Screw lifetime prediction based on wavelet neural network and empirical mode decomposition

3.1. Dynamic clustering data processing technology

Screw performance degradation is a long process. Condition monitoring of the performance degradation in total life cycle of the ball screw is necessary. To avoid the omission of information, data acquisition should be working during the total life cycle. This paper adopts dynamic clustering data processing technology to reduce redundant data. The basic idea is designing a flexible rectangular window to process data online [6]. By continuously moving the rectangular window in the signal, the signal can be compressed so the redundant data are eliminated.

The original signal can be expressed as {$a\left(i\right)$; $i=$ 1, 2,…, $N$}. New signal after clustering can be defined as {$b\left(j\right)$; $j=$1, 2,…, $M$}, where $M\ll N$. $k$ and $h$ are used to express the length and width of the rectangular window respectively. In the premise of $\text{1}\le k\le k_{max}$, the new signal after clustering can be expressed as:

When $\left|a\right(i+k)-a(i\left)\right|>h$ or $k=k_{max}$, $b\left(j\right)=a(i+k)$. Where $k_{max}$ is the maximum length of the rectangular window. The maximum length $k_{max}$ and width $h$ can be adjusted according to the experimental environment.

3.2. Feature extraction

The original features are extracted by following methods. With the method of wavelet analysis, the vibration signal is decomposed into 5 levels selecting wavelet “db1”, and distinct time-frequency features based on wavelet packet energy are extracted as a part of the original features. Time domain or frequency domain features are also been applied to rich the original features. Considering the certain correlation among the high dimensional original features, principal component analysis (PCA) is applied to eliminate redundant information.

4.1. Wavelet neural network

Topology of WNN is shown in Fig. 2. $x_1$, $x_2$,…, $x_n$ is the input of WNN, while $y_1$, $y_2$,…, $y_m$ is the output of WNN. $w_{ij}$ and $w_{jk}$ are the weight of WNN [7].

The output of hidden layer can be expressed as:

where $j=$ 1, 2,…, $l$ and $l$ is the number of hidden nodes. $h_j$ is wavelet basis function. $b_j$ and $a_j$ are the displacement factor and scale factor of the wavelet basis function, respectively.

The output of WNN can be defined as:

where $k=$ 1, 2,…, $m$ and $y_k$ is the output of neural network.

Fig. 2Topology of wavelet neural network

4.2. Empirical mode decomposition

Since screw remaining lifetime is a kind of trend variable, this paper adopts EMD to extract predicting trend from the output of WNN. The decomposition principle of EMD is defined as follows:

Step 1: searching all the local peaks of raw data sequence $s\left(t\right)$. Using cubic spline to envelope all local maximum points and minimum points, respectively. Sequence $m\left(t\right)$ is the mean value sequence of two envelope curves.

Step 2: subtracting $m\left(t\right)$ from $s\left(t\right)$ and getting a sequence $h_1\left(t\right)$. Repeating above steps until $h_1\left(t\right)$ is an intrinsic mode function, then $h_1\left(t\right)$ is the first intrinsic mode function and defined as $c_1\left(t\right)$.

Step 3: subtracting $c_1\left(t\right)$ from $s\left(t\right)$ and getting a residual sequence $r_1\left(t\right)$.

Step 4: treating $r_1\left(t\right)$ as a new raw data sequence. Repeating above steps until $r_n\left(t\right)$ is a monotonic function. The raw data sequence $s\left(t\right)$ can be expressed by some intrinsic mode functions and a mean trend:

where $c_k\left(t\right)$ are the intrinsic mode functions and $r_n\left(t\right)$ is the mean trend.

4.3. The structure of lifetime prediction model

Ball screw rated lifetime $L_h$ can be defined as:

where $L$ is the rated revolution of ball screw and $n_m$ is the rotational speed of ball screw. Then:

where $L_e$ is the expected remaining lifetime of ball screw and $L_w$ is the servicing lifetime.

Fig. 3 shows the flow chart of building screw lifetime prediction model. By different feature extraction methods, high dimension features are extracted from sensor signals. Then feature vectors, which dimensions are reduced by PCA, are obtained and stored in feature vectors sample library together with their expected remaining lifetime. Training samples are randomly selected from the sample library to train WNN, while EMD is adopted to extract the output tendency of GNN. Testing samples are used to test the tendency accuracy. By means of increasing samples and modifying network parameters, prediction accuracy is ensured. Then screw lifetime prediction model can be obtained.

Fig. 3Flow chart of building prediction model