cutting force model

8/2/2019 Cutting Force Model

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Modeling of Cutting Forces for High-speed

Milling of Titanium Alloys

Dept of Mechanical Engineering

Prof. M. Rahman and Wang Zhigang


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Overview

Introduction

Literature review

Modeling of milling process

Modeling of cutting forces for high-speed milling of Ti-6Al-4V

Verification of the cutting force model Conclusions


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Cutting temperature as a result of cutting speed

Introduction


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Definition of high-speed machining (HSM)

Introduction

fibre-reinforced

plastics

bronze,brass

cast iron

steel

titaniumalloys

convent

ional

range

aluminum

alloys

cutting speed vc [m/min]

10 100 1000 10000

HSC-ran

ge

transitio

nrange

nickel basedalloys

Cutting Speed Area Depends on Material


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Higher productivity

Generate high-quality surfaces, burr-free

edges and stress-free components.

Cutting forces are lower

Minimize the heat effect on machined parts;eliminate the usage of cutting fluids

Significant advantages of HSM

Introduction (contd.)


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Aircraft & aerospace production, tool and die mold

manufacturing (high productivity)

Optical industry, fine mechanical parts (high surface

quality)

Precision mechanics, magnesium alloys (Cutting

heat taken away by chips)

Automotive industry, household equipments (low

cutting forces)

Applications of HSM


HSM is a strategic part of making F/A-18E/F tactical fighters


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their good strength-to-weight ratio

superior corrosion resistance.


Titanium alloys have been widely

used in the aerospace, biomedical,

automotive and petroleum industriesbecause of

Among all titanium alloys, Ti-6Al-4V

is most widely used.

Their machinability is very poor.


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The performance of conventional tools is poor

Advanced tool materials, such as cubic boronnitride (CBN), polycrystalline diamond (PCD)

Binderless CBN used in this study

High-speed machining of Ti-6Al-4V

Experiments are costly and time-consuming


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15 topics related to modeling of machining

operations

Six major operational groups: single straight edge

orthogonal, single straight edge oblique, turning,

milling, drilling and form-tool machining

Survey of recent research on modeling by CIRP

working group

Literature review

Over 55 major research group are currently active in

modeling efforts


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Experimental/empirical modeling (43% of

research groups)

Analytical modeling (32% of research group)

Numerical modeling (18% of research group)

Three categories of cutting force modeling

Literature review (contd.)


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Taylor, the father of metal cutting science,

firstly used empirical approach to propose

the well-known Taylors equation

The power-law form of Taylor equation

extended to predict cutting forces

Empirical model


vfCaF=


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Only valid for particular cutter geometry and

workpiece combination;

Large numbers of empirical experiments are

required;

Empirical model


This method is not suitable for cutting tool

design purpose


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Mechanistic model

Shear plane model

Shear zone model

Predictive machining theory

Analytical model



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Based on the assumption that the magnitude of

cutting forces depend on the uncut area

Tangential force in milling is given as:

Mechanistic model


AKF tt =

whereKtis the specific cutting pressure,

A is the uncut chip area;

Limits: Ktdepend on specific combination of tool/workpiece


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Initiated by Merchants shear plane theory

Use the minimum energy principle to

determine the shear angle

Shear plane model


Shear plane theory assumes that thin shear

zone is a plane, and that work material

deforms at constant flow stress


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Strain hardening properties of the work

material had a profound effect on the

hydrostatic stress distribution in the chip

formation zone.

Oxley and Welsh (1963) introduced the

parallel-sided shear zone model of chipformation.

Shear zone model



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Shear zone model


VC

Chip

Tool

A

BD

F

E

C

Between the boundary of CD and EF


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Based on the parallel-sided shear zone

model

The flow stress of a metal is influenced by:

Oxleys predictive machining theory


properties of work material

effective strain

effective rate of deformation or strain rate

cutting temperature


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Mostly use the finite element method (FEM)

More accurate than the analytical model

Predict cutting forces, strain, strain rate and

temperature, etc.

