Application of K-Core analysis based on complex network community structure to rolling bearing fault diagnosis

A complex network is a method to analyze nonlinear time series on multiple temporal and spatial scales. K-core analysis of several typical nonlinear series was studied, and then based on this, a fast algorithm method was employed to divide the community structure. According to the definition of k-core, the rolling bearings complex network was divided into different cores at the router level, and the main characteristic quantities such as degree distribution and clustering coefficient of every k-core can be analyzed. The results show that there is a corresponding relationship between the characteristic parameters and the community structure of the bearing. The analysis results show that compared with the traditional method, the method of complex network community structure and k-core analysis has a better diagnosis effect.


1.
Introduction Rolling bearings have a wide range of applications in rotating machinery, and bearing failure is also an important cause of mechanical failure.Therefore, the diagnostic analysis of rolling bearing faults is of great practical importance.Rolling bearing fault diagnosis focuses on the inner ring, outer ring, and rolling elements [1][2][3] .The traditional diagnostic method is resonance demodulation analysis of bearing fault vibration signals [4] .However, the collected vibration signals usually contain many useless noise signals, which seriously affects the extraction of fault characteristics of the vibration signals.In recent years, many scholars have carried out diagnostic analyses of rolling bearing faults using wavelet transform, but due to limitations such as single diagnostic conditions, it lacks operability in reality [5][6][7][8][9] .
The essence of fault diagnosis is the identification of patterns in the state of the diagnosed object.The time series of the output of a chaotic system identifies its dynamical equations and parameters which is an inverse problem to the study of dynamics.In the traditional case, system identification involves estimating the parameters of a system when the structure of its equations is known for a single non-linear or chaotic system.However, the equations of the identified system are not the true dynamical equations of the original system and it is difficult to characterize the dynamics of complex systems [10][11] .In the event of a bearing failure, partially damaged components generate high-frequency vibrations which will excite the intrinsic frequency of the bearing, and the amplitude of the highfrequency vibrations is also modulated by the impulse excitation force.As a result, the vibration signal of the bearing is non-linear and non-smooth.Traditional methods make it difficult to reveal their kinetic characteristics and cannot highlight their essence.Prof. Hao Berlin proposed that the connection between kinematic orbits and formal languages could be established through symbolic dynamics, and the complexity could be portrayed with the help of syntactic complexity theory, which is conducive to highlighting the essential features of dynamical systems.Therefore, the use of complex networks to study complex non-linear dynamical systems has received widespread attention.
This paper combines complex network community structure and k-core analysis to make further research into chaotic sequences of rolling bearing fault vibration signals [12][13][14][15][16][17][18] .Research shows that efficient resolution of complex networks of bearings can make the fault information be identified quickly.
Remark 1.The k-core analysis is an effective way to reduce the complexity of a system, revealing the nature and hierarchy of its structure.Compared to the degree distribution, k-core analysis is more capable of describing the internal connectivity relationships between nodes in a network while reducing the complexity of the network.By using k-core analysis, accurate diagnostic analysis of bearing faults can be conducted based on characteristics such as degree distribution and clustering.
Remark 2. The k-core analysis allows us to determine the k-core decomposition in the network: the nodes in the network are classified into different core subsets according to their k-core values, which are stable and hierarchical and can be used to identify the community structure and important nodes in the network.

Acquisition of data
The equipment in Figure 1 is a test stand for collecting bearing vibration signals, type SKF6205, including a motor, acceleration sensor, control unit, signal collector, etc.The collected vibration signals include signals from the normal operating state of the bearing, as well as signals from the bearing's fault states in the inner ring, rolling elements, and outer ring.Among them, the fault damage is a single damage caused by electrical discharge machining, with a diameter of 0.178mm and a depth of 0.279 mm.The test bearing is connected directly to a motor with a load of 0 to 3 hp and a speed of 1797 r/min.The vibration data collection uses an accelerometer sensor, which is vertically fixed above the induction motor.The sampling frequency of the vibration signals is 12,000 Hz, 10 groups of each fault signal are collected, and 10,000 points are sampled.

Vibration mechanism of rolling bearings
Factors influencing the vibration mechanism: internal factors are its structural characteristics, processing and assembly faults that occur during operation, etc. External factors are the movement of other components on the drive shaft and the action of forces, etc.When the drive shaft is running at a certain speed under a certain load, it generates excitation to the vibration system consisting of the bearing, housing, and shell, thus causing the system to vibrate, its vibration generation mechanism is shown in Figure 2.

