Digital synchronous decomposition and period-N bifurcation size identification in dynamic systems: application to a milling process

Period-N bifurcation signal commonly contains the typical vibration and period-N bifurcation components. The period-N bifurcation size can be defined based on the period-N bifurcation component. It is challenging to extract the period-N bifurcation component with high accuracy in real-time since many frequencies may be contained in the period-N bifurcation signal. We proposed a digital synchronous decomposition (DSD) method to address this issue. Mathematical proof and numerical simulations validated that the DSD accurately decomposes the period-N bifurcation signal into the typical vibration and period-N bifurcation components in real-time. Moreover, an identification method and a bifurcation diagram were proposed to identify the period-N bifurcation size and depict period-N bifurcation evolution based on the DSD, respectively. We applied the proposed identification method to a benchmark milling process model and an actual milling process. Comparisons with the fast Fourier transform (FFT) highlight the proposed identification method's advantage. Besides, the proposed bifurcation diagram overcomes the limitation of the existing bifurcation diagrams and can quantitatively depict the period-N bifurcation severity and evolution.