A new angle-domain cepstral method for generalised gear diagnostics under constant and variable speed operation

Cepstrum is generally an uncommon term. In engineering, specifically Fourier analysis, it is defined as the result obtained when the inverse Fourier transform of a logarithmic signal spectrum is computed. In other words, it is a useful technique for investigating and detecting the periodic structures in the frequency spectra, but it also makes forcing and transfer functions additive. In practice, one of its main applications is to analyze and recognize human speech, machine vibration analysis and address reflection inference as in radar and earth seismology. Cepstrum operations can be labeled as cepstral analysis, or quefrency analysis used in denoting the cepstrum independent variable and liftering used to edit the cepstrum.

One engineering application of the cepstrum is to eliminate speed effects in gear vibrations caused by varying frequency components (orders) traversing fixed resonance frequencies. Order-tracking the signal with respect to the reference signal is one way of obtaining a cepstrum with discrete components in variable speed conditions. Previous studies recommended obtaining close estimates of the amplitude of the transfer function between the acceleration measurement and gearmesh force by using a short-pass exponential lifter before order tracking. While different approaches like subtraction in the cepstrum have been employed to remove the (amplitude of) transfer functions, they have various limitations.

Mark’s average log ratio (ALR) approach, a popular concept in gear diagnostics, assumes that the transfer function between the vibration responses measured externally and the static transmission error (STE) forcing function correspond to linear differential equations such that the STE can be decomposed into additive components. Whereas the cepstrum liftering can remove the dynamic transfer function amplitude, it fails to remove the phase effects. This could compromise the practical cepstrum applications, by limiting them to spectra, not waveforms.

Inspired by previous studies, Professor Robert Bond Randall, Dr. Wade Smith, Associate Professor Pietro Borghesani and Professor Zhongxiao Peng from UNSW Sydney proposed an angle cepstrum comb liftering (ACCL) gear diagnostic method to achieve gear fault diagnosis under variable speed conditions by removing the dynamic modal effects, defined in the frequency domain, as well as Mark’s mesh transfer functions, defined in the order domain. It employs the concept of Mark’s ALR more efficiently by giving a single value for different frequency ranges with enhanced noise suppression capabilities. The work is currently published in the peer-reviewed journal, Mechanical Systems and Signal Processing.

The research team showed that the resulting log spectrum was periodic over several gearmesh frequency periods, providing a representation of the static transmission error elements without modification by the mesh transfer functions. Therefore, this technique could remove both the dynamic transfer function and mesh transfer function effects independently of speed to produce a forcing function closely related to the components of the combined elemental transmission error in terms of geometric and elastic deviations from ideal.

Furthermore, the new approach proved to be a powerful diagnostic tool because it provided similar results at different speeds, even with large-speed variations up to ± 20%. It was mostly considered to be effective for local faults, including relatively large localized spalls, tooth root cracks and uniformly distributed pitting (as sums over the individual pits). This made it possible to obtain parameters correlating well with crack size and pitting severity indicators. The presented approach provided alternatives to commonly used residual and classical TSA signals (different at different speeds) to provide repeatable health indicators and enable their evaluation for widely varying speeds.

In summary, UNSW Sydney reported a novel method for gear diagnosis based on applying an adjustable comb lifter to the order-tracked acceleration signal cepstrum in the angle domain. The feasibility of the modified log amplitude spectra in tracking fault development was demonstrated using three case studies. Repeatable time waveforms were also achieved when the modified amplitude spectra were combined with the original phase spectra. Overall, the applied method proved natural and appropriate in evaluating the results and tracking fault development. In a statement to Advances in Engineering, the corresponding and lead author Professor Randall explained their findings will enhance fault diagnosis in gears under different operating conditions.