# Dynamic Stability Assessment of High-Speed Railway Bridges Using Numerical Model Updating
## 4 Dynamic Stability Assessment for High-Speed Railway Bridges  
### Algorithm for Numerical Model Updating Using the Univariate Search Method  
The univariate search method, one of the unconstrained optimization techniques, is a one-dimensional search method that alters only one variable at a time while fixing the other variables in order to update the approximate value. The univariate search method defines the error function, which is defined as Equation (6), as the object function, and performs optimization in a way that minimizes the error function,  
\[\text{Error}=\sqrt{\sum_{n=1}^{N}\left|\text{D}_{n}^{A}-\text{D}_{n}^{M} \right|^{2}} \tag{6}\]  
where \(\text{D}_{n}^{M}\) is the \(n\)-th dynamic property estimated from the measurement and \(\text{D}_{n}^{A}\) is the \(n\)-th dynamic property calculated by using numerical analysis. If the objective function is set as a simple error function, as shown in Equation (6), the convergence of optimization can be difficult in the presence of many target values because all objective functions cannot be satisfied. Therefore, the objective function was set as the relative error of variables in order to perform numerical model updating using many target values, as shown in Equation (7). Here, \(\text{P}_{n,k}\) shows variables for the \(n\)-th dynamic property for the \(k\)-th calculation. \[\text{Relative Error},d_{k}=\frac{\text{P}_{n,k}-\text{P}_{n,k-1}}{\text{P}_ {n,k}-1} \tag{7}\]  
In order to perform numerical model updating based on the univariate search method, each target value is updated through different variables. The variables can be sequentially updated only if the target values other than the \(n\)-th variable have extremely low sensitivity for the \(n\)-th variable when each target value is updated after the target values and variables are set. When the sensitivity of multiple target values to multiple variables is not zero for complex structures such as bridges, the sequential updating of variables will cause errors in the updating results. To prevent such errors, variables were set for each target value, and all the variables were updated using the univariate search method. \begin{table}
\begin{tabular}{l c c c c} \hline \hline \multirow{2}{*}{**Target Bridge**} & \multirow{2}{*}{**Mode Type**} & \multicolumn{2}{c}{**Natural Frequency (Hz)**} & \multicolumn{2}{c}{**Effective Span**} & \multicolumn{1}{c}{**Damping Ratio**} \\  & & **Frequency (Hz)** & **Length (m)** & **(\%)** \\ \hline \multirow{2}{*}{Imgi 2 nd Bridge} & vertical & 4.96 & 40.75 & 2.88 \\  & torsional & 25.64 & 37.78 & - \\ \multirow{2}{*}{Maeaji Bridge} & vertical & 10.54 & 25.07 & 1.66 \\  & torsional & 26.29 & 26.43 & - \\ \multirow{2}{*}{Hwalchun Bridge} & vertical & 9.44 & 23.56 & 1.93 \\  & torsional & 31.36 & 23.45 & - \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Dynamic properties estimated from the target bridges. The measured value, analyzed value, and mode contribution of the \(n\)-th dynamic property for numerical model updating can be expressed as Equations (8)-(10),  
\[\mathrm{D}_{n}^{M}=\left[d_{n,1}^{M},\ d_{n,2}^{M},\ \cdots,d_{n,i}^{M}\right] \tag{8}\]  
\[\mathrm{D}_{n,k}^{A}=\left[d_{n,k,1}^{A},\ d_{n,k,2}^{A},\ \cdots,d_{n,k,i}^{A}\right] \tag{9}\]  
\[\mathrm{F}_{k,i}=\left[f_{k,1},\ f_{k,2},\cdots,f_{k,i}\right] \tag{10}\]  
where \(d_{n,i}^{M}\) is the value of the \(i\)-th mode of the \(n\)-th dynamic property measured from the bridge, \(d_{n,k,i}^{A}\) is the value of the \(i\)-th mode of the \(n\)-th dynamic property obtained from the \(k\)-th forward analysis using the numerical model, and \(f_{k,i}\) is the mode contribution of the \(i\)-th mode obtained from the \(k\)-th forward analysis.