Chatter stability for micromilling processes with flat end mill

Abstract

A mechanistic model is developed to predict micromilling forces with flat end mill for both shearing and ploughing-dominant cutting regimes. The model assumes that there is a critical chip thickness that determines whether a chip will form or not. Numerical method is extended to predict the chatter stability in micro end milling, which is performed based on the proposed cutting force model. The simulating procedure for predicting stability and cutting forces is presented in detail, and the stability diagram is constructed. The validation experiments are conducted to verify the simulation results. Both experimental cutting forces measured and machined workpiece surface scanned through digital microscope are analyzed and used to verify the proposed model.

Appendix

• 1. If h > h _c, transforming the cutting forces (the first equation in Eq. (12)) from the cutter coordinates to the global coordinates, obtained

  $$ \begin{array}{l}\left\{\begin{array}{c}\hfill \varDelta {F}_x^j\left(z,t\right)\hfill \\ {}\hfill \varDelta {F}_y^j\left(z,t\right)\hfill \\ {}\hfill \varDelta {F}_z^j\left(z,t\right)\hfill \end{array}\right\}=\left[\begin{array}{ccc}\hfill - \sin \varphi \left(t,j,z\right)\hfill & \hfill - \cos \varphi \left(t,j,z\right)\hfill & \hfill 0\hfill \\ {}\hfill - \cos \varphi \left(t,j,z\right)\hfill & \hfill \sin \varphi \left(t,j,z\right)\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right]\left\{\begin{array}{c}\hfill \varDelta {F}_{\mathrm{r}}^j(t)\hfill \\ {}\hfill \varDelta {F}_{\mathrm{t}}^j(t)\hfill \\ {}\hfill \varDelta {F}_{\mathrm{z}}^j(t)\hfill \end{array}\right\}\hfill \\ {}=\left(\left[\begin{array}{cc}\hfill {\mathrm{DK}}_{11}^j\left(z,t\right)\hfill & \hfill {\mathrm{DK}}_{12}^j\left(z,t\right)\hfill \\ {}\hfill {\mathrm{DK}}_{21}^j\left(z,t\right)\hfill & \hfill {\mathrm{DK}}_{22}^j\left(z,t\right)\hfill \\ {}\hfill {\mathrm{DK}}_{31}^j\left(z,t\right)\hfill & \hfill {\mathrm{DK}}_{32}^j\left(z,t\right)\hfill \end{array}\right]\left\{\begin{array}{c}\hfill A(t)\hfill \\ {}\hfill B(t)\hfill \end{array}\right\}+\left[\begin{array}{cc}\hfill {\mathrm{DC}}_{11}^j\left(z,t\right)\hfill & \hfill {\mathrm{DC}}_{12}^j\left(z,t\right)\hfill \\ {}\hfill {\mathrm{DC}}_{21}^j\left(z,t\right)\hfill & \hfill {\mathrm{DC}}_{22}^j\left(z,t\right)\hfill \\ {}\hfill {\mathrm{DC}}_{31}^j\left(z,t\right)\hfill & \hfill {\mathrm{DC}}_{32}^j\left(z,t\right)\hfill \end{array}\right]\left\{\begin{array}{c}\hfill \overset{.}{A}(t)\hfill \\ {}\hfill \overset{.}{B}(t)\hfill \end{array}\right\}+\left\{\begin{array}{c}\hfill {\mathrm{DE}}_1^j\left(z,t\right)\hfill \\ {}\hfill {\mathrm{DE}}_2^j\left(z,t\right)\hfill \\ {}\hfill {\mathrm{DE}}_3^j\left(z,t\right)\hfill \end{array}\right\}\right)\varDelta z\hfill \end{array} $$

