王志堅(jiān)，常 雪，王俊元※，杜文華，段能全，黨長(zhǎng)營(yíng)

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排列熵優(yōu)化改進(jìn)變模態(tài)分解算法診斷齒輪箱故障

王志堅(jiān)1，常 雪2，王俊元1※，杜文華1，段能全1，黨長(zhǎng)營(yíng)1

（1. 中北大學(xué)機(jī)械工程學(xué)院，太原 030051；2. 重慶大學(xué)機(jī)械工程學(xué)院，重慶 400044）

為了準(zhǔn)確提取齒輪箱中復(fù)合故障特征，該文選用變模態(tài)分解（variational mode decomposition，VMD）對(duì)振動(dòng)信號(hào)進(jìn)行處理，它能夠?qū)⑿盘?hào)分解為多個(gè)固有模態(tài)函數(shù)（intrinsic mode function，IMF），但需預(yù)設(shè)分解層數(shù)和懲罰因子；因此，為了能夠自適應(yīng)地確定分解層數(shù)，該文提出了排列熵優(yōu)化算法(permutation entroy optimization，PEO)，該算法可以根據(jù)待分解信號(hào)的特點(diǎn)自適應(yīng)的確定分解層數(shù)；同時(shí)，為了解決VMD算法對(duì)噪聲的敏感性，該文根據(jù)噪聲輔助數(shù)據(jù)分析的思想，提出了改進(jìn)VMD算法（modified variable modal decomposition，MVMD），該算法首先添加成對(duì)符號(hào)相反的高斯白噪聲到原始信號(hào)，再利用VMD算法對(duì)其進(jìn)行分解，經(jīng)過(guò)多次循環(huán)，原始信號(hào)中的噪聲相互抵消，而后將每次循環(huán)得到的每層IMF分別進(jìn)行集成平均。利用該算法分別對(duì)含有多故障特征的齒輪箱仿真信號(hào)及實(shí)測(cè)信號(hào)進(jìn)行處理，均提取出了故障特征。該文所提方法對(duì)封閉式功率流試驗(yàn)臺(tái)進(jìn)行復(fù)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(fù)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(tài)分解

0 引 言

滾動(dòng)軸承和齒輪等在農(nóng)用機(jī)械如變速箱等旋轉(zhuǎn)機(jī)構(gòu)中起著重要作用，與其他零件相比，發(fā)生故障的概率較高，不可預(yù)測(cè)性較強(qiáng)[1]。由于齒輪箱內(nèi)部結(jié)構(gòu)較為復(fù)雜，當(dāng)發(fā)生故障時(shí)，其故障類(lèi)型多為復(fù)合故障，且其故障特征常常被淹沒(méi)在強(qiáng)背景噪聲中，因此，需要開(kāi)發(fā)一種有效的自適應(yīng)故障提取方法[2-3]。

經(jīng)過(guò)國(guó)內(nèi)外諸多科研工作者的不斷探索，復(fù)合故障特征提取的方法也層出不窮?，F(xiàn)階段，非參數(shù)型降噪方法如經(jīng)驗(yàn)?zāi)B(tài)分解和局部均值分解、參數(shù)型降噪方法如總體經(jīng)驗(yàn)?zāi)B(tài)分解，已經(jīng)被運(yùn)用于復(fù)合故障診斷當(dāng)中，但都會(huì)由于噪聲干擾導(dǎo)致模態(tài)混疊現(xiàn)象[4-6]。

2014年，Dragomiretskiy等提出了一種新的信號(hào)處理算法，即變分模態(tài)分解[7](variational mode decomposition，VMD)。VMD具有堅(jiān)實(shí)的理論基礎(chǔ)，分解精度較高[8]。但該算法需要預(yù)先設(shè)定分解層數(shù)，而值往往只能憑經(jīng)驗(yàn)而定，因此，分解結(jié)果很容易受到人為因素的影響而出現(xiàn)過(guò)分解或者欠分解現(xiàn)象，即當(dāng)取值過(guò)大時(shí)，會(huì)產(chǎn)生過(guò)分解現(xiàn)象，分解出異常的白噪聲分量；而當(dāng)取值過(guò)小時(shí)則會(huì)出現(xiàn)欠分解現(xiàn)象，導(dǎo)致部分故障特征未被提取出來(lái)。除此之外，VMD算法對(duì)噪聲比較敏感[9-11]，即分解結(jié)果容易受到背景噪聲的影響，特別是在強(qiáng)背景噪聲環(huán)境下，更容易產(chǎn)生由噪聲引起的虛假分量，而對(duì)于后續(xù)的故障識(shí)別，虛假分量的產(chǎn)生容易導(dǎo)致誤診斷[12-15]。

