Sufficient conditions for the stability of single and multiple regenerative chatter

Sufficient conditions ensuring stable and unstable machining when chatter is accompanied with periodic, nonlinear and noise perturbations have been derived. Single and multiple regenerative chatter leading to delay differential equations (DDEs) are considered. Transcendental characteristic equations of the linearized delay equations for the determination of boundaries of stable and unstable machining through Hopf bifurcation at equilibrium are analysed. A generalized centre manifold is constructed in order to reduce the infinite-dimensional character of the governing DDEs to ordinary differential equations (ODEs) with single and multiple time delays. The ODEs are further written into amplitude and phase equations, and then averaged over a period of [0,2π] to obtain constant solutions. Conditions for super- and subcritical stability in the deterministic sense are derived from the scalar bifurcation equations of the trivial and nontrivial solutions, while those for stable machining in the stochastic sense are obtained by the evaluation of the largest Lyapunov exponent.