Multiple operational mode prediction at milling tool-tip based on transfer learning

Abstract

Understanding the tool-tip dynamics is crucial for evaluating the performance in milling and essential for chatter prediction; obtaining and predicting tool-tip modes efficiently and accurately is thus essential, especially when the milling parameters or tool-holder assembly change. However, there is currently no such efficient and explainable method with high generalization ability for obtaining and predicting the tool-tip modes considering the above change. To address this issue, the stochastic subspace identification (SSI) method is initially used to acquire multiple operational modes more efficient and cost-effective than traditional methods under varying milling parameters. Subsequently, machine learning (ML) models are trained to predict the above modes under varying spindle speeds and axial cutting depth. Moreover, when changes occur in the tool-holder assembly, a transfer learning (TL) model based on receptance coupling substructure analysis (RCSA) theory is proposed to re-establish the modes prediction model efficiently with the above data. The TL model has a modal frequency prediction error below 2% and a damping ratio prediction error below 10%, thereby demonstrating robust generalization capabilities. Finally, predicting milling stability with the above modes prediction model, which can provide a stability lobe diagram with higher accuracy than the traditional method, is introduced. In conclusion, the multiple operational modes are acquired more efficiently with the SSI method, and the ML model or TL model with RCSA theory is thus established efficiently when milling parameters or tool-holder assembly change. The obtained model is used for chatter prediction as follows and performs better in prediction accuracy.

Appendices

Appendix A. Euler-Bernoulli beam fit method

Below is the parameterized model for translational receptance ${H_{11}}$

$${H_{11}}{\text{=}}\frac{{\sin (\lambda L)\cosh (\lambda L) - \cos (\lambda L)\sinh (\lambda L)}}{{{\lambda ^3}EI(1+i\eta )(\cos (\lambda L)\cosh (\lambda L) - 1)}}$$

(A.1)

where

$${\lambda ^4}{\text{=}}{\omega ^2}((\rho A)/(EI(1+i\eta ))),A=\pi {d^2}/4,I=\pi {d^4}/64$$

(A.2)

where w represents angular frequency.ρ, E represents the density and elastic modulus of the object being studied, respectively.$\eta$represents the damping factor. d represents the diameter of the beam and L represents the length of the beam.

The undetermined parameters in the equation (A.1) and (A.2) are d, L and$\eta$. By fitting these three values based on experimental results, their specific values can be obtained.

Once these model parameters are obtained, other three FRFs can be calculated using the following equations

$${L}_{11}={N}_{11}=\frac{-{sin}(\lambda L\left){sin}h(\lambda L\right)}{{\lambda }^{2}EI(1+i\eta )\left({cos}(\lambda L\right){cos}h(\lambda L)-1)}$$

$${P}_{11}=\frac{{cos}(\lambda L\left){sin}h(\lambda L\right)+{sin}(\lambda L\left){cos}h(\lambda L\right)}{\lambda EI(1+i\eta )\left({cos}(\lambda L\right){cos}h(\lambda L)-1)}$$

(A.3)

Appendix B. Equivalent parameter for simplification of the tool flute

The equation of parameters a and r

$${a}_{i}=\left\{\begin{array}{c}\frac{2{D}^{2}{f}_{d}+2D{{f}_{d}}^{2}}{3{D}^{2}+4D{f}_{d}-{{f}_{d}}^{2}},i=3\\ \frac{D{f}_{d}}{D+{f}_{d}},i=4\end{array}\right.$$

$${r}_{i}=\left\{\begin{array}{c}\frac{3{D}^{3}-5D{{f}_{d}}^{2}}{2(3{D}^{2}+4D{f}_{d}-{{f}_{d}}^{2})},i=3\\ \frac{{D}^{2}-D{f}_{d}}{2(D+{f}_{d})},i=4\end{array}\right.$$

(B.1)

where i represents the number of tool flute.

