2.1. Chip Geometry and Cutting Forces
Figure 2 shows the dynamic cutting model for ball end milling of a convex surface. A similar model is used for a concave surface. In this figure R is the radius of curvature of the workpiece, rb is the ball radius, h is the nominal depth of cut,f is the feed rate and VF is the angle of rotation of a cutting edge element. k; and C; are the stiffiless and damping coefficients of the tool-spindle system, respectively. The angle 77 defines direction of the resultant tool displacement.
In a stable (chatter-free) cutting condition, a point on the cutting edge follows a path defined by the following equation
x p = -r(¢) cos('?+ 2;rn)
Yp = r(¢)sin(VF + 2;rn)+ L,ntf sin(() +ndO) (l) n
zp =-rb cos¢+ L,ntf cos(O+nd())
n
144 B. W. IKUA ET AL.
where n, is the number of cutting edges, n is the number of revolutions, </> is the locational angle of the point, and r( ¢) is the radius of the arc traced by the cutting point.
Equation (1) is used to detennine the chip thickness tin the horizontal (x-y) plane.
The instantaneous depth of cut s( fll,¢) is defined in the radial direction from the ball center 0, and it is evaluated in terms of h, f,,, t, R, rb and fJ.
The radial and tangential elemental cutting forces are calculated as8 d'F _ TsL1Ac sin(/J-ae) k d"'
r - + rrb 'f'
sin ¢ls cos(¢s + {J-ae)
(2)
where L1Ac is the elemental chip area given by L1Ac=s(¢,f11)rbd¢, r, is the shearing strength of the workpiece material, ¢,is the shear angle, a. is the effective rake angle, fl
is the friction angle and k, and k, are the radial and tangential edge force coefficients, respectively. Since the helix angle of a ball end mill varies along the cutting edge, the shear angle and the effective rake angle have to be calculated at all points along the cutting edge. The shear angle ¢,is obtained from Merchant's rule and the effective rake angle a e is obtained from Stabler's rule.
The force components Fx and Fy can be calculated by integrating the elemental cutting forces and projecting them onto the Cartesian coordinate system.
2.2. Simulation of Chatter Vibration
The cutting forces excite tool vibrations according to the following equation
(3)
where t;; are the damping ratios and (J) ni are the natural angular frequencies of the tool- spindle system. The tool displacements in the x- and y-directions are calculated from Eq.
(3) by Euler integration method using sufficiently small time steps. The resultant tool displacements q (t) are then evaluated. The direction of the tool displacement is defined by the angle 17, which is given by 17 = tan -l (y Ix) .
Since the cutting edges of the ball end mill are helical, the tool displacements have different effects on the instantaneous depth of cut at different points along the cutting edge. This presents some difficulties in the computation of the change in the uncut chip thickness along the cutting edge. The difficulty can, however, be overcome if we consider the apparent changes inf, h and f,,, due to the tool displacements. The uncut chip thickness can then be evaluated using the apparent values at each time step.
CHATTER IN RAMPING
The tool displacement q(t) causes apparent changes inf, h, andf,, given by
,1/ (t) = q(t) sin TJ(I) sin()}
'1h(t) = q(t) sin TJ(I) cos() '1f P (t) = q(t) cos TJ(I)
The modified value feed ratej{t} at a certain instant in kth cut is, given by f(t) = f +'1f(t) lk.
145
(4)
(5) The modified cross-feed and nominal depth of cut are calculated in the same way. The instantaneous depth of cut at a point on the cutting edge is then calculated by including the effect of dynamic deflection in the previous cut as follows
S ã ã k l,j, = s' l,jã , k + 0 s ã ã l,j, k -I - 0 s Iã ã ,), k ' (6) where s' is the instantaneous depth of cut calculated at zero tool deflection, os is the change in instantaneous depth of cut due to the tool deflection, i is the time index and j represents the location of the point.
The cutting forces are then calculated based on the new instantaneous depth of cut, and the dynamic displacements in the next time step are evaluated. This process is repeated for a number of spindle revolutions. If the amplitudes of the displacements grow with time, chatter is likely to occur, and the process is said to be unstable. If the vibrations decay with time, the process is stable.
The nominal depth of cut is increased in steps of 0.05 mm, until when the amplitudes of the vibrations are seen to increase. The simulation algorithm assumes that the process is unstable when the peak values of vibrations grow by at least 10% in five consecutive tooth passes.
The dynamic properties of the tool-spindle system are given in Table 1. These values were determined experimentally. The standard cutting conditions used in the simulations and experiments are given in Table 2. Unless otherwise stated, the calculations and the experiments were carried out with these conditions.
The local helix angle yofthe ball end mill was measured and found to be related with the position angle ¢according to the following equation; r = 0.2904 + 0.10163¢.
Table 1. Dynamic properties of tool-spindle system
Stiffuess kr 4.49MN/m
ky 4.29MN/m Natural frequency
Damping ratio
1875 Hz 1900 Hz
0.02 'ã 0.02
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Table 2. Cutting conditions
Workpiece radius, R 40mm
Ball radius, rb 8mm
Feed rate,/ 0.05 mm/tooth
Cross-feed,J;, 0.5mm
Nominal depth of cut, h 0.5mm Milling position angle, (} 15-75°
Spindle speed, N 2000minã1
Cutting mode L-U
Unstable region
Unstable region 0.8
§
::;0.6 .:::
to.4 -8
t 0.4
-8 -;
] 0.2
ãE
.!:l 0.2
u ãE Stable region u
2000 3000 4000 5000 Spindle speed N min -I
(a) Convex surface
6000 2000 3000 4000 5000
Spindle speed N min ã1 (b) Concave surfuce
Figure 3. Computed stability lobes .