排列熵優化改進變模態分解算法診斷齒輪箱故障

王志堅，常 雪，王俊元※，杜文華，段能全，黨長營

?

排列熵優化改進變模態分解算法診斷齒輪箱故障

王志堅1，常 雪2，王俊元1※，杜文華1，段能全1，黨長營1

（1. 中北大學機械工程學院，太原 030051；2. 重慶大學機械工程學院，重慶 400044）

為了準確提取齒輪箱中復合故障特征，該文選用變模態分解（variational mode decomposition，VMD）對振動信號進行處理，它能夠將信號分解為多個固有模態函數（intrinsic mode function，IMF），但需預設分解層數和懲罰因子；因此，為了能夠自適應地確定分解層數，該文提出了排列熵優化算法(permutation entroy optimization，PEO)，該算法可以根據待分解信號的特點自適應的確定分解層數；同時，為了解決VMD算法對噪聲的敏感性，該文根據噪聲輔助數據分析的思想，提出了改進VMD算法（modified variable modal decomposition，MVMD），該算法首先添加成對符號相反的高斯白噪聲到原始信號，再利用VMD算法對其進行分解，經過多次循環，原始信號中的噪聲相互抵消，而后將每次循環得到的每層IMF分別進行集成平均。利用該算法分別對含有多故障特征的齒輪箱仿真信號及實測信號進行處理，均提取出了故障特征。該文所提方法對封閉式功率流試驗臺進行復合故障提取，160和360 Hz的故障頻率分別被提取出。該方法為齒輪箱復合故障診斷提供新思路。

齒輪；算法；噪聲；多故障；排列熵；變模態分解

0 引 言

滾動軸承和齒輪等在農用機械如變速箱等旋轉機構中起著重要作用，與其他零件相比，發生故障的概率較高，不可預測性較強[1]。由于齒輪箱內部結構較為復雜，當發生故障時，其故障類型多為復合故障，且其故障特征常常被淹沒在強背景噪聲中，因此，需要開發一種有效的自適應故障提取方法[2-3]。

經過國內外諸多科研工作者的不斷探索，復合故障特征提取的方法也層出不窮。現階段，非參數型降噪方法如經驗模態分解和局部均值分解、參數型降噪方法如總體經驗模態分解，已經被運用于復合故障診斷當中，但都會由于噪聲干擾導致模態混疊現象[4-6]。

2014年，Dragomiretskiy等提出了一種新的信號處理算法，即變分模態分解[7](variational mode decomposition，VMD)。VMD具有堅實的理論基礎，分解精度較高[8]。但該算法需要預先設定分解層數，而值往往只能憑經驗而定，因此，分解結果很容易受到人為因素的影響而出現過分解或者欠分解現象，即當取值過大時，會產生過分解現象，分解出異常的白噪聲分量；而當取值過小時則會出現欠分解現象，導致部分故障特征未被提取出來。除此之外，VMD算法對噪聲比較敏感[9-11]，即分解結果容易受到背景噪聲的影響，特別是在強背景噪聲環境下，更容易產生由噪聲引起的虛假分量，而對于后續的故障識別，虛假分量的產生容易導致誤診斷[12-15]。

對于分解層數的自適應確定方法，Yi等[16]利用粒子群優化算法（particle swarm optimization，PSO）確定了VMD算法中的分解層數；Zhang等[17]利用蝗蟲優化算法（grasshopper optimization algorithm，GOA）對VMD算法中的參數進行了優化。除此之外，還有學者利用蟻群算法[18]、人工魚群算法[19]等其他優化算法對VMD算法中的參數進行優化。相比于憑經驗確定值，這些優化算法能夠根據原始信號自動確定值，具有很好的自適應性。但這些優化算法的弊端也十分明顯，都存在計算量大、冗余度高、計算效率低等問題[20]。

基于此本文提出一種基于排列熵的優化算法。從噪聲輔助數據分析[21]的角度改進VMD，進一步提高信號的信噪比。同時為了減小重構誤差，使所添加的白噪聲被完全中和，每次循環時，添加2個幅值相等、符號相反的白噪聲到原始信號，然后再利用VMD算法分別對其進行分解，最終經過多次循環，使原始信號中的噪聲相互抵消；將每次循環得到的各層IMF（intrinsic mode function）分別進行集成平均，再根據集成均值的結果對信號進行重構[22]；對重構信號再次進行VMD分解，作為MVMD算法最終的結果輸出。