Numerical modeling


FEM requires much more computation time,

especially for 3-D simulation


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Relative motion between the cutting tool and

workpiece for face milling:

Modeling of milling process

Rotation of the spindle

Translational motion of

feed


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Traditional trochoid curve

The motion of the cutter is like the trace of a point fixed on a

circle that rolls along a line

Modeling of milling process (contd.)


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Traditional undeformed chip thickness

h

f

f

O

O

A

R

For high speed milling or micro-milling, there is a great need

for higher accuracy, so the circular tooth-pass could not meet

the requirement


)sin()( fh =


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The true tool trajectory during slot milling


O O DA

B

C

0

Tool tip

Workpiece

Chip

thickness

True tooltrajectory


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Analytical solution to undeformed chip thickness

where


02)sin(2 = aa

)2

sin(

=R

fa

RCDRBCh

)cos(

)cos()cos()(

===

)(24

83

12

23

2

1 543

23

2

22 aOaaaa +

++

+++=

where

and = 0 -


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Effects of nose radius


dFt

2

1

dFr

r-a

r

d

d

OO0

B

A

C

0

DII

E

I F

h() h()


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Non-uniform chip area


+

=+= 1

0

2

1

1

0

)sin

)((

2

1)(

2

1)(

2

12

22

22

22

d

arrdAOrdOBrS

+2

1 ]cos)(sin)(cos)(2sin)([2

1 2222222

dhhrhh

It needs to establish a model about 3-D milling

process to simulate the cutting process around

the tool tip


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Equivalent element representation


Represent the uneven uncut chip area with

the equivalent rectangular contact area

Represent the uneven intersection surfacewith the equivalent one which is suitable for

axisymmetric deformation simulation (Ozel,

1998)


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Equivalent element representation

Equivalent element representation with a parallelogram


Ft

er-a

r

OO0

B

A

C

Fr

he

3

FZ

D

Fr

h() h()

When chip thickness is less

than 0.05mm, the size effect is

very obvious.


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Overview of this section

Modeling of cutting force

Brief review of Oxleys theory

Deformation behavior of Ti-6Al-4V

Hybrid cutting force between Oxleys

theory and FEM simulation


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His model has been widely used by many researchers

Modeling of cutting force (contd.)

Two assumptions: cutting edge is perfectlysharp; uniform normal stress distributes at

the tool/chip interface

His work was mainly focused on the carbon

steel work material

Oxleys predictive cutting force theory

Two limitations:


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Cutting force diagram based on shear zone model


VC

ChipTool

Fc

FSFN

A

B

G

l

lc

t2

Fn

FR

FR FT

Ff

t1

D

F

E

C

Based on the parallel-sided shear zone model


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Geometric and cutting forces relations:


+=

=

=

=

sin/)cos()cos(

cos

)cos(sin

12 tt

VV

VV

CS

Cchip

cossin

)sin()sin(

cossin

)cos()cos(

cos

1

1

==

==

=

wtkFF

wtkFF

FF

ABRT

ABRC

SR

kAB, , n and C ?????

Cn+= )4/(21tan


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Oxleys predictive cutting forces theory:


3/1n

ABABk =

Shear flow stresskAB along the shear plane

where 1 is initial stress constant, and n is strain-hardening

index, AB is effective strain along the shear plane. 1 and nvary with strain rate and temperature

3/ABAB =

Effective strain and strain-rate are calculated as

3/ABAB && =

where AB and are maximum strain and strain rate along

ABAB&


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Maximum strain and strain rate alongAB

where lis the length ofAB, and Cis strain-rate constant.

Velocity modified temperature

)cos(sin

cos

2

1

=AB

l

VC SAB =&

)]/lg(1[ 0mod &&vTT AB =

the constants v and 0& are taken as 0.09 and 1/sec

where TAB is the cutting temperature at shear planeAB,


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Temperature at shear plane AB

where TW is the initial workpiece temperature and TSZcanbe calculated from the equation:

SZWAB TTT +=

)cos(

cos1

1

=S

SZ

F

wStT

TAB depends on cutting forces and thermal properties of

workpiece material.


