The correlation analysis of gear tooth broken-pitting compound fault and single fault based on Laplacian Eigenmaps

2.1. Principle of Laplacian eigenmaps algorithm

Laplacian eigenmaps algorithm can be relatively intuitive to achieve dimensionality reduction. The principle is that the high-dimensional data in the space after the very close to the point of mapping to low-dimensional space can also maintain a close distance from each other, and the Laplacian-Belatrami is used to construct the objective function corresponding to low-dimensional embedding, so as to achieve the goal of optimal low-dimensional embedding of high-dimensional data.

2.2. Laplacian eigenmaps algorithm

The LE, as a manifold learning algorithm that maintains local features, is mainly used to deal with data classification problems. It can be used to redraw high-dimensional images in low dimensional space and keep the essential features as much as possible. The Laplacian Eigenmaps core: if the very close distance $d_{ij}$ between the two sample points $x_i$and $x_j$ is less than the threshold of a certain set, you can conclude that the two points have edges in the neighborhood graph, and the simplified method is that gives it the value of weight is one, that is, $w_{ij}=$ 1, or $w_{ij}=$ 0. There are some basic steps of LE algorithm:

1) Selecting the number of neighbourhood points to construct the neighbourhood image $G$. It is necessary to obtain the Euclidean distance between any sample point and the other point, assuming that the Euclidean distance $d_{ij}$ between the sample point $x_i$ and the sample point $x_j$ is less than the threshold of a certain set, which can define the sample point $x_j$ is the nearest neighbor of the sample point $x_i$, or in turn, the sample point $x_i$ is the nearest neighbor of the sample point $x_j$, next, both $x_i$ and $x_j$ have edges in Fig. 1.

2) Given weight $w$. If the $x_i$ and $x_j$ is neighbor, so we can define $w_{ij}=e^{-{‖x_i-x_j‖}^2/t^2}$, or $w_{ij}=$ 0.

3) Calculate low-dimensional embedding $Y$. For $Y$, there is:

where $L$ is Laplacian matrix, $L=D-W$, $D$ is the sum of the rows or columns of the value of weight in diagonal matrix, so $D_{ij}={\sum }_j{w}_{ij}$. Add the constraint $YD{Y}^T=$ 1 to eliminate the effect of the scale factor. The problem of LE algorithm is transformed into optimization problem:

AS result, the low-dimensional embedding $Y$ is the matrix made of eigenvector $v_1\dots v_{d+1}$ corresponding to the smallest $d+1$ eigenvalues:

3.1. Based on ADAMS dynamic model of gear tooth broken-pitting

It is a key point to calculate the meshing stiffness of gears when it builds dynamic model of gear transmission system in ADAMS [16]. Many scholars have a deep research [17] and they all considered that it can effectively simulate the gear bending and torsional deformation through adding auxiliary gear and torsional spring [18-21].

As shown in Fig. 1, the dynamic model of gear transmission system was built with normal state in ADAMS [22].

Fig. 1Dynamic model of gear transmission system

The gear transmission system was designed for double-stage driving. It concludes three shafts which contain input shaft, intermediate shaft, output shaft, and two pairs of gears where gear 1 meshed with 2 and gear 3 meshed with 4.

It will generate gear bending, torsional, and contact deformation in the mesh movement, and the contact deformation of the tooth surface can cause the change of meshing stiffness and damping. Then, based on the dynamic model of the gear transmission system in Fig. 1, the fault geometry is implanted in the solid model of the gear transmission system, which can effectively simulate the dynamic effects of gear tooth breakage and pitting failure through the contact algorithm in ADAMS software. It established dynamics model of single fault to broken teeth, pitting, and broken teeth pitting composite fault through changing the geometry in the three-dimensional entity model. Specifically, as shown in Fig. 2, four different degrees of broken gear failure include 1/5 broken teeth, 2/5 broken teeth, 3/5 broken teeth, and 4/5 broken teeth; four different kinds of gear pitting failure include pitting 1 which is to form a circle with a diameter of 4 mm, pitting 2 which is based on pitting 1 to form a diameter of 5 mm circle, pitting 3 which is based on pitting 2 more than a diameter of 6 mm circle, and pitting 4 which is in pitting 3 on the basis of more than a diameter of 7 mm circle. The composite fault is obtained by combining the broken tooth with the pitting failure, composite failure 1 is the combination of the broken tooth 1 and the pitting corrosion 1, and then the compound fault of different degrees can be obtained by analogy.

