Elsevier

Journal of Sound and Vibration

Volume 435, 24 November 2018, Pages 36-55
Journal of Sound and Vibration

Initial center frequency-guided VMD for fault diagnosis of rotating machines

https://doi.org/10.1016/j.jsv.2018.07.039Get rights and content

Highlights

  • The variation features of iteration times are firstly investigated with relevant to different ICFs.

  • A phenomenon of convergent CF is found in the U shape region of the iteration times curve.

  • A novel ICF guided VMD method is proposed to extract the weak features of rotating machines.

  • The superiorities of proposed method are demonstrated by the numerical and experimental cases.

Abstract

Variational mode decomposition (VMD), an effective signal decomposing technique, has attracted considerable attention in recent years. The successful applicability of the VMD method mainly depends on the selection of decomposition parameters. Most of the existing VMD-like fault diagnostic methods only aim to optimize the number of either decomposed modes or balance parameters (or both) by adopting qualitative analysis, priori criteria, or genetic-like algorithms. However, the types of initial center frequencies (ICFs) in VMD are seldom discussed alongside the fault diagnosis of rotating machines. In actual scenarios, the selected ICFs have significant effects on the analytical results. Thus, in this study, the variation features of the center frequency (CF) of extracted modes are investigated with different ICFs, in which the converging U-shape phenomenon is found. Motived by this phenomenon, a novel ICF-guided VMD method is proposed to extract accurately the weak damage features of rotating machines. In particular, the proposed method is composed of two steps. First, an energy fluctuation spectrum is presented to rapidly highlight the ICF of the latent mode. Second, a CF-guided VMD optimization strategy is constructed to extract the optimal mode by adaptively refining the balance parameter. The simulations and experimental verifications confirm the effectiveness of the proposed method to enhance the fault diagnosis of rotating machines. A comparison with existing methods demonstrates the superiority of the proposed method to detect weak faults.

Introduction

The detection of weak local damage has been widely explored in the fault diagnosis of rotating machines, mainly because localized damages can increasingly cause surface contact and accelerate degradation, and may even rapidly lead to catastrophic failure. Vibration analysis techniques seem to be one of the effective paths for the health monitoring of mechanical systems [1]. Such techniques include time series analysis [[2], [3], [4]], relevant spectral kurtosis (SK) [[5], [6], [7]], envelope demodulation [8,9], time–frequency analysis [[10], [11], [12]], sparse representation [[13], [14], [15]], frequency response analysis [16], the adaptive signal decomposition [[17], [18], [19], [20]], and so on.

Given that prior transform basis are seldom considered, a number of adaptive signal decomposition methods have also become popular in fault-related feature extraction and extensively studied in fault diagnosis of rotating machines [17]. A new adaptive signal decomposition method named variational mode decomposition (VMD) has been recently proposed [21] to split multi-component signals into individual meaningful modes, which can excellently conduct feature extraction in the fault diagnosis of rotating machines [[22], [23], [24]]. In this paper, we mainly focus on the analytical properties of VMD and devote our efforts in making the VMD method more convenient and reliable in detecting the weak faults of rotating machines.

The VMD, which is based on a clear mathematical theory with a variational model, is generally regarded a good alternative to the empirical mode decomposition (EMD). In this paper, VMD is compared with the intrinsic mode of EMD. The known advantages of using VMD are as follows: noise resistance due to the Wiener filter-like procedure, the balancing of backward errors due to non-recursive sifting, and a narrow-band definition of meaningful modes. Wang et al. [22] utilized the VMD to extract rub-impact features and then validated the superiority of VMD over the empirical wavelet transform [25], EMD, and ensemble EMD [26] in detecting rubbing-caused multiple signatures. An et al. [27] demonstrated the better performance of VMD for pedestal looseness fault diagnosis compared with the EMD and the wavelet transform. Zhang et al. [28] explored the ability of the VMD in the rolling-bearing fault diagnosis of a multi-stage centrifugal pump. Li et al. [29] proposed a multi-dimensional variational decomposition method to detect the bearing-crack of wind turbines by introducing VMD into convolutive blind-source separation. Lv et al. [30] utilized a hybrid method based on VMD and classification models to determine bearing fault types. Zhang et al. [31] used energy entropy based on VMD to detect the chatter degree of a milling process. In our previous work [23], a hybrid method that combines VMD with a multi-Teager energy operator was proposed to detect weak local bearing defects. Recently, Bagheri et al. [32] extended the applicability of VMD to structural system identification.

