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Corrigendum to: “An efficient linear approximation of acceleration method for milling stability prediction” [Int. J. Mach. Tools Manuf. 74 (2013) 56–64]
International Journal of Machine Tools & Manufacture 92 (2015) 97–99
Contents lists available at ScienceDirect
International Journal of Machine Tools & Manufacture journal homepage: www.elsevier.com/locate/ijmactool
Corrigendum
Corrigendum to: “An efficient linear approximation of acceleration method for milling stability prediction” [Int. J. Mach. Tools Manuf. 74 (2013) 56–64] Tao Huang, Xiaoming Zhang n, Xiaojian Zhang, Han Ding School of Mechanical Science and Engineering, State Key Laboratory of Digital Manufacturing Equipment and Technology, Huazhong University of Science and Technology, Wuhan 430074, China
The authors regret there are some typographical errors in Appendix: A. Matlab code for the second form in the paper cited above, where the brackets (mm,:) for matrices Tpn and Tpn1 are lost at some lines in the original version. This can lead to the Matlab codes unsuccessfully executed, but does not affect in any way the results and conclusions of the paper. The corrected codes are shown below. The modified lines are highlighted with red. The authors would like to apologise for any inconvenience caused. clc;clear tic; N ¼2; Kt ¼ 6e8; Kn ¼ 2e8; w0 ¼922*2*pi; zeta ¼0.011; wd ¼w0*sqrt(1-zeta\widehat2); m_t ¼0.03993; c ¼2*zeta*w0*m_t; k ¼w0\widehat2*m_t; aD ¼0.05; E ¼10; up_or_down¼-1; if up_or_down¼ ¼1 fist¼0; fiex ¼acos(1-2*aD); elseif up_or_down¼ ¼-1 fist¼acos(2*aD-1); fiex ¼pi; end % specific cutting force coefficient based on Equation (23) for ii ¼1:Eþ 1 dtr ¼(fiex-fist)/E; h(ii) ¼ 0; for jj¼ 1:N fi ¼ fistþ(ii-1)*dtr þ(jj-1)*2*pi/N; if (fi 4 ¼fist)*(fi o ¼fiex) g¼ 1;
n
DOI of original article: http://dx.doi.org/10.1016/j.ijmachtools.2013.07.006 Corresponding author. Tel.: þ 86 2787559842; fax: þ 86 2787559416. E-mail address: [email protected] (X. Zhang).
http://dx.doi.org/10.1016/j.ijmachtools.2015.02.006 0890-6955/
% % % % %
number of teeth tangential cutting force coefficient normal cutting force coefficient angular natural frequency (rad/s) relative damping
% % % % % % % % % % % %
mass (kg) damping stiffness radial depth of cut number of discretization interval 1: up-milling, 1: down-milling up-milling start angle exit angle down-milling start angle exit angle
% the angle interval % loop for tooth jj
% tooth is in the cut
98
T. Huang et al. / International Journal of Machine Tools & Manufacture 92 (2015) 97–99
