Hybrid Constraints Embedding Variational Nonlinear Chirp Mode Decomposition for Rotating Machine Fault Diagnosis Under Time-Varying Speed Conditions

Accurate extraction of time-varying instantaneous frequency (IF) is crucial for fault diagnosis in rotating machinery operating under time-varying speed conditions, as real-world signals often exhibit nonlinearity and complexity. Variational nonlinear chirp mode decomposition (VNCMD) demonstrates significant capability and adaptability in extracting time-varying IFs by converting broadband signals into narrowband signals through a demodulation technique. However, its accuracy is limited by the fact that it only enforces smoothness on the demodulated signals, and it does not consider the inherent proportional relationships among IFs in rotating machine vibration signals. These limitations hinder VNCMD from accurately extracting IFs and result in poor noise robustness, a strong dependence on the initial IFs, and difficulties in handling nonlinear chirp signals (NCSs) with closely spaced IFs. To overcome these limitations, this article proposes hybrid constraints embedding VNCMD (HCVNCMD), which places further constraints on demodulated signals and IFs. Specifically, a sparse constraint is first applied to the demodulated signals. This improves their estimation accuracy and reduces errors in the subsequent IF estimates. Second, a low-rank constraint is applied to the IFs, effectively capturing their proportional relationships and optimizing inaccurate initial IFs to achieve the true IFs. The hybrid constraints embedded in HCVNCMD significantly enhance noise robustness and improve IF estimation accuracy, enabling effective processing of NCSs with closely spaced IFs. Both simulation and experimental cases demonstrate that the HCVNCMD outperforms existing time–frequency analysis (TFAs) methods in accurately estimating the IFs of NCSs. Furthermore, it successfully identifies fault features in gear and bearing under time-varying speed conditions, confirming its effectiveness and practicability.