HHT-based nonlinear signal processing method for parametric and non-parametric identification of dynamical systems

This paper presents a time–frequency signal processing method based on Hilbert–Huang transform (HHT) and a sliding-window fitting (SWF) technique for parametric and non-parametric identification of nonlinear dynamical systems. The SWF method is developed to reveal the limitations of conventional signal processing methods and to perform further decomposition of signals. Similar to the short-time Fourier transform and wavelet transform, the SWF uses windowed regular harmonics and function orthogonality to extract time-localized regular and/or distorted harmonics. On the other hand, HHT uses the apparent time scales revealed by the signal's local maxima and minima to sequentially sift components of different time scales, starting from high- to low-frequency ones. 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. Methods are developed to reduce the end effect caused by Gibbs’ phenomenon and other mathematical and numerical problems of HHT analysis. For parametric identification of a nonlinear one-degree-of-freedom system, the method processes one free damped transient response and one steady-state response and uses amplitude-dependent dynamic characteristics derived from perturbation analysis to determine the type and order of nonlinearity and system parameters. For non-parametric identification, the method uses the maximum displacement states to determine the displacement–stiffness curve and the maximum velocity states to determine the velocity-damping curve. Moreover, the SWF method and a synchronous detection method are used for further decomposition of components extracted by HHT to improve the accuracy of parametric and non-parametric estimations. Numerical simulations of several nonlinear systems show that the proposed method can provide accurate parametric and non-parametric identifications of different nonlinear dynamical systems.     

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.     

基于信号时频分析理论识别时变模态参数的实验

为研究温度对结构模态参数的影响设计了一套温度可控的实验设备。在这套实验设备提供的可控温度环境中采集悬臂梁结构的加速度响应信号,利用基于信号时频分析的模态参数辨识算法处理实验数据,得到其时变模态参数,包括固有频率和振型,以此研究温度对其模态参数的影响。分析结果显示了基于信号时频分析的模态参数辨识算法在处理非平稳信号以得到结构的时变模态参数上的应用前景,更重要的是实验数据的分析结果较好地反映了温度对结构模态参数的影响,为热环境下结构振动特性分析提供了可靠而且有价值的分析方法和实验依据。     

IDENTIFICATION OF TIME-VARYING MODAL PARAMETERS USING LINEAR TIME-FREQUENCY REPRESENTATION

A new method of parameter identification based on linear time-frequency representation andHilbert transform is proposed to identify modal parameters of linear time-varying systems frommeasured vibration responses. Using Gabor expansion and synthesis theory measured responses arerepresented in the time-frequency domain and modal components are reconstructed by time-frequencyfiltering. The Hilbert transform is applied to obtain time histories of the amplitude and phase angle ofeach modal component, from which time-varying frequencies and damping ratios are identified. The     

基于Hilbert-Huang变换与理想带通滤波器的系统识别

针对经验模态分解存在模态混叠现象,提出基于Hilbert-Huang变换与理想带通滤波器的系统识别方法。该方法利用傅里叶变换得到结构加速度响应频响函数,粗略估计固有频率范围,通过半功率带宽法设计理想带通滤波器,定量化确定通带带宽,使信号在经过滤波器后频域内零相移,同时不改变其幅值谱。结构响应通过指定频带的理想带通滤波器产生若干窄带信号,利用经验模态分解获取结构模态响应,经Hilbert变换构造模态响应解析信号,并通过线性最小二乘拟合提取结构模态参数与物理参数。结果表明:半功率带宽法可实现带通滤波器频带的定量化设计,理想带通滤波器的零相移特点较好契合Hilbert-Huang变换用于系统识别的要求,两者结合可有效地解决模态混叠现象,减少虚假模态,大大提高结构系统识别精度。     

On the modal curve of nonlinear normal modes

In a nonlinear holonomic conservative system having two-degree-of-freedom, the modal curves of normal mode vibrations are investigated investimgated by the harmonic balance method. The general procedure to compute the modal curve is suggested. Even if the linearized frequencies of the system are satisfied with the commensurability condition under which the approaches using the perturbation method have the problem of small divisor, the modal curve can be obtained by this method, provided that the fundamental harmonics are dominant when the normal modes are expanded in Fourier series in time domain. As an example, in a system with cubic nonlinearity the modal curves are computed analytically and numerically to compare both results.     

