A new weak fault diagnosis method based on multi-scale wavelet noise tuning cascaded multi-stable stochastic resonance

3.1. Discrete wavelet transform (DWT)

Wavelet transform has good localization ability, and can extract weak signal components from signal by multi-resolution decomposition [30].

A wavelet basic function $\phi \left(t\right)$ is given as:

where ${\phi }_{a,b}\left(t\right)$ is obtained by the dilation and translation of $\phi \left(t\right)$, $a$ is dilation factor and $b$ is frequency shift factor. Suppose $x\left(t\right)\in L^2\left(R\right)$, $x\left(t\right)$ can be transformed into discrete wavelet by wavelet bases function $\phi \left(t\right)$. The wavelet transform coefficient is shown as:

where $a$, $b$ and $t$ are continuous variables. Thus Eq. (9) is continuous wavelet transform.

Discrete wavelet function can be obtained by discreting the parameter of continuous wavelet function [31]. First, taking $a=a_1^j\text{,}$$a_1>0\text{,}$$j\in Z\text{,}$ the correspond wavelet function is $a_1^{-j/2}\phi (a_1^{-j}t-a_1^{-j}b)$, $j\in Z$. After taking $b=a_1^{j}k{b}_1$, then the intervals of time axis is $a_1^j{b}_1$. Usually, discrete wavelet is binary discrete. When $a_1=\text{2}$, and we can get $a=2^j$, $b=2^{j}k{b}_1$, where $j\in Z$ and $k\in Z$. Then the correspond wavelet function is $2^{-j/2}\phi (2^{-j/2}t-k{b}_1)$, where $j\in Z$. Further $b_1$ is normalized, we can get the following function shown as:

Transform coefficient can be written as:

3.2. Multi-scale wavelet decomposition and reconstruction

Binary orthogonal discrete wavelet transform decomposes signal into uncorrelated wavelet coefficients, and divides the wide band signal into a plurality of narrow bands. The schematic diagram of decomposition is shown in Fig. 2.

The decomposed wavelet coefficients include approximate coefficients $c_J\left(k\right)$ and detail coefficients $d_j\left(k\right)$ written in the form [32]:

where $⟨\cdot ,\cdot ⟩$ represents an inner product operator, $j$ is decomposition level, $J$ is cut-off decomposition level. The functions ${\phi }_{j,k}\left(t\right)$ and ${\psi }_{j,k}\left(t\right)$ are expressed respectively as:

where $\phi \left(t\right)$ is the base functions of wavelet decomposition, ${\phi }_{j,k}\left(t\right)$ is obtained by the dilation and translation of $\phi \left(t\right)$, $\psi \left(t\right)$ is scaling function, ${\psi }_{j,k}\left(t\right)$ is obtained by the dilation and translation of $\psi \left(t\right)$.

The inverse transform of DWT can be realized by the following function:

Adjusting the size of different scale signal’s amplitudes and reconstructing the signal, the signal can be expressed as:

where $K_i$ is scale contraction factor, and it can adjust amplitude of each scale component. In order to improve SNR, we usually turn the approximate coefficients of scale contraction factor to higher, while turn the detail coefficients of scale contraction factor to lower.

Fig. 2Using DWT to decompose the signal x(t)

4.1. Multi-scale multi-stable stochastic resonance method

Wavelet transform has good localization properties in time-frequency. The signal can be decomposed into different scale frequency components by multi-scale wavelet decomposition. Due to the strong selection characteristics of noise intensity and frequency in nonlinear multi-stable system, each scale signal need to be adjusted by scaling factor. Then the decomposed signal is used as the input of the cascaded multi-stable systems. When the signal is processed by multi-scale wavelet decomposition, the noise intensity of input signal is reduced after decomposition. The schematic of multi-scale cascaded multi-stable SR system is shown in Fig. 3.

Fig. 3The schematic of multi-scale SR system

Firstly, the signal is reorganized by the approximate signal with each scale noise. The noise is equivalent to noise reduction signal. Then, it is used as the new input signal of each cascaded multi-stable system to optimize SR effect of each unit. And the output signal of entire system is the average value of output of all units [33]. In addition, we select the decomposition scale by the traversing method.

4.2. Multi-scale cascaded multi-stable SR system

We proposed a new SR system named multi-scale cascaded multi-stable SR system (MCMSR) shown in Fig. 4. The MCMSR model comprises a plurality of multi-stable SR systems. The output signal of first multi-stable system is used as the input signal of the second multi-stable system. The output signal of the second multi-stable system is used as the input signal of the third multi-stable system, and so on. At last, the output of the last level multi-stable SR system is the output of entire MCMSR model.

