## 3 Space--Time Dynamic Analysis Method Based on DCM
### Stability Analysis of Transfer Matrix \(\mathbf{Q}\)  
The stability of recurrence process is analyzed based on the spectral radius \(R(\mathbf{Q})\) of transfer matrix in Equation (26). According to the construction of \(\mathbf{Q}\), if the attribute parameters of the beam are determined, the transfer matrix is only related to the DC weight coefficient and the time step \(\Delta T\). Orthogonal grids are selected as discrete schemes, so that the number of boundary nodes in \(\xi\) and \(\tau\) directions is \(m\) and \(n\) respectively. When the basis function scheme is determined, the DC weight coefficients are determined by \(m\) and \(n\). The specific property parameters of the beam are as follow: rigidity EI = 4.7726 \(\times\)\(10^{7}\) N/m; distributed mass \(\rho\) = 420 kg/m; length L = 0.01 m; and shear coefficient \(\zeta=30/\pi^{2}\). Firstly, by controlling the time step \(\Delta T\) = 0.25 s, the relationship between \(R(\mathbf{Q})\) and orthogonal grids of \(m\) and \(n\) under different conditions is explored, where the uniform distribution of nodes and the nodes of CGL distribution form are considered. Figure 4 shows the curves of \(R(\mathbf{Q})\) versus \(n\) for \(m\) = 3, 4, 5, respectively. Each case includes two distributions for comparison. Figure 4: The relationship between spectral radius and mesh form. It can be seen from Figure 4 that \(R(\mathbf{Q})\) in the case of CGL distribution increases gradually to 1 with the increase of \(n\), and then remains stable around 1. In the case of uniform distribution, \(R(\mathbf{Q})\) will have a mutation at the position of \(n=6\) or 7, and the trends expect above mutation are roughly the same as for the CGL distribution. It can be discovered that the spectral radius under the two distributions is generally less than or equal to 1, but the range of spectral radius close to 1 under the CGL distribution is wider. Then the effect of time step \(\Delta T\) on spectral radius \(R(\mathbf{Q})\) of transfer matrix is studied. As shown in Figure 5, the relationship curves between the spectral radius and the time step length calculated under different grids of \(m\) and \(n\) are given. In general, \(R(\mathbf{Q})\) first maintains a process near 1 with the increase of \(\Delta T\), then begins to decline rapidly at a certain point, and, finally, gradually converges to a smaller value. The position of the sudden drop varies with the number of nodes in the time domain, that is, when \(m\) is the same, the larger \(n\), the later the drop, and the wider the range of \(R(\mathbf{Q})\) close to 1. In addition, by comparing these three figures, it can be found that increasing the number of nodes \(m\) in the spatial direction alone has little effect on the value of \(R(\mathbf{Q})\). The spectral radius of the transfer matrix \(\mathbf{Q}\) is close to or less than 1 on orthogonal grids, where the main difference is the range width of \(R(\mathbf{Q})\) near 1. In order to study the time-history displacement of the transmission process under different values of \(R(\mathbf{Q})\), several specific conditions are selected according to the grid form and time step length to do the stability verification analysis. The specific conditions are shown in Table 1. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & & \multicolumn{2}{c}{\(m=5\), \(n=5\)} & \multicolumn{3}{c}{\(m=5\), \(n=7\)} \\ \cline{2-7}  & **Case 1** & **Case 2** & **Case 3** & **Case 4** & **Case 5** & **Case 6** \\ \hline distribution & & & & & & \\ form & CGL & CGL & CGL & CGL & equally & CGL \\ \(\Delta T\)(s) & 0.064 & 0.16 & 0.2 & 0.16 & 0.25 & 0.35 \\ \(R(\mathbf{Q})\) & 1.0015 & 0.9229 & 0.6477 & 0.9984 & 1.5524 & 0.6521 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: \(R(\mathbf{Q})\) under different mesh forms and time steps. Figure 5: The relationship between spectral radius and time step.