# Chaotic Characteristic Analysis of Spillway Radial Gate Vibration under Discharge Excitation
## 4 Results and Discussion
### Discussion  
It can be seen from Figure 9b that high-frequency components appear in the NPS of the original signal, which may result from the high-order local vibration highlighted by the gate and high-frequency white noise collected by the sensor. The vibration energy values of the high-frequency components are low and the NPS amplitude is less than 0.5, which is not the overall vibration value of the gate. Similarly, [PERSON] et. al [12] supposed that the vibration sources for a pumping station pipe mainly included the low-frequency flow-water fluctuation caused by flow water and the blade frequency, rotation frequency, and frequency doubling caused by the unit's operation. The low-frequency fluctuating load of flow water was the main cause of the chaotic characteristics of the pumping station pipeline. However, the vibration source was relatively simple when the gate was opened, that is, the low-frequency dynamic water load acting on the gate panel. In order to further clarify the vibration excitation characteristics that caused the gate to exhibit chaotic characteristics, the CEEMDAN decomposition results of the vibration signal in the point 3# \(x\)-direction on the lower main beam mentioned above were used as an example. Three components, IMF1, IMF2, and IMF3-14, were compared. As previously mentioned, Figure 9 shows the comparison of the vibration acceleration values before and after the noise reduction stages at point 3# in the \(x\)-direction. Figure 17 shows the time variances and normalized power spectral density curves of the three components.  
It can be seen from Figure 17b that the vibration energy is concentrated between 40 and 100 Hz in acceleration components IMF1 and IMF2, respectively, of point 3# along the \(x\)-direction, and the NPS is less than 0.5. The main energy values of the IMF3-14 components are mainly concentrated between 0.1 and 20 Hz. According to statistics [8], 93% of the gate flow-water fluctuation main frequency variations were below 20 Hz, and few exceeded 20 Hz. Due to the limitations of the observation conditions, although the fluctuating pressure sensors were not installed on the gate panel, it could be inferred from the existing studies that the IMF3-14 component was mainly the low-frequency forced vibration component of the gate induced by the flow-water fluctuation. Therefore, the analyses of the chaotic characteristics of IMF1, IMF2, and IMF3-14 representing different components can further clarify the exhibition of chaotic gate characteristics.  
Figure 17: Time and frequency domain diagrams of point 3# in the \(x\)-direction for different IMF components: (**a**) time histories of each IMF component and (**b**) NPS of each IMF component.  
As shown in Figure 18, the maximum Lyapunov exponents, \(\lambda_{1}\), of IMF1 and IMF2 are 0.0192 and 0.0348, while the maximum Lyapunov exponent, \(\lambda_{1}\), of IMF3-14 is 0.0744. The maximum Lyapunov exponents of IMF1 and IMF2 are 0.0681 after the IWTD was denoised and combined with the remaining components. If the vibration excitation components (IMF1 and IMF2) with weak chaotic characteristics were removed, the maximum Lyapunov exponents, \(\lambda_{1}\), of IMF3-14 increased from 0.0681 to 0.0744, and the chaotic characteristics were more obvious. The above-mentioned results show that the forced vibration of the gate is chaotic due to the fluctuating load acting on the gate panel.