## 4 Results of Oscillation Mode under Different Parameters
### Different Section Aspect Ratios  
Several previous studies [17; 18; 19] have shown that section aspect ratio \(\alpha\) has a great impact on the attack angle of the triangular prism, thus causing the change in the lift instability of the prism. In this section, four complete responses of triangular prisms with different \(\alpha\) varying from 0.5 to 1.5 are compared in Figure 14. Especially the cross-sectional shape is the regular triangular when \(\alpha\) = 0.866. In this figure, the damping coefficients are \(c\) = 100.92 N\(\cdot\)m\(\cdot\)s\({}^{-1}\), the system stiffness \(K\) = 1200 N/m, and the system mass \(m\) = 32.81 kg. The descriptions of \(A^{*},f^{*}\), \(U_{r}\), LL, UL, and CV are listed in the Abbreviations.  
According to Figure 14, the following characteristics can be found.  
For \(0.5\leq\alpha\leq 0.866\), the prism undergoes the HG responses which consist of the complete VIV branches and incomplete galloping branches. In the responses, the critical velocities (CV) are observed while the VIV-galloping transition branches and the self-excited regions cannot be observed. Moreover,  
Figure 14: Oscillation modes under different section aspect ratios \(\alpha\): (**a**) Amplitude ratio \(A^{*}\) versus reduced velocity \(U_{r}\); (**b**) Frequency ratio \(f^{*}\) versus reduced velocity \(U_{r}\).  
as \(\alpha\) increases, the CV drops from a high velocity (which cannot be measured due to the limitation of the velocity range) to 11.537, and the galloping branch is gradually widened and develops into a low velocity, thereby indicating that the ability of a triangular prism to be externally excited from VIV to galloping is enhanced as the section aspect ratio increases.  
When \(\alpha\) enters into the range of \(1\leq\alpha\leq 1.5\), the oscillation mode of the prism transitions from HG to SG. In the responses, the complete galloping branches and the VIV-galloping transition branches are observed and the VIV branches are incomplete. Moreover, as \(\alpha\) increases, the LL decreases from a high velocity (which cannot be measured due to the limitation of the velocity range) to 11.846 while the UL increases from 7.208 to 9.104, leading to an expansion of the self-excited region and indicating that the ability for a triangular prism to be self-excited from VIV to galloping is enhanced as the section aspect ratio increases.  
It is noted that CG is not observed at this case because of the limitation of the triangular prisms. According to the developments of the HG and SG with the variation of \(\alpha\), we can infer that CG should occur at the range of \(0.866<\alpha<1\).  
According to the above analysis, the impacts of section aspect ratio \(\alpha\) on the oscillation modes of triangular prisms can be summarized as follows. When the section aspect ratio is low, the oscillation mode of triangular prism is HG; as the ratio gradually increases, the oscillation mode gradually transitions from HG to CG, followed by SG. Therefore, the ability of the triangular prism to be excited (including the self-excited and externally excited) from VIV to galloping is gradually enhanced as the section aspect ratio increases.