High-order full-discretization method using Lagrange interpolation for stability analysis of turning processes with stiffness variation

In turning processes, chatter is an unstable vibration which adversely affects surface finish and machine tool components. Stiffness variation (SV) is an effective strategy for chatter suppression by periodically modulating the stiffness around a nominal value. The dynamics of SV turning is governed by a time periodic delay differential equation (DDE) where the time-period/time-delay ratio (TPTDR) can be arbitrary. Recently, first-, second- and higher-order full-discretization methods (FDMs) have been reported as a popular class of methods for milling stability prediction. However, these FDMs can only deal with time periodic DDE where the TPTDR equals one. In this paper, two high-order FDMs using Lagrange interpolation (HLFDMs) are proposed for stability analysis of SV turning. On each discrete time interval, the time delay term is interpolated by the second-degree Lagrange polynomial, and the time periodic term is linearly interpolated. The state term is approximated using linear interpolation and second-degree Lagrange polynomial interpolation, achieving the first- and second-order HLFDM, respectively. Finally, the transition matrix over a single period is deduced for stability analysis via the Floquet theory. Benchmark examples of damped delay Mathieu equations are used to verify the proposed algorithm, which demonstrates that HLFDMs are highly efficient and accurate. In addition, the second-order HLFDM is used to investigate the effects of SV amplitude and frequency parameters. These results provide theoretical insights for the selection of SV parameters.