  (A.1)

  where

  $$ \begin{array}{c}\hfill \left\{\begin{array}{c}\hfill {\mathrm{DK}}_{11}^j\left(z,t\right)\hfill \\ {}\hfill {\mathrm{DK}}_{12}^j\left(z,t\right)\hfill \\ {}\hfill {\mathrm{DK}}_{21}^j\left(z,t\right)\hfill \\ {}\hfill {\mathrm{DK}}_{22}^j\left(z,t\right)\hfill \\ {}\hfill {\mathrm{DK}}_{31}^j\left(z,t\right)\hfill \\ {}\hfill {\mathrm{DK}}_{32}^j\left(z,t\right)\hfill \end{array}\right\}=\left[\begin{array}{ccccc}\hfill -{k}_1{K}_{\mathrm{tc}}\hfill & \hfill -\left({k}_2{K}_{\mathrm{tc}}+{k}_1{C}_{\mathrm{p}}\varOmega \right)\hfill & \hfill -{k}_2{C}_{\mathrm{p}}\varOmega \hfill & \hfill 0\hfill & \hfill 0\hfill \\ {}\hfill {k}_1{C}_{\mathrm{p}}\varOmega \hfill & \hfill \left({k}_2{C}_{\mathrm{p}}\varOmega -{k}_1{K}_{\mathrm{tc}}\right)\hfill & \hfill -{k}_2{K}_{\mathrm{tc}}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ {}\hfill {k}_2{K}_{\mathrm{tc}}\hfill & \hfill \left({k}_2{C}_{\mathrm{p}}\varOmega -{k}_1{K}_{\mathrm{tc}}\right)\hfill & \hfill -{k}_1{C}_{\mathrm{p}}\varOmega \hfill & \hfill 0\hfill & \hfill 0\hfill \\ {}\hfill -{k}_2{C}_{\mathrm{p}}\varOmega \hfill & \hfill \left({k}_2{K}_{\mathrm{tc}}+{k}_1{C}_{\mathrm{p}}\varOmega \right)\hfill & \hfill -{k}_1{K}_{\mathrm{tc}}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill {k}_3{K}_{\mathrm{tc}}\hfill & \hfill {k}_3{C}_{\mathrm{p}}\varOmega \hfill \\ {}\hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -{k}_3{C}_{\mathrm{p}}\varOmega \hfill & \hfill {k}_3{K}_{\mathrm{tc}}\hfill \end{array}\right]\left\{\begin{array}{c}\hfill { \sin}^2\varphi \left(t,j,z\right)\hfill \\ {}\hfill \frac{1}{2} \sin 2\varphi \left(t,j,z\right)\hfill \\ {}\hfill { \cos}^2\varphi \left(t,j,z\right)\hfill \\ {}\hfill \sin \varphi \left(t,j,z\right)\hfill \\ {}\hfill \cos \varphi \left(t,j,z\right)\hfill \end{array}\right\}\hfill \\ {}\hfill =\varLambda \left\{\begin{array}{c}\hfill { \sin}^2\varphi \left(t,j,z\right)\hfill \\ {}\hfill \frac{1}{2} \sin 2\varphi \left(t,j,z\right)\hfill \\ {}\hfill { \cos}^2\varphi \left(t,j,z\right)\hfill \\ {}\hfill \sin \varphi \left(t,j,z\right)\hfill \\ {}\hfill \cos \varphi \left(t,j,z\right)\hfill \end{array}\right\}\hfill \end{array} $$

  (A.2)
  $$ \left\{\begin{array}{c}\hfill {\mathrm{DC}}_{11}^j\left(z,t\right)\hfill \\ {}\hfill {\mathrm{DC}}_{12}^j\left(z,t\right)\hfill \\ {}\hfill {\mathrm{DC}}_{21}^j\left(z,t\right)\hfill \\ {}\hfill {\mathrm{DC}}_{22}^j\left(z,t\right)\hfill \\ {}\hfill {\mathrm{DC}}_{31}^j\left(z,t\right)\hfill \\ {}\hfill {\mathrm{DC}}_{32}^j\left(z,t\right)\hfill \end{array}\right\}=\left[\begin{array}{ccccc}\hfill -{k}_1{C}_{\mathrm{p}}\hfill & \hfill -{k}_2{C}_{\mathrm{p}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill -{k}_1{C}_{\mathrm{p}}\hfill & \hfill -{k}_2{C}_{\mathrm{p}}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ {}\hfill {k}_2{C}_{\mathrm{p}}\hfill & \hfill -{k}_1{C}_{\mathrm{p}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill {k}_2{C}_{\mathrm{p}}\hfill & \hfill -{k}_1{C}_{\mathrm{p}}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill {k}_3{C}_{\mathrm{p}}\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill {k}_3{C}_{\mathrm{p}}\hfill \end{array}\right]\left\{\begin{array}{c}\hfill { \sin}^2\varphi \left(t,j,z\right)\hfill \\ {}\hfill \frac{1}{2} \sin 2\varphi \left(t,j,z\right)\hfill \\ {}\hfill { \cos}^2\varphi \left(t,j,z\right)\hfill \\ {}\hfill \sin \varphi \left(t,j,z\right)\hfill \\ {}\hfill \cos \varphi \left(t,j,z\right)\hfill \end{array}\right\}=\varGamma \left\{\begin{array}{c}\hfill { \sin}^2\varphi \left(t,j,z\right)\hfill \\ {}\hfill \frac{1}{2} \sin 2\varphi \left(t,j,z\right)\hfill \\ {}\hfill { \cos}^2\varphi \left(t,j,z\right)\hfill \\ {}\hfill \sin \varphi \left(t,j,z\right)\hfill \\ {}\hfill \cos \varphi \left(t,j,z\right)\hfill \end{array}\right\} $$

