## 2 Theoretical Background
### Double Coordinate Free-Interface Mode Synthesis Method  
The double coordinate free-interface mode synthesis method adopts both displacement and force compatible conditions. A structure with A and B substructures is used to illustrate the principle of this method. The low-order free interface main modes and the residual adhesion modes of substructures are used to construct presumed mode sets. First, transformation from the physical coordinates to modal coordinates is conducted for each substructure, as follows:  
\[\left\{u^{\left(z\right)}\right\}=\left\{ \begin{array}{c}u_{i}^{\left(z \right)}\\ u_{j}^{\left(z\right)}\end{array} \right\}=\left[\phi^{\left(z\right)}\right] \left\{p^{\left(z\right)}\right\}=\left[ \begin{array}{cc}\phi_{l}^{\left(z \right)}&\psi_{s}^{\left(z\right)}\end{array} \right]\left\{ \begin{array}{c} p_{l}^{\left(z\right)}\\ p_{s}^{\left(z\right)}\end{array} \right\}=\left[ \begin{array}{cc}\phi_{l}^{ \left(z\right)}&\phi_{l}^{\left(z\right)}\\ \phi_{l}^{\left(z\right)}&\psi_{j}^{\left(z\right)}\end{array} \right]\left\{ \begin{array}{c}p_{l}^{\left(z\right)}\\ p_{s}^{\left(z\right)}\end{array} \right\} \tag{1}\]  
where \(z=A,B\); \(\left\{u_{i}^{\left(z\right)}\right\}\), \(\left\{u_{j}^{\left(z\right)}\right\}\) are the physical coordinates of the internal and interfacial degree of freedoms (DOFs), respectively; \(\left\{p_{l}^{\left(z\right)}\right\}\) and \(\left\{p_{s}^{\left(z\right)}\right\}\) are independent and dependent modal coordinates, respectively; \(\left[\phi^{\left(z\right)}\right]\) is the coordinate transformation matrix; \(\left[\phi_{l}^{\left(z\right)}\right]\) is the main mode set; and \(\left[\psi_{s}^{\left(z\right)}\right]\) is the residual adhesion mode set. \[\left[\psi_{s}^{\left(z\right)}\right]=\left(\left[G^{\left(z\right)}\right]- \left[\phi_{l}^{\left(z\right)}\right]\left[\lambda_{l}^{\left(z\right)} \right]^{-1}\left[\phi_{l}^{\left(z\right)}\right]^{T}\right)\left[ \begin{array} []{c}O_{l\times j}\\ I_{j\times j}\end{array} \right] \tag{2}\]  
where \(\left[G^{\left(z\right)}\right]\) is the flexibility matrix of the substructure; \(\left[\lambda_{l}^{\left(z\right)}\right]\) is the low-order eigenvalue matrix; \(\left[O_{l\times j}\right]\) is the zero matrix and \(\left[I_{j\times j}\right]\) is the unit matrix; \(i\) represents the number of internal DOFs; \(j\) is the number of interfacial DOFs. The modal coordinate set \(\left\{p\right\}\) of the system can be divided as follows. \[\left\{p\right\}=\left[ \begin{array}{cc}\left\{p_{l}^{\left(A\right)} \right\}^{T}&\left\{p_{s}^{\left(A\right)}\right\}^{T}&\left\{p_{l}^{\left(B \right)}\right\}^{T}&\left\{p_{s}^{\left(B\right)}\right\}^{T}\end{array}  \right]^{T} \tag{3}\]  
The modal coordinate set \(\left\{p\right\}\) is transformed by a positioning matrix to rearrange the elements  
\[\left\{\overline{p}\right\}=\left[ \begin{array}{cc}\left\{p_{s}^{\left(A \right)}\right\}^{T}&\left\{p_{s}^{\left(B\right)}\right\}^{T}&\left\{p_{l}^{ \left(A\right)}\right\}^{T}&\left\{p_{l}^{\left(B\right)}\right\}^{T}\end{array}  \right]^{T}=\left[ \begin{array}{cc}p_{s}^{T}&p_{l}^{T}\end{array} \right]^{T} \tag{4}\]  
\[\left\{p\right\}=\left[E\right]\left\{\overline{p}\right\} \tag{5}\]  
where \(\left[E\right]\) is the positioning matrix, which can be represented by  
\[\left[E\right]=\left[ \begin{array}{cccc}O&O&I&O\\ I&O&O&O\\ O&O&O&I\\ O&I&O&O\end{array} \right]. \tag{6}\]  
The node force of a substructure is equal to the interface force between substructures when the structure is in low-order free vibration. \[\left\{F^{\left(z\right)}\right\}=\left\{ \begin{array}{c}O_{i}\\ f_{j}^{\left(z\right)}\end{array} \right\} \tag{7}\]By substituting the modal coordinates into the motion equation, and ignoring the influence of inertial force, it can be obtained that  
\[\Big{[}\psi_{s}{}^{(z)}\Big{]}^{T}\Big{[}k^{(z)}\Big{]}\Big{[}\psi_{s}{}^{(z)} \Big{]}\Big{\{}p_{s}{}^{(z)}\Big{\}}=\Big{[}\psi_{s}{}^{(z)}\Big{]}^{T}\Bigg{\{} \begin{array}{c}O_{i}\\ f_{j}{}^{(z)}\end{array}