1 排列熵和變分模態分解基本理論

1.1 排列熵算法的原理

排列熵（permutation entroy，PE）是由Bandt等[23]提出的一種可以檢測時間序列隨機性和動力學突變的方法，該算法具有原理簡單、計算效率高、魯棒性好等優點，適用于非線性數據分析[24]，該算法的具體步驟如下

式中P表示第個符號出現的概率；H表示時間序列的復雜和隨機程度，H越大，說明時間序列越隨機，H越小，說明時間序列越規則。

1.2 變分模態分解

VMD算法中的每一個IMF分量的中心頻率以及帶寬在迭代求解過程中不斷更新，最終的分解結果將根據原始信號頻域特性進行自適應分解，得到個IMFs，而模型的約束條件是這個IMFs之和等于輸入的原信號。約束變分模型的具體構造步驟如下

2 VMD算法改進

該算法的具體步驟如下

1）設定的初始值為2，排列熵的閾值取經驗值0.6；

為了提高信噪比，本文提出一種基于VMD的降噪方法，即改進的VMD算法（modified VMD，MVMD）。根據文獻[26]可知，CEEMD為了能夠減小重構誤差，使所添加的白噪聲被完全中和，所以在向待分解信號中添加白噪聲時，所添加的白噪聲為正負白噪聲對，通過該文獻中的仿真與試驗分析可知，相比于EEMD中單純的添加正白噪聲，CEEMD中添加正負白噪聲對的方法達到了降低重構誤差、促進白噪聲相互中和的目的。因此，基于正負白噪聲對在降低重構誤差方面的顯著效果，本文所提出的MVMD在添加輔助白噪聲時，也采取添加正負白噪聲對的思想，即每次循環時所添加的白噪聲為2個幅值相等、符號相反的正負白噪聲對，加上正負白噪聲之間的中和作用，在實現降噪目的的同時又不引入新的噪聲。加入輔助白噪聲后，將得到2個待分解信號，然后再利用VMD算法分別對其進行分解，經過多次循環，原始信號中的噪聲將相互抵消，最終將每次循環得到的各層IMF分別進行集成平均，根據集成均值的結果對信號進行重構，對重構信號再次進行VMD分解的具體步驟如下

4）重復步驟2)、3)，且每次循環開始時加入新的高斯白噪聲對；

PEO-MVMD的流程如圖1所示。

注：k為分解層數；N為循環次數。下同。

3 仿真信號分析

齒輪箱發生復合故障時，其振動信號往往是多調制源共存的。因此，采用齒輪故障仿真信號和滾動軸承故障仿真信號進行分析，構造如下

圖2 仿真信號的時域波形

注：IMF為固有模態函數，下同。

MVMD算法中需要設置循環次數和所添加的白噪聲幅值std。兼顧信號處理的效率，本文取循環次數=100；當白噪聲幅值std取0.15時，重構信號的信噪比最高，即降噪效果最好，因此，對于仿真信號，取MVMD算法中所添加白噪聲幅值為0.15。

VMD分解結果如圖5所示，原始信號中30 Hz的低頻成分被成功的提取出來；但中頻的120 Hz信號，由于受到強背景噪聲的干擾，被分解到了IMF2和IMF3這2個模態中，發生了模態混疊現象，且頻譜特征十分微弱，易造成誤診斷。MVMD分解結果如圖6所示，在IMF1中，原始信號中30 Hz的低頻信號的頻譜特征十分明顯；在IMF2中，調幅信號的120 Hz中心頻率以及2個調制頻率也都成功的從含有噪聲的原始信號中分離出來，且邊頻帶均勻對稱的分布在主頻兩側；在IMF3中，280 Hz的中心頻率以及均勻分布在其兩側的10 Hz多條邊頻帶也十分突出，雖然在500 Hz附近出現了殘余噪聲，但相比于280 Hz的主要頻率成分，噪聲成分十分微弱，對故障特征的識別影響不大。對MVMD分解后的信號與原仿真信號進行重構，結果如圖7所示，盡管第3層有少量的殘余噪聲存在，但是重構效果很好。