  (A.3)
  $$ \left\{\begin{array}{c}\hfill {\mathrm{DE}}_1^j\left(z,t\right)\hfill \\ {}\hfill {\mathrm{DE}}_2^j\left(z,t\right)\hfill \\ {}\hfill {\mathrm{DE}}_3^j\left(z,t\right)\hfill \end{array}\right\}=\left[\begin{array}{ccc}\hfill -{K}_{\mathrm{re}}\hfill & \hfill -{K}_{\mathrm{te}}\hfill & \hfill 0\hfill \\ {}\hfill {K}_{\mathrm{te}}\hfill & \hfill -{K}_{\mathrm{re}}\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill 0\hfill & \hfill {K}_{\mathrm{ze}}\hfill \end{array}\right]\left\{\begin{array}{c}\hfill \sin \varphi \left(t,j,z\right)\hfill \\ {}\hfill \cos \varphi \left(t,j,z\right)\hfill \\ {}\hfill 1\hfill \end{array}\right\}=\varTheta \left\{\begin{array}{c}\hfill \sin \varphi \left(t,j,z\right)\hfill \\ {}\hfill \cos \varphi \left(t,j,z\right)\hfill \\ {}\hfill 1\hfill \end{array}\right\} $$

  (A.4)

  Integrating Eq. (A.1) with respect to z, obtained

  $$ \left\{\begin{array}{c}\hfill {F}_x^j(t)\hfill \\ {}\hfill {F}_y^j(t)\hfill \\ {}\hfill {F}_z^j(t)\hfill \end{array}\right\}={\displaystyle {\int}_{z_1}^{z_2}\left\{\begin{array}{c}\hfill \mathrm{d}{F}_x^j\left(z,t\right)\hfill \\ {}\hfill \mathrm{d}{F}_y^j\left(z,t\right)\hfill \\ {}\hfill \mathrm{d}{F}_z^j\left(z,t\right)\hfill \end{array}\right\}}=\left[\begin{array}{cc}\hfill {K}_{11}^j(t)\hfill & \hfill {K}_{12}^j(t)\hfill \\ {}\hfill {K}_{21}^j(t)\hfill & \hfill {K}_{22}^j(t)\hfill \\ {}\hfill {K}_{31}^j(t)\hfill & \hfill {K}_{32}^j(t)\hfill \end{array}\right]\left\{\begin{array}{c}\hfill A(t)\hfill \\ {}\hfill B(t)\hfill \end{array}\right\}+\left[\begin{array}{cc}\hfill {C}_{11}^j(t)\hfill & \hfill {C}_{12}^j(t)\hfill \\ {}\hfill {C}_{21}^j(t)\hfill & \hfill {C}_{22}^j(t)\hfill \\ {}\hfill {C}_{31}^j(t)\hfill & \hfill {C}_{32}^j(t)\hfill \end{array}\right]\left\{\begin{array}{c}\hfill \overset{.}{A}(t)\hfill \\ {}\hfill \overset{.}{B}(t)\hfill \end{array}\right\}+\left\{\begin{array}{c}\hfill {E}_1^j(t)\hfill \\ {}\hfill {E}_2^j(t)\hfill \\ {}\hfill {E}_3^j(t)\hfill \end{array}\right\} $$

