When a local fault in the rolling element bearing appears during operational running, the induced bearing vibration signal is essentially equivalent to a mixed multi-component signal, which usually includes multiple frequency components (e.g., periodic impulses, random impulses, discrete harmonics, and heavy environmental noise). This indicates that it is difficult to detect the weak periodic impulses related to bearing faults by traditional signal processing methods. Variational mode extraction (VME) as a new signal processing method with band-pass filtering properties can separate the desired periodic impulse component from the mixed multi-component signal due to its similar basis as the popular variational mode decomposition (VMD). However, the filtering performance of VME is significantly influenced by two parameters (i.e., penalty factor and center-frequency). To address this issue, all the previous work focused on single-domain and single-objective parameter optimization from the filtering signal or its envelope, whereas the objective function with a single-criterion cannot depict bearing fault signatures from the impulsiveness and cyclostationarity simultaneously in general cases. Therefore, to improve the bearing fault feature extraction ability and avoid the problem of artificial parameter selection of VME, this paper proposes an adaptive variational mode extraction method based on multi-domain and multi-objective optimization (AVME-MDMO), in which two key parameters of VME are automatically determined by multi-objective gray wolf optimizer (MOGWO) with a novel multi-objective function named integrated measure of sparsity-impact (IMSI) of square envelope (SE) and square envelope spectrum (SES). As a result, the optimal VME is employed to obtain the specific mode component and its corresponding SES for achieving bearing fault diagnosis. The effectiveness of the proposed method has been validated on simulation signal and experiment data of rolling element bearing. Besides, the comparison results demonstrate its superiority and robustness in excavating fault signatures over the VME or VMD with single-objective function, and the fast kurtogram.
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