# Reduction of Coupling Interface Degrees of Freedom in Mixed-Interface Component Mode Synthesis
## 2 A Brief Review of the Traditional MCMS
\right\} \tag{4}\]  
where, \(\mathbf{g}\) is the reaction force vector relative to the coupling DOFs of the substructure. Figure 1: Scheme of a mixed-substructure. The first coordinate transformations that transforming the physical coordinates \(\mathbf{u}\) to generalized coordinates \(\mathbf{q}\) are done in each substructure according to the different interconnection conditions. That is,  
\[\left\{\mathbf{u}_{f}\right\}=\left\{ \begin{array}{l}\mathbf{u}_{f}^{I}\\ \mathbf{u}_{f}^{B}\end{array} \right\}= \begin{bmatrix}\mathbf{\Phi}_{k}^{I}\\ \mathbf{\Phi}_{k}^{B}\end{bmatrix} \left\{\mathbf{q}_{k}\right\}=\left[\mathbf{T}_{f}^{I} \right]\left\{\mathbf{q}_{k}\right\} \tag{5}\]  
\[\left\{\mathbf{u}_{c}\right\}=\left\{ \begin{array}{l}\mathbf{u}_{c}^{I}\\ \mathbf{u}_{c}^{B}\end{array} \right\}= \begin{bmatrix}\mathbf{\Phi}_{m}&\mathbf{\Phi}_{c}\\ \mathbf{0}&\mathbf{I}\end{bmatrix} \left\{ \begin{array}{l}\mathbf{q}_{m}\\ \mathbf{u}_{c}^{B}\end{array} \right\}=\left[\mathbf{T}_{c}^{1}\right]\left\{ \begin{array} []{l}\mathbf{q}_{m}\\ \mathbf{u}_{c}^{B}\end{array} \right\} \tag{6}\]  
where \(\mathbf{\Phi}_{k}\) and \(\mathbf{\Phi}_{m}\) are the truncated normal mode involving in mode synthesis; \(\mathbf{\Phi}_{c}\) is the constrained mode; \(\mathbf{T}^{1}\) is the first coordinate transformation matrix. The corresponding generalized stiffness and mass matrix can be written as  
\[\left[\mathbf{K}_{f}^{(1)}\right]=\left[\mathbf{T}_{f}^{1}\right]^{\mathrm{T}}\left[ \mathbf{K}_{f}\right]\left[\mathbf{T}_{f}^{1}\right]\quad,\quad\left[\mathbf{M}_{f}^{(1)} \right]=\left[\mathbf{T}_{f}^{1}\right]^{\mathrm{T}}\left[\mathbf{M}_{f}\right]\left[ \mathbf{T}_{f}^{1}\right] \tag{7}\]  
and  
\[\left[\mathbf{K}_{c}^{(1)}\right]=\left[\mathbf{T}_{c}^{1}\right]^{\mathrm{T}}\left[ \mathbf{K}_{c}\right]\left[\mathbf{T}_{c}^{1}\right]\quad,\quad\left[\mathbf{M}_{c}^{(1)} \right]=\left[\mathbf{T}_{c}^{1}\right]^{\mathrm{T}}\left[\mathbf{M}_{c}\right]\left[ \mathbf{T}_{c}^{1}\right] \tag{8}\]  
In the B-H method, only the displacement-compatibility is involved in mode synthesis. That is,  
\[\left\{\mathbf{u}_{f}^{B}\right\}=\left\{\mathbf{\Phi}_{k}^{B}\right\}\left\{\mathbf{q}_ {k}\right\}=\left\{\mathbf{u}_{c}^{B}\right\} \tag{9}\]  
Substituting Equation (9) into Equation (6), the second coordinate transformation for the slave substructure and the corresponding generalized stiffness and mass matrix can be expressed as  
\[\left\{ \begin{array}{l}\mathbf{q}_{m}\\ \mathbf{u}_{c}^{B}\end{array} \right\}= \begin{bmatrix}\mathbf{I}_{m}&\mathbf{0}\\ \mathbf{0}&\mathbf{\Phi}_{k}^{B}\end{bmatrix} \left\{ \begin{array}{l}\mathbf{q}_{m}\\ \mathbf{q}_{k}\end{array} \right\}=\left[\mathbf{T}_{c}^{2}\right]\left\{ \begin{array}[] {l}\mathbf{q}_{m}\\ \mathbf{q}_{k}\end{array} \right\} \tag{10}\]  
\[\left[\mathbf{K}_{c}^{(2)}\right]=\left[\mathbf{T}_{c}^{2}\right]^{\mathrm{T}}\left[ \mathbf{K}_{c}^{(1)}\right]\left[\mathbf{T}_{c}^{2}\right]\quad,\quad\left[\mathbf{M}_{c}^ {(2)}\right]=\left[\mathbf{T}_{c}^{2}\right]^{\mathrm{T}}\left[\mathbf{M}_{c}^{(1)} \right]\left[\mathbf{T}_{c}^{2}\right] \tag{11}\]  
Assembling all the generalized stiffness and mass matrices of the master and slave substructures, the global reduced stiffness and mass matrices are  
\[\left[\overline{\mathbf{K}}\right]=\left[\mathbf{K}_{f}^{(1)}\right]+\left[\mathbf{K}_{c}^ {(2)}\right]\quad,\quad\left[\overline{\mathbf{M}}\right]=\left[\mathbf{M}_{f}^{(1)} \right]+\left[\mathbf{M}_{c}^{(2)}\right] \tag{12}\]  
The uncoupled eigenvalue problem is shown in Equation (13), and the order number of the eigenvalue problem equals to \((m+k)\). \[\left(\left[\overline{\mathbf{K}}\right]-\omega^{2}\left[\overline{\mathbf{M}}\right] \right)\left\{ \begin{array}{l}\mathbf{q}_{m}\\ \mathbf{q}_{k}\end{array} \right\}=\mathbf{0} \tag{13}\]