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Forces and Mechanics of Cutting
• Why should we know?– Power requirement for the machine tool can be
calculated– Design of stiffness, etc. for the machine
tolerances– Whether workpiece can withstand the cutting
force
• Ernst and Merchant (1941) did the first scientific analysis– Normal = N– Along the tool = F
• FC and FT along and normal to cutting along the direction of tool movement with velocity, ‘v’.– –
N
F
sincos NSC FFF sincos SNT FFF
(1)
(2)
sin
cos
cos
cossin
sincos
sincos
cossin
RF
RF
FR
FFF
FFF
FFN
FFF
T
C
S
TCN
TCS
TC
TC(3)
(4)
(5)
(6)
(7)
(8)
(9)
• We can measure FC and FT using force dynamometer.
• FS, FN, F, and N can be found.
– FS and FN from equations 5 & 6.
– F and N from 3 & 4
• tan
N
F( friction angle)
sincos
cossin
TC
TC
FF
FF
Eq. 10
• Cutting Force FC depends on
– FC increases as t0 increases
– FC decreases as rake angle increases and as speed increases
• Why FC is affected by speed:
– As speed goes up, shear angle goes up, and friction reduces.
• Forces can also be affected by the nose radius. Large nose radius increases force. (Blunt tool)
• Large nose radius can create positive rake angle and cause rubbing and create plastic deformation.
• Coefficient of friction in metal cutting range from 0.5 to 2.0
• Shows how high friction can rise on the chip-tool interface
• Forces on the tool tip are very high because of small contact area.
Stresses• Average shear stress
• Average normal stress
• The area where the stress acts (area of shear plane)
• AS can be increased by increasing t0.• is independent of rake angle• decreases with increase in rake angle.
S
S
A
F
S
N
A
F
sin0wtAS
• Consequently, normal stress in the shear plane has no effect on the magnitude of shear stress.
• Problems in finding stresses on the rake face:– Hard to find the contact on the rake face.– Stresses in practice is not uniformly distributed
on the rake face.
Shear-angle relationship• Let’s take friction angle as
• Assume is independent of .Differentiate with respect to and equate to 0
(zero).
0
0
sincossec
sin
cos
cos
t
F
A
F
wtA
RF
RF
C
S
S
S
S
C
In the previous slides wecalled this
)(
• The equation (A) shows that
If decreases and/or increases then decreases.• In practice this analysis is corrupted because of
several reasons like:– Shear stress is effected by normal stresses.
– is effected by etc.
– (see graph 8.19)
2245
90tancottan
o (A)
Specific Energy• Total power = FCV
• If width of unit = ‘W’
• Total energy/unit volume of material removal.
• Frictional specific energy:00 t
F
Vt
VFu CCt
000
cossin
t
FFr
t
Fr
Vt
FVu tCCf
• Power required to shear along the plane
• As uf increases, shear angle decreases, and hence us goes up directly.
• Thus friction plays an important part in metal cutting.
sft
SSs
uuu
Vt
VFu
0
cos
sin
cos
sin
t
f
u
u
Problem• t0=0.005 in, V=400 ft/min, α=10o, =0.25, tc=0.009,
Fc=125 lb, Ft=50 lb.
• What % of total energy is consumed in friction?
Summary• Velocity triangle• Merchants circle• Compute Forces and obtain Fs, Fn etc based on
measuring Fc and Ft (Equations 1 to 10 of this slide set)• Calculate Shear stress and normal Stress• Specific energy• Shear angle relationships• Relationships between rake angle, velocity, shear angle
and cutting force• Effect on Ft due to –ve and +ve rake angle.
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