Rotating machinery fault diagnosis based on improver resonance sparse decomposition

Fault diagnosis of rotating machinery plays an important role for the reliability and safety of modern industrial systems, and it is challenging to detect the weak character signal in the noisy background. This article presents an improved resonance sparse decomposition method for fault diagnosis of rolling bearings. The PSO algorithm is used to optimise the quality factor of the resonance sparse decomposition, which can overcome the deficiency caused by the manual given quality factor in the traditional resonance sparse decomposition method and achieve the effective separation of resonance components. An envelope analysis of the low-resonance decomposition including the major fault components was performed to extract the fault feature frequency. Based on this, the two decomposition methods are applied to the fault diagnosis of the inner and outer rings of the bearing, respectively. The experimental results show that the improved resonance sparse decomposition method can reduce the interference components when extracting the frequency of fault features and improve the accuracy of fault diagnosis.


Introduction
Rotating machinery is a significant part of major equipment. The malfunction of rotating machinery usually causes a decrease in performance. In the worst case, it results in break-down of whole system, or even catastrophic failure with enormous economic losses and safety problems [1]. Condition monitoring and fault diagnosis of machines are gaining more importance [2]. In the early stage of the fault, the vibration signal which influenced by the strong noise and continuous oscillation signal shows non-stationary and non-linear characteristics and it is difficult to identify the transient transient impact signal. How to extract the useful fault impact information in the signal has always been a hot spot and difficulty in the field of fault diagnosis.
In 2011, Selesnick [3] proposed the using of the resonancebased sparse signal decomposition method (RSSD). Considering the effect of empirical mode decomposition on signal decomposition, the resonance-based sparse signal decomposition is based on the improvement of wavelet transform. It is a non-linear signal analysis method based on signal resonance properties. That is, according to the quality factor Q difference of impact signal and harmonic signal, the complex signal is decomposed by TQWT to obtain high-quality factor and low-quality factor, and then passed morphological component analysis (MCA), splitting the signal by the augmented Lagrangian Shrinkage Algorithm (SALSA) to obtain high resonance component with high-quality factors and low resonances with low-quality factors component. Among them, the high-resonance component is composed of sustained oscillation cycle signals and the low-resonance component is composed of non-oscillatory transient impulsions that does not exhibit sustained oscillatory behaviour. Therefore, the transient impact signal and the continuous oscillation periodic signal can be successfully separated according to different Q-factors. Chen [4] introduced it into fault diagnosis of rotor impact grinding, and successfully extracted transient information of fault information. Cui [5] applies parallel base-tracking sparse decomposition to weak gearbox fault diagnosis and successfully extracts fault impact components. Zhang [6] combined ensemble empirical mode decomposition (EEMD) with resonance-based sparse signal decomposition to achieve early fault diagnosis of rolling bearings. However, among the most methods, the high and low Q-factors are manually specified, and it is not guaranteed that the most appropriate Qfactors will be given, which will affect the final decomposition effect. In order to avoid the randomness of the manual parameter selection, the selection of the quality factor needs to be optimised.
The particle swarm optimisation (PSO) algorithm [7,8] is a representative cluster intelligence method proposed by Eberhart and Kennedy of Purdue University in the USA which is inspired by the behavioural patterns of the birds and fish groups. Unlike the genetic algorithm based on the evolutionary idea of 'survival of the fittest and survival of the fittest', the PSO algorithm evolves through the cooperation and sharing of information among individuals in the population, and then finds the optimal solution to the problem.
The PSO algorithm has the advantages of simple principle, fast convergence rate and large search range, and it is suitable for the problem of processing parameter optimisation. Here, the PSO algorithm is applied to the Q-factor selection process of resonancebased sparse signal decomposition. The kurtosis of the lowresonance component is used as the objective function to optimise the Q-factor.

Resonance properties of the signal:
The resonance properties of the signal are defined by the quality factor Q (Q = f c /BW where f c is the signal frequency, BW is bandwidth) [9]. The signal collected by the sensor is mainly composed of three parts: the harmonic components generated by the vibration of the mechanical system itself, the noise components, and the transient impact component caused by the fault. The former two of them have larger quality factors and higher frequency aggregation, the rest have a smaller quality factor, higher time aggregation. The resonance properties of the signal are shown in Fig. 1 transformation, but their oscillation behaviour does not change, that is they have the same quality factor.

