5.2.1. Background Description

In order to monitor the operation status of engine, vibration sensors are set up for amassing the vibration signal of engine [27]. Then, the vibration signal can be processed to ge<sup>t</sup> the time-domain characteristics, as shown as Figures 16–18. The mean, variance, and kurtosis, which reflect the center, degree of dispersion, and degree of convex of signal, are selected as the health status indicators of the engine [28]. The real status of the engine is shown in Figure 19. The assessment result grades of engine are defined as three statuses according to the different gap between the crankshaft and bearing connecting rod: First, a gap of 0.08 mm to 0.1 mm belongs to "Health"; a gap of 0.18 mm to 0.2 mm belongs to "Fault"; a gap of 0.32 mm to 0.34 mm belongs to "Failure". Thus, a frame of discernment Φ is defined as follows.

$$\Phi = \{Health, Failure, Failure\} = \{\mathbf{D}\_1, \mathbf{D}\_2, \mathbf{D}\_3\} \tag{36}$$

**Figure 16.** The mean of vibration signal.

**Figure 17.** The variance of vibration signal.

**Figure 18.** The kurtosis of vibration signal.

**Figure 19.** The health status of engine.

There are 150 sets of data, including the status of "*Health*", "*Fault*", and "*Failure*". The mean of vibration signal ranges from 0.802 to 0.1761. The variance of vibration signal ranges from 0.0038 to 0.0191, and the kurtosis of vibration signal ranges from 2.2159 to 7.6801 as shown in Figures 16–18, respectively.

### 5.2.2. Construction and Optimization of Assessment Model

To construct a health assessment model of engine, first the indicators reference grades are determined as follows.

$$H\_{\text{mean}} = H\_{\text{variance}} = \{ \text{Small}, \text{Medium}, \text{ Slight large}, \text{Large} \} = \{ S, M, SL, L \} \tag{37}$$

$$H\_{kurtosis} = \{Average, High\} = \{A, H\} \tag{38}$$

where *Hmean*, *Hvariance*, and *Hkurtosis* denote the reference grades of mean, variance, and kurtosis.

Due to the reference grades are disaccord with assessment result grades, transformation matrixes are introduced to transform input information, and the initial values of transformation matrix and reference values are determined based on expert's knowledge in Table 13.


**Table 13.** The initial values of transformation matrix.

The Table 13 can be expressed as a form of matrix as Formulas (39) and (40). Then, the initial evidence is given by using the rule-based transformation technique. According to the implement process of example 1 in Section 5.1, the optimized simulated status is introduced in Figure 18. In the process of optimization, 75 sets of data are selected alternately from 150 sets of data as training data, and the whole sets of data are taken as test data.

$$A\_1 = \begin{bmatrix} 0.9 & 0.85 & 0.15 & 0.05 \\ 0.1 & 0.1 & 0.45 & 0.15 \\ 0 & 0.05 & 0.4 & 0.75 \end{bmatrix} \tag{39}$$

$$A\_{2} = \begin{bmatrix} 0.7 & 0.35 & 0.05 & 0\\ 0.25 & 0.5 & 0.2 & 0.25\\ 0.05 & 0.15 & 0.75 & 0.75 \end{bmatrix}, A\_{3} = \begin{bmatrix} 0.7 & 0\\ 0.2 & 0.3\\ 0.1 & 0.7 \end{bmatrix} \tag{40}$$

It is shown in Figure 20 that contrasting with the optimized simulated status, the error between the initial status and real status is rather large, especially in the first and third stages. By calculating the root mean square, the accuracy of the optimized status is 41.7% higher than the initial status. The optimized and calculated model parameters are given in Tables 14 and 15.

**Figure 20.** The comparison between the initial and optimized status.

**Table 14.** The parameters of optimized model parameters.


