A novel high-order discretization method for the milling stability prediction considering the differential of directional cutting coefficient and vibration velocities

Abstract

A novel high-order discretization method for the prediction of milling stability is proposed in this paper to increase the accuracy and efficiency considering the differential of directional cutting coefficient and vibration velocities. Same to the existing full-discretization method (FDM) and semi-discretization method (SDM), the milling system is expressed as a linear time-periodic equation and the time period is discretized into discrete time intervals to approximate the solution. In this algorithm, the cutting force coefficient matrix of the whole integrand is reconstructed to lay the foundation for the fast and accurate approximation. Then, the monodromy matrix (or the Floquet matrix) is calculated by the method used in the temporal finite element analysis (TFEA) instead of the multiple recursive algorithms which can improve the computational time. Finally, the computational efficiency is defined in a new way which gives quantitative comparisons for different discretization methods. It is shown that the proposed method can reduce the computational cost by 91–95% when the same errors are required.

Appendix

1.1 MATLAB code for the single-DOF millling model

close all
clear all
clc
tic
% the parameters
N = 2;	% number of teeth
Kt = 6e8;	% tangential cutting force coefficient
Kn = 2e8;	% normal cutting force coefficient
w0 = 922*2*pi;	% angular natural frequency (rad/s)
zeta = 0.011;	% relative damping
m_t = 0.03993;	% modal mass (kg)
aD = 1.0;	% radial immerison
mot = -1;	% 1: up-milling; -1: down-milling
if mot =  = 1
fist = 0;	% start angle for up-milling
fiex = acos(1—2*aD);	% exit angle for up-milling
elseif mot =  = -1
fist = acos(2*aD—1);	% start angle for down-milling
fiex = pi;	% exit angle for down-milling
end
stx = 400;	% number of steps for spindle speed
sty = 200;	% number of steps for depth of cut
w_st = 0.1e-3;	% starting depth of cut (m)
w_fi = 10e-3;	% final depth of cut (m)
o_st = 5e3;	% starting spindle speed (rpm)
o_fi = 25e3;	% final spindle speed (rpm)
m = 40;	% number of discretization interval
d = ones(m + m,1);
D = diag(d,-2);	% matrix D
A = zeros(2, 2);	% matrix A
A(1, 2) = 1;
A(2, 1) = -w0^2;
A(2, 2) = -2*zeta*w0;
I = [1 0;0 1];
A0 = 2*A;	% matrix 2*A
A2 = A;
MM = pi-acos(2*aD—1);	% angular immerison
% construction of monodromy matrix
for x = 1:stx + 1
I1 = eye(m + m + 2);
C1 = zeros(m + m + 2);
D1 = zeros(m + m + 2);
E1 = zeros(m + m + 2);
o = o_st + (x-1)*(o_fi—o_st)/stx;
tau1 = 60/o/N;
tau = 60/o/N*(MM/pi);
tau2 = tau1-tau;
dt = tau/m;	%time interval
dt1 = 2*tau/m;
for k = 1: m + 1
dtr = 2*MM/N/m;
h(k) = 0;
h1(k) = 0;
for j = 1:N
fi = 0;
fi = k*dtr + (j-1)*2*pi/N + fist;
if (fi >  = fist)*(fi <  = fiex)