Newman's fast algorithm
The GN algorithm, although more accurate, has a large complexity and has limitations for large complex networks.Therefore, Newman proposes a faster algorithm based on the GN algorithm [19] .This fast algorithm is a cohesive algorithm based on the idea of a greedy algorithm, which is as follows: (1) The network contains n communities, which are initialized such that each node is a separate community, and the initial ij e and i a satisfy the following conditions: Nodes i and j are connected by edges 1/2 0 Other i k is the degree of node i; m is the total number of connected edges in the network.
(2) Merge the communities with edges connected in turn and calculate the increment of the merged module degree.
Qee a a ea a Χ < * , <, (3) (3) Repeat step 2. and keep merging communities so that the whole complex network is merged into one community, this process is performed at most n-1 times.
At the end of this algorithm, different community structures can be obtained by choosing to disconnect at different locations.Among these community structures, the best quality network community structure is obtained by selecting a Q value with the largest local modularity.Figure 3 shows the community structure of the Zachary network obtained from the analytical construction using Newman's fast algorithm.

Community structural analysis of complex network for bearing vibrations
According to the Newman fast algorithm, the community structure of the bearing vibration is plotted using UCINET software and NETDRAW software, as shown in Figures 4 and 5.

k-core analysis of complex networks of bearing vibrations
This paper selects the degree distribution of nodes in some cores for fitting, as shown in Figure 6.The horizontal axis represents the degree value d of the nodes, and the vertical axis represents the number of nodes n with a degree value of d.It can be seen that in the normal operating state and the fault state of the bearing, different types of vibration signals are generated, which leads to an irregular distribution of degrees.Therefore, the degree distribution of each core cannot be fitted to a straight line.However, there is a certain similarity in the degree distribution of each core: not all nodes with large degrees have a higher core number.Some nodes have high degrees but belong to smaller cores.This characteristic is consistent with the definition of the k-core and indicates that the degree distribution of each core node is not affected by the decomposition [20] .Figure 7 is a distribution plot of the local clustering coefficients for some cores in the complex network of bearings.The horizontal axis represents the degree value d of nodes in each core, and the vertical axis represents the local clustering coefficient C(d) for the nodes with a degree value of d in each core.It can be seen from the figure, that as the degree value d of the core increases, the size of the local clustering coefficient C(d) varies differently, generally showing a decreasing trend.The clustering coefficients are larger for normal operating conditions of the bearings and smaller for fault conditions.There is some similarity between the cores: each core has more points in the normal state than in the fault state.This is because the clustering is better in the normal state than in the faulty state, which is consistent with the community structure in Figure 5.In the inner ring failure, with the rotation of the bearing, the location of the damage point in the inner ring is also changing, the load borne is not uniform, and the degree value of the k-core is overall larger than that of the normal operating condition, while the clustering coefficient is relatively small.However, due to the uneven loads applied, the internal community structure is more complex than the normal operating condition, which is consistent with the distribution trend of Community B in Figure 5 and Figure 4 (b).
In rolling body failures, the damage point is in direct contact with the inner ring, and the impulse force propagated by the inner ring acts on the outer ring through the rolling body, with some energy loss in the propagation process.Therefore, the k-core presents a larger degree value and a smaller clustering coefficient, which is consistent with the distribution trend of Figure 5 Community B and Figure 4 (b).
In the outer ring failure, the static load density distributed in the outer ring is constant, and the corresponding k-core degree value is larger than the inner ring failure degree value and the rolling body failure degree value, but the clustering coefficient is smaller, which is consistent with the trend of the distribution of Community D in Figure 5.However, the vibration signals are also more complex because the sensor is located where the radial load density is the highest, which is in line with the distribution trend as shown in Figure 4 (d).

Conclusion
Faults in bearings can be accurately diagnosed through complex networks, whereas traditional diagnostic methods such as wavelet transforms have many inconveniences in practice.
Remark 3: The wavelet transform method needs to know the information about the structural parameters of the bearings, the rotational speed of the machine, and the theoretical formula for calculating the characteristic frequency before it is applied; moreover, the rotational speed of the machine is changing during operation, and the characteristic frequency related to the rotational speed will also fluctuate to a certain extent or even jump to a larger extent.
Remark 4: The wavelet transform method has a single diagnostic condition and a unique rule, which cannot be diagnosed if the test information is incomplete or missing.
1.There is a certain correspondence between the community structure and the k-core for several states of the bearing: The network complexity is relatively low in a normal operational state, corresponding to smaller degree values in each core of the k-core, larger clustering coefficients, and a more tightly connected network.
2. For the faulty state, because the faulty surface hits other parts in the vibration case, which makes the vibration signal fluctuates to a certain extent, thus affecting the whole clustering nature.In the faulty case, the degree values are larger, and the clustering coefficients in the k-core are smaller.
3. Each node in the network corresponds to a state of vibration signals, which also respond to different statistical properties in each core.A certain hierarchical division of the whole network can provide a more intuitive understanding of the network structure inside the bearing.

Figure 2 .
Figure 2. The mechanism of vibration generation in rolling bearings.

Figure 3 .
Figure 3. Newman fast algorithmic analysis of results in Zachary networks.

Figure 4 Figure 4 .
Figure 4 illustrates the complex network community structure of four states of bearings, and each state contains 42 nodes.In this case, the line between two nodes indicates a transition from one vibration signal to another, with fluctuating modes occurring during the transition, resulting in different vibration states.

Figure 5 .
Figure 5. Complex network community structure for bearings.