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机械原理基于局部平均分解的阶次跟踪分析及其在齿轮故障诊断中的应用Junsheng Cheng, Kang Zhang, Yu Yang关键词： 阶次跟踪分析 局部平均分解 解调 齿轮 故障诊断摘要：局部平均分解(LMD)是一种新的自适应时频分析方法,这种方法特别适合处理多分量的调幅信号和调频(AM-FM)信号。通过使用LMD方法,可以将任何复杂的信号分解为一系列的产品功能PF分量(PFs),每个PF分量都是纯调频信号和包络信号的乘积，且通过纯调频信号可以获得具有物理意义的瞬时频率。从理论上讲,每个PF分量都是一个单分量的AM-FM信号。 因此,可以将LMD的过程看作是信号解调的过程。齿轮发生故障时,振动信号呈现明显的AM-FM特征。因此，针对齿轮升降速过程中故障振动信号为多分量的调制信号，以及故障特征频率随转速变化的特点，提出了一种基于LMD和阶次跟踪分析的齿轮故障诊断方法。齿轮箱的故障诊断实验表明本文提出的方法能有效地提出齿轮故障诊断特征。1 引言齿轮传动是机械设备中常见的传动方式, 故对齿轮进行故障诊断具有重要意义。齿轮故障诊断的关键一步是故障特征的提取。一方面，传统的齿轮故障诊断方法的重点在一个固定的旋转速度检测振动信号的频谱分析。 而齿轮作为一种旋转部件, 其升降速过程的振动信号往往包含了丰富的状态信息, 一些在平稳运行时不易反映的故障特征在升降速过程中可能会充分地表现出来1,此外，来自齿轮振动信号的暂态过程中，速度依赖性总是显示非平稳特征。如果频谱分析直接应用于非平稳振动信号，混频将不可避免的发生，这将对故障特征提取带来不良影响。在以往的研究中，为了跟踪技术，通常利用振动信号中添加旋转机械轴转速信息，已经成为一个在旋转机械故障诊断2,3的重要途径。从本质上讲，阶次跟踪分析技术可以在时域非平稳信号转换成角域静止，可以突出的旋转速度相关的振动信息和抑制无关的信息。因此，阶次跟踪分析是在助跑过程中齿轮的故障特征提取和运行了一个可取的方法 另一方面，当发生故障的齿轮振动信号，拿起在运行和运行过程中始终存在的振幅特性调制和频率调制（AMFM）。为了提取齿轮故障振动信号的调制特征，解调分析是最流行的方法之一 4,5 。然而，传统的解调方法，如希尔伯特变换解调和传统包络分析有其自身的局限性 6 。这些缺点包括两个方面：（1）在实践中大多数的齿轮故障振动信号都是多组分是调频信号。这些信号，在传统的解调方法，他们通常是通过带通滤波器分解成单组分是调频信号的解调，然后提取的频率和振幅信息。然而，这两个数载波频率的载波频率成分和幅值都难以在实践中被确定，所以带通滤波器的中心频率的选择具有主体性，将解调误差和使它提取机械故障振动信号的特征是无效的；（2）由于希尔伯特不可避免的窗口效应变换，当使用希尔伯特变换提取调制信息，目前的非瞬时响应特性，即，在调制信号被解调以及打破中间部分的两端会再次产生调制，使振幅指数衰减的方式得到的波动，然后解调误差将增加 7 。为了克服第一个缺点，一个合适的分解方法应寻找独立的多分量信号为多个单组分是调频信号的包络分析之前。由于EMD（经验模态分解）自适应复杂多分量信号分解为一系列固有模态函数（IMF）的瞬时频率的物理意义 8,9 ，基于EMD的阶比跟踪方法已广泛应用于齿轮故障诊断 13 。然而，仍然存在许多不足之处 14 ，如在EMD的端点效应和模态混 15 ，仍在进行。此外，对原信号通过EMD分解，产生了由希尔伯特变换（上面提到的）缺点是不可避免的在IMF进行希尔伯特变换的包络分析。此外，有时无法解释的负瞬态频率时会出现瞬时频率计算每个IMF进行希尔伯特变换 16 局部均值分解（LMD）是一种新型的解调分析方法，特别适合于处理多组分的幅度调制和频率调制（AM调频）信号 16 。用LMD，任何复杂的信号可以分解成许多产品功能（PFS），每一种产品的包络线信号（获得直接由分解）的PF瞬时振幅可以得到一个纯粹的频率调制信号从一个良好定义的瞬时频率可以计算。在本质上，每个PF正是一种单组分我调频信号。因此，LMD的程序可以，事实上，作为解调过程。调制信息可以通过频谱分析的瞬时振幅（包络信号，直接获得通过分解）每个PF分量进行希尔伯特变换，而不是由PF分量。因此，当LMD和EMD方法分别应用到解调分析，与EMD，LMD的突出优点是避免希尔伯特变换。此外，LMD迭代过程中所采用的手段和当地的幅度不平滑的地方用EMD的三次样条的方法，这可能带来的包络的误差和影响的精度瞬时频率和振幅。此外，与EMD端点效应相比并不明显，因为在LMD方法更快的速度和算法的迭代次数更少 17 。 基于以上分析，阶次跟踪和解调技术，LMD最近的发展，科学相结合，并应用于齿轮故障诊断过程中各轴速度。首先，订单跟踪技术被用于将从时间域的齿轮振动信号角域。其次，分解角域重采样信号的PF系列LMD，因此组件和相应的瞬时振幅和瞬时频率可以得到的。最后，进行频谱分析的故障信息含有显性PF分量的瞬时幅值。从实验的振动信号，表明该方法能有效地提取故障特征和分类准确齿轮工作状态的分析结果。 本文的组织如下。第2节是一个给定的LMD方法理论。在第3节中的齿轮故障诊断方法中，以技术和LMD跟踪相结合的提出和实践应用表明，提出的方法。此外，LMD和基于EMD的比较也在第3节提到了基础的方法。最后，我们得出了第4部分的结论。2 LMD 方法 LMD方法的本质是通过迭代从原始信号中分离出纯调频信号和包络信号,然后将纯调频信号和包络信号相乘便可以得到一个瞬时频率具有物理意义的PF分量,循环处理直至所有的PF分量分离出来对任意信号x(t),其分解过程如16: ( 1) 确定原始信号第i个局部极值及其对应的时刻,计算相邻两个局部极值和的平均值 (1) 将所有平均值点mi在其对应的时间段,内伸一线段,然后用滑动平均法进行0平滑处理,得到局均值m11(t) 。 ( 2) 采用局部极值点计算局部幅值 : =| -|/2 (2)将所有局部幅值点ai在其对应的时间段,内伸成一条线段,然后采用滑动平均法进行平滑处理,得到包估计函数a11(t) 。 ( 3) 将局部均值函数m11(t)从原始信号x(t)中分离来, 即去掉一个低频成分,得到 h11(t)=x(t)-m11(t) (3) ( 4)用h11(t)除以包络估计函数A11( t)以对h11(t)进行解调,得到 s11(t)=h11(t)/A11(t) (4) 对s11( t)重复上述步骤便能得到s11(t)的包络估计函数A12(t),若A12(t)不等于1,则s11( t)不是一个纯调频信号需要重复上述迭代过程n次,直至s1n(t)为一个纯调频信号,即 s1n(t)的包络估计函数 A1(n+1)(t)=1,所以,有 (5) (6)为理论上, 迭代终止的条件 (7) 在实践中,一种变体会提前确定。如果1a1(n + 1)(t)1 +and1s1n(t)1,然后迭代过程将停止( 5) 把迭代过程中产生的所有包络估计函数相乘便可以得到包络信号( 瞬时幅值函数) : （8）( 6) 将包络信号A1(t)和纯调频信号s1n(t)相乘便可以得到原始信号的第一个PF分量: PF1(t）=a1(t)s1n(t) ( 9)PF1(t)包含了原始信号中频率值最高的成分,是一个单分量的调幅-调频信号,PF1(t)的瞬时幅值就是包络信号A1(t),PF1(t)的瞬时频率f1(t)则可由纯调频信号s1n(t)求出,即: (10) ( 7)将第一个PF分量PF1(t)从原始信号x(t)中分离出来, 得到一个新的信号u1(t),将u1( t)作为原始数据重复以上步骤,循环k次,直到 uk为一个单调函数为止,即: (11) 原始信号x(t)能够被所有的PF分量和uk重构,即: (12) 产品功能p的数量在哪里.此外,相应的完整的时频分布可以通过组装瞬时幅度和瞬时频率的PF组件。3 基于阶次跟踪分析与 L M D 的齿轮故障诊断3.1 阶次跟踪分析 阶次跟踪分析首先根据参考轴的转速信息对时域信号进行等角度重采样, 将时域非平稳信号转换为角域平稳信号, 再对角域平稳信号进行谱分析得到阶次谱。阶次跟踪分析能够提取信号中与参考轴转速有关的信息, 同时抑制与转速无关的信号, 因此非常适合分析旋转机械在变转速过程下的振动信号。实现阶次跟踪分析技术的关键在于, 如何实现被分析信号相对于参考轴的等角度重采样, 即阶次重采样。常用的阶次重采样方法有硬件阶次跟踪法 6、计算阶次跟踪法 7和基于瞬时频率估计的阶次跟踪法 8等。硬件阶次跟踪法直接通过专用的模拟设备实现信号的等角度重采样,实时性好,但只适用于轴转速较稳定的情况,且成本很高;基于瞬时频率估计的阶次跟踪法不需要专门的硬件设备,无需考虑硬件安装问题,且成本较低, 但是不适用于分析多分量信号,而实际工程信号大多为多分量信号, 因此其实际应用意义不大;COT法通过软件的形式实现等角度重采样,分析精度高, 对被分析的信号没有特别的要求,并且无需特定的硬件, 因此是一种应用广泛的阶次跟踪分析方法。根据试验条件采用COT法实现信号的阶次重采样,其具体步骤如下:1. 对振动信号和转速信号分两路同时进行等时间间隔(间隔为$t)采样,得到异步采样信号;2. 通过转速信号计算等角度增量 $H 所对应的时间序列ti ;3. 根据时间序列ti的值,对振动信号进行插值,求出其对应的幅值,得到振动信号的同步采样信号,即角域平稳信号;4.使用LMD分解平衡角重采样信号,因此sPF系列组件和相应的瞬间振幅和瞬时频率可以获得5.光谱分析应用于每个PF的瞬时振幅组件,然后我们有订单谱3.2 齿轮故障诊断实例升降速过程中的齿轮故障振动信号通常是多分量的调幅-调频信号,并且故障特征频率会随着转速的变化而改变。