# Incipient Fault Diagnosis of Rolling Bearings Based on Impulse-Step Impact Dictionary and Re-Weighted Minimizing Nonconvex Penalty Lq Regular Technique

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

- (1)
- When a spalling defect or pitting corrosion is induced, a series of successive impulses will be generated during subsequent operation. However, most dictionaries and optimal wavelet-basis constructed in the previous method only use single pulse or single impact frequencies, e.g., the optimal Laplace wavelet, single-side Morlet wavelet basis, transient impulse atoms, etc. Therefore there is no guarantee that the sparse-basis construction can match the natural waveform structure of the vibration signal well.
- (2)
- In practice, due to the fluctuation of the load and speed, and the interference of the harsh working environment, some random variations will be generated between an impulse and its neighboring impulses. The traditional sparse reconstruction methods such as greedy pursuit, orthogonal matching pursuit (OMP), L1-norm regularization and iterative shrinkage algorithm ignore those time-varying physical characteristics, which leads to a lower success rate of the transient impulse reconstruction. On the other hand, the traditional sparse reconstruction approaches also treat all vibration signal values equally and thus ignore the fact that the vibration peak value may have more useful information about periodical transient impulses and should be preserved at a larger weight value.

## 2. Impulse-Step Impact Dictionary and Its Simulation

_{o}is defect size on the outer race, d is the rolling element diameter, D

_{0}is the diameter of outer race, i.e., D

_{0}= D

_{p}+ d, D

_{p}is the pitch diameter and f

_{r}is the shaft rotation frequency, f

_{c}is the bearing cage frequency, i.e., ${f}_{c}=\frac{{f}_{r}}{2}(1-\frac{d}{{D}_{p}}\mathrm{cos}\alpha )$, α is the contact angle. As a matter of fact, when the defect size l

_{o}(mm) is smaller than the rolling element diameter d, so the rolling element cannot come into contact with the bottom of the pitting failure, the distance of the rolling element entering and then exiting from the fault region is half of the defect size l

_{o}(mm). Thus the period time $\Delta t$ becomes:

_{i}(mm) on the inner race, the corresponding period time $\Delta {t}_{i}$ can be expressed as follows:

_{n}is the system natural frequency, u the time when the impulse-like impact occurs, τ is system damping and a is the peak value ratio of the impulse-like response to the step-like response [27].

_{n}= 10,000 Hz, the impulse-like response happened u is 0.005, the rotor speed rotation frequency f

_{r}is 800 rpm. The time-domain waveform of the impulse-like signal with noise is shown in Figure 2e. The signal-noise ratio (SNR) is 20 dB. It can be seen that the similarities between the measured signal and the simulated signal with noise presented in Figure 1b is quite apparent.

## 3. Re-Weighted Minimizing Nonconvex Penalty Lq Regular Technique

#### 3.1. Review of Sparse Representation

- (1)
- Designing a redundant dictionary D. The first important issue is how to construct a redundancy dictionary D that suitable for the transient behavior of fault impulse components.
- (2)
- Recovering sparse coefficients $\alpha $. Another important issue is how to design an optimization algorithm to calculate the sparse coefficients of vibration fault signal.

#### 3.2. Re-Weighted Minimizing Nonconvex Penalty Lq Regular and Its Simulation Experiment

Algorithm 1 Re-weighted minimizing nonconvex penalty Lq regular (R-NSMLq) | |

1: | Input: Matrix D, measurement vector b and estimated sparsity level s; |

2: | Choose appropriate parameters $\lambda (\lambda >0)$, q (0 < q ≤ 1); |

3: | Initialize ${\alpha}^{(0)}$ such that $D{\alpha}^{(0)}=b$, and ${\epsilon}_{0}=1$; |

4: | For k = 0; |

5: | Solve the following linear system for ${\alpha}^{(k)}$, |

6: | $(\frac{q{\alpha}^{(k+1)}[i]}{{({\epsilon}_{k}^{2}+{\Vert {\alpha}^{(k)}[i]\Vert}_{2}^{2})}^{1-q/2}}{)}_{1\le i\le M}+\frac{1}{\lambda}{D}^{T}(D{\alpha}^{(k+1)}-b)=0$ (11) |

7: | Or |

8: | $({D}^{T}D+diag{(\frac{q\lambda}{{({\epsilon}_{k}^{2}+{\Vert {\alpha}^{(k)}[i]\Vert}_{2}^{2})}^{1-q/2}})}_{1\le i\le M}){\alpha}^{(k+1)}={D}^{T}b$ (12) |

9: | When the required reconstruction precision is obtained, the coefficients ${\alpha}^{(k)}$ will be considered as the output value assigned to $\alpha $, meanwhile end to this algorithm, otherwise execute next steps. |

