One method of obtaining the Taylor Series Expansion Coefficients, which are suitable for engineering application, are presented: Artificial neural network(ANN) ， by virtue of its high nonlinear and learning abilities. The Taylor Series ( TS ) can be represented as a standard 3-layers feed-forward neural network after transforming in which the weights are correspond to the Taylor Coefficients. Therefore, the Taylor Coefficients can be determined by using Back Propagation(BP)algorithm. In the methods, we only need the sample space of the original function. So the method have the value of application in industries.

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Fault characteristic frequency is the main basis for rolling element bearing diagnostics but finding a suitable frequency band for demodulation and searching for the fault characteristic frequencies consume a lot of time and manpower in practice. A data-driven method based on recursive variational mode decomposition (RVMD), and an envelope order capture is proposed to realize the automatic fault diagnosis of bearing under different operating conditions. The process starts with a new proposed RVMD of the vibration signal, where the mode with maximum kurtosis of the unbiased autocorrelation of the envelope is selected to get envelope order spectrum. Thereafter, an order capture algorithm is designed to automatically search for the fault characteristic orders in theory, which are used for constructing feature vectors for diagnosis. The proposed method is tested on two test-beds which both contain the same type of bearing (SKF6205) but operate in different conditions, and gets good performance in bearing diagnosis. In addition, the fault diagnosis of test-bed two using training samples that are from test-bed one is investigated. This method reveals well generalization capability in the fault diagnosis of the same type of rolling element bearing under different operating conditions. INDEX TERMS Rolling element bearing, automatic fault diagnosis, recursive variational mode decomposition, envelope order capture.