SNR is used to judge whether the system produces stochastic resonance. The calculation formula of SNR is written in the form:

where $A_d$ is the amplitude value corresponding to the driving signal frequency, and $A_n$ is the sum of all the amplitude values except $A_d$ in the amplitude spectrum [34].

Fig. 4Multi-scale cascaded multi-stable SR system

4.3. Simulation experimental verification

In order to conform to the adiabatic approximation theory, we select the small parameters signal. The input signal $s\left(t\right)$ is $A\mathrm{s}\mathrm{i}\mathrm{n}\left(2\pi f_{0}t\right)$. The amplitude of signal is equal to 0.3 and the frequency of signal $f_0$ is equal to 0.01 Hz. Let $u\left(t\right)=s\left(t\right)+\eta \left(t\right)$, in which $u\left(t\right)$ is noisy signal, $\eta \left(t\right)$ is Gaussian white noise and its strength is 0.6. The system parameters are $b$, $c$ and $d$, the value of each are 0.72, –0.3 and 0.08, respectively.

The waveform and spectrum of a noisy sinusoidal signal are shown in Fig. 5. Sampling frequency $f_s$ is 5 Hz. It can be observed in the frequency domain in Fig. 5, where there is a uniform distribution, and the white noise and frequency has a minor peak nearby 0.01 Hz, the amplitude is 0.3212.

Fig. 6(a) shows decomposition result of the processed signal after DWT. The db1 wavelet is used for decomposition and the scale is set to $J=\text{4}$. The traversing method is used to select decomposition scale. The decomposition scale is set as $j=$1, $j=$2, $j=$3, $j=$4, $j=$5, $j=$6, and then the best decompose effect value of j is found. It can be seen from Fig. 6(a), $s$ is the original signal, $a_4$ is the signal’s low frequency part, $d_1$ to $d_4$ are the signal’s high frequency parts. Fig. 6(b) is spectrum after Fast Fourier Transform (FFT) of the original and the reconstructed signal of approximation signal and each scale noise signal. The size of each scale signal components of reconstructed signal is to be adjusted by the scale factor. From Fig. 6(b) we can find that there are distinct peaks at 0.01 Hz and the peak values in Fig. 6(bb) to Fig. 6(be) are much higher than that in Fig. 6(ba). In Fig. 6(be) there has the maximum peak value, but the peak value not exceeding 1. At the same time, there are many noise signal components in Fig. 6(bb) to Fig. 6(be). It shows that the result of wavelet decomposition is seriously affected by the noise.

Fig. 5The original signal’s waveform and spectrum

a) Waveform of original signal

b) Spectrum of original signal

Fig. 6The decomposition results of original signal after DWT

a) The time-domain components

b) The spectrum components

After the input signals were processed by stochastic resonance systems in Fig. 6, we could obtain the output signal’s waveform and spectrum shown in Fig. 7. Signal detection results for a single multi-stable SR system and MCMSR system were compared in Fig. 7.

Observing the Fig. 7(a-c), we can find the original signal on the time-domain plot is difficult to identify. When it is processed by a multi-scale stochastic resonance system, we can clearly discern the periodic signal in the time-domain plot, and noise is also decreased. Observing the spectrum in Fig. 7(d-f), it is apparent that peak value at 0.01 Hz has constantly become larger. In the frequency domain of MCMSR system output, the amplitude at $f_0$ is equal to 2.053 which is 4 times than the original signal and 1.35 times than single stochastic resonance system output signal. This indicates that the MCMSR system has better signal enhancement and desorption capability than the multi-stable stochastic resonance system. The input SNR curves of original signal, single SR model and MCMSR model are plotted in Fig. 8 altogether. We can find that the target signal can be amplified by SR effect in the single multi-stable model and the MCMSR model. Especially, the SNR is promoted greatly in the low noise intensity region in the MCMSR model. Therefore, the MCMSR system is very effective on detection signal, and greatly improves the ability to detect weak signals.

Fig. 7Comparison of signal detection between single SR model and MCMSR model

Fig. 8SNR curves of original signal, single SR model and MCMSR model

5.1. The bearing data

The experimental platform of Case Western Reserve University in US is shown in Fig. 9 [35]. The rolling bearing fault vibration signals are collected by acceleration sensors. Motor housing drive end bearing is SKF6205 in the experiment, bearing speed $N=$1797 rpm, and the transfer frequency $f_r$ is 29.95 Hz. Rolling generally consists of an inner ring, an outer ring, rolling elements and cage, if the rolling bearing malfunctions, the fault frequency is predictable. We can assume that outer ring of the rolling bearing is fixed, and the inner ring is rotating with the working shaft. The pitch diameter of the bearing is $D$, the ball diameter of the bearing is $d$, the contact angle is $\alpha$, the number of rolling elements is $n$, then the fault characteristic frequency of inner ring is as follows:

The bearing’s details geometry is provided in Table 1.