10: | Let β be a constant, where 0 < β < 1. Update $\alpha $ by formula ${\epsilon}_{k+1}=\mathrm{min}\{{\epsilon}_{k},\beta \cdot r{({\alpha}^{(k+1)})}_{s+1}\}$, where $r(\alpha )$ represents the rearrangement of absolute values of $r({\alpha}^{(k+1)})$ in the decreasing order, and $r{(\alpha )}_{s+1}$ is the (s + 1) th component value of $r(\alpha )$. Note that, if ${\epsilon}_{k+1}=0$, choose ${\alpha}^{(k+1)}$ to be an approximation of sparse solution and stop this iteration. |

11: | Let k = k + 1, and return to step 4 to continue. |

12: | Output: Sparse coefficients α; |

13: | End |

**Theorem**

**1.**

^{o}is a sparse signal with sparsity level s which satisfies Dx

^{o}= b. Without loss of generality, here the sparse coefficient $\alpha $ is substituted by vector x. The smooth parameter ${\epsilon}_{k}\to {\epsilon}_{*}$ with $k\to \infty $. Matrix D satisfies the restricted isometry property (RIP) [30,31,33] of order 2 s with ${\delta}_{2s}<1$, when ${\epsilon}_{*}>0$, the sequence {x

^{(k)}} has at least one convergent subsequence. Suppose that the limit ${\epsilon}_{k}={\epsilon}_{*}$ is a local optimal solution for Equation (10), we have:

^{o}, it satisfies,

_{1}, C

_{2}and C

_{3}are independent positive constants. To prove the Theorem 1, the following two lemmas (i.e., Lemmas 1 and 2) [35,36] are required.

**Lemma**

**1**

**.**For all $x,y\in {R}^{N}$ and 0 < q ≤ 1, if ${\epsilon}_{k}\ge {\epsilon}_{k+1}\ge 0$, it satisfies:

**Proof.**

**Lemma**

**2**

**.**Let ${L}_{q}(x,\epsilon ,\lambda )={\displaystyle \sum _{j=1}^{N}{[{\alpha}_{j}^{2}+{\epsilon}^{2}]}^{q/2}}+\frac{1}{2\lambda}{\Vert D\alpha -b\Vert}_{2}^{2}$, if be the solution of ${L}_{q}(x,\epsilon ,\lambda )$ for k = 0 ,1, 2,…N, then:

_{4}is an independent positive constant.

**Proof.**

## 4. Experimental Evaluation

^{−2}. Figure 5a shows the vibration raw signal of the whole life-cycle of bearing 1. Figure 5b shows the Kurtosis curve over the whole life-cycle of bearing 1 and indicates that there is a long time with normal operation in whole life-cycle, but the period of fault from incipient stage to severe stage is relatively short. As shown in Figure 5, there is an obvious transient feature at point 647 in the incipient fault. However, due to the interference of harsh working environment and background noises, the engineer cannot sure whether the fault is happened before point 647 or not. Hence, to verify the effectiveness of the proposed method for bearing incipient fault diagnosis, the experimental data at point 535 was chosen which has no obviously wave phenomenon in whole life-cycle.

_{o}, 4f

_{o}and 5f

_{o}) are clearly detected, therefore, the proposed method is exactly suitable for incipient fault bearing signal.

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A

**Proof**

**of Theorem 1**

**.**It should be noted from Equation (18) that the $\left\{{x}^{(k)}\right\}$ is monotonically decreasing sequence because ${\Vert {x}^{(k+1)}-{x}^{(k)}\Vert}_{2}^{2}\to 0$. Thus we might as well set $\left\{{x}^{(k)}\right\}$ converges to ${x}^{{\epsilon}_{*},\lambda}$. Letting $i\to \infty $, the Equation (11) can be rewritten as:

^{o}and T* be an index dataset of s largest entries in L2-norm of $\left\{{x}^{{\epsilon}_{*},\lambda}\right\}$, since ${\Vert {x}^{0}\Vert}_{0}\le s$, we get:

_{1}and C

_{2}are as follows:

_{0}, and ${x}^{({k}_{0})}$ is a s-sparse signal. Otherwise, there is a sequence $\left\{{x}^{({n}_{k})}\right\}$ satisfies ${\epsilon}_{{n}_{k}}=\alpha \cdot r{\left({x}^{({n}_{k})}\right)}_{s+1}>0$. In the former case, ${x}^{({k}_{0})}$ is a s-sparse signal, and we get ${x}^{0,\lambda}={x}^{(k)}$. In the latter case, due to $\left\{{x}^{({n}_{k})}\right\}$ is bounded with limit point ${x}^{0,\lambda}$, and without loss of generality, we assume the convergent sub-sequence of $\left\{{x}^{({n}_{k})}\right\}$ is also ${x}^{({n}_{k})}$, ${x}^{0,\lambda}=\underset{k\to \infty}{\mathrm{lim}}{x}^{({n}_{k})}$, and then $\underset{k\to \infty}{\mathrm{lim}}r{({x}^{({n}_{k})})}_{s+1}=\underset{k\to \infty}{\mathrm{lim}}\frac{{\epsilon}_{{n}_{k}}}{\alpha}=0$, that is, the sub-sequence of ${x}^{({n}_{k})}$ is a s-sparse signal. Therefore, based on the above two cases, and without loss of generality, we assume the s-sparse signal is ${x}^{(k)}$, we have $\underset{k\to \infty}{\mathrm{lim}}{x}^{(k)}={x}^{*}$, using the RIP of D, we have,