针对升降速过程齿轮故障振动信号的这些特点, 提出了基于阶次跟踪分析和 LM D 的齿轮故障诊断方法。首先采用阶次跟踪分析将齿轮升降速过程的时域振动信号转换成角域平稳信号;然后对角域信号进行LMD分解,得到一系列PF分量,以及各个PF分量的瞬时幅值和瞬时频率; 最后对各个PF分量的瞬时幅值进行频谱分析,便可以有效地提取出齿轮故障特征。为了验证方法的正确性,在旋转机械试验台上进行了齿轮正常和齿根裂纹两种工况的试验。该系统中, 电机输入轴齿轮齿数z1=55, 输出轴齿轮齿数z2 = 75。在输入轴齿轮齿根上加工出小槽,以模拟齿根纹故 障, 因此齿轮啮合阶次xm=55,故障特征阶次xc=1。图1和图2所示分别为由转速传感器测得的输入轴瞬时转速n(t),以及由振动传感器测得的齿轮故障 振动加速度a(t),其中采样频率为8192H z,采样时间为20s从图1可以看出,输入轴转速首先从150r/min逐渐加速至1410r/min, 然后再减速到820r/min,而加速度信号的幅值也随着作出了相应的变化。不失一般性,截取图2中5 7s升速过程的信号 a1(t)进行分析。 图 1 输 入轴的瞬时转速 n ( t )图 2 齿轮故障振动加速度信号 a( t )值在秩序O=55和O=110相应的齿轮啮合秩序和双。因此这意味着频率混淆现象已经在很大程度上消除。然而,为j1()仍然是一个多个组件MA-MF信号。因此,一边频带反映故障特征频率模糊。有效地提取故障特征,应用LMD j - 1(),因此七PF组件和残渣可以得到图6所示,这意味着LMD解调的进展。因此,它是可以提取齿轮故障特性,利用频谱分析的瞬时振幅PF组件包含主要故障信息。通过分析,我们知道失败的主要信息包括在第一个PF组件。因此,无花果。7和8给瞬时振幅a1()的第一个PF组件PF 1()和相应的秩序光谱的a1(),很明显,有不同的光谱峰值在第一顺序(O = 1)对应齿轮阶次跟踪功能,符合齿轮的实际工况。图9和图10显示转速信号的n(t)和振动加速度信号的时域波形s(t)齿轮分别与破碎的牙齿,采样率为8192 Hz和总样品时间是20年代。断齿故障引入输入轴上的齿轮与激光切割槽的牙根。首先,一段信号s1(t)5 s-7年代为进一步分析的进步是拦截;其次,假设样本点每旋转400;第三,角域信号为j1()图11所示可以通过执行命令重采样s1(t);第四,LMD适用于j-1();最后,相应的秩序频谱图12所示的瞬时振幅首先PF组件PF 1()可以了,很明显,有不同的光谱峰值(比在图8)在第一顺序(O = 1)阶次跟踪分析对应于齿轮故障功能,符合齿轮的实际工况。同样的,我们同样可以做正常的齿轮。转速信号n(t)和振动的时域波形加速度信号s(t)的正常齿轮分别列在无花果。13和14,采样率为8192 Hz和总样品时间是20多岁。在上述相同的方法应用于原始信号图14所示,结果无花果所示。15和16。图15显示了角域j - 1()执行顺序重采样后的信号部分(5s-7年代在筹备进展)的原始信号。图16显示了相应的瞬时振幅谱第一个PF组件,很难找到齿轮故障特征,也符合实际的工作状态的装备。目前,多组分的另一个竞争解调方法AM-FM信号,即经验模式分解(EMD)存在,已经被广泛应用于信号解调分析(7、22)。为了比较两个EMD方法,取代LMD,我们能做的同样使用EMD进行重采样信号无花果所示。图4、11和15 图 3 齿轮故障振动加速度信号的频谱图 4 阶次重采样后的齿轮故障振动 加速度信号图5 j1()的阶次谱分别,因此可以获得一系列国际货币基金组织(IMF)组件。此外,相应的瞬时振幅和国际货币基金组织每个组件的瞬时频率可以通过希尔伯特变换计算。通过分析,我们知道,IMF主要特征信息包含在第一个组件。因此,只有应用于瞬时频谱分析第一个国际货币基金组织(IMF)组件的振幅。无花果。17日至19日给订单频谱对应三种振动信号的破解断层、断齿故障和正常的齿轮,分别,很明显,订单跟踪分析基于EMD也可以提取齿轮故障特性,确定齿轮的工作状态。尽管EMD和LMD都可以分解原始信号实际上,两种方法之间的差异仍然存在。EMD方法比较,如第一节中所述,LMD有更多迭代次数少等优点,不明显的效果和更少的瞬时频率的虚假成分,可以使用更多的应用在实践中。 图 6 角域信号j1( )的LMD分解结果图 7 PF1()的瞬时幅值A1()图 8 第1个PF分量的幅值谱图 9 输入轴的瞬时转速 n(t)图 1 0 正常齿轮的振动加速度信号 a(t)图11 阶次重采样后的正常齿轮振动加速度信号j1(）图 12 第一个PF分量的幅值谱图13 输入轴转速r(t)正常齿轮前和过程中图图14 齿轮的振动加速度信号(t)在正常状态图15 相应的振动加速度信号为j1()角域通过应用顺序重采样tos(t)图14所示。图17 第一个IMF分量的幅值谱 图 18 第一个IMF分量的幅值谱3 结论 在齿轮故障诊断技术、阶次跟踪是一个著名的技术,可用于故障检测的旋转机器采用振动信号。针对齿轮故障振动信号的调制特点在助跑和破败的和缺点在齿轮经常可以发相关轴转速在瞬态过程中,阶次跟踪和技术LMD相结合用于齿轮故障诊断。从理论分析和实验结果以下几点得出结论: ( 1) 在分析齿轮变转速状态下的振动信号时,转速波动会引起频谱图出现频率混叠, 而阶次跟踪分析通过对信号进行阶次重采样能够在很大程度上消除频率混叠, 使频谱图的谱线清晰可读。 ( 2) 齿轮故障时的振动信号为一多分量的调幅- 调频信号, 采用LMD方法能将其分解为若干个PF分量之和,同得到各个PF分量的瞬时幅值和瞬时频率, 实现了原信号的解调。对含有齿轮故障特征的PF分量的瞬时幅值进行频谱分析, 能够准确地提取出齿轮故障特征信息。 图19 阶次的第一个国际货币基金组织(IMF)组件的正常使用EMD齿轮 ( 3) 对齿轮正常和齿根裂纹两种工况的振动信号进行了分析,分析结果表明, 本文方法能够准确地反映出齿轮的实际工况。References1 S.K. 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Batista, On the HHT, its problems, and some solutions, Mechanical Systems and Signal Processing 22 (6) (2008) 13741394.An order tracking technique for the gear fault diagnosis using local meandecomposition methodJunsheng Cheng, Kang Zhang, Yu YangState Key Laboratory of Advanced Design and Manufacturing for Vehicle Body, Hunan University, Changsha, 410082, PR ChinaCollege of Mechanical and Vehicle Engineering, Hunan University, Changsha, 410082, PR Chinaa r t i c l ei n f oa b s t r a c tArticle history:Received 17 November 2010Received in revised form 13 December 2011Accepted 30 April 2012Available online 28 May 2012Local mean decomposition (LMD) is a new self-adaptive timefrequency analysis method,which is particularly suitable for the processing of multi-component amplitude-modulatedand frequency-modulated (AMFM) signals. By using LMD, any complicated signal can bedecomposed into a number of product functions (PFs), each of which is the product of anenvelope signal and a purely frequency modulated signal from which physically meaningfulinstantaneous frequencies can be obtained. Theoretically, each PF is exactly a mono-componentAMFM signal. Therefore, the procedure of LMD can be regarded as the process of demodulation.While fault occurs in gear, the vibration signals would exactly present AMFM characteristics.Therefore, targeting the modulation feature of gear fault vibration signal in run-ups and run-downs and the fact that fault characteristics found in gear vibration signal could often be relatedto revolution of the shaft in the transient process, a gear fault diagnosis method in which ordertracking technique and local mean decomposition is put forward. The analysis results from thepractical gearbox vibration signal demonstrate that the proposed algorithm is effective in gearfault feature extraction. 