High-order semi-discretization methods for stability analysis in milling based on precise integration

Abstract

The third-order and fourth-order semi-discretization methods (3rd SDM and 4th SDM) are presented to predict milling stability based on precise integration. Firstly, the milling system is mathematically represented by a delay differential equation (DDE) and re-expressed in the state space. After dividing the tooth passing period into free and forced vibration time intervals, the forced vibration time interval is equally discretized into many small-time intervals. Then, the DDE is integrated on any small-time interval and solved in the framework of the semi-discretization algorithm. Subsequently, the 3rd SDM and 4th SDM are derived for predicting the stability limits in milling. In the 3rd SDM and 4th SDM, the third-order and fourth-order interpolation polynomials are used to approximate the delayed term, respectively. The periodic-coefficient matrix is approximated by linear interpolation polynomial. The convergence rates of the one degree of freedom (one-DOF) milling system and the milling system with multiple modes are conducted. The results indicate the convergence rates of the 3rd SDM and 4th SDM are higher than those of the benchmark methods. Besides, no matter for large or low radial immersion conditions, the stability lobe diagrams (SLDs) obtained by the proposed 3rd SDM and 4thSDM are closer to the reference than those obtained by the benchmark methods. The computational efficiency of the 3rd SDM and 4th SDM is also evaluated. Compared with the benchmark methods, the presented 3rd SDM and 4th SDM are superior in computational efficiency. Generally, the main advantage of the proposed methods is the improvement of computational efficiency. The proposed 3rd SDM and 4th SDM can use a smaller discrete number for achieving the desired accuracy to reduce the size of the state transition matrix and improve the computational efficiency.