3.2. Time domain and frequency domain analysis

The simulation conditions are set up on the basis of establishing the compound fault dynamic model, both the driver and load are set to the step function, step (time, 0, 0, 0.2, driver and load), That is, 0 to 0.2 s is the speed and load loading process, after 0.2 s to achieve stability, the simulation time is 1 s. The step number is set to 16384, which means the sampling frequency is 16384 Hz. In the case of the same load and speed that the motor speed is 30 Hz and load is 1790 N·m, the time-frequency maps of the different-degree tooth-pitting compound faults are shown in Fig. 3-6.

Fig. 2Dynamics model of the gear broken teeth-pitting compound fault

a) Composite fault 1 kinetic model

b) Composite fault 2 kinetic model

c) Composite fault 3 kinetic model

d) Composite fault 4 kinetic model

Fig. 3Compound fault 1 in time domain and frequency domain

a)

b)

Fig. 4Compound fault 2 in time domain and frequency domain

a)

b)

As can be seen from the time domain diagram, the vibration signal of the gear has obvious periodicity. In the gear meshing process, there will be relatively large peak value, which shows that there is a shock component in the meshing process. With the deepening of the broken tooth-pitting compound, the amplitude of the signal is also significantly increased, indicating that the impact of the simulation signal has become more serious.

Fig. 5Compound fault 3 in time domain and frequency domain

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b)

Fig. 6Compound fault 4 in time domain and frequency domain

a)

b)

4.1. Fault data processing

On the basis of the above-mentioned data of broken tooth-pitting compound fault and single fault simulation data, the manifold learning method (LE) is used to reduce the dimension of the data, obtain the low-dimensional eigenvalue, and then fault cluster to get the fault circle. The low-dimensional distribution between the compound fault and the single fault is obtained, and the change of the overlapping area between the fault circle of the compound fault and the single fault and the change of the center of the fault clustering are observed. Repeat the above steps, you can get varying degrees of fault circle overlapping area and the center of the change, and then analyze the situation at different speeds, the final comparative analysis, get the corresponding results.

4.2. The correlation analysis of gear tooth broken-pitting compound fault and tooth broken single fault

At the simulation speed of 10 Hz, 20 Hz, 30 Hz, according to the above steps to deal with broken teeth-pitting compound fault and broken single fault simulation data, the results shown below, in which the center of each figure 1, 2, 3, 4 correspond to the four severities of the fault.

The low-dimensional distribution of the vibration data analysis results of the broken tooth-pitting compound fault and the single-fault simulation of the broken tooth is shown in Fig. 7(a)when the simulation input speed is 10 Hz. It is obvious from Fig. 7(a) that the compound fault has a significant change with the fault level, and the fault circle is relatively concentrated for the tooth broken fault, and the fault feature is obvious, which is consistent with the fact that this is due to the single fault of the broken tooth only tooth area changes, the extraction of the fault feature value of the degree of little impact.

In the simulation speed of 20 Hz, the broken tooth-pitting compound fault and broken single fault simulation vibration data analysis results of the low-dimensional distribution summary map, as shown in Fig. 7(b). It can be seen from Fig. 7(b) that the overlapping area of the fault circle has obvious change, and a tendency to become smaller until there is no overlapping area with the increase of the fault degree. It is shown that the degree of correlation between the single fault and the compound fault is weakened, that is, the degree of influence of the single fault on the compound fault becomes smaller, which is similar to the previous conclusion.

Fig. 7The Summary of low-dimensional distribution of compound faults and single faults under different fault levels when the simulation input speed is 10 Hz, 20 Hz, 30 Hz

a) Rotate speed is 10 Hz

b) Rotate speed is 20 Hz

c) Rotate speed is 30 Hz

It is obvious that the curve curvature at 20 Hz is greater than the curvature of the curve when the speed is 10 Hz, and the center of the fault circle changes more in comparison with the change of the center line of the compound fault and single fault between Fig. 7(a) and Fig. 7(b), that is, changes in fault characteristics are more obvious. When the speed increases, the impact of the gear fault on the gear is significantly increased, the energy generated also increased, the extracted fault feature value changes more obvious.

When the simulation input speed is 30 Hz, the low-dimensional distribution of the vibration analysis data of the broken tooth-pitting compound fault and the single fault of the broken tooth is shown in Fig. 7(c). From the figure we can see that in the fault of a single fault, the fault circle is relatively concentrated, the fault feature is more obvious, which is the previous two kinds of speed to get the conclusion is consistent.

It is obvious that the curve curvature at 20 Hz is greater than the curvature of the curve at 10 Hz, but less than when the rotate speed is 30 Hz, and the change of the single fault is more obvious, because as the speed increases, the impact of the gear fault on the gear increases, resulting in increased energy, extracted the fault feature value is more obvious, so the center of fault circle changes more obvious, that curve curvature changes are more obvious.

