mechanics of cutting - iit kanpurhome.iitk.ac.in/~vkjain/l6-ta-202 mechanics of cutting.pdf ·...
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![Page 1: Mechanics of Cutting - IIT Kanpurhome.iitk.ac.in/~vkjain/L6-TA-202 MECHANICS OF CUTTING.pdf · Mechanics of Cutting &, sin(90 ( )) cos( ) sin ... Forces in Orthogonal Cutting:](https://reader030.vdocuments.mx/reader030/viewer/2022012315/5aed89e87f8b9a66259013cd/html5/thumbnails/1.jpg)
Mechanics of Cutting
![Page 2: Mechanics of Cutting - IIT Kanpurhome.iitk.ac.in/~vkjain/L6-TA-202 MECHANICS OF CUTTING.pdf · Mechanics of Cutting &, sin(90 ( )) cos( ) sin ... Forces in Orthogonal Cutting:](https://reader030.vdocuments.mx/reader030/viewer/2022012315/5aed89e87f8b9a66259013cd/html5/thumbnails/2.jpg)
&
,sin(90 ( )) cos( )
sincos( )
u
c c
u
c
ABC ABDtAB
sint talso AB
tt
φ
φ α φ αφ
φ α
∆ ∆
=
= =− − −
=−
: : :
: :
:
c
u
f
s
c
t Chip thicknesst Uncut chip thicknessV Chip Sliding VelocityV Shear VelocityV Cutting Velocity
Shear Angleφ
Fig: Schematic of Geometry of chip formation
Geometry of chip Formation:
φ
90-ф+α = 90-(ф-α)
![Page 3: Mechanics of Cutting - IIT Kanpurhome.iitk.ac.in/~vkjain/L6-TA-202 MECHANICS OF CUTTING.pdf · Mechanics of Cutting &, sin(90 ( )) cos( ) sin ... Forces in Orthogonal Cutting:](https://reader030.vdocuments.mx/reader030/viewer/2022012315/5aed89e87f8b9a66259013cd/html5/thumbnails/3.jpg)
How to determine φ & rc ? tc should be determined from the chip. tu (= feed) and α are already known.To determine tc with micrometer, is difficult and not so because of
uneven surface. How? (say, f=0.2 mm/rev. An error of even 0.05 mm will cause an error of 25 % in the measurement of tc)
Volume Constancy Condition: Volume of Uncut chip = Volume of cut chip
: /
1
1(1 ) tan
ctan1
uc
c
c
c c
c c
c
c
tr Chip thickness Ratio Coeffinicienttcos cos sin sin
r sinr cot cos r sin
r cos r sin
r osr sin
φ α φ αφ
φ α αα α φ
αφα
=
+=
= += −
∴ = −
φ
90-ф+α= 90-(ф-α)
Substitute the value of tu /tc
from earlier slide and simplify to get:
SHEAR ANGLE AND CHIP THICKNESS RATIO EVALUATION
3Dr. V.K.jain, IIT Kanpur
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u u c cL t b L t b=
c c u uL t L t∴ =
, u cc
c u
t Lor rt L
= =
c
u
L = Chip length L = Uncut chip lengthb = Chip width (2-D Cutting)
SHEAR ANGLE AND CHIP THICKNESS RATIO EVALUATION
LENGTH OF THE CHIP MAY BE MANY CENTIMETERS HENCE THE ERROR IN EVALUTION OFrc WILL BE COMPARATIVELY MUCH LOWER.
