Stability of systems with time-periodic delay: application to variable speed boring process

Abstract

The stability of systems with fluctuating delay is studied. The aim is to demonstrate that, greater depths of cut may be achieved in a boring process, when the speed of the spindle is modulated sinusoidally instead of being kept constant. Since the variation of spindle speed is small and independent of the tool motion, by expanding the delay terms about a finite mean delay and augmenting the system, the time-dependent delay system can be written as a system of non-linear delay equations with fixed delay. The augmented system of equations is autonomous and has two pairs of pure imaginary eigenvalues without resonance. The centre-manifold and normal form methods are then used to obtain an approximate and simpler four-dimensional system. Analysis of this simpler system shows that periodic variations in the delay lead to larger stability boundaries.