In general: with the deepening of the fault, the overlapping area between the fault circle of the broken tooth-pitting compound fault and the single fault is gradually becoming smaller, and the degree of association between the faults is gradually weakened, compound faults and broken tooth faults can achieve good recognition results in the middle and late stages. With the increase of the rotate speed, the change of the center line of the fault circle with different degree of broken tooth-pitting compound fault and that of single fault is more obvious, that is, the curvature of the curve increases gradually, and the change of fault characteristic becomes more obvious. The change of the whole fault characteristic is more obvious for the single fault of the broken teeth, especially in the early stage of the fault and the later, the fault feature is more obvious. Finally, combined with eigenvalue of fault, as long as you know eigenvalue of fault and where the fault circle in the area and the distances of center of fault circle, we can estimate the compound fault or a single fault and fault degree, so as to provide a new method for the diagnosis and prediction of compound fault and single fault.

4.3. The correlation analysis of gear tooth broken-pitting compound fault and pitting single fault

In the simulation speed of 10 Hz, 20 Hz, 30 Hz, according to the above steps to deal with tooth-pitting compound fault and pitting single fault simulation data, the results shown below, in which the center of each figure 1, 2, 3, 4 correspond to the four severities of the fault.

Fig. 8The Summary of low-dimensional distribution of compound faults and pitting single faults under different fault levels when the simulation input speed is 10 Hz, 20 Hz, 30 Hz

a) Rotate speed is 10 Hz

b) Rotate speed is 20 Hz

c) Rotate speed is 30 Hz

Fig. 8. The Summary of low-dimensional distribution of compound faults and pitting single faults under different fault levels when the simulation input speed is 10 Hz, 20 Hz, 30 Hz.

In the simulation of 10 Hz, the low-dimensional distribution of the analysis results of the vibration data of the broken tooth-pitting compound fault and pitting single fault simulation is shown in Fig. 8(a). It can be seen from Fig. 8(a) that the broken teeth-pitting compound faults and pitting faults all deepen with the degree of fault, the overlapping area of the fault circle has obvious change, from the presence of overlap to the last overlapping area is zero. The fault circle area is similar to the overall variation of the faulty circle area of the compound fault for the pitting failure, the fault circle of single fault is more concentrated than the compound fault. It shows that the pitting failure and the compound fault are related greatly, that is to say, pitting has a great influence on the compound faults. Due to the change of pitting failure degree is the depth and quantity of pitting holes. The fault characteristic value of pitting is larger than that of tooth broken single fault.

In the simulation speed of 20 Hz, the low dimensional distribution of the compound vibration and pitting failure simulation vibration data analysis results are shown in Fig. 8(b). It can be seen from Fig. 8(b) that with the increase of the fault degree, the overlapping area of the fault circle has obvious change, and the overlapping area tends to become smaller gradually until there is no overlapping area, it shows that the degree of association between pitting single fault and compound fault is weakened, that is, the effect of pitting single fault on compound fault becomes smaller. Compared with the fault circle of tooth broken single fault, the overall variation of the faulty circular area of the pitting single fault is obvious, which is similar to the change of the compound fault and it consistent with the conclusion when the speed is 10 Hz.

Compared with the fault center curve of the compound fault and pitting single fault in Fig. 8(a) and Fig. 8(b), it can be seen that the curvature change of the compound fault curve is increased, which is consistent with the previous analysis. The curve curvature at 20 Hz and the curve curvature change at 10 Hz are not obvious for pitting a single fault, indicating that the effect of speed on the pitting is not obvious.

When the simulation input speed is 30 Hz, the low-dimensional distribution of the analysis results of the vibration data of the broken tooth-pitting compound fault and pitting single fault simulation is shown in Fig. 8(c). It can be seen from Fig. 8(c) that the variation of the fault circle area of the pitting single fault is similar to that of the compound fault. In addition to pitting single fault 1, the curvature change of the compound fault curve is consistent with the previous analysis in comparison with the change of the center of the fault in the compound fault and pitting single fault in Fig. 8(a), Fig. 8(b) and Fig. 8(c). The curvature changes of the curves among the three different speeds are basically the same for pitting single fault, and there is no obvious change.

In general, with the deepening of the degree of fault, the overlapping area of the fault circle of tooth broken-pitting compound fault and pitting single fault decrease gradually to zero, the correlation degree between them gradually become weak, and can well identify fault in late fault. With the increase of the speed, the curvature of the center line of the fault circle increases gradually, and the change of the fault characteristics is more obvious. The curvature of the center line of fault circle is not obvious for pitting failure, but in the early stage of the fault, the single fault feature of pitting is more obvious than the compound fault. Finally, the combination of eigenvalue of fault, as long as you know eigenvalue of fault and where the fault circle in the area and the distances of center of fault circle, you can determine whether the compound fault or a single fault and the degree of failure.