Study of the effective cutter radius for end milling of free-form surfaces using a torus milling cutter

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Study of the effective cutter radius for end milling of

free-form surfaces using a torus milling cutter

Jean-Max Redonnet, Sonia Djebali, Stéphane Segonds, Johanna Senatore,

Walter Rubio

To cite this version:

Jean-Max Redonnet, Sonia Djebali, Stéphane Segonds, Johanna Senatore, Walter Rubio. Study of the

effective cutter radius for end milling of free-form surfaces using a torus milling cutter.

Computer-Aided Design, Elsevier, 2013, 45 (6), pp.951-962. �10.1016/j.ad.2013.03.002�. �hal-01244681�

using a torus milling utter

Jean-Max REDONNET, Sonia DJEBALI, Stéphane SEGONDS,

JohannaSENATORE and Walter RUBIO

UniversitédeToulouse,Institut ClémentAder

UPS,118rtedeNarbonne,31062ToulouseFran e

Computer-AidedDesign,vol45(6), pp951-962,2013

DOI:10.1016/j. ad.2013.03 .00 2

Abstra t

Whenend millingfree-formsurfa es using atorus milling utter, thenotionof utteree tive radiusis

oftenusedtoaddressthepro edureforremovalofmaterialfromapurelygeometri alperspe tive. Using

anoriginalanalyti alapproa h,thepresentstudyestablishesarelationenablingthevalueofthisee tive

radius to be easily omputed. The limits of validity of this relation are then dis ussed and pre isely

dened.

Byway ofan illustration,an exampleofhowthis relation anbeused to generateanumeri altoolfor

analysisofthepossibilitiesforma hiningfree-formsurfa esonmulti-axisma hine-toolsisalsopresented.

Keywords: free-formsurfa e; CNCma hine-tool;end-mill; toroidal utter; ee tivetoolradius; swept

urve

Contents

1 Introdu tion 2

1.1 Previousworksontheee tive utterradius . . . 2

1.2 Thepresentarti le's ontribution . . . 2

2 Cal ulatingthe ee tiveradius 3 2.1 Introdu tion. . . 3

2.2 Demonstrationof lemma1. . . 3

2.3 Demonstrationof lemma2. . . 8

2.4 Cal ulatingtheee tiveradius . . . 9

3 Dis ussion 12 3.1 Analysisoftheexpressionofee tiveradius . . . 12

3.2 Studyintolimitsofvalidityoftheexpressionoftheee tiveradius. . . 12

4 Exampleof anappli ation 13 4.1 Introdu tion. . . 13

4.2 Relationbetweenstepoverdistan eandee tiveradius . . . 14

4.3 Comparisonmethodology . . . 15

4.4 Results. . . 15

End milling of free-form surfa es is essentially used

tomanufa turemouldsanddieswhereitisoften

ex-tremely ostlyintermsofprodu tiontime onsumed.

From apurely geometri al standpoint, pre ise

mod-elling of the movements of the utter and its

posi-tioning in relation to the surfa e are indispensable

to be able to propose improvements to boost

pro-du tivity. From this perspe tive, an approa h

om-monlyadoptedisbasedonnotionsofee tiveradius

and/orsweep urve. Indeed,goodknowledgeofthese

geometri alentitiespavesthewayforapre ise

anal-ysis of the tra e left by the utter in theworkpie e

and through that, the quantity of material a tually

removed.

1.1 Previous works on the ee tive

ut-ter radius

Toolpath planning, optimisation of utter

position-ingandnon-interferen eissuesareoftenthefo usof

resear h ondu ted in the eld of free-form surfa e

ma hining on multi-axis ma hine tools [1℄. Among

theseworks, manystudiesrefer tothe notionof

ut-ter ee tive radius. The rst to introdu e this

on- ept were Vi kers and Quan in 1989. In [2℄, they

show how a at-end mill tilted to the front an be

moreprodu tivethanaball-endmill. Todoso,they

introdu e thenotionofee tiveradiusinthe aseof

theat-end mill:

R

ef f

=

R

sin(φ)

where

R

is the utter radius and

φ

its tilt angle in the plane formed by its feed dire tion and axis of

rotation.

The relative e ien y of at-end mills and

ball-end mills is also analysed in [3℄, [4℄ and [5℄. These

worksare also basedon theee tive radius on ept

to showthat, all otherparameters being equal,

at-end utters, when orre tly used, produ e a lower

s allop height than that produ ed by ball-end

ut-ters. In [4℄ and [5℄, the authors also show that at

end mills leave pronoun ed marks in thefeed

dire -tion leading to a greater roughness of the surfa es

obtained (forthesame feed pertooth).

Following theseworksa numberof authors have

arguedinfavour ofusing torus utterswhenmilling

free-formsurfa es. Indeed,torusmills allowa

signif-i ant ee tive radius to be retained while avoiding

byat-endmills[6℄. Manystudiesarriveatthesame

on lusions,whethertheyadoptapro edureto

opti-misethe utter position[7,8℄or seekrather to

elim-inateinterferen e [9,10℄.

