Research on High-Order Vese–Osher Model Based on Partial Differential Equation and Its Application in Planetary Gearbox Fault Diagnosis

Planetary gearbox is a crucial variable-speed transmission component in wind turbines, and the solar wheel is prone to local faults under harsh environmental conditions. Firstly, to avoid the influence of time-varying transmission paths, this paper preprocesses the fault vibration signal of solar wheel using window function and further transforms it to the angular domain.. Then, a high-order Vese-Osher model based on partial differential equations is subsequently developed to effectively identify the fault features of solar wheel. By introducing the mathematical expression of second derivative, this approach avoids the over-smoothing problem associated with the first derivative. While the Euler equation is commonly used to solve the total variation model, it cannot handle the G-norm, therefore, the divergence norm is considered to reformulate the model. Additionally, two weighting parameters are introduced between different regularization terms to optimize the solution, enhancing the ability of model to suppress noise in the fault-windowed vibration signal of solar wheel in the angular domain and thereby making the fault spectral lines more distinguishable. The Split Bregman algorithm is adopted to avoid directly solving high-order nonlinear partial differential equations. Furthermore, the method leverages the fast Fourier transform and generalized soft-thresholding equation to decompose the global optimization problem into several subproblems. Since higher-order cumulant exhibit superior noise suppression, the diagonal slices of third-order cumulant for the model output are extracted, and further obtain the 1.5-dimensional demodulation spectrum. This makes the fault feature of solar wheel completely prominent. Finally, The robustness of the above method in fault diagnosis of solar wheel is verified by fault test data.