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Computation about the cutting temperature in

the shear zone


Assume TAB=Tw

Calculate for planeAB, thermal properties SandK;

Tmod, kAB;FS= kABlw;RTand ; TSZand TAB

Compare new TAB and old TAB

Calculate ; ;R=FS/cos,F,NandFC

TAB = new TAB

a small given value

a small given value


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Mean chip temperature calculation:


Assume mean chip temperature TC=Tw + TSZ

Calculate the chip thermal properties S and K from

appropriate equations; TC; TC=Tw + TSZ+ TC

Compare new TCand old TC

Calculate for the toolchip interface shear flow stress kchip = 1/3

TC= new TC

a small given value

a small given value


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Shear flow stress at the tool-chip interface

where 1 corresponds to the value ofTmod. This equation

neglects the influence of strain on the flow stress above astrain of one.

Average shear stress at the tool-chip interface

3

1=chipk

wl

F

c

f=int

++=

])4/(21[31

sincos

sin1

Cn

Cntlc

where contact length


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Solution to shear angle Modeling of cutting force (contd.)

The solution to is taken at the intersection of the two curves.

Shear angle (deg)

kchip

int

0 45

600

Shears

tress(MPa)


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Determination of strain-rate constant


Based on the assumption of uniform normal stress

along the tool-chip interface, the normal stress N atB

is also given by

CnkAB

N 222

1'

+=

c

NNwl

F

=

The tool-chip interface is assumed to be a direction of

maximum shear stress, the normal stress N at B isgiven by

Ccan be found by fulfilling the condition N = N


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Summary of the machining calculations


For a given and C, the equilibrium values of

are found (when int is equal to the value ofkchip).

Then, the required value of C is determinedfrom the stress boundary condition.

The above procedure is iterated for a given range

ofand C until all the equilibrium conditions are

fulfilled.


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Modeling of cutting force (contd.)Given: cutting conditions: , U, t1, w, Tw and material properties; Assign values for 1, 2, , final

Assume the initial value ofC(say C= 5)

Calculate l= t1/sin; VS, AB andAB;

Calculate the flow stress int at tool/chip interface

Calculate ; ;R=FS/cos,F,NandFC

Calculate the tool/chip interface shear flow stress kchip

=45o?

Plot int and kchip versus and selectsolution point where int=kchip

Print the final results, such as , forces etc

Compare Nand N

=min?

PlotFc versus

and determine =min for minimumFc

= i

Assume (say = 5o)

=final?No

Yes

No

Yes

Yes

No

a small given value

a small given value

Calculation of temperature at shear plane

Calculate the mean temperature at the chips

= + 0.1oEstimate new C

=min

M d li f i f


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Summary of the machining calculations


Five iterative procedures involved in the

computation: determination of temperature at

AB, and at tool-chip interface, possible ranges

of C, and . These three parameters are veryimportant for its accuracy.

Such a procedure is extremely time-consuming.

M d li f i f


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Modeling of flow stress properties of Ti-6Al-4V


Johnson-Cook (JC) strength model representsthe flow stress of a material as the product of

strain, strain-rate and temperature:

])(1)][ln(1][)([0

m

rm

rn

TTTTCBA

++= &

&

ABn

AB

n

ABm

rm

rn

ABBA

nB

TT

TTCnB

d

d

])([

)(])(1)][ln(1[)(

1

0

1

+=

+=

&

&

Then, differentiate with respect to :

M d li f tti f


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20oC


700oC

M d li f tti f


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Modeling of cutting forces


In Oxleys model (1989), flow stress in the shear plane zone,

kAB can be calculated according to:

This is replaced by:

3/1n

ABABk =

3/=ABk

where is the effective flow stress alongAB, which can be

calculated using the constitutive equation.

