First part of the thesis deals with the dynamics and stability of nonlinear delay gyroscopic systems with periodically varying delay. The aim is to demonstrate that greater depths of cut may be achieved in a boring process (2 DOF) 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 state dependent delay system. The augmented system of equations is autonomous and has two pairs of pure imaginary eigenvalues without resonance. The center 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. In [1], the authors present asymptotic expansion for the top Lyapunov exponent of a scalar linear delay differential equation driven by a two state Markov process. We extend their result to a vector case. In the case when driving noise is small, we construct an asymptotic expansion for the top Lyapunov exponent, which determines the almost-sure stability of the system. In [2] authors present a technique to suppress chatter, where spindle speed is varied as piecewise constant uniform noise. Using the results of vector case, we attempt to see whether stabilization can be achieved by varying spindle speed as a two state Markov chain. We find that the noise considered has destabilizing effect.
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Machine tool chatter suppression using spindle speed variation
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