Runge-Kutta methods for a semi-analytical prediction of milling stability

On the basis of Runge-Kutta methods, this paper proposes two semi-analytical methods to predict the stability of milling processes taking a regenerative effect into account. The corresponding dynamics model is concluded as a coefficient-varying periodic differential equation with a single time delay. Floquet theory is adopted to predict the stability of machining operations by judging the eigenvalues of the state transition matrix. This paper firstly presents the classical fourth-order Runge-Kutta method (CRKM) to solve the differential equation. Through numerical tests and analysis, the convergence rate and the approximation order of the CRKM is not as high as expected, and only small discrete time step size could ensure high computation accuracy. In order to improve the performance of the CRKM, this paper then presents a generalized form of the Runge-Kutta method (GRKM) based on the Volterra integral equation of the second kind. The GRKM has higher convergence rate and computation accuracy, validated by comparisons with the semi-discretization method, etc. Stability lobes of a single degree of freedom (DOF) milling model and a two DOF milling model with the GRKM are provided in this paper.