## 3 Principle of Diagnosis Based on Optimized VMD-RF Algorithm  
### The Basic Principle of VMD Algorithm  
VMD is a new non-stationary and non-linear method in the signal decomposition and calculation field, which is based on the theoretical basis of classical Wiener filtering, Hilbert transform, and heterodyne demodulation. The essence is to iteratively solve the variational problem. VMD has a strict mathematical theory as the basis for the adaptive decomposition of signals [17]. According to the preset number of modal components, VMD can decompose the signal into K modes with center frequency and limited bandwidth. Through continuous iteration, the center frequency and bandwidth are updated to finally determine an ensemble of IMFs, written as \(u_{k}(t)\), \(k~{}=~{}1,2,\ldots,K\). Input signal \(x(t)\) can be restored by simple summation. The decomposition of the algorithm is as follows [18; 19; 20]:  
1. The analytic signal of each mode function \(u_{k}(t)\) is computed by the Hilbert transform method, thus a unilateral spectrum of the analytic signal is obtained: \[\left[\delta(t)+\frac{j}{\pi t}\right]u_{k}(t).\] (1) \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Cylinder**} & \multicolumn{2}{c}{**Normal Valve Clearance**} & \multicolumn{2}{c}{**Abnormal Valve Clearance**} \\ \cline{2-5}  & **Intake Valve/mm** & **Exhaust Valve/mm** & **Intake Valve/mm** & **Exhaust Valve/mm** \\ \hline B4 & 0.3 & 0.5 & 0.6 & 0.5 \\ B1 & 0.3 & 0.5 & 0.9 & 0.5 \\ B5 & 0.3 & 0.5 & 0.3 & 0.8 \\ B2 & 0.4 & 0.4 & 0.4 & 1.1 \\ B6 & 0.3 & 0.5 & 0.6 & 0.8 \\ B3 & 0.3 & 0.4 & 0.9 & 1.1 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: The experimental parameters of valve clearance faults. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Performance Specication** & **Unit** & **Value** \\ \hline Incremental scale factor & V/mm & 3.94 \\ Frequency Response & Hz & 0–8000 \\ Temperature range & \({}^{\circ}\)C & 0–45 \\ Linear Range & mm & 4 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The main parameters of the eddy current sensor. Since it is necessary to adjust the center frequency of the analytical signal corresponding to each mode component, an exponent term \(e^{-j\omega_{k}t}\) is added to modulate the spectrum of each mode component to the baseband: \[\left[\left(\delta(t)+\frac{j}{\pi t}\right)u_{k}(t)\right]e^{-j\omega_{k}t}.\] (2)
3. The bandwidth of each component is estimated through the Gaussian smoothness of the demodulated signal, that is, the squared \(L^{2}\)-norm of the gradient. According to the constraint conditions, the optimal variational mode is  
information and there is a significant pulse in the waveform, the signal has strong sparse characteristics and the power spectrum entropy is small. Therefore, we need to have the IMF component with the smallest power spectral entropy value. At the same time based on the above two points, we propose a criterion for optimal selection:  
\[T\mathrm{ar}=p(\alpha)+\lambda q(\alpha)\, \tag{5}\]  
where \(p\) represents the energy leakage ratio, \(q\) represents the power spectrum entropy increment ratio, \(\alpha\) represents the penalty function, and \(\lambda\) represents the weighting factor. This paper takes \(\lambda=0.75\). Taking the vibration signal of large exhaust valve clearance increment (fault 4) at 1500 rpm as an example, we analyze the penalty function \(\alpha\) as an independent variable, here the penalty function \(\alpha\in[10,3000]\). According to the above criteria for optimal selection, we can get the parabolic curve of Tar changing with \(\alpha\), as shown in the Figure 3.  
From a partially enlarged image, we can find that when the penalty function \(\alpha\) takes a value of 80, Tar can get the minimum value. Therefore, we decided to choose the penalty function \(\alpha=80\) in this paper. The program code involved in this paper is all achieved in the MATLAB R2015b environment.