# Chaotic Characteristic Analysis of Spillway Radial Gate Vibration under Discharge Excitation
## 3 SRG Analysis of the Prototype Test Data
### Signal Denoising  
In the processes of transmission and acquisition, the vibration signal of the discharge structure is easily affected by high-frequency white noise and low-frequency water flow, which is usually represented by a low SNR and non-stationary random signal [22]. In addition, the presence of noise affected the results derived from the chaos theory. Therefore, it was necessary to reduce the noise level before analyzing the chaotic characteristics of the data. After the original signals were decomposed by CEEMDAN to obtain the IMF components, the intensity of the noise in each IMF component could be determined by calculating the multi-scale permutation entropy (MPE) value. This measured the randomness of the time variance values of the signals at different scales [23], with a magnitude approaching 1 indicating relatively high randomness and non-stationarity values. The MPE threshold was set to 0.6 [24] in this work, and the MPE values of IMF components higher than or equal to 0.6 were regarded as noisy components and needed further filtering out by the wavelet threshold denoising (WTD) method. The principle of the WTD method is that the wavelet coefficients generated by the signal through the wavelet transform contains important information regarding the signal. Figure 5: Flow pattern diagram of the discharge flow. Figure 6: Contours of velocity magnitudes near the bottom edge of SRG under three working conditions: (**a**) condition 1 (h = 3.5 m); (**b**) condition 2 (h = 24 m); (**c**) condition 3 (h = 1.0 m). The real signal possessed coefficients with relatively high magnitudes, while the noise signal presented relatively low coefficients. By setting a reasonable threshold for the wavelet coefficient, the wavelet coefficient lower than the threshold was perceived as noise and set to 0 to eliminate the effect of noise. The steps performed were as follows:  
1. Select the appropriate wavelet basis and the number of decomposition layers to conduct the wavelet decomposition of noisy signals;
2. Select the appropriate threshold method for the threshold processing of wavelet coefficients at each decomposition scale;
3. Reconstruct the wavelet coefficient to complete the noise reduction process. The common threshold function is divided into soft and hard threshold functions [25, 26], which are given in Equations (19) and (20), respectively:  
\[W_{j,k}=\left\{ \begin{array}{l}sgn(w_{j,k}\big{(}\big{|}w_{j,k}\big{|}-\lambda )),\big{|}w_{j,k}\big{|}\geq\lambda\\ 0,\big{|}w_{j,k}\big{|}<\lambda\end{array} \right. \tag{19}\]  
\[W_{j,k}=\left\{ \begin{array}{l}w_{j,k},\big{|}w_{j,k}\big{|}\geq\lambda\\ 0,\big{|}w_{j,k}\big{|}<\lambda\end{array} \right. \tag{20}\]  
Here, Equation (19) denotes the soft threshold function and Equation (20) denotes the hard threshold function. \(W_{j,k}\) denotes the wavelet decomposition coefficient and \(W_{j,k}\) is the wavelet coefficient after decomposition. \(\lambda\) is the threshold value. However, the discontinuity of the hard threshold function at the threshold point and the constant error of the soft threshold function still affected the denoising to some extent. Therefore, in this paper, a new threshold function was introduced to overcome these shortcomings. It combines the characteristics of the soft and hard threshold values, which is expressed as the following equation:  
\[W_{j,k}=\left\{ \begin{array}{l}sgn(w_{j,k})\times\left|\left|w_{j,k}\right| \right.\\ 0,\big{|}w_{j,k}\big{|}<\lambda\end{array} \right.-\left.\frac{\lambda}{\sqrt[ \big{|}w_{j,k}\big{|}^{\beta}-\big{|}\lambda\big{|}^{\beta}+1}\right.\\ \left.\frac{1}{\epsilon^{\sqrt{\big{|}w_{j,k}\big{|}-\lambda\big{|}}}}\right| \end{array}\right.,\left|w_{j,k}\right|\geq\lambda \tag{21}\]  
The equation above was continuous at the threshold point, so it could overcome the discontinuity of the hard threshold function at the threshold and meet the condition of an odd function.