瞬态激励、应变片测量、非线性最优化识别振动系统模态参数

本文提出了用力锤激励、应变片测量响应、用非线最优化方法来识别振动系统模态参数的理论和方法。本文提出了“撼应系数”这一概念。通过撼应系数及柔度系数和刚度系数,找到了应变和位移的关系。文中导出了用应变片测量响应时系统的应变导纳函数公式。采用本文所述方法可识别出振动系数的固有频率、位移振型、应变振型、应力振型、广义阻尼及主刚度和主质量等模态参数。     

一种模态弱响应且模态密集的参数识别方法

针对运行模态分析中响应信号数据样本较短、模态密集,以及弱响应信号淹没在大噪声中,系统模态难以全部辨识的问题,提出了联合相关函数与传递率识别系统模态参数的方法(联合方法),该方法先应用传递率近似频响函数获取系统弱响应频率特征函数,然后再通过小波变换进行模态识别。运用随机激励下的GARTEUR飞机模型仿真运行状态的输出进行了数值仿真实验。结果表明:相比于多参考最小二乘复频域法,联合方法不仅能提高模态频率的识别精度,而且还能极大地提高阻尼比的识别精度,尤其是传递率对模态密集的弱响应模态识别结果良好。     

Optimization of the End Effect of Hilbert-Huang transform (HHT)

In fault diagnosis of rotating machinery, Hilbert-Huang transform(HHT) is often used to extract the fault characteristic signal and analyze decomposition results in time-frequency domain. However, end effect occurs in HHT, which leads to a series of problems such as modal aliasing and false IMF(Intrinsic Mode Function). To counter such problems in HHT, a new method is put forward to process signal by combining the generalized regression neural network(GRNN) with the boundary local characteristic-scale continuation(BLCC).Firstly, the improved EMD(Empirical Mode Decomposition) method is used to inhibit the end effect problem that appeared in conventional EMD. Secondly, the generated IMF components are used in HHT. Simulation and measurement experiment for the cases of time domain,frequency domain and related parameters of HilbertHuang spectrum show that the method described here can restrain the end effect compared with the results obtained through mirror continuation, as the absolute percentage of the maximum mean of the beginning end point offset and the terminal point offset are reduced from 30.113% and27.603% to 0.510% and 6.039% respectively, thus reducing the modal aliasing, and eliminating the false IMF components of HHT. The proposed method can effectively inhibit end effect, reduce modal aliasing and false IMF components, and show the real structure of signal components accurately.     

Identification of nonlinear boundary effects using nonlinear normal modes

Local nonlinear effects due to micro-slip/slap introduced in boundaries of structures have dominant influence on their lower modal model. This paper studies these effects by experimentally observing the behavior of a clamped–free beam structure with local nonlinearities due to micro-slip at the clamped end. The structure is excited near one of its resonance frequencies and recorded responses are employed to identify the nonlinear effects at the boundary. The nonlinear response of structure is defined using an amplitude-dependent nonlinear normal mode identified from measured responses. A new method for reconstructing nonlinear normal mode is represented in this paper by relating the nonlinear normal mode to the clamped end displacement-dependent stiffness parameters using an eigensensitivity analysis. Solution of obtained equations results equivalent stiffness models at different vibration amplitudes and the corresponding nonlinear normal mode is identified. The approach results nonlinear modes with efficient capabilities in predicting dynamical behavior of the structure at different loading conditions. To evaluate the efficiency of the identified model, the structure is excited at higher excitation load levels than those employed in identification procedures and the observed responses are compared with the predictions of the model at the corresponding input force levels. The predictions are in good agreement with the observed behavior indicating success of identification procedure in capturing the physical merits involve in the boundary local nonlinearities.     

Modal testing of nonlinear vibrating structures based on nonlinear normal modes: Experimental demonstration

Realizing that nonlinearity is a frequent occurrence in engineering structures and that linear experimental modal analysis (EMA) is of limited usefulness in this context, the present paper is an attempt to develop nonlinear EMA by targeting the extraction of nonlinear normal modes (NNMs) from time series of nonlinear mechanical systems. Based on a nonlinear extension of phase resonance testing, the proposed methodology excites the structure to isolate a single NNM during the experiments. Thanks to the invariance principle, the energy dependence of that nonlinear mode (i.e., the NNM modal curves and their oscillation frequencies) can be extracted from the resulting free decay response using time-frequency analysis. This paper is devoted to the experimental demonstration and robustness of this procedure. To this end, an experimental cantilever beam with a geometrical nonlinearity is considered, and the ability of the proposed methodology to extract its NNMs from the measured responses is assessed.     