Fig. 9The experimental platform

If bearing inner ring happens fault, then the characteristic frequency $f_0$ is equal to 162 Hz. Fig. 10 are the fault signals’ waveform and spectrum maps of bearing inner ring. Fig. 10(a) is the time domain waveform where exists a cyclical shock. Fig. 10(b) is the spectrum map where we cannot see the obvious vibration characteristics in original low frequency spectrum, and there are a lot of noise signal components, that is to say, from the original vibration signal, it is very difficult to diagnose the inner ring fault of rolling bearing.

Fig. 10The waveform and spectrum of actual signal

a) Waveform of actual signal

b) Spectrum of actual signal

5.2. Analysis of rolling bearing faults

In the actual signal detection and fault diagnosis, the case which meets the adiabatic approximation theory is rare. If the frequency of signal cannot occur stochastic resonance under the condition of big argument, the signal can be transformed into a frequency of small parameters; we can use the method of stochastic resonance for weak signal amplification. Based on the above ideas, scale decomposition stochastic resonance method has been studied in detail [36]. The Scale decomposition stochastic resonance is shown in Fig. 11. The specific process is as follows: First, the input signal $s\left(t\right)$ with noise signal $\eta \left(t\right)$ carry out the integration scale decomposition in order to obtain the condition of occurring stochastic resonance. Then output signal $y\left(t\right)$ is obtained by after input signal $s\left(t\right)$ processed by Multi-stable SR system $U\left(x\right)$.

Fig. 11Scale decomposition stochastic resonance

By above analysis, the method of rotating machinery fault diagnosis based on MCMSR is proposed, which is presented in Fig.12. First, we get vibration signal from the rotating machine shown in Fig. 12(a). Second, in this scenario, the spectrum of vibration signal is extracted first by FFT shown in Fig. 12(b). Third, in order to get a better effect of stochastic resonance, the second sampling signal processing through a multi-scale wavelet transform is shown in Fig.1 2(c). In this section, the high frequency energy constantly is shifted toward low-frequency part. Last, the new signal is fed back to the MCMSR model, and the system output signal is processed by Fourier transform. If the spectrum of input signal contains the fault feature frequency, the corresponding periodic component of fault will certainly be enhanced by MCMSR system, which is conducive to us to make the diagnosis of mechanical failure.

Fig. 13 shows the specific process of vibration signal by scale wavelet decomposition. Fig. 13(a) shows the decomposition result of processed signal after DWT. The scale $J$ is 4. In Fig. 13(a), $s$ is the vibration signal, $a_4$ is the signal’s low frequency part, $d_1$ to $d_4$ are the signal’s high frequency parts, and $s=a4+d1+d2+d3+d4$. Approximation signal and each scale noise signal are reconstructed respectively, and Fig. 13(b) is the spectrum of the vibration signal and the reconstructed signal. From Fig. 13 we can see that it is difficult to identify the fault characteristic frequency.

Fig. 12Principle diagram of the proposed method

Fig. 13The decomposition results of actual signal after DWT

a) The time-domain components

b) The spectrum components

Fig. 14The analyzed results of bearing inner race fault using signal SR model and MCMSR model

When the input signals in Fig. 13(b) are processed by stochastic resonance systems, we can obtain the results of output signal. The spectra of output signals by the single multi-stable SR method and MCMSR method are shown in Fig. 14(a) and (b), respectively. The amplitude of frequency domain output signal in Single SR system is 0.06458 shown in Fig. 14(a). In the multi-scale SR system, the amplitude of frequency domain output signal is 0.3316 which is 5.13 times than the single SR system output. The experimental results indicate that the MCMSR can effectively extract fault diagnosis characteristic from heavy background noise. The different output SNR curves of MCMSR model, single SR model and original signal are depicted in Fig. 15 together. It can be seen that the SNR of MCMSR model is higher than single SR model and original signal. At the same time, there is a wider parameter interval to keeping higher SNR for the noise intensity D. From the above analysis, multi-scale SR method has more advantages than the traditional single SR method in enhancing the fault message.

Fig. 15SNR curves of original signal, single SR model and MCMSR model