Influence of characteristic parameters of signal on fault feature extraction of singular value method

2.1. The basic principle of singular value decomposition

The noise reduction method using singular value decomposition is separable for the vibration energy of the effective signal and the noise signal. For a matrix containing noise signals, first separate it according to the principle of singular value decomposition. Then, the singular value of the effective signal is preserved, and the singular values of the noise signal are all set to 0. Finally reconstructed according to the appropriate order, and then the process of obtaining the effective signal after noise reduction.

According to the above research, the singular value decomposition method has a good effect in the field of fault characteristic signal de-noising and fault identification. In mathematical theory, singular value decomposition [21, 22] (SVD) is a method of orthogonalization calculation of a real matrix ${\mathbf{H}}^{m\times n}$. No matter whether the row and column of the matrix are related, there are two matrices ${\mathbf{U}}^{m\times m}$ and matrices ${\mathbf{V}}^{n\times n}$ such that Eq. (1) holds [23]:

where $\mathbf{S}$ singular value matrix is a real matrix $\mathbf{H}$, $\mathbf{S}=\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}({\sigma }_1,{\sigma }_2,\cdot \cdot \cdot ,{\sigma }_r)$, ${\sigma }_i$ is the singular value of the real matrix $\mathbf{H}$, and make ${\sigma }_1\ge {\sigma }_2\ge \cdot \cdot \cdot \ge {\sigma }_i\ge {\sigma }_r\ge 0$, $r=rank\left(\mathbf{H}\right)$.

Set a set of discrete random time domain signals $\mathbf{X}$, its signal sequence can be represented by $X=(x_1,x_2,\cdot \cdot \cdot ,x_m)$. By using the singular value decomposition method, the noise in the signal $\mathbf{X}$ can be filtered out. The operation process is as follows:

1) According to the characteristics of the random signal $\mathbf{X}$, a real matrix, namely Hankel matrix $\mathbf{H}$ is constructed:

where, $m$ is dimension of the matrix $\mathbf{H}$.

2) Singular value decomposition of matrices $\mathbf{H}$ according to Eq. (2). Orthogonal matrix $\mathbf{S}=\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}({\sigma }_1,{\sigma }_2,\cdot \cdot \cdot ,{\sigma }_r)$, $\mathbf{S}$ is the singular value matrix of the matrix $\mathbf{H}$.

3) Since the Hankel matrix is full rank matrix, the elements ${\sigma }_1$, ${\sigma }_2$,⋅⋅⋅, ${\sigma }_r$ of the singular value matrix $\mathbf{S}$ are sorted. Satisfy the condition: ${\sigma }_1\ge {\sigma }_2\ge \cdot \cdot \cdot \ge {\sigma }_k\gg {\sigma }_{k+1}\ge {\sigma }_{k+2}\ge \cdot \cdot \cdot \ge {\sigma }_r>0$.

4) Find the mutation point $k$ of the singular value in step (3), reserved point $k$ previous singular value $({\sigma }_1,{\sigma }_2,\cdot \cdot \cdot ,{\sigma }_k)$, and set all the singular values after the point $k$ to 0, such as $({\sigma }_{k+1},{\sigma }_{k+2},\cdot \cdot \cdot ,{\sigma }_r)=(\mathrm{0,0},\cdot \cdot \cdot ,0)$.

5) According to formula $d_i={\sigma }_i-{\sigma }_{i+1}$, the difference spectrum of singular values in step (4) is calculated, and the order $i$ corresponding to the maximum value is found from the difference spectrum of singular values. Then, the singular value is reconstructed according to the order $i$, that is, the signal obtained after the reconstruction is the characteristic signal after noise reduction.

2.2. Determine the time delay of the signal sequence

In the process of signal reconstruction, the selection of the delay time parameter $\tau$ is very critical. When the value of $\tau$ is small, it will cause a serious correlation between the elements in the delay time series. It also leads to no difference between the elements, and the trajectory of the phase space is compressed to the main diagonal, so that the noise dominates the phase space of the reconstructed characteristic signal [24, 25]. When $\tau$ value is large, the correlation between elements in the phase space of the reconstructed feature signal is reduced, resulting in serious loss of reconstruction information. Therefore, when reconstructing the effective feature signal, a reasonable delay time value must be selected to ensure the validity of the reconstructed signal.

