The Rational Spline Interpolation Based-LOD Method and Its Application to Rotating Machinery Fault Diagnosis

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.

On the approximation of $ \ exp (-x) $ on the half-axis by spline functions in three-point rational interpolants

For the function $f(x)=\exp(-x)$, $x\in [0,+\infty)$ on grids of nodes $\Delta: 0=x_0<x_1<\dots $ with $x_n\to +\infty$ we construct rational spline-functions such that $R_k(x,f, \Delta)=R_i(x,f)A_{i,k}(x)\linebreak+R_{i-1}(x, f)B_{i,k}(x)$ for $x\in[x_{i-1}, x_i]$ $(i=1,2,\dots)$ and $k=1,2,\dots$ Here $A_{i,k}(x)=(x-x_{i-1})^k/((x-x_{i-1})^k+(x_i-x)^k)$, $B_{i,k}(x)=1-A_{i,k}(x)$, $R_j(x,f)=\alpha_j+\beta_j(x-x_j)+\gamma_j/(x+1)$ $(j=1,2,\dots)$, $R_j(x_m,f)=f(x_m)$ при $m=j-1,j,j+1$; we take $R_0(x,f)\equiv R_1(x,f)$. Bounds for the convergence rate of $R_k(x,f, \Delta)$ with $f(x)=\exp(-x)$, $x\in [0,+\infty)$, are found.

Wind Velocity Data Interpolation Using Rational Cubic Spline

Wind velocity data is always having positive value and the minimum value approximately close to zero. The standard cubic spline interpolation (not-a-knot and natural) as well as cubic Hermite polynomial may be produces interpolating curve with negative values on some subintervals. To cater this problem, a new rational cubic spline with three parameters is constructed. This rational spline will be used to preserve the positivity of the wind velocity data. Numerical results shows that the proposed scheme work very well and give visually pleasing interpolating curve on the given domain.