# Design and Evaluation of Novel Submerged Floating Tunnel Models Based on Dynamic Similarity
## 4 Cauchy Similarity Test
### Free Decay Test  
The free decay test was conducted to evaluate the natural frequencies of the scale model, which is an inherent dynamic characteristic of a structure, significantly affecting its dynamic response. Accelerometers were placed at the quarter-span, three-eighths of the span, the mid-span, three-quarters of the span, and seven-eighths of the span of the model. Initially, a weight was suspended from the mid-span of the model using a thin string to induce an initial deformation. After cutting the string, the model started to undergo free vibration. Two sets of free decay tests were conducted, with initial loads of 10 kg and 20 kg applied, respectively, to enhance the reliability of the test results. Figure 8 depicts the free vibration acceleration time history curve of the model. After the free vibration starts, the acceleration amplitude decays to zero rapidly, with the entire process lasting approximately 1.25 s. The larger the initial load applied, the more pronounced the acceleration response. By performing a Fast Fourier Transform on the time history data of the free vibration acceleration, the power spectral density was obtained, as shown in Figure 9. The first-order natural frequency of the model was 16.0 Hz, and the second-order natural frequency was 40.0 Hz. The first-order natural frequency, where most of the energy is concentrated, dominates the vibration process, and its peak value significantly exceeds that of the second-order natural frequency. Figure 6: Deformation contour in numerical simulations. (**a**) High-fidelity numerical model; (**b**) simplified numerical model. Figure 7: Load–deflection response curve for the scale model. (**a**) Mid-span deflection; (**b**) quarter-span deflection. The analysis of frequency distribution at various measurement points, as shown in Figure 9, reveals the acceleration responses along the axial direction of the model, as illustrated in Figure 10. The experimental results indicate that the first modal shape of the model displays a \"V\" shape, with the maximum acceleration amplitude at the mid-span measurement point and smaller amplitudes at the endpoints. Conversely, the second modal shape is an \"S\" shape, where the mid-span response is nearly zero, and the responses at symmetric positions on either side are equal in magnitude but opposite in direction. The left and right axes in Figure 10, respectively, present the acceleration amplitudes for the first and second modal responses. The peak in the first mode is approximately seven times that of the second, demonstrating its dominance in the structural dynamic response. Figures 11 and 12 display the modal contour maps of the numerical models. The overall shape of the model is a hollow cylinder symmetrical about the X and Y axes. There is a good consistency between the numerical and experimental modal shapes. Using the logarithmic decrement method, as outlined in Equation (24) and based on the free vibration time history curve (Figure 8), the damping ratio of the structure was calculated to be 0.051. \[\zeta=\frac{1}{2\pi k}\ln\frac{x_{n}}{x_{n+k}} \tag{24}\]  
where \(x_{n}\) represents the amplitude of the \(n\)th peak in the time history cure, and \(x_{n+k}\) is the amplitude of the \((n+k)\)th peak; \(k\) denotes the number of cycles between these two peaks. Figure 8: Temporal analysis of free decay tests. (**a**) Time history with 10 kg initial load; (**b**) time history with 20 kg initial load. Figure 9: Power spectrum density analysis from free decay tests. (**a**) Under 10 kg initial load condition; (**b**) under 20 kg initial load condition. Table 3 compares the first two natural frequencies of the model. The results of the numerical model are close to the model test results, with an error of around 7%. Compared with the numerical models, the first-order natural frequency of the physical model is slightly higher, and the second-order natural frequency is slightly lower. This deviation could be due to the influence of counterweight steel rings on the local stiffness of the structure.