Improved ensemble local mean decomposition based on cubic trigonometric cardinal spline interpolation and its application for rotating machinery fault diagnosis

Ensemble local mean decomposition has been gradually introduced into mechanical vibration signal processing due to its excellent performance in electroencephalogram signal analysis. However, an unsatisfactory problem is that ensemble local mean decomposition cannot effectively process vibration signals of complex mechanical system due to the constraints of moving average. The process of moving average is time-consuming and inaccurate in complex signal analysis. Therefore, an improved ensemble local mean decomposition method called C-ELMD with modified envelope algorithm based on cubic trigonometric cardinal spline interpolation is proposed in this article. First, the shortcomings in sifting process of ensemble local mean decomposition is discussed and, furthermore, advantages and disadvantages of the common interpolation methods adopted to improve ensemble local mean decomposition are compared. Then, cubic trigonometric cardinal spline interpolation is employed to construct the local mean and envelope curves in a more precise way. In addition, the influence of shape-controlling parameter on envelope estimation accuracy in cubic trigonometric cardinal spline interpolation is also discussed in detail to select an optimal shape-controlling parameter. The effectiveness of cubic trigonometric cardinal spline interpolation for improving the accuracy of ensemble local mean decomposition is demonstrated using a synthetic signal. Finally, the proposed cubic trigonometric cardinal spline interpolation is tested to be effective in gear and bearing fault detection and diagnosis.

The Cardinal Spline Methods for the Numerical Solution of Nonlinear Integral Equations

In this study, an effective technique is presented for solving nonlinear Volterra integral equations. The method is based on application of cardinal spline functions on small compact supports. The integral equation is reduced to a system of algebra equations. Since the matrix for the system is triangular, it is relatively straightforward to solve for the unknowns and an approximation of the original solution with high accuracy is accomplished. Several cardinal splines are employed in the paper to enhance the accuracy. The sufficient condition for the existence of the inverse matrix is examined, and the convergence rate is analyzed. We compare our method with other methods proposed in recent papers and demonstrated the advantage of our method with several examples.

Evaluation of Methods of Interpolation (Existing and Proposed) using Monte Carlo Simulation Technique and Delete-D Jackknife Method

It is possible to encounter missing value in a research. Missing value may be as a result of none response in primary data collection or unavailability of data in the case of secondary data. It may occur within a set of observations or at the tail end of the observations. The paper addresses missing value within a set of observations (interpolation). Existing methods considered are linear, log-linear, Catmull-Rom spline and cardinal spline and a method of estimating missing value is presented. For evaluation of the methods, random data are simulated using Monte Carlo simulation approach and analytical approach is used to determine the most effective method. Among the findings, linear, log-linear and the proposed method give high precision estimate compare to the CS and CSR methods. With the use of tension parameter, CS is better than CSR method.

Research on Influences of Employment in Manufacturing Industry on that in the Service Industry Based on Bayes Model

Using the data of 285 prefectural and the above-level cities from 2004 to 2016, this thesis reveals the impact of employment in China's urban manufacturing industry on the employment of service industries with the Bayesian model. Under the Bayesian framework, partial linear semi-parametric model is proposed. The nonlinear functions are fitted by using truncation base cardinal spline and considering the random error terms of mixed normal fitting models. The results show that: employment in the urban manufacturing industry in China has significant influence on the employment in the service industry. When the number of employees in the manufacturing industry changes from 0 to 650,000, the manufacturing industry has less influence. When the number of the employees in the manufacturing industry changes from 650,000 to 900,000, the employees of the service industry will dramatically increase. When the number of the employees in the manufacturing industry is more than 900,000, the employees in the service industry will be prone to stable growth.

Curvature Variation Minimizing Cardinal Spline Curves

Since the free parameter and the two auxiliary points have effect on the shape of the Cardinal spline curves, a natural idea arises to find the optimal parameter and auxiliary points to obtain the smoothest curves. We use curvature variation minimization to achieve this goal in this paper. By minimizing an appropriate approximation of the curvature variation energy, the unique solution can be easily obtained. Some examples show that the Cardinal spline curves with minimum curvature variation are better than the Catmull-Rom spline curves in dealing with interpolation problems.

Warp-Knitted Fabrics Simulation Using Cardinal Spline and Recursive Rotation Frame

To realize 3–D simulation of warp-knitted fabrics, this paper proposes a 3–D loop modeling method based on Cardinal spline and recursive rotation frame method. Fabric structure and loop geometry are studied to form a visible lapping detection model and loop geometrical key points obtained according to chain links and yarn arrangements. In this paper, eight points are set for stitched loop and three points for the inlay. To connect all the control points via a curved line to form a knitted loop shape, the Cardinal spline method is adopted because it is simple to calculate and adjustable in radians. A recursive rotation frame is established to calculate the coordinates of the yarn cross section, eliminating the problem of yarn discontinuity caused by excessive rotation of the moving frame. With the 3–D loop modeling method, some specific mesh and spacer structures are simulated to demonstrate applicability.