Suppose a measured time series is $x\left(t\right)$, and the autocorrelation function of the sequence after normalization is shown in Eq. (3):

where $N$ is the length of the sample; $\tau$ is the delay time; $\overline{x}$ is the average value of the sample, $\overline{x}=\frac{1}{N}\sum _{m=1}^{N}x\left(m\right)$.

The research shows that the autocorrelation function is used to select the delay time to make the autocorrelation function value exactly attenuate to the time corresponding to $1/e$, which is the optimal value of the effective phase space eigenvector delay time.

2.3. Determine the embedding dimension of the signal sequence

The Cao’s algorithm, which is the Improved False Nearest Neighbors (IFNN) method, it is used to calculate the embedding dimension of the characteristic signal. The calculation principle as follows [26, 27]:

1) Assume that in an $m$ dimension ‘Embedding Space’, the vector ${\overrightarrow{X}}_{m_{}}\left(i\right)$ time series at the $i$ phase point is shown in Eq. (4):

where $x\left(i\right)$ is the signal corresponding to the $i$ time series; $i$ is the time series number, $i=\mathrm{1,2},3,\cdots ,N-(m-1)\tau$; $\tau$ is the delay time. Calculate the Euclidean distance $R_m\left(i\right)$ from the sequence to the nearest neighbor ${\overrightarrow{X}}_m^{NN}\left(i\right)$, as shown in Eq. (5):

where $‖•‖$ is the $\infty$-norm of the signal sequence.

2) When the $m$ dimensional phase space is extended to the $m+$1 dimension, the same vector sequence at the $i$ phase point is ${\overrightarrow{X}}_{m+1}\left(i\right)$. Calculate the Euclidean distance $R_{m+1}\left(i\right)$ from the sequence to the nearest neighbor ${\overrightarrow{X}}_{m+1}^{NN}\left(i\right)$, as in Eq. (6):

3) Note that ${\overrightarrow{X}}_{m_{}}\left(i\right)$ means the $i$th reconstructed vector with embedding dimension $m$ from Eq. (4). Similar to the idea of the false neighbor method, and we define $a\left(i,m\right)=R_{m+1}\left(i\right)/R_m\left(i\right)$. From the definition of $a\left(i,m\right)$, one can see that the threshold value should be determined by the derivative of the underlying signal, therefore, for different phase points $i$, $a\left(i,m\right)$ should have different threshold values at least in principle. Furthermore, different time series data may have different threshold values. These imply that it is very difficult and even impossible to give an appropriate and reasonable threshold value which is independent of the dimension $m$ and each trajectory’s point, as well as the considered time series data. To avoid the above problem, we instead define the relationship between $E\left(m\right)$ and $m$, as shown in Eqs. (7) and (8):

4) Continue to increase the dimension $m_1$ of the phase space, and make $m_1>m$, and repeat the calculation of the first three steps. According to the result of each calculation, draw a diagram of the relationship between $E_1\left(m_i\right)$ and $m_i$. When a value in the graph tends to be stable within $E_1\left(m_i\right)$ certain threshold, the $m_i$ corresponding to the value is the optimal embedding dimension of the feature signal.

5) However, in the finite sequence of practical applications, it is difficult to discriminate whether the change in $E_1\left(m_i\right)$ value is stable as $m_i$ increases. Therefore, it is necessary to add a criterion, as shown in Eqs. (9) and (10):

Studies have shown that for a random time series, the value of $E_2\left(m\right)$ is always equal to 1; For a deterministic time series, the value of $E_2\left(m\right)$ is not equal to 1 within a certain threshold.

3.1. Relationship between SVD and signal frequency

According to the above analysis of the singular value of the analog signal, it is found that there is a certain correlation between the signal frequency and the singular value. In this section, the coupled signals formed by different frequency segment signals and combined in different ways are simulated and calculated to study the variation law between signal frequency and singular value.

