Biomedical Engineering Reference
In-Depth Information
The idea that the drag of the chip on the rake phase of the cutter causes the
drill to experience a stick-slip friction effect has been incorporated into the
model as well, and the law has been mathematically defined. The friction law
used to incorporate the effect of friction in the derivations is defined as:
2
u
(
u
)
(
) exp
1
2
1
U
o
where: u - chip speed, Uo - friction cutting speed, μ
1
, μ
2
- threshold friction
coefficients.
For obtaining the dynamic model of the drill reverse engineering has been
used, where the drill has been laser scanned and the obtained 3D model was
imported into ABAQUS program for analysis. The motion of the tip during
vibration has been obtained from ABAQUS.
The equation of motion has been expressed as:
..
.
m
c
k
F
where: m, c, k, - mass, damping, stiffness, F
η
-forcing function.
The vibration mode with regenerative chatter which includes the chip
thickness variation has been expressed as:
"
'
'
' 2
(1
(
) cos
)(
p
p
p
)
0
T
where: η is the rescaled amplitude variable, and η' is the derivative with
respect to the rescaled time variable, γ - damping coefficient, β = wη/k and
, are the rescaled p
1
, p
2
, η
T
= η (t-T), T - time to complete one
revolution of the workpiece.
The dependence on the friction model is explained through the coefficient
p
1
of G (force penetration rate function), which depends on the rake angle ʱ
and vibration angle θ and on the form of the friction relation μ=h(u):
p
p
2
'
h
h
cos(
)
,
2
2 cos(
)
1
cos(
)
p
0
0
0
1
3
'
sin
h
V
cos
0
2
0
1
h
cos (
)
