The chatter model presented in this paper is based on that of Budak and Altintas [5].
While the complete derivation of the model is not shown here, the equations required for calculating stability limit are presented, with the modifications necessary to model rougher endmills.
KINEMATICS AND DYNAMICS OF ROUGHING END MILLS 133
The chatter model for regular cutters is first considered. The structural dynamics model and the block diagram describing dynamic milling are shown in the Laplace domain in Figure 3. The cutting forces excite the structure in the X (feed) and Y (normal) directions, causing a dynamically changing chip thickness. The dynamic chip thickness, {~(s),~y(s)}, is the difference between the cutter dynamic displacement, {x(s),y(s)}, and the displacement during the previous tooth pass, {x0(s),y0(s)}, at time (t-r). For regular, uniform pitch cutters, this time delay is equal to the tooth period; r = T. It is assumed that self-excited vibrations will develop at a specified chatter frequency, me. Depending on the frequency of chatter, and on the time delay, r , a phase shift between vibrations of successive tooth passes can cause the system to become stable, or unstable. This mechanism, known as the regeneration effect, is the primary cause of chatter vibrations.
@
{
d¥(s )} Regenerative Dynamic 11.y(s) Chip Load
Dynamic Displacement of previous flute
Regeneration Delay
eã"
Milllng Force and Directional Coefficients
.!_aK fa.(*) a.,.(t)]
2 '[a•(*) a,,(*)
Dynamic Displacement
{x(s)}
y(s)
{F.(s)} Dynamic
F,(s) Cutting Force
Transfer Function [
G.(s) G.,.(s)l G.(s) G,,(s)
Figure 3. Milling dynamics model and block diagram representing the regeneration effect causing chatter.
For the chatter model, a zero helix cutter is assumed, and a simplified force model is used, where edge forces are ignored and only the dynamically changing component of the chip load is considered.
From the model of Budak and Altintas [5], the characteristic equation in the frequency domain describing the system in Figure 3 is:
Or in simplified matrix form,
1 .
F =-abK1A(t/J)G(l{J)JF 2
(6)
(7)
134 M. L. CAMPOMANES
where a is the axial depth of cut, bis the regeneration coefficient. For a regular edge cutter, b = 1-e-i(J)cr. K1 is the tangential cutting pressure coefficient, G(iwc) is the structure's transfer functions at the chatter frequency, we, and A(</>) is the dynamic milling force coefficients matrix, defined as:
N
axx(¢) = L - g(¢1 >[sin(2¢1) + K,(1-cos(2¢1 ))]
j N
axy(¢) = L -g(¢1)[1 + cos(2¢1 )+ K, sin(2¢1)]
j N
ayx(¢) = LK(¢jl- cos(2¢1)-K, sin(2¢1 )]
j N
ayy(¢) = LK(¢J >[sin(2¢J) + K,(1 +cos(2¢J n]
j
(8)
where K, is the ratio of radial to tangential cutting pressure coefficients, and g(</>i) is a step function, which considers when the tooth j is immersed in the workpiece; i.e.:
(9)
</>,, and <f>.x are the start and exit immersion angles of the cutter in and out of the workpiece. The matrix A(</>) varies as the cutter rotates, but for the purpose of determining stability, its average value over one complete revolution of the cutter may be used, which can be easily evaluated numerically.
For a regular edge, uniform pitch cutter, the regeneration delay, r, is a constant and is equal to the tooth period, T. For a rougher endmill, however, the time between tooth contact at any defined location on the cutter varies along the cutting edge based on the edge geometry, the angular position of the cutter, and on the feed rate, as shown in Figure I. Again, the cutter is discretized into numerous axial elements. At any angular position of the cutter, </>,for each flute and axial element, r11(</>) is calculated numerically as follows. The number of preceding flutes, k , which passed since the last contact with the surface currently being cut is identified by finding k with the largest value of the expression
[&J-k.I -ks, sin¢1] ~max k = 1,2, ... , N (IO)
The & term represents the sinusoidal edge serration for tooth j - k at axial element I , from Eq. (3). For flute j, the time delay between cuts at element I, r 11 (</>) , is then evaluated as:
KINEMATICS AND DYNAMICS OF ROUGHING ENDMILLS 135
k I/Im
r:1,(t/J)=I-, m=modU+N-i,N)
i=J n
(11)
The regeneration coefficient, bJ1(¢,i(J)c) is now defined as:
(12)
For the purpose of determining stability, the average of the coefficient over the length of the cutter is used; i.e. an average of every axial element, I . The coefficients are then summed for each flute, giving the effective regeneration coefficient, beff ( ¢, i (J)c) .
For roughing endmills, beff(t/J) and A(t/J) still vary with the angular position of the cutter. Again, for the purpose of stability calculations, the average of the combined coefficients over one revolution of the cutter is sufficient, and can easily be evaluated numerically.
Defining matrix C(i(J)J as:
(13)
The simplified dynamic milling equation for roughing endmills is:
(14)
which has a non-trivial solution when its determinant is zero:
(15)
The eigenvalue of the characteristic equation, defined as A= _J_aK1 , can be evaluated, 2
and the axial depth of cut at the border of stability is solved for as:
a = - -2A K,
(16)
136 M. L. CAMPOMANES
This gives a complex value for axial depth of cut limit at a given chatter frequency, Ole.
The frequency at which chatter will occur is the frequency at which the system is most unstable. Furthermore, the axial depth of cut limit must be a real value.
The frequency spectrum is discretized for a range of possible chatter frequencies, and the value a is calculated at each interval. The smallest real value of a is the axial depth of cut limit, a1;m , and the corresponding frequency, Ole, is the frequency at which chatter will develop.
This is repeated for a range of spindle speeds to obtain the chatter stability lobe diagram.
200
0.002 0.004 0.006 0.008 0.01 0.012 0.002 0.004 0.006 0.008 0,01 0.012
Time[s] Time[s]
(A) (B)
Figure 4. X and Y Cutting force for roughing cut with 6 flute regular and 6 pitch rougher endmill, r=l9.05rnrn, rF5000 rpm; A) a=20mm, half immersion, st=O.lmm/tooth, rF5000 rpm, B) a=IOOmm, r=l9.05rnrn, b=l.Omm, st=0.05mm/tooth