An early fault feature extraction method for rolling bearings based on variational mode decomposition and random decrement technique

2.1. Variational Mode Decomposition

According to the Reference [11], we get the complete algorithm for VMD, summarized in following algorithm.

Initialize$\left\{{\widehat{u}}_k^1\right\}$, $\left\{{\widehat{\omega }}_k^1\right\}$, $\left\{{\widehat{\lambda }}^1\right\}$, $n\leftarrow 0$

Repeat $n\leftarrow n+1$

for$k=1:K$do

Update${\widehat{u}}_k^{n+1}$for all$w\ge 0$:

Update${\omega }_k$:

end for

Dual ascent for all$\omega \ge 0$:

until convergence:

The reconfiguration signal can be expressed as:

2.2. Random decrement technique

RDT can be used to describe the impulse response of the system, and its advantage is that it can extract free impact response from the stationary random response in the system. The core is to assume that a system is subjected to stationary random excitation, and the response is the superposition of deterministic response determined by the initial condition and random response determined by the initial external load. Under the same initial conditions, stationary random response is divided into several sections, and the general mean of the intercept segments is calculated, so as to extract the free attenuation response.

The response signal is divided into $L$ segments, and each segment of the signal is expressed as $x_i\left(t\right)$, whose length is $\tau$. Each response signal has the same trigger value, and the trigger value can be expressed as:

The overall mean of the $L$ segment signal is calculated, and the random decrement function can be expressed as:

where $x_i\left(t_i\right)=x_s,i=\mathrm{1,2},\cdots ,L$.

2.3. Proposed method

The variational mode decomposition is used to decompose the collected vibration signals, and the component with the larger correlation coefficient is selected as the fault component. Then the fault component is processed by random decrement technology, and the Hilbert envelope spectrum of the fault component is made. According to the proposed method, the early fault characteristic of rolling bearing outer ring is extracted.

3.1. Experiment condition

The experimental system is shown in Figure 1, including vibration sensor, coupling, driving motor, torque decoder/encoder and dynamometer. The type of the test bearing is SKF 6205-2RS, and the technical parameters of the bearing are shown in Table 1.

Fig. 1The experimental system

Electric spark is used to process faults on bearing outer ring. The fault of diameter 0.18 mm and depth 0.28 mm is used to simulate the slight faults of outer ring. The speed of the driving motor is 1750 r/min, the sampling frequency is 12000 Hz, and the number of sampling points is 32768. Through theoretical calculation, the rotation frequency of the bearing can be obtained $f_r=$ 29.16 Hz; the fault characteristic frequency of the bearing outer ring is $f_{Oi}=$ 104.3Hz.

3.2. Experimental data processing

The time-domain and frequency domain waveform of the early fault vibration signal of the rolling bearing outer ring is shown in Fig. 2. It can be seen from Fig. 2 that the vibration signal has obvious impact components and there is a lot of background noise. The spectrum components are complex and can not extract the rotation frequency (29.16 Hz) and outer ring fault characteristic frequency (104.3 Hz).

Fig. 2The time-domain and frequency domain waveform of the early fault vibration signal of the rolling bearing outer ring

a) Time domain waveform

b) Frequency domain waveform

In order to highlight the fault features, the vibration signal is processed by the method proposed in this paper. The number of decomposed modes $K$ is 4. The vibration signal is decomposed by VMD, and the component U2 and U3 with the largest correlation coefficient of vibration signal are selected as fault components. Then the fault component is processed by random decrement method and its Hilbert envelope spectrum is calculated, as shown in Fig. 3. It can be seen from Fig. 3 that the fault component of rolling bearing outer ring contains many characteristic frequencies, including 2 times the frequency 58.6 Hz, the outer ring fault characteristic frequency 100.1 Hz (the reason which is not exactly consistent with the theoretical value 104.3 Hz is the effect of the random sliding of the rolling element) and its 2 frequency doubling 200.2 Hz. In addition, there are also modulation sidebands, such as 158.7 Hz ($\text{2}{f}_r+f_{Oi}=$ 158.7 Hz), 258.8 Hz ($\text{2}{f}_r+\text{2}{f}_{Oi}=$ 258.8 Hz), 358.9 Hz ($\text{2}{f}_r+\text{3}{f}_{Oi}=$ 358.8 Hz), and 459 Hz ($\text{2}{f}_r+\text{4}{f}_{Oi}=$ 459 Hz). Therefore, the proposed method successfully extracted the characteristic frequency of the early fault of the outer ring of the rolling bearing.

Fig. 3The Hilbert envelope spectrum of the fault component

a) U2

b) U3