## 4 Results and Discussion
### Flow Induced Vibration of Two Staggered Cylinders
#### 4.2.2 Amplitude and Frequency Responses  
Amplitude and frequency are very important parameters to describe the motion characteristics and flow-induced vibration response of oscillators. The transition of the different FIV states of the cylinder is accompanied by changes in the response of different frequencies. The amplitude adopted in this paper was obtained by taking the root mean square value of the absolute value of all the displacements in the stable vibration phase of the cylinder. The root mean square of displacement reflected the intensity of vibration of a cylinder. The vibration frequency was obtained by FFT. The amplitude and frequency responses of the two cylinders under different spacing ratios are presented in Figures 10 and 11. From Figure 10, it can be seen that the two cylinders underwent initial, upper and lower branches within the calculated reduced velocity. At a low reduced velocity (\(U^{*}\leq 5\)), the amplitude of the upstream cylinder increased with the increase of the reduced velocity. At this time, the upstream cylinder vibration was located in the initial branch, in which the flow-induced vibration of the cylinder could not be clearly observed to be strengthened or suppressed. At this time, the velocity was low, and the gap distance had little effect on the upstream cylinder amplitude. When \(5<U^{*}<11\), the FIV of the upstream cylinder was in the upper branch with high amplitude, and the maximum amplitude was obtained at \(U^{*}=6\). The vibration of the upstream cylinder became chaotic, and the vibration was located in the lower branch when \(U^{*}>11\). For the downstream cylinder, the initial branch, the upper branch and the lower branch were located at \(U^{*}\leq 7\), \(7<U^{*}<11\) and \(U^{*}>11\), respectively, and the maximum amplitude occurred at \(U^{*}=11\). Due to the influence of the upstream cylinder, the upper branch region of the downstream cylinder was narrower than the upper branch region of the upstream cylinder. When \(U^{*}>11\), the vibration of the downstream cylinder became chaotic, and at this time, the vibration of the downstream cylinder tended to develop towards the lower branch. Besides, it was found that the upstream cylinder amplitude decreased slowly with the increase of the reduced velocity when the reduced velocity reached to 6, while the downstream cylinder vibration amplitude increased. In the vicinity of \(U^{*}=7\), the amplitude curves of the upstream and downstream cylinders intersected. In the case of \(U^{*}<7\), the amplitude of the upstream cylinder was larger than that of the downstream cylinder. After \(U^{*}>7\), the amplitude of the upstream cylinder was smaller than that of the downstream cylinder. This is because when the reduced velocity was large, the vibration between the two cylinders was stimulated, and the interaction between the two cylinders was increased. The downstream cylinder was affected by the upstream cylinder vortex shedding, and the vibration of the downstream cylinder was strengthened. Therefore, there was a critical reduced velocity. Before reaching the critical reduced velocity, the amplitude of the upstream cylinder was higher than that of the downstream cylinder. After the critical reduced velocity exceeded, the amplitude of the upstream cylinder was lower than that of the downstream cylinder. As can be seen from Figure 11, the frequency ratio of the two cylinders was close, and the frequency ratio of the downstream cylinder was smaller than that of the upstream cylinder under the same working condition. With the increase of the spacing ratio, the difference of frequency ratios for the two cylinders decreased gradually. The vibration of the two cylinders basically coincided at \(T/D=1\). The interaction between the two cylinders decreased at a large spacing ratio, and the two cylinders vibrated at almost the same frequency. The upstream cylinder frequency locking phenomenon appeared at \(U^{*}=5\)-\(6\), and the amplitude corresponded to the maximum.