(rc = Lc / Lu)
4Dr. V.K.jain, IIT Kanpur
![Page 5: Mechanics of Cutting - IIT Kanpurhome.iitk.ac.in/~vkjain/L6-TA-202 MECHANICS OF CUTTING.pdf · Mechanics of Cutting &, sin(90 ( )) cos( ) sin ... Forces in Orthogonal Cutting:](https://reader030.vdocuments.mx/reader030/viewer/2022012315/5aed89e87f8b9a66259013cd/html5/thumbnails/5.jpg)
Clearance Angle
Work
ToolChip
Ft
Fc
F
N
Fn
Fs
α
α
β
∅
R
79
α
G
EA
B
∅
D
Fα
Δ FAD = (β - α)Δ GAD = φ + (β - α)
FORCE CIRCLE DIAGRAM
![Page 6: Mechanics of Cutting - IIT Kanpurhome.iitk.ac.in/~vkjain/L6-TA-202 MECHANICS OF CUTTING.pdf · Mechanics of Cutting &, sin(90 ( )) cos( ) sin ... Forces in Orthogonal Cutting:](https://reader030.vdocuments.mx/reader030/viewer/2022012315/5aed89e87f8b9a66259013cd/html5/thumbnails/6.jpg)
Forces in Orthogonal Cutting:• Friction force, F• Force normal to Friction force, N• Cutting Force, FC • Thrust force, Ft • Shear Force, FS •Force Normal to shear force, Fn •Resultant force, R
'
'S N c t
R F N
R F F F F R
→ →
→ → → → →
= +
= + = + =
Force Analysis
A
C
F
D
G
Force Circle Diagram
6
FREE BODY DIAGRAM
Dr. V.K.jain, IIT Kanpur
![Page 7: Mechanics of Cutting - IIT Kanpurhome.iitk.ac.in/~vkjain/L6-TA-202 MECHANICS OF CUTTING.pdf · Mechanics of Cutting &, sin(90 ( )) cos( ) sin ... Forces in Orthogonal Cutting:](https://reader030.vdocuments.mx/reader030/viewer/2022012315/5aed89e87f8b9a66259013cd/html5/thumbnails/7.jpg)
c st cF F os F inα α= +
c sc tN F os F inα α= −
( )Coefficient of Friction µ
c stan c
t c
c t
F os F inFN F os F sin
α αµ βα α+
= = =−
Friction Angleβ =
tan tan
t c
c t
F FF F
αµα
+=
−1, tan ( )also β µ−=
7Dr. V.K.jain, IIT Kanpur
FORCE ANALySIS
DIVIDE R.H.S. BY Cos α
![Page 8: Mechanics of Cutting - IIT Kanpurhome.iitk.ac.in/~vkjain/L6-TA-202 MECHANICS OF CUTTING.pdf · Mechanics of Cutting &, sin(90 ( )) cos( ) sin ... Forces in Orthogonal Cutting:](https://reader030.vdocuments.mx/reader030/viewer/2022012315/5aed89e87f8b9a66259013cd/html5/thumbnails/8.jpg)
c sS c tF F os F inφ φ= −
c sN t cF F os F inφ φ= +
,also
c ( )CF R os β α= −
S c ( )F R os φ β α= + −
( )( )
C
S
F cosF cos
β αφ β α
−∴ =
+ −
( ) u uS
t b tShearPlaneArea A bsin sinφ φ
= = ×
8Dr. V.K.jain, IIT Kanpur
Foce Analysis
Δ FAD = (β - α)Δ GAD = φ + (β - α)
![Page 9: Mechanics of Cutting - IIT Kanpurhome.iitk.ac.in/~vkjain/L6-TA-202 MECHANICS OF CUTTING.pdf · Mechanics of Cutting &, sin(90 ( )) cos( ) sin ... Forces in Orthogonal Cutting:](https://reader030.vdocuments.mx/reader030/viewer/2022012315/5aed89e87f8b9a66259013cd/html5/thumbnails/9.jpg)
Let be the strength of work materialτ
uS S
t bF Asin
τ τφ
= =
C( )F
( )ut b cos
sin cosτ β αφ φ β α
−= + −
,and 1( )
ut bRsin cosτφ φ β α
= × + −
( ) s ( )( )
ut
t b sinF R insin cosτ β αβ αφ φ β α
−= − = ×
+ −
tan( )t
c
FF
β α= −
9Dr. V.K.jain, IIT Kanpur
Foce Analysis
![Page 10: Mechanics of Cutting - IIT Kanpurhome.iitk.ac.in/~vkjain/L6-TA-202 MECHANICS OF CUTTING.pdf · Mechanics of Cutting &, sin(90 ( )) cos( ) sin ... Forces in Orthogonal Cutting:](https://reader030.vdocuments.mx/reader030/viewer/2022012315/5aed89e87f8b9a66259013cd/html5/thumbnails/10.jpg)
