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A novel approach with smallest transition matrix for milling stability prediction

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Abstract

The introduction of chatter limits the processing efficiency of milling. Chatter prediction is an off-line strategy to select chatter-free cutting parameters. This paper presents a novel method for prediction of milling chatter which aims to reduce the dimension of transition matrix of the discrete map, where the eigenvalues of the transition matrix determine the system stability by using Floquet theory. A linear weight function is introduced when the weighted residual method is applied to the delay differential equation on discrete time intervals. Thus the displacement item can be removed from the state vectors. When the number of discrete intervals is fixed, it is concluded that the transition matrix obtained by the proposed method is the smallest among the time-domain methods. Meanwhile, the acceleration continuity condition is naturally satisfied on discrete time nodes which endows the method with competitive accuracy.

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Authors and Affiliations

  1. School of Mechanical Science and Engineering, State Key Laboratory of Digital Manufacturing Equipment and Technology, Huazhong University of Science and Technology, Wuhan, 430074, China

    Tao Huang, Xiaoming Zhang & Han Ding

Authors
  1. Tao Huang
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  2. Xiaoming Zhang
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  3. Han Ding
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Corresponding author

Correspondence to Xiaoming Zhang.

Appendix

Appendix

1.1 A. Matrices in Eq. (17)

1.2 B. Program Code

figure b
$$\begin{aligned}&\mathbf {F}_m=\left[ \begin{array}{c} \mathbf {f}_m^1 \\ \mathbf {f}_m^2 \\ \vdots \\ \mathbf {f}_m^E\\ \mathbf {0}\\ \mathbf {0}\\ \mathbf {0} \end{array}\right] ,\quad \mathbf {F}_c=\left[ \begin{array}{c} \mathbf {f}_c^1 \\ \mathbf {f}_c^2 \\ \vdots \\ \mathbf {f}_c^E\\ \mathbf {0}\\ \mathbf {0}\\ \mathbf {0} \end{array}\right] ,\quad \mathbf {F}_k=\left[ \begin{array}{c} \mathbf {f}_k^1 \\ \mathbf {f}_k^2 \\ \vdots \\ \mathbf {f}_k^E\\ \mathbf {0}\\ \mathbf {0}\\ \mathbf {0} \end{array}\right] \\&\mathbf {F}_a=\mathrm{diag}\left[ \begin{array}{l} K_c^1\\ K_c^2\\ \vdots \\ K_c^E\\ 0\\ 0 \end{array}\right] \mathbf {F}_K,\quad \mathbf {F}_1=\left[ \begin{array}{rrrrrr} 0&{}0&{}0&{}\cdots &{}0 \\ \vdots &{}\vdots &{} \vdots &{}\cdots &{} 0\\ 0&{}0&{}0&{}\cdots &{}0\\ 1&{}0&{}0&{}\cdots &{}0\\ 0&{}1&{}0&{}\cdots &{}0 \end{array}\right] \\&\mathbf {F}_{e}=\left[ \begin{array}{c} \mathbf {0} \\ \vdots \\ \mathbf {0}\\ \Lambda \cdot \mathrm{diag} \left[ \begin{array}{l} e^{\lambda _1 t_f}\\ e^{\lambda _2 t_f} \end{array} \right] \cdot \Lambda ^{-1} \left[ \begin{array}{l} \mathbf {v}_{E+1}\\ \mathbf {a}_{E+1} \end{array}\right] \end{array}\right] \end{aligned}$$
figure c
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Huang, T., Zhang, X. & Ding, H. A novel approach with smallest transition matrix for milling stability prediction. Nonlinear Dyn 90, 95–104 (2017). https://doi.org/10.1007/s11071-017-3649-0

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