  (A.5)

  where

  $$ \left[\begin{array}{cc}\hfill {K}_{11}^j(t)\hfill & \hfill {K}_{12}^j(t)\hfill \\ {}\hfill {K}_{21}^j(t)\hfill & \hfill {K}_{22}^j(t)\hfill \\ {}\hfill {K}_{31}^j(t)\hfill & \hfill {K}_{32}^j(t)\hfill \end{array}\right]=\varLambda \left\{\begin{array}{c}\hfill \mathrm{ss}\left(t,j\right)\hfill \\ {}\hfill \mathrm{sc}\left(t,j\right)\hfill \\ {}\hfill \mathrm{cc}\left(t,j\right)\hfill \\ {}\hfill s\left(t,j\right)\hfill \\ {}\hfill c\left(t,j\right)\hfill \end{array}\right\},\left[\begin{array}{cc}\hfill {C}_{11}^j(t)\hfill & \hfill {C}_{12}^j(t)\hfill \\ {}\hfill {C}_{21}^j(t)\hfill & \hfill {C}_{22}^j(t)\hfill \\ {}\hfill {C}_{31}^j(t)\hfill & \hfill {C}_{32}^j(t)\hfill \end{array}\right]=\varGamma \left\{\begin{array}{c}\hfill \mathrm{ss}\left(t,j\right)\hfill \\ {}\hfill \mathrm{sc}\left(t,j\right)\hfill \\ {}\hfill \mathrm{cc}\left(t,j\right)\hfill \\ {}\hfill s\left(t,j\right)\hfill \\ {}\hfill c\left(t,j\right)\hfill \end{array}\right\},\left\{\begin{array}{c}\hfill {E}_1^j(t)\hfill \\ {}\hfill {E}_2^j(t)\hfill \\ {}\hfill {E}_3^j(t)\hfill \end{array}\right\}=\varTheta \left\{\begin{array}{c}\hfill s\left(t,j\right)\hfill \\ {}\hfill c\left(t,j\right)\hfill \\ {}\hfill \varDelta z\hfill \end{array}\right\} $$

  (A.6)

  here

  $$ \begin{array}{c}\hfill \mathrm{ss}\left(t,j\right)={\displaystyle {\int}_{z_1}^{z_2}{ \sin}^2\varphi \left(t,j,z\right)\mathrm{d}z}, sc\left(t,j\right)={\displaystyle {\int}_{z_1}^{z_2}\frac{1}{2} \sin 2\varphi \left(t,j,z\right)\mathrm{d}z}, cc\left(t,j\right)={\displaystyle {\int}_{z_1}^{z_2}{ \cos}^2\varphi \left(t,j,z\right)\mathrm{d}z},\hfill \\ {}\hfill s\left(t,j\right)={\displaystyle {\int}_{z_1}^{z_2} \sin \varphi \left(t,j,z\right)\mathrm{d}z},c\left(t,j\right)={\displaystyle {\int}_{z_1}^{z_2} \cos \varphi \left(t,j,z\right)\mathrm{d}z}\hfill \end{array} $$

  (A.7)

