Incipient fault signatures in rolling bearings are often compromised by intense background noise and stochastic impulses. Conventional resonance demodulation frequently relies on rigid frequency partitioning, which tends to disrupt the physical continuity of resonance bands and results in the incomplete capture of essential diagnostic information. Furthermore, the robustness of prevailing optimal demodulation frequency band (ODFB) selection indicators remains limited under heavy noise interference. This study develops the WLERgram framework, which utilizes regularized Fourier series to capture the global morphology of the vibration spectrum. By anchoring filter boundaries at natural energy troughs, the method mitigates spectral truncation based on inherent signal characteristics. The framework integrates an Adaptive Morphological Consensus (AMC) strategy, employing multi-scale operators to extract rotation-correlated components and enhance resistance to incoherent interference. By incorporating a Weighted Logarithmic Energy Ratio (WLER) metric, the method utilizes a nonlinear operator to implement differential mapping between coherent fault harmonics and stochastic noise, enabling autonomous optimization of the demodulation band. Validations using synthetic simulations and experimental benchmarks (CWRU and UORED) suggest that WLERgram offers reliable feature extraction performance and diagnostic robustness under harsh noise environments.

To address the challenge of effectively extracting early-stage failure features in rolling bearings, this paper proposes a two-stage harmonic optimization-gram method based on spectral amplitude modulation (SAM-TSHOgram). The method first employs amplitude spectra with varying weighting exponents to preprocess the signal, performing nonlinear adjustments to the vibration signal’s spectrum to enhance weak periodic impact characteristics. Subsequently, a two-stage evaluation strategy based on spectral coherence (SCoh) was designed to adaptively identify the optimal frequency band (OFB). The first stage employs the Periodic Harmonic Correlation Strength (PHCS) metric, based on autocorrelation, to coarsely screen candidate bands with strong periodic structures. The second stage utilizes the Sparse Harmonic Significance (SHS) metric, based on spectral negative entropy, to refine the candidate set, selecting bands with the most prominent harmonic features. Finally, SCoh is integrated over the selected OFB to generate an Improved Envelope Spectrum (IES). The proposed method was validated using both simulated and experimental vibration signals from bearings and gearboxes. The results demonstrate that SAM-TSHOgram significantly outperforms conventional approaches such as EES, Fast Kurtogram, and IESFOgram in terms of signal-to-noise ratio (SNR) enhancement, harmonic clarity, and diagnostic robustness. These findings confirm its potential for reliable early fault detection in rolling bearings.

To improve the effectiveness of Multiscale Range Entropy (MRE) in extracting features from rolling bearing faults, this paper proposes a novel bearing fault diagnosis method that combines the Northern Goshawk Optimization (NGO) algorithm, MRE, and a Deep Extreme Learning Machine (DELM). First, the raw vibration signal is decomposed using Variational Mode Decomposition (VMD) to obtain its Intrinsic Mode Functions (IMFs). Second, the NGO algorithm is employed to optimize the MRE parameters, using the minimum crest factor as the objective function. Then, using these optimized parameters, MRE is applied to extract fault features. The resulting feature set undergoes dimensionality reduction to create the final sample set. Finally, an NGO-optimized DELM is used for fault classification. The experimental results demonstrate that this method effectively extracts the characteristic signals of rolling bearing faults through robust parameter optimization, thereby improving the accuracy of fault diagnosis.