Modal parameter identification of stay cables from output-only measurements

Stay cables are one of the most critical structural components in modern cable-stayed bridges and the cable tension plays an important role in the construction, control and monitoring of cable-stayed bridges. We propose a time domain and a time-frequency domain approaches for modal parameter identification of stay cables using output-only measurements. The time domain approach uses the subspace algorithm which is improved with a new modal coherence indicator. The time-frequency approach uses the wavelet transform of signals which is improved with a new analyzing wavelet. The wavelet transform is applied to the free response of ambient vibration which is obtained using the random decrement technique. Two experiments of stay cables are presented. The first experiment concerns a stay cable in laboratory where the external load is applied through an impact hammer and the vibratory signals are acquired through four accelerometers. The second experiment concerns the Jinma cable-stayed bridge that connects Guangzhou and Zhaoqing in China. It is a single tower, double row cable-stayed bridge supported by 112 stay cables. Ambient vibration of each stay cable is carried out using accelerometers. From output-only measurements, the modal parameters of stay cables are extracted. Once the eigenfrequencies and the damping coefficients are obtained, the cable forces and the Scruton number are derived. In a continuous monitoring and modal analysis process, the tension forces and Scruton numbers could be used to assess the health of stay cables in cable-stayed bridges.     

基于AVMD与改进能量算子的非稳态谐波分析

针对非稳态谐波分析中时频参数检测精度较低的问题,提出一种基于自适应变分模态分解(AVMD)与改进能量算子的非稳态电力谐波分析方法。首先,采用AVMD对非稳态谐波信号进行分解,其中采用波形特征匹配法对非稳态谐波信号进行延拓以减轻边界效应影响,并提出能量差和相关系数作为AVMD中模态分解个数的判据;结合模态分量,提出改进间隔采样能量算子快速提取谐波的瞬时幅值和频率,根据差分和信号完成其起止时刻的定位,实现非稳态谐波时频参数的快速准确测量。仿真与实测结果表明,本文方法能够在电网工频波动、间谐波以及噪声干扰等情况下有效完成非稳态谐波的准确检测,实现暂态谐波的精确定位,且对非稳态谐波频率、幅值的最大检测误差分别为0.094 9%和0.931 4%。     

多自由度振动系统非线性动力特性的HHT辨识方法研究

HHT方法(包括EMD方法)已成功地用于线性振动系统的辨识研究。本文针对多自由度非线性振动系统,利用小波分析方法对非线性振动响应进行预处理,把信号分解成一系列的窄带信号,应用EMD方法使得各阶内在模函数(IMF)均为单一成份信号,然后运用HT方法辨识自由振动响应的非线性特征。典型的线性振动系统和非线性振动系统的辨识结果说明了该方法的有效性。     

Modal identification of spindle-tool unit in high-speed machining

The accurate knowledge of high-speed motorised spindle dynamic behaviour during machining is important in order to ensure the reliability of machine tools in service and the quality of machined parts. More specifically, the prediction of stable cutting regions, which is a critical requirement for high-speed milling operations, requires the accurate estimation of tool/holder/spindle set dynamic modal parameters. These estimations are generally obtained through Frequency Response Function (FRF) measurements of the non-rotating spindle. However, significant changes in modal parameters are expected to occur during operation, due to high-speed spindle rotation.The spindle's modal variations are highlighted through an integrated finite element model of the dynamic high-speed spindle-bearing system, taking into account rotor dynamics effects. The dependency of dynamic behaviour on speed range is then investigated and determined with accuracy. The objective of the proposed paper is to validate these numerical results through an experiment-based approach. Hence, an experimental setup is elaborated to measure rotating tool vibration during the machining operation in order to determine the spindle's modal frequency variation with respect to spindle speed in an industrial environment. The identification of natural frequencies of the spindle under rotating conditions is challenging, due to the low number of sensors and the presence of many harmonics in the measured signals. In order to overcome these issues and to extract the characteristics of the system, the spindle modes are determined through a 3-step procedure. First, spindle modes are highlighted using the Frequency Domain Decomposition (FDD) technique, with a new formulation at the considered rotating speed. These extracted modes are then analysed through the value of their respective damping ratios in order to separate the harmonics component from structural spindle natural frequencies. Finally, the stochastic properties of the modes are also investigated by considering the probability density of the retained modes. Results show a good correlation between numerical and experiment-based identified frequencies. The identified spindle-tool modal properties during machining allow the numerical model to be considered as representative of the real dynamic properties of the system.     

Hilbert-Huang Transform (HHT) transient analysis of composite panel undergoing high-velocity impact

Fast Fourier transform (FFT) has been widely used to analyze distribution patterns of frequency components in dynamic response signals. Given a stationary dynamic response signal, a fixed frequency distribution pattern can be obtained efficiently using FFT. If the system of concern is not stationary, however, the frequency distribution pattern varies with time, and the variation in that pattern cannot be effectively determined via FFT. To overcome this weakness, time-frequency dual-domain signal analysis methods such as wavelet transform and Hilbert-Huang transform (HHT) have been introduced. HHT has been shown to be particularly effective in analysis of non-stationary signals obtained from non-linear as well as linear systems. In the present study, the transient characteristics of a composite panel undergoing high-velocity impact were investigated. The composite panel, along with the colliding bullet, were modeled using the finite element method. To verify the reliability of the analysis model, an impact experiment was carried out, which proved that the model provides reliable, similar-to-experimental results.     

