# Dynamic Stability Assessment of High-Speed Railway Bridges Using Numerical Model Updating
## 3 Estimation of the Dynamic Properties of the High-Speed Railway Bridges
\[\overset{\rightarrow}{x}(k,t)=\sum_{r=1}^{L}\phi_{r}(k)q_{r}(t) \tag{2}\]  
\[[X]=\left[ \begin{array}{cccc}x_{1}(t_{1})&x_{2}(t_{1})&\cdots&x_{L}(t_{1})\\ x_{1}(t_{2})&x_{2}(t_{2})&\cdots&x_{L}(t_{2})\\ \vdots&\vdots&\cdots&\vdots\\ x_{1}(t_{N})&x_{2}(t_{N})&\cdots&x_{L}(t_{N})\end{array} \right]=\left[\{q\}_ {1}\{\phi\}_{1}^{T}+\cdots+\{q\}_{L}\{\phi\}_{L}^{T}\right] \tag{3}\]  
Figure 4: Cross-correlation signal measured in the Imgi 2 nd Bridge: (**a**) Time domain; (**b**) Frequency domain. In Equation (2), \(\overrightarrow{x}(k,t)\) is the vector at time \(t\) of the \(k\)-th mode, \(\phi_{r}(k)\) is the \(r\)-th normal mode, \(q_{r}(t)\) is the \(r\). -th eigenvibration vector, and \(L\). is the number of modes to be considered. In Equation (3), \(x_{L}(t_{N})\) is the value measured at time \(t_{N}\). at the \(L\)-th measurement position, \(\left\{q\right\}_{L}\) is the time history function \(q_{L}(t)\) with an array of \(N\times 1\), and \(\left\{\phi\right\}_{L}\) is the mode shape at the measurement position of the structure. Among the measurement positions, the final measurement position is the \(L\). -th measurement position, and the number of data measured at one point is \(N\). If the correlation matrix of the measured response is expressed as Equation (4), Equation (5) can be obtained by multiplying both sides of Equation (4) by the \(i\)-th mode shape \(\left\{\phi\right\}_{i}\). \[[\text{C}]=\frac{1}{N}[\text{X}]^{T}[\text{X}] \tag{4}\]  
\[[\text{C}]\{\phi\}_{i}=\lambda_{i}\{\phi\}_{i} \tag{5}\]  
The eigenvalue \(\lambda_{i}\) can be expressed as \((1/N)\{q\}_{i}^{T}\{q\}_{i}\) in the mode coordinate system \(\{q\}_{i}\), while \(\lambda_{i}\) and the mode shape \(\{\phi\}_{i}\) can be calculated by using Equation (5). If proper orthogonal decomposition is applied using the cross-correlation signal, the mode shape can be estimated even for the response in which time synchronization is not performed for each group. Figure 5 shows the mode shape estimated by using the acceleration response measured from the target bridges. Figure 5: The mode shape estimated by applying proper orthogonal decomposition at the target bridges: (**a**) Imgi 2 nd Bridge; (**b**) Maeaji Bridge; (**c**) Hwalchun Bridge. In Figure 4, the first vertical mode of the Imgi 2 nd Bridge is very dominant, and the first torsional mode, which is the second mode, can be seen below 50 Hz. Since it is difficult to identify a clear mode after 35 Hz, only dynamic characteristics below 35 Hz were used as target values for numerical model updating. Table 3 shows the estimated dynamic properties as target values for numerical model updating. For the mode shape, it is difficult to directly use the response ratio at each position as a target value. As a result, the mode shape was assumed to be the same sine function as the mode shape of the result calculated in the numerical analysis, and half of the period of the sine function was used as the target value. Some of the effective span lengths estimated in Table 3 are longer than the span lengths of the existing bridges. The effective span length is not the actual span length of a bridge; rather, it can be considered as the theoretical value of the length between the intercepts of the mode shape.