2. Cutting force components and their
significances
10/24/2023
Cutting force
2
Cutting force R resolved into PX, PY and PZTurning force resolved into PZ, PX and PY
R
3. Significance of PZ, PX and PY
10/24/2023
Cutting force
3
PZ : called the main or major component as it is the
largest in magnitude. It is also called power component
as it being acting along and being multiplied by VC
decides cutting power (PZ.VC) consumption.
Py : may not be that large in magnitude but is
responsible for causing dimensional inaccuracy and
vibration.
PX : It, even if larger than PY, is least harmful and hence
least significant.
5. The cutting force cont’d…
10/24/2023
F
F – Friction force, Fn– Normal force
Forces on the tool side
Fn
5
Cutting force
6. The cutting force cont’d…
10/24/2023
Fco
Fs – Shear force, Fco– Compressive force
Forces on the chip side,
Fs
6
Cutting force
7. The cutting force cont’d…
10/24/2023
Fco
Fs
F
Fn
R
Ft
Fc
Merchant’s Circle
β
β – friction angle
α
α
β -α
Merchant’s theory
2
1
4
7
Cutting force
8. Cutting force cont’d…
10/24/2023
2
1
2
2
t
c F
F
R
2
1
2
2
n
f F
F
R
2
1
2
2
co
s F
F
R
The shear force
s
F To shear a work material can be expressed in terms
of
cutting force c
F
sin
cos t
c
s F
F
F
Area of shear is given by
s
A
sin
c
s
A
A
8
Cutting force
As
9. The cutting force cont’d…
10/24/2023
The shear strength of a material on the shear plane
may be obtained by
s
c
t
c
s
s
s
A
F
F
A
F
sin
)
sin
cos
(
Ernest and merchant theory suggests that is formed
in such a way that the energy required to shear a material
will be minimum.
Fco
Fs
R
)
cos(
R
s F
F
sin
c
s
s
s
s
A
A
F
s
s
A
C
A
Shear strength
Area of shear
Area of uncut chip
9
Cutting force
10. The cutting force cont’d…
10/24/2023
cos
1
sin
c
s
R
A
F
)
cos(
R
C F
F
cos
cos
sin
c
s
c
A
F
The equation my now be differentiated with respect to
equal to zero to find the value of for which is
minimum.
c
F
0
d
dFc
2
1
4
Solution
10
Cutting force
11. The cutting force cont’d…
10/24/2023
The result didn’t agree with the experimental, since the assump
didn’t considered the dependency of shear stress with normal s
Merchant reconsidered the assumption and he included his new
Assumption
11
Cutting force
12. The cutting force cont’d…
10/24/2023
)
sin(
R
co F
F
Fco
Fs
R
sin
c
s
s
s
co
A
A
F
))
sin(
(
sin
R
c
s F
A
s
s
)
cot(
)
cot(
s
s
)
cot(
s
so
s k
)
tan(
1
k
so
s
12
Cutting force
13. The cutting force cont’d…
10/24/2023
)
,
,
(
f
s
))
tan(
1
)(
cos(
sin
)
cos(
k
A
F c
so
C
The new equation for c
F
• It is now assumed that and are constants for a partic
Material
K so
• and are constants for the cutting operation
c
A
0
d
dFc
c
2
1 c Is constant
13
Cutting force
14. Lee and Shafer’s solution
10/24/2023
The theory of lee and Shafer's was the result of
an attempt to apply the plasticity and slip lines
theory to orthogonal metal cutting
Assumptions
The material is rigid
The behavior of the material is independent of the
rate of deformation
The effect of temperature during deformation is
neglected
14
Cutting force
15. Lee and Shafer’s solution
cont’d…
10/24/2023
• ABC is deformation zone
•AB shear plane
•AC is stress free so that slip
lines
meet this line at an angle of 45
•Slip lines :
•Parallel to AB
•Perpendicular to AB
4
The theory of lee and Shafer's is not applied for tools
with negative rake angle .
