Sparse time series modeling of the baseline vibration from a gearbox under time-varying speed condition

•A sparse FP-AR model is proposed for modeling non-stationary gearbox vibrations.•Both simulation and experimental signals are used to validate the proposed model.•The proposed model has higher modeling accuracy than the conventional models. Time series model-based approach (TSMBA) is promising in processing vibration signals and assessing the health condition of gearboxes. Accurate time series modeling of the baseline vibration is critical to the TSMBA. Gearboxes often operate under time-varying speed condition, which makes the baseline vibration non-stationary. To accurately model such signals, non-stationary time series models are in demand. Conventional functional pooled autoregression (FP-AR) model is a good option. However, conventional FP-AR assumed 1) consecutive AR terms and 2) identical functional space that describes the dependency between AR parameters and rotating speed, which limited its modeling accuracy. To improve modeling accuracy, this paper proposes a sparse FP-AR model that uses sparse AR terms and non-identical functional spaces. To obtain such a sparse FP-AR model, a new model selection procedure is developed by adopting the least absolute shrinkage and selection operator. The sparse FP-AR model has been validated using simulation signals from a simulation model for a fixed-axis gearbox and experimental signals from two independent fixed-axis gearbox test-rigs. The modeling accuracy was measured by mean squared errors and randomness tests of the modeling residuals, goodness-of-fit between the one-step ahead prediction and real gear vibration, and time-frequency spectra. Results have shown that the proposed sparse FP-AR model has higher modeling accuracy than the conventional one. Meanwhile, TSMBA that uses the sparse FP-AR model was applied for detecting gear tooth crack faults under time-varying speed condition. Results have shown that the proposed method benefits the fixed-axis gearbox in early detection of faults and better assessment of fault progressions.