The equation of equivalent radius R_eq is:

$${{R}_{eq}}_{i}=\left\{\begin{array}{c}{a}_{i}{cos}(\theta )+\sqrt{({{r}^{2}}_{i}-{{a}_{i}}^{2}{{sin}}^{2}\theta )}\\ 0<\theta <2\pi /3,i=3\\ {a}_{i}{sin}(\theta )+\sqrt{({{r}^{2}}_{i}-{{a}_{i}}^{2}{{sin}}^{2}\theta )}\\ 0<\theta <\pi /2,i=4\end{array}\right.$$

(B.2)

The equations of cross-sectional moment of inertia with single tool flute ${I_{xx}}_{{_{i}}}$and ${I_{yy}}_{{_{i}}}$ are

$${I_{xx}}_{{_{i}}}=\left\{ \begin{gathered} \int\limits_{0}^{{2\pi /3}} {\frac{{{R_{\text{e}\text{q}}}_{{_{i}}}(\theta )}}{4}{{\sin }^2}(\theta )d\theta - \frac{\pi }{8}{{(\frac{{{f_d}}}{2})}^2},i=3} \hfill \\ \int\limits_{0}^{{\pi /2}} {\frac{{{R_{\text{e}\text{q}}}_{{_{i}}}(\theta )}}{4}{{\sin }^2}(\theta )d\theta - \frac{\pi }{8}{{(\frac{{{f_d}}}{2})}^2},i=4} \hfill \\ \end{gathered} \right.$$

(B.3)

$${{I}_{yy}}_{i}=\left\{\begin{array}{c}{\int }_{0}^{2\pi /3}\begin{array}{c}\frac{{{R}_{eq}}_{i}(\theta {)}^{4}}{4}{{cos}}^{2}(\theta )d\theta -[\frac{\pi }{8}(\frac{{f}_{d}}{2}{)}^{4}\\ +\frac{\pi ({f}_{d}/2{)}^{2}}{2}({r}_{i}+{a}_{i}-\frac{{f}_{d}}{2})],i=3\end{array}\\ {\int }_{0}^{\pi /2}\begin{array}{c}\frac{{{R}_{eq}}_{i}(\theta {)}^{4}}{4}{{cos}}^{2}(\theta )d\theta -[\frac{\pi }{8}(\frac{{f}_{d}}{2}{)}^{4}\\ +\frac{\pi ({f}_{d}/2{)}^{2}}{2}({r}_{i}+{a}_{i}-\frac{{f}_{d}}{2})],i=4\end{array}\end{array}\right.$$

(B.4)

The equation of cross-sectional moment of inertia in x or y direction is

$${I_{xx\_tot}}_{{_{i}}}={I_{yy\_tot}}_{{_{i}}}=\left\{ \begin{gathered} 1.5({I_{xx}}_{{_{i}}}+{I_{yy}}_{{_{i}}}),i=3 \hfill \\ 2({I_{xx}}_{{_{i}}}+{I_{yy}}_{{_{i}}}),i=4 \hfill \\ \end{gathered} \right.$$

(B.5)

Appendix C. The expressions of rotational receptances

The expressions of rotation-to-force, displacement-to-moment,rotation-to-moment FRFs.

${G_{21}}$,${G_{12}}$,${G_{22}}$respectively are

$${G_{21}}=\frac{{{U_2}}}{{{Q_1}}}={R_{{v_3}{v_3}}}\frac{{{q_{{v_3}}}}}{{{Q_1}}}+{R_{{v_3}{v_4}}}\frac{{{q_{{v_2}}}}}{{{Q_1}}}{\text{+}}{R_{{v_3}{v_5}}}\frac{{{q_{{v_1}}}}}{{{Q_1}}}$$

(C.1)

where

$$\frac{1}{{{Q_1}}}\left[ \begin{gathered} {q_{{v_3}}} \hfill \\ {q_{{v_2}}} \hfill \\ {q_{{v_1}}} \hfill \\ \end{gathered} \right]=\left[ \begin{gathered} {R_{{v_3}{v_3}}}+{R_{{u_3}{u_3}}}{\text{ }}{R_{{v_3}{v_2}}}+{R_{{u_3}{u_2}}}{\text{ }}{R_{{v_3}{v_1}}}+{R_{{u_3}{u_1}}} \hfill \\ {R_{{v_2}{v_3}}}+{R_{{u_2}{u_3}}}{\text{ }}{R_{{v_2}{v_2}}}+{R_{{u_2}{u_2}}}{\text{ }}{R_{{v_2}{v_1}}}+{R_{{u_2}{u_1}}} \hfill \\ {\text{ }}{R_{{v_1}{v_3}}}+{R_{{u_1}{u_3}}}{\text{ }}{R_{{v_1}{v_2}}}+{R_{{u_1}{u_2}}}{\text{ }}{R_{{v_1}{v_1}}}+{R_{{u_1}{u_1}}} \hfill \\ \end{gathered} \right]\cdot \left[ \begin{gathered} {R_{{u_3}{u_1}}} \hfill \\ {R_{{u_2}{u_1}}} \hfill \\ {R_{{u_1}{u_1}}} \hfill \\ \end{gathered} \right]$$

