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Currently, most of the vibration signal analysis methods for bearing fault diagnosis under varying speed conditions are based on the resampling technology with shaft rotational frequency (SRF). However, the SRF obtained by a fixed tachometer or time-frequency (TF) ridge detection reduces measuring flexibility or introduces errors inevitably. In this paper, a multi-band feature extraction method using coarse TF ridge-guided variational nonlinear chirp mode decomposition (VNCMD) is proposed for bearing fault diagnosis under varying speed conditions. Specifically, the proposed method is conducted as follows. First, the low-frequency component (LFC) and resonance component are extracted by the low-pass filtering and the fast kurtogram method, respectively, to alleviate the noise interference. Second, the coarse TF ridges are identified by a tractable ridge estimation method that is based on the TF representation for preliminary selection of the initial instantaneous frequency. Third, the coarse TF ridge-guided VNCMD is constructed to track the SRF and instantaneous fault characteristic frequency (IFCF) from the envelope signals of the LFC and the resonance component, respectively. Finally, the characteristic frequency ratio is computed on the basis of the values of SRF and IFCF to determine the fault type of ball bearing without resampling. The simulation studies and experimental verifications confirm that the proposed method can accurately locate bearing defect types and outperforms some existing methods. INDEX TERMS Bearing fault diagnosis, varying speed condition, ridge extraction, adaptive signal decomposition. NOMENCLATURE ADMM Alternate direction method of multipliers ARE Average relative error CFR Characteristic frequency ratio EMD Empirical mode decomposition GD Generalized demodulation IF Instantaneous frequency IFCF Instantaneous fault characteristic frequency LFC Low-frequency component NCM Nonlinear chirp mode VMD Variational mode decomposition VNCMD Variational nonlinear chirp mode decomposition The associate editor coordinating the review of this manuscript and approving it for publication was Baoping Cai.

Currently, study on the relevant methods of variational mode decomposition (VMD) is mainly focused on the selection of the number of decomposed modes and the bandwidth parameter using various optimization algorithms. Most of these methods utilize the genetic-like algorithms to quantitatively analyze these parameters, which increase the additional initial parameters and inevitably the computational burden due to ignoring the inherent characteristics of the VMD. From the perspective to locate the initial center frequency (ICF) during the VMD decomposition process, we propose an enhanced VMD with the guidance of envelope negentropy spectrum for bearing fault diagnosis, thus effectively avoiding the drawbacks of the current VMD-based algorithms. First, the ICF is coarsely located by envelope negentropy spectrum (ENS) and the fault-related modes are fast extracted by incorporating the ICF into the VMD. Then, the fault-related modes are adaptively optimized by adjusting the bandwidth parameters. Lastly, in order to identify fault-related features, the Hilbert envelope demodulation technique is used to analyze the optimal mode obtained by the proposed method. Analysis results of simulated and experimental data indicate that the proposed method is effective to extract the weak faulty characteristics of bearings and has advantage over some advanced methods. Moreover, a discussion on the extension of the proposed method is put forward to identify multicomponents for broadening its applied scope.

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