Amongtheworksthataddressthe utteree tive

radius on ept,thosemostfrequentlyen ounteredin

theliteratureutilisetheenvelope urve on ept. For

a given utter position, the envelope urve

materi-alises the tra e left by the utter in the material.

In[11℄,itisapproximated,foratorusmilling utter,

bytheproje tionofa ir leinaplanenormal tothe

feed. In [12℄, it is given in the impli it form for an

APT utter.

Within the s ope of ma hining simulation [13℄,

manystudies usethis on ept to determine the

vol-ume of swarf a tually removed by the utter, but

most of these works [14 16℄ address this issue

nu-meri ally,whi hdoesindeedallowthesweptvolume

to be omputed, but pre ludes an analyti al study

of the ee tive utter radius. The sweep urve and

ee tive utter radius notions are also largely used

inworksaddressing onstant s allopheight

ma hin-ingplanning. Thistoolpath planning te hnique was

initiallyintrodu ed in [17℄ and [18℄ using a ball-end

utter. Subsequently it was adapted for a at-end

mill [19,20℄ and for thetorus milling utter [21,22℄,

tools for whi h the ee tive radius assumes its full

signi an e.

Analysisofthemainstudiespublishedintheeld

shows that most works overing the ee tive radius

ofthetorus milling utterrelyongeometri

approx-imations (with non-negligible onsequen es) or use

a numeri al approa h that, ompared with an

ana-lyti alapproa h,provesto belessexible andmu h

moretime- onsuming in omputation.

1.2 The present arti le's ontribution

Thepresentarti le willintrodu eanewstudyofthe

torus milling utter ee tive radius. Its originality

liesin its totally analyti al approa h that

neverthe-lessrefrainsfromanygeometri approximation. The

mainresultofthisworkisthedenitionofarelation

authorizing an analyti al al ulation ofthe ee tive

utter radius.

Thisstudyis alsoa ompaniedbyananalysisof

this relation and its limits, thus allowing the s ope

for itsvalidity to be learly determined.

i al tools potentially useful within the s ope of end

milling of free-form surfa es on multi-axis ma hine

tools. The aim with this example is not to dene

a omplete pro edure to plan tool paths, but

sim-plytoemphasisethepossibilitiesoeredbyusingan

analyti al formula where numeri al pro edures are

usually applied.

Thearti le on ludeswithareminderofthemain

results obtained and some remarks on forth oming

workson thissubje t.

2 Cal ulating the ee tive radius

2.1 Introdu tion

It will be shown how it is possible to al ulate

an-alyti ally, at the point of utter/workpie e onta t,

theee tiveradius ofa torusmilling utter

ma hin-ing a free-form surfa e on a multi-axis NC ma hine

tool. This al ulationis basedon two mathemati al

demonstrations thatwill be introdu ed prior to the

omputation itself.

A torus milling utter dened by

R

and

r

,

R

being theouterradius ofthat utter and

r

beingits torus radius, is onsidered. The tra e left by that

utter in the material at a given instant is a urve

that will be referred to as the envelope urve. It is

thesu essionofsu henvelope urvesthatformsthe

envelope surfa e generated by the utter movement

inthematerial. At ea hinstant, theenvelope urve

isdened by

F

t

· n = 0

, where

F

t

isa ve tor in the utter feed dire tion and

n

a ve tor normal to the surfa e ofthe utter.

In what follows in the present study, the ve tor

F

t

will be assumed to be onstant for all points of the utter; this is equivalent to asserting that the

utter moves in translation, at least lo ally.

More-over,onlythepartoftheenvelope urveofthe utter

ontained inthetoruspartofthe utterwill be

on-sidered. Indeed, thegreat majority of torus milling

uttersusedinindustry areround insert uttersand

only thatpart is a tive. Also, studying theparts of

the envelope urve ontained in the ylindri al and

dis oid portions of the utter isunproblemati and,

even in the ase of solid torus milling utters, these

parts of the utter are normally ina tive when

re-moving material, espe iallywhen ondu ting

nish-ingoperations.

areasfollows:

Lemma 1 Let

P

be the mathemati aloperation for

proje tionalong thefeed dire tion

F

t

inaplane nor-malto

F

t

. Let

T

p

(v)

,bethe urveresultingfromthe proje tion along

P

of the utter envelope. Let

E(t)

be the ellipse resulting from the proje tion along

P

of the utter entre-torus ir le, and

oE(t)

an oset exterior to that ellipse with a value equal to the

ra-diusof the utter torus. Then the two urves

T

p

(v)

and

oE(t)

are oin ident.

Lemma 2 The radius of urvature of a plane

o-set urve is equal to the radius of urvature of the

original urveaugmented by the oset value.

