Centrifugal pump fault detection based on SWT and SVM

2.1. Description of SWT

Synchrosqueezing wavelet transform is a new time-frequency signal analysis method proposed by Daubeche in 2011 [7]. By redistributing the wavelet coefficients, the signal energy in the spectrum is gathered near the real frequency, which improves the resolution of the signal in time-frequency.

For $x\left(t\right)$, its Fourier transform can be expressed as:

where the $w$ represents the angular frequency. The reverse transformation of $w$ is:

Thus, the signal $x\left(t\right)$ can be expressed as the sum of a set of harmonics with different frequencies, as in Eq. (3):

where $m_k\left(t\right)=A_k\cos \left({\phi }_k\left(t\right)\right)$ represents the $k$th harmonic component. $A_k$ represents the instantaneous amplitude of the signal, ${\phi }_k\left(t\right)$ represents the instantaneous phase of the $k$th harmonic component. And $e\left(t\right)$ represents the noise.

The continuous wavelet coefficients of a single harmonic signal $m\left(t\right)$ are obtained by wavelet transform as follows:

If the main frequency of the parent wavelet is ${\omega }_0$, then theoretically the transformed $W_m\left(a,b\right)$ will approach $a={\omega }_0/\omega$ in the time-scale plane. In fact, the wavelet coefficients are divergent on the scale, which makes the time-frequency spectrum of the wavelet transform relatively fuzzy. For arbitrary $\left(a,b\right)$ and $W_m\left(a,b\right)\ne 0$, the instantaneous frequency is calculated as follows:

The mapping of $\left(a,b\right)$ to $\left({\omega }_m\left(a,b\right),b\right)$ transforms the wavelet coefficients into the time-frequency plane. SWT compresses the wavelet coefficients in the interval:

Near the central frequency omega in the time-frequency plane. By processing the signal in this way, the frequency components are compressed in the frequency domain, making it easier for the frequency components to be recognized in the time-frequency plane. The discrete calculation formula of the compression wavelet coefficients is as follows:

The continuous calculation formula is as follows:

2.2. Description of SVM

Support Vector Machine (SVM) is a supervised learning algorithm, which is widely used in data classification and discrimination problems. SVM can effectively solve the problem of two-class classification by finding the best separation hyperplane in the feature space to maximize the interval between positive and negative samples on the training set. In order to solve the nonlinear problem, kernel function is introduced into SVM.

SVM for nonlinear classification is generally divided into two steps:

1. A nonlinear map is used to transform the data in the low-dimensional space X into a high-dimensional feature space F.

2. The linear classifier is used to realize the classification in the high-dimensional feature space F.

Consider the following data set form:

${\varphi }_i\left(x\right)$ is the mapping from space X to space F, but the explicit representation of the nonlinear mapping is difficult to reckon. Since the dual form is a property of SVM, the classification decision rules can be represented by the inner product of the test set and the training set:

Select a kernel function $K$, map the data to a high-dimensional space F, find the optimal separation hyperplane in space F, and distinguish the non-linear data:

3.1. Case study description

The data in the experiment is collected from the centrifugal pump in Fig. 1. Four common faults of the pump were set: bearing roller wearing, bearing inner race wearing, bearing outer race wearing and pump impeller wearing. The centrifugal pump is driven by the motor with a stable working speed of 2900 r/min. Vibration data of the centrifugal pump is obtained by accelerometers with a sampling frequency of 10.24 kHz.

Fig. 1Centrifugal pump

3.2. SWT-SVM based fault diagnosis

The collected centrifugal pump vibration signals are intercepted into a plurality of one-second long signal segments. SWT is used to analyze these signal segments. The results of frequency spectrum are in Fig. 2. It can be seen from the frequency spectrum graph that the amplitude of signal frequency domain is quite different under different fault modes.

Fig. 2Frequency spectrum

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e)

Given that the frequency domain characteristics of different vibration signals are quite different in the fixed frequency band, the energy of each peak part of the spectrum graph is selected as the feature. Through dimension reduction, 12000 sets of three-dimensional feature sets were obtained, including bearing roller wearing, bearing inner race wearing, bearing outer race wearing and centrifugal pump impeller wearing. There are also 2400 feature sets under normal conditions. The distribution of feature sets in three-dimensional space is shown in Fig. 3. Notably, feature sets corresponding to different states can be clearly distinguished.

Fig. 3Distribution of feature sets

SVM is chosen as classifier in this experiment. The feature set is divided into 10000 training sets and 2000 test sets, in which the training set is used to train the SVM network. After the training, we input the test sets data into classifier. Comparing the predicted values of the model output with the true values, we can get a prediction accuracy of 93.8 % for the network. The forecast result is shown in Fig. 4.

Fig. 4The forecast result of SVM