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 rege nerative effect, linear structural damping, quadratic and cubic nonlin ear stiffness bf the machine tool, and linear, quadratic, and cubic re generative terms. Using the width of cut was a bifurcation parameter, we find, using linear theory, that disturbances decay with time and he nce chatter does not occur if w < w(c) and disturbances grow exponenti ally with time and hence chatter occurs if w > w(c). In other words, a s w increases past w(c), a Hopf bifurcation occurs leading to the birt h of a limit cycle. Using the method of multiple scales, we obtained t he normal form of the Hopf bifurcation by including the effects of the quadratic and cubic nonlinearities. This normal form indicates that t he bifurcation is supercritical; that is, local disturbances decay for w < w(c) and result in small limit cycles (periodic motions)for w > w (c). Using a six-term harmonic-balance solution, we generated a bifurc ation diagram describing the variation of the amplitude of the fundame ntal harmonic with the width of cut. Using a combination of Floquet th eory and Hill's determinant, we ascertained the stability of the perio dic solutions. There are two cyclic-fold bifurcations, resulting in la rge-amplitude periodic solutions, hysteresis, jumps, and subcritical i nstability. As the width of cut w increases, the periodic solutions un dergo a secondary Hopf bifurcation, leading to a two-period quasiperio dic motion (a two-torus). The periodic and quasiperiodic solutions are verified using numerical simulation. As w increases further, the toru s doubles. Then, the doubled torus breaks down, resulting in a chaotic motion. The different attractors are identified by using phase portra its, Poincare' sections, and power spectra. The results indicate the i mportance of including the nonlinear stiffness terms.