## 2 Theoretical Background
### Decoupling of Equations of Motion  
The classical differential equation of the system dynamics model with \(n\) nodes, each with six degrees of freedom is as follows:  
\[\mathbf{M}\tilde{\mathbf{z}}+\mathbf{C}\dot{\mathbf{z}}+\mathbf{K}\mathbf{z}=F(t) \tag{1}\]  
where \(\mathbf{M}\), \(\mathbf{C}\), and \(\mathbf{K}\) are the mass, damping, and stiffness matrices of the structure, respectively, and their dimensions are all \(6\)\(n\)\(\times\)\(6\)\(n\). \(F(t)\) is an input vector with dimensions of \(6\)\(n\)\(\times\)\(1\) and \(\tilde{\mathbf{z}}/\dot{\mathbf{z}}/\mathbf{z}\) are acceleration, velocity, and displacement vectors, respectively, with the dimensions of \(6\)\(n\)\(\times\)\(1\). When the system is in undamped free vibration, its dynamic model is rewritten as follows:  
\[M\tilde{\mathbf{z}}+\mathbf{K}\mathbf{z}=0 \tag{2}\]  
Assuming a small deformation of the structure, the structure vibrates slightly near the equilibrium position and the solution of the above equations can be set as follows:  
\[\mathbf{z}_{i}=\mathbf{z}_{mi}sin(w_{i}t+\Phi_{i}) \tag{3}\]  
where \(\mathbf{z}_{i}\) is the displacement vector of all degrees of freedom at the \(i\)th order frequency, \(\mathbf{z}_{mi}\) is the \(i\)th order eigenvector, \(w_{i}\) is the \(i\)th order eigenvalue, and \(\phi_{i}\) is the initial phase. Substituting Equation (3) into Equation (2):  
\[M\Big{[}-w_{i}^{2}\mathbf{z}_{mi}sin(w_{i}t+\phi_{i})\Big{]}+\mathbf{K}[\mathbf{z}_{mi}sin (w_{i}t+\phi_{i})]=0 \tag{4}\]  
Through the above solution, the eigenvalue \(w_{i}\) (\(i\) = 1,..., m) can be obtained. The corresponding characteristic column vector \(\mathbf{z}_{mi}\) = (\(z_{m1}\)\(\dots\)\(z_{mm}\)) is the inherent matrix of the system. Taking a three-degree-of-freedom system as an example, the equation for the \(i\)th-order eigenvalue is as follows [16]:  
\[\left[ \begin{array}{c}z_{1i}\\ \dot{z}_{1i}\\ z_{2i}\\ \dot{z}_{2i}\\ z_{3i}\\ \end{array} \right]=z_{mi}sin(w_{i}t+\phi_{i}) \tag{5}\]  
Generalized coordinates \(z_{1}(t)\), \(z_{2}(t)\), and \(z_{3}(t)\) are used to represent the horizontal displacements of three discrete masses \(m_{1}\), \(m_{2}\), and \(m_{3}\), respectively. \(F_{1}\), \(F_{2}\), and \(F_{3}\), respectively, represent the corresponding generalized external forces (one unit acceleration is applied to one degree of freedom at a time, while the other degrees of freedom remain in state). The three-degree-of-freedom system is shown in Figure 2. Three discrete mass motion equations are written using [PERSON]'s law. The mass, damping, and stiffness matrices of the system are obtained by Equations (7) and (8). The equation of motion can be written in the form of Equation (6):  
\[\left[ \begin{array}{ccc}m_{1}&0&0\\ 0&m_{2}&0\\ 0&0&m_{3}\end{array} \right]\left[ \begin{array}{c}\ddot{z}_{1}\\ \ddot{z}_{2}\\ \ddot{z}_{3}\end{array} \right]+\left[ \begin{array}{ccc}c_{1}&-c_{1}&0\\ -c_{1}&(c_{1}+c_{2})&-c_{2}\\ 0&-c_{2}&c_{2}\end{array} \right]\left[ \begin{array}{c}\dot{z}_{1}\\ \dot{z}_{2}\\ \dot{z}_{3}\end{array} \right]+\left[ \begin{array}{ccc}k_{1}&-k_{1}&0\\ -k_{1}&(k_{1}+k_{2})&-k_{2}\\ 0&-k_{2}&k_{2}\end{array} \right]\left[ \begin{array}{c}z_{1}\\ z_{2}\\ z_{3}\end{array} \right]=\left[ \begin{array}{c}F_{1}\\ F_{2}\\ F_{3}\end{array} \right] \tag{6}\]  
To simplify the system, let \(m_{1}=m_{2}=m_{3}=m\), \(k_{1}=k_{2}=k\), and \(c_{1}=c_{2}=0\). According to modal analysis theory, we can calculate the eigenvalues \(w_{1}^{2}=0\), \(w_{2}^{2}=\frac{k}{m}\), and \(w_{3}^{2}=\frac{3k}{m}\). After the eigenvalues are calculated, eigenvectors \(z_{1}\), \(z_{2}\), and \(z_{3}\) are obtained according to Equation (4):  
\[\frac{z_{2}}{z_{1}}=\frac{k-w_{i}^{2}}{k} \tag{7}\]  
\[\frac{z_{3}}{z_{1}}=\frac{m^{2}w_{i}^{2}-3 kmw_{i}^{2}+k^{2}}{k^{2}} \tag{8}\]  
Each column is the modal vector \(z_{1}\), \(z_{2}\), and \(z_{3}\).