對(duì)于分解層數(shù)的自適應(yīng)確定方法，Yi等[16]利用粒子群優(yōu)化算法（particle swarm optimization，PSO）確定了VMD算法中的分解層數(shù)；Zhang等[17]利用蝗蟲(chóng)優(yōu)化算法（grasshopper optimization algorithm，GOA）對(duì)VMD算法中的參數(shù)進(jìn)行了優(yōu)化。除此之外，還有學(xué)者利用蟻群算法[18]、人工魚(yú)群算法[19]等其他優(yōu)化算法對(duì)VMD算法中的參數(shù)進(jìn)行優(yōu)化。相比于憑經(jīng)驗(yàn)確定值，這些優(yōu)化算法能夠根據(jù)原始信號(hào)自動(dòng)確定值，具有很好的自適應(yīng)性。但這些優(yōu)化算法的弊端也十分明顯，都存在計(jì)算量大、冗余度高、計(jì)算效率低等問(wèn)題[20]。

基于此本文提出一種基于排列熵的優(yōu)化算法。從噪聲輔助數(shù)據(jù)分析[21]的角度改進(jìn)VMD，進(jìn)一步提高信號(hào)的信噪比。同時(shí)為了減小重構(gòu)誤差，使所添加的白噪聲被完全中和，每次循環(huán)時(shí)，添加2個(gè)幅值相等、符號(hào)相反的白噪聲到原始信號(hào)，然后再利用VMD算法分別對(duì)其進(jìn)行分解，最終經(jīng)過(guò)多次循環(huán)，使原始信號(hào)中的噪聲相互抵消；將每次循環(huán)得到的各層IMF（intrinsic mode function）分別進(jìn)行集成平均，再根據(jù)集成均值的結(jié)果對(duì)信號(hào)進(jìn)行重構(gòu)[22]；對(duì)重構(gòu)信號(hào)再次進(jìn)行VMD分解，作為MVMD算法最終的結(jié)果輸出。

1 排列熵和變分模態(tài)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(cè)時(shí)間序列隨機(jī)性和動(dòng)力學(xué)突變的方法，該算法具有原理簡(jiǎn)單、計(jì)算效率高、魯棒性好等優(yōu)點(diǎn)，適用于非線(xiàn)性數(shù)據(jù)分析[24]，該算法的具體步驟如下

式中P表示第個(gè)符號(hào)出現(xiàn)的概率；H表示時(shí)間序列的復(fù)雜和隨機(jī)程度，H越大，說(shuō)明時(shí)間序列越隨機(jī)，H越小，說(shuō)明時(shí)間序列越規(guī)則。

1.2 變分模態(tài)分解

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

2 VMD算法改進(jìn)

該算法的具體步驟如下

1）設(shè)定的初始值為2，排列熵的閾值取經(jīng)驗(yàn)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(jìn)的VMD算法（modified VMD，MVMD）。根據(jù)文獻(xiàn)[26]可知，CEEMD為了能夠減小重構(gòu)誤差，使所添加的白噪聲被完全中和，所以在向待分解信號(hào)中添加白噪聲時(shí)，所添加的白噪聲為正負(fù)白噪聲對(duì)，通過(guò)該文獻(xiàn)中的仿真與試驗(yàn)分析可知，相比于EEMD中單純的添加正白噪聲，CEEMD中添加正負(fù)白噪聲對(duì)的方法達(dá)到了降低重構(gòu)誤差、促進(jìn)白噪聲相互中和的目的。因此，基于正負(fù)白噪聲對(duì)在降低重構(gòu)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(shí)，也采取添加正負(fù)白噪聲對(duì)的思想，即每次循環(huán)時(shí)所添加的白噪聲為2個(gè)幅值相等、符號(hào)相反的正負(fù)白噪聲對(duì)，加上正負(fù)白噪聲之間的中和作用，在實(shí)現(xiàn)降噪目的的同時(shí)又不引入新的噪聲。加入輔助白噪聲后，將得到2個(gè)待分解信號(hào)，然后再利用VMD算法分別對(duì)其進(jìn)行分解，經(jīng)過(guò)多次循環(huán)，原始信號(hào)中的噪聲將相互抵消，最終將每次循環(huán)得到的各層IMF分別進(jìn)行集成平均，根據(jù)集成均值的結(jié)果對(duì)信號(hào)進(jìn)行重構(gòu)，對(duì)重構(gòu)信號(hào)再次進(jìn)行VMD分解的具體步驟如下

4）重復(fù)步驟2)、3)，且每次循環(huán)開(kāi)始時(shí)加入新的高斯白噪聲對(duì)；

PEO-MVMD的流程如圖1所示。

注：k為分解層數(shù)；N為循環(huán)次數(shù)。下同。

3 仿真信號(hào)分析

齒輪箱發(fā)生復(fù)合故障時(shí)，其振動(dòng)信號(hào)往往是多調(diào)制源共存的。因此，采用齒輪故障仿真信號(hào)和滾動(dòng)軸承故障仿真信號(hào)進(jìn)行分析，構(gòu)造如下

圖2 仿真信號(hào)的時(shí)域波形

注：IMF為固有模態(tài)函數(shù)，下同。

MVMD算法中需要設(shè)置循環(huán)次數(shù)和所添加的白噪聲幅值std。兼顧信號(hào)處理的效率，本文取循環(huán)次數(shù)=100；當(dāng)白噪聲幅值std取0.15時(shí)，重構(gòu)信號(hào)的信噪比最高，即降噪效果最好，因此，對(duì)于仿真信號(hào)，取MVMD算法中所添加白噪聲幅值為0.15。

