Spectral method for prediction of chatter stability in low radial immersion milling

The aim of this paper is to develop an integral equation based spectral method for prediction of chatter stability in low radial immersion milling. First, the delay-differential equation with time-periodic coefficients governing the dynamic milling process is transformed into the integral equation. Then, the duration of one tooth period is divided into the free vibration and the forced vibration processes. While the former one has an analytical solution, the discretization technique is explored to approximate the solution of the latter one. After the forced vibration duration being equally discretized, the Gauss-Legendre formula is used to discretize the definite integral, in the meantime the Lagrange interpolation is adopted for approximating the state item and the time-delay item by using the corresponding discretized state points and time-delay state points. The approximate Floquet transition matrix is thereafter constructed to predict the milling stability based on the Floquet theory. The benchmark examples are utilized to verify the proposed method. Compared with previous time domain methods, the proposed method enables higher rate of convergence. The results also demonstrate that the proposed method is high-effective.