2012 Elsevier Ltd. All rights reserved.Keywords:Order tracking techniqueLocal mean decompositionDemodulationGearFault diagnosis1. IntroductionGears are the important and frequently encountered components in the rotating machines that find widespread industrialapplications. Therefore, the corresponding gear fault diagnosis has been the subject of extensive research.The key step of gear fault diagnosis is the extraction of fault feature. On the one hand, the conventional gear fault diagnosismethods focus on examining the frequency spectrum analysis of vibration signal at a fixed rotation speed. Unfortunately, theinformation obtained thus is only partial because some faults maybe do not respond significantly at the fixed operation speed.Since faults commonly found in gear could often be related to revolution of the shaft, more comprehensive information may beacquired by measuring the gear vibration signal in the process of run-up and run-down 1. In addition, vibration signals derivedfrom gear in the transient process that are speed-dependent always display non-stationary feature. If frequency spectrum analysisis directly applied to the non-stationary vibration signal, frequency mixing would occur inevitably, which will bring undesirableeffect to the fault feature extraction. In past research, order-tracking technique, which normally exploits a vibration signalsupplemented with information of shaft speed of rotating machinery, has become one of the significant approaches for faultdiagnosis in rotating machinery 2,3. Essentially, order-tracking technique can transform a non-stationary signal in time domaininto stationary one in angular domain, which can highlight the vibration information related to rotation speed and restrain theunrelated information. Therefore, order tracking is a desirable method to extract gear fault feature in the process of run-up andrun-down.Mechanism and Machine Theory 55 (2012) 6776 Corresponding author at: State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body, Hunan University, Changsha, 410082, PR China.Tel.: +86 731 88664008; fax: +86 731 88711911.E-mail address: signalp (J. Cheng).0094-114X/$ see front matter 2012 Elsevier Ltd. All rights reserved.doi:10.1016/j.mechmachtheory.2012.04.008Contents lists available at SciVerse ScienceDirectMechanism and Machine Theoryjournal homepage: /locate/mechmtOn the other hand, while faults occur in gears, the vibration signal picked up in run-up and run-down process always presentthe characteristics of amplitude-modulated and frequency-modulated (AMFM). In order to extract the modulation featureof gear fault vibration signals, demodulation analysis is one of the most popular methods 4,5. However, conventionaldemodulation approaches such as Hilbert transform demodulation and traditional envelope analysis have their own limitations6. These drawbacks include two aspects: (1) in practice most gear fault vibration signals are all multi-component AMFMsignals. For these signals, in conventional demodulation approaches, they are usually decomposed into single component AMFMsignals by band-pass filter and then demodulated to extract frequencies and amplitudes information. However, both the numberof the carrier frequency components and the magnitude of the carrier frequency are hard to be determined in practice, so theselection of central frequency of band-pass filter carries great subjectivity that would bring demodulation error and make itineffective to extract the characteristic of machinery fault vibration signal; (2) owing to the inevitable window effect of Hilberttransform, when Hilbert transform is used to extract the modulate information, the demodulation results present non-instantaneous response characteristic, that is, at the two ends of the modulated signal which has been demodulated as