Time-frequency characterization of nonlinear normal modes and challenges in nonlinearity identification of dynamical systems

Presented here is a new time-frequency signal processing methodology based on Hilbert-Huang transform (HHT) and a new conjugate-pair decomposition (CPD) method for characterization of nonlinear normal modes and parametric identification of nonlinear multiple-degree-of-freedom dynamical systems. Different from short-time Fourier transform and wavelet transform, HHT uses the apparent time scales revealed by the signal's local maxima and minima to sequentially sift components of different time scales. Because HHT does not use pre-determined basis functions and function orthogonality for component extraction, it provides more accurate time-varying amplitudes and frequencies of extracted components for accurate estimation of system characteristics and nonlinearities. CPD uses adaptive local harmonics and function orthogonality to extract and track time-localized nonlinearity-distorted harmonics without the end effect that destroys the accuracy of HHT at the two data ends. For parametric identification, the method only needs to process one steady-state response (a free undamped modal vibration or a steady-state response to a harmonic excitation) and uses amplitude-dependent dynamic characteristics derived from perturbation analysis to determine the type and order of nonlinearity and system parameters. A nonlinear two-degree-of-freedom system is used to illustrate the concepts and characterization of nonlinear normal modes, vibration localization, and nonlinear modal coupling. Numerical simulations show that the proposed method can provide accurate time-frequency characterization of nonlinear normal modes and parametric identification of nonlinear dynamical systems. Moreover, results show that nonlinear modal coupling makes it impossible to decompose a general nonlinear response of a highly nonlinear system into nonlinear normal modes even if nonlinear normal modes exist in the system.     

Instantaneous frequency of an arbitrary signal

This paper defines the non-negative pointwise instantaneous frequency (pIF) and pointwise instantaneous amplitude (pIA) of an arbitrary time signal to be the circular frequency and radius of curvature of the signal’s instantaneous trajectory on the complex plane consisting of the signal and its conjugate part from the Hilbert transform. One analytical and three computational methods are derived to prove and validate this concept. The analytical method is derived based on the definition of pIF and circle fitting. A five-point frequency tracking method is developed to eliminate the incapability of the original four-point Teager–Kaiser algorithm (TKA) for obtaining pIF of signals with moving averages. A three-point conjugate-pair decomposition (CPD) method is derived based on circle fitting using a pair of conjugate harmonic functions for frequency tracking. Moreover, the Hilbert–Huang transform (HHT) uses the empirical mode decomposition (EMD) to sift a signal’s instantaneous dynamic component from its sectional moving average (sMA) as the first intrinsic mode function, and then Hilbert transform is used to compute the first IMF’s frequency and amplitude as the sectional instantaneous frequency (sIF) and sectional instantaneous amplitude (sIA). Because finite difference is used in the five-point TKA, its accuracy is easily destroyed by noise. On the other hand, because CPD uses a constant and a pair of windowed regular harmonics to fit data points and estimate pIF and pIA, noise filtering is an implicit capability of CPD and its accuracy increases with the number of processed data points. Numerical simulations confirm that pIF and pIA are non-negative and physically meaningful and can be used for frequency tracking and accurate characterization of complex signals. However, sIF and sIA from HHT are more useful for system identification because the IMFs sifted by EMD often correspond to actual vibration modes.     

Gear fault identification based on Hilbert–Huang transform and SOM neural network

Gear vibration signals always display non-stationary behavior. HHT (Hilbert–Huang transform) is a method for adaptive analysis of non-linear and non-stationary signals, but it can only distinguish conspicuous faults. SOM (self-organizing feature map) neural network is a network learning with no instructors which has self-adaptive and self-learning features and can compensate for the disadvantage of HHT. This paper proposed a new gear fault identification method based on HHT and SOM neural network. Firstly, the frequency families of gear vibration signals were separated effectively by EMD (empirical mode decomposition). Then Hilbert spectrum and Hilbert marginal spectrum were obtained by Hilbert transform of IMFs (intrinsic mode functions). The amplitude changes of gear vibration signals along with time and frequency had been displayed respectively. After HHT, the energy percentage of the first six IMFs were chosen as input vectors of SOM neural network for fault classification. The analysis results showed that the fault features of these signals can be accurately extracted and distinguished with the proposed approach.     

