8th ASCE Specialty Conference on Probabilistic Mechanics and Structural Reliability PMC2000-265 ON THE STATIONARY STABILITY OF MACHINE-TOOL CHATTER WITH ERGODIC DAMPING PROCESS

A direct method of obtaining sufficient conditions for the stability behaviour of machining subjected to ergodic coefficient and nonlinear regenerative chatter has been presented. By a combined use of Hopf bifurcation theorem, the centre manifold, the integral stochastic averaging method, and the Lyapunov exponent, explicit analytical expressions for the limiting condistions of deterministic and stochastic stability boundaries have been derived. Problem Formulation The overall quality of any machine-tool is determined by many di¤erent factors, and among which the irregular regeneration waviness on the workpiece induced by chatter instabilities is the most signi…cant. Chatter is a vibratory phenomenon that is manifested when the tool is cutting a surface x(t) from a workpiece at time t that is already modulated from the pre-machined surface pro…le x(t¡h) at time t¡h, where h ´ T = 2= is the time delay between successive tool passes, is the angular velocity of the workpiece and T is the time period for one complete revolution. The standard equation describing the vibratory motion is the nonlinear delay di¤erential equation Äx + 2±0!0 _x +!20x +!20f(x ¡ x(t ¡ h)) + ±1T _xg = ¡¢f; which was originally formulated by Tobias, 1965. In this equation