Measurement, identification and modelling of damping in pneumatic tyres

This paper deals with a research approach specifically designed for the measurement, identification and modelling of damping in pneumatic tyres. As a less rigorous and more global approach of characterizing the engineering property of tyre damping, experimental modal analysis and methodologies based on both real and complex modes are introduced. In the context of the real modal analysis, the flexible ring tyre model is also investigated. A standard testing procedure is employed for the collection of vibration data in pneumatic tyres, and the linearity and reciprocity of the tyre structure and the orthogonality of its modes is proven at the initial stage of investigation. After demonstrating the disadvantages of the proportional damping assumption in describing the tyre vibration, the first-order complex modal interpretation with general viscous damping assumption is implemented and discussed, including an appropriate normalization of the complex modes. As a further extension to the analysis, the second-order Rayleigh's small-damping approximation is introduced into the complex modes framework, in order to obtain an accurate global estimation of the damping distribution.As a natural evolvement of the applied damping models in this paper, from the simplest single-valued proportional damping in flexible ring tyre model, proportional viscous damping in real modal interpretation, up to the generally distributed viscous damping in second-order approximation of the complex modal interpretation, the finally identified damping matrices, both in modal and physical coordinates, present a reasonable explanation of damping effect in pneumatic tyres. Agreement between the theory and experimentally identified results also proves the suitability of this approach for damping identification in complex structures.     

Instantaneous Frequency Estimation for Motion Echo Signal of Projectile in Bore Based on Polynomial Chirplet Transform

Microwave interferometer is one of the devices for measuring the movement travel–varying or time-varying velocity of projectile in bore. Microwave interferometer first obtains the Doppler echo signal including the motion information of the projectile in bore, then the velocity is measured based on instantaneous frequency estimation (IFE) of the processed and transformed signal. The parametric time-frequency analysis method can make spectral energy of nonlinear frequency modulation (FM) signal concentrate at some range in the new transform domain. As the motion echo signal of projectile in bore (MSPB) is a nonlinear FM signal, it could be described by polynomial chirplet, one of polynomial FM signal modes, which is used to construct transform kernel for the signal. In this paper, Polynomial chirplet transform (PCT) method is proposed to analyze the simulation and experiment echo signals of projectile in bore. The estimation error and Renyi entropy are used to measure quantify of the time-frequency distribution. Compared with short-time Fourier transform (STFT) method and Wigner-Ville distribution (WVD) method, our results show that the PCT method has most powerful anti-interference performance and highest accuracy of instantaneous frequency estimation for the simulation signal, and lowest Renyi entropy of the instantaneous frequency estimation for the experiment signal. In general, the PCT method has powerful anti-interference performance and high time-frequency concentration and accuracy of instantaneous frequency estimation for the motion echo signal of projectile in bore.     

齿轮系统谐波共振的多尺度分析方法

考虑齿轮啮合动态刚度、传递误差、齿侧间隙等非线性因素,将时变刚度按5次谐波展开,齿侧间隙按3次多项式拟合,运用多尺度方法分析了单对直齿轮传动系统的谐波共振响应特性,讨论了系统在非共振硬激励下消去长期项的条件,给出了系统中存在的多种频率因子,发现了系统中存在2阶、3阶超谐波共振和1／2阶、1／3阶次谐波共振,推导了稳态振动下的频率响应方程,并绘制了频率响应曲线,分析了静态激励、动态激励、参数激励以及系统中阻尼对稳态响应的不同影响作用。     

Identification of modal parameters of unmeasured modes using multiple FRF modal analysis method

Current modal analysis methods seek to identify the modal parameters of some or all of the modes in the measured frequency range of interest. In many applications however, it will be very useful if modal parameters of some of the out-of-range modes can be identified during modal analysis. Such a goal is obviously theoretically possible since the raw measured frequency response functions (FRFs), upon which modal analysis is performed, do contain adequate information about the out-of-range modes in the form of residue contributions. In this paper, a new method for the estimation of modal parameters using multiple FRFs analysis is presented. In the process of modal identification, the proposed method not only presents accurate modal parameters of the modes which are present in the measurement frequency range, but also quite accurately identifies some of the modes which are not measured. The method calculates the required modal parameters by solving eigenvalue problem of an equivalent eigensystem derived from those measured FRF data. All measured FRFs are used simultaneously to construct the equivalent eigensystem matrices from which natural frequencies, damping loss factors and modeshape vectors of interest are solved. Since the identification problem is reduced to an eigenvalue problem of an equivalent system, natural frequencies and damping loss factors identified are consistent. Applications of the method to both numerically simulated and practically measured FRF data are given to demonstrate the practicality of the proposed method and the results have shown the method is capable of accurately identifying modal parameters of out-of-range modes.