## 4 Pressure Characteristics on the Gate Panel
### The Pressure Distribution Characteristics on the Gate Panel
#### 4.1.2 Ihhtr Analysis of Pressure
As the decomposition order increased,  
Figure 9: IMF component and spectrogram of fluctuating pressure at the P2 measuring point: (a) ICEEMDAN; (b) FFT. the bandwidth of higher-order components narrowed significantly, and the amplitude of the Hilbert spectrum gradually decreased, along with a notable decline in Hilbert energy. Hence, the initial high-frequency components encompassed the majority of turbulent energy, albeit with low-frequency elements corresponding to the continuous intermixing and stochastic motion of vortices of varying scales within the turbulence. The instantaneous frequencies of individual components varied randomly over time, with Hilbert energy oscillating similarly. Sporadic abrupt changes were more pronounced in higher-order components. For instance, the primary frequency band of IMF1 was below 25 Hz (with a high-energy region below 20 Hz). However, the high-energy region of IMF2 rapidly narrowed to less than 10 Hz, and subsequently, the frequency band continued to narrow as the decomposition order increased. After IMF6, the bandwidth approached zero, and the amplitude of the Hilbert spectrum decreased to less than 5% of that of lower-order components. Nevertheless, the random oscillations in energy were significantly attenuated, reflecting the extremely weak non-stationary characteristics of the water flow pressure fluctuations. Based on the analysis in Figure 11, which integrated the results from other measurement points, the decomposition and Hilbert transformation of the fluctuating pressure signals on the gate panel revealed distinct frequency components. This process can be interpreted as decomposing the turbulent water flow into vortices of varying scales. The modal aliasing phenomenon of fluctuating pressure in each component further indicated the presence of multi-scale vortices within the turbulent vortex structure. The random diffusion and mixing of these vortices collectively contributed to the turbulence observed in the water flow. The preceding analysis demonstrates that the typical time series data of fluctuating water pressure exhibit weak non-stationarity, yet they can still be considered stationary to some extent. Furthermore, under specific operating conditions with e values of 10%, 25%, and 75%, the fluctuating pressure acting on the gate panel can be regarded as an ergodic stationary random process. The characteristic value of the ergodic stationary random process for fluctuating pressure acting on the gate can be obtained through random function theory and spectral analysis. This approach serves as a prerequisite for various other gate FIV analysis studies. The presence of energy at a specific frequency in the Hilbert marginal spectrum indicates the potential occurrence of vibration at that frequency. The frequency corresponding to the maximum amplitude of the marginal spectrum represents the most likely vibration frequency, as well as the dominant frequency of primary interest in this study. This study utilized the Hilbert marginal spectrum to investigate the energy propagation patterns  
Figure 10: Hilbert spectra of the components of eigenpoints. (**a**–**h**): IMF 1–8. of fluctuating pressure signals. Figure 12 illustrates the Hilbert marginal spectrum at characteristic points along the surface of the radial gate panel. The dominant frequencies at the seven measurement points on the gate panel were all below 20 Hz. Notably, the high-frequency components were more pronounced at P1 and P2 near the bottom edge, exhibiting significantly higher marginal spectrum amplitudes compared to the middle and upper sections of the gate. According to turbulence theory, low- and high-frequency components correspond to large- and small-scale vortices, respectively. Therefore, it can be inferred that the dominant large vortices in the turbulence gradually transform into small vortices here.