15
Cutting force
17. Friction force
10/24/2023
Sticking friction
Sliding friction
r
A a
A
real area Apparent area
r
f
f A
F
y
n
r
F
A
n
f
F
F
f
F
f
Shear strength
Friction force
n
F Normal force
17
Cutting force
18. Model of chip–tool friction in
orthogonal cutting
10/24/2023
18
Cutting force
19. Friction force cont’d…
10/24/2023
It is assumed that the stress distribution on the tool face could
be represented by the expression
f
f
y
f qx
The maximum normal stress occurs when
and therefore
max
f
f
l
x
y
f qx
m
ax
y
f
f l
q
max
y
l
x
f
f f
)
(
max
19
Cutting force
20. Force of friction cont’d…
10/24/2023
)
1
(
)
( 1
max
y
l
l
l
a
l
a
F y
f
y
st
f
w
f
st
w
st
f
)
1
(
max
y
l
a
F
f
w
f
n
f
st
f
st
n
f
l
l
y
F
F
1
tan
max
20
Cutting force
21. Cutting forces on turning
10/24/2023
Cutting force
Feed force
Radial force
Thrust force
c
F
f
F
r
F
D
F
•No motion on the radial
No work done by radial
motion
• The feed force is negligible
•Power(P) needed for turning is :
60
c
cV
F
P
Watt
21
Cutting force
22. Specific cutting Energy
10/24/2023
The rate of energy consumption during machining
is
m
P
V
F
P c
m
The energy consumed per unit volume of metal
removed, specific cutting energy is
s
P
c
c
w
m
s
A
F
Z
P
P
m
P
c
F
s
P
w
Z
Power consumed
Cutting force
Specific cutting energy
Metal removal rate
Un deformed chip cross
section
c
A
22
Cutting force
25. Forces on Shaping
10/24/2023
Cutting force
25
sin
cos
T
n
T
f
F
F
F
F
T
c
F
F cutting component acts against v
Thrust component perpendicular to transient
surface
n
f
F
F Feed component
Normal component
26. Forces in Drilling
10/24/2023
Cutting force
26
2
/
f
f Feed per revolution
Feed per cutting
edge
sin
)
2
/
(
sin
)
2
/
(
1
D
w
f
t
sin
tan
)
/
2
(
tan 1 D
r
r Radius of the point on the cutting
rake angle
Helix angle
2
2
27. Force on drilling cont’d…
10/24/2023
Cutting force
27
Development of torque and thrust
during drilling
friction
ch
T F
F
F
F
sin
2
ch
F
F Total Thrust force
Force on chisel edge
friction
ch
c M
M
z
F
M
ch
M
M
Total Moment
Moment due to chisel
edge
80%M
8%M
12%M
sin
5
6
.
0
T
c
F
F
D
F
M
28. Forces on Milling
10/24/2023
Cutting force
28
D
d
Nn
V
t
c
f
c
2
max
D
d
Nn
V
t
c
f
ave
ave
c
t
t max Max. uncut chip thickness
Average chip thickness
t
f
c
t d
2
D
29. Forces on milling cont’d…
10/24/2023
Cutting force
29
Thrust force doesn't cause any torque
The total torque is caused by the cutting force
Components of milling force
C
F
T
F )
2
/
(
_
_ d
F
M c
tooth
cutting
one
d Diameter of the cutter
Arbor rotation and torque
Variation of torque due to single cutting
30. Forces on milling cont’d…
10/24/2023
Cutting force
30
Z
2
)
i
Z
2
)
ii
Z
2
)
iii
Type of tooth Nature of torque
31. Power consumed by machine
tool
10/24/2023
Cutting force
31
m
m
e
P
P
m
e
m
P
P
Power consumed
Electric Power consumed by machine tool spindle
rotor
Machine tool transmission efficiency
4
.
0
t
P
P o
s
t
P
Ps
0
Specific cutting energy
A constant in (J/mm3) depends on tool material
Uncut chip thickness in mm