(C.2)

$${G_{12}}=\frac{{{U_1}}}{{{Q_2}}}={R_{{u_1}{u_3}}}\frac{{{q_{{u_3}}}}}{{{Q_1}}}+{R_{{u_1}{u_2}}}\frac{{{q_{{u_2}}}}}{{{Q_1}}}{\text{+}}{R_{{u_1}{u_1}}}\frac{{{q_{{u_1}}}}}{{{Q_1}}}$$

(C.3)

where

$$\frac{1}{{{Q_2}}}\left[ \begin{gathered} {q_{{u_3}}} \hfill \\ {q_{{u_2}}} \hfill \\ {q_{{u_1}}} \hfill \\ \end{gathered} \right]=\left[ \begin{gathered} {R_{{v_3}{v_3}}}+{R_{{u_3}{u_3}}}{\text{ }}{R_{{v_3}{v_2}}}+{R_{{u_3}{u_2}}}{\text{ }}{R_{{v_3}{v_1}}}+{R_{{u_3}{u_1}}} \hfill \\ {R_{{v_2}{v_3}}}+{R_{{u_2}{u_3}}}{\text{ }}{R_{{v_2}{v_2}}}+{R_{{u_2}{u_2}}}{\text{ }}{R_{{v_2}{v_1}}}+{R_{{u_2}{u_1}}} \hfill \\ {\text{ }}{R_{{v_1}{v_3}}}+{R_{{u_1}{u_3}}}{\text{ }}{R_{{v_1}{v_2}}}+{R_{{u_1}{u_2}}}{\text{ }}{R_{{v_1}{v_1}}}+{R_{{u_1}{u_1}}} \hfill \\ \end{gathered} \right]\cdot \left[ \begin{gathered} {R_{{v_3}{v_3}}} \hfill \\ {R_{{v_2}{v_3}}} \hfill \\ {R_{{v_1}{v_3}}} \hfill \\ \end{gathered} \right]$$

(C.4)

$${G_{22}}=\frac{{{U_2}}}{{{Q_2}}}={R_{{v_3}{v_3}}}\frac{{{q_{{v_3}}}}}{{{Q_2}}}+{R_{{v_3}{v_2}}}\frac{{{q_{{v_2}}}}}{{{Q_2}}}{\text{+}}{R_{{v_3}{v_1}}}\frac{{{q_{{v_1}}}}}{{{Q_2}}}$$

(C.5)

where

$$\frac{1}{{{Q_2}}}\left[ \begin{gathered} {q_{{v_3}}} \hfill \\ {q_{{v_2}}} \hfill \\ {q_{{v_1}}} \hfill \\ \end{gathered} \right]=\left[ \begin{gathered} {R_{{v_3}{v_3}}}+{R_{{u_3}{u_3}}}{\text{ }}{R_{{v_3}{v_2}}}+{R_{{u_3}{u_2}}}{\text{ }}{R_{{v_3}{v_1}}}+{R_{{u_3}{u_1}}} \hfill \\ {R_{{v_2}{v_3}}}+{R_{{u_2}{u_3}}}{\text{ }}{R_{{v_2}{v_2}}}+{R_{{u_2}{u_2}}}{\text{ }}{R_{{v_2}{v_1}}}+{R_{{u_2}{u_1}}} \hfill \\ {\text{ }}{R_{{v_1}{v_3}}}+{R_{{u_1}{u_3}}}{\text{ }}{R_{{v_1}{v_2}}}+{R_{{u_1}{u_2}}}{\text{ }}{R_{{v_1}{v_1}}}+{R_{{u_1}{u_1}}} \hfill \\ \end{gathered} \right]\cdot \left[ \begin{gathered} {R_{{u_3}{u_3}}} \hfill \\ {R_{{u_2}{u_3}}} \hfill \\ {R_{{u_1}{u_3}}} \hfill \\ \end{gathered} \right]$$

(C.6)