# Dynamic Impact Factor and Resonance Analysis of Curved Intercity Railway Viaduct
## 4 Numerical Analysis
### Resonance Condition  
The effect of vehicle load on the curved bridge is similar to the excitation of a certain frequency. When the excitation frequency of the vehicle is close to the natural vibration frequency of the bridge, it is easy to cause bridge resonance and endanger the service \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Value** & **Unit** \\ \hline Elastic modulus & \(3.45\times 10^{10}\) & Pa \\ Sectional area & 3.6552 & m\({}^{2}\) \\ Poisson’s ratio & 0.2 & - \\ _ly_ & 1.3514 & m\({}^{4}\) \\ _lz_ & 9.1108 & m\({}^{4}\) \\ Density & \(2.65\times 10^{3}\) & kg/m\({}^{3}\) \\ Damping ratio & 2\% & - \\ Span length & 30 and 25 & m \\ Curve radius & 400 & m \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Bridge parameters. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Span Length** & **Order of Vibration** & **Frequency/Hz** & **Mode** \\ \hline \multirow{4}{*}{30 m} & First & 3.83 & Vertical bending \\  & Second & 9.95 & Transverse bending \\  & Third & 15.324 & Vertical bending \\  & Fourth & 39.789 & Transverse bending \\ \hline \multirow{4}{*}{25 m} & First & 5.52 & Vertical bending \\  & Second & 14.32 & Transverse bending \\  & Third & 22.066 & Vertical bending \\  & Fourth & 57.297 & Transverse bending \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Natural vibration characteristics of bridge. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Parameter** & **Symbol** & **Unit** & **Value** & **Parameter** & **Symbol** & **Unit** & **Value** \\ \hline Mass of wheel & \(m_{w}\) & kg & 1420 & \begin{tabular}{c} Lateral stiffness of \\ primary suspension \\ \end{tabular} & \(k_{\text{sp}}\) & N/m & \(10.4\times 10^{6}\) \\ \hline Mass of bogie & \(m_{t}\) & kg & 2550 & \begin{tabular}{c} vertical stiffness of \\ Secondary suspension \\ \end{tabular} & \(k_{\text{ss}}\) & N/m & \(1.7\times 10^{6}\) \\ \hline Mass of car body & \(m_{c}\) & kg & 21,920 & \begin{tabular}{c} Lateral stiffness of \\ secondary suspension \\ \end{tabular} & \(k_{\text{gs}}\) & N/m & \(10.4\times 10^{6}\) \\ \hline Roll mass moment of car & \(l_{cx}\) & kg\(\cdot\)m\({}^{2}\) & 14,890 & \begin{tabular}{c} Vertical damping of \\ primary suspension \\ \end{tabular} & \(c_{\text{sp}}\) & N\(\cdot\)s/m & \(1.7\times 10^{6}\) \\ \hline Pitch mass moment of & \(l_{xy}\) & kg\(\cdot\)m\({}^{2}\) & 617,310 & \begin{tabular}{c} Lateral damping of \\ primary suspension \\ \end{tabular} & \(c_{\text{sp}}\) & N\(\cdot\)s/m & \(10.4\times 10^{6}\) \\ \hline Yaw mass moment of & \(l_{cz}\) & kg\(\cdot\)m\({}^{2}\) & 617,310 & \begin{tabular}{c} Vertical damping of \\ secondary suspension \\ \end{tabular} & \(c_{\text{ss}}\) & N\(\cdot\)s/m & \(1.7\times 10^{6}\) \\ \hline Roll mass moment & \(l_{\text{rx}}\) & kg\(\cdot\)m\({}^{2}\) & 1050 & \begin{tabular}{c} Lateral damping of \\ secondary suspension \\ \end{tabular} & \(c_{\text{gs}}\) & N\(\cdot\)s/m & \(10.4\times 10^{6}\) \\ \hline Pitch mass moment & \(l_{yy}\) & kg\(\cdot\)m\({}^{2}\) & 1750 & Length of carriage & dv & m & 19 \\ \hline \begin{tabular}{c} Yaw mass moment \\ of bogie \\ \end{tabular} & \(l_{tz}\) & kg\(\cdot\)m\({}^{2}\) & 1980 & \
\begin{tabular}{c} Distance between \\ wheelset of one bogie \\ \end{tabular} & \(L_{1}\) & m & 2.2 \\ \hline Vertical stiffness of & \(k_{\text{sp}}\) & N/m & \(1.7\times 10^{6}\) &
\begin{tabular}{c} Distance between bogie \\ of one carriage \\ \end{tabular} & \(L_{2}\) & m & 12.5 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Vehicle parameters. performance. According to [PERSON] et al. [41], when the vehicle speed is close to the speed calculated by Equation (5), the bridge resonance occurs. \[v_{res,n,i}=3.6\frac{f_{n}^{B}d_{V}}{i}\ \mathrm{km/h},\ \mathrm{where}\ i=1,2,3\cdots \tag{5}\]  
where \(v_{res,n,i}\) is the \(i\)th order resonance velocity of the \(n^{\mathrm{th}}\) order bridge frequency and \(f_{n}^{B}\) is the \(n^{\mathrm{th}}\) order bridge natural vibration frequency in Hz.