Itwillthusbeshowninitiallythattheproje tion

ofthe utterenvelope urveinaplanenormaltothe

feed dire tion

F

t

an be dened by an ellipse aug-mentedbyanoset equal to the utter torus radius

(se tion2.2).

Itwillthenbeshownthattheradiusof urvature

of an oset to this ellipse is equal to the radius of

urvature of the original ellipse augmented by the

osetvalue (se tion2.3).

Based on these results, it will then be possible

to al ulate analyti ally the ee tive radius of the

utter

R

ef f

onsidering the radius of urvature of the ellipse

E(t)

to whi h is added the utter torus radius the utter(se tion 2.4).

All these al ulations were veried using the

al-gebrai omputation softwareMaxima [23℄.

2.2 Demonstration of lemma 1

2.2.1 Statement of the problem

Firstlytheproje tionofthe utterenvelope urvein

aplanenormalto

F

t

is onsidered, thenanosetby

r

oftheellipsedened bytheproje tionofthetorus major radius ir leof the utter ( entreof thetorus

tube)in thesameplane (Fig. 1 and2).

Thepurposeofthisdemonstrationistoshowthat

thesetwo urves oin ide.

2.2.2 Denitions

Naming

R

t

theradiusofthe uttertorus entre ir le (

R

t

= R − r

), thetoroid surfa e dening the utter

in itsreferen e frame an bedened by:

T(u, v) =





(R

t

+ r cos(v)) cos(u)

(R

t

+ r cos(v)) sin(u)

r sin(v)





(1) with

u ∈ [0, 2π]

and

v ∈

−

π

2

, 0



Let

F

,beaunitve torinthema hiningdire tion

F

t

:

F

=

F

t

kF

t

k

Thetra e leftbythe utter(envelope urve) an

thenbedened by

F

· n = 0

,where

n

isthe normal to the utter surfa e.

Consider theproje tion along thefeed dire tion

F

inaplane

P

perpendi ularto

F

. Naming

a

,

b

and

c

the oordinates of

F

, theplane

P

is expressed by equation:

a x + b y + c z = d

with

d ∈ R

C(t)

C

p

(t)

S(t, w)

P

x

y

z

F

Figure1: Proje tionofaparametri urveinaplane

Let

C(t)

be a urve dened in three dimensions by:

C(t) =





C

x

(t)

C

y

(t)

C

z

(t)





The urve

C

p

(t)

resultingfromtheproje tion of

C(t)

in

P

alongthedire tion

F

isthendenedbythe interse tion of the plane

P

and the surfa e dened by

S(t, w) = C(t) + f (w) F

where

f (w)

is a s alar

fun tion of the parameter

w

dened in

[−∞, +∞]

(Fig. 1). This surfa e is the ruled surfa e dened

from

C(t)

and

F

. Theproje ted urve

C

p

(t)

isthus dened bythesystem:















a x + b y + c z = d

x = C

x

(t) + a f (w)

y = C

y

(t) + b f (w)

z = C

z

(t) + c f (w)

where

x

,

y

and

z

represent thethree oordinates of the urve

C

p

(t)

.

Resolving this system in relation to

x

,

y

,

z

and

f (w)

,theexpressionofthese oordinatesisobtained asa fun tion of

t

thatwill be referred to as

C

px

(t)

,

C

py

(t)

and

C

pz

(t)

:















C

px

(t) =

−a c C

z

(t)−a b C

y

(t)+c

2

C

x

(t)+b

2

C

x

(t)+a d

c

2

+b

2

+a

2

C

py

(t) =

−b c C

z

(t)+c

2

C

y

(t)+a

2

C

y

(t)−a b C

x

(t)+b d

c

2

_+b

2

_+a

2

C

pz

(t) =

b

2

C

z

(t)+a

2

C

z

(t)+c (−b C

y

(t)−a C

x

(t))+c d

c

2

_+b

2

_+a

2

asalso

f (w) =

−c C

z

(t) − b C

y

(t) − a C

x

(t) + d

c

2

_{+ b}

2

_{+ a}

2

Giventhattheve tor

F

isunitary,thisgives

a

2

₊

b

2

+ c

2

= 1

, when e the equation for the proje ted urve:

C

p

(t) =





−a c C

z

(t) − a b C

y

(t) + c

2

C

x

(t) + b

2

C

x

(t) + a d

−b c C

z

(t) + c

2

C

y

(t) + a

2

C

y

(t) − a b C

x

(t) + b d

b

2

C

z

(t) + a

2

C

z

(t) + c (−b C

y

(t) − a C

x

(t)) + c d





(2) 2.2.3 Contextualisation

Within the s ope of the present study,the utter is

dened within its own referen e frame, theaxis

o-in iding with its axis of rotation. As the utter is

a surfa e of revolution, whatever the movement of

translation driving it, the envelope urve resulting

planeofsymmetry ontainingthe axis

z

ofthe refer-en eframe. Furthermore, a proje tion

P

ina plane normalto thefeed

F

is onsidered. Thisproje tion thus orresponds to a ve tor ontained within the

planeofsymmetryof the envelope urve. The

prob-lem isthus axisymmetri . Consequently, theresults

obtainedinthe aseofaparti ularproje tion(i.e. in

a given radial dire tion) are true whateverthe

pro-je tion

P

onsidered, meaning whatever translation movement drives the utter. It an thus be

onsid-ered that the results obtained in the ase of a

pro-je tion along a ve tor ontained inthe plane

x = 0

(or

y = 0

) an beextended tothegeneral ase.

