Chatter frequently occurs during cutting operations, which seriously restricts the machining productivity and workpiece accuracy. Consequently, accurate and efficient stability prediction is of great significance to determine stable machining parameters. A cubic Hermite–Newton approximation method which can determine the chatter stability boundaries more efficiently is presented in this paper. The milling dynamic system can be expressed as time-periodic delay differential equations (DDEs) with consideration of the regeneration effect. A typical benchmark example is provided to assess the convergence feature and stability lobes of the cubic Hermite–Newton approximation method and several existing methods. The results indicate that the cubic Hermite–Newton approximation method can achieve satisfactory results. For the sake of developing the cubic Hermite–Newton approximation method with higher convergence rate and computational efficiency, the tooth-passing period is further separated into two distinct phases according to whether the value of coefficient matrix equals to zero. Meanwhile, the linear interpolation polynomial is used to predict milling stability, and then piecewise polynomial interpolation was utilized in two adjacent time intervals to correct this prediction. By adopting the two benchmark examples, the effectiveness of the two new methods can be analyzed using existing methods. The results demonstrate that the two new methods have superior accuracy and efficiency.

Chatter vibration in machining operations has been identified as one of the major obstacles to improving surface quality and productivity. Therefore, efficiently and accurately predicting stable cutting regions is becoming increasingly important, especially in high-speed milling processes. In this study, on the basis of a predictor–corrector scheme, the following three error correction methods are developed for milling stability analysis: the Correction Hamming–Milne-based method (CHM), the Correction Adams–Milne-based method (CAM) and the Predictor–Corrector Hamming–Adams–Milne-based method (PCHAM). Firstly, we employ the periodic delay differential equations (DDEs), which are usually adopted to describe mathematical models of milling dynamics, and the time period of the coefficient matrix is divided into two unequal subintervals based on an analysis of the vibration modes. Then, the Hamming method and the fourth-order implicit Adams–Moulton method are separately utilized to predict the state term, and the Milne method is adopted to correct the state term. Based on local truncation error, combining the Hamming and Milne methods creates a CHM that can more precisely approximate the state term. Similarly, combining the fourth-order implicit Adams–Moulton method and the Milne method creates a CAM that can more accurately approximate the state term. More importantly, the CHM and the CAM are employed together to acquire the state transition matrix. Thereafter, the effectiveness and applicability of the three error correction methods are verified by comparing them with three existing methods. The results demonstrate that the three error correction methods achieve higher prediction accuracy without sacrificing computational efficiency. Compared with the 2nd SDM, the calculation times of the CHM, CAM and PCHAM are reduced by around 56%, 56% and 58%, respectively. Finally, verification experiments are carried out using a CNC machine (EMV650) to further validate the reliability of the proposed methods, where ten groups of cutting tests illustrate that the stability lobes predicted by the three error correction methods exhibit better agreement with the experimental results.

Regenerative chatter is an unfavorable phenomenon that severely affects machining efficiency and surface finish in milling operations. The prediction of chatter stability is an important way to obtain the stable cutting zone. Based on implicit multistep schemes, this paper presents the third-order and fourth-order implicit exponentially fitted methods (3rd IEM and 4th IEM) for milling stability prediction. To begin with, the delay differential equations (DDEs) with time-periodic coefficients are employed to describe the milling dynamics models, and the principal period of the coefficient matrix is firstly decomposed into two different subintervals according to the cutting state. Subsequently, the fourth-step and fifth-step implicit exponential fitting schemes are applied to more accurately estimate the state term. Two benchmark milling models are utilized to illustrate the effectiveness and advantages of the high-order implicit exponentially fitted methods by making comparisons with the three typical existing methods. Under different radial immersion conditions, the numerical results demonstrate that the 3rd IEM and the 4th IEM exhibit both faster convergence rates and higher prediction accuracy than the other three existing prediction methods, without much loss of computational efficiency. Finally, in order to verify the feasibility of the 3rd IEM and the 4th IEM, a series of experimental verifications are conducted using a computer numerical control machining center. It is clearly visible that the stability boundaries predicted by the 3rd IEM and the 4th IEM are mostly consistent with the cutting test results, which indicates that the proposed high-order exponentially fitted methods achieve significantly better prediction performance for actual milling processes.