This paper presents a numerical scheme to predict the milling stability based on the integral equation and numerical integration formulas. First, the milling dynamics taking the regenerative effect into account is represented in the form of integral equation. Then, the tooth passing period is precisely divided into the free vibration phase during which the analytical solution is available and the forced vibration phase during which an approximate solution is needed. To obtain the numerical solution of the integral equation during the forced vibration phase, the time interval of interest is equally discretized. Over each small time interval, Newton-Cotes integration formulas or Gauss integration formulas are employed to approximate the integral term in the integral equation. After establishing the state transition matrix of the system in one period, the milling stability is predicted by using Floquet theory. The benchmark examples are utilized to verify the proposed approach. The results demonstrate that it is highly efficient and accurate.
This paper analyzes the stability of milling with variable pitch cutter and tool runout cases characterized by multiple delays, and proposes a new variable-step numerical integration method for efficient and accurate stability prediction. The variable-step technique is emphasized here to expand the numerical integration method, especially for the low radial immersion cases with multiple delays. First, the calculation accuracy of the numerical integration method is discussed and the variable-step algorithm is developed for milling stability prediction for single-delay and multiple-delay cases, respectively. The milling stability with variable pitch cutter is analyzed and the result is compared with those predicted with the frequency domain method and the improved full-discretization method. The influence of the runout effect on the stability boundary is investigated by the presented method. The numerical simulation shows that the cutter runout effect increases the stability boundary, and the increasing stability limit is verified by the milling chatter experimental results in the previous research. The numerical and experiment results verify the validity of the proposed method.variable-step, numerical integration method, milling, chatter, stability prediction, multiple delays, variable pitch cutter, cutter runout Citation:Zhang X J, Xiong C H, Ding Y, et al. Variable-step integration method for milling chatter stability prediction with multiple delays.
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