  Summing the cutting forces over all of the N number of teeth,

  $$ \begin{array}{l}\left\{\begin{array}{c}\hfill {F}_x(t)\hfill \\ {}\hfill {F}_y(t)\hfill \\ {}\hfill {F}_z(t)\hfill \end{array}\right\}={\displaystyle \sum_{j=1}^N\left\{\begin{array}{c}\hfill {F}_x^j(t)\hfill \\ {}\hfill {F}_y^j(t)\hfill \\ {}\hfill {F}_z^j(t)\hfill \end{array}\right\}}={\displaystyle \sum_{j=1}^N\left[\begin{array}{cc}\hfill {K}_{11}^j(t)\hfill & \hfill {K}_{12}^j(t)\hfill \\ {}\hfill {K}_{21}^j(t)\hfill & \hfill {K}_{22}^j(t)\hfill \\ {}\hfill {K}_{31}^j(t)\hfill & \hfill {K}_{32}^j(t)\hfill \end{array}\right]\left\{\begin{array}{c}\hfill A(t)\hfill \\ {}\hfill B(t)\hfill \end{array}\right\}}+{\displaystyle \sum_{j=1}^N\left[\begin{array}{cc}\hfill {C}_{11}^j(t)\hfill & \hfill {C}_{12}^j(t)\hfill \\ {}\hfill {C}_{21}^j(t)\hfill & \hfill {C}_{22}^j(t)\hfill \\ {}\hfill {C}_{31}^j(t)\hfill & \hfill {C}_{32}^j(t)\hfill \end{array}\right]\left\{\begin{array}{c}\hfill \overset{.}{A}(t)\hfill \\ {}\hfill \overset{.}{B}(t)\hfill \end{array}\right\}}+{\displaystyle \sum_{j=1}^N\left\{\begin{array}{c}\hfill {E}_1^j(t)\hfill \\ {}\hfill {E}_2^j(t)\hfill \\ {}\hfill {E}_3^j(t)\hfill \end{array}\right\}}\hfill \\ {}=\left[\begin{array}{cc}\hfill {K}_{11}(t)\hfill & \hfill {K}_{12}(t)\hfill \\ {}\hfill {K}_{21}(t)\hfill & \hfill {K}_{22}(t)\hfill \\ {}\hfill {K}_{31}(t)\hfill & \hfill {K}_{32}(t)\hfill \end{array}\right]\left\{\begin{array}{c}\hfill A(t)\hfill \\ {}\hfill B(t)\hfill \end{array}\right\}+\left[\begin{array}{cc}\hfill {C}_{11}(t)\hfill & \hfill {C}_{12}(t)\hfill \\ {}\hfill {C}_{21}(t)\hfill & \hfill {C}_{22}(t)\hfill \\ {}\hfill {C}_{31}(t)\hfill & \hfill {C}_{32}(t)\hfill \end{array}\right]\left\{\begin{array}{c}\hfill \overset{.}{A}(t)\hfill \\ {}\hfill \overset{.}{B}(t)\hfill \end{array}\right\}+\left\{\begin{array}{c}\hfill {E}_1(t)\hfill \\ {}\hfill {E}_2(t)\hfill \\ {}\hfill {E}_3(t)\hfill \end{array}\right\}\hfill \\ {}=\left[\begin{array}{cc}\hfill {K}_{11}(t)\hfill & \hfill {K}_{12}(t)\hfill \\ {}\hfill {K}_{21}(t)\hfill & \hfill {K}_{22}(t)\hfill \\ {}\hfill {K}_{31}(t)\hfill & \hfill {K}_{32}(t)\hfill \end{array}\right]\left\{\begin{array}{c}\hfill {q}_x(t)-{q}_x\left(t-{\tau}_1\right)+{q}_u(t)-{q}_u\left(t-{\tau}_1\right)+{\tau}_1f\hfill \\ {}\hfill {q}_y(t)-{q}_y\left(t-{\tau}_2\right)+{q}_v(t)-{q}_v\left(t-{\tau}_2\right)\hfill \end{array}\right\}\hfill \\ {}+\left[\begin{array}{cc}\hfill {C}_{11}(t)\hfill & \hfill {C}_{12}(t)\hfill \\ {}\hfill {C}_{21}(t)\hfill & \hfill {C}_{22}(t)\hfill \\ {}\hfill {C}_{31}(t)\hfill & \hfill {C}_{32}(t)\hfill \end{array}\right]\left\{\begin{array}{c}\hfill {\overset{.}{q}}_x(t)-{\overset{.}{q}}_x\left(t-{\tau}_1\right)+{\overset{.}{q}}_u(t)-{\overset{.}{q}}_u\left(t-{\tau}_1\right)\hfill \\ {}\hfill \overset{.}{q_y}(t)-\overset{.}{q_y}\left(t-{\tau}_2\right)+\overset{.}{q_v}(t)-\overset{.