Research on iterated Hilbert transform and its application in mechanical fault diagnosis

Multicomponent AM–FM demodulation is an available method for machinery fault vibration signal analysis, so a new method for mechanical fault diagnosis based on iterated Hilbert transform (IHT) is proposed. The principle of computing the asymptotically exact multicomponent sinusoidal model for an arbitrary signal by iterating Hilbert transform is introduced, and some properties of IHT are analyzed. Theoretical analysis for the generic two-component signal shows that there are limitations in the direct estimation of instantaneous frequencies via the phase signals of the previously obtained model. Therefore, a smoothed instantaneous frequency estimation (SIFE) method based on difference operator and zero-phase digital low-pass filtering is proposed, and then the accuracy and validity of this method have been proved by the simulation results. The analysis results of the mechanical fault signals show that the weak features of these signals can be efficiently extracted with the proposed approach.     

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。     

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.     

基于希尔伯特变换的核电站松动部件定位方法

针对当前的核电站松动部件定位方法在估算信号到达时间差时存在起始点难以准确判定、误差大、抗噪声性能差的缺点,提出一种基于希尔伯特变换的时间差估计方法,从而根据时间差估算松动部件位置实现定位。希尔伯特变换可提取信号轮廓线,以轮廓线的第一个峰值到达时间作为能量最大弯曲波到达传感器的时间;分析信号的主频率可以计算出弯曲波的传播速度,根据波速和信号到达不同传感器之间的时间差使用三角形定位法实现松动部件的定位分析。搭建平板试验台,在不同信噪比条件下和不同方法之间进行对比试验,结果显示这种方法的平均估计偏差为34.1 mm,在信噪比为1 dB时仍然能够得到比较准确的结果,具有定位精度高、抗噪声性能好、运算速度快等特点。     

Extraction of microseismic waveforms characteristics prior to rock burst using Hilbert–Huang transform

This study provides a new research idea concerning rock burst prediction. The characteristics of microseismic (MS) waveforms prior to and during the rock burst were studied through the Hilbert–Huang transform (HHT). In order to demonstrate the advantage of the MS features extraction based on HHT, the conventional analysis method (Fourier transform) was also used to make a comparison. The results show that HHT is simple and reliable, and could extract in-depth information about the characteristics of MS waveforms. About 10 days prior to the rock burst, the main frequency of MS waveforms transforms from the high-frequency to low-frequency. What’s more, the waveforms energy also presents accumulation characteristic. Based on our study results, it can be concluded that the MS signals analysis through HHT could provide valuable information about the coal or rock deformation and fracture.     

A robust condition monitoring methodology for grinding wheel wear identification using Hilbert Huang transform

Grinding is a finishing operation performed to obtain the desired finish on the component. Wheel wear is one of the primary constraints in achieving the desired productivity in grinding. A new methodology is proposed for accurate and timely identification of wheel wear in cylindrical grinding using Hilbert Huang transform and support vector machine. During the grinding of EN31 carbon steel, the condition of the wheel and its wear was monitored with an accelerometer and power cell. Both vibration and power signals captured were used to identify the condition of the wheel and its wear. An exhaustive feature set is generated in the frequency and the time-frequency domain. Hilbert Huang transform, an adaptive time-frequency analysis technique, was used to extract the features of tool wear in the time-frequency domain. The first three IMF constituents were further chosen for feature extraction of statistical parameters based on their mean energy. Random forests algorithm was used to identify the relevant features. The methodology was validated with several grinding experiments and, is found to give an accuracy of 100% with both low and high cutting depths. The results indicated the robust and reliable wheel wear detection in cylindrical grinding with the use of relatively cheap sensors like accelerometers. The proposed method can be widely used in many applications in the industry where grinding is predominantly used as the finishing operation.     

Hilbert versus Concordia transform for three-phase machine stator current time-frequency monitoring

This paper deals with mechanical fault diagnosis in three-phase induction machines from stator current measurements. According to machine models, mechanical faults lead to amplitude and/or phase modulations of the measured stator current with possibly time varying carrier frequency. The modulation diagnosis requires a univocal definition of the instantaneous phase and amplitude. This is performed by associating a complex signal to the real measured one. For a convenient separate modulation diagnosis, the complex signal instantaneous phase and amplitude are expected to carry, respectively, information about the phase and amplitude modulations. The complex signal is classically obtained through the Hilbert transform. Under Bedrosian conditions, the so-called analytic signal allows a separate modulation diagnosis. However, mechanical faults may also produce fast modulations violating the Bedrosian conditions. This study proposes an alternative complex signal representation which takes advantage of the three stator current measurements available in a three-phase machine. From two stator current measurements, the Concordia transform builds a complex vector, the so-called space vector, which unconditionally allows separate modulation diagnosis. This paper applies and compares the Hilbert and Concordia transforms, theoretically and in case of simulated and experimental signals with various modulation frequency ranges.     

The Huang Hilbert Transform for evaluating the instantaneous frequency evolution of transient signals in non-linear systems

Traditional approaches for the analysis of transient signals are generally based on the apriori knowledge of the system under test for choosing a preliminary set of waveforms; consequently, they use a mathematical algorithm to decompose the signal itself into a suitable combination of the chosen waveforms. Conversely, this work is aimed to investigate the possibility of extracting the features of transient signals through the evaluation of their instantaneous frequency evolution. For this aim, the Huang Hilbert Transform (HHT) has been exploited (i) to decompose the input signal into a set of Intrinsic Mode Functions (IMFs), (ii) to extract the IMFs analytical signals, (iii) to evaluate their amplitude and phase evolutions, (iv) to compute the instantaneous frequency of the input signal and (v) to extract the signal information searched for. In order to evaluate its performance, the proposed approach has been firstly applied to a synthesized signal with known instantaneous amplitude and frequency evolution. Successively, in order to assess the reliability of HHT results with signals acquired on experimental circuits, the current flowing in an actual RLC circuit during its free natural oscillation has been analyzed. With the aim of analyzing the performance gained also in the presence of evident non-linearities, a saturable inductor has been introduced in the test circuit. Also in this case, by comparing the achieved results with those shown by different traditional approaches, great advantages have been experienced in terms of accuracy. Furthermore, beyond the accurate frequency representation, the experimental results evidenced the intrinsic ability of the proposed approach to extract meaningful information related to the knowledge of the underlying process. Finally, it is worth noting that the results reported in this paper requested no apriori knowledge about the signal/process under test.     

A signal decomposition or lowpass filtering with Hilbert transform?

Recently, Chen and Wang discovered an explicit formula that makes use of the Hilbert transform for accurate decomposition of a lower harmonic from a signal composition. This letter presents another proof with a new interpretation for the formula using the Bedrosian identity for overlapping signals. This new and simpler proof is based only on the Hilbert transform and does not involve presentation of the Fourier transform. As a result the discovered formula is introduced as a lowpass filter suitable for non-stationary signals.     

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.