Time–frequency interpretation of multi-frequency signal from rotating machinery using an improved Hilbert–Huang transform

The Hilbert–Huang transform (HHT) has proven to be a promising tool for the analysis of non-stationary signals commonly occurred in industrial machines. However, in practice, multi-frequency intrinsic mode functions (IMFs) and pseudo IMFs are likely generated and lead to grossly erroneous or even completely meaningless instantaneous frequencies, which raise difficulties in interpreting signal features by the HHT spectrum. To enhance the time–frequency resolution of the traditional HHT, an improved HHT is proposed in this study. By constructing a bank of partially overlapping bandpass filters, a series of filtered signals are obtained at first. Then a subset of filtered signals, each associated with certain energy-dominated components, are selected based on the maximal-spectral kurtosis–minimal-redundancy criterion and the information-related coefficient, and further decomposed by empirical mode decomposition to extract sets of IMFs. Furthermore, IMF selection scheme is applied to select the relevant IMFs on which the HHT spectrum is constructed. The novelty of this method is that the HHT spectrum is just constructed with the relevant, almost monochromatic IMFs rather than with the IMFs possibly with multiple frequency components or with pseudo components. The results on the simulated data, test rig data, and industrial gearbox data show that the proposed method is superior to the traditional HHT in feature extraction and can produce a more accurate time–frequency distribution for the inspected signal.     

Fault diagnosis of a gearbox based on the analysis of a fractional energy gathering band

To extract the weak fault feature of the accelerating process from a gearbox, a fractional energy gathering band time–frequency aggregated spectrum (FETFAS) is proposed to achieve a fast time–frequency analysis of a large signal and to highlight target components. The best order of the fractional Fourier transform (FRFT) is determined according to the rotating speed signal and transmission ratio. The vibration signal from the accelerating process of a gearbox is processed using the best order FRFT. The energy gathering band (EGB) is determined from the modulus spectrum of the FRFT. Then, the result of the FRFT within the EGB is analyzed using time–frequency analysis, and the energy from this result is aggregated to form the FETFAS. The experimental results show that the method to determine the best order of the FRFT from the rotating speed signal is fast and accurate. The time–frequency analysis of the FRFT’s results in the EGB requires less computation and has a high resolution. The FETFAS has the ability to focus and zoom and is able to highlight the target components and restrain noise. Therefore, the FETFAS is an effective method to extract weak fault feature from the signal of gearbox’s accelerating process.     

Target location method for pipeline pre-warning system based on HHT and time difference of arrival

The increasing serious pipeline leakage accidents are caused by third part damage. The third party damage activity around pipeline can generate seismic wave, and the seismic wave can be used for target identification and damage activity location. This paper presents a novel passive location method based on arrival time difference of specific seismic wave characteristic frequencies. The seismic signals are typically non-stationary and the conventional methods cannot analyze them well. Hilbert–Huang Transform (HHT), including empirical mode decomposition (EMD) and Hilbert transform, is a new time–frequency analysis method and can be used for seismic signals analyzing. Firstly, EMD is applied to process the signals and obtain the intrinsic mode functions (IMFs) features. The kurtosis features are used to identify targets and characteristic frequencies are selected as principal components according to IMFs energy features. These principal components are processed by windowed harmonic wavelet transform and then instantaneous features of seismic signals can be extracted. TDOA can be deduced from the arrival time difference of principal frequency components. Finally, target location can be achieved by the time difference analyzing, sensors layout and the relative position between sensors and targets. The seismic signals acquired from field experiment are analyzed and the results are discussed.     

基于HHT的导弹发射冲击时频谱分析

为研究弹载部件在导弹发射过程中的冲击响应及冲击信号的传递特性,进行了基于希尔伯特-黄变换(Hilbert-Huang transform,简称HHT)的导弹发射冲击时频谱分析。由于经验模态分解(empirical mode decomposition,简称EMD)结果易受白噪声的影响,研究了总体经验模态分解(ensemble empirical mode decomposition,简称EEMD)技术。以弹体不同位置的实测冲击信号为对象,应用HHT技术进行分析,准确得到了导弹发射冲击信号的固有模态函数(intrinsic mode function,简称IMF)和时间-频率-能量谱特征,并研究了两次冲击的频率分布和各阶IMF与原始信号的相关性。结合边际谱分析对比了两个舱段能量在中低频和高频的传递特性,进一步验证了HHT方法在分析非线性和非平稳冲击信号中的优越性。     

Feature extraction of AC square wave SAW arc characteristics using improved Hilbert–Huang transformation and energy entropy

In order to extract the arc feature information related to welding quality in alternating current square wave submerged arc welding (AC Square Wave SAW), an improved Hilbert–Huang transform (HHT) is put forward to investigate the time–frequency distribution of arc current, and the energy entropy is employed to quantitatively judge the arc characteristics. The empirical mode decomposition (EMD) is used to decompose the collected current signal into a number of Intrinsic Mode Functions (IMFs). The method for removing the high frequency and undesirable low-frequency IMFs is proposed by using the correlation coefficient of the IMF and the original signal as criterion, and the valid IMFs are selected for the Hilbert transform and energy entropy calculation. The improved HHT combining with energy entropy can quantitatively describe the time–frequency energy distribution characteristics of the arc current signal at different duty cycle, frequency and welding speed. Experimental results are provided to confirm the effectiveness of this approach to extract the arc physical information related to welding quality.     

