## 4 Results of Oscillation Mode under Different Parameters
### Different Damping Coefficients  
The complete responses of the regular triangular prism at different damping coefficients \(c\) are plotted in Figure 11. Different damping coefficients were obtained by varying the load resistances of PTO. In this figure, the system stiffness \(K\) = 1400 N/m, the system mass \(m\) = 32.81 kg, the descriptions of \(A^{*}\),\(f^{*}\), \(U_{r}\), LL, UL, UL and CV are listed in the Abbreviations. For 52.22 N\(\cdot\)s\(\cdot\)m\({}^{-1}\)\(\leq\)\(c\)! 92.52 N\(\cdot\)s\(\cdot\)m\({}^{-1}\), the complete galloping branches, the VIV-galloping transition branches and the self-excited regions can be clearly observed, whereas the VIV upper branches cannot be clearly observed. Moreover, as \(c\) increases, the VIV branches gradually become complete while the VIV-galloping branches get narrower and eventually disappear. Correspondingly, the LL (lower limit of self-excited region) increases from 6.625 to 7.875, but the UL (upper limit of self-excited region) decreases from 13.5 to 11, leading to the reduction of the self-excited region. It indicates that the prism in these damping cases experiences SG responses. Meanwhile, as the damping increases, the ability of triangular prism to be self-excited from VIV to galloping is weakened. For 92.52 N\(\cdot\)s\(\cdot\)m\({}^{-1}\)\(<\)\(c\)\(\leq\) 225.85 N\(\cdot\)s\(\cdot\)m\({}^{-1}\), the complete VIV branches--which consist of the initial branch, upper branch and lower branch, the incomplete galloping branches, and the critical velocity (CV)--can be clearly observed. On the contrary, the VIV-galloping transition branch and the self-excited region both disappear. Moreover, as \(c\) increases, the CV increases from 10.375 to 14.125, and the galloping branch is gradually narrowed and eventually disappears, thus revealing that the prism in these damping cases experiences the HG responses. Meanwhile, the ability of the triangular prism to be externally excited from VIV to galloping is also weakened as the damping increases. For \(c\) = 92.52 N\(\cdot\)s\(\cdot\)m\({}^{-1}\), an interesting response consists of the complete VIV branches, and the complete galloping branch can be clearly observed. In the response, VIV and galloping branches intersected at \(U_{r}\) = 9.125, accompanied by \(A^{*}\), essentially reaching the peak (0.876) of VIV branch and \(f^{*}\) before starting to drop down to 0.832. In addition, exclusively at \(U_{r}\) = 9.125, the prism is self-initiated from VIV to galloping; for \(U_{r}\)! 9.125, the prism can be initiated to galloping only by an appropriate external excitation. \(U_{r}\) = 9.125 is actually the coincidence velocity of UL, LL and CV. According to the definitions of oscillation modes in [PERSON]'s [28] research, in the present experiments, the oscillation mode at 92.52 N\(\cdot\)s\(\cdot\)m\({}^{-1}\)\(<\)\(c\)\(<\) 225.85 N\(\cdot\)s\(\cdot\)m\({}^{-1}\) is HG2, while the oscillation mode at \(c\) = 92.52 N\(\cdot\)s\(\cdot\)m\({}^{-1}\) is HG1. In the present study, the SG responses (52.22 N\(\cdot\)s\(\cdot\)m\({}^{-1}\)\(\leq\)\(c\)\(<\) 92.52 N\(\cdot\)s\(\cdot\)m\({}^{-1}\)) are shown to have the VIV lower branch, and the VIV branches gradually become complete as the damping increases. Meanwhile, the HG responses (92.52 N\(\cdot\)s\(\cdot\)m\({}^{-1}\)\(<\)\(c\)\(\leq\) 225.85 N\(\cdot\)s\(\cdot\)m\({}^{-1}\)) exhibit the complete VIV branches. We can thus conclude that the HG1 mode is actually the critical oscillation mode between the SG mode and HG (or HG2) mode. This critical oscillation mode can be observed only at a certain condition. Therefore, HG1 can be renamed as critical galloping (CG). According to the above analysis, the impacts of damping coefficients \(c\) on the oscillation modes of triangular prisms can be summarized as follows. When the damping is low, the oscillation mode of triangular prism is SG; as the damping gradually increases, the oscillation mode gradually transitions from SG to CG, followed by HG.