## 4 Verifications  
### Vibration Correlation Analyses by XWT Method  
As recommended in the introduction, cross-wavelet transform (XWT) is an efficient method to analyze the correlation between two signals. Suppose the wavelet coefficients \(W_{y}(a,\tau)\) of a finite time signal \(y(t)\)  
\[W_{y}(a,\tau)=\frac{1}{\sqrt{a}}\int_{-\infty}^{+\infty}y(t)\psi^{\prime}(t)dt \tag{26}\]  
with  
\[\psi^{\prime}(t)=\frac{1}{\sqrt{a}}\psi\!\left(\frac{t-\tau}{a}\right)(a>0,\tau \in\mathbb{R}) \tag{27}\]  
where \(a\) represents the scaling parameter; \(t\) represents time; \(\tau\) represents the location of \(t\); \(\psi(t)\) denotes the mother wavelet; and \(\psi^{\prime}(t)\) extends the mother wavelet to a wavelet family. In practice, the mother wavelet \(\psi(t)\) is always considered as a Morlet wavelet:  
\[\psi(t)=\pi^{-1/4}e^{-i\omega t}e^{-t^{2}/2} \tag{28}\]  
where \(\omega\) is the nondimensional frequency, which is considered to be six to satisfy the admissibility condition [41]. For two signals, \(y(t)\) and \(z(t)\), the XWT is defined as follows:  
\[XW_{y,z}(a,\tau)=W_{y}(a,\tau)W_{z}{}^{*}(a,\tau) \tag{29}\]  
the asterisk \({}^{*}\) denotes the complex conjugate. In engineering practice, the XWT is always transferred to the form of the cross-wavelet scalogram (XWS) to analyze the non-stationary signal:  
\[XWS_{y,z}(a,\tau)=XW_{y,z}(a,\tau)XW_{y,z}{}^{*}(a,\tau)=\left|XW_{y,z}(a,\tau )\right|^{2} \tag{30}\]  
By defining \(XW_{y,z}(a,\tau)\) and \(XWS_{y,z}(a,\tau)\), the correlation between two signals in both time and frequency domains is revealed, and the value of \(XWS_{y,z}(a,\tau)\) represents the correlation level. Applying the XWT technique for signals obtained from 1# dam block and eight guide wall blocks, the correlation level in the form of XWS is illustrated in Figure 18. The figures show that the vibration of the dam block and guide wall blocks demonstrate clear synchronized correlation for the frequency band of about 2.5-7 Hz. It can be interpreted that under these synchronized vibration frequencies, AV tends to occur. For the 1# and 2# guide wall block, the correlations of both time and frequency domains are weaker, and the dominant synchronized frequency is located around 2.8 Hz, which may represent the natural frequency of the large-sized dam. It can be deduced that the vibration of these guide wall blocks may derive from the dam block, i.e., AV is dominant in these guide wall blocks. Meanwhile, in downstream guide wall blocks (3#-8# blocks), the correlative frequency bands are broader, and the dominant synchronized frequency is gradually increased to about 6.2 Hz, which represents the natural frequency of the small-sized guide wall. The natural frequency of the dam becomes less evident, from which it can be inferred that the influence of AV is attenuated in downstream guide wall blocks. The guide wall vibration transforms into independent FV, and the broad correlative frequency bands are derived from the broad frequency band of flow fluctuation that acts on both the dam and the guide wall. Figure 18: Contour of the cross-wavelet scalogram (XWS) between 1# dam block and corresponding guide wall blocks. (**a**) 1# guide wall block; (**b**) 2# guide wall block; (**c**) 3# guide wall block; (**d**) 4# guide wall block; (**e**) 5# guide wall block; (**f**) 6# guide wall block; (**g**) 7# guide wall block; (**h**) 8# guide wall block. Combined with the theoretical model of the AFCV system in Section 3, the above phenomena can be elucidated. Generally speaking, the damping ratio of a built-up structure is fixed, which can infer that parameters \(\xi_{1}\) and \(\xi_{3}\) are constant; each guide wall block can be considered as being the same structures, therefore, the parameters \(\lambda_{13}\) and \(\mu\) are fixed in this case.