## 1 Introduction  
Rotating machinery, such as gears and rolling bearings, is an important component of mechanical equipment whose fault will have a serious impact on the safe operation of the mechanical device. Therefore, it is of great significance to realize real-time online monitoring of the running condition of rotating machinery [1]. When rotating machinery breaks down, its vibration signals are usually nonlinear and non-stationary. Consequently, traditional time-domain or frequency-domain analysis methods are not suitable for the direct analysis of these signals [2]. The time-frequency analysis method is often used to extract the fault features of the vibration signals of rotating machinery because it can provide both the time-domain and frequency-domain information of signals, and has been adopted to solve multiple fault diagnosis problems [3; 4]. However, classic time-frequency analysis methods have the inherent limitation of insufficient self-adaptability. Considering short-time Fourier transform (STFT) as an example, the time-frequency structure no longer changes once its window function is fixed [5]. Another example is the wavelet transform (WT), whose basis function and time-frequency structure are also fixed and are not self-adaptive in nature [6]. As real self-adaptive time-frequency analysis methods, empirical mode decomposition (EMD) [7] and local mean decomposition (LMD) [8] can decompose a multi-component amplitude-modulation and frequency-modulation (AM-FM) signal into a plurality of single-component signals and a residue in a self-adaptive manner, and therefore, they have been extensively applied [9; 10; 11; 12; 13; 14]. However, there are still some theoretical limitations with these two methods. For EMD, these problems include the end effects [15], over- and undershoot [16], mode-mixing [17], and unexplainable negative frequencies generated by the Hilbert transform [18]. For LMD, the use of a moving average to derive the local mean function and the envelope estimation function leads to a large amount of calculation. In addition, the selection of moving step lengths is subjective, where wrongly selected step lengths will lead to a poor decomposition outcome [19], and local mutation and other problems may occur to a pure FM signal obtained by means of the division [20]. Recently, the author proposed a new self-adaptive time-frequency analysis method: local oscillatory-characteristic decomposition (LOD) [21]. This method is based on the local characteristic time scale of signals, and by means of operations such as the differential, coordinate domain transformation, and piecewise linear transformation, a multi-component AM-FM signal is decomposed into a series of mono-oscillation components (MOCs) whose instantaneous frequencies have physical meanings and a residue. The method is highly efficient in the calculation and more suitable for real-time online processing and analysis of data. The LOD method has been applied to fault analysis of gears and rolling bearings since its proposal and has achieved a good analysis outcome [22; 23], but some limitations have been encountered in further applications. For example, although the piecewise linear transformation operation improves the calculation efficiency, the linear connection may lead to the unsmoothness of the decomposed MOC components. In particular, owing to a continuous decrease in the number of extreme points as decomposition proceeds, the errors caused by unsmoothness gradually accumulate, which may seriously affect the accuracy of decomposition. Rational spline interpolation [24] is a spline interpolation method with good convergence and continuity. The shape of an interpolation curve can be changed by adjusting the introduced pole parameter \(p\); owing to this characteristic, this interpolation method can theoretically avoid the limitations of over- and undershoot, low smoothness, etc. generated by cubic spline, linear interpolation, and other interpolation curves with fixed shapes [25; 26].