Variational problem of VMD construction in gear calculation of variable speed gearbox

The signal f is decomposed into k IMF components, each of which is a finite bandwidth with different center frequency. Then the variational problem can be described as minimizing the sum of the bandwidth of each component, and the constraint condition is that the sum of each component is equal to the original signal.

After Hilbert Demodulation, the analytic signals of each IMF component UK (T) are obtained

An estimated center frequency E-J ω KT is set for each analytic signal, and the spectrum of modal component is modulated to the fundamental frequency band

By calculating the square L2 norm of the above analytic signal gradient and estimating the bandwidth of each modal component, the constrained variational problem is obtained

Where UK {} = U1,…, UK {} and ω K {} = ω 1,…, ω K {} denote the center frequencies of K IMF components and IMF components respectively.