Stability and bifurcation analyses of chatter vibrations in a nonlinear cylindrical traverse grinding process

In this study, stability and bifurcation analyses are performed on a cylindrical traverse grinding process in order to investigate its nonlinear chatter behaviors. The grinding model system under consideration appears to be a set of autonomous doubly regenerative delay differential equations. The linear stability boundaries of this grinding system are first evaluated by performing an eigen-analysis on the linearized system. In this stability analysis, a boundness condition for the chatter frequency is obtained and is used to avoid difficulties in identifying the stability boundary caused by the infinite-dimensional nature of the delayed system. The resulting linear stability diagrams are illustrated in the parametric windows of interest. Based on these stability diagrams, bifurcation analyses are conducted by the methods of multiple scales and harmonic balance (MMS and MHB) in order to investigate the local and global chatter behaviors. The criticality of Hopf bifurcation is analytically determined based on the normal form equations of the grinding system through the MMS. Furthermore, the codimension-two bifurcations of equilibrium, such as the Bautin and Hopf–Hopf bifurcations, are also identified in this study. For large-amplitude chatter behaviors, the periodic solutions bifurcated from the critical equilibrium are calculated by the MHB. The cyclic fold bifurcation of the limit cycle is newly identified. In these bifurcation analyses, the nonlinear chatter behaviors of the present grinding system are examined and discussed through the resulting bifurcation diagrams of limit cycles, the Bautin bifurcation diagram, and the corresponding phase portraits. These results are validated by comparison with those obtained through direct numerical integration.