Tunable Q-factor wavelet transform:
TQWT is similar to the rational expansion wavelet transform, using discrete Fourier transform to decompose and reconstruct the signal through dualchannel filters, enabling full dispersion and perfect reconstruction. The dual-channel filter bank principle as shown in Figs. 2a and b.
In Fig. 2, H 0 (ω) and H 1 (ω) are low-pass and high-pass filter, respectively, α and β are low-pass and high-scale. Once the scale factor (α, β) is given, the low-pass filter H 0 (ω) and the high-pass filter H 1 (ω) are also uniquely determined. The Q-factor Q is related to the scale factor β. The formula is: In order to ensure the decomposition effect, it is necessary to perform oversampling processing during signal decomposition. TQWT is set to oversampling with redundancy r, and the relationship between r and the scale factor (α, β) is shown in (2). At the same time, the maximum number of L max layers for a given set of (Q, r) can be obtained from (3).

Q-factor optimisation:
In TQWT, after the quality factor Q and the redundancy r are determined, the corresponding filter's scaling factor can be determined, which can be obtained from (1) and (2).
From (4), it can be seen that if Q and r are increased at the same time, the frequency resolution of the filter is improved, and makes L max larger, which increasing the amount of calculation. At the same time, excessive Q and r appear strange signals and affect the decomposition effect. Therefore, how to weigh the filter's resolution and calculation, choose the appropriate Q and r is the key to resonance sparse signal decomposition. When r ≥ 3, TQWT has obtained better localised performance. In order to reduce the amount of calculation, r = 3 is taken in the text. So how to get the most suitable quality factor becomes the key.

Particle swarm optimisation
Particle swarm optimisation (PSO) is inspired by the behavioural patterns of flock and fish groups. Unlike the genetic algorithm based on the evolutionary idea of 'survival of the fittest and survival of the fittest', the PSO algorithm evolves through the collaboration and sharing of information among individuals in a population and then get the optimal solution to the problem. The PSO algorithm is simple in principle, fast in convergence, and large in search range, and is suitable for processing parameter optimisation problems. The PSO algorithm describes the problem's optimisation process as the bird's foraging process. The location of the food is the parameter corresponding to the optimal solution. The PSO algorithm abstracts the solution of each optimisation problem into a bird in the search space. The bird flies at a certain speed. The speed and location of the flight are dynamically adjusted according to the bird's own flying experience and the companion's flying experience.
In the ideal space, birds are abstracted into particles with no mass or volume. Suppose there are N particles, and each particle has two feature quantities: position and velocity. Then, the speed and position of all particles in the space can be indicated According to the objective function of the optimisation problem, each particle has a fitness value and judges the position of the particle according to it. During the flight, the particles know their current position and the best position pbest they find. These two positions are the flight experience of the particle itself. At the same time, each particle knows the best position gbest found in the current entire population, which comes from the experience of the particle's companion. Particles adjust their speed and position based on the following four parameters: (1) current position; (2) current speed; (3) distance between current position and own optimal position; and (4) distance between current position and group optimal position.
In the PSO algorithm, the speed and position of the particles are updated through iterations. During the Kth iteration, each particle is updated as follows: In the formula, i = 1, 2, …, n, n is the total number of particles; d = 1, 2, …, D, D is the dimensionality of the space; v id k is the ddimensional component of particle i velocity at the Kth iteration; x id k is the d-dimensional component of particle i location at the K-th iteration; P id k is the d-dimensional component of the optimal position pbest currently passed by particle i; P gd k is the ddimensional component of the optimal position gbest currently passed by all particles in the population; c 1 , c 2 are the learning factors and make c 1 = c 2 = 2; rand() is a random function between (0,1); ω is the inertia weight factor, which is generally between0.1-0.9.