A proje tion

P

is hosen whose dire tion

F

is ontained in the plane of equation

x = 0

, with o-ordinate

a

of

F

thus being null. The plane

P

nor-mal to this proje tion will then have for equation

b y + c z = d

and the oordinates of ve tor

F

are:

F

=





0

b

c





Inwhatfollowsinthepresentdemonstration,

b 6=

0

and

c 6= 0

will be onsidered. Indeed, instan es

where

b = 0

and

c = 0

orrespond to horizontal

or verti al utter pathsthat onstitute spe ial ases

thatwill be addressedinse tion 3.2.

Furthermore, as ve tor

F

is unitary, it an be assertedthat

b

2

_{+ c}

2

_{= 1}

.

Also,insofarasthefo usison urvesproje ted

orthogonally ina plane normalto the milling

dire -tion, anyplane normal tothatdire tion an be

ho-sen without impairing generality. To simplify

om-putation, a plane

P

passing through the origin is hosen, that is a plane with equation

b y + c z = 0

. Thisgives

d = 0

.

Takingthese onsiderationsintoa ount,the

equa-tion (2) for a urve transformed along proje tion

P

be omes:

C

p

(t) =





C

x

(t)

−b c C

z

(t) + c

2

C

y

(t)

b

2

C

z

(t) − b c C

y

(t)





(3)

In the following demonstration, will be

onsid-ered the proje tion

P

, dened by equation (3) en-ablinga urvetobeproje tedinaplane

P

a ording toave tor

F

,withplane

P

goingthroughtheorigin and being normalto the ve tor

F

that is ontained intheplaneof equation

x = 0

.

2.2.4 Demonstration

First of all,theproje tion

P

isapplied to the ir le

C(t)

,the uttertorus entre,denedbytheequation

C(t) =





R

t

cos(t)

R

t

sin(t)

0





with

t ∈ [0, 2π]

Using(3),theorthogonalproje tionofthat ir le

anbedened intheplaneofequation

b y + c z = 0

. This proje tion is an ellipse that will be referred to

as

E(t)

,andwhose equation is:

E(t) =





R

t

cos(t)

c

2

R

t

sin(t)

−b c R

t

sin(t)





Inwhatfollows,onlythelowerpartoftheellipse

E(t)

will be onsidered, that is the part dened by

t ∈ [−π, 0]

(Fig. 2).

The unit ve tor

nE(t)

normal to

E(t)

and on-tainedin theplane

P

an thenbe dened by:

nE(t) =

dE(t)

dt

× F

dE(t)

dt

× F

thatis

nE(t) =

1

pb

2

_sin

2

_{(t) + c}

2





c cos(t)

c sin(t)

−b sin(t)





(4)

Given thepreviously established restri tions (

c 6= 0

and

b 6= 0

), this expression is dened whatever

t ∈

[−π, 0]

.

In the plane

P

,

oE(t)

is dened, an oset with value

r

to theellipse

E(t)

:

oE(t) = E(t) + r nE(t)

This urve isexpressedasfollows:

oE(t) =











cos(t) R

t

+

√

c r cos(t)

b

2

_sin

2

(t)+c

2

c

2

sin(t) R

t

+

√

c r sin(t)

b

2

_sin

2

(t)+c

2

−b c sin(t) R

t

−

√

b r sin(t)

b

2

_sin

2

(t)+c

2











(5)

Se ondly,theenvelope urveis onsidered,thatis

thetra eleftbythe utter inthematerialatagiven

view onplane

(y, z)

z

y

r

R

R

t

P

T

r

(v)

F

view onplane

P

E(t)

oE(t)

T

p

(v)

Figure 2: Proje tion oftheenvelope urve andthe torus entre ir le inaplane normal to

F

is given by the equation (1), this envelope urve is

dened by

F

· nT(u, v) = 0

, where

nT(u, v)

is a ve tor normalto

T(u, v)

.

nT(u, v) =

∂T(u, v)

∂u

×

∂T(u, v)

∂v

thatis

nT(u, v) =





r cos(u) cos(v) (R

t

+ r cos(v))

r sin(u) cos(v) (R

t

+ r cos(v))

r sin(v) (R

t

+ r cos(v))





with

u ∈ [0, π]

and

v ∈

−

π

2

, 0



The equation

F

· nT(u, v) = 0

an then be ex-pressed

r (c sin(v) + b sin(u) cos(v)) (R

t

+ r cos(v)) = 0

(6)

when e it an be dedu edthat

sin (u) = −

c sin (v)

_{b cos (v)}

(7)

for

v ∈

−

π

2

, 0



. Se tion 3.2 adresses the ase where

v = −

π

2

. In what follows in the demonstration, it willbe onsideredthat

−

π

2

< v 6 0

.

