Variational multi-harmonic duality mode pursuit method for extracting repetitive transient components from vibration signals

The repetitive transient component (RTC) in vibration signals is a typical symptom reflecting the localised defects of rotating machinery, which can be mathematically defined by feature parameters, such as the inter-arrival interval, maximum amplitude, resonance frequency, and attenuation ratio. In particular, the inter-arrival interval and maximum amplitude reveal the location and severity of failure and provide critical information for condition monitoring and fault diagnosis. However, there are some shortcomings of applicability, anti-noise, and fidelity, which are unfavourable for pursuing fault-induced RTC and enabling reliable recognition of key feature parameters. Hence, a novel method called variational multi-harmonic duality mode pursuit (VMHDMP) is proposed to extract the RTC accurately. A multi-harmonic duality mode function (MHDMF) that comprehensively depicts the critical feature parameters of the RTC was first defined in the frequency domain. Based on the defined MHDMF, the pursuit of RTC is formulated as a frequency-domain variational optimisation problem with a local compact time duration constraint, which optimally searches for the mode with an explicit MHDMF from the complex-valued frequency signal. A strategy involving the mutual conversion of real- and complex-valued signals was incorporated into the solution algorithm to satisfy the decomposability of complex-valued frequency signals. Finally, an inverse Fourier transform is performed on the analytical form of the extracted MHDMF to obtain a faithful RTC. Essentially, VMHDMP provides an optimal ‘transient-pass’ inverse filter bank with compact passage only near the occurrence moment of each transient in RTC. Thus, the vital parameters closely related to the faults are fully retained, whereas the frequency-overlap noise and large-amplitude interference occurring at different moments can be effectively eliminated. Simulated and experimental examples were used to test the effectiveness of VMHDMP.