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.

提出了一种广义数值微分方法，用于具有周期系数和离散时滞的非自治时滞微分方程（DDE）的稳定性预测。首先，以状态空间形式描述周期性DDE，并平均离散系统的周期。然后，基于有限差分公式，通过多个相邻采样网格点处状态函数值的线性组合来近似离散的一阶导数与时间的关系。这样，原始DDE近似为一系列代数方程，并且可以在一个周期上构造Floquet转换矩阵。最后，根据Floquet理论，通过检查特征值来确定系统稳定性。时滞阻尼Mathieu方程被认为是验证所提方法有效性和效率的典型案例。与通过基准算法（半离散化方法）的稳定性图和收敛速度进行比较。作为一种应用，该方法可用于预测变螺距铣刀的铣削稳定性，计算结果与实验验证的实例吻合良好。