圖4 EEMD分解后的IMFs與其對應的頻譜

圖5 VMD分解后的IMFs與其對應的頻譜 Fig.5 IMFs and spectrum after VMD

圖6 MVMD分解后的IMFs與其對應的頻譜 Fig.6 IMFs and spectrum after MVVM

圖7 MVMD分解得到的IMF與組成信號對比

4 驗證試驗

4.1 試驗方法與設置

1. 調速電機2. 聯軸器3. 陪試齒輪箱4. 轉速扭轉儀5. 扭力桿 6. 試驗齒輪箱7. 三向加速度傳感器1 8. 三向加速度傳感器2

1. Speed regulating motor 2. Clutch 3. Companion gearbox 4. Rotating speed torsion meter 5. Torsion bar 6. Test gear box 7. Triaxial acceleration sensor 1 8. Triaxial acceleration sensor 2

圖8 齒輪傳動試驗臺

Fig.8 Gear transmission test bench

如圖9所示，本試驗中齒輪箱的復合故障包括齒輪剝落和軸承外圈故障。其中齒輪剝落通過齒輪疲勞試驗產生，外圈故障通過電火花加工方法人為植入。外圈故障160.2 Hz、齒輪嚙合頻率360 Hz。

a. 齒輪剝落 a. Spalling failure of gearb.電火花加工外圈裂縫 b.Electric spark machining bearing outer ring crack

4.2 VMD分解結果

為驗證本文所提算法的有效性，分別采用VMD和基于PEO-MVMD對上述復合故障信號進行分解。圖10為VMD的分解結果，有混疊現象，第二層含有160 Hz的特征信息，此外第二層中齒輪的嚙合頻率360 Hz峰值較小。

圖10 VMD分解后的IMFs與其對應的頻譜

4.3 MVMD分解結果

采用PEO算法確定分解層數，設定初始值為2，根據是否出現過分解進行循環迭代，查找的最優值。最終PEO算法輸出的的最優值為2；此外，取白噪聲幅值std為0.85，循環次數=100。MVMD算法對上述齒輪箱復合故障信號進行分解。結果如圖11所示，齒輪箱中外圈故障頻率160 Hz以及齒輪故障特征頻率360 Hz及其2倍頻720 Hz均被成功提取出來，而且頻率幅值遠大于VMD提取結果，相對于VMD效果更佳。再次驗證了文中所提方法的有效性。

圖11 MVMD分解后的IMFs與其對應的頻譜

5 結 論

本文提出了基于變模態分解的改進算法，即首先采用PEO算法根據待分解信號的特點自適應地確定所需要分解的層數，再利用降噪效果優異的MVMD對原始信號進行分解。通過對齒輪箱試驗信號進行分析，試驗結果表明，相比于VMD，本文提出的基于PEO的MVMD具有自適應性和強降噪性能，診斷出了封閉式功率流試驗臺中的軸承外圈故障頻率和齒輪嚙合頻率，分別為160和360 Hz，能夠自適應確定VMD的值，PEO算法輸出的分解層數最優值為2，并成功提取出齒輪故障特征的2倍頻720 Hz。

[1] Liang X, Zuo M J, Feng Z. Dynamic modeling of gearbox faults: A review[J]. Mechanical Systems & Signal Processing, 2018, 98: 852－876.

[2] 王志堅，韓振南，劉邱祖，等. 基于MED-EEMD的滾動軸承微弱故障特征提取[J]. 農業工程學報，2014，30(23)：70－78. Wang Zhijian, Han Zhennan, Liu Qiuzu, et al. Feature extraction of rolling bearing's weak fault based on MED-EEMD[J]. Transactions of the Chinese Society of Agricultural Engineering (Transactions of the CSAE), 2014, 30(23): 70－78. (in Chinese with English abstract)

[3] Wang Z, Wang J, Zhao Z, et al. A novel method for multi-fault feature extraction of a gearbox under strong background noise[J]. Entropy, 2017, 20(1): 10－26.

[4] 周士帥，竇東陽，薛斌. 基于LMD和MED的滾動軸承故障特征提取方法[J]. 農業工程學報，2016，32(23)：70－76. Zhou Shishuai, Dou Dongyang, Xue Bin. Fault feature extraction method for rolling element bearings based on LMD and MED[J]. Transactions of the Chinese Society of Agricultural Engineering (Transactions of the CSAE), 2016, 32(23): 70－76. (in Chinese with English abstract)

[5] Tang G, Luo G, Zhang W, et al. Underdetermined blind source separation with variational mode decomposition for compound roller bearing fault signals[J]. Sensors, 2016, 16(6): 897－914.