g = 1;	% tooth is in the cut
else
g = 0;	% tooth is out of cut
end
fi1(j,k) = fi*g;
h(k) = h(k) + g*(Kt*cos(fi) + Kn*sin(fi))*sin(fi);	%cutting coefficients h(t)
h1(k) = h1(k) + g*((Kt*cos(fi)^2-Kt*sin(fi)^2) + 2*Kn*sin(fi)*cos(fi))*2*pi*o/60;	%diferentials of cutting coefficients
end
end
Fi0 = expm(A2*dt1);
Fi01 = expm(A2*dt1);
Fi011 = expm(A2*tau2);
Fi0111 = expm(A2*dt);
invA0 = inv(A0);
invA2 = inv(A2);
Fi1 = invA2 * (Fi01—I);
Fi2 = invA2 * (Fi01*dt1—Fi1);
Fi3 = invA2 * (Fi01*dt1*dt1—2*Fi2);
Fi4 = invA2 * (Fi01*dt1*dt1*dt1—3*Fi3);
Fi5 = invA2 * (Fi01*dt1*dt1*dt1*dt1—4*Fi4);
Fi6 = invA2 * (Fi01*dt1*dt1*dt1*dt1*dt1—5*Fi5);
Fi10 = invA2 * (Fi0111—I);
Fi20 = invA2 * (Fi0111*dt—Fi10);
Fi30 = invA2 * (Fi0111*dt*dt—2*Fi20);
Fi40 = invA2 * (Fi0111*dt*dt*dt—3*Fi30);
for y = 1:sty + 1
w = w_st + (y-1)*(w_fi—w_st)/sty;
Fi = eye(m + m + 2,m + m + 2);	%Floquet matrix
B0K = [0 0;w*h(1)/m_t 0];
B1K = [0 0;w*h1(1)/m_t 0];
B2K = [0 0;0 w*h(1)/m_t];
C0K = [0 0;w*h(2)/m_t 0];
C1K = [0 0;w*h1(2)/m_t 0];
C2K = [0 0;0 w*h(2)/m_t];
BB = (dt*(Fi20/dt—2*Fi30/dt/dt + Fi40/dt/dt/dt))*C2K;
AA1 = (Fi10—3*Fi30/dt/dt + 2*Fi40/dt/dt/dt)*C0K;
AA2 = (dt*(Fi20/dt—2*Fi30/dt/dt + Fi40/dt/dt/dt))*C1K;
AA = AA1—AA2;
FFK = AA—BB;
DD = (dt*(—Fi30/dt/dt + Fi40/dt/dt/dt))*B2K;
CC1 = (3*Fi30/dt/dt—2*Fi40/dt/dt/dt)*B0K;
CC2 = (dt*(—Fi30/dt/dt + Fi40/dt/dt/dt))*B1K;
CC = CC1—CC2;
FFK1 = CC—DD;
for i = 2:m
B0K = [0 0;w*h(i-1)/m_t 0];
B1K = [0 0;w*h1(i-1)/m_t 0];
B2K = [0 0;0 w*h(i-1)/m_t];
C0K = [0 0;w*h(i)/m_t 0];
C1K = [0 0;w*h1(i)/m_t 0];
C2K = [0 0;0 w*h(i)/m_t];
D0K = [0 0;w*h(i + 1)/m_t 0];
D1K = [0 0;w*h1(i + 1)/m_t 0];
D2K = [0 0;0 w*h(i + 1)/m_t];
%hermite coefficients
ZZ1 = ((3*Fi6/dt-18*Fi5 + 39*dt*Fi4-36*dt*dt*Fi3 + 12*dt*dt*dt*Fi2) +
(Fi5-6*dt*Fi4 + 13*dt*dt*Fi3-12*dt*dt*dt*Fi2 + 4*dt*dt*dt*dt*Fi1))/4/dt/dt/dt/dt;
ZZ2 = (Fi6-6*dt*Fi5 + 13*dt*dt*Fi4-12*dt*dt*dt*Fi3 + 4*dt*dt*dt*dt*Fi2)/4/dt/dt/dt/dt;
ZZ3 = (Fi5-4*dt*Fi4 + 4*dt*dt*Fi3)/dt/dt/dt/dt;
ZZ4 = ((Fi6-4*dt*Fi5 + 4*dt*dt*Fi4)/dt/dt/dt/dt-(Fi5-4*dt*Fi4 + 4*dt*dt*Fi3)/dt/dt/dt);
ZZ5 = (7*(Fi5-2*dt*Fi4 + dt*dt*Fi3)/4/dt/dt/dt/dt-3*(Fi6-2*dt*Fi5 + dt*dt*Fi4)/4/dt/dt/dt/dt/dt);
ZZ6 = ((Fi6-2*dt*Fi5 + dt*dt*Fi4)/4/dt/dt/dt/dt-2*(Fi5-2*dt*Fi4 + dt*dt*Fi3)/4/dt/dt/dt);
%coefficient terms
YY2 = ZZ1*D0K-ZZ2*D1K;
YY21 = ZZ2*D2K;
XX2 = YY2-YY21;
YY3 = ZZ3*C0K-ZZ4*C1K;
YY31 = ZZ4*C2K;
XX3 = YY3-YY31;
YY4 = ZZ5*B0K-ZZ6*B1K;
YY41 = ZZ6*B2K;
XX4 = YY4-YY41;
E1(1:2,m + m + 1:m + m + 2) = Fi011;
C1(3:4,1:2) = Fi0111;
C1(2*i + 1:2*i + 2,2*i-3:2*i-2) = Fi0;
D1(3:4,1:2) = -FFK1;
D1(3:4,3:4) = -FFK;
D1(2*i + 1:2*i + 2,2*i-3:2*i-2) = -XX4;
D1(2*i + 1:2*i + 2,2*i-1:2*i) = -XX3;
D1(2*i + 1:2*i + 2,2*i + 1:2*i + 2) = -XX2;
end
Fi = inv(I1-C1-D1)*(-D1 + E1);	% transition matrix
ss(x,y) = o;
dc(x,y) = w;
ei(x,y) = max(abs(eig(Fi)));	% matrix of eigenvalues
end
stx + 1-x
end
toc
figure;
contour(ss,dc,ei, [1],'b'),xlabel('rpm'),ylabel('w')