well as themiddle part with break would produce modulation again, which makes the amplitude get fluctuation in an exponentialattenuation way, and then the demodulation error would increase 7. In order to overcome the first drawback, an appropriatedecomposition method should be looked for to separate multi-component signal into a number of single component AMFMsignals before the envelope analysis. Since EMD (Empirical mode decomposition) could adaptively decompose a complicatedmulti-component signal into a sum of intrinsic mode functions (IMFs) whose instantaneous frequencies have physicalsignificance 8,9, order tracking method based on EMD has been widely used in the gear fault diagnosis 1013. However, therestill exist many deficiencies in EMD such as the end effects 14 and modes mixing 15 that are still underway. In addition, afterthe original signal is decomposed by EMD, the drawback produced by Hilbert transform (above mentioned) is inevitable whenIMF is performed envelope analysis by Hilbert transform. Moreover, sometimes the unexplainable negative instantaneousfrequency would appear when calculating instantaneous frequency by performing Hilbert transform to each IMF 16.Local mean decomposition (LMD) is a novel demodulation analysis method, which is particularly suitable for the processing ofmulti-component amplitude-modulated and frequency-modulated (AMFM) signals 16. By using LMD, any complicated signalcan be decomposed into a number of product functions (PFs), each of which is the product of an envelope signal (obtaineddirectly by the decomposition) from which instantaneous amplitude of the PF can be obtained and a purely frequency modulatedsignal from which a well-defined instantaneous frequency could be calculated. In essence, each PF is exactly a mono-componentAMFM signal. Therefore, the procedure of LMD could be, in fact, regarded as the process of demodulation. Modulationinformation can be extracted by performing spectrum analysis to the instantaneous amplitude (envelope signal, obtained directlyby the decomposition) of each PF component rather than by performing Hilbert transform to the PF components. Hence, whenLMD and EMD are applied to the demodulation analysis respectively, compared with EMD, the prominent advantage of LMD is toavoid the Hilbert transform. In addition, the LMD iteration process which uses smoothed local means and local magnitudes avoidsthe cubic spline approach used in EMD, which maybe bring the envelope errors and influence on the precision of theinstantaneous frequency and amplitude. Moreover, compared with EMD the end effect is not obvious in LMD approach because offaster algorithm speed and less iterative times 17.Based upon the above analysis, order-tracking analysis and the recent development of demodulation techniques, LMD, arecombined and applied to the gear fault diagnosis of various shaft speeds process. Firstly, order tracking technique is used totransform the gear vibration signals from time domain to angular domain. Secondly, decompose the re-sampling signal of angulardomain by LMD, thus s series PF components and corresponding instantaneous amplitudes and instantaneous frequencies can beobtained. Finally, spectrum analysis is carried out to the instantaneous amplitudes of the PF component containing dominant faultinformation. The analysis results from the experimental vibration signal show that the proposed method can extract fault featureof the gear effectively and classify working condition accurately.This paper is organized as follows. A theory of the LMD approach is given in Section 2. In Section 3 a gear fault diagnosisapproach in which order tracking technique and LMD are combined is put forward and the practice applications of proposedmethod are demonstrated. In addition, the comparison between LMD-based and EMD-based method is also given in Section 3.Finally, we offer the conclusion in Section 4.2. LMD analysis methodAs mentioned above, the nature of LMD is to demodulate AMFM signals. By using LMD a complicated signal can bedecomposed into a set of product functions, each of which is the