VMD分解結(jié)果如圖5所示，原始信號(hào)中30 Hz的低頻成分被成功的提取出來(lái)；但中頻的120 Hz信號(hào)，由于受到強(qiáng)背景噪聲的干擾，被分解到了IMF2和IMF3這2個(gè)模態(tài)中，發(fā)生了模態(tài)混疊現(xiàn)象，且頻譜特征十分微弱，易造成誤診斷。MVMD分解結(jié)果如圖6所示，在IMF1中，原始信號(hào)中30 Hz的低頻信號(hào)的頻譜特征十分明顯；在IMF2中，調(diào)幅信號(hào)的120 Hz中心頻率以及2個(gè)調(diào)制頻率也都成功的從含有噪聲的原始信號(hào)中分離出來(lái)，且邊頻帶均勻?qū)ΨQ(chēng)的分布在主頻兩側(cè)；在IMF3中，280 Hz的中心頻率以及均勻分布在其兩側(cè)的10 Hz多條邊頻帶也十分突出，雖然在500 Hz附近出現(xiàn)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(duì)故障特征的識(shí)別影響不大。對(duì)MVMD分解后的信號(hào)與原仿真信號(hào)進(jìn)行重構(gòu)，結(jié)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(gòu)效果很好。

圖4 EEMD分解后的IMFs與其對(duì)應(yīng)的頻譜

圖5 VMD分解后的IMFs與其對(duì)應(yīng)的頻譜 Fig.5 IMFs and spectrum after VMD

圖6 MVMD分解后的IMFs與其對(duì)應(yīng)的頻譜 Fig.6 IMFs and spectrum after MVVM

圖7 MVMD分解得到的IMF與組成信號(hào)對(duì)比

4 驗(yàn)證試驗(yàn)

4.1 試驗(yàn)方法與設(shè)置

1. 調(diào)速電機(jī)2. 聯(lián)軸器3. 陪試齒輪箱4. 轉(zhuǎn)速扭轉(zhuǎn)儀5. 扭力桿 6. 試驗(yàn)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(dòng)試驗(yàn)臺(tái)

Fig.8 Gear transmission test bench

如圖9所示，本試驗(yàn)中齒輪箱的復(fù)合故障包括齒輪剝落和軸承外圈故障。其中齒輪剝落通過(guò)齒輪疲勞試驗(yàn)產(chǎn)生，外圈故障通過(guò)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(jié)果

為驗(yàn)證本文所提算法的有效性，分別采用VMD和基于PEO-MVMD對(duì)上述復(fù)合故障信號(hào)進(jìn)行分解。圖10為VMD的分解結(jié)果，有混疊現(xiàn)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(duì)應(yīng)的頻譜

4.3 MVMD分解結(jié)果

采用PEO算法確定分解層數(shù)，設(shè)定初始值為2，根據(jù)是否出現(xiàn)過(guò)分解進(jìn)行循環(huán)迭代，查找的最優(yōu)值。最終PEO算法輸出的的最優(yōu)值為2；此外，取白噪聲幅值std為0.85，循環(huán)次數(shù)=100。MVMD算法對(duì)上述齒輪箱復(fù)合故障信號(hào)進(jìn)行分解。結(jié)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(lái)，而且頻率幅值遠(yuǎn)大于VMD提取結(jié)果，相對(duì)于VMD效果更佳。再次驗(yàn)證了文中所提方法的有效性。

圖11 MVMD分解后的IMFs與其對(duì)應(yīng)的頻譜

5 結(jié) 論

本文提出了基于變模態(tài)分解的改進(jìn)算法，即首先采用PEO算法根據(jù)待分解信號(hào)的特點(diǎn)自適應(yīng)地確定所需要分解的層數(shù)，再利用降噪效果優(yōu)異的MVMD對(duì)原始信號(hào)進(jìn)行分解。通過(guò)對(duì)齒輪箱試驗(yàn)信號(hào)進(jìn)行分析，試驗(yàn)結(jié)果表明，相比于VMD，本文提出的基于PEO的MVMD具有自適應(yīng)性和強(qiáng)降噪性能，診斷出了封閉式功率流試驗(yàn)臺(tái)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(yīng)確定VMD的值，PEO算法輸出的分解層數(shù)最優(yōu)值為2，并成功提取出齒輪故障特征的2倍頻720 Hz。

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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(jiān)，常 雪，王俊元，杜文華，段能全，黨長(zhǎng)營(yíng). 排列熵優(yōu)化改進(jìn)變模態(tài)分解算法診斷齒輪箱故障[J]. 農(nóng)業(yè)工程學(xué)報(bào)，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(guó)家自然科學(xué)基金（59975064）

王志堅(jiān)，博士，副教授，主要研究方向?yàn)樾D(zhuǎn)機(jī)械復(fù)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