[6] He S, Chen J, Zhou Z, et al. Multifractal entropy based adaptive multiwavelet construction and its application for mechanical compound-fault diagnosis[J]. Mechanical Systems & Signal Processing, 2016, 76: 742－758.

[7] Dragomiretskiy K, Zosso D. Variational mode decomposition[J]. IEEE Transactions on Signal Processing, 2014, 62(3): 531－544.

[8] 王新，閆文源. 基于變分模態分解和SVM的滾動軸承故障診斷[J]. 振動與沖擊，2017，36(18)：252－256. Wang Xin, Yan Wenyuan. Fault diagnosis of roller bearings based on the variational mode decomposition and SVM[J]. Journal of Vibration & Shock, 2017, 36(18): 252－256. (in Chinese with English abstract)

[9] Zhao X, Zhang S, Zhishen L I, et al. Application of new denoising method based on VMD in fault feature extraction[J]. Journal of Vibration Measurement & Diagnosis, 2018, 38(1): 1－12.

[10] Xu A, Yan J, Wu X, et al. Rotating machinery fault feature extraction based on VMD and MP algorithm[J]. Foreign Electronic Measurement Technology. DOI:10.16450/j.cnki. issn.1004-6801.2018.02

[11] Mohanty S, Gupta K K, Raju K S. Hurst based vibro-acoustic feature extraction of bearing using EMD and VMD[J]. Measurement, 2018, 117: 200－220.

[12] Chen L, Xiao C, Li X, et al. A seismic fault recognition method based on ant colony optimization[J]. Journal of Applied Geophysics, 2018, 152: 1－8.

[13] Wang Z, Jia L, Qin Y. Adaptive diagnosis for rotating machineries using information geometrical kernel-ELM based on VMD-SVD[J]. Entropy, 2018, 20(1): 73－91.

[14] Wang ZJ, Wang J Y. A novel method for multi-fault feature extraction of a gear box under Strong background noise[J]. Entropy, 2017, 20(1): 10.

[15] Li Z, Chen J, Zi Y, et al. Independence-oriented VMD to identify fault feature for wheel set bearing fault diagnosis of high speed locomotive[J]. Mechanical Systems & Signal Processing, 2017, 85: 512－529.

[16] Yi C, Lv Y, Dang Z. A fault diagnosis scheme for rolling bearing based on particle swarm optimization in variational mode decomposition[J]. Shock and Vibration, 2016:9372691. http://dx.doi.org/10.1155/2016/9372691.

[17] Zhang X, Miao Q, Zhang H, et al. A parameter adaptive VMD method based on grasshopper optimization algorithm to analyze vibration signals from rotating machinery[J]. Mechanical Systems & Signal Processing, 2018, 108: 58－72.

[18] Zhou Y, Fazhi H E, Qiu Y. Dynamic strategy based parallel ant colony optimization on GPUs for TSPs[J]. Science China (Information Sciences), 2017, 60(6): 260－262.

[19] Shen W, Guo X, Wu C, et al. Forecasting stock indices using radial basis function neural networks optimized by artificial fish swarm algorithm[J]. Knowledge-Based Systems, 2011, 24(3): 378－385.

[20] Huang N, Chen H, Cai G, et al. Mechanical fault diagnosis of high voltage circuit breakers based on variational mode decomposition and multi-layer classifier[J]. Sensors, 2016, 16(11): 1887. DOI: 10.3390/s16111887

[21] Lee D H, Ahn J H, Koh B H. Fault detection of bearing systems through EEMD and optimization algorithm[J]. Sensors, 2017, 17(11): 2477. DOI:10.3390/s17112477

[22] Singh J, Darpe A K, Singh S P. Bearing damage assessment using Jensen-Rényi Divergence based on EEMD[J]. Mechanical Systems & Signal Processing, 2017, 87: 307－339.

[23] Bandt C, Pompe B. Permutation entropy: A natural complexity measure for time series[J]. Physical Review Letters, 2002, 88(17): 174102. DOI:10.1103/PhysRevLett. 8.174102

[24] Frank T D. Delay fokker-planck equations, perturbation theory, and data analysis for nonlinear stochastic systems with time delays[J]. Physical Review E Statistical Nonlinear & Soft Matter Physics, 2005, 71(3): 1－14.