}{q_v}\left(t-{\tau}_2\right)\hfill \end{array}\right\}+\left\{\begin{array}{c}\hfill {E}_1(t)\hfill \\ {}\hfill {E}_2(t)\hfill \\ {}\hfill {E}_3(t)\hfill \end{array}\right\}\hfill \\ {}=\left[\begin{array}{cccc}\hfill {K}_{11}(t)\hfill & \hfill {K}_{12}(t)\hfill & \hfill {K}_{11}(t)\hfill & \hfill {K}_{12}(t)\hfill \\ {}\hfill {K}_{21}(t)\hfill & \hfill {K}_{22}(t)\hfill & \hfill {K}_{21}(t)\hfill & \hfill {K}_{22}(t)\hfill \\ {}\hfill {K}_{11}(t)\hfill & \hfill {K}_{12}(t)\hfill & \hfill {K}_{11}(t)\hfill & \hfill {K}_{12}(t)\hfill \\ {}\hfill {K}_{21}(t)\hfill & \hfill {K}_{22}(t)\hfill & \hfill {K}_{21}(t)\hfill & \hfill {K}_{22}(t)\hfill \end{array}\right]\left\{\begin{array}{c}\hfill {q}_x(t)\hfill \\ {}\hfill {q}_y(t)\hfill \\ {}\hfill {q}_u(t)\hfill \\ {}\hfill {q}_v(t)\hfill \end{array}\right\}-\left[\begin{array}{cccc}\hfill {K}_{11}(t)\hfill & \hfill {K}_{12}(t)\hfill & \hfill {K}_{11}(t)\hfill & \hfill {K}_{12}(t)\hfill \\ {}\hfill {K}_{21}(t)\hfill & \hfill {K}_{22}(t)\hfill & \hfill {K}_{21}(t)\hfill & \hfill {K}_{22}(t)\hfill \\ {}\hfill {K}_{11}(t)\hfill & \hfill {K}_{12}(t)\hfill & \hfill {K}_{11}(t)\hfill & \hfill {K}_{12}(t)\hfill \\ {}\hfill {K}_{21}(t)\hfill & \hfill {K}_{22}(t)\hfill & \hfill {K}_{21}(t)\hfill & \hfill {K}_{22}(t)\hfill \end{array}\right]\left\{\begin{array}{c}\hfill {q}_x\left(t-{\tau}_1\right)\hfill \\ {}\hfill {q}_y\left(t-{\tau}_2\right)\hfill \\ {}\hfill {q}_u\left(t-{\tau}_1\right)\hfill \\ {}\hfill {q}_v\left(t-{\tau}_2\right)\hfill \end{array}\right\}+\left\{\begin{array}{c}\hfill {K}_{11}(t)\hfill \\ {}\hfill {K}_{21}(t)\hfill \\ {}\hfill {K}_{11}(t)\hfill \\ {}\hfill {K}_{21}(t)\hfill \end{array}\right\}{\tau}_1f\hfill \\ {}+\left[\begin{array}{cccc}\hfill {C}_{11}(t)\hfill & \hfill {C}_{12}(t)\hfill & \hfill {C}_{11}(t)\hfill & \hfill {C}_{12}(t)\hfill \\ {}\hfill {C}_{21}(t)\hfill & \hfill {C}_{22}(t)\hfill & \hfill {C}_{21}(t)\hfill & \hfill {C}_{22}(t)\hfill \\ {}\hfill {C}_{11}(t)\hfill & \hfill {C}_{12}(t)\hfill & \hfill {C}_{11}(t)\hfill & \hfill {C}_{12}(t)\hfill \\ {}\hfill {C}_{21}(t)\hfill & \hfill {C}_{22}(t)\hfill & \hfill {C}_{21}(t)\hfill & \hfill {C}_{22}(t)\hfill \end{array}\right]\left\{\begin{array}{c}\hfill \overset{.}{q_x}(t)\hfill \\ {}\hfill \overset{.}{q_y}(t)\hfill \\ {}\hfill \overset{.}{q_u}(t)\hfill \\ {}\hfill \overset{.}{q_v}(t)\hfill \end{array}\right\}-\left[\begin{array}{cccc}\hfill {C}_{11}(t)\hfill & \hfill {C}_{12}(t)\hfill & \hfill {C}_{11}(t)\hfill & \hfill {C}_{12}(t)\hfill \\ {}\hfill {C}_{21}(t)\hfill & \hfill {C}_{22}(t)\hfill & \hfill {C}_{21}(t)\hfill & \hfill {C}_{22}(t)\hfill \\ {}\hfill {C}_{11}(t)\hfill & \hfill {C}_{12}(t)\hfill & \hfill {C}_{11}(t)\hfill & \hfill {C}_{12}(t)\hfill \\ {}\hfill {C}_{21}(t)\hfill & \hfill {C}_{22}(t)\hfill & \hfill {C}_{21}(t)\hfill & \hfill {C}_{22}(t)\hfill \end{array}\right]\left\{\begin{array}{c}\hfill \overset{.}{q_x}\left(t-{\tau}_1\right)\hfill \\ {}\hfill \overset{.}{q_y}\left(t-{\tau}_2\right)\hfill \\ {}\hfill \overset{.}{q_u}\left(t-{\tau}_1\right)\hfill \\ {}\hfill \overset{.}{q_v}\left(t-{\tau}_2\right)\hfill \end{array}\right\}+\left\{\begin{array}{c}\hfill {E}_1(t)\hfill \\ {}\hfill {E}_2(t)\hfill \\ {}\hfill {E}_3(t)\hfill \end{array}\right\}\hfill \\ {}=\mathbf{K}(t)\mathbf{q}(t)-\mathbf{K}(t)\mathbf{q}\left(t-{\tau}_{1,2}\right)+\widehat{\mathbf{K}}(t){\tau}_1f+\mathbf{C}(t)\overset{.}{\mathbf{q}}(t)-\mathbf{C}(t)\overset{.}{\mathbf{q}}\left(t-{\tau}_{1,2}\right)+\mathbf{E}(t)\hfill \end{array} $$

