# Chaotic Characteristic Analysis of Spillway Radial Gate Vibration under Discharge Excitation
## 4 Results and Discussion
### Chaotic Characteristic Analysis  
#### 4.2.1 Correlation Dimension Analysis  
The G-P algorithm [28] was selected to calculate the saturation correlation dimension. According to the optimal time delay, \(\tau\), the embedding dimension gradually increased from \(m\) = 2 to 30, and the relationship between \(C(r)\) and \(r\) was calculated according to Equation (9); then, the double logarithmic relation curve of different \(m\) values was obtained.  
For the measuring points on the supporting arm, the RMS amplitude of the vibration acceleration of point 7# was higher than those of points 5# and 6#. Taking the results from the point 7 \(z\)-direction as an example, Figure 12 shows that the double logarithmic relation curve with an arrow indicates the increased m value from 2 to 30. Select the section of the curve with an approximately linear variation and take its slope as the correlation dimension under the corresponding embedding dimension. The correlation dimension in Figure 13 increases with the increase in the embedding dimension till it reaches the saturated correlation dimension, \(D_{2}\). Figure 13 shows the relationship between \(D_{2}\) and \(m\) of the acceleration time series. According to this step, the law of the correlation dimension variation of each measuring point under three working conditions was calculated, as shown in Figure 14.  
Figure 12: ln\(C(r)\) versus ln\(r\) plots for the vibration series of point 7 in the \(z\)-direction.  
Figure 13: Relationship between \(D_{2}\) and \(m\) at point 7 in the \(z\)-direction.  
It can be concluded from Figure 14 that:  
1. The results show that the correlation dimension reaches the saturation value of the embedding dimension, which agrees well with the calculated results attained by the AFN method, indicating that the acceleration signal after the noise reduction stage can better reflect the chaotic characteristics of the SRG.
2. Overall, the correlation dimension varies from 5.2561 to 9.1092. The correlation dimension of point 1 in the \(z\)-direction under conditions 1 and 3 and point 7 in the \(x\)-direction present the lowest magnitudes, and the remaining points are concentrated in the range of 6.1916-9.1092. All of the correlation dimensions are fractional numbers, which indicates that the FIV response in each direction on the SRG exhibit chaotic and fractal characteristics.
3. At the same measuring points, the correlation dimension, \(D_{2}\), of each measuring point under condition 1 is larger than those under conditions 2 and 3, indicating that the uncertainty of the gate vibration increases when the opening is large.
4. The saturation correlation dimension of point 3 presents the highest magnitude of the correlation dimension in the direction along the \(x\)-axis, indicating that the lower main beam has the most complex vibration.
5. The \(y\)-direction (vertical) correlation dimensions of the three measuring points on the supporting arm are large and the \(x\)-direction (axial) measurement point on the supporting arm is obviously small, indicating that the axial direction of the supporting arm presents a lower-dimensional chaotic attractor. The complexity and nonlinear law of the vibration process of the axial direction of the supporting arm can be determined by using fewer independent control variables.  
The reference [8] results show that, when the dynamic instability zone-induced parametric vibration of the supporting arm is generated under the action of the fluctuating load, the out-of-plane vibration of the supporting arm has a remarkable influence on the dynamic instability of the SRG. Therefore, more attention should be paid to the vertical (\(y\)-direction) vibration of the supporting arm when monitoring the gate vibration.