Compiling the milling load spectrum of a machining center based on an L-moment ratio diagram mixture distribution method

The load spectrum is an important component in the reliability testing of machining centers. The statistical characteristics of a load must be accurately estimated to evaluate the capacity of a machining center to withstand the applied load. Due to the complexity of the processing technology, diversity of the processed materials, and uncertain factors, several types of cutting loads present single or bimodal distributions for which the conventional one-component probability density function (PDF) is inappropriate. Mixture distributions are regarded as an appropriate alternative for modeling such load data. In addition, parameter estimation for these mixture models requires an advanced optimization method, which takes a substantial amount of time to process. However, machining centers typically involve a variety of processing methods, each giving rise to various load types, resulting in a significant increase in the amount of calculation in parameter estimation. Accordingly, rapid and accurate selection of the best matching hybrid model is a great challenge. To solve this problem, a method for compiling a two-component distribution function based on the linear moment ratio diagram (L-MRD) method is proposed to rapidly and accurately model multiple types of load distributions. First, the one-component and the two-component mixture models consisting of normal distribution (Norm), Gumbel or extreme value type I (E), generalized extreme value (GEV), Pearson type III (P3), and 3-parameter Lognormal (LN3) distributions are selected as candidate models, and the mean and amplitude are applied to milling total eight load sets. Thereafter, the best two-component hybrid model for each load set is determined by taking the estimated value of the load sample data in L-MRD and the relative position of the candidate distribution in L-MRD as the basis for selecting the best matching distribution. The estimation of the parameter of the mixture models is obtained with the least-squares and maximum likelihood methods. The optimization of the objective functions related to these estimation methods is carried out with a genetic algorithm (GA) that is more adapted to mixture distributions. Finally, the results obtained with the L-MRD are compared with those obtained with goodness-of-fit statistics through model selection criteria. Analysis results indicate that L-MRD quickly and effectively selects the optimal mixture distribution component type on a single graph and offers more information than a goodness-of-fit criterion because it provides knowledge about such characteristics as the skewness and kurtosis of the load data samples. In addition, the mixture models improve the modeling flexibility, minimize statistical errors, enhance the fitting accuracy, and generate load spectra that better reflect the actual load characteristics.