  (A.8)

• 2. If 0 < h ≤ h _c, transforming the cutting forces (the first equation in Eq. (12)) from the cutter coordinates to the global coordinates, obtained

  $$ \begin{array}{l}\left\{\begin{array}{c}\hfill \varDelta {F}_x^j\left(z,t\right)\hfill \\ {}\hfill \varDelta {F}_y^j\left(z,t\right)\hfill \\ {}\hfill \varDelta {F}_z^j\left(z,t\right)\hfill \end{array}\right\}=\left[\begin{array}{ccc}\hfill - \sin \varphi \left(t,j,z\right)\hfill & \hfill - \cos \varphi \left(t,j,z\right)\hfill & \hfill 0\hfill \\ {}\hfill - \cos \varphi \left(t,j,z\right)\hfill & \hfill \sin \varphi \left(t,j,z\right)\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right]\left(\left\{\begin{array}{c}\hfill {K}_{\mathrm{rpc}}\hfill \\ {}\hfill {K}_{\mathrm{tpc}}\hfill \\ {}\hfill {K}_{\mathrm{apc}}\hfill \end{array}\right\}{A}_{\mathrm{p}}+\left\{\begin{array}{c}\hfill {K}_{\mathrm{re}}\hfill \\ {}\hfill {K}_{\mathrm{te}}\hfill \\ {}\hfill {K}_{\mathrm{ae}}\hfill \end{array}\right\}\right)\varDelta z\hfill \\ {}=\left(\left\{\begin{array}{c}\hfill {\mathrm{DP}}_1^j\left(z,t\right)\hfill \\ {}\hfill {\mathrm{DP}}_2^j\left(z,t\right)\hfill \\ {}\hfill {\mathrm{DP}}_3^j\left(z,t\right)\hfill \end{array}\right\}{A}_{\mathrm{p}}+\left\{\begin{array}{c}\hfill {\mathrm{DE}}_1^j\left(z,t\right)\hfill \\ {}\hfill {\mathrm{DE}}_2^j\left(z,t\right)\hfill \\ {}\hfill {\mathrm{DE}}_3^j\left(z,t\right)\hfill \end{array}\right\}\right)\varDelta z\hfill \end{array} $$

