High-Dimensional Harmonic Balance Analysis for a Turning Process with State-Dependent Delay

Abstract

The present study considers the state-dependent delay differential equations (SD-DDEs) for the turning process. In general, series expansion of the SD-DDEs turning system is essential in the nonlinear analysis such as the conventional methods of multiple scales and harmonic balance. Unfortunately, the mathematical theory of SD-DDEs, especially for those with an implicit function of delay, was just recently developed and any rigorous mathematical theory has not yet been proven. As one approach for the nonlinear analysis of the SD-DDEs, physically reasonable results could be obtained by extending the general theory of DDEs to the SD-DDEs through the use of the series expansion in conjunction with the implicit function, although there still remains an open issue of its mathematical rigorousness. The other approach may be treating the original SD-DDEs directly. To this end, the high-dimensional harmonic balance (HDHB) analysis is performed in this study in order to investigate the nonlinear behaviors of the turning system in the form of SD-DDEs without its series expansion. The results obtained by HDHB analysis are validated by comparing results with those of direct time integration. Using the resulting bifurcation diagrams, nonlinear chatter behaviors of the turning system are examined and discussed.