Usingthisrelation(7)intheexpressionof

T(u, v)

equation(1)theequationoftheenvelope urve,

referredto as

T

r

(v)

,isobtained:

T

r

(v) =









q

1 −

c

b

2

2

_cos

sin

2

2

(v)

_(v)

(R

t

+ r cos(v))

−

c sin(v) (R

t

+r cos(v))

b cos(v)

r sin(v)









for

u ∈

0,

π

2



,and

T

r

(v) =









−

q

1 −

b

c

2

2

_cos

sin

2

2

(v)

_(v)

(R

t

+ r cos(v))

−

c sin(v) (R

t

+r cos(v))

b cos(v)

r sin(v)









for

u ∈



_π

2

, π



. Asthe urve

T

r

(v)

issymmetri alin relationtotheplaneofequation

x = 0

orresponding

to parameter

u =

π

2

, only the part dened by

0 6

u 6

π

₂

will be onsidered in what follows, with the samereasoningbeingappli ablebysymmetryforthe

partdened by

π

2

6

u 6 π

.

As previously, using (3) the proje tion of that

urve anbedened intheplane

P

. Thusthe urve

T

p

(v) =









q

1 −

_b

c

2

2

sin

2

(v)

cos

2

(v)

(R

t

+ r cos(v))

−

c

3

sin(v) (R

t

+r cos(v))

b cos(v)

− b c r sin(v)

c

2

sin(v) (R

t

+r cos(v))

cos(v)

+ b

2

r sin(v)









(8)

Theproblemposed anthusberedu edto

show-ingthat

oE(t) = T

p

(v)

.

Byidentifyingthe oordinatesofthesetwo urves

(equations5and8)memberbymember,3equations

areobtained:

cos(t) R

t

+

c r cos(t)

pb

2

_sin

2

_{(t) + c}

2

=

s

1 −

c

2

sin

2

_(v)

b

2

_cos

2

_(v)

(R

t

+ r cos(v))

(9)

c

2

sin(t) R

t

+

c r sin(t)

pb

2

_sin

2

_{(t) + c}

2

= −

c

3

_{sin(v) (R}

t

+ r cos(v))

b cos(v)

− b c r sin(v)

(10)

−b c sin(t) R

t

−

b r sin(t)

pb

2

_sin

2

_{(t) + c}

2

=

c

2

sin(v) (R

t

+ r cos(v))

cos(v)

+ b

2

_{r sin(v)}

(11)

Analysingtheseequations,it learlyemergesthat

the last two, (10) and (11), are equivalent. Indeed,

by multiplying ea h term of equation (10) by

−b/c

, equation(11) isobtained.

To show that the two urves are equal, all one

needstodo isnda hangeinvariablelinking

t

and

v

su hthattheequationofoneofthetwo urves an be transformedinto the equationof theother urve.

Todoso,thetermson

R

t

and

r

between therst equation(9) andone of the two others (forexample

(10))are identiedmemberbymember.

From equation (9), identifying the terms on

R

t

, thefollowing isobtained:

cos(t) =

s

1 −

c

2

sin

2

_(v)

b

2

_cos

2

_(v)

(12) andidentifyingtheterms on

r

,this gives:

c cos (t)

pb

2

_sin

2

_{(t) + c}

2

=

s

1 −

c

2

sin

2

_(v)

b

2

_cos

2

_(v)

cos (v)

(13) whi h,after simpli ation (see 5), results in

return-ingto equation(5):

cos(t) =

s

1 −

c

2

_sin

2

_(v)

b

2

_cos

2

_(v)

From equation(10),identifying thetermson

R

t

, thefollowing isobtained:

c

2

_{sin(t) = −}

c

3

_sin(v)

b cos(v)

when e it an be dedu ed

sin(t) = −

c sin(v)

b cos(v)

1 − cos

2

(t) =

c

2

_sin

2

_(v)

b

2

_cos

2

_(v)

whi hresults inreturningto equation (12):

cos(t) =

s

1 −

c

2

_sin

2

_(v)

b

2

_cos

2

_(v)

From this same equation (10), identifying the

termson

r

,thefollowing isobtained:

c sin(t)

pb

2

_sin

2

_{(t) + c}

2

= −

 c

3

b

sin(v) + b c sin(v)



(14)

whi h, after simpli ation (see 5), again gives the

equation(12):

cos(t) =

s

1 −

c

2

_sin

2

_(v)

b

2

_cos

2

_(v)

Identi ationof thetermson

R

t

and

r

for equa-tions(9)and(10)thusleadstothesamerelation(12)

linking parameters

t

and

v

. Using this relation asa hange in variable, it proves possible to pass from

equation(5)to equation(8). There isthusa hange

invariable to go from one urve to theother. This

leadsto on ludingthat urves

T

p

(v)

and

oE(t)

o-in ide.

