Abstract
According to the chaotic features and typical fractional order characteristics of the bearing vibration intensity time series, a forecasting approach based on long range dependence (LRD) is proposed. In order to reveal the internal chaotic properties, vibration intensity time series are reconstructed based on chaos theory in phase-space, the delay time is computed with C-C method and the optimal embedding dimension and saturated correlation dimension are calculated via the Grassberger–Procaccia (G-P) method, respectively, so that the chaotic characteristics of vibration intensity time series can be jointly determined by the largest Lyapunov exponent and phase plane trajectory of vibration intensity time series, meanwhile, the largest Lyapunov exponent is calculated by the Wolf method and phase plane trajectory is illustrated using Duffing-Holmes Oscillator (DHO). The Hurst exponent and long range dependence prediction method are proposed to verify the typical fractional order features and improve the prediction accuracy of bearing vibration intensity time series, respectively. Experience shows that the vibration intensity time series have chaotic properties and the LRD prediction method is better than the other prediction methods (largest Lyapunov, auto regressive moving average (ARMA) and BP neural network (BPNN) model) in prediction accuracy and prediction performance, which provides a new approach for running tendency predictions for rotating machinery and provide some guidance value to the engineering practice.
Highlights
Rolling bearings are widely used as the most common and key components in rotating machinery systems
The statistical time series modeling approaches include auto regressive (AR), auto regressive moving average (ARMA), auto regressive integrated moving average (ARIMA), and generalized auto regressive conditional heteroskedasticity (GARCH) [4,5,6] methods, the use of those models is often restricted by linearity, normality and stationarity characteristics and they are very well suited to capture short range dependence and stable periodicity, they are not well suited to deal with nonlinearity, resulting in poor prediction accuracy
Chaos time series prediction methods, such as adaptive prediction and local adaptive prediction, Volterra series method and largest Lyapunov Exponent (LLE) [13,14] are based on the theory of chaos phase space reconstruction, and the weight parameters are regulated automatically, In addition, reference [15] proposed a method for reconstructing dynamical systems based on time-delay embedding and use it on the bearing fault diagnostics, compares the reconstructed bearing vibration data with a baseline reconstruction from normal bearing data to generate a severity index over time with Wasserstein distance and the earth mover’s distance (EMD), but these studies can't make full use of the advantages of the adaptive algorithm for its inability to adjust the filter parameters owing to the unknown real-value during multi-step prediction
Summary
Rolling bearings are widely used as the most common and key components in rotating machinery systems. Chaos time series prediction methods, such as adaptive prediction and local adaptive prediction, Volterra series method and largest Lyapunov Exponent (LLE) [13,14] are based on the theory of chaos phase space reconstruction, and the weight parameters are regulated automatically, In addition, reference [15] proposed a method for reconstructing dynamical systems based on time-delay embedding and use it on the bearing fault diagnostics, compares the reconstructed bearing vibration data with a baseline reconstruction from normal bearing data to generate a severity index over time with Wasserstein distance and the earth mover’s distance (EMD), but these studies can't make full use of the advantages of the adaptive algorithm for its inability to adjust the filter parameters owing to the unknown real-value during multi-step prediction Despite these preliminary efforts mentioned above, all the statistical models and conventional methods were found to be insufficient to address the characteristics of bearing vibration intensity.
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