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Local oscillatory-characteristic decomposition (LOD) is a relatively new self-adaptive time-frequency analysis methodology. The method, based on local oscillatory characteristics of the signal itself uses three mathematical operations such as differential, coordinate domain transform, and piecewise linear transform to decompose the multi-component signal into a series of mono-oscillation components (MOCs), which is very suitable for processing multi-component signals. However, in the LOD method, the computational efficiency and real-time processing performance of the algorithm can be significantly improved by the use of piecewise linear transformation, but the MOC component lacks smoothness, resulting in distortion. In order to overcome the disadvantages mentioned above, the rational spline function that spline shape can be adjusted and controlled is introduced into the LOD method instead of the piecewise linear transformation, and the rational spline-local oscillatory-characteristic decomposition (RS-LOD) method is proposed in this paper. Based on the detailed illustration of the principle of RS-LOD method, the RS-LOD, LOD, and empirical mode decomposition (EMD) are compared and analyzed by simulation signals. The results show that the RS-LOD method can significantly improve the problem of poor smoothness of the MOC component in the original LOD method. Moreover, the RS-LOD method is applied to the fault feature extraction of rotating machinery for the multi-component modulation characteristics of rotating machinery fault vibration signals. The analysis results of the rolling bearing and fan gearbox fault vibration signals show that the RS-LOD method can effectively extract the fault feature of the rotating mechanical vibration signals.

The rolling bearing and gear fault features are generally shown as modulation characteristics of their vibration signals. The empirical envelope (EE) method is an accordingly common demodulation method. However, the EE method has the defects of over- and undershoot, which may lead to demodulation error. According to this, an envelope optimization algorithm -- empirical optimal envelope (EOE) is introduced into the EE method, and an improved empirical envelope (IEE) method is obtained to calculate the instantaneous amplitude and instantaneous frequency of mono-component modulation signal. Furthermore, aiming at the actual measured mechanical vibration signal has multi-component modulation feature, the IEE method is combined with an adaptive signal decomposition method -- local oscillatory characteristic decomposition (LOD) proposed by the author, thereby a new multi-component signal demodulation method based on LOD and IEE is proposed. The proposed method is compared with Hilbert transform (HT) and Teager energy operator (TEO) demodulation methods by the simulation signal and actual measured mechanical vibration signal. The results show that the demodulation effects including edge effects, negative frequency, over- and undershoot of the proposed method are significantly improved and can extract the rolling bearing and gear fault feature information clearly.