else g¼0; end h(ii) ¼h(ii) þ g*(Kt*cos(fi) þKn*sin(fi))*sin(fi); end end kw ¼zeros(Eþ3,E þ3); for mm ¼ 1:E kw(mm,mm) ¼ 1/2*(h(mm) þh(mm þ1)); end A ¼ [1,1;0,1]; B ¼[1/3,1/6;1/2,1/2]; I ¼eye(2,2); Z ¼zeros(2*Eþ2,E þ3); Z(1,1) ¼ 1; Z(2,2) ¼ 1; Tpn1 ¼zeros(Eþ3,E þ3); Tpn ¼ zeros(E þ3,E þ3); for mm ¼ 1:E Tpn(mm,mm þ2) ¼1; Tpn1(mm,mm þ 3) ¼1; end % calculation for Z 2i in Eq. (30) for mm ¼ 2:Eþ1 TT0¼zeros(2,1); Z(2*mm-1:2*mm,1:2) ¼ A\widehat(mm-1); for nn ¼3:mm þ1 TT ¼A\widehat(mm-nn þ1)*B; Z(2*mm-1:2*mm,nn) ¼ TT(:,1) þTT0; TT0 ¼TT(:,2); end Z(2*mm-1:2*mm,mm þ2) ¼TT0; end Fm ¼zeros(E þ3,E þ3); Fc ¼zeros(Eþ3,E þ3); Fk ¼zeros(Eþ3,E þ3); for mm ¼ 1:E Fm(mm,:) ¼ [0,0]*Z(2*mm-1:2*mm,:) þ1/2*Tpn(mm,:) þ 1/2*Tpn1(mm,:); Fc(mm,:) ¼[0,1]*Z(2*mm-1:2*mm,:) þ1/3*Tpn(mm,:) þ1/6*Tpn1(mm,:); Fk(mm,:) ¼[1,1/2]*Z(2*mm-1:2*mm,:) þ1/8*Tpn(mm,:) þ1/24*Tpn1(mm,:); end S_st ¼5e3; S_ed ¼25e3; DOC_st ¼0e-3; DOC_ed ¼ 10e-3; stx ¼ 400; sty ¼200; % start of computation for SP ¼1:stx þ1 S¼S_st þ(SP-1)*(S_ed-S_st)/stx; T ¼60/S/N; tf ¼T-T*(fiex-fist)*N/2/pi; tj¼ (T-tf)/E; Lmck ¼ m_t/tj*Fm þc*Fc þ k*tj*Fk; % calculation of free vibration from Eq. (32) lamda1 ¼ -zeta*w0þ wd*i; lamda2 ¼ -zeta*w0-wd*i; expla1 ¼exp(lamda1*tf); expla2 ¼exp(lamda2*tf); ladt1 ¼lamda1*tj; ladt2 ¼lamda2*tj; Lf ¼zeros(Eþ 3,Eþ 3); Rf ¼zeros(Eþ3,E þ3); Lf(Eþ 1:Eþ3,1:3) ¼eye(3);
% tooth is out of cut % Eq. (2)
% Equation (23) % matrix A2 , Eq. (25) % matrix B2
n % Eq. (27), location of item q€ i Δt 2 n % Eq. (27), location of item q€ i þ 1 Δt 2
% Z(2*mm-1:2*mm,1:2) ¼Z 2m
% Eq. (29) %Fmði; :Þ ¼ Fi2M %Fcði; :Þ ¼ Fi2C %Fkði; :Þ ¼ Fi2K % % % % % %
starting spindle speed (rpm) final spindle speed (rpm) starting depth of cut (m) final depth of cut (m) steps of spindle speed steps of depth of cut
% % % % % %
loop for spindle speeds spindle speed period duration time of free vibration time step part of Eq. (34)
% the eigenvalue of free vibration
% Lf ðE þ1 : E þ 3; :Þ ¼ F n21 , Eq. (32) % Rf ðE þ 1 : E þ 3; :Þ ¼ F n2ðE þ 1Þ % calculation of F n21
T. Huang et al. / International Journal of Machine Tools & Manufacture 92 (2015) 97–99
Fla¼ [expla1,expla2; ladt1*expla1,ladt2*expla2; ladt1\widehat2*expla1,ladt2\widehat2*expla2]; Fc1¼ 1/(ladt2-ladt1)*[ladt2,-1;-ladt1,1]; FF¼ Fla*Fc1; Rf(E þ1:Eþ 3,:)¼ FF*Z(2*Eþ 1:2*E þ2,:); for DOC ¼1:sty þ1 b ¼DOC_st þ(DOC-1)*(DOC_ed-DOC_st)/sty; Lb ¼tj*b*kw*Fk; L1 ¼Lmckþ LbþLf; R1 ¼Lb þRf; Q1 ¼inv(L1)*R1; ei(SP,DOC)¼ max(abs(eig(Q1))); ss(SP,DOC) ¼S; dc(SP,DOC) ¼b; end end toc; figure contour(ss,dc,ei,[1,1],‘k’)
% Eq. (32)
% Eq. (32) % % % % % %
calculation of F n2ðE þ 1Þ loop for depth of cuts depth of cut part of Eq. (34) construction of L1 ¼ P2 , Eq. (34) construction of R1 ¼ Q 2 , Eq. (34)
