# The Vibration Response Characteristics of Neighboring Tunnels Induced by Shield Construction
## 4 Numerical Model Validation
### Analysis of Simulation Results  
On the construction tunnel side, a monitoring point is selected at every 1 m interval to measure the vibration displacement of the segment unit at each position, as shown in Figure 15.  
As shown in Figure 15, as the distance from the vibration source increases, the vibration displacement of the tunnel segment decreases most significantly. Within the tunnel spacing range of 4 m, the vibration attenuation is more pronounced. From 1 m to 4 m, the vibration displacement on the adjacent side decreases by 80.29%. To better validate the impact of different frequencies on the vibrational response of tunnel structures, vibration sources at frequencies of 2, 4, 6, 8, 10, 15, and 20 Hz were selected for input. Monitoring was conducted at a point 3 m away from the vibration source, with the specific displacement data shown in Figure 16. From the figure, it is evident that, as the frequency increases, the vibrational response progressively weakens. At 2 Hz, the peak vibration displacement is approximately 0.05 mm, while, at 4 Hz, it reduces to 0.015 mm. Under high-frequency vibration at 20 Hz, the peak vibration displacement is only 0.0005 mm. The displacement time-history data were first converted into acceleration time-history data through reverse calculation, and then Fourier transform was performed based on the formula. The acceleration, calculated using Formula (1), was used to determine the power density. The peak power density was observed at 5 Hz, with an exponential decrease as the propagation distance increased. Simulation results align with the theoretical formula, and Figure 17 illustrates the acceleration probability density curves for the 0-50 Hz and 4-6 Hz frequency ranges. \begin{table}
\begin{tabular}{c c c c} \hline
**Abbreviation** & **Symbol** & **Explanation Explained** & **Assignment** \\ \hline kn & kn & Normal stiffness & \(2\times 10^{7}\) \\ ks & ks & Tangential stiffness & \(8\times 10^{6}\) \\ fric & \(\mu\) & Coefficient of friction & 0.5 \\ rgap & gr & Contact clearance & \(0.5\times 10^{-4}\) \\ emod & E & Effective modulus of soil & \(4\times 10^{7}\) \\ emod & E & Effective modulus of concrete & \(3\times 10^{10}\) \\ kratio & K & The ratio of stiffness in the normal to tangential directions & 1.5 \\ lin\_force & Fl & Linear force (contact surface coordinate system) & 0 \\ dp nratio & \(\xi\) & Normal critical damping ratio & 0.05 \\ pb\_ten & 5. & Tensile strength (stress) & \(1\times 10^{5}\) \\ pb coh & e & Cohesion (stress) & \(1\times 10^{7}\) \\ pb fa & \(\varphi\) & Friction angle (degrees) & 0.0 \\ pb emod & E & Bonding effective modulus & \(1\times 10^{5}\) \\ pb kratio & K & The ratio of the stiffness of the bonding normal to the tangential direction & 1.5 \\ \hline \end{tabular}
\end{table} 
Table 1: Numerical model parameter table. Figure 14: Simulated displacement results. Figure 16: Vibration displacement data at 3 m from the vibration source with different frequencies of vibration wave input. Figure 15: Vibration displacement of segment at different positions. The vibration acceleration power density data were processed using the formula in Equation (2), resulting in a calculated damping ratio of 0.059062. The corresponding fitting curve is presented in Figure 18.  
This value is very close to the initial damping ratio of 0.05 set in the model parameters, indicating that the numerical simulation results are highly accurate. Furthermore, the attenuation trend of the acceleration power density generally follows an exponential decay pattern. However, a detailed analysis of the simulation results reveals that the primary deviations in the power density curve are concentrated within the 2-3 m range. Specifically, within this interval, a significant attenuation of acceleration power density is observed. For instance, at a distance of 2 m from the vibration source, the acceleration power density is 1.68 \(\times\) 10\({}^{-7}\) ((mm/s\({}^{2}\))/Hz), while, at 3 m, this value rapidly decreases to 4.9 \(\times\) 10\({}^{-8}\) ((mm/s\({}^{2}\))/Hz).