ISSN: 2180-1053 Vol. 3 No. 1 January-June 2011 29
FIGURE 2: CAD based material removal model for machining.
3.1 Machining load model for helical endmill
As shown in Figure 3, the machining loads acting on a helical lute endmill are equally discretized into a inite number of elements along
the tool axis. The total cuting loads F
x
, F
y
and F
z
acting on the tool at a particular instant are obtain by summing the force components acting
on each individual discretized element [16, 17, 18].
FIGURE 3: Cutting force model for helical endmill.
+ =
φ φ
+ =
φ φ
+ =
φ φ
φ φ
φ = φ
χ
γ χ =
FIGURE 3: Cuting force model for helical endmill.
z K
z h
K z
dF
te j
tc tj
d ]
, [
, +
= φ
φ ,
z K
z h
K z
dF
re j
rc rj
d ]
, [
, +
= φ
φ ,
z K
z h
K z
dF
ae j
ac aj
d ]
, [
, +
= φ
φ are differential forces corresponding to discretiz
al and axial directions. The coefficients K , K , K and φ
φ φ =
φ χ
γ χ
=
+ =
φ φ
+ =
φ φ
+ =
φ φ
1 ment thickness in
are the spec φ
φ φ =
φ χ
γ χ
=
where dF
tj
, dF
rj
and dF
aj
are diferential forces corresponding to discretized element thickness in the tangential, radial and axial
directions. The coeicients K
tc
, K
rc
, K
ac
and K
te
, K
re
, K
ae
are the speciic cuting force coeicients and speciic edge cuting force coeicients
to each tangential, radial and axial direction, determined from the experimental analysis.
+ =
φ φ
+ =
φ φ
+ =
φ φ
ng force coefficien tal analysis.
φ is th cut chip thickness for the fl
φ φ =
φ χ
γ χ
=
is the tool’s immersion angle start from positive y-axis and
h
j
is the instantaneous uncut chip thickness for the lute j and can be deine as:
ISSN: 2180-1053 Vol. 3 No. 1 January-June 2011 Journal of Mechanical Engineering and Technology
30 +
z f
z h
j t
j
φ φ
sin ,
= tooth and
z
j
φ is the entry and exit angl χ
γ χ
=
φ φ =
2 φ
at certain position χ
γ χ
=
where f
t
is the feed per tooth and
φ φ =
per tooth and z
j
φ is t rection. Since this study using a h
the same instant and χ
γ χ
=
is the entry and exit angle for lute j at certain position in the axial direction. Since this study using a
helical cuter the full length of the cuting edge does not enter or exit the cut at the same instant and the angular delay between disretize
elements
φ φ
φ = φ
tting edge does n elements
χ can be
γ χ
=
can be approximated as follows:
φ φ
φ
=
φ χ
oximated as follows:
FIGURE 4: Discretized unrolled helical endmill geometry.
γ χ
=
FIGURE 4: Discretized unrolled helical endmill geometry.
φ φ =
φ χ
rad r
db
γ χ
tan =
φ φ =
φ χ
γ χ
=
3
where r is the tool diameter,
diameter,
γ
is h
φ
for
χ
yields:
γ φ
=
[ ]
φ φ
φ φ
φ γ
φ −
+ −
− =
[ ]
φ φ
φ φ
φ γ
φ +
− −
=
[ ]
φ φ
φ γ
φ − =
φ φ
δ δ δ
is helix angle and db is the element thickness. Rearranging Eqs. 3 and substituting
γ
bstituting
φ
d
for
χ
γ φ
=
[ ]
φ φ
φ φ
φ γ
φ −
+ −
− =
[ ]
φ φ
φ φ
φ γ
φ +
− −
=
[ ]
φ φ
φ γ
φ − =
φ φ
δ δ δ
for
γ
ting
φ
for
χ
yie
γ φ
=
[ ]
φ φ
φ φ
φ γ
φ −
+ −
− =
[ ]
φ φ
φ φ
φ γ
φ +
− −
=
[ ]
φ φ
φ γ
φ − =
φ φ
δ δ δ
yields:
γ
φ χ
γ φ
tan .d
r db
=
[ ]
φ φ
φ φ
φ γ
φ −
+ −
− =
[ ]
φ φ
φ φ
φ γ
φ +
− −
=
[ ]
φ φ
φ γ
φ − =
φ φ
δ δ δ
γ
φ χ
γ φ
=
4
[ ]
φ φ
φ φ
φ γ
φ −
+ −
− =
[ ]
φ φ
φ φ
φ γ
φ +
− −
=
[ ]
φ φ
φ γ
φ − =
φ φ
δ δ δ
By substituting and integrating the diferential cuting forces from Eqs. 1 to 4 within the lower and upper boundaries of the lute which is in
cut. The tangential, radial and axial forces can be transformed in x, y, z
Cartesian directions and becomes:
γ
φ χ
γ φ
=
[ ]
φ φ
φ φ
φ γ
φ
Zju Zjl
j j
r j
t t
xj
z z
K r
f K
F 2
sin 2
2 cos
tan 4
− +
− −
=
[ ]
φ φ
φ φ
φ γ
φ
Zju Zjl
j r
j j
t t
yj
z K
z z
r f
K F
2 cos
2 sin
2 tan
4 +
− −
=
[ ]
φ φ
φ γ
φ
Zju Zjl
j t
t a
zj
z r
f K
K F
cos tan
− =
φ φ
δ δ δ
γ
φ χ
γ φ
=
[ ]
φ φ
φ φ
φ γ
φ −
+ −
− =
[ ]
φ φ
φ φ
φ γ
φ
+ −
− =
[ ]
φ φ
φ γ
φ
− =
5
φ φ
δ δ δ
where Zjl
+ =
φ φ
+ =
φ φ
+ =
φ φ
ng force coefficie tal analysis.
φ is th cut chip thicknes
φ φ =
φ χ
γ χ
=
and Zju
+ =
φ φ
+ =
φ φ
+ =
φ φ
ng force coefficien tal analysis.
φ is th cut chip thicknes
φ φ =
φ χ
γ χ
=
are the lower and upper axial engagement limits of the in cut immersion of the lute j. From Eqs. 5 the instantaneous
cuting forces acting on the whole endmill can be obtained, which are used as the input for fea to compute the delection of the workpiece.
ISSN: 2180-1053 Vol. 3 No. 1 January-June 2011 31
3.2 finite element modelling of thin-wall workpiece
Kategori