% % % %
transition matrix P2 1 Q 2 matrix of eigenvalues matrix of spindle speeds matrix of depth of cuts
99
Contents lists available at ScienceDirect
International Journal of Machine Tools & Manufacture journal homepage: www.elsevier.com/locate/ijmactool
Corrigendum
Corrigendum to: “An efficient linear approximation of acceleration method for milling stability prediction” [Int. J. Mach. Tools Manuf. 74 (2013) 56–64] Tao Huang, Xiaoming Zhang n, Xiaojian Zhang, Han Ding School of Mechanical Science and Engineering, State Key Laboratory of Digital Manufacturing Equipment and Technology, Huazhong University of Science and Technology, Wuhan 430074, China
The authors regret there are some typographical errors in Appendix: A. Matlab code for the second form in the paper cited above, where the brackets (mm,:) for matrices Tpn and Tpn1 are lost at some lines in the original version. This can lead to the Matlab codes unsuccessfully executed, but does not affect in any way the results and conclusions of the paper. The corrected codes are shown below. The modified lines are highlighted with red. The authors would like to apologise for any inconvenience caused. clc;clear tic; N ¼2; Kt ¼ 6e8; Kn ¼ 2e8; w0 ¼922*2*pi; zeta ¼0.011; wd ¼w0*sqrt(1-zeta\widehat2); m_t ¼0.03993; c ¼2*zeta*w0*m_t; k ¼w0\widehat2*m_t; aD ¼0.05; E ¼10; up_or_down¼-1; if up_or_down¼ ¼1 fist¼0; fiex ¼acos(1-2*aD); elseif up_or_down¼ ¼-1 fist¼acos(2*aD-1); fiex ¼pi; end % specific cutting force coefficient based on Equation (23) for ii ¼1:Eþ 1 dtr ¼(fiex-fist)/E; h(ii) ¼ 0; for jj¼ 1:N fi ¼ fistþ(ii-1)*dtr þ(jj-1)*2*pi/N; if (fi 4 ¼fist)*(fi o ¼fiex) g¼ 1;
n
DOI of original article: http://dx.doi.org/10.1016/j.ijmachtools.2013.07.006 Corresponding author. Tel.: þ 86 2787559842; fax: þ 86 2787559416. E-mail address: [email protected] (X. Zhang).
http://dx.doi.org/10.1016/j.ijmachtools.2015.02.006 0890-6955/
% % % % %
number of teeth tangential cutting force coefficient normal cutting force coefficient angular natural frequency (rad/s) relative damping
% % % % % % % % % % % %
mass (kg) damping stiffness radial depth of cut number of discretization interval 1: up-milling, 1: down-milling up-milling start angle exit angle down-milling start angle exit angle
% the angle interval % loop for tooth jj
% tooth is in the cut
98
T. Huang et al. / International Journal of Machine Tools & Manufacture 92 (2015) 97–99
