胡茑庆
论文题目：转子碰摩非线性行为与故障辨识的研究
作者简介：胡茑庆，男，1967年1月出生，1997年09月师从国防科技大学温熙森教授，于2001年12月获博士学位。
摘
要
世界是非线性的，线性只是其近似描述。20世纪下半叶兴起的混沌理论为非线性动力学系统的研究开创了新途径。混沌的发现表明：某些确定性的非线性系统具有内在的随机行为。过去20多年来，对无所不在的混沌现象研究几乎遍及每个学科领域。对复杂机械系统中可能出现的强非线性行为（如混沌）展开研究并探讨混沌理论在机械故障诊断中的应用，对于复杂机械系统的设计、使用、诊断与维修，特别是早期故障预示具有重要意义。
随着机械运转速度的日益提高以及各种新型材料在高速机械中的广泛应用，机械系统的非线性将更加突出，可能直接（或间接）导致机械系统的故障，其中的许多关键的理论和技术问题有待于进一步深入研究。因此，从理论和实验上对这个问题进行研究意义重大，而对非线性行为特别是混沌行为的预测与机器运行状态早期检测，以及利用基于混沌理论的信号/信息处理方法的研究显得尤为突出。正是在这样的背景下并受探究世界本原的科学精神的感召，本文在国家自然科学基金项目（“转子系统混沌行为与故障的关联关系及预测方法研究”）的资助下，较早地对转子系统这种典型的机械系统中可能出现混沌现象进行了深入探索；同时，对基于混沌理论的系统辨识与信号处理技术进行了较早深入的研究。主要是从机械故障诊断和信号处理的角度切入和展开的，旨在回答：实际的转子系统运动响应中是否存在复杂的非线性行为（混沌行为）？如何通过观测时间序列数据来判断、辨识系统的复杂非线性行为？表征非线性行为的特征指数与故障是否有一定的关联关系？如何将混沌理论（如混沌振子模型、随机共振原理等）应用于转子系统故障的早期诊断与预测？
本论文主要完成两个方面的研究工作：从碰摩转子实验系统中观测混沌现象并进行辨识与分析；基于非线性科学理论与技术对转子系统的行为进行分析、预测与早期辨识。提出了一些新的见解。论文的主要研究工作和创新成果如下：
1．深入系统地研究了碰摩转子的非线性行为与特征规律。首先采用并改进了已有的转子尖锐碰摩模型，通过定量和定性的理论分析，获得了尖锐碰摩转子振动响应形式。设计了尖锐碰摩转子试验台并开展了细致的实验研究，获得了不同碰摩情况下的振动响应特征规律。通过理论和实验分析，得出在尖锐碰摩情况下，早期碰摩在一定条件下会出现1/3
、2/3
的分频成分的结果（
表示工频）。通过理论和实验分析归纳出转子尖锐碰摩不同阶段下的特征规律及相应的辨识方法，这些结果对这类故障的早期检测具有重要的参考价值和指导作用。其次，建立了基于局部碰摩力变化且具有转、定子偏心的Jeffcott非线性转子的动力学模型，大量的数值仿真表明局部碰摩转子存在分频现象和一定的分叉规律，获得了大不平衡、小阻尼、高转速条件下，局部碰摩容易产生拟周期或混沌振动的结果。基于数值分析结果，设计并建立了局部碰摩转子系统试验台，在大范围的转速里进行了细致的实验研究，观察到了包括周期、拟周期、次谐波与超谐波以及混沌振动在内的丰富的振动现象，观测与仿真结果定性一致。最后，采用基于观测时间序列的重构相空间分析方法对转子系统的非线性行为进行辨识与分析，获得了系统出现强非线性行为的统计意义上的证据。理论与实验分析获得的碰摩转子振动响应的特征规律对于碰摩的预测具有一定的参考价值。
2．从非线性动力学分析的全新角度，系统地完善并发展了基于观测数据的混沌识别技术。即在只能获得短数据集条件下，提出了具有工程化前景的相空间重构技术和统计特征指数算法，可以有效而快速地评判碰摩转子观测数据所隐含的动力学行为。主要包括：1)
在短数据集情况下，为了快速、合理地选择嵌入空间参数，提出了延迟时间选择的交叉位移改进法和嵌入维数选择的伪近邻距离统计增长法，其特点是速度快、重复性好；2)
对影响关联维数计算的各种因素进行了深入分析，提出了在短数据集约束下估计关联维数的具体方法；3)
提出了在短数据集条件下，通过最大瞬时Lyapunov指数来估计最大Lyapunov指数的方法，并依据Lyapunov指数之和与系统的能量耗散机制相关联这一结果，从理论上分析并提出了Lyapunov指数之和的变化规律可用来监测强非线性系统的阻尼变化，从而可以监测系统状态变化的新策略和新方法。研究表明关联维数和最大Lyapunov指数对非线性动力学行为的辨识是行之有效的，其有效性在转子碰摩的各种状态的分类与辨识中获得了证实。
3．提出了通过观测数据的不可长期预测性并结合特征指数分析对信号的混沌特性进行综合判别的新方法。主要研究工作包括：1)
