c}^{B}\end{array} \right\}+ \begin{bmatrix}\mathbf{C}_{c}^{I}&\mathbf{0}\\ \mathbf{0}&\mathbf{C}_{c}^{B}\end{bmatrix} \left\{ \begin{array}{c}\mathbf{u}_{c}^{I}\\ \mathbf{u}_{c}^{B}\end{array} \right\}+ \begin{bmatrix}\mathbf{K}_{c}^{I}&\mathbf{0}\\ \mathbf{0}&\mathbf{K}_{c}^{B}\end{bmatrix} \left\{ \begin{array}{c}\mathbf{u}_{c}^{I}\\ \mathbf{u}_{c}^{B}\end{array} \right\}=\left\{ \begin{array}{c}\mathbf{f}_{c}^{I}\\ \mathbf{f}_{c}^{B}\end{array} \right\}+\left\{ \begin{array}{c}\mathbf{0}\\ \mathbf{g}_{c}\end{array}