Orthogonal polynomial approximation method for stability prediction in milling

Based on orthogonal polynomial approximation scheme, this paper presents several stability prediction methods using different kinds of orthogonal polynomials. The milling dynamics with consideration of the regenerative effect is described by time periodic delay-differential equations (DDEs). Firstly, this work employs the classical Legendre and Chebyshev polynomials to approximate the state term, delayed term, and periodic-coefficient matrix. With the help of direct integration scheme (DIS), the state transition matrixes which indicate the mapping relations of the dynamic response between the current tooth pass and the previous tooth pass are obtained. The stability lobe diagrams for single degree of freedom (DOF) and two DOF milling models are generated by using the Legendre and Chebyshev polynomial approximation-based methods. The rate of convergence of the Legendre and Chebyshev polynomial-based methods is compared with that of the benchmark first-order semi-discretization method (1stSDM). The comparison results indicate that the rate of convergence and the numerical stability of the Legendre and Chebyshev polynomial-based methods are both need to be improved. In order to develop new methods with high rate of convergence and numerical stability base on DIS, the monic orthogonal polynomial sequences are constructed by using Gram-Schmidt orthogonalization to approximate the state term, delayed term, and periodic-coefficient matrix. The rate of convergence and the computational efficiency of the monic orthogonal polynomial-based methods are evaluated by comparing with those of the benchmark 1stSDM. The results turn out that the monic orthogonal polynomial-based methods are advantageous in terms of the rate of convergence and numerical stability. The stability lobe diagrams for single DOF and two DOF milling models obtained by the monic orthogonal polynomial-based methods are compared with those obtained by the 1stSDM. Finally, the monic orthogonal polynomial-based methods are proved to be the effective and efficient methods to predict the milling stability.