HHT在Lamb波检测信号分析中的应用

将一种新的超声信号处理技术用于Lamb波波形中多个模式到达时间的提取。通过将希尔伯特-黄变换(Hilbert-Huang transform,简称HHT)与快速傅里叶变换(fast Fourier transform,简称FFT)、小波变换(wavelettransform,简称WT)在时频分辨率方面的比较,表明HHT能够精确识别信号中两种频率分量突变的时刻,显示了HHT方法的优越性。将HHT方法的特性用于Lamb波模式到达时间的提取,从HHT的能量-时间图上可以看出,能量峰值时刻对应着各Lamb波模式的到达时间。试验结果与理论值具有较好的一致性。     

改进的HHT方法在旋转机械不对中故障特征提取中的应用

HHT(希尔伯特-黄变换)能够将振动信号分解为有限的具有实际物理意义的模态分量,并由此可对机械故障信号进行特征提取,但噪声的干扰对分解过程和分解结果影响却很大。针对这一不足,本文提出了先利用小波变换技术对含噪故障信号进行消噪处理,再作HHT分析的方法;利用此方法对实测的不对中振动信号进行了故障特征提取和分析。结果表明,该方法克服了直接运用HHT分解方法由噪声带来的不必要的干扰,提高了参数提取的准确性,并由此提高了机械故障诊断率。     

Time–frequency analysis of tribological systems—part II: tribology of head–disk interactions

Head–disk interface processes operating in contact and near contact recording generate signals that have a distinct frequency for short time intervals and these processes are known as non-stationary. Time–frequency representation displays time, frequency, and amplitude to characterize such processes. Examples drawn from practical head–disk interface signals are analyzed by adapting the fast Fourier transform algorithm to illustrate the dynamic features jointly in time and frequency. Time–frequency analysis of laser Doppler vibrometer (LDV), friction, and acoustic emission (AE) signals give evidence of non-stationary signals obtained from head–disk dynamics experiments. Novel results depicted by the time–frequency analysis technique not reported elsewhere demonstrate the benefit and usefulness of the proposed techniques.     

Hilbert transform in vibration analysis

This paper is a tutorial on Hilbert transform applications to mechanical vibration. The approach is accessible to non-stationary and nonlinear vibration application in the time domain. It thrives on a large number of examples devoted to illustrating key concepts on actual mechanical signals and demonstrating how the Hilbert transform can be taken advantage of in machine diagnostics, identification of mechanical systems and decomposition of signal components.     

温度场中桥面矩形薄板的主共振—主参数共振

为了研究温度场中桥面矩形薄板受简谐激励的主共振-主参数共振问题,应用弹性力学理论建立其动力学方程,应用Galerkin方法将其转化为非线性振动方程.利用非线性振动的多尺度分析方法求得系统主共振-主参数共振的近似解,并进行数值计算.分析温度、激励等对系统主共振-主参数共振的影响.指出系统主共振-主参数共振幅频响应曲线具有双峰特点并呈M形状.     

一种Hilbert变换法在非线性系统分析中的应用

Hilbert变换是信号处理领域常用工具之一,将Hilbert变换法改进并应用到信号分解和非线性系统振动分析中。振动信号的多谐波性,使得Hilbert变换法提取信号的瞬时频率和瞬时相角可以通过滤波的方法分离成快变和慢变两部分,从而提取系统的振动分量;通过迭代计算,依次获得振动信号中所有谐波分量;将非线性振动方程用瞬态幅值相关的瞬态参数表示,从而求解系统的频响关系曲线方程。通过相应的数值模拟计算,验证了改进的Hilbert变换法在非线性振动分析的有 效性。     

HHT端点效应的最大Lyapunov指数边界延拓方法

针对HHT(Hilbert-Huangtransfrom)的端点效应问题,提出基于最大Lyapunov指数预测模型的HHT边界延拓方法.该方法通过相空间重构,并利用时间序列相似点的演化行为,采用最大Lyapunov指数预测模型来对时间序列的端点进行预测,有效避免了不同边界条件的三次样条插值和Hilbert变换频谱泄露对...     