[25] 鄭近德，程軍圣，楊宇. 改進的EEMD算法及其應用研究[J]. 振動與沖擊，2013，32(21)：21－26.

[26] Yeh J R, Shieh J S, Huang N E. Complementary ensemble empirical mode decomposition: A novel noise enhanced data analysis method[J]. Advances in Adaptive Data Analysis, 2011, 2(2): 135－156.

Gearbox fault diagnosis based on permutation entropy optimized variational mode decomposition

Wang Zhijian1, Chang Xue2, Wang Junyuan1※, Du Wenhua1, Duan Nengquan1, Dang Changying1

(1.030051,; 2.400044,)

gearbox composite fault diagnosis has received extensive attention. The composite fault is that 2 or more faults occur simultaneously in the mechanical equipment. Due to the different degrees of damage of the composite fault, the complicated transmission path of the fault characteristic signal, and the interference of the background noise, the strength between the fault components is not balance. The weak fault features are usually overwhelmed by strong faults or noise and the strong faults are weakened by the high-frequency energy in the process of transmission, it is easy to be missed or misdiagnosis, especially in the case of variable speed and variable load, the coupling of composite fault features poses great challenge to the healthy and reasonable diagnosis of mechanical equipment. With the development of computer technology, some new novel adaptive noise reduction methods are proposed, including parametric decomposition methods and nonparametric decomposition methods, but they are more or less affected by noise interference and modal aliasing. Variational mode decomposition(VMD) decompose a complex signal into several different time scales, and each time scale contains a center frequency, which can overcome the modal aliasing phenomenon, variational mode decomposition is widely applied to gearbox composite fault diagnosis, and has achieved amazing results, but it needs to preset the decomposition layersand penalty factor, and is sensitive to the background noise. In order to adaptively determine the number of decomposition layers, this paper proposed permutation entropy optimization algorithm, which can adaptively determine the number of decomposition layersaccording to the characteristics of the signal to be decomposed. In order to solve the sensitivity of VMD to noise, this paper proposed modified variational mode decomposition(MVMD) based on the idea of noise aided data analysis. The algorithm first added the opposite gauss white noise to the original signal, and then used VMD to decompose it. After repeated cycles, the noise in the original signal would offset each other, then the ensemble average is generated for each IMF（intrinsic mode function） in each cycle, and the signal was reconstructed according to the result of ensemble mean. The VMD decomposition of the reconstructed signal was taken as the final output result of MVMD. This algorithm was used to process the gear box simulation signal and the measured signal with multiple fault features respectively, and the decomposition results showed that the algorithm can not only improve the signal to noise ratio(SNR) of the signal effectively, but also successfully extract the multiple fault features of the gear box in the strong noise environment, the fault frequencies of 160 and 360 Hz were extracted respectively which correspond to the bearing outer ring frequency and the gear meshing frequency. This method provides a new idea for gearbox composite fault diagnosis, it can not only overcome the interference of strong noise, but also accurately extract fault characteristics. In the future work, the research group will introduces the intelligent algorithm into the variational mode decomposition to determine the number of layers decomposed adaptively, at the same time, the combination of variational mode decomposition and support vector machine or neural network can improve the efficiency of intelligent fault diagnosis, this is a new idea for the healthy operation of agricultural machinery.

gears; algorithm; noises; multi-fault; permutation entropy; variable modal decomposition

王志堅，常 雪，王俊元，杜文華，段能全，黨長營. 排列熵優化改進變模態分解算法診斷齒輪箱故障[J]. 農業工程學報，2018，34(23)：59－66. doi：10.11975/j.issn.1002-6819.2018.23.007 http://www.tcsae.org

Wang Zhijian, Chang Xue, Wang Junyuan, Du Wenhua, Duan Nengquan, Dang Changying. Gearbox fault diagnosis based on permutation entropy optimized variational mode decomposition[J]. Transactions of the Chinese Society of Agricultural Engineering (Transactions of the CSAE), 2018, 34(23): 59－66. (in Chinese with English abstract) doi：10.11975/j.issn.1002-6819.2018.23.007 http://www.tcsae.org

2018-05-26

2018-9-30

國家自然科學基金（59975064）

王志堅，博士，副教授，主要研究方向為旋轉機械復合故障診斷。Email：wangzhijian1013@163.com

10.11975/j.issn.1002-6819.2018.23.007

TN911.72；TP206

A

1002-6819(2018)-23-0059-08