## 4 Case Study and Analysis of Vibration Control Effect
### Design of Periodic Row Piles  
According to the previous analysis, ground vibrations can be significantly suppressed after laying down a steel-spring FST. However, the ground vibration increased in the frequency range of 6-12 Hz owing to the resonance effect of the FST. Periodic vibration isolation row piles were designed and used to control ground vibration in the full-frequency \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **CFST** & **SFST** \\ \hline slab thickness/m & 0.4 & 0.54 \\ slab length/m & 25 & 25 \\ slab width/m & 3 & 3 \\ vibration isolator stiffness/(MN/m) & 6.9 & 5.5 \\ vibration isolator damping/(MNs/m) & 0.03 & 0.05 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Parameters of two types of FST. Figure 10: Ground vertical vibration accelerations for different cases. band to further control the vibration in this frequency band, using a collaborative approach of vibration source reduction and propagating path isolation. The distribution of the local resonance-type row piles was designed using the principle of phononic crystals, as shown in Figure 11. This type of pile consists of a rubber sleeve and a concrete pile, which can be prefabricated in the factory. As for the prefabrication and practical engineering of the row piles, the concrete part could be first inserted into the rubber sleeve and prefabricated in the factory to obtain the row piles with the proposed form. Then, according to practical application requirements and on-site conditions, the application of row piles in engineering can be achieved via an open excavation and backfilling method or by drilling holes with a width greater than the pile diameter before inserting the piles. The design goal of the row piles was to generate a bandgap that could cover the natural frequency of the FST, such that the row piles could be set according to this parameter and arranged to block vibration propagation in the soil at a frequency of approximately 8 Hz. Based on the Bloch-Floquet theory and plane wave expansion method, the inner and outer diameters \(r_{1}\) and \(r_{2}\) of the cylindrical piles and the pile arrangement period constant \(a\) were calculated. The detailed calculation principles and process can be found in [63; 64]. COMSOL Multiphysics 5.3a (a commercial FE software) was used to solve the bandgaps in the row piles. The parameters of the rubber sleeve were as follows: \(E=0.3\) MPa, \(v=0.463\), \(\rho=1300\) kg/m\({}^{3}\), \(V_{\rm P}=969\) m/s (P-wave velocity) and \(V_{\rm s}=39\) m/s (S-wave velocity). The parameters of the concrete pile were \(E=35\) GPa, \(v=0.25\), \(\rho=2500\) kg/m\({}^{3}\), \(V_{\rm P}=4545\) m/s and \(V_{\rm s}=2688\) m/s. When the pile was arranged in three rows using a square lattice and the period constant a = 1.5 m, the inner radius of the concrete and the outer radius of the rubber were r\({}_{1}=0.4\) m and r\({}_{2}=0.5\) m, respectively, and the bandgap was generated at 7-9 Hz, as shown in Figure 12. The lines in this figure means the vibration modes of the soil-piles coupled system. When there is no vibration mode in a certain frequency band, the vibration waves in that frequency band cannot propagate within this system. That is, the row of piles can be set according to these parameters and arrangements to prevent vibration propagation in the 7-9 Hz frequency band of the soil. To determine the length of the pile, two steps were considered.