  (A.9)

  where

  $$ \left\{\begin{array}{c}\hfill {\mathrm{DP}}_1^j\left(z,t\right)\hfill \\ {}\hfill {\mathrm{DP}}_2^j\left(z,t\right)\hfill \\ {}\hfill {\mathrm{DP}}_3^j\left(z,t\right)\hfill \end{array}\right\}=\left[\begin{array}{ccc}\hfill -{K}_{\mathrm{rpc}}\hfill & \hfill -{K}_{\mathrm{tpc}}\hfill & \hfill 0\hfill \\ {}\hfill {K}_{\mathrm{tpc}}\hfill & \hfill -{K}_{\mathrm{rpc}}\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill 0\hfill & \hfill {K}_{\mathrm{apc}}\hfill \end{array}\right]\left\{\begin{array}{c}\hfill \sin \varphi \left(t,j,z\right)\hfill \\ {}\hfill \cos \varphi \left(t,j,z\right)\hfill \\ {}\hfill 1\hfill \end{array}\right\} $$

  (A.10)

  Integrating Eq. (A.8) with respect to z, obtained

  $$ \left\{\begin{array}{c}\hfill {F}_x^j(t)\hfill \\ {}\hfill {F}_y^j(t)\hfill \\ {}\hfill {F}_z^j(t)\hfill \end{array}\right\}={\displaystyle {\int}_{z_1}^{z_2}\left\{\begin{array}{c}\hfill \mathrm{d}{F}_x^j\left(z,t\right)\hfill \\ {}\hfill \mathrm{d}{F}_y^j\left(z,t\right)\hfill \\ {}\hfill \mathrm{d}{F}_z^j\left(z,t\right)\hfill \end{array}\right\}}=\left\{\begin{array}{c}\hfill {P}_1^j(t)\hfill \\ {}\hfill {P}_2^j(t)\hfill \\ {}\hfill {P}_3^j(t)\hfill \end{array}\right\}{A}_{\mathrm{p}}+\left\{\begin{array}{c}\hfill {E}_1^j(t)\hfill \\ {}\hfill {E}_2^j(t)\hfill \\ {}\hfill {E}_3^j(t)\hfill \end{array}\right\} $$

  (A.11)

  where

  $$ \left\{\begin{array}{c}\hfill {P}_1^j(t)\hfill \\ {}\hfill {P}_2^j(t)\hfill \\ {}\hfill {P}_3^j(t)\hfill \end{array}\right\}=\varUpsilon \left\{\begin{array}{c}\hfill s\left(t,j\right)\hfill \\ {}\hfill c\left(t,j\right)\hfill \\ {}\hfill \varDelta z\hfill \end{array}\right\} $$

  (A.12)

  Summing the cutting forces over all of the N number of teeth,

  $$ \begin{array}{l}\left\{\begin{array}{c}\hfill {F}_x(t)\hfill \\ {}\hfill {F}_y(t)\hfill \\ {}\hfill {F}_z(t)\hfill \end{array}\right\}={\displaystyle \sum_{j=1}^N\left\{\begin{array}{c}\hfill {F}_x^j(t)\hfill \\ {}\hfill {F}_y^j(t)\hfill \\ {}\hfill {F}_z^j(t)\hfill \end{array}\right\}}={\displaystyle \sum_{j=1}^N\left\{\begin{array}{c}\hfill {P}_1^j(t)\hfill \\ {}\hfill {P}_2^j(t)\hfill \\ {}\hfill {P}_3^j(t)\hfill \end{array}\right\}{A}_{\mathrm{p}}}+{\displaystyle \sum_{j=1}^N\left\{\begin{array}{c}\hfill {E}_1^j(t)\hfill \\ {}\hfill {E}_2^j(t)\hfill \\ {}\hfill {E}_3^j(t)\hfill \end{array}\right\}}\hfill \\ {}=\left\{\begin{array}{c}\hfill {P}_1(t)\hfill \\ {}\hfill {P}_2(t)\hfill \\ {}\hfill {P}_3(t)\hfill \end{array}\right\}{A}_{\mathrm{p}}+\left\{\begin{array}{c}\hfill {E}_1(t)\hfill \\ {}\hfill {E}_2(t)\hfill \\ {}\hfill {E}_3(t)\hfill \end{array}\right\}=\mathbf{P}(t){A}_{\mathrm{p}}+\mathbf{E}(t)\hfill \end{array} $$

  (A.13)