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The occurrence of machining chatter may undermine the workpiece surface quality, accelerate the tool wear, and even result in serious damage to the machine tools. Consequently, it is of great importance to predict and eliminate the presence of such unstable and detrimental vibration. In this paper, we present an extended Adams-Moulton-based method for the stability prediction of milling processes with multiple delays. Taking the nonuniform pitch cutters or the tool runout into account, the regenerative chatter for milling operations can be formulated as delay differential equations with multiple delays. The dynamics model for milling regenerative chatter is rewritten in the state-space form. Dividing the spindle rotation period equally into small time intervals, the delay terms are approximated by Lagrange interpolation polynomials, and the Adams-Moulton method is adopted to construct the Floquet transition matrix. On this basis, the milling stability can be derived from the spectral radius of the transition matrix based on Floquet theory. The calculation efficiency and accuracy of the proposed algorithm are verified through making comparisons with the semidiscretization method (SDM) and the enhanced multistage homotopy perturbation method (EMHPM). The results show that the proposed method has both high computational efficiency and accuracy.

In machining operations, chatter vibration is still one of the main constraints to high productivity and part quality. It is a typical kind of self-excited vibration between the cutter and the workpiece and can occur in almost every machining process. The onset of such detrimental instability may result in poor surface roughness, rapid tool wear, and large reduction of tool life. Therefore, many researches on the modelling, prediction, and avoidance of milling chatter have been conducted [

To achieve stable milling operations, one effective and significant technique is selecting proper cutting parameters based on the stability lobe diagrams, which can be acquired via the milling stability prediction. Therefore, many approaches have been proposed to approximate the DDEs to derive the milling stability lobe diagrams, such as numerical methods [

Nevertheless, the above works were mostly conducted based on the ideal milling operations with regular uniform pitch cutters, in which there exists only single time delay. Taking the nonuniform pitch cutters or the case tool runout into account, the regenerative chatter models for milling operations are described by delay differential equations with multiple delays. As a consequence, many efforts have been made to extend the above algorithms to the multiple delays case. Altintaş et al. [

In recent years, efficient and accurate milling stability prediction has been a key issue both in academic and industrial fields. However, it is difficult to achieve both high computational accuracy and efficiency simultaneously. Based on our previous work [

Theoretically speaking, when the constant pitch cutter is employed and the cutter runout is neglected, there exists only one delay term in the system dynamics equation. Based on [

However, the actual milling cutters cannot be always completely symmetrical. Therefore, there may exist a certain deviation between the spindle rotation axis and the tool geometry axis, which constitutes the so-called cutter runout. In addition, when the variable pitch cutter without cutter runout is considered, any cutting point will always remove the surface left by the first previous tooth. In this case, due to the unevenly pitched space angle, the delays that equal relevant tooth passing period will be different. Consequently, in regard to the practical milling process, the governing dynamics equation should be modelled as delay differential equations with multiple delays instead. Based on [

Without loss of generality, we utilize milling with nonconstant pitch cutter to illustrate the mathematical model of milling with multiple delays. Based on [

The screen function

For more details of the milling dynamics models, one can refer to [

To numerically determine the milling stability, the governing equation (

By applying the state-space theory, the analytical response of (

Firstly, the spindle rotation period

During the

By substituting

In order to simplify the derivation process, we will use some abbreviated expressions; that is,

When the time delay

Similarly, the delay terms

By substituting (

In addition,

Let

Obviously,

By combining (

Finally, the transition matrix

According to Floquet theory, the stability of periodic systems depends on the spectral radius of the transition matrix. Consequently, the stability of milling operations can be obtained from the following criterion:

It should be noted that the construction of the Floquet transition matrix should be based on the period of the coefficient matrix rather than on that of the delay terms. For milling processes with single delay, the time period

In this section, the computation efficiency and accuracy of the proposed method are verified by a two-DOF milling operation with variable pitch cutter in [

Based on [

In (^{2} and ^{2}.

With the same matrix transformation presented in Section

To verify the feasibility of the proposed method, both large and low radial immersion milling conditions need to be investigated. First, the radial immersion ratio

Stability lobe diagrams computed by the SDM, the EMHPM, and the EAMM with the radial immersion ratio

Stability lobe diagrams computed by the SDM, the EMHPM, and the EAMM with the radial immersion ratio

Comparisons of relative errors among the SDM, the EMHPM, and the EAMM with the radial immersion ratio

Meanwhile, we set the radial immersion ratio

Stability lobe diagrams computed by the SDM, the EMHPM, and the EAMM with the radial immersion ratio

Stability lobe diagrams computed by the SDM, the EMHPM, and the EAMM with the radial immersion ratio

Comparisons of relative errors among the SDM, the EMHPM, and the EAMM with the radial immersion ratio

In this work, an efficient and accurate semianalytical algorithm is proposed for the stability prediction of milling processes with multiple delays. Firstly, the milling dynamics model for regenerative chatter is rewritten in the state-space form. After the spindle rotation period is equally discretized, the delay terms are approximated by Lagrange interpolation polynomials, and the Adams-Moulton method is employed to construct the Floquet transition matrix. Finally, the stability of milling operations can be predicted by examining the spectral radius of the Floquet transition matrix. A two-DOF milling model with variable pitch tool has been adopted to demonstrate the proposed method. The numerical results demonstrate that under the same computational condition the proposed method achieves a higher computational efficiency than the SDM and the EMHPM. Compared with the SDM and the EMHPM, the computation time of the proposed method can be reduced by 67–70% and 45–53%, respectively. In general, the accuracy of the EAMM is higher than the SDM and the EMHPM. In addition, the accuracy of the stability lobe diagrams computed by the proposed method can be improved significantly over the spindle speed range from 2000 rpm to 5000 rpm.

The authors declare that they have no conflicts of interest.

This work was partially supported by the Innovation Fund of National Business Aircraft Manufacturing Engineering Technology Research Center (Grant no. SAMC14-JS-15-046), the National Natural Science Foundation of China (NSFC) (Grant no. 51375297), and the Program of Shanghai Subject Chief Scientist (Grant no. 14XD1402000).