改进了局部线性拟合的非线性预测方法；2)
发展了非线性时间序列预测的相轨迹方法；3)
提出了利用观测数据的短期、长期可预测性可对动力学行为进行辨识的新方法。研究表明，上述预测方法结合特征指数分析，可以对非线性行为进行综合分类与辨识，通过多指数、多角度地对观测数据进行分析，使获得的辨识结果更为可信。该预测方法在转子碰摩非线性行为的分类与辨识中的应用表明是行之有效的。
4．以理论和实验分析所获得的碰摩故障特征规律为基础，提出采用Duffing振子微弱信号检测方法对转子系统碰摩故障特征进行早期检测的新方法。主要包括：理论分析了Duffing方程的全局解和全局分叉规律并讨论了分叉值随阻尼、外部激励幅值的变化规律，从中获得Duffing方程外轨解的最大轨道所对应的分叉阈值特性可用来进行微弱信号检测的结果，给出了Duffing振子进行早期故障特征微弱信号检测的实现模型，并对相关问题进行了深入讨论。以此为基础，提出了检测信号存在性策略、确定微弱信号幅值的方法和估计信号频率的方法。针对机械故障诊断应用环境，提出了利用该方法进行故障早期检测的决策策略。特别地，为了在噪声环境中准确判定混沌到周期响应突变结果，首次提出了采用源于符号动力学的符号序列分析技术（STSA）来对振子的相变进行自治地快速辨识的方法，给出了辨识的具体流程与检验阈值的确定方法。综合上述工作，数值分析与实验表明，Duffing振子方法可以将信噪比为-25dB左右的微弱信号进行可靠检测。该方法对诸如碰摩故障特征的早期检测是行之有效的。
5．首次将随机共振理论与方法引入机械故障早期诊断领域，提出了采用随机共振原理来检测微弱的特征信号的新方法，以尽早而准确地捕捉表征转子碰摩早期的特征信号。首先，从哲学辩证法角度，分析了噪声在信号处理中的正、反作用，从而引出了随机共振用于信号处理的基本思想。其次，给出了基于随机共振的弱特征信号处理模型，分析了模型的响应及描述指数（如局部信噪比）、模型数值算法，特别分析了多个特征信号检测的可行性。最后将该模型用于早期碰摩特征信号的检测。研究表明该方法简单、稳健、可靠，能把信噪比较低的表征碰摩故障的周期信号从强背景噪声中可靠地提取出来。该方法是一种具有实际应用价值的故障早期特征检测方法，特别适合于在实时的应用场合中，在短数据记录的情况下，从很强的噪声中检测微弱的表征故障特征的有用信号。这种基于非线性科学的新颖微弱信号处理技术将有可能在复杂机械状态监测与故障诊断以及维修系统中得到广泛应用。顺便指出，该方法深入的理论与应用研究工作又获得了国家自然科学基金的资助（“基于随机共振的机械故障早期检测方法研究”
）。
值得一提的是，受国家863高技术课题资助，本论文研究工作所获得的模型与算法，正移植到某发动机健康监控与故障诊断系统中。
关键词：转子碰摩，相空间重构，非线性特征描述指数，相轨迹演化，非线性时间序列预测，Duffing混沌振子，随机共振，符号时间序列分析，早期故障预示
Abstract
The real world is nonlinear whereas the linearity is its approximation.
Chaos theory developed in the late half of 20th century gives a new
approach to the research on nonlinear dynamical system. The discovery of chaos
shows that some deterministic nonlinear system can exhibit random-like behavior.
Chaos is studied in many diverse fields
within the last twenty years. The researches on strong nonlinear behavior
such as chaos in complex mechanical system and application of chaos theory in
machinery fault diagnosis are of significance to design, operation, diagnosis
and maintenance of complex mechanical system. Especially, it is very important
for prognosis of incipient fault of complex system.