( ) Schip
S
FMean Shear Stress tA
= ( )On Chip
( c s )sin
c t
u
F os F inb t
φ φ φ−=
( ) Nchip
S
FMean Normal StressA
σ = ( )On Chip
( c s )sin =
t c
u
F os F inb t
φ φ φ+
10Dr. V.K.jain, IIT Kanpur
Foce Analysis
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VELOCITY ANALYSIS
: cV Cutting velocity of tool relative to workpiece
: fV Chip flow velocity
: sV Shear velocity
Using sine Rule:
sin(90 ( )) sin sin(90 )fc sVV V
φ α φ α= =
− − −
cos( ) sin cosfc sVV V
φ α φ α= =
−
sin cos( )
cf c c
Vand V V rφφ α
= = ⋅−
cos coscos( ) cos( )
c ss
c
V VVV
α αφ α φ α
= ⇒ =− − 11
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Shear Strain & Strain Rate
Two approaches of analysis:
Thin Plane Model:- Merchant, PiisPanen, Kobayashi & Thomson
Thick Deformation Region:- Palmer, (At very low speeds) Oxley, kushina, Hitoni
Thin Zone Model: Merchant
ASSUMPTIONS:-
• Tool tip is sharp, No Rubbing, No Ploughing
• 2-D deformation.
• Stress on shear plane is uniformly distributed.
• Resultant force R on chip applied at shear plane is equal, opposite and
collinear to force R’ applied to the chip at tool-chip interface. 12
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Expression for Shear StrainThe deformation can be idealized as a process of block slip (or preferred slip planes)
( ) deformationShearStrainLength
γ =
s AB AD DBy CD CD CD
γ ∆= = = +∆
tan( ) cotφ α φ= − +
sin( )sin cos cos( ) ,sin cos( )
φ α φ ϕ φ αφ φ α
− + −−
cossin cos( )
αγφ φ α
∴ =−
13Dr. V.K.jain, IIT Kanpur
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In terms of shear velocity (V s ) and chip velocity (Vf), it can be written as
cos sincesin cos( )
s s
c c
V VV V
αγφ φ α
∴ = = −
( )Shear strain rate γ•
1syd s
dt dt y yγγ
•
∆ ∆ ∆ = = = ∆ ∆
s c coscos( )
V Vy y
αφ α
= =∆ − ∆
, : where y Mean thickness of PSDZ∆14Dr. V.K.jain, IIT Kanpur
Expression for Shear Strain rate
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Shear angle relationship
• Helpful to predict position of shear plane (angle φ)
• Relationship between-Shear Plane Angle (φ)Rake Angle (α)Friction Angle(β)
Several TheoriesEarnst-Merchant(Minimum Energy Criterion):Shear plane is located where least energy is required for shear.
Assumptions:-• Orthogonal Cutting.• Shear strength of Metal along shear plane is not affected by
Normal stress.• Continuous chip without BUE.• Neglect energy of chip separation.
15Dr. V.K.jain, IIT Kanpur
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Assuming No Strain hardening:
Condition for minimum energy, 0cdFdφ
=
u 2 2
cos cos( ) sin sin( ) cos ( )sin cos ( )
cdF t bd
φ φ β α φ φ β ατ β αφ φ φ β α
+ − − + −= − + − 0=
cos cos( ) sin sin( ) 0φ φ β α φ φ β α∴ + − − + − =
cos(2 ) 0φ β α+ − =
22πφ β α+ − =
1 ( )4 2πφ β α∴ = − −
16Dr. V.K.jain, IIT Kanpur
Shear angle relationship
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17
THANK YOU
Dr. V.K.jain, IIT Kanpur