Effect of regenerative process on the sample stability of a multiple delay differential equation

Abstract

In this paper, explicit analytical expressions for the stability behaviour of a single degree of freedom machining model have been derived. The regenerative chatter due to chip thickness and feed rate variations, a delay in the spring force and the probabilistic property of the displacement process of the chatter induced by the earlier tool cuts in the undeformed chip thickness have been taken into account. A characteristic equation for the linearized stability at equilibrium machining is presented, and regions of stable and unstable machining for multiple fixed time delays are captured in the phase plane of two model parameters. By a combined use of the classical Hopf bifurcation theorem and the centre manifold, equations governing stochastic chattering, which are infinite in character, are reduced to two-dimensional ordinary differential equations. The integral averaging method and the Lyapunov exponent have been employed to explicitly derive the required analytical expressions.