Perturbation Methods in Nonlinear Dynamics—Applications to Machining Dynamics

The role of perturbation methods and bifurcation theory in predicting the stability and complicated dynamics of machining is discussed using a nonlinear single-degree-of-freedom model that accounts for the regenerative effect, linear structural damping, quadratic and cubic nonlinear stiffness of the machine tool, and linear, quadratic, and cubic regenerative terms. Using the width of cut w as a bifurcation parameter, we find, using linear theory, that disturbances decay with time and hence chatter does not occur if w < wc and disturbances grow exponentially with time and hence chatter occurs if w > wc. In other words, as w increases past wc, a Hopf bifurcation occurs leading to the birth of a limit cycle. Using the method of multiple scales, we obtained the normal form of the Hopf bifurcation by including the effects of the quadratic and cubic nonlinearities. This normal form indicates that the bifurcation is supercritical; that is, local disturbances decay for w < wc and result in small limit cycles (periodic motions) for w > wc. Using a six-term harmonic-balance solution, we generated a bifurcation diagram describing the variation of the amplitude of the fundamental harmonic with the width of cut. Using a combination of Floquet theory and Hill’s determinant, we ascertained the stability of the periodic solutions. There are two cyclic-fold bifurcations, resulting in large-amplitude periodic solutions, hysteresis, jumps, and subcritical instability. As the width of cut w increases, the periodic solutions undergo a secondary Hopf bifurcation, leading to a two-period quasiperiodic motion (a two-torus). The periodic and quasiperiodic solutions are verified using numerical simulation. As w increases further, the torus doubles. Then, the doubled torus breaks down, resulting in a chaotic motion. The different attractors are identified by using phase portraits, Poincare´ sections, and power spectra. The results indicate the importance of including the nonlinear stiffness terms.