Delay dynamical systems with applications to machine-tool chatter

The analyses of machine-tool chatter have shown qualitative differences between the theoretical and experimental results. The qualitative disagreements include, among other things, chatter stability charts, onset of chatter, jump phenomenon, finite amplitude instability, sub- and super-harmonic resonances. A major cause of this disagreement may stem from the assumption of negligible or small time delay actions and from ignoring the effect of stochastic noise on machine-tool systems. Additional causes are the inability to describe precisely the nature of the polynomial degree of nonlinearities existing in such systems and the prevailing mechanisms under which chatter may grow or decay.; In this thesis, the Lyapunov-Schmidt Reduction (LSR) along with the stochastic averaging method and the concept of the Lyapunov exponent have been employed in an attempt to analytically identify the regions of stable, finite amplitude instability and unstable chatter cutting amplitudes. The author hastens to add that although the applications considered here are mainly mechanical engineering-oriented, the work is totally theoretical. The present research focus is a rigorous theoretical presentation of the application of the LSR, deterministic and stochastic averaging methods and Lyapunov exponents in chatter lathe investigations. It is hoped that the deterministic and stochastic results of the lathe dynamical system, obtained without the assumption of small time delay, will provide some guidance in assessing the various mechanisms met in other machine-tool systems.; The equations of motion of the lathe dynamical system are represented by a series of third and fifth degree nonlinear deterministic and stochastic delay equations which are widely considered to adequately describe the phenomenon of machine-tool chatter. The largest Lyapunov exponent, obtained by the analytical method based on deterministic and stochastic averaging principle, characterizes the stability of the trivial and nontrivial solutions of the lathe dynamical system and their corresponding boundaries or critical points at which deterministic and stochastic chatter vibration bifurcates.; The Taylor series technique is typical of the fashion in which several engineers replace the governing infinite-dimensional delay equations with the traditional finite ordinary differential equations. Some typical delay equations are studied in order to illustrate some of the salient discrepancies between the application of the mathematical techniques: Lyapunov-Schmidt Reduction, Successive integration and Taylor series expansions. (Abstract shortened by UMI.)...