product of an envelope signal and a purely frequency modulatedsignal. Furthermore, the completed timefrequency distribution of the original signal can be obtained. For any signal x(t), it can bedecomposed as follows 16:(1) Determine all local extrema niof the original signal x(t), and then the mean value miof two successive extrema niand ni+1can be calculated bymini ni121All mean value miof two successive extreme are connected by straight lines, and then local mean function m11(t)can be formed by using moving averaging to smooth the local means mi.68J. Cheng et al. / Mechanism and Machine Theory 55 (2012) 6776(2) A corresponding envelope estimate aiis given byainini1?22Similarly, the envelope estimate aiis smoothed in the same way and the corresponding envelope function a11(t) isformed.(3) The local mean function m11(t) is subtracted from the original signal x(t) and the resulting signal h11(t) is given byh11t x t m11t 3(4) h11(t) can be amplitude demodulated by dividing it by envelope function a11(t)s11t h11t =a11t 4Ideally, s11(t) is a purely frequency modulated signal, namely, the envelope function a12(t) of s11(t) should satisfya12(t)=1. If a12(t)1, then s11(t) is regarded as the original signal and the above procedure needs to be repeateduntil a purely frequency modulated signal s1n(t) that meets 1s1n(t)1 is derived. In other words, envelopefunction a1(n+1)(t) of the resulting s1n(t) should satisfy a1(n+1)(t)=1. Thereforeh11t x t m11t h12 s11t m12t h1nt s1 n1t m1nt 8:5in which,s11t h11t =a11t s12t h12t =a12t s1nt h1nt =a1nt 8:6where the objective is thatlimna1nt 17In practice, a variation can be determined in advance. If 1a1(n+1)(t)1+ and 1s1n(t)1, then iterativeprocess would be stopped.(5) Envelope signal a1(t), namely, instantaneous amplitude function, can be derived by multiplying together the successiveenvelope estimate functions that are acquired during the iterative process described above.a1t a11t a12t a1nt nq1a1qt 8where q is the times of the iterative process.(6) Multiplying envelope signal a1(t) by the purely frequency modulated signal s1n(t) the first product function PF1of theoriginal signal can be obtained.PF1t a1t s1nt 9PF1contains the highest frequency oscillations of the original signal. Meantime, it is a mono-component AMFMsignal, whose instantaneous amplitude is exactly the envelope signal a1(t) and instantaneous frequency is definedfrom the purely frequency modulated signal s1n(t) asf1t 12d arccos s1nt ?dt10(7) Subtract the first PF component PF1(t) from the original signal x(t) and we have a new signal u1(t), which becomes the neworiginal signaland the whole of the above procedure is repeated,i.e. up tok times,until ukbecomes monotonic functionu1t x t PF1t u2t u1t PF2t ukt uk1t PFkt 8:1169J. Cheng et al. / Mechanism and Machine Theory 55 (2012) 6776Thus, the original signal x(t) was decomposed into k-product and a monotonic function ukx t Xkp1PFpt ukt 12where p is the number of the product function.Furthermore, the corresponding complete timefrequency distribution could be obtained by assembling the instantaneousamplitude and instantaneous frequency of all PF components.3. The gear fault diagnosis method based on order tracking technique and LMD3.1. Order tracking analysis and the corresponding fault diagnosis methodOrder-tracking technique could transform a non-stationary signal in time domain into a stationary signal in angular domain byapplying equi-angular re-sampling to vibration signal with reference to shaft speed. Furthermore, order spectrum can be obtainedby using spectrum analysis to stationary signal in angular domain, thus the information related to rotation speed can behighlighted and the unrelated one could be restrained. Therefore, order-tracking is suitable for the vibration signal analysis ofrotation machine.There are three popular techniques for producing synchronously sampled data: a traditional hardware solution, computedorder tracking (COT) and order tracking based on estimation of instantaneous frequency 1820. The traditional hardwareapproach, which uses specialized hardware to dynamically adapt the