3.1.1. Sampling summation of signal frequencies

The simulated signal $f_{(1,i)}\left(t\right)$ of the sampled signal frequency is composed of two sets of sinusoidal signals of different frequencies and a random noise signal ${\xi }_{G-noise}\left(t\right)$, shown in Eq. (11). There are five groups of frequency ${\omega }_i=\{i=1~5|5.6\pi ,8\pi ,12\pi ,18\pi ,28\pi \}$, and some simulation parameters are set to, $A=$1, $\phi =$0; The simulation parameters of the random white noise signal ${\xi }_{G-noise}\left(t\right)$: the mean is 0, and the variance is 0.7:

11

$f_{\left(1,i\right)}\left(t\right)=\left[{\sum }_{i=1}^{n}A\mathrm{s}\mathrm{i}\mathrm{n}\left({\omega }_{i}t+\varphi \right)\right]+{\xi }_{G-noise}\left(t\right).$

1) Relationship between autocorrelation function and delay time.

According to the calculation principle of the autocorrelation function method, the five sets of similarity signals of Eq. (11) are respectively subjected to autocorrelation simulation calculation, and the results shown in Fig. 1 are obtained. From the relationship between the autocorrelation function of each signal and the maximum delay time in the figure, it is concluded that the more the number of frequencies including the sub-signal in the analog signal, the attenuation of the white noise signal is more significant with the delay time delay. In addition, the delay time values of the five sets of analog signals are large from the figure.

Fig. 1Relationship between autocorrelation function and delay time

2) Relationship between $E\left(m\right)$ and embedded dimension $m$.

According to the analysis results of Fig. 1, the modified Cao’s algorithm is used to simulate the relationship between the $E\left(m\right)$ values of the five sets of analog signals and the embedded dimension $m$, and the results are shown in Fig. 2. It can be seen from the figure that when the number of sub-signals containing different frequencies in the analog signal is increasing, the frequency of $E_2\left(m\right)$ fluctuates up and down in 1, and the embedding dimension $m$ corresponding to the wobble region also becomes larger. The values of $E_2\left(m\right)$ are not always 1, indicating that these five sets of analog signals have a certain and unique mode of motion. In addition, the number of sub-signals of different frequencies superimposed on the analog signal has little effect on the value of $E_1\left(m\right)$. Therefore, the optimal embedding dimension of the five sets of analog signals is roughly reasonable between 10 and 15.

Fig. 2Relationship between Em and embedded dimension m

a)$f\left(\mathrm{1,1}\right)$

b)$f\left(1,2\right)$

c)$f\left(1,3\right)$

d)$f\left(1,4\right)$

e)$f\left(1,5\right)$

The singular value of the analog signal increases as the number of frequency signals superimposed in the signal increases, and it is also proportional to the frequency value in Fig. 3. In addition, the fluctuation of the number of frequencies or the magnitude of the frequency has a great influence on the singular value of the effective characteristic signal, and has no influence on the phase space of the noise signal.

Fig. 3The singular values of different frequency signals

In order to more clearly reflect the difference of the singular values of each signal, the cloud diagram shown in Fig. 4 is used to illustrate the distribution of singular values in its space. As the number of sub-signal products in the analog signal increases, the distribution of the largest singular values is concentrated on the left most top of the graph. The trend of the distribution gradient from the upper right to the lower left gradually decreases, and the width of each gradient gradually increases from large to small. In addition, in the distribution space of the entire singular value, the singular value of 99 % or more is below 54.88.