It was shown that in the ase of a proje tion

P

in a planegoing through the originand along a ve tor

F

ontained intheplane ofequation

x = 0

,thetwo urves

T

p

(v)

and

oE(t)

oin ide. As the problem isaxisymmetri , what istruein thisinstan e isalso

truewhatevertheplaneofproje tion, providedthat

theproje tionismadealonganormaltothatplane.

Inthe ase ofa translationmovement, the urve

re-sultingfromthe proje tion planeofthetra e leftby

the utter thus oin ides with the urve parallel to

the ellipse, lo ated at a distan e

r

outside the lat-terthatitselfresultsfromtheproje tionofthetorus

entre ir le.

2.3 Demonstration of lemma 2

This demonstration's obje tive is to show that the

radius of urvature of a plane oset urve is equal

to theradius of urvatureof the original urve

aug-mentedbythevalue ofthe oset.

Let

C

beaplane urve whoseparameters areset by its urvilinear abs issa

s

. Let

C

o

be an oset urve derived from

C

:

C

o

= C + r n

where

r

isthes alarvalueoftheosetand

n

theunit normalto

C

orientedtowardsthe entreof urvature. Deriving the previous expression in relation to

the urvilinear abs issa

s

,the following isobtained:

dC

o

ds

=

dC

ds

+ r

dn

ds

where

dC

ds

istheunitve tor

t

,tangent to

C

. Calling

s

o

the urvilinear abs issa of the urve

C

o

one obtains:

dC

o

ds

o

ds

o

ds

= t + r

dn

ds

Frenet formulae give:

dn

ds

= τ b − κ t

where

κ

isthe urvatureof

C

atthepoint onsidered. As the urve

C

is plane, twisting

τ

is null and thus:

dn

ds

= −κ t

dC

o

ds

o

ds

o

ds

= t − r κ t = (1 − r κ) t

(15) By denition

dC

o

ds

o

is the unit ve tor tangent to

C

o

. As the urve

C

o

is the oset of

C

,for a given valueof

s

,both urveshavethesametangent. Thus

dC

o

ds

o

= t

When e, in(15):

t

ds

o

ds

= (1 − r κ) t

whi h an besimplied:

ds

o

ds

= 1 − r κ

(16)

Moreover, theFrenet formulae give:

dt

ds

o

= κ

o

n

when e

dt

ds

o

=

dt

ds

ds

ds

o

or

dt

ds

= κ n

and,a ording to (16)

ds

ds

o

=

1

1 − r κ

Thus

κ

o

n

= κ n

1

1 − r κ

When e it an bededu ed that:

κ

o

=

κ

1 − r κ

(17)

Let

ρ

be theradius of urvature of

C

and

ρ

o

the radius of urvature of

C

o

. These magnitudes are relatedto urvatures

κ

and

κ

o

by:

ρ =

1

κ

et

ρ

o

=

1

κ

o

When e, in(17):

1

ρ

o

=

1

ρ

1 −

r

ρ

whi h an be expressed:

1

ρ

o

=

1

ρ − r

andnally:

ρ

o

= ρ − r

Inthe aseofaplane urve

C

,theradius of ur-vature of an oset to

C

is equal to the radius of urvatureof

C

redu edbythealgebrai valueof the oset. It anthereforebe on ludedthatinthe ase

ofanosetremoterfromthe entreof urvaturethan

theoriginal urve,the radiusof urvatureofthe

o-set urve will be equal to the radius of urvature of

the initial urve in reased by the absolute value of

theoset.

2.4 Cal ulating the ee tive radius

During milling (Fig. 3), the utter axis oin ides

withthe

z

-axisofthe referen eframeand the

x

and

y

axesofthereferen eframearesetaspreviously(see se tion2.2.3), i.e. the feed dire tion

F

t

is ontained intheplane

x = 0

.

Furthermore, let

D

be the unit ve tor ontained intheplane

(x, y)

thatindi atesthe dire tionof the greatest slope at the utter/workpie e point of

on-ta t (point

C

c

). The normal to the surfa e at that point (

n

cc

) isthen ontained inthe plane

(D, z)

.

O

z

D

r

R

R

t

v

n

cc

C

c

S

Figure 3: Dening angle

S

S

designates the slope of the surfa e ma hined at the utter/workpie e point of onta t

C

c

. This angle

S

is ontained in the plane

(D, z)

. Only the

ase where

S 6= 0

will be onsidered in the present al ulation. Indeed,milling ina dire tion ontained

withintheplane

(x, y)

onstitutesaspe ial asethat willbe addressedinse tion3.2. Thus

S > 0

.

