# A Multiple and Multi-Level Substructure Method for the Dynamics of Complex Structures
## 2 Component Mode Synthesis of Fixed Interface  
For undamped systems, the dynamic system equation can be written as follows:  
\[\mathbf{M\ddot{u}}+\mathbf{Ku}=\mathbf{F} \tag{1}\]  
In the formula, \(\mathbf{M}\) is the mass matrix, \(\mathbf{K}\) is the stiffness matrix, \(\mathbf{u}\) is the displacement vector, \(\mathbf{F}\) is the external force vector, and \(\mathbf{\dot{u}}\) represents the second derivative of displacement. All the degrees of freedom of the system are divided into two parts:  
1. The part that is not connected with other substructures (internal degrees of freedom), denoted as \(\{\mathbf{u}_{\mathrm{o}}\}\). 2. The part connected with other substructures (external degree of freedom or boundary degree of freedom) is denoted as \(\{\mathbf{u}_{\mathrm{a}}\}\). According to the partition of external and internal degrees of freedom, Equation (1) can be written as follows:  
\[\left[ \begin{array}{cc}\mathbf{M}_{\mathrm{oo}}&\mathbf{M}_{\mathrm{oa}}\\ \mathbf{M}_{\mathrm{ao}}&\mathbf{M}_{\mathrm{aa}}\end{array} \right]\left[ \begin{array}{c}\mathbf{\ddot{u}}_{\mathrm{o}}\\ \mathbf{\ddot{u}}_{\mathrm{a}}\end{array} \right]+\left[ \begin{array}{cc} \mathbf{K}_{\mathrm{oo}}&\mathbf{K}_{\mathrm{oa}}\\ \mathbf{K}_{\mathrm{ao}}&\mathbf{K}_{\mathrm{aa}}\end{array} \right]\left[ \begin{array}{c}\mathbf{u}_{\mathrm{o}}\\ \mathbf{u}_{\mathrm{a}}\end{array} \right]=\left[ \begin{array}{c}\mathbf{F}_{ \mathrm{o}}\\ \mathbf{F}_{\mathrm{a}}\end{array} \right] \tag{2}\]  
In the problem of solving the eigenvalues of the system, \(\mathbf{F}_{\mathrm{O}}=\mathbf{0}\), \(\mathbf{F}_{\mathrm{a}}=\mathbf{F}_{\mathrm{oa}}\mathbf{e}^{\mathrm{i}\omega t}\), \(\mathbf{u}=\mathbf{U}\mathbf{e}^{\mathrm{i}\omega t}\). \(\mathbf{F}_{0}\) represents the amplitude of the external force vector, \(\mathbf{U}\) is the amplitude of the displacement response. In this regard, Equation (2) can be written as follows:  
\[\left(\left[ \begin{array}{cc}\mathbf{K}_{\mathrm{oo}}&\mathbf{K}_{\mathrm{oa }}\\ \mathbf{K}_{\mathrm{ao}}&\mathbf{K}_{\mathrm{aa}}\end{array} \right]-\omega^{2} \left[ \begin{array}{cc}\mathbf{M}_{\mathrm{ao}}&\mathbf{M}_{\mathrm{oa}} \\ \mathbf{M}_{\mathrm{ao}}&\mathbf{M}_{\mathrm{aa}}\end{array} \right]\right)\left[ \begin{array}{c}\mathbf{U}_{\mathrm{o}}\\ \mathbf{U}_{\mathrm{a}}\end{array} \right]=\left[ \begin{array}{c}\mathbf{0} \\ \mathbf{F}_{\mathrm{0a}}\end{array} \right] \tag{3}\]  
In the formula, \(\omega\) represents the circular frequency (in rad/s) of the structure. When the interface surface is constrained, \(\mathbf{U}_{\mathrm{a}}=0\) is substituted into the first equation of Equation (3) to obtain the following eigenvalue equation of the interface:  
\[\left(\mathbf{K}_{\mathrm{oo}}-\omega^{2}\mathbf{M}_{\mathrm{oo}}\right) \mathbf{U}_{\mathrm{o}}\mathbf{=}\mathbf{0} \tag{4}\]  
Therefore, the eigenvalue equation of the interface corresponding to Equation (4) is \(\overline{\varphi}_{\mathrm{bL}}\), and the eigenvalue matrix is \(\mathbf{A}_{\mathrm{bL}}\), then the main mode \(\varphi_{\mathrm{bL}}\) is as follows  
\[\varphi_{\mathrm{bL}}\mathbf{=}\left[ \begin{array}{c}-\\ \overline{\varphi}_{\mathrm{bL}}\\ \mathbf{0}\end{array} \right] \tag{5}\]  
Assuming that \(\overline{\mathbf{N}}_{0}\) is made up of \(\varphi_{\mathrm{bL}}\) and \(\mathbf{\Phi}_{\mathrm{c0}}\), then \(\overline{\mathbf{N}}_{0}\) can be written as follows  
\[\mathbf{N}_{0}\mathbf{=}\left[ \begin{array}{cc}\mathbf{\Phi}_{\mathrm{c0}} &\varphi_{\mathrm{bL}}\end{array} \right] \tag{6}\]  
Among them, \(\overline{\mathbf{N}}_{0}\) is the matrix of the first coordinate change of the substructure, \(\varphi_{\mathrm{bL}}\) is the main mode reserved by the substructure constrained interface, and \(\mathbf{\Phi}_{\mathrm{c0}}\) is the constrained mode of all the interface degrees of freedom. Ignoring the inertial force term in Formula (3), the static force equation is given as follows:  
\[\left[ \begin{array}{cc}\mathbf{K}_{\mathrm{oo}}&\mathbf{K}_{\mathrm{oa}}\\ \mathbf{K}_{\mathrm{ao}}&\mathbf{K}_{\mathrm{aa}}\end{array}