With the increase of machinery operating speed and wide application of
various new-style materials in high-speed machinery, nonlinear problem of
mechanical system which may cause abnormal state even fault directly or
indirectly becomes more and more obvious. But many underlying pivotal problems
remain to be done. Theoretical and experimental studies focusing on this problem
are very important. Particularly, it is more important to study the prediction
of nonlinear behavior and chaotic behavior, early detection of operating state
and signal/information processing method based on chaos theory. Just under such
circumstance, this thesis is impelled by scientific spirit to explore the
essence of nature and is supported by the NSFC project “research on
relationship between chaotic behavior and fault in rotors and its prediction
method”. The thesis has firstly explored the possible chaotic phenomena of
rotors deeply. Meantime, the system identification and signal processing
techniques based on chaos theory are studied deeply. The starting point of
research work is mechanical fault diagnosis and signal processing and is asked
to answer the following questions: 1) Is there any complex nonlinear behavior
(chaotic behavior) in dynamical response of rotors? 2) How to estimate and
identify such behavior using measured time series? 3) Is there certain
relationship between indices describing nonlinear behavior and system fault? 4)
How to apply the chaotic theory to early detection and diagnosis of fault of
rotors?
Subsequently,
this dissertation mainly includes, observation and identification of chaotic
phenomena from rub-impact rotor rig, analysis and prediction for nonlinear
behavior of rotor rub-impact based on nonlinear signal processing, early
detection and recognition of rub-impact fault based on nonlinear theory and
chaos theory. Some new models and approaches are proposed. The specific works
finished and main innovative contributions of this dissertation are as follows:
1. Nonlinear behavior and characteristic rule of rub-impact rotor are
deeply and systematically studied. First, Combined with quantitative and
qualitative analysis, solutions of vibration response of sharp rub-impact rotor
are obtained by the improved sharp rub-impact model of rotor. Test rig of sharp
rub-impact rotor is designed and meticulous experiment has been accomplished.
Characteristic rule of vibration response is obtained in various cases of
rub-impact. The result that the 1/3
,2/3
sub-harmonic components (
denotes operating frequency component) occur in inception of rub-impact in the
case of sharp rub-impact under certain condition, is obtained via theoretical
and experimental analysis. According
to theoretical and experimental analysis, the characteristic rule and
corresponding identification method are concluded under different acute rubbing
phases. These results are of significance to detect corresponding incipient
fault. Second, Dynamic model of Jeffcott nonlinear
rotor with eccentric between stator and rotor is built based on rub-impact
force. Numerical simulation demonstrates that local rub-impact has sub-harmonic
phenomena and bifurcation phenomena. The rub-impact rotor response includes
quasi-periodic or chaotic vibration when severe unbalance, small damping and
high rotating speed. Based on the result of numerical analysis, rotor test rig
about local rub-impact is designed and built. Experimental research has been
done within broad range of rotating speeds. Very rich and complicated vibration
phenomena including not only periodic (synchronous and non-synchronous)
components but also quasi-periodic and chaotic motions, are observed. The
observed result is qualitatively consistent with that of simulation. At last,
Phase space reconstruction analysis method based on observed time series is used
to analyze and identify nonlinear dynamics of rotor system. The evidence with
statistical meaning representing strong nonlinear behavior is obtained. The
above results show that sub-harmonic phenomena produced by local rub-impact
provide mechanism and evidence for the early diagnosis of this fault. Vibration
response characteristics of rub-impact rotor obtained by theoretical and
experimental analysis are of significance to prediction of rub-impact.
2. From
the new point of view of nonlinear dynamical analysis, this thesis develops the
chaotic identification technique based on measured data. Namely, when only the
short data set is obtained, the phase space reconstruction technique and
characteristic indices algorithm with wide prospects of engineering application
are presented. These methods can be used to distinguish the underlying nonlinear
dynamical behavior of measured data from rub-impact rotor. The main work
finished includes: 1) In the case of short data set, for the purpose of
selecting embedding space parameters as fast and exact as possible, the improved
across displacement method for selecting time delay and the relative gain ratio
of false neighbors distances method for selecting embedding dimension are
presented. These methods have high
computational speed and good repeatable capacity. 2) Various factors related to
the estimation of correlation dimension are analyzed deeply and the implemented
method for estimating correlation dimension is also presented under the
constraint of short data set. 3) The method for estimating largest Lyapunov
exponent via instantaneous largest Lyapunov exponent is presented under the
condition of short data set. An important property is concluded that the sum of
Lyapunov exponents is related to the generalised divergence of the flow in phase
space of the system, and related to the energy dissipation mechanism of a
system. Theoretical analysis demonstrates that the sum of Lyapunov exponents
must be related to the damping of a mechanical system, and can be utilised to
monitor any damping change of the system. Consequently, a new strategy is
presented to monitor changes of states of complex nonlinear system. The above
research shows that identification of nonlinear behavior using correlation
dimension and the largest Lyapunov exponent synchronously is feasible and their
effectiveness is validated on classification and identification of various
states of rub-impact rotor.
