## 3 Analysis of Vertical Vibrations in Track-Bridge Structures Under Short-Wavelength Irregularities
### Model Validation  
In this study, a high-speed railway and a simply supported box girder bridge are used for validation, consisting of five spans, each 32 m long, for a total length of 160 m. The bridge is equipped with a CRTS II-type slab ballastless track system. The structural parameters are listed in Table 1. The pier support stiffness is \(k_{y4}=10^{15}\) N/m. The vehicle considered is a single CRH3 high-speed train car, with parameters obtained from Reference [21]. The train travels at a speed of 80 m/s. Short-wavelength irregularities, based on the standard Chinese track irregularity spectrum, are introduced in the wavelength range of 0.01 m to 1 m. The corresponding time-domain samples of track irregularities are generated using the method described in Reference [22]. The resulting excitation frequencies range from 80 to 8000 Hz, meeting the requirements for high-frequency vibration analysis in this study. The modal configuration is as follows: 120 global modes and 100 local modes are used for the rail, and the degrees of freedom for the track slab, base layer, and bridge are set to 120, 120, and 50, respectively. To accurately capture high-frequency vibration responses, the time step of the Newmark method is set to \(5\times 10^{-5}\) s. The mid-span point of the rail is selected as the observation point. The vibration displacement and acceleration responses of the rail are calculated using both the proposed hybrid method and the conventional FEM approach. The corresponding time-history curves are plotted for comparison in Figure 3a,b. Additionally, the time-domain response of the track slab calculated using the proposed method is shown in Figure 3c,d. The acceleration amplitudes of the rail and track slab are \(1.0056\times 10^{3}\) m/s\({}^{2}\) and \(43.9355\) m/s\({}^{2}\), respectively. \begin{table}
\begin{tabular}{c c c c} \hline
**Structural Parameters** & **Values** & **Structural Parameters** & **Values** \\ \hline Cross-sectional area of & \(7.745\times 10^{-3}\) m\({}^{2}\) & Cross-sectional area of & \(8.75\) m\({}^{2}\) \\ rail \(A_{r}\) & \(7800\) kg/m\({}^{3}\) & Density of bridge \(\rho_{b}\) & \(2500\) kg/m\({}^{3}\) \\ Young’s modulus of & \(2.1\times 10^{11}\) Pa & Young’s modulus of & \(3.945\times 10^{10}\) Pa \\ rail \(E_{r}\) & \(2.1\times 10^{11}\) Pa & bridge \(E_{b}\) & \(3.945\times 10^{10}\) Pa \\ Moment of inertia of & \(3.217\times 10^{-5}\)m\({}^{4}\) & Moment of inertia of & \(9.25\) m\({}^{4}\) \\ \(\text{Cross-sectional area of}\) & \(3.217\times 10^{-5}\)m\({}^{4}\) & Moment of inertia of & \(9.25\) m\({}^{4}\) \\ \(\text{Cross-sectional area of}\) & \(0.51\) m\({}^{2}\) & Fastener stiffness \(k_{y1}\) & \(6\times 10^{7}\) N/m \\ \(\text{Slab }A_{s}\) & \(0.51\) m\({}^{2}\) & Fastener stiffness coefficient of & \(6\times 10^{7}\) N/m \\ \(\text{Density of slab }\rho_{s}\) & \(2500\) kg/m\({}^{3}\) & Stiffness coefficient of & \(9\times 10^{8}\) N/m \\ \(\text{Young’s modulus of}\) & \(3.9\times 10^{10}\) Pa & Stiffness coefficient of & \(1\times 10^{8}\) N/m \\ \(\text{Moment of inertia of}\) & \(1.7\times 10^{-3}\) m\({}^{4}\) & Damping coefficient of & \(4.77\times 10^{4}\) N \(\cdot\) s/m \\ \(\text{Cross-sectional area of}\) & \(0.885\) m\({}^{2}\) & Damping coefficient of & \(8.3\times 10^{4}\) N \(\cdot\) s/m \\ \(\text{Density of base }\rho_{f}\) & \(2500\) kg/m\({}^{3}\) & Damping coefficient of & \(2.48\times 10^{5}\) N \(\cdot\) s/m \\ \(\text{Young’s modulus of}\) & \(3.62\times 10^{10}\) Pa & Fastener spacing \(dl\) & \(0.64\) m \\ \(\text{Moment of inertia of}\) & \(6.6375\times 10^{-3}\) m\({}^{4}\) & & \\ \hline \end{tabular}
\end{table} 
Table 1: Track–bridge structural parameters. In the FEM model, each layer of the track-bridge system is discretized into 251 elements, totaling 2008 elements. The comparison shows that the hybrid method greatly improves computational efficiency, completing the simulation in approximately 13 s, compared to around 120 s for the FEM model.