sample rate, is only suitable for the case that rotating speedof shaft is relatively smooth, thus resulting to a high cost. The method of order tracking based on estimation of instantaneousfrequency has no need for specialized hardware and thus cost is relatively low, however, it has failed to analyze multiplecomponent signal. While in practice most gear fault vibration signals exactly present the characteristic of multi-component.Therefore, this technique has little practice significance. COT technique realized equi-angular re-sampling by software, thereforeit not only requires no specialized hardware, but also have no limitation for analysis signal that means it is more flexible and moreaccurate. Just for this reason, COT is introduced into the gear fault detection in this paper.The step of the gear fault diagnosis method based on order tracking technique and LMD can be listed as follows:(1) The vibration signals and a tachometer signal are asynchronously sampled, that is, they are sampled conventionally atequal time incrementst;(2) Calculate the time series ticorresponding to equi-angular increments by tachometer signals;(3) According to the time series ti, apply interpolation to the vibration signals, thus the synchronous sampling signal, namely,stationary signal in angular domain, can be obtained;(4) Use LMD to decompose the equi-angular re-sampling signal, thus s series PF components and corresponding instantaneousamplitudes and instantaneous frequencies can be acquired;(5) Apply spectrum analysis to the instantaneous amplitude of each PF component, and then we have the order spectrum.3.2. ApplicationSince the gear fault vibration signal in run-up and run-down process are always multiple component AMFM signals and faultfeature frequency would vary with rotation speed, the fault diagnosis method in which order tracking technique and LMD arecombined would be suitable for gear fault detection.To verify the effectiveness of the proposed method, the fault diagnosis method based on order tracking technique and LMDwas applied to the experimental gear vibration signals analysis. An experiment has been carried out on the rotating machinerytest rig that is used for modeling different gear faults 21. Here we consider three working conditions that are gear with normalcondition, with cracked tooth and with broken tooth. Standard gears with teeth number z=55 and z=75 are used on input andoutput shafts respectively, in which the crack fault is introduced into the gear on the input shaft by cutting slot with laser in theroot of tooth, and the width of the slot is 0.15 mm, as well as its depth is 0.3 mm. Therefore, the mesh order is xm=55 and thefault feature order is xc=1. Figs. 1 and 2 give the rotation speed signal r(t) picked up by a tachometer and vibration accelerationsignal s(t) of the gear with crack fault collected by a piezoelectric acceleration sensor respectively, in which the sample frequencyis 8192 Hz and total sample time is 20 s, and from which we know the speed of input shaft increased gradually from 150 rpm to1410 rpm, then decreased to 820 rpm. Meantime, the amplitude of vibration acceleration signal accordingly changed, from whicha section of signal s1(t) of 5 s7 s in the run-up progress is intercepted for further analysis. Fig. 3 gives the spectrum of s1(t) byapplying spectrum analysis directly to vibration signal. For the rotation speed changes with time, the frequency mixing arises.Therefore, it is impossible to find meshing frequency and fault feature frequency in Fig. 3. As a result, actual gear workingcondition cannot be identified. Replace direct spectrum analysis by the order tracking method. Firstly, assume sample point perrotation is 400, namely, the maximum analysis order is 200. Secondly, angular domain signal j1() shown in Fig. 4 can be obtainedby performing order re-sampling to s1(t), in which horizontal ordinate has changed from time to radian. Thirdly, thecorresponding order spectrum of j1() can be calculated that is illustrated in Fig. 5, from which we can find obvious spectral peak70J. Cheng et al. / Mechanism and