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Mechanics of Orthogonal Cutting

t1 is un-cut chip thickness
t2 is cut chip thickness
r is chip thickness ratio
r = t1/t2 < 1 ( t1 < t2)
k = 1/r = chip reduction coefficient
α is rake angle
φ is shear angle
Assumptions
1. No contact at the flank.
2. Width of chip remains constant.
3. Uniform cutting velocity.
4. A continues chip is produced.
5. Volumetric changes of material during machining is zero.
That is
Volume before cutting = volume after cutting
t1 *b*l1 = t2*b*l2 t1/t2 = l2/l1 = r
Also we can say that volumetric flow rate is also equal
t1*b*Vc = t2*b*Vf t1/t2 = Vf/ Vc
Vc is cutting velocity
Vf is chip flow velocity
Vs is shear velocity
WORKPIECE
t1
α
Shear
plane
Friction
planeVertical
plane
Vs
Vc
Vf
TOOL
width
φ

WORKPIECE
TOOL
t1
α
Shear
plane
Friction
plane
Vertical
plane
Vs
Vc
Vf
φ
α
φ
AC
B
t1
sinФ = t1 / AB
90-Ф
sin(90- Ф+ α ) = sin (90-(Ф-α)) = cos(Ф-α) = t2/AB
t1 = AB*sinФ
t2 = AB cos (Ф-α)
Therefore
t1/t2 = (AB*sinФ) / (AB*cos(Ф-α) )
r = sinФ / cos (Ф-α)
r = sinФ / ( cosФ*cosα+ sinФ*sinα )
r = ( sinФ /cos Ф ) / [( cosФ*cosα+ sinФ*sinα ) / cosФ]
r = tanФ/ (cosα + tanФ*sinα )
rcosα + r*tanФ*sinα = tanФ
tanФ -r*tanФ*sinα = rcosα
tanФ (1- rsinα) = rcos α
tanФ = rcosα / (1- rsinα)
From triangle ABC & ACD
D
90-Ф+α
Ф-α
RELATION BETWEEN R, Φ AND α

WORKPIECE
TOOL
t1
α
Shear
plane
Friction
plane
Vertical
plane
Vs
Vc
Vf
φ
Vc
Vf Vs
φ90-α
90-(Ф-α)
By applying SINE rule
(Vf / sinФ) =[Vs / sin(90-α)] = [Vc/sin(90-(Ф-α)]
(Vf / sinФ) = (Vs / cosα) = [Vc/cos(Ф-α)]
Vf = [Vc*sin α /cos(Ф-α)]
Vf = Vc*r
Vs = [(Vc*cos α /cos(Ф-α)]
VELOCITY RELATIONSHIPS

work piece
tool
R2
F
N
Fc
Ft
R1
FS
Fn
φ
α
FS
R
F
N
β
φ
Ft
Fc
Fn
α
β-α
Fc is Cutting Force
Ft is Thrust Force
R1 is Resultant Force of Fc & Ft
F is Friction Force
N is Normal Force of F
R2 is Resultant Force of F& N
Fs is Shear Force
Fn Normal Force to Fs
R1 isalso Resultant Force of Fs & Fn
Weknow
F = µN
From diagram
tanβ = F/N
F = tanβ*N
Therefore
µ= tanβ
β is Angle of friction
µ is coefficient of friction

Ft
Fc
β-α
R
FS
R
F
N
β
φ
Ft
Fc
Fn
α
β -α
R = (Fc ^2) + (Fv^2)
Tan(β-α) = Fc /Ft
R
φ
Fn R = (Fs^2) + (Ns^2)
FS
Tan(β-α+φ) = Fs/Ns
β
β-α
α
R = (F^2) + (N^2)
R
F
N
Theories of Angles
Lee & Shaffer theory : Φ+β-α = 45
Stabler theory : Φ+β-(α/2)= 45
Merchant Constant (Cm) : 2Φ+β-α
Energy for Cutting (Ec) = Fc * VC
Energy for friction (Ef) = F * VF
Energy for shearing (Es) = Fs * Vs
Percentage of energy loss in friction
= (Ec/Ef)*100
Percentage of energy loss in shearing
= (Ec/Ef)*100

R
F
N
β
φ
Ft
Fc
α
β -α
A
O
D
C
E
G
B
α
9O-α
9O-α
Relationship of Fs & Fn with Fc & Ft
Fn = AE = AD+DE = DE+CB = Fc sin φ + Ft cos φ
Fs = OA = OB-AB = OB-BC = Fc cos φ - Ft sin φ
Relationship of F & N with Fc & Ft
F= OA = CB = CG+GB = ED+GB = Fc sin α + Ft cos α
N= AB = OD –CD = OD- GE = Fc COS α - Ft sinα
FS
R
Φ
Ft
Fc
Fn
α
β -α
O
B
A
C
G
E
9O-Φ
9O-Φ

Shear area = As = W*t1 / sin Φ
Shear stress = τ = Fs / As τ = Fs sinΦ / (w*t1)
Shear strain = Ƴ = Cot Φ + tan (Φ-α) = cos / [sin Φ*cos (Φ-α)]
Shear strain rate = Vs/ts
WORKPIECE
t1
Fs
φ W
Vs
The minimum value of shear strain when rake angle is zero
Shear strain = Ƴ = Cot Φ + tan Φ
(d/dΦ) {cot Φ + tan Φ } = 0
-cosec^2 Φ + sec^2 Φ =0
-(1/sin^2 Φ)+ (1/cos^2 Φ) =0
cos^2 Φ-sin^2 Φ=0
[(1-cos 2Φ)/2] - [(1-sin 2Φ)/2] =0
2cos 2 Φ = 0
2Φ =90
Φ =45
For minimum value
2Φ - α =90
For orthogonal cutting
Depth of cut = t1 = feed*θ ( θ is side cutting edge angle )
Width of cut = t1/ sin θ

Fc
Ft
N
F
µ = Tanβ = F/N = Ft/Fc
But when Ft > Fc, β > 45 µ > 1
In this case use formulae for finding µ
The classical friction theory
µ = [ln ( 1/r)] / [( π/2) – α]
Actuvally the value of µ is always comes less than one
Ft < Fc, β < 45 µ < 1

• Taylorstool life equation:-
•VTn=C
• Tool life equation (generalized)–
•VTnfn1dn2=C
• Tool life exponents n,n1, n2 are found by plotting
experimental data on log V – log T,log T– log fand
log T– log d scales.

Determination of toollife constants n,n1, n2
• Longand expensive test, involves considerableamount
of material, labor and machiningtime.
• Recourseis taken to experimental design techniques
suchasfactorial design, multiple regression analysis and
response surface methodology to reduce cost and no. of
observations.

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Mechanics of Orthogonal Cutting

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Mechanics of Orthogonal Cutting