Thisangle

S

isalsothatbetweenthe utteraxis

z

andthenormaltothesurfa eatthepointof onta t

n

cc

. Theve tor

n

cc

anbe expressedin theform:

n

cc

= − sin(S) D + cos(S) z

(18)

α

willdesignatetheangleseparatingve tor

D

of the

y

-axis. Thefollowing al ulationswillbelimited tothe asewhere

−

π

2

< α <

π

2

. Where

α = ±

π

2

,the dire tion withthegreatest slope

D

isperpendi ular to the feed dire tion

F

. Thesevalues orrespond to spe ial asesthat will be studied inse tion3.2.

Inthereferen eframe

(O, x, y)

theve tor

D

an be expressed:

D

= sin(α) x + cos(α) y

(19) Thus,inthereferen eframe

(O, x, y, z)

,the ve -tor

n

cc

anbe expressed:

n

cc

=





− sin(S) sin(α)

− sin(S) cos(α)

cos(S)





(20)

Besides,for ea h point

C

c

,theve tor

F

is deter-mined su h that it belongs to the plane tangent to

the surfa e at this point (Fig. 4). In the referen e

frame

(O, x, y, z)

,

F

anbe expressedin theform :

F

=





0

cos(ψ)

sin(ψ)





(21)

where

ψ

designates the angle formed by ve tor

F

and its proje tion in the plane

(x, y)

. This angle is ontained intheplane madebyve tors

z

and

y

.

Toobtainadispla ementofthe uttertangentto

thesurfa e at the utter/workpie e point of onta t

(point

C

c

),the urvedenedbythetra eleftbythe utter in the material, referred to as the envelope

urve, veries theequation

F

· n

cc

= 0

. Using (20) and(21) inthisequation, thefollowing isobtained:

− cos(ψ) sin(S) cos(α) + sin(ψ) cos(S) = 0

Thisgives, for ea h utter onta t point

view onplane

P

normal to

F

y

x

O

D

y

z

F

t

F

ψ

x

E

y

E

R

t

C

c

C

c

E

cc

F

α

E(t)

Figure4: Denition ofelements usedto al ulate theee tive radius

R

ef f

Infurther al ulations, only the ases where

0 <

ψ <

π

₂

are onsidered. Caseswhere

ψ = 0

and

ψ =

π

2

orrespond to spe ial instan es that will be studied

in se tion 3.2 (indeed, for

ψ = 0

the utter moves horizontally and for

ψ =

π

2

,itmovesverti ally).

Inits ownreferen e

(O; x

E

, y

E

)

,anellipse is de-nedbytheparametri equation:

E(t) =





µ cos(t)

η sin(t)

0





(23)

wherevalues

µ

and

η

representrespe tivelythe semi-major axis and the semi-minor axis of the ellipse

E(t)

.

Inthe aseoftheellipseresultingfromproje tion

of the torus entre ir le of radius

R

t

, in a plane normalto

F

(Fig. 4),values

µ

and

η

aredened by:



µ = R

t

η = R

t

sin(ψ)

(24)

Using thefa tthat:

sin(ψ) =

tan(ψ)

p1 + tan

2

_(ψ)

and equation(22) inequation (24), thevalue ofthe

semi-minor axis

η

an be expressedby:

η = R

t

tan(S) cos(α)

p1 + tan

2

_{(S) cos}

2

_(α)

(25)

In its ownplane, theellipse

E(t)

is thus dened by:

E(t) =







R

t

cos(t)

R

t

√

tan(S) cos(α)

1+tan

2

(S) cos

2

(α)

sin(t)

0







(26)

C(t)

isdened by:

ρ

C

=

dC(t)

dt

3

dC(t)

dt

×

d

2

C(t)

dt

2

Fromequation(23), the following anbe al ulated:

dE(t)

dt

=





−µ sin(t)

η cos(t)

0





and

d

2

E(t)

dt

2

=





−µ cos(t)

−η sin(t)

0





Theradiusof urvatureoftheellipse

E(t)

isthus equal to:

ρ

E

=

µ

2

sin

2

(t) + η

2

cos

2

(t)

3

/

2

η µ

and is only dened for

η 6= 0

and

µ 6= 0

. Now,

a ording to equations (24),

µ = 0

implies

R

t

= 0

,

whi h annot be and

η = 0

implies

ψ = 0

. The

asewhere

ψ

isnull orrespondsto ma hininginthe plane

(x, y)

(Fig. 4) whi h is equivalent to saying that

S = 0

(Fig. 3) or

α = ±

π

2

( f. equation (22)); now, ashasalreadybeenstated,theseinstan es will

be analysedinse tion 3.2.

