Economics of Metal Cutting 1

8/10/2019 Economics of Metal Cutting 1

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Analysis: Machining Economics

How can we optimizethe machining of a part ?

Identify the objective, formulate a model, solve for optimality

Typical objectives: maximum production rate, and/or minimum cost

Are these objectives compatible (satisfied simultaneously) ?

Formulating model: observations hypothesis theory model


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Analysis: Machining Economics..

Formulating model: observations hypothesis theory model

Observation:

A given machine, tool, workpiece combination has finite max MRR

Hypothesis:

Total volume to cut is minimumMaximum production rate

Model objective:

Find minimum volume stockfor a given part

-- Near-net shape stocks (use casting, forging, )

-- Minimum enclosing volumes of 3D shapes

Models:

- minimum enclosing cylinder for a rotational part

- minimum enclosing rectangular box for a milled part

Solving:

-- requires some knowledge of computational geometry


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Analysis: Machining Economics..

Model objective:

Find optimum operations plan and toolsfor a given part

Model: Process Planning

- Machining volume, tool selection, operations sequencing

Solving:

- in general, difficult to optimize

Example:

or

or

??


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Analysis: process parameters optimization

Model objective:Find optimum feed, cutting speedto [maximize MRR]/[minimize cost]/

Feed:

Higher feedhigher MRR

Fin ish cutting:

surface roughness feed

Given surface finish, we can find maximum allowed feed rate


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Process parameters optimization: feed

Rough cutting:

MRR cutting speed, VMRR feed, f

cannot increase V and f arbitrarily

V MRR;

Tool temperature V, f;

For a given increase in MRR: Vlower tool life than f

Optimum feed: maximum allowed for tool [given machine power, tool strength]


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Process parameters optimization: Speed

provided upper limits, but not optimum

Need a relationbetween tool life and cutting speed(other parameters being constant)

Model objective:

Given optimum feed, what is the optimumcutti ng speed

Taylors model (empirically based): V tn = constant


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One batch of large number,Nb, of identical parts

Replace tool by a new one whenever it is worn

Total non-productive time =Nbtltl= time to (load the stock + position the tool + unload the part)

Nbbe the total number of parts in the batch.

Total machining time =Nbtmtm= time to machine the part

Total tool change time =Nttctc= time to replace the worn tool with a new one

Nt= total number tools used to machine the entire batch.

Cost of each tool = Ct,

Cost per unit time for machine and operator =M.

Average cost per item:t

b

t

c

b

t

mlpr C

N

Nt

N

NMMtMtC


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Average cost per item: tb

t

cb

t

mlpr CN

N

tN

N

MMtMtC

Let: total length of the tool path =L

VLtm

1 MLVVLM

t= tool life Nt= (Nbtm)/t Nt/ Nb = tm/ t

Taylors model Vtn= C t = C 1/n/ V1/n = C/V1/n

C

VL

C

V

V

L

t

t

N

N nnnm

b

t/)1(/1


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Average cost per item: tb

t

cb

t

mlpr CN

N

tN

N

MMtMtC

1 MLVV

LM

C

VL

N

N nn

b

t

/)1(

nn

tclpr VCtMC

LMLVMtC /)1(1 )(


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nn

tclpr VCtMC

L

MLVMtC /)1(1

)(

nn

tc

prV

n

nCtMC

LMLVdV

dC /)21(2 )1()(0

Optimum speed (to minimize costs)

n

tc n

n

CtM

MCV

)1()(

*

Optimum speed (to minimize time)

cb

t

mlpr tN

N

ttt

Average time to produce part:


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Optimum speed (to minimize costs)n

tc nn

CtMMCV

)1()(*

Optimum speed (to minimize time)

c

b

t

mlpr tN

Nttt Average time to produce part:

load/unload time

machining timetool change time

V

L

tm

c

b

t

mlpr tN

Nttt

C

VL

N

N nn

b

t

/)1(

Substitute, differentiate, solve for V*

Economics of Metal Cutting 1

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