_{3}is represented as follows:

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**Figure 1.**The time-domain waveform of the fault signal for a single pitting failure. (

**a**) The physical model; (

**b**) Time-domain waveform of the fault signal.

**Figure 2.**The time-domain waveform of (

**a**) impulse-like impact atom; (

**b**) step-impulse impact atom; (

**c**) impulse-step impact atom; (

**d**) impulse-step impact signal without noise and; (

**e**) impulse-step impact signal with a SNR of 20 dB.

**Figure 3.**Comparison results of recoverability with different q. (

**a**) Random signal with 32 non-zero pulses; (

**b**) Comparison results of RSR with different q.

**Figure 5.**The vibration raw signal and the Kurtosis curve of the whole life-cycle of bearing 1. (

**a**) The vibration raw signal of the whole life-cycle of bearing 1; (

**b**) The Kurtosis curve of the wholse life-cycle of bearing 1.

**Figure 6.**Original vibration signal and its time-frequency analysis. (

**a**) Original vibration signal; (

**b**) Time-frequency distribution of original vibration signal; (

**c**) Amplitude spectrum of the original vibration signal; (

**d**) Hilbert envelope spectrum of original vibration signal.

**Figure 7.**The comparison of amplitude spectrum of the IMF modes. (

**a**) The amplitude spectrum of IMF modes with K = 20 and α = 2000; (

**b**) The amplitude spectrum of IMF modes with K = 21 and α = 2000.

**Figure 8.**The 20-IMF components of original signal decomposed by VMD method. (

**a**) IMF1-IMF10; (

**b**) IMF11-IMF20.

**Figure 9.**The identified results using the proposed method. (

**a**) The reconstructed signal; (

**b**) Time-frequency distribution of the reconstructed signal; (

**c**) Hilbert envelope spectrum of the reconstructed signal.

**Figure 10.**The identified results using the proposed method. (

**a**) The reconstructed signal using the OMP method; (

**b**) Time-frequency distribution of the reconstructed signal using the OMP method; (

**c**) Hilbert envelope spectrum of the reconstructed signal using the OMP method; (

**d**) The reconstructed signal using the L1-Norm regularization method; (

**e**) Time-frequency distribution of the reconstructed signal using the L1-Norm regularization method; (

**f**) Hilbert envelope spectrum of the reconstructed signal using the L1-Norm regularization method.

**Figure 11.**Diagnosis result using the spectral kurtogram method. (

**a**) Kurtogram of 19th IMF model component; (

**b**) The Hilbert envelope spectrum of the band-pass filtered signal.

Parameter | Outside Diameter (mm) | Pitch Diameter (mm) | Contact Angle | Element Number | Pitting Defect Size |
---|---|---|---|---|---|

Value | d_{0} = 7.95 | D_{p} = 45.14 | α = 0° | 14 | l_{0} = 1.28 |

Regular Operator-q | Smoothing Parameter | Penalty Parameter | Maximum Iterations Number | Stopping Threshold |
---|---|---|---|---|

0.5 |
ε_{0} = 0 | $\lambda ={10}^{-6}$ | 1000 | ${\Vert {x}^{k}-{x}^{0}\Vert}_{2}/{\Vert {x}^{0}\Vert}_{2}\le {10}^{-3}$ |

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## Share and Cite

**MDPI and ACS Style**

Li, Q.; Liang, S.Y.
Incipient Fault Diagnosis of Rolling Bearings Based on Impulse-Step Impact Dictionary and Re-Weighted Minimizing Nonconvex Penalty Lq Regular Technique. *Entropy* **2017**, *19*, 421.
https://doi.org/10.3390/e19080421

**AMA Style**

Li Q, Liang SY.
Incipient Fault Diagnosis of Rolling Bearings Based on Impulse-Step Impact Dictionary and Re-Weighted Minimizing Nonconvex Penalty Lq Regular Technique. *Entropy*. 2017; 19(8):421.
https://doi.org/10.3390/e19080421

**Chicago/Turabian Style**

Li, Qing, and Steven Y. Liang.
2017. "Incipient Fault Diagnosis of Rolling Bearings Based on Impulse-Step Impact Dictionary and Re-Weighted Minimizing Nonconvex Penalty Lq Regular Technique" *Entropy* 19, no. 8: 421.
https://doi.org/10.3390/e19080421