# Vibration Response Characteristics of Prefabricated Frame Structures Around the Subway
## 4 Multipoint Vibration Excitation Input Method  
This section systematically evaluates vibration excitation methods by contrasting tunnel-soil-structure coupled systems with base-uniform excitation approaches, aiming to establish an effective multipoint input strategy for subway-induced vibrations. The baseline precast concrete frame structure (\(\gamma\) = 0.5) was integrated with the site vibration model using coupling techniques, forming a 3D tunnel-soil-structure interaction system depicted in Figure 5. Infinite element boundaries were added around the model. These infinite element boundaries can absorb the vibrations propagating to the model boundaries, thereby simulating the soil mass extending to infinity. A 6 m diameter shield-constructed tunnel was modeled, featuring 0.4 m thick C50-grade lining segments, C30 concrete track bed, and T60 rail profiles. For the targeted 1-100 Hz frequency spectrum (\(\omega_{1}=6.28\) rad/s, \(\omega_{2}=628\) rad/s), site-specific Rayleigh damping parameters were calculated: \(\alpha\) = 0.124 and \(\beta\) = 3.15 \(\times\) 10\({}^{-5}\). Geotechnical parameters are provided in Table 3. The model employed C3D8R elements for soil, lining segments, and track bed (0.5-2.5 m mesh size, 507,288 elements), B31 beam elements for \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \(\gamma\) & 1.0 & 0.9 & 0.8 & 0.7 & 0.6 & 0.5 & 0.4 & 0.3 & 0.2 & 0.1 \\ \hline \(K_{r}\) N-m/rad & \(3.66\times 10^{8}\) & \(3.29\times 10^{8}\) & \(2.93\times 10^{8}\) & \(2.56\times 10^{8}\) & \(2.19\times 10^{8}\) & \(1.83\times 10^{8}\) & \(1.46\times 10^{8}\) & \(1.09\times 10^{8}\) & \(7.32\times 10^{7}\) & \(3.66\times 10^{7}\) \\ \hline \end{tabular}
\end{table} 
Table 2: Values of rotational stiffness under different \(\gamma\). Figure 4: Elevation diagram of the numerical baseline model. Figure 3: Plan layout diagram of the baseline structure. rails (3000 elements), and CIN3D8 infinite elements for boundary conditions. The loading protocol implemented wheel forces through ABAQUS DLOAD subroutine using 2.5 ms time steps. Figure 6 compares the one-third octave band results between the computational model and experimental measurements from Reference [10]. The model's vibration responses exhibited consistent growth-attenuation trends with field data across center frequencies. Prior to 40 Hz, vibration acceleration magnitudes exhibited frequency-proportional amplification, whereas post-40 Hz spectral components displayed graded high-frequency attenuation. Both model and measurements showed peak energy at 40 Hz, with a 4.8% discrepancy. Within the 10-50 Hz dominant frequency range for subway-induced environmental vibrations, the model demonstrated superior simulation fidelity. Vertical vibrations were extracted from roof corner nodes in both tunnel-soil-structure coupled models and base-consistent input baseline structures. The spectral distribution characteristics of vertical vibrations showed remarkable consistency between the coupling model and the base-uniform excitation method, maintaining amplitude discrepancies  
Figure 5: Three-dimensional tunnel–soil–structure interaction system. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Soil** & **Thickness m** & \begin{tabular}{c} **Dynamic Elastic** \\ **Modulus Mpa** \\ \end{tabular} & \begin{tabular}{c} **Poisson’s** \\ **Ratio** \\ \end{tabular} & \
\begin{tabular}{c} **Density** \\ **kg/m\({}^{3}\)** \\ \end{tabular} &
\begin{tabular}{c} **Rayleigh** \\ **Parameters** \\ \end{tabular} \\ \hline Layer 1 & 10 & 141 & 0.443 & 1650 & \(\alpha=0.124\) \\ Layer 2 & 12 & 175.3 & 0.439 & 1940 & \(\beta=3.15\times 10^{-5}\) \\ Layer 3 & 18 & 353 & 0.439 & 1985 & \(\beta=3.15\times 10^{-5}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Soil layer parameters. Figure 6: One-third octave band results between the computational model and experimental measurement. within 5% across frequency bands. This indicated comparable computational mechanisms between both methods.