# Design and Evaluation of Novel Submerged Floating Tunnel Models Based on Dynamic Similarity
## 5 Foude Similarity Test
### Hydrodynamic Characteristics Test  
The inherent frequency of submerged tunnels in water is influenced by various factors, such as the buoyancy-weight ratio, the geometric shape of the tunnel tube, and the configuration of the mooring system. The white noise wave test is a common method for assessing the hydrodynamic characteristics of marine structures. [PERSON] et al. [35] conducted a comparative analysis of the hydrodynamic responses of two types of cage systems by deriving transfer functions through the input of regular white noise waves. [PERSON] et al. [36] investigated the hydrodynamic characteristics of a Floating Liquefied Natural Gas (FLNG) system under the influence of white noise. [PERSON] [37] obtained the Response Amplitude Operators (RAOs) for the motion response of side-by-side vessels through white noise testing. Figure 12: Mode shapes of the simplified numerical model. (**a**) First-order bending mode about the \(x\)-axis; (**b**) first-order bending mode about the \(y\)-axis; (**c**) second-order bending mode about the \(x\)-axis; (**d**) second-order bending mode about the \(y\)-axis. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Modal**} & \multicolumn{3}{c}{**Numerical Simulation**} & \multirow{2}{*}{**Model Test**} & \multirow{2}{*}{**Relative Error**} \\  & **Fidelity Model** & & & \\ \hline First-order & 14.87 & 15.07 & 16.0 & 7.1\% \\ Second-order & 42.15 & 42.69 & 40.0 & 6.7\% \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Comparative analysis of natural frequencies for scale model and numerical models. In the experimental setup within the water tank, a six-meter scaled model was employed. This model has the same configuration as the previously used two-meter model, differing solely in its length. The diameter of the six-meter model was 20 cm, with a buoyancy-weight ratio (BWR) of 1.4. The tunnel body model was connected by anchor cables to form a complete SFT system. The total water depth was 1.0 m. The center of the tunnel body was situated 0.3 m below the free surface, maintaining the free boundary conditions at both ends. An underwater vibration table drove a wave-making device to generate white noise waves, as shown in Figure 13. In the diagram, yellow represents the coordinate axes, white represents the experimental setup, and red indicates the direction of the incident waves. Figure 14 illustrates the incident wave direction, as well as the arrangement of the sensors and the anchor cables. The waves approached perpendicular to the axis of the tunnel body. Five pairs of anchor cables were evenly spaced at intervals of 1.4 m along this axis, each set at an inclination angle of 45 degrees. The cables featured an elastic modulus of 206 GPa and a diameter of 3.8 mm. Table 4 presents the parameters of the sensors and data acquisition systems used in this experiment. Acceleration measurements were taken using piezoelectric accelerometers (YX1211) characterized by their compact size and light weight. These accelerometers were magnetically attached to the counterweight steel rings within the tunnel body. A dynamic data acquisition system (TST 3000) was employed to capture acceleration data at a sampling rate of 2000 Hz. Roll angle measurements of the tunnel body utilized three-dimensional motion sensors based on MEMS technology (WT9011G4K), offering a sampling rate of 200 Hz and an inclination measurement accuracy of 0.2 degrees. These sensors were directly connected to a computer via a serial port, and angle data were collected by the upper computer software. Given the structure of this new model, both the accelerometers and tilt sensors can be strategically placed inside the tunnel body to facilitate the waterproofing of the sensors and minimize disruption to the external flow field.