Bifurcation analysis on a turning system with large and state-dependent time delay

Stability and bifurcation analyses were performed in this study on the turning process with a state-dependent and large time delay using the method of multiple scales (MMS). The turning system tool was modeled as an oscillator with two degrees of freedom, and both the cubic nonlinear stiffness and the nonlinear cutting force were considered. The nonlinear cutting force was appropriately expanded in a Taylor series considering the state-dependency of the time delay. The time delay and parameters were scaled through the proper ordering process to reflect the large delay effect on an asymptotic formulation of the MMS. Asymptotic solutions were then obtained by the MMS in the large delay regime and used to calculate the linear stability boundaries (i.e., Hopf bifurcation points) and coexisting one-period periodic solutions (i.e., limit cycles) of the turning system. To investigate the local and global behaviors of the tool chatter, bifurcation diagrams were obtained at various workpiece rotating speeds. The validity of the results was examined by comparison with those obtained through the method of harmonic balance and direct numerical integration. Additionally, using the bifurcation diagrams, the effects of the state-dependent time delay and nonlinear stiffness on the chatter vibration behaviors were examined.