## 3 Results and Discussion
When _f_/_f_s = 0.75, the scale and strength of the vortices shedding in the wake almost maintains the same level. The vortex shedding frequency is only impacted by the oscillation frequency _f_e as per the PSD at _A_/_D_ = 0.5 and _f_/_f_s = 0.75 in Figure 8, thereby being in the \"lock-in\" range. With an increase in the frequency ratio _f_/_f_s, the strength of the vortex shedding also increases. But the vortex shedding mode keeps a 2S mode in the range of 0.75 \(\leq\)_f_/_f_s \(\leq\) 1.35, and the dominant frequency is always the excitation frequency _f_e. Additionally, the boundary of the cylinder moves faster as the excitation frequency increases, thereby accelerating the nearby fluid more and causing the wake vortices large lateral displacement in downstream. Along with an increase in the oscillating frequency ratio, the amplitudes of the lift and drag coefficients also increase. Additionally, the temporal evolutions of the lift and drag coefficients show different levels of fluctuation and no significant periodicity at _f_/_f_s \(\geq\) 0.95. It could be that the time of the vortex shedding is not consistent per cycle under the condition of small excitation amplitude or it could be that the vortex shedding is a lasting process, leading the lift and drag coefficients to fluctuate irregularly. Similarly, Figure 9 shows the vorticities around and behind the oscillating cylinder at _A_/_D_ = 1 and _f_/_f_s = 0.5 to 1.35. Figure 10 shows the temporal evolutions of the lift and drag coefficients (left) and the PSD of the lift coefficient of the oscillating cylinder (right) in the corresponding parameters. However, the vortex shedding mode and the characteristics of the lift and drag coefficients at _A_/_D_ = 1 are very different from that at _A_/_D_ = 0.5. When \(f_{\textit{eff}}\)/\(f_{\textit{s}}=0.5\), the vortex shedding mode is a similar \"P + S\" mode, and the lift and drag coefficients also show the \"beat\" characteristics. However, its vortex shedding mode actually is a 2S mode placed out of the \"lock-in\" range. This is due to the fact that the vortex shedding is also influenced by the excitation frequency and the Stroulhal frequency at \(A\)/\(D=1\) and \(f_{\textit{eff}}\)/\(f_{\textit{s}}=0.5\), which is in line with the case at \(A\)/\(D=0.5\) and \(f_{\textit{eff}}\)/\(f_{\textit{s}}=0.5\). The detachment of the vortices from the cylinder surface is dominated by a joint action of these two frequencies, leading to the complexity of the vortex shedding in the wake. The temporal evolutions of the lift and drag coefficients exhibit significant fluctuations, which correspond to the vortex shedding mode. When \(A\)/\(D=1\) and \(f_{\textit{eff}}\)/\(f_{\textit{s}}=0.75\), the vortex shedding exhibits a 2S mode, and the vortices shed behave regularly with a higher dominant frequency (0.08772) than that of its excitation frequency \(f_{\textit{e}}\) (0.06748). Thus, the vortices shed in the wake closely to the condition of the same excitation frequency. Additionally, the temporal evolutions of the lift and drag coefficients also perform periodicity. The  
Figure 10: Temporal evolutions of the lift and drag coefficients (left) and the PSD of the lift coefficient (right) of the oscillating cylinder at \(A\)/\(D=1\) and different frequencies. (a) \(f_{\textit{eff}}\)\({}_{\textit{s}}=0.5\); (b) \(f_{\textit{eff}}\)/\(f_{\textit{s}}=0.75\); (c) \(f_{\textit{eff}}\)\({}_{\textit{s}}=0.95\); (d) \(f_{\textit{eff}}\)\({}_{\textit{s}}=1.05\); (e) \(f_{\textit{eff}}\)\({}_{\textit{s}}=1.2\); and (f) \(f_{\textit{eff}}\)\({}_{\textit{s}}=1.35\). case of \(f_{c}\)/\(f_{s}=0.75\) is an exception to the \"lock-in\" range, which could be due to the fact that it is in an intermediate state from the 2S mode to the 2P\({}_{0}\) mode or it could be that the dominant frequency is a fusion between the excitation frequency \(f_{c}\) and the Stroulhal frequency \(f_{s}\). In the case of \(f_{c}\)/\(f_{s}=0.95\), the vortex shedding mode is transformed into a 2P mode.