Non-parametric simultaneous identification of both the nonlinear damping and restoring characteristics of nonlinear systems whose dampings depend on velocity alone

This paper presents a general method, which is aimed at identifying both the nonlinear damping and restoring characteristics of nonlinear oscillation systems in which the nonlinear damping is characterized as a function of velocity alone. The method developed for this simultaneous identification involves the non-parametric identification of nonlinear systems. Both system displacement and velocity responses are required for its implementation. However, the numerical approach to this method results in the instability of the numerical solutions, which also means that the solutions identified lack of stability properties. This difficulty is solved by employing a stabilization technique (or regularization). Although the method presented herein is built on the basis of the measurement of the system displacement and velocity responses, a conceptual systematic procedure is also proposed to describe how the system’s acceleration response can be used for simultaneous identification. Finally, an example involving a highly nonlinear system is presented to demonstrate the proposed method’s workability for simultaneous nonlinear system identification.     

A wavelet-based approach for the identification of damping in non-linear oscillators

A special class of non-linear damping models is studied in which the damping force is proportional to the product of positive powers of the absolute values of displacement and velocity. For a single degree of freedom system, the Krylov–Bogoliubov averaging method is used to determine the approximate free response. The wavelet transform of this response is used as a time-scale representation for parameter identification: two methods based on this wavelet transform are presented to estimate instantaneous frequency, damping and envelope of the system. The first method uses cross-sections of the wavelet transform. The second method uses ridges and skeletons of the wavelet transform. This second method is general and gives accurate results in the case of noisy non-linear oscillators. These methods are illustrated using a simulated example.     

基于局部特征尺度分解的经验包络解调方法及其在机械故障诊断中的应用

提出一种基于局部特征尺度分解(Local characteristic-scale decomposition, LCD)和经验包络(Empirical envelope method, EE)解调的非平稳信号分析方法。该方法通过局部特征尺度分解将一个复杂信号自适应地分解为若干个内禀尺度分量之和,对得到的各个内禀尺度分量进行经验包络解调,得到各个分量信号的瞬时幅值和瞬时频率信息,从而得到原始信号完整的时频分布。采用仿真信号将基于LCD和EE解调的时频分析方法和希尔伯特黄变换方法进行对比,结果表明,新提出的信号分解和解调方法在抑制端点效应和迭代所需时间,瞬时特征的精确性等方面优于希尔伯特黄变换方法。针对滚动轴承和齿轮故障振动信号的调制特点,将基于LCD和EE的时频分析方法引入机械故障诊断中,对试验信号的分析结果表明,基于LCD和EE的时频分析方法能有效地提取机械故障振动信号的特征。     

Hilbert-Huang变换中的模态混叠问题

希尔伯特-黄变换(Hilbert-Huang transform,简称HHT)存在的模态混叠现象严重影响了实际应用效果。在分析研究HHT原理及模态混叠产生机理的基础上,提出了基于形态滤波预处理与端点延拓相结合的方法抑制模态混叠现象。与集合经验模态分解(ensemble empirical mode decomposition,简称EEMD)方法比较,所提出的方法能够更快速、准确地分解出表征信号的本征模态函数(intrinsic mode function,简称IMF)分量。将该方法应用于滚动轴承的实测信号分析,结果表明,该方法在实际应用中同样具有很好的模态混叠抑制效果。     

Bearing fault diagnosis using FFT of intrinsic mode functions in Hilbert–Huang transform

A number of techniques for detection of faults in rolling element bearing using frequency domain approach exist today. For analysing non-stationary signals arising out of defective rolling element bearings, use of conventional discrete Fourier transform (DFT) has been known to be less efficient. One of the most suited time–frequency approach, wavelet transform (WT) has inherent problems of large computational time and fixed-scale frequency resolution. In view of such constraints, the Hilbert–Huang Transform (HHT) technique provides multi-resolution in various frequency scales and takes the signal's frequency content and their variation into consideration. HHT analyses the vibration signal using intrinsic mode functions (IMFs), which are extracted using the process of empirical mode decomposition (EMD). However, use of Hilbert transform (HT)-based time domain approach in HHT for analysis of bearing vibration signature leads to scope for subjective error in calculation of characteristic defect frequencies (CDF) of the rolling element bearings. The time resolution significantly affects the calculation of corresponding frequency content of the signal. In the present work, FFT of IMFs from HHT process has been incorporated to utilise efficiency of HT in frequency domain. The comparative analysis presented in this paper indicates the effectiveness of using frequency domain approach in HHT and its efficiency as one of the best-suited techniques for bearing fault diagnosis (BFD).