Prediction of chatter stability in high speed milling using the numerical differentiation method

Abstract

A numerical differentiation method is presented to predict the high speed milling stability of a two degrees of freedom (DOF) system based on the finite difference method and extrapolation method. The milling dynamics taking the regenerative effect into account are represented as linear periodic delayed differential equations (DDE) in the state space form. Then, each component of the first derivative of the state function versus time at the discretized sampling grids is approximated as a weighted linear sums of the state function values at its neighboring grid points, where the weight coefficients are calculated based on the extrapolation method. As such, the DDE on the forced vibration duration is approximately discretized as a series of algebraic equations. Thereafter, the Floquet transition matrix can be constructed on one tooth passing period by combining the analytical solution of the free vibration and the algebraic equations of the forced vibration. Finally, the milling stability is determined according to Floquet theory. The stability diagrams and convergence of critical eigenvalues in comparison with the benchmark algorithms (the semi-discretization method and numerical integration method) via experimentally verified examples are utilized to demonstrate the effectiveness and efficiency of the proposed method.

An improved numerical integration method to predict the milling stability based on the Lagrange interpolation scheme

“…the trapezoidal NIM (Trap-NIM) [31], and the Simpson NIM (Simp-NIM) [34]. The convergence rate and stability lobe diagrams via the single-DOF and two-DOF milling systems are calculated by using these calculated methods.…”

“…Subsequently, based on the Simpson integration formula, the Simpson method was developed to analyze the stability of milling operation, whose convergence rate was faster than the first-order SDM and second-order FDM [33]. A numerical differentiation method based on the Taylor series expansion was presented to predict the chatter stability as well [34]. According to the linear multistep method, the Milne-Simpson-based-corrector method was investigated to improve the calculated performance [35].…”

An updated Simpson-Based Method for Chatter Stability Prediction in Milling

“…Dai et al [37] presented a milling stability prediction method using the precise integration method. By approximating the differential term of the DDE, several methods were proposed for milling stability analysis, such as the differential quadrature method [38], the numerical differentiation method [39], and the Chebyshev-wavelet-based method [40]. Based on the numerical solution of ordinary differential equations, Qin et al presented the Adams-Moulton-based method (AMM) [41] and the Adams-Simpson-based method (ASM) [42].…”

Stability Analysis for Milling Process with Variable Pitch and Variable Helix Tools by High-Order Full-Discretization Methods

“…en, on the framework of FDM, Ding et al [8] introduced the second-order FDM (2nd FDM), Guo et al [9] proposed the third-order FDM (3rd FDM), and Ozoegwu reported the least squares approximation methods [10] and hyper-thirdorder full-discretization methods [11]. Besides, the numerical integration method [12] and differential quadrature method [13] developed by Ding and his coworkers, the Runge-Kutta-based methods proposed by Niu et al [14], the improved precise integration method proposed by Li et al [15], the Simpson-based method presented by Zhang et al [16], numerical differentiation method reported by Zhang et al [17], and so on are proposed for the stability prediction in the milling process. More recently, Olvera et al [18] presented the stability analysis for a single degree of freedom down-milling operation in a thin-walled workpiece by using the enhanced multistage homotopy perturbation (EMHP) method.…”