## 2 Methodology
### Feature Extraction Based on Hilbert-Huang Transform (HHT)  
Hilbert-Huang Transform is a time-frequency analysis method presented by [PERSON] et al. [57]. This method is suitable for non-stationary signals. HHT contains two processing processes: Empirical mode decomposition (EMD) and Hilbert Transform (HT). EMD decomposes a complex signal into a series of Intrinsic Mode Function (IMF). Then, Hilbert Transform is used to obtain the instantaneous frequency and amplitude. The Hilbert spectrum obtained by this method has a high time-frequency resolution. In addition, each IMF component obtained by EMD has a clear physical meaning, and the processed signal retains the local characteristics of the original signal.  
The decomposed component of EMD must satisfy two conditions so as to be considered as IMFs. First, the difference between the number of extrema and zero crossings is less than or equal to 1. Second, the mean value of the envelope determined by local maxima and local minima must equal zero.  
The specific process of HHT (EMD and HT) are as follows:  
Step 1: For an original signal \(x(t)\), the local maxima and local minima are computed and upper and lower envelop are obtained by connected them using a cubic spline, respectively.  
Step 2: The mean of the upper and lower envelop \(m_{1}(t)\) is calculated and is subtracted from the original signal \(x(t)\):  
\[h_{1}(t)=x(t)-m_{1}(t). \tag{19}\]  
If \(h_{1}(t)\) satisfies the conditions of the IMF definition, then \(h_{1}(t)\) is the first IMF component, and let \(h_{1}=h_{1}(t)\). Otherwise, \(x(t)\) is replaced by \(h_{1}(t)\) and repeat the previous process \(k\) times until meet the IMF conditions, and let \(h_{1}=h_{1,k}(t)\).  
\[h_{1,j+1}(t)=h_{1,j}(t)-m_{1,j+1}(t),j=1,2,\ldots,k. \tag{20}\]  
Step 3: After the first IMF is found, the residue is calculated by subtracted \(h_{1}\) from the \(x(t)\):  
\[r_{1}=x(t)-h_{1}. \tag{21}\]  
Step 4: Take \(r_{1}\) as the new data, repeat step 1, step 2 and step 3, and obtain the \(n\)th IMF component \(h_{n}\) and \(n\)th residue \(r_{n}\), until the residue \(r_{n}\) becomes too small or \(r_{n}(t)\) a monotonic function.  
Step 5: For each IMF component \(h_{n}(t)\), Hilbert Transform is performed as follows:  
\[y_{n}(t)=\frac{P}{\pi}\int_{-\infty}^{+\infty}\frac{h_{n}(\tau)}{t-\tau}d\tau, \tag{22}\]  
where \(P\) is the Cauchy principal value.  
Step 6: Coupling \(h_{n}(t)\) and \(y_{n}(t)\), a signal \(z_{n}(t)\) can be obtained [58]:  
\[z_{n}(t)=h_{n}(t)+iy_{n}(t)=a_{n}(t)e^{i\varphi_{n}(t)}, \tag{23}\]where \(a_{n}(t)=\sqrt{h_{n}(t)+y_{n}(t)}\) and \(\varphi_{n}(t)=arctan(y_{n}(t)/h_{n}(t))\) are instantaneous amplitude and instantaneous phase. Instantaneous frequency can be obtained by derivation of the instantaneous phase:  
\[\omega_{n}(t)=\frac{d\varphi_{n}(t)}{dt}. \tag{24}\]  
Step 7: The time-frequency distribution of amplitude is called Hilbert spectrum and is expressed as:  
\[H(\omega,t)=\text{Re}\bigg{\{}\sum a_{n}(t)\exp[i\!\int\omega_{n}(t)dt]\bigg{\}}. \tag{25}\]  
Step 8: The Hilbert marginal spectrum is obtained by integrating the Hilbert spectrum:  
\[h(\omega)=\int_{0}^{T}H(\omega,t)dt. \tag{26}\]  
According to the Hilbert marginal spectrum, the amplitude of the fundamental frequency, second-order harmonic and third-order harmonic can be obtained. Then, the nonlinear coefficient can be calculated to evaluate the damage state.