In actual scenarios, the performance analysis of VMD is strictly subjected to its initial parameters. As a result, the selection of suitable decomposition parameters has become a critical problem for VMD-based signal decomposition methods. Some researchers have paid attention on the selection of decomposition parameters. Wang et al. [33] investigated the filter bank properties of VMD by extensively conducting numerical simulation case studies and qualitatively verifying that the performance of VMD relies on the initial center frequencies (ICFs) of its modes and the selection of balance parameter α. To solve the improper selection of the number K of the decomposed modes, a set of criteria, such as the correlation coefficient [34,35], center frequency (CF) difference [36], scale–space distribution [37], and scaling exponent [38], among others, was introduced into VMD to enhance the analysis results. Considering that parameters α and K are related to the diagnostic results, Pu et al. [39] employed the envelope peak and frequency magnitude ratio of the Fourier spectrum to find the corresponding optimal values of the two parameters. However, the effective fault information that can be derived from the local damage may be buried in the weak band, and the selected K based on the envelope peak of the Fourier spectrum will likely neglect the fault-related components. In our previous work [40], the two parameters of α and K in VMD were also explored to track the intricate trend component buried in the measured signals. Intelligence optimization algorithms, such as genetic algorithm [41], particle swarm optimization [42], artificial fish swarm algorithm [43], and grasshopper optimization algorithm [44], were employed to simultaneously define the two abovementioned parameters based on different fitness functions. However, the batch agents, iteration number, and other parameters should be determined in advance for the optimization algorithms. Moreover, the batch agents and iteration number can significantly affect the decomposing efficiencies of VMD, while some parameters can influence the final decomposition results of VMD.

According to the literature review, past efforts have mainly focused on determining the parameters α and K for the fault diagnosis of rotating machines by adopting qualitative analysis, priori criteria, or genetic-like algorithms. However, the selection strategy of the initial parameters in VMD remains largely unexplored and thus requires further enhancement to fully advance the application of VMD. For instance, ICFs have significant influences on the filter bank and analysis results, but they are seldom reported or discussed alongside the fault diagnosis of rotating machines in the accessible literature. In addition, only one balance parameter α is employed to extract all modes, but the theoretical bandwidths of the individual hidden modes are unspecified in the vibration data. In this study, the variation signatures of CF of the extracted modes with the different ICFs are investigated by analyzing the simulated fault-related signal, which implies that the two types of ICFs (i.e., uniformly spaced distribution and zero initialization) can be neglected. Motived by the findings on the variation characteristics of CF, a novel ICF-guided VMD method is proposed to detect the weak local damage of rotating machines. In the proposed method, a sensitive index named energy fluctuation spectrum (EFS) is first presented to rapidly select the ICFs in the informative frequency band. Then, the CF-guided optimization strategy is proposed to optimize the individual mode extracted from the raw vibration data. Finally, the fault condition of the rotating machine is determined by conducting an envelope spectral analysis of the optimal mode. The known advantages of the proposed method are as follows. (1) Setting the number of decomposed modes via the prior experience is not necessary due to the benefit from the coarse ICF indicated by the sensitive index of EFS. (2) A given coarse ICF can enhance the decomposing efficiency of VMD and avoid extracting some interference modes from raw vibration data. (3) The proposed CF-guided optimization strategy can adjust the balance parameter to automatically match the best bandwidth for fault diagnosis as a result of the CF guidance.

The remainder of this paper is organized as follows. In Section 2, the basic theory of the VMD is reviewed. In Sections 3, the motivations of this study are given and the novel ICF-guided VMD method for feature extraction of weak local damages is proposed. The simulated and experimental cases employed to validate the proposed method are presented in Section 4. The conclusions are drawn in Sections 5.

Section snippets

Brief introduction of the VMD principle

VMD is a recently developed adaptive signal decomposition method, and it can decompose a real-valued signal z(t) into K meaningful modes uk(t), k(1,2,K) with non-recursively sifting structure by using a variational constraint model [21]. The extracted meaningful modes are quasi-orthogonal band-limited signals with specific sparsity properties in the frequency domain and most compact around their CFs ωk. The procedure of VMD for extracting mode uk(t) from the measured data can be briefly

Motivation of the present study

In the VMD algorithm, the decomposition parameters (i.e., number K of the decomposed modes, balance parameter α, Lagrangian multiplier coefficient λ(t), and ICFs ωk0 of the decomposed modes) have significant effects on the analysis results. In general, the coefficient λ(t) in the Lagrangian multiplier term can be set to zero even without strictly enforcing the constraint requirements, particularly under the situation in which the noise component should be excluded in the decomposition [21,43].

Simulation analysis case

To evaluate the performance of the proposed method, a simulated signal that imitates the vibrations of a rotating machine [49] is constructed as follows:x(t)=lAlcos(2πlf0t+θl)Rotorvibrationr(t)+kBkcos(2πkZf0t+ϕk)Gearmeshingg(t)+iCi·S(tiTfτi)Repetitivetransientsz(t)+jRj·S(tTrj)Randomimpulsesri(t)+n(t).

In Eq. (10), the first term stands for the vibration components from rotors or shafts, where f0 denotes the fundamental rotating frequency, and Al and θl are the amplitude and initial

Conclusion

VMD is a recently proposed novel technique for adaptive signal decomposition. The CF variation features of the extracted modes by using VMD are initially investigated with different ICFs. Then, a notable phenomenon termed as U-shape converging property is observed from the CF variation curves of the numerical simulation in this study, which means that the expected mode can be obtained reliably provided that a coarse ICF is preset around the U-shaped region. Subsequently, a novel ICF-guided VMD

Acknowledgments

The research is supported by the National Natural Science Foundation of China (Grant No. 51705349, 51605319), the China Postdoctoral Science Foundation (Grant No. 2017M621811), the Natural Science Foundation of Jiangsu Province (Grant No. BK20150339), and the Natural Science Fund for Colleges and Universities in Jiangsu Province (Grant No.17KJB460012), which are highly appreciated by the authors.

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