Machine Theory 55 (2012) 6776values at order O=55 and O=110 corresponding to gear meshing order and the double. Thus it means that frequency aliasingphenomenon has been eliminated to a large degree. However, j1() is still a multiple component MAMF signal. Therefore, sidefrequency band reflecting fault feature frequency is indistinct. To extract fault characteristic effectively, apply LMD to j1(), thusseven PF components and a residue can be obtained shown in Fig. 6, which means LMD is a demodulation progress. Therefore, it ispossible to extract gear fault feature by utilizing spectrum analysis to the instantaneous amplitude of PF component containingdominant fault information. By analysis, we know that the main failure information is included in the first PF component.Therefore, Figs. 7 and 8 give instantaneous amplitude a1() of the first PF component PF1() and the corresponding orderspectrum of a1(), from which it is clear that there are distinct spectral peak value at the 1st order (O=1) corresponding to gearfault feature order xc, which accords with the actual working condition of the gear.Figs. 9 and 10 show the rotation speed signal n(t) and the time domain waveform of vibration acceleration signal s(t) of thegear with broken tooth respectively, in which the sample rate is 8192 Hz and total sample time is 20 s. The broken tooth fault isintroduced into the gear on the input shaft by cutting slot with laser in the root of tooth. Firstly, a section of signal s1(t) of 5 s7 sin the run-up progress is intercepted for further analysis; secondly, assume sample point per rotation is 400; thirdly, angulardomain signal j1() shown in Fig. 11 can be obtained by performing order re-sampling to s1(t); fourthly, apply LMD to j1();finally, the corresponding order spectrum shown in Fig. 12 of instantaneous amplitude of the first PF component PF1() can beacquired, from which it is clear that there are distinct spectral peak value (it is bigger than that in Fig. 8) at the 1st order (O=1)corresponding to gear fault feature order xc, which accords with the actual working condition of the gear.Similarly, we can do likewise for the normal gear. The rotation speed signal n(t) and the time domain waveform of vibrationacceleration signal s(t) of the normal gear are listed in Figs. 13 and 14 respectively, in which the sample rate is 8192 Hz and totalsample time is 20 s. After the same method mentioned above is applied to the original signal shown in Fig. 14, the results areshown in Figs. 15 and 16. Fig. 15 shows the angular domain signal j1() after performing order re-sampling to the section (5 s7 sin the run-up progress) of the original signal. Fig. 16 shows the corresponding order spectrum of instantaneous amplitude of thefirst PF component, from which it is difficult to find gear fault feature order, which also accords with the actual working conditionof the gear.At present, another competing demodulation method for multi-component AMFM signal, namely, empirical modedecomposition (EMD), already exist and have been widely used in signal demodulation analysis7,22. In order to compare twoapproaches, replacing LMD by EMD, we can do likewise using EMD for the re-sampling signals shown in Figs. 4, 11 and 15Fig. 2. The vibration acceleration signal s(t) of the gear with crack fault.Fig. 3. The spectrum of vibration signal of the gear with crack fault.Fig. 1. The input shaft speed r(t) of the cracked gear in the run-up and run-down process.71J. Cheng et al. / Mechanism and Machine Theory 55 (2012) 6776respectively, thus a series IMF component can be obtained. Furthermore, the corresponding instantaneous amplitude andinstantaneous frequency of each IMF component can be calculated by Hilbert transform. By analysis, we know that the dominantfeature information is included in the first IMF component. Therefore, spectrum analysis is only applied to the instantaneousamplitude of the first IMF component. Figs. 1719 give the order spectrum corresponding to three vibration signals of crackedfault, broken tooth fault and normal gear, respectively, from which it is clear that order