Statistical modelling of rolling bearing cage fault vibrations through conditional heteroscedastic time-series analysis

Rolling element bearings play an important role in rotating machinery and are the source of non-stationary and heteroscedastic vibration responses caused by localized defects which are impulsive. This study proposes and tests a conditional-volatility modelling approach for characterizing bearing cage-faults based on autoregressive (AR), generalized autoregressive conditional heteroskedasticity (GARCH) and exponential generalized autoregressive conditional heteroskedasticity (EGARCH) models. The tri-axial vibration measurements were taken on a bearing test rig provided by Spectra Quest and were analysed under healthy and cage-fault operating conditions. The modelling procedure starts by modelling the deterministic mean response with the axial and tangential vibration components and testing if there is any time-varying conditional variance in the remaining stochastic structure. 40,000 synchronized observations were assessed using a comparative model assessment based on parameter significance, likelihood-based criteria, regression error and volatility diagnostics. The AR baseline fit the best, with an R² of 0.9220, but it does not capture the conditional variance dynamics. The GARCH (1,1) model seemed to have captured the shock sensitivity and high volatility persistence with the highest value of α + β = 0.9564 as well as the best likelihood-based statistics among the volatility models: AIC = -0.4994, BIC = -0.4981. In addition, the EGARCH (1,1) model exhibited substantial magnitude and persistence effects; although the sign of the persistence effect was opposite for positive and negative shocks, it was not statistically significant (γ = -0.0025, p = 0.7766) for the dataset of tested cages. The results show that the GARCH family conditional variance modelling is a useful model to describe the bearing vibration when there is a fault, but currently, no evidence suggests that EGARCH is the best model for the present data over GARCH.