# Chaotic Characteristic Analysis of Spillway Radial Gate Vibration under Discharge Excitation
## 4 Results and Discussion  
### Phase-Space Reconstruction of the Vibration Acceleration Data  
As can be seen from the layout of _the_ gate vibration acceleration points in Figure 2, points 2# and 4# are at symmetrical locations with respect to points 1# and 3#, respectively. Therefore, this study focused on analyzing the vibration acceleration characteristics in the three directions of measuring point 1# on the upper main beam, point 3# on the lower main beam, and points 5#, 6#, and 7# on the upper supporting arm. The denoised signal was reconstructed in the phase space and the AMI method was used to calculate the optimal time delay, \(\tau\). Figure 10 shows the AMI calculation diagram of point 3# in the \(x\)-direction (the radial direction of the gate leaf) under condition 1, and the results derived from this method are summarized in Table 3. After determining the optimal time delay, the optimal embedding dimension was selected using the AFN method. The variations in \(E_{1}\) and \(E_{2}\) of the point 3# \(x\)-directional data with \(m\) are shown in Figure 11. According to Figure 11a, the \(E_{2}\) curve before the noise reduction always fluctuates at a small range of around 1 for any m value, indicating that the random noise sequence is mixed into the time series. As shown in Figure 11b, the apparent deviation of the \(E_{2}\) curve after the noise reduction from 1 exists at a relatively low \(m\) value, and finally approaches 1 as \(m\) increases, indicating that \(E_{2}\) is obviously related to m. This method can be used to determine whether the time series exhibits chaotic characteristics. When the value of the \(E_{1}\) function no longer changes significantly with the embedding dimension, and the value of the \(E_{2}\) function approaches 1, the corresponding \(m\) value is the optimal embedding dimension. \begin{table}
\begin{tabular}{c c} \hline \hline
**Method** & _dnSNR_ \\ \hline CEEMDAN-MPE-HWTD & 1.95 \\ CEEMDAN-MPE-SWTD & 1.85 \\ CEEMDAN-MPE-IWTD & 1.83 \\ \hline \hline \end{tabular} Note: the CEEMDAN-MPE-HWTD method represents a combined denoising method based on the complete ensemble empirical mode decomposition with adaptive noise (CEEMDAN), multi-scale permutation entropy (MPE), and hard wavelet threshold denoising (HWTD) methods. Moreover, SWTD denotes soft wavelet threshold denoising, and IWTD denotes improved wavelet threshold denoising.  
\end{table} 
Table 2: Comparison results of the _dnSNR_ values. Figure 9: Comparison of vibration acceleration values before and after noise reduction stages at measuring point 3# in the \(x\)-direction: (**a**) time history curves and (**b**) normalized power spectrum (NPSs). Figure 11: Point 3# \(x\)-direction vibration acceleration embedding dimension before (**a**) and after (**b**) noise reduction stages under condition 1. Figure 10: \(\tau\) calculation of point 3# in the \(x\)-direction (the radial direction of the gate leaf) under condition 1. \begin{table}
\begin{tabular}{c c c c c c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Condition**} & \multirow{2}{*}{**Point**} & \multicolumn{4}{c}{**1**} & \multicolumn{4}{c}{**3**} & \multicolumn{4}{c}{**5**} & \multicolumn{4}{c}{**6**} & \multicolumn{4}{c}{**7**} \\ \cline{3-14}  & & \(x\) & \(y\) & \(z\) & \(x\) & \(y\) & \(z\) & \(x\) & \(y\) & \(z\) & \(x\) & \(y\) & \(z\) & \(x\) & \(y\) & \(z\) \\ \hline \multirow{2}{*}{1} & \(\tau\) & 5 & 5 & 6 & 5 & 8 & 5 & 4 & 5 & 5 & 4 & 4 & 5 & 4 & 4 & 4 \\  & \(m\) & 21 & 21 & 20 & 20 & 21 & 22 & 22 & 25 & 22 & 24 & 24 & 25 & 22 & 22 & 24 \\ \hline \multirow{2}{*}{2} & \(\tau\) & 4 & 7 & 7 & 5 & 5 & 6 & 4 & 5 & 8 & 4 & 4 & 5 & 4 & 4 & 4 \\  & \(m\) & 21 & 20 & 20 & 20 & 22 & 22 & 21 & 23 & 21 & 19 & 19 & 22 & 22 & 24 \\ \hline \multirow{2}{*}{3} & \(\tau\) & 5 & 7 & 8 & 5 & 5 & 5 & 5 & 6 & 5 & 5 & 5 & 5 & 5 & 4 \\  & \(m\) & 22 & 19 & 19 & 21 & 20 & 20 & 22 & 22 & 21 & 22 & 21 & 21 & 20 & 21 & 22 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: \(\tau\) and \(m\) values of vibration acceleration time series of measuring points.