Resonance sparse signal decomposition based on PSO optimisation
(i) Optimise Q-factor. According to given rolling bearing fault signal, use PSO optimisation algorithm to optimise Q-factor: Using Q 1 and Q 2 as particles, the kurtosis of the low-resonance component as the objective function, the kurtosis value of the low-resonance component corresponding to the different Q-factor as the fitness value, and the particle population by searching for the maximum value of the objective function. Iterative updates are performed to finally obtain the optimised Q 1 * and Q 2 * . (ii) Resonance Sparse Signal Decomposition. Using the optimal Q 1 * and Q 2 * , the sparse decomposition of the fault signal is performed to obtain the low-resonance component that characterises the fault component.
(iii) Spectrum analysis. Extracting rolling bearing fault characteristic frequency.

Experimental verification
The experimental data of the rolling bearing faults of the Case Western Reserve University in the United States is used. The model of the bearing to be analysed is SKF6205. Its size parameters are shown in Table 1(1 in = 2.54 cm). The bearing speed is set to 1797 rpm, so the rotation frequency is 29.95 Hz, the sampling frequency is 12, 000 Hz, and the number of sampling points is 4096. The fault frequencies of the inner ring f IR = 162.19 Hz and outer ring of the rolling bearing f OR = 107.36 Hz are obtained by formula (6) and (7).
The time-domain waveform of the rolling bearing inner ring failure is shown in Fig. 3, and the noise content in the time-domain waveform makes it difficult to extract fault information. Using the PSO algorithm optimise resonance sparse signal decomposition method, the inner ring fault signal in Fig. 3 is decomposed, and the optimised Q-factor is Q 1 = 9.31 and Q 2 = 1.46. The corresponding low-resonance components shown in Fig. 4a, and the corresponding kurtosis value is 8.53. Envelope analysis of the lowresonance components is shown in Fig. 4b. From the figure, the fault frequency f IR and its frequency multiplication can be identified, and the interference components with fewer components can be identified.
Using the traditional resonance sparse decomposition method, the signal in Fig. 3 is decomposed and make Q 1 = 4, Q 2 = 1. The envelope analysis of the low-resonance component obtained by the decomposition is performed. As shown in Fig. 5, the frequency of interference in the spectrum is more than in Fig. 4, and only the frequency of the fault signature can be identified.  The time-domain waveform of the rolling bearing outer ring failure is shown in Fig. 6. From the figure, it is not obvious that the fault impact component is observed, and the signal needs to be further decomposed to extract the effective low-resonance component. Using the PSO algorithm optimise resonance sparse signal decomposition method and the optimised Q-factor is Q 1 = 19.65 and Q 2 = 1.10. The corresponding low-resonance components shown in Fig. 7a, and the corresponding kurtosis value is 17.76. Envelope analysis of the low-resonance components is shown in Fig. 7b. From the figure, the fault frequency f oR and its frequency multiplication 2 f oR , 3 f oR , 4 f oR can be identified and fully demonstrate the effectiveness of the proposed method.
For comparative analysis, we use the traditional resonance sparse decomposition method, the signal in Fig. 6 is decomposed and make Q 1 = 4, Q 2 = 1. The envelope analysis of the lowresonance component obtained by the decomposition is performed. We cannot find the outer ring fault characteristic frequency from

Conclusion
Fault diagnosis for rotating machinery is vital to reducing maintenance costs, operation downtime, and safety hazards. The characteristics of early failure of rolling bearings and the influence of environmental noise make it difficult to extract fault feature information effectively. In view of this problem, this paper presents a diagnosis method resonance sparse signal decomposition based on PSO.
According to given rolling bearing fault signal, use PSO algorithm to optimise Q-factor, the kurtosis of the low-resonance component as the objective function, the kurtosis value of the lowresonance component corresponding to the different Q-factor as the fitness value, and the particle population by searching for the maximum value of the objective function to obtain the optimised Q-factor, then use the optimal Q-factor, the sparse decomposition of the fault signal is performed to obtain the low-resonance component that characterises the fault component, extracting rolling bearing fault characteristic frequency from the spectrum analysis at last. The experimental results show that the improved resonance sparse decomposition method can reduce the interference components when extracting the frequency of fault features and improve the accuracy of fault diagnosis.

Acknowledgments
This work is supported by the National Natural Science Foundation of China (No. 51575055).