To address the strong variability, complex cyclic characteristics, and difficulty in characterizing the main shaft torque load of vertical ring-die biomass briquetting machines, this study proposes a random load spectrum generation method based on kernel density estimation (KDE) and a Copula function. The measured torque time-history signal was processed using wavelet-threshold denoising and rainflow counting to extract cycle mean and cycle amplitude samples. KDE was used to estimate their marginal distributions, and a Copula function was introduced to construct the joint distribution model. The random load spectrum was then reconstructed through two-dimensional probability integration based on the fitted joint density function. The results show that the Frank Copula best describes the dependence structure between the cycle mean and the cycle amplitude. The reconstructed load spectrum agrees well with the measured load spectrum in terms of marginal frequency distribution and main peak intervals, with an RMSE of 6.3161 and an NRMSE of 6.94%. Compared with the KDE-independent baseline model, the proposed KDE–Frank Copula model reduces the RMSE by 12.74%. These results indicate that the proposed method can effectively characterize the statistical features of the random torque load of the main shaft and provide methodological support for load spectrum generation, fatigue life prediction, and reliability design of vertical ring-die biomass briquetting machines.
Na et al. (Fri,) studied this question.