Fig. 4Distribution of singular value space of analog signal

3.1.2. Sample product of signal frequency

This simulation signal contains the product of a plurality of sinusoidal signals of different frequencies, and a certain amount of random noise signals $f_{(2,i)}\left(t\right)$ are combined to form an analog signal ${\xi }_{G-noise}\left(t\right)$. The expression of the analog signal is shown in Eq. (12): This frequency ${\omega }_i=\{i=1~5|5.6\pi ,8\pi ,12\pi ,18\pi ,28\pi \}$ is five groups, the other parameters are the same as above:

12

$f_{\left(2,i\right)}\left(t\right)=\left[{\prod }_{n=1}^{n}A\mathrm{s}\mathrm{i}\mathrm{n}\left({\omega }_{i}t+\varphi \right)\right]+{\xi }_{G-noise}\left(t\right).$

Fig. 5Relationship between autocorrelation function and delay time

Fig. 6Relationship between Em and embedded dimension m

a)$f\left(\mathrm{2,1}\right)$

b)$f\left(2,2\right)$

c)$f\left(2,3\right)$

d)$f\left(2,4\right)$

e)$f\left(2,5\right)$

1) Relationship between autocorrelation function and delay time.

In the same way, the analog signal composed of the products of different frequency sub-signals is used in the same way, and the relationship between the analog signal and the delay time is calculated by the autocorrelation function method, and the result is shown in Fig. 5. From this figure, the more the number of sub-signals of different frequencies in the analog signal, the lower the autocorrelation of the signal. In addition, the number of products of different frequency sub-signals has little effect on the attenuation amplitude of the noise signal over the entire delay time scale.

2) Relationship between $E\left(m\right)$ and embedded dimension $m$.

In the same way, the Cao’s algorithm is used to calculate embedded dimension the analog signals of Eq. (16) respectively, and the calculation results are obtained in Fig. 6.

As the number of sub-signals constituting the analog signal increases, the value of $E_2\left(m\right)$ fluctuates slightly above and below 1, and the fluctuation trend of the value of $E_2\left(m\right)$ remains stable on the scale of the embedding dimension $m$. At the same time, it is explained that each analog signal has a certain motion characteristic. In addition, the value of $E_1\left(m\right)$ is basically 1 in the dimension with the embedding dimension of 10-15, which also shows that the optimal embedding dimension of this set of analog signals can be more reasonable in the dimension value of 10-15.

The sampling product of a plurality of frequency signals, the singular value of the analog signal decreases as the number of frequencies increases is shown in Fig. 7. The fluctuation of the number of frequencies or the frequency value has a great influence on the singular value of the effective signal, and the influence on the noise signal is small.

Fig. 7The singular values of different frequency signals

The larger value group of the singular value of the analog signal is mainly concentrated before the embedding dimension is 5 in Fig. 8, and each gradient stripe line approximate to the vertical line is sequentially expanded to the right side. In addition, the singular value is 26.3-33.8, which accounts for about half of the entire singular value space. Therefore, when the feature signal is composed of a plurality of sets of sub-signals of different frequencies, the more the number of sub-signals, the more severe the interference of the effective characteristic information.

3.2. Relationship between SVD and signal amplitude

In this section, the influence of the change of signal amplitude on the singular value of the signal is studied by using the method of simulation. The simulation signal $f_{(3,i)}\left(t\right)$ consists of a sinusoidal signal and a random white noise signal ${\xi }_{G-noise}\left(t\right)$. The expression of this signal is shown in Eq. (13). The parameter is set to: the values of the amplitude $A_i$ are $A_i=\left\{i=1~5|\mathrm{0.45,0.86,1},\mathrm{2.3,6}\right\}$, the values of the frequency are ${\omega }_i=\left\{i=\mathrm{1,2}|28\pi ,5.6\pi \right\}$, the values of the initial phase are ${\varphi }_i=\left\{i=\mathrm{1,2}|3.08\pi ,1.5\pi \right\}$, other parameters are the same as above:

Fig. 8Distribution of singular value space of analog signal

1) Relationship between autocorrelation function and delay time.

In the same way, the variation law between the autocorrelation function and the delay time of each signal is studied by changing the amplitude of the signal, and the result shown in Fig. 9 is obtained. As the amplitude of the analog signal increases, the autocorrelation of each signal increases. At the same time, the autocorrelation of the signal has a high degree of regularity over the entire delay time scale. In addition, as the amplitude of the signal increases, the noise signal is gradually masked.

Fig. 9Relationship between autocorrelation function and delay time

2) Relationship between $E\left(m\right)$ and embedded dimension $m$.

