## 4 Numerical Verification of Innovative Designs
### Dynamic Characteristic Analysis  
The main purpose of the dynamic analysis model of the bridge is to analyze the self-vibration frequency and period of the structure and control the overall stiffness. The main difference between the dynamic analysis model and the static analysis model is that the dynamic model adds mass data. What plays a decisive role in the dynamic analysis is generally the first few orders of self-vibration frequency and mode. The first six orders of self-vibration characteristics of the structure calculated by MIDAS/CIVIL software using the subspace iterative method are shown in Table 8. The first-order frequency of the structure is 0.53 Hz, and the human sense frequency range is 2.5-3.5 Hz; the structure frequency is not in the human sense frequency range, and the frequency distribution of each order is relatively reasonable.  
The vibration modes reflect the influence of boundary constraints and component stiffness on the vibration condition, and the vibration modes of the first six orders of the structure are shown in Figure 13. Figure 13 shows that the first four orders of arch rib mode are mainly flexural vibration, and the fifth order mode has bending and twisting vibration; the frequency of bending and torsional vibration, 0.95, is 1.79 times the first order of flexural vibration frequency of 0.53, which indicates that the anti-bending and torsional stability performance of the main and secondary arch collaborative Y-shaped thin-walled steel box arch bridge is better than the anti-flexural stability performance. In the first six orders of modes, the main girder modes are all deflection and drift, and no torsional vibration occurs, indicating that the main girder has good stability performance against torsion. In the high-order formation, the deflection vibration of the main girder no longer occurs alone, and the deflection vibration of the main girder occurs simultaneously with the deflection vibration of the arch ribs. The reason for the simultaneous vibration of the main girder and the arch ribs is the connecting force transmission effect of the suspenders. Structural mode is mainly manifested as flexure and drift, and special attention should be paid to flexural vibration in the design. The torsional stability of the structure is rather good, mainly due to the stable tetrahedral structure formed by the three bifurcated arch ribs, the contribution of the secondary arch ribs and connecting ribs to increasing the torsional stiffness, the wind bracing and crossbeams between the double arch ribs increasing the torsional stiffness of the structure, and the asymmetry of the distribution angle of the suspenders between the main girder and the main arch rib increasing the torsional stability.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Vibration Mode** & **Frequency** & **Vibration Mode Characteristics** \\
**Serial Number** & **(Hz)** & \\ \hline
1 & 0.53 & Arch rib flexure in cross-bridge direction \\
2 & 0.65 & Arch rib flexure in vertical direction \\  & & Main girder drift in along-bridge direction \\
3 & 0.72 & Main girder drift in cross-bridge direction \\
4 & 0.73 & Antisymmetric flexure of main girder in vertical direction \\
5 & 0.95 & Antisymmetric bending and twisting of arch ribs in \\  & & cross-bridge direction \\
6 & 1.12 & Arch rib flexure in vertical direction \\  & & Symmetrical flexure of main girder in vertical direction \\ \hline \hline \end{tabular}
\end{table}
Table 8: Calculation results of self-vibration characteristics.  
Figure 13: The first six orders of vibration mode characteristics of Y-shaped steel box arch bridge: (**a**) Mode 1; (**b**) Mode 2; (**c**) Mode 3; (**d**) Mode 4; (**e**) Mode 5; (**f**) Mode 6.