## 3 Results and Discussion
However, the vortex shedding has a larger lateral displacement in downstream as compared with the case at \(A\)/\(D=1\) and \(f_{c}\)/\(f_{s}=0.95\), although the vortex shedding mode is the same. Additionally, this phenomenon also indicates that the transition of the vortex shedding mode is more likely to occur at high excitation amplitude. In the case of \(f_{e}\)/\(f_{s}\) = 0.95 and \(f_{e}\)/\(f_{s}\) = 1.05, the vortex shedding modes are also in an intermediate state from the 2P\({}_{0}\) mode to the P + S mode. However, the vortex shedding exhibits a small lateral displacement in downstream, converging towards the wake centerline. The temporal evolution of the lift coefficient at \(f_{e}\)/\(f_{s}\) = 0.95 shows a similar irregular \"jumping\" in the process of the downward and upward oscillation, respectively, i.e., double \"jumpings\". The temporal evolution of the lift coefficient at \(f_{e}\)/\(f_{s}\) = 1.05 also exhibits the double \"jumpings\". Additionally, the strength of the vortex shedding remains the same in the process of the downward and upward oscillation, although the shedding vortex can be a single vortex or a vortex pair. When \(f_{e}\)/\(f_{s}\) = 1.2 and \(f_{e}\)/\(f_{s}\) = 1.35, the vortex shedding modes keep a P + S mode. The single vortex is still weaker and is dissipated quickly in the downstream. However, the vortex pair enjoys stronger strength, and spreads upward. Therefore, the temporal evolution of the lift coefficient also shows an irregular \"jumping\" on its right side per cycle. Figure 13 shows a diagram of the vortex formation modes in the above computed range at \(Re\) = 4000. The results feature complex vortex formation transitions. A \"lock-in\" line at \(f_{e}\)/\(f_{s}\) = 0.75 determines the components of the vortex shedding frequency for a series of simulations of \(A\)/\(D\) = 0.5 to 1.5 and \(f_{e}\)/\(f_{s}\) = 0.5 to 1.35. The vortex shedding is dominated by the excitation frequency and the Stroulhal frequency when \(f_{e}\)/\(f_{s}\) \(<\) 0.75, located in the left of the \"lock-in\" line. However, the vortex shedding is only determined by a dominated vortex shedding frequency when \(f_{e}\)/\(f_{s}\) \(\geq\) 0.75, which is within the range of the \"lock-in\" phenomenon. For a lower excitation amplitude, the vortex shedding mode shows a 2S mode. With an increase in the excitation amplitude, the vortex shedding mode turns into a 2P\({}_{0}\) or 2P mode, corresponding to lower oscillation frequency ratios and higher ones,  
respectively. The mentioned 2P mode characterizes a combined mode including a strong vortex pair and a weak one, which occur in higher oscillation frequency ratios. In addition, the strength of the vortex shedding also increases with the increase of the oscillation frequency ratio. As the excitation amplitude increase to \(A/D=1.5\), the vortex shedding mode exhibits a P + S mode in high oscillation frequency ratios. An unstable status exists in the mode transition from the 2P\({}_{0}\) mode to the P + S mode, which is also considered as an intermediate state. Additionally, the case of a higher excitation amplitude and oscillating frequency ratio has a vortex shedding mode featuring a strong vortex pair and a single weak vortex. Figure 12: Temporal evolutions of the lift and drag coefficients (left) and the PSD of the lift coefficient (right) of the oscillating cylinder at \(A/D=1.5\) and different frequencies. (**a**) \(f_{c}/f_{s}=0.5\); (**b**) \(f_{c}/f_{s}=0.75\); (**c**) \(f_{c}/f_{s}=0.95\); (**d**) \(f_{c}/f_{s}=1.05\); (**e**) \(f_{c}/f_{s}=1.2\); and (**f**) \(f_{c}/f_{s}=1.35\).