else g¼0; end h(ii) ¼h(ii) þ g*(Kt*cos(fi) þKn*sin(fi))*sin(fi); end end kw ¼zeros(Eþ3,E þ3); for mm ¼ 1:E kw(mm,mm) ¼ 1/2*(h(mm) þh(mm þ1)); end A ¼ [1,1;0,1]; B ¼[1/3,1/6;1/2,1/2]; I ¼eye(2,2); Z ¼zeros(2*Eþ2,E þ3); Z(1,1) ¼ 1; Z(2,2) ¼ 1; Tpn1 ¼zeros(Eþ3,E þ3); Tpn ¼ zeros(E þ3,E þ3); for mm ¼ 1:E Tpn(mm,mm þ2) ¼1; Tpn1(mm,mm þ 3) ¼1; end % calculation for Z 2i in Eq. (30) for mm ¼ 2:Eþ1 TT0¼zeros(2,1); Z(2*mm-1:2*mm,1:2) ¼ A\widehat(mm-1); for nn ¼3:mm þ1 TT ¼A\widehat(mm-nn þ1)*B; Z(2*mm-1:2*mm,nn) ¼ TT(:,1) þTT0; TT0 ¼TT(:,2); end Z(2*mm-1:2*mm,mm þ2) ¼TT0; end Fm ¼zeros(E þ3,E þ3); Fc ¼zeros(Eþ3,E þ3); Fk ¼zeros(Eþ3,E þ3); for mm ¼ 1:E Fm(mm,:) ¼ [0,0]*Z(2*mm-1:2*mm,:) þ1/2*Tpn(mm,:) þ 1/2*Tpn1(mm,:); Fc(mm,:) ¼[0,1]*Z(2*mm-1:2*mm,:) þ1/3*Tpn(mm,:) þ1/6*Tpn1(mm,:); Fk(mm,:) ¼[1,1/2]*Z(2*mm-1:2*mm,:) þ1/8*Tpn(mm,:) þ1/24*Tpn1(mm,:); end S_st ¼5e3; S_ed ¼25e3; DOC_st ¼0e-3; DOC_ed ¼ 10e-3; stx ¼ 400; sty ¼200; % start of computation for SP ¼1:stx þ1 S¼S_st þ(SP-1)*(S_ed-S_st)/stx; T ¼60/S/N; tf ¼T-T*(fiex-fist)*N/2/pi; tj¼ (T-tf)/E; Lmck ¼ m_t/tj*Fm þc*Fc þ k*tj*Fk; % calculation of free vibration from Eq. (32) lamda1 ¼ -zeta*w0þ wd*i; lamda2 ¼ -zeta*w0-wd*i; expla1 ¼exp(lamda1*tf); expla2 ¼exp(lamda2*tf); ladt1 ¼lamda1*tj; ladt2 ¼lamda2*tj; Lf ¼zeros(Eþ 3,Eþ 3); Rf ¼zeros(Eþ3,E þ3); Lf(Eþ 1:Eþ3,1:3) ¼eye(3);
% tooth is out of cut % Eq. (2)
% Equation (23) % matrix A2 , Eq. (25) % matrix B2
n % Eq. (27), location of item q€ i Δt 2 n % Eq. (27), location of item q€ i þ 1 Δt 2
% Z(2*mm-1:2*mm,1:2) ¼Z 2m
% Eq. (29) %Fmði; :Þ ¼ Fi2M %Fcði; :Þ ¼ Fi2C %Fkði; :Þ ¼ Fi2K % % % % % %
starting spindle speed (rpm) final spindle speed (rpm) starting depth of cut (m) final depth of cut (m) steps of spindle speed steps of depth of cut
% % % % % %
loop for spindle speeds spindle speed period duration time of free vibration time step part of Eq. (34)
% the eigenvalue of free vibration
% Lf ðE þ1 : E þ 3; :Þ ¼ F n21 , Eq. (32) % Rf ðE þ 1 : E þ 3; :Þ ¼ F n2ðE þ 1Þ % calculation of F n21
T. Huang et al. / International Journal of Machine Tools & Manufacture 92 (2015) 97–99
Fla¼ [expla1,expla2; ladt1*expla1,ladt2*expla2; ladt1\widehat2*expla1,ladt2\widehat2*expla2]; Fc1¼ 1/(ladt2-ladt1)*[ladt2,-1;-ladt1,1]; FF¼ Fla*Fc1; Rf(E þ1:Eþ 3,:)¼ FF*Z(2*Eþ 1:2*E þ2,:); for DOC ¼1:sty þ1 b ¼DOC_st þ(DOC-1)*(DOC_ed-DOC_st)/sty; Lb ¼tj*b*kw*Fk; L1 ¼Lmckþ LbþLf; R1 ¼Lb þRf; Q1 ¼inv(L1)*R1; ei(SP,DOC)¼ max(abs(eig(Q1))); ss(SP,DOC) ¼S; dc(SP,DOC) ¼b; end end toc; figure contour(ss,dc,ei,[1,1],‘k’)
% Eq. (32)
% Eq. (32) % % % % % %
calculation of F n2ðE þ 1Þ loop for depth of cuts depth of cut part of Eq. (34) construction of L1 ¼ P2 , Eq. (34) construction of R1 ¼ Q 2 , Eq. (34)
% % % %
transition matrix P2 1 Q 2 matrix of eigenvalues matrix of spindle speeds matrix of depth of cuts
99