In the same way, the results of Fig. 10 are obtained through simulation calculation. As the number of signal amplitudes increases, the values of $E_1\left(m\right)$ and $E_2\left(m\right)$ are basically unchanged. However, as the magnitude of the signal amplitude increases, the value of $E_2\left(m\right)$ varies greatly. In addition, the value range of the signal optimal embedding dimension $m$ is concentrated between 5 and 10.

Fig. 10Relationship between Em and embedded dimension m

a)$f\left(\mathrm{3,1}\right)$

b)$f\left(3,2\right)$

c)$f\left(3,3\right)$

d)$f\left(3,4\right)$

e)$f\left(3,5\right)$

Fig. 11Relationship between amplitude and singular values

The singular value of the analog signal increases as the amplitude increases is shown in Fig. 11. As the amplitude increases, the transition point of the singular value of the analog signal shifts back slightly. In addition, the change of amplitude has the greatest influence on the singular value of the effective signal, and the influence on the noise is weak.

Similarly, the signal amplitude increases monotonically in the forward direction, the value of the useful singular value also increases in the same direction in Fig. 12. However, singular values that may be useful in the entire singular value space account for only about 2 %.

Fig. 12Distribution of singular value space of analog signal

3.3. Relationship between SVD and initial phase of signal

In this section, the simulation analysis method is used to study the relationship between signal phase and signal singular value. The simulation signal $f_{(4,i)}\left(t\right)$ consists of an initial sinusoidal signal and a random white noise signal ${\xi }_{G-noise}\left(t\right)$, and the mathematical expression of the simulated signal is shown in Eq. (14). Parameter setting: The values of the frequency are ${\omega }_i=\left\{i=\mathrm{1,2}|28\pi ,5.6\pi \right\}$; the values of the initial phase are ${\varphi }_i=\left\{i=1~5|\mathrm{0,1.5}\pi ,3\pi ,8\pi ,16\pi \right\}$, and the other parameters are the same as above:

1) Relationship between autocorrelation function and delay time.

This part is studied the influence of the initial phase on the phase space of the signal by changing the initial phase of the analog signal. According to the set parameters, the results shown in Fig. 13 are obtained through simulation calculation. from this graph, the initial phase also affects the autocorrelation of the signal to a small extent. During a certain delay period, the autocorrelation of the signal increases significantly with the increase of the initial phase. However, on the entire delay time scale, the initial phase of the signal has no effect on the noise attenuation amplitude.

2) Relationship between $E\left(m\right)$ and embedded dimension $m$.

Similarly, by calculating the change of the initial phase value of the signal, the result shown in Fig. 14 is obtained. It can be obtained from the modification that the initial phase of the signal changes, and the value of $E_2\left(m\right)$ fluctuates slightly between 1 and 5, and its value always 1 after the dimension 5. Explain that the initial state of the signal changes without affecting the distribution of the signal phase space. In addition, the value of $E_2\left(m\right)$ is always 1 after the dimension 5. This shows that the optimal embedding dimension of this set of analog signals is 5.

Fig. 13Relationship between autocorrelation function and delay time

Fig. 14Relationship between Em and embedded dimension m

a)$f\left(\mathrm{4,1}\right)$

b)$f\left(4,2\right)$

c)$f\left(4,3\right)$

d)$f\left(4,4\right)$

e)$f\left(4,5\right)$

The singular value of the analog signal is substantially unchanged as the initial phase increases in Fig. 15. However, the change in the initial phase has a significant effect on the noise signal.

Fig. 15Relationship between initial phase and singular value

The fluctuation of the initial phase value of the analog signal has less influence on the magnitude of the useful singular value is shown in Fig. 16. However, the distribution that can lead to useful singular values becomes more complicated. In addition, the fluctuation of the initial phase mainly affects the characteristic information of singular values within 24.57-30.95. At the same time, the singular value of this kind of feature accounts for about 80 %, and the gradient of its distribution is more complicated. This shows that when the state of the initial phase of the signal changes, it does not cause interference to the main feature information, but it affects the characteristic state of the noise signal.