tracking analysis based on EMD can alsoextract gear fault feature and identify gear working condition. Although both EMD and LMD can decompose the original signaleffectively, the difference between two methods still exists. Comparing to EMD method, as mentioned in Section 1, LMD have moreadvantages such as less iterative times, unobvious end effect and less phoniness components of the instantaneous frequency, whichmake it possible to use for more applications in practice.Fig. 4. The corresponding vibration acceleration signalj1() in angular domain by applying order re-sampling tos(t) shown in Fig. 2.Fig. 5. The order spectrum of j1().Fig. 6. The decomposition results of j1() by LMD.72J. Cheng et al. / Mechanism and Machine Theory 55 (2012) 6776Fig. 8. The order spectrum of the first PF component shown in Fig. 6.Fig. 9. The input shaft speed r(t) of the gear with broken tooth in the run-up and run-down process.Fig. 7. The instantaneous amplitude a1() of PF1().Fig. 10. The vibration acceleration signal s(t) of the gear with broken tooth.Fig. 11. The corresponding vibration acceleration signalj1() in angular domain by applying order re-sampling tos(t) shown in Fig. 10.73J. Cheng et al. / Mechanism and Machine Theory 55 (2012) 6776Fig. 12. The order spectrum of the first PF component of the broken gear fault vibration signal.Fig. 13. The input shaft speed r(t) of the normal gear in the process of the run-up and run-down.Fig. 14. The vibration acceleration signal s(t) of a gear under normal state.Fig. 15. The corresponding vibration acceleration signal j1() in angular domain by applying order re-sampling to s(t) shown in Fig. 14.Fig. 16. The order spectrum of the first PF component of the normal gear vibration signal.74J. Cheng et al. / Mechanism and Machine Theory 55 (2012) 67764. ConclusionIn gear fault diagnostic technology, order tracking is a well-known technique that can be used for fault detection of rotationmachinery by using vibration signals. Targeting the modulation feature of gear fault vibration signal in run-ups and run-downsand the fact that faults found in gear could often be related to shaft speed in the transient process, order tracking technique andLMD are combined to use for the gear fault diagnosis. From the theory analysis and experiment results the following points can beconcluded:(1) When vibration signal under various shaft speed condition is processed, frequency mixing resulted from conventionalspectrum analysis can be overcome by introducing order tracking technique, which make the resulting spectrum linereadable.(2) Considering that the corresponding vibration signal often displays the AMFM feature when faults occur in gear, LMDapproach is applied to demodulation. By using LMD, signal can be decomposed into a number of product functions.Meantime instantaneous amplitude and instantaneous frequency of each PF component can be obtained, thusdemodulation of original signal eventually is realized. Furthermore, gear fault feature can be extracted accurately byapplying spectrum analysis to the instantaneous amplitude of certain PF component including dominant featureinformation. In the proposed method, since the instantaneous amplitude can be obtained directly from the process of LMDFig. 17. The order spectrum of the first IMF component of the gear with cracked tooth by using EMD.Fig. 18. The order spectrum of the first IMF component of the gear with broken tooth by using EMD.Fig. 19. The order spectrum of the first IMF component of the normal gear by using EMD.75J. Cheng et al. / Mechanism and Machine Theory 55 (2012) 6776other than using Hilbert transform, the limitations which is produced by Hilbert transform (mentioned in Section 1) can beavoided.(3) The analysis results from experimental signals with normal and defective gears show that the diagnosis approach proposedcould identify gear status-with or without fault accurately and effectively.AcknowledgmentsThis work was supported by the Chinese National Science Foundation Grant (no. 50775068, no. 51075131), Hunan ProvincialNaturalScienceFoundationofChina(no.11JJ2026)andHigh-Te