The reconstruction of a semi-discretization method for milling stability prediction based on Shannon standard orthogonal basis

Abstract

In order to increase the calculation speed of the semi-discretization method (SDM) without accuracy loss, this paper reconstructs the SDM for predicting the stability lobes of the dynamic milling process, mainly considering the regenerative effect. The model of the dynamic milling process is expressed as the linear delay-differential equations (DDE). The fast calculation method is established by reconstructing the SDM based on the Shannon standard orthogonal basis (SSOB). First, the delay term of DDE is constructed without information loss based on Shannon interpolation functions, and SSOB is derived. Secondly, the closed form expression for the transition matrix of the system is constructed based on the SSOB, and the stability limit is predicted based on the Floquet theory. The transition matrix-based SDM and SSOB are theoretically compared, and it shows that the SDM is a special case of the method based on SSOB when the SSOB is regarded as the average in the sampling interval. The fast calculation method is established by using the variable sampling numbers during the period of the delay time in which the variable sampling numbers are determined by the condition which is used to construct the SSOB. Finally, this proposed fast method is used to the one and two degrees of freedom milling model, and the results show that the calculation accuracy is not reduced, and the calculation speed based on the proposed method can be improved nearly five times on the one degree of freedom model and 2.6 times on the two degrees of freedom model, compared to the semi-discretization method.