Cylindrical roller bearing fault diagnosis based on VMD-SVD and Adaboost classifier method

2.1. Signal decomposition by VMD algorithm

VMD is an adaptive signal decomposing method, which can overcome the shortcomings like endpoint effects, modal aliasing as reported in EMD research. Meanwhile, the instantaneous frequency of each basic component by decomposing has its practical meaning in physical.

Its process could be divided into two steps: the establishment part and the solution to the constrained variational problem. The mentioned problem was written as follows:

Subjected to $\sum _k{u}_k=f$.

In order to transform the constrained variational problem into an unconstrained variational problem, the quadratic penalty and Lagrangian multipliers were introduced to Eq. (1). The augmented Lagrangian was expressed as follows:

The Eq. (2) was solved by iterate search with alternating direction method of multipliers (ADMM).

The mode $u_k$ and the century frequency ${\omega }_k$ were written respectively as follows:

2.2. Feature extraction by SVD algorithm

The singular value decomposition theorem and its characteristics can be described as follows:

Theorem: (Singular Value Decomposition Theorem) If $A\in R^{m\times n}$ (without losing generality, setting $m\ge n$) and $rank\left(A\right)=r$, there are two orthogonal matrices:

and a diagonal matrix:

that make the following formula establish, which is:

where the ${\lambda }_i^2$ is the eigenvalue of $A^{T}A$ and $A{A}^T$ and $u_i$, $v_i$ are the eigen vectors of $A^{T}A$ and $A{A}^T$ corresponding to the eigenvalue ${\lambda }_i^2$. The elements on the diagonal of $S$ are singular values.

2.3. Pattern recognition by Adaboost classifier

Adaboost is an accurate classifier construction algorithm that the strong learning algorithm is formed by a family of weak learning algorithms through certain rules, which can get a high recognition accuracy of the classified through the sample training. The specific description of the algorithm is as follows:

1) The weak learning algorithm is given and the training set is shown as {($x_1$, $y_1$), ($x_2$, $y_2$),…, ($x_n$, $y_n$)}, where $x_i$ represents the training sample vector and $y_i$ represents the classification mark.

2) The initialized weight is defined as $D_1\left(x_i\right)=1/n$, $i=$ 1,…, $n$.

3) For $t=$1, 2,…, $T$:

a) The weak learning algorithm is trained under the weight value $D_t$ and the predictive function $h_t$: $x_i\to \left\{-1,+1\right\}$ can be obtained.

b) The error rate of the predictive function is calculated by:

c) Updating the weights based on the above error rate:

where $a_t=\mathrm{l}\mathrm{n}\left(\left(1-e_t\right)/e_t\right)/2$ and $Z_i$ is a normalization factor.

d) The final prediction function can be obtained after training:

3.1. Feature extraction based on EMD-SVD

In order to extract the eigenvector of the data sufficiently, we first decomposed the data by VMD in each mode and obtained its components. In this paper, the component number $K$ was set to 4, and the center frequency evolution schematic diagram of the component was shown in Fig. 4. The data in normal state and its four components after the decomposition of VMD were shown in Fig. 5. After that, the four components were constructed into a matrix $D_{102400\times 4}$. And then, we took the first 10,000 rows and cut them into 100 matrices in same size (written as $D{i}_{100\times 4}$).

Fig. 4Evolution of center frequencies

The singular value vector Vector_SVD_4×1 of each matrix Di was obtained by SVD. And the singular value vector was used as the eigenvector, so there were 100 eigenvectors under each condition. Finally, 300 eigenvectors were totally obtained with all the data.

Fig. 5The VMD decomposition diagram of normal data

3.2. State classification based on Adaboost classifier

The 300 eigenvectors were entered as training data into the Adaboost classifier. The label value of all eigenvectors extracted from normal data was set to 1, and the other 200 eigenvectors was set to –1. The Adaboost classifier consists of a number of weak classifiers that dynamically adjust threshold values. As can be seen from Fig. 6, with the increase of the number of weak classifiers, the model gradually converged. Finally, a strong classifier was obtained based on these weak classifiers.

Fig. 6Classification error versus number of weak classifier

Because the adaboost classifier in this paper is a binary classifier, the above model could be trained to distinguish normal data from fault data. However, there was no distinction between the rolling element fault and the outer ring fault. Therefore, this paper built another Adaboost classifier in the same way. The training data adopted the eigenvectors of rolling element fault and outer ring fault. In which, the label value of the rolling element fault was set to –1, and the outer ring fault was 1. After training, this classifier could distinguish between these two fault modes.

The accuracy of these classifiers was calculated by classifying the test data. By entering 300 samples (including 100 normal samples, 100 outer ring fault samples, and 100 scroll-fault samples) into the constructed classifier, the fault detection rate of distinguishing between normal data and fault data was 97 % and the fault diagnosis rate of distinguishing between the fault modes was 95 %. The results showed that the method proposed in this paper was practical and effective for fault diagnosis of cylindrical roller bearing.