## 4 Examples of Application
### Feature Extraction  
These are considered as a pattern recognition problem which consists of three phases namely, feature extraction, feature selection and feature classification. Among them, feature extraction is a key step in fault diagnosis. It usually involves two important aspects: one is that the extracted features should be sensitive to the target fault, and the other is that they should be insensitive or less sensitive to other conditions. For this article, ideal characteristics should be able to effectively identify the fault of abnormal valve clearance under different rotational speed conditions of the diesel engine, that is, stability to the rotational speed conditions. The main purpose of this work was to develop a method to obtain desirable features for detecting and diagnosing abnormal valve clearance.  
When the valve clearance became abnormal, the force of the valve impacting on valve seat changed, and the spectral energy of each frequency component in the signal changed accordingly. The IMFs produced by VMD represent a set of stationary signals in feature scales, of which the spectrum energy could be used to characterize different working states of valves. Therefore, the original signal was decomposed into D21, D22, D23, and D24 by the two-time decomposition strategy, and their spectral energies were taken as characteristic parameters.  
Diesel engine's vibration signal has the problems of local impact, wide frequency band distribution, and being susceptible to external noise. If we only used the VMD method to decompose the vibration signal, the characteristic information of valve fault could not be completely extracted. In order to fully reflect the working state of the valve, RMS value, peak value, peak-to-peak value, waveform index, pulse index, K-factor, crest factor, kurtosis, and skewness might be taken as time domain characteristic parameters representing the state of the vibration signal. For the valve, when the clearance was abnormal, the main frequency of the cylinder head's vibration signal might move to high or low frequency, and its distribution was also an effective parameter reflecting the working state of the valve. Therefore, the center of gravity frequency, mean square frequency, frequency variance, RMS frequency, and standard deviation of frequency were used as frequency domain characteristics to reflect different valve's operating conditions.  
Figure 7: Sketch of valve train.  
To sum up, this paper proposed to extract the spectral energy characteristics of IMFs, and combined the time domain and frequency domain features to comprehensively select feature parameters for establishing the feature vectors. The specific steps are as follows:  
1. Acquiring experimental data for normal and fault conditions of diesel engine's valves.
2. Studying the frequency components of the vibration signal. After determining the penalty function \(\alpha\) by the criterion for optimal selection, each group of vibration signals was decomposed by the VMD method according to the two-time decomposition strategy.
3. The IMFs performed Fourier transform to calculate the corresponding spectral energy \(E_{i}\) and extracted the spectral energy characteristics. The formula is as follows: \[E_{i}=\sum\
olimits_{n=1}^{N}\left|x_{i}(n)\right|^{2},i=1,2,3.\] (7)
4. Calculating the RMS value, peak value, peak-to-peak value, waveform index, pulse index, K factor, crest factor, kurtosis, and skewness of each group of signals.
5. Calculating the gravity frequency, mean square frequency, frequency variance, RMS frequency, and standard deviation of frequency of each group of signals.
6. Integrating spectrum energy of IMFs, nine time domain features and five frequency domain features, a total of 18-dimensional features were used to construct feature vectors.