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.