M d li f tti f


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The change rate of flow stress (dk/ds2) normal toAB can be

assumed to be only related to the actual strain-rate

222 ds

dt

dt

d

d

dk

ds

d

d

dk

ds

dk

==

Then, the first term on the right-hand side of above equation

can be obtained as:

ABn

AB

n

AB kBA

nBd

dddk

])([3)(

33/

1

+==

(1)

(2)

M d li f tti f


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The second term on the R.H.S of Eq. (1) is the strain-rate

)cos(

cos

=

l

VC

dt

d C

The last term is the reciprocal of the cutting speed normal

toAB, which can be presented as

)sin(

1

2 CVds

dt=

Modeling of cutting force


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])([

)(2

sin

1

)cos(

cos

])([3

)( 1

2

n

AB

n

ABAB

C

C

n

AB

n

ABAB

BAl

CnBk

Vl

CV

BA

nBk

ds

dk

+=

+=

Eq. (1) can be simplified as:

1

2

dsdsdkdp =

According to the stress equilibrium equation alongAB

from Oxley (1989), the following relation exists

(3)

Modeling of cutting force ( d )


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n

ABAB

n

ABBA

nBk

BAppC

)(2

])()[(

+=

Finally the unknown parameter Cis given by

By applying the equation alongAB, substituting for

dk/ds2 from Eq. (3), the next equation is given

])([

)(2n

AB

n

ABABBA

BA

CnBkpp

+=

wherepA andpB are the hydrostatic stresses at pointsA andB

Modeling of cutting force ( d )


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In Oxleys theory, the angle made by theresultant forceR withAB is expressed as

22

)

4

(21tan

s

l

k

k

AB

+=

The following equation is obtained

n

AB

n

AB

BABCn

++=)

4(21tan

Modeling of cutting force ( td )


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Based on above description, for given values of toolrake angle , the cutting speed VC, the thickness t1 and width of cut w of the undeformed chip,

together with the thermal and flow stress properties

of the workpiece material and the initial

temperature of the work Tw (say, 20oC in all

calculations), FEM can be employed to simulate

the metal deformation process .



Verification of the model


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Experimental setup

Verification of the model

workpiece

dynamometer

Work table

Cutter

Verification of the model (contd )


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Cutting tools: binderless cubic boron nitride

Cutting conditions

Experimental design and cutting conditions

Cutting speed: 300, 350 and 400 (m/min)

Feed rate: 0.075, 0.100 and 0.125 (mm/r)

Depth of cut: 0.075, 0.010 and 0.125 (mm)

Verification of the model (contd.)



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Deformation zone of FEM simulation


Effects of cutting

edge radius has

been considered



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Temperature distribution on the tip of the tool




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Temperature simulation




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(a) Estimated cutting forces

at a = 0.075mm,f= 0.075mm/r and v = 350m/min

-20

-10

0

10

20

30

40

0 45 90 135 180

Angular position (deg)

Cuttingf

orces(N)

Fx (N)

Fy (N)

Fz (N)

-20

-10

0

10

20

30

40

50

0 45 90 135 180


Cuttingforces(N)

Fx (N)

Fy (N)

Fz (N)

(b) Experimental cutting forces




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at a = 0.10mm,f= 0.1mm/r and v = 350m/min

-20

-10

0

10

20

30

40

50

60

70

80

0 45 90 135 180


Cutting

forces(N)

Fx (N)

Fy (N)

Fz (N)

-40

-20

0

20

40

60

80

100

0 45 90 135 180


Cutting

forces(N)

Fx (N)

Fy (N)

Fz (N)


(a) Estimated cutting forces (b) Experimental cutting forces



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a = 0.10mm,f= 0.1mm/r and v = 400m/min

-30

-20

-10

0

10

20

30

40

50

60

70

0 45 90 135 180


Cutting

forces(N)

Fx (N)

Fy (N)

Fz (N)

-20

-10

0

10

20

30

40

50

60

70

80

0 45 90 135 180


Cutting

forces(N)

Fx (N)

Fy (N)

Fz (N)


(a) Estimated cutting forces (b) Experimental cutting forces

Conclusions


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Analytical solution to undeformed chip thickness

is derived; JC model is used to describe the deformation

behavior of the workpiece material;

After FEM simulation, a new cutting force model

for high-speed milling of Ti-6Al-4V is proposed;

The cutting forces can be predicted withreasonable accuracy for all three directions.


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Thank you for your attention!!!

cutting force model

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