## 3 Results and Discussion
Moreover, the secondary vortex of each vortex pair has lower strength than that of the primary vortex. This vortex shedding mode corresponds to a 2P overlap mode, 2P\({}_{0}\), which is consistent with the vortex shedding mode at \(f_{c}\)/\(f_{s}\approx 1\) obtained by [PERSON] and [PERSON] [3]. Additionally, the vortex shedding mode is within the range of the \"lock-in\" phenomenon as a stable state and the vortex shedding frequency of the oscillating cylinder approaches the natural shedding frequency, which can lead to the approximate symmetry of the wake vortices. As the frequency ratio \(f_{c}\)/\(f_{s}\) increases to 1.05, the vortex shedding mode steps in an intermediate state from a 2P mode to a P + S mode. Two vortex pairs exist per cycle. The strength of one vortex pair is significantly stronger than that of the other, and the weak vortex pair dissipates faster when it moves downstream. The strong and weak vortex pairs are separated in the process of the downward and upward oscillation, respectively. More importantly, the asymmetry of the strength of the vortex shedding leads in the asymmetry of the temporal evolution of the lift coefficient per cycle. In other ways, an irregular \"jumping\" phenomenon occurs at the right side of the temporal evolution of the lift coefficient per cycle when the cylinder moves to near a negative extreme displacement. However, the strong vortex pair is separated in the process of the upward oscillation at \(f_{c}\)/\(f_{s}\approx 1.2\), which is different from that at \(f_{c}\)/\(f_{s}\approx 1.05\), although they have the same vortex shedding modes. Accordingly, an irregular \"jumping\" phenomenon occurs at the left side of the temporal evolution of the lift coefficient per cycle. From \(f_{c}\)/\(f_{s}=1.05\) to \(f_{c}\)/\(f_{s}=1.2\), the discrepancy in time of separation of the strong vortex pair mainly derives from the difference in the oscillation frequency. Additionally, the irregular \"jumping\" occurs when the oscillating direction switches. When \(f_{c}\)/\(f_{s}=1.35\), its vortex shedding mode is in line with that at \(f_{c}\)/\(f_{s}=1.2\), but the lateral displacement of the vortex shedding is greater than that at \(f_{c}\)/\(f_{s}=1.2\). From \(f_{c}\)/\(f_{s}=1.05\) to \(f_{c}\)/\(f_{s}=1.35\), the temporal evolutions of the lift coefficient all show a periodic fluctuation, and locations of the irregular \"jumping\" on them mainly depend on the time of the separation of the vortices. As above, Figure 11 shows the vorticities around and behind the oscillating cylinder at \(A\)/\(D=1.5\) and \(f_{c}\)/\(f_{s}=0.5\) to 1.35. Figure 12 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. As \(A\)/\(D\) further increases to 1.5, the lateral displacement of the vortex shedding is also increased. Similarly, the vortex shedding mode also shows a similar \"P + S\" model at \(A\)/\(D=1.5\) and \(f_{c}\)/\(f_{s}=0.5\), and the lift and drag coefficients also show the \"beat\" characteristics. The temporal evolutions of the lift and drag coefficients also fluctuate drastically. The PSD of the lift coefficient at \(f_{c}\)/\(f_{s}=0.5\) shows that the vortex shedding frequency is affected not only by \(f_{c}\) and \(f_{s}\), but also by other nondominant frequencies, as shown in Figure 12a. The vortex shedding mode shows a 2P\({}_{0}\) mode placed out of the \"lock-in\" range, because the vortex shedding in the wake is either a single vortex or a vortex pair (including a strong vortex and a weak vortex). The phenomenon is also caused by the joint action of the excitation frequency and the Stroulhal frequency. When \(f_{c}\)/\(f_{s}=0.75\), the vortex shedding mode has transformed into a 2P\({}_{0}\) mode. The PSD of the lift coefficient at \(f_{c}\)/\(f_{s}=0.75\) also verifies that the vortex shedding frequency only rests with the excitation frequency \(f_{c}\).