## 5 Discussion on Occurrence of SG and HG
### Occurrence Conditions of SG and HG  
It can be seen from the previous analysis that the self-excited characteristics of the galloping strongly depend on the amplitude of the vortex-induced vibration. Meanwhile, the amplitude of the vortex-induced vibration depends on the damping coefficient, stiffness, the mass of the oscillation system, and the section aspect ratio of the prism. In order to reveal the occurrence condition of soft galloping and hard galloping, a series of experiments with different configurations were conducted. The results are plotted in Figure 16.  
In this figure, the ordinates are all damping coefficient \(c\), the abscissas are the system stiffness \(K\), the mass of oscillation \(m\), and the section aspect ratio \(\alpha\), respectively. In the legends, H(V) and H  
Figure 16: Occurrence maps of SG, CG and HG: (**a**) Map with variations of \(c\) and \(K\); (**b**) Map with variations of \(c\) and \(m\); (**c**) Map with variations of \(c\) and \(\alpha\).  
represent the hard galloping. In H(V) region, no galloping branch can be observed in the range of test velocity, due to the limitation of the experimental setup. S represents the soft galloping and C represents the critical galloping. In the S region, the galloping can be self-excited by VIV. In the H and H(V) regions, the galloping cannot be self-excited by VIV, but can be excited by a given threshold's initial displacement. In the C region, the galloping can be self-excited by VIV only at the critical velocity.  
In the figure, as \(K\) increases from 800 to 1600 N/m, the corresponding \(c\) of CG gradually increases from 68 to 116 N\(\cdot\)s\(\cdot\)m\({}^{-1}\). Meanwhile, as \(m\) increases from 32 to 45 kg, the corresponding \(c\) of CG gradually decreases from 96 to 72 N\(\cdot\)s\(\cdot\)m\({}^{-1}\). Additionally, as \(\alpha\) increases from 0.5 to 1.5, the corresponding \(c\) of CG rapidly increases from 48 to 192 N\(\cdot\)s\(\cdot\)m\({}^{-1}\). These results indicate that a higher stiffness and section aspect ratio, or a lower mass will cause SG damping range expansion.  
By analyzing the test data, the experiential expression to determine the occurrence of CG is obtained as follows  
\[c_{\text{c}}=1.5m_{\text{d}}\left(\frac{K}{m}\right)^{0.766} \tag{16}\]  
\(c_{\text{c}}\) is the critical damping coefficient. Once the section aspect ratio of the prism, stiffness and mass of the oscillation system are determined, the critical damping coefficient \(c_{\text{c}}\) can be confirmed. It is noted that the displaced mass \(m_{\text{d}}\) is dependent on section aspect ratio (\(m_{\text{d}}=0.5\rho DHL=0.5\rho\alpha D^{2}L\)).  
Based on Equation (16), the calculated critical damping coefficients \(c_{\text{c-cal}}\) are obtained and listed in Table 7. Meanwhile, the test critical damping coefficients \(c_{\text{c-mea}}\) are listed in Table 7 as well. It is observed that the calculated results agree well with the test results, indicating Equation (16) is reasonable.  
Therefore, the oscillation mode of triangular prism can be predicted by Equation (16). While \(c>c_{\text{c}}\), HG responses can be observed; while \(c<c_{\text{c}}\), SG responses can be observed; while \(c=c_{\text{c}}\), CG responses can be observed.