Fig. 16Distribution of singular value space of analog signal

4.1. Experimental setup

In this section, the fixed-axis gear unit structure and the tooth surface pitting failure of the gear teeth are tested to verify the effectiveness of the singular value decomposition de-noising method. In the gearbox under test, spur gears were used as the research object for this experiment. The layout of the test bench is shown in Fig. 17. On the outside of the gearbox bearing support, four test points are set: measuring point I, measuring point II, measuring point III and measuring point IV. Four similar uniaxial acceleration sensors are installed at the four measuring points to collect the acceleration value of the fault vibration.

The input speed is 1500 r/min; the input current of the magnetic powder loader is 0.1 A; the experimental object is the spur gear, the modulus $m=$22 mm, the number of teeth $z=$55, and the tooth width $b=$20 mm. The material of the gear is 18CrNiMo7 steel.

Fig. 17Structural layout of the test bed: 1 – motor input shaft; 2 – measured gear box; 3 – torque transducer; 4 – magnetic powder loader

4.2. Analysis of experimental results

1) Experimental data.

The time domain values of the vibration acceleration at the four measuring points were collected by experiment, as shown in Fig. 18. It can be seen from the figure that the experimental data at the four measuring points are affected by a large number of interference signals, so the vibration characteristics at the test position are not obvious, and the noise reduction processing of the measured experimental data is required.

Fig. 18Time domain diagram of vibration acceleration at points before noise reduction

2) Calculate the delay time.

According to the calculation principle of the autocorrelation function method, the delay time of the time series of the fault vibration signals at the four measuring points is calculated separately, and the calculation results is shown in Table 1.

3) Calculate the embedded dimension.

According to the improved Cao’s algorithm, the optimal embedding dimension of the vibration signal at the four test positions on the gearbox is calculated separately in Fig. 19. It can be seen from the figure that the value of $E_2\left(m\right)$ is not equal to 1, indicating that the four measuring points have certain vibration characteristics. In addition, $E_1\left(m\right)$ is close to a constant value of 1 for the first time when the embedding dimension $m$ is equal to 19. Therefore, the optimal embedding dimension of the vibration signal at these four measuring points is equal to 19.

Fig. 19Relationship between Em and m of experimental signals

a) Measuring point I

b) Measuring point II

c) Measuring point III

d) Measuring point IV

4) Singular value decomposition of experimental data.

According to the singular value decomposition theory, combined with the simulation principle of singular value decomposition method. In this section, use the same analysis method to perform singular value decomposition on the experimental data at the measuring point I, the measuring point II, the measuring point III and the measuring point IV. The singular value decomposition parameter is set to: hysteresis time $\tau =$ 2, initial order $m=$ 90. After the singular value decomposition, the calculation result is shown in Fig. 20. The singular values of the four measuring points all have abrupt changes at the position of 1000 in Fig. 20(a). Theoretically, the value is the boundary point between the effective signal and the noise signal, but the singular value after the point is still high. In addition, the singular value is gradually reduced gradually, but the singular value energy is still high; and the singular value difference spectrum is obtained before the 5th order in Fig. 20(b), and the spectrum value is the largest. At the same time, the difference spectrum values at positions of 10, 20, 64, etc. are approximately 100. It is indicated that the part after the singular value of 1000 also contains a certain amount of valid information.

5) Analysis of fault vibration characteristics.

The experimental data after the noise reduction is subjected to Fourier calculation, and the frequency response diagram at each measurement point is obtained in Fig. 21. The measuring points II and III are coaxial with the input power source, and the vibration of the power source is greatly affected. The vibration at the position of the measuring point is complicated. The measuring points I and IV are coaxial with the load.

Fig. 20Singular values of fault signals at each points

Fig. 21Frequency of fault signals at each point

However, the measuring point I is far away from the load and less affected by the load vibration. Therefore, the vibration characteristics of the measuring point I are only related to the fault vibration characteristics, and the low frequency vibration amplitude is large. The measuring point IV close to the load, and the vibration characteristics at the measuring point are affected by the load excitation. In addition, it is possible to derive the fault vibration at the four measuring points on the gearbox, and a large-cycle frequency doubling resonance phenomenon is generated with the transmission system.