3. A new integrated method is presented to identify chaotic property of
signal by short-term predictability, non-long-term predictability of observed
data and the result of characteristic indices including correlation dimension
and the largest Lyapunov exponent. The main work finished includes: 1) Local
linear fitting method for predicting nonlinear time series is improved. 2)
Prediction method based on phase trajectory (PBPT) for predicting nonlinear time
series is presented. 3) A new idea on identification of dynamical behavior is
presented via short-term predictability and long-term predictability of observed
data. The research shows that the above predicting method combined with
characteristic indices analysis can classify and identify nonlinear
behavior. Experimental data are analyzed with several indices and from
multi-angle. Consequently, the identified result will be more reliable. The
effectiveness of the above method has been validated on classification and
identification for nonlinear behavior of rub-impact rotor system.
4. On the basis of characteristic pattern of rub-impact fault from
theoretical and experimental analysis, a new idea is presented for the early
detection of rub-impact fault of rotor using weak signal detection method via
Duffing oscillator. The main work is given as follows. All forms of solutions and global bifurcations of Duffing equation are
analyzed in theory. Various bifurcations
thresholds varying with damping and external exciting amplitude are discussed.
The analysis concludes that the bifurcation threshold corresponding to the
maximum orbit of solutions outside homoclinic orbit of Duffing equation can be
used to detect weak signal. The implemented model of Duffing oscillator for
weak signal detection is presented and some aspects about weak signal detection
via Duffing oscillator for practical reasons are discussed deeply.
Based on the above analysis, the
approaches for testing existence of characteristic signal and estimating the
amplitude and frequency of certain signal are presented.
In view of practical application environment of
machinery fault diagnosis, a decision-making strategy is presented for early
fault detection via Duffing oscillator method.
Particularly, in order to accurately distinguish the transition from chaos to
periodic response under noisy environment, it is for the first time that a quick
identification method and implemented procedure using symbolic time series
analysis technique (STSA) based on symbolic dynamics are presented to determine
the phase transition of oscillator autonomously. The implementation flow and
approach of determining threshold about transition are given. Based on all the
above investigation, numerical and experimental analysis demonstrates that the
method can reliably detect weak signal with signal-noise ratio as small as
-25dB. This method is feasible for characteristics detection of early fault such
as rub-impact fault of rotor.
5. It is the first time that the theory and method of stochastic
resonance (SR) are introduced to machinery fault diagnosis. A new detection
method for weak characteristic signal based on SR is presented to catch the
characteristic signal of mechanical faults as early and accurately as possible.
First, from the point of view of philosophic dialectic, the positive and
negative effects of noise on signal processing are analyzed. Then the basic idea
of SR applied to detect weak signal in heavy noise is introduced. Second, the
detecting model of weak signal based on SR is given. Then the response of model
and its characteristic indices (such as local signal-to-noise ratio) are
analyzed and the numerical algorithm of model is obtained. In particular, the
feasibility for detecting multiple weak signals is analyzed. At last, this model
is applied to detect characteristic signal of incipient rub-impact. The result
shows that this method is simple, robust and reliable. The weak sinusoid signal
of lower signal-to–noise ratio can be reliably extracted from heavy noise.
This method is valuable in practical engineering for characteristics detection
of early fault. Especially, it is applied to reliably detect weak signal
describing incipient fault buried in very heavy noise from short data records.
The new weak signal processing technique based on nonlinear filtering using
bistable system will find wide applicability in complex condition monitoring and
fault diagnosis system and information maintenance system in the near future. By
the way, the further investigation of theory and application about SR is also
supported by the NSFC project “on methodology of early detection of machinery
fault based on stochastic resonance”.
Spoken with emphasis, the models and algorithms finished in this dissertation are now transplanted to engine health monitoring and fault diagnosis system supported by national 863 high-tech research program.
Key
words: rub-impact of rotor system, phase space reconstruction, nonlinear
characteristic indices, phase trajectory evolution, nonlinear time series
prediction, Duffing chaotic oscillator, stochastic resonance, symbolic time
series analysis, prognosis for incipient fault
