VMD for solving variational problems in gear calculation of variable speed gearbox

By introducing the quadratic penalty factor α and Lagrange operator λ (T), the constrained variational problem is transformed into the unconstrained variational problem, and the following augmented Lagrange function expression is formed

The eigenmode components and their central frequencies are updated iteratively by using the alternating direction of multiplier algorithm (ADMM). The saddle point obtained by the final formula is the optimal solution of the original problem. Each IMF component is obtained by the following formula

Where u ^ n + 1K is the result of Wiener filtering of the current residual f ^ (T) – ∑ I ≠ Ku ^ I (ω). The center frequency of each IMF component is updated by the following formula ω n + 1K