Generalized Numerical Differentiation Method for Stability Calculation of Periodic Delayed Differential Equation: Application for Variable Pitch Cutter in Milling

This paper proposes a generalized numerical differentiation method for stability prediction of the non-autonomous delayed differential equations (DDEs) with periodic coefficients and discrete delays. Firstly, the periodic DDE is described in state-space form and the period of a system is equally discretized. Then, the discrete first derivatives versus time are approximated by a linear combination of the state function values at multiple neighboring sampling grid points based on the finite-difference formulas. Such that, the original DDE is approximated as a series of algebraic equations and the Floquet transition matrix can be constructed on one period. At last, the system stability is determined according to the Floquet theory by checking the eigenvalues. The delayed damped Mathieu equation is regarded as a typical case to verify the effectiveness and efficiency of the presented method. The stability diagrams and rate of convergence are computed in comparison with those via the benchmark algorithm (the semi-discretization method). As an application, the presented method is used to predict the stability of milling with variable pitch cutter, and the computational result agrees well with the experimentally verified example.