Theradiusof urvatureoftheellipseisthusgiven

by:

ρ

E

=

µ

2

_{1 − cos}

2

(t)
+ η

2

_cos

2

_(t)

3

/

2

η µ

=

µ

2

η



1 − cos

2

(t) +

η

2

µ

2

cos

2

_(t)



3

/

2

=

µ

2

η



1 + cos

2

(t)

 η

2

µ

2

− 1



3/2

(27)

Equation(27) anbeusedto al ulatetheradius

of urvatureoftheellipse

E(t)

asafun tionofthe pa-rameter

t

ofthat urve. Let

E

cc

bethepointofthat ellipse orrespondingtothepointof onta t

C

c

(Fig. 4). To determine theradius of urvature of

E(t)

at point

E

cc

, the value of parameter

t

at that point has to be known. To nd it, equation (12) is used

again. In this equation

b

and

c

are the oordinates of the unit feed ve tor

F

and an be expressed by

c = sin(ψ)

and

b = cos(ψ)

(see equation21 andFig. 4). In addition,at point

C

c

,thevalueof parameter

v

isgiven by

v = −

π

2

+ S

(Fig. 3), when e it anbe dedu edthat

sin(v) = − cos(S)

and

cos(v) = sin(S)

. Applying these onsiderations to equation (12), the

following is obtained:

cos(t) =

s

1 −

sin

2

_{(ψ) cos}

2

_(S)

cos

2

_{(ψ) sin}

2

_(S)

when e ittranspiresnaturally that:

cos(t) =

s

1 −

tan

2

_(ψ)

tan

2

_(S)

Now, a ordingto equation(22),

tan(ψ)

tan(S)

= cos(α)

It an thereforebe onrmed that at point

E

cc

, thereis

cos(t) = sin(α)

. Usingthisresultinequation (27)givingtheradius of urvatureof theellipse,the

radiusof urvature

ρ

E

atthatpoint anbeexpressed:

ρ

E

=

µ

2

η



1 + sin

2

(α)

 η

2

µ

2

− 1



3/2

Usingexpressionsof

µ

and

η

establishedin equa-tions (24) and (25), theexpression of

ρ

E

be omes:

ρ

E

=

R

2

_t

R

t

tan(S) cos(α)

√

1+tan

2

_{(S) cos}

2

_(α)











1 + sin

2

(α)













R

t

tan(S) cos(α)

√

1+tan

2

_{(S) cos}

2

_(α)



2

R

_t

2

− 1





















3

_/

₂

=

R

t

p1 + tan

2

_{(S) cos}

2

_(α)

tan(S) cos(α)

1 + sin

2

_(α)



tan

2

(S) cos

2

(α)

1 + tan

2

_{(S) cos}

2

_(α)

− 1

ρ

E

=

R

t

p1 + tan

2

(S) cos

2

(α)

tan(S) cos(α)



1 −

sin

2

_(α)

1 + tan

2

_{(S) cos}

2

_(α)



3

/

2

=

R

t

tan(S) cos(α)

tan

2

(S) cos

2

(α) + cos

2

(α)

3

/

2

1 + tan

2

_{(S) cos}

2

_(α)

=

R

t

cos

2

_{(α) 1 + tan}

2

_(S)

3

/

2

tan(S) (1 + tan

2

(S) cos

2

_(α))

Given that

1 + tan

2

_{(S) =}

1

cos

2

_(S)

, it an stated that:

ρ

E

=

R

t

cos

2

(α)

cos

3

_{(S) tan(S) (1 + tan}

2

_{(S) cos}

2

_(α))

Also, given therestri tions establishedon

α

and

S

(that is

−

π

2

< α <

π

2

and

S 6= 0

), this expression an be simpliedas:

ρ

E

=

R

t

cos

2

(α)

cos

2

_{(α) sin}

3

_{(S) + cos}

2

_{(S) sin(S)}

(28) or again:

ρ

E

=

R

t

cos

2

(α)

sin(S) 1 − sin

2

(α) sin

2

(S)

(29) Thisexpression allows theradius of urvatureof

the ellipse

E(t)

resulting from the proje tion of the torusmajorradius ir leofthe utterinaplane

nor-mal to the feed dire tion to be al ulated, and this

forthetorus entrepointofthat urve orresponding

to thepoint of onta twiththe ma hined surfa e.

Based on lemma 2 applied to theellipse, the

ra-diusof urvature

R

ef f

on

C

c

an be expressedby:

R

ef f

=

(R − r) cos

2

_(α)

sin(S) 1 − sin

2

(α) sin

2

(S)

+ r

(30) This expression an be used to al ulate the

ef-fe tive radius of the utter at the utter/workpie e

point of onta t inthe ase of end millingof a

free-form surfa e with a torus milling utter moving in

translationon amulti-axis CNC ma hine.

3 Dis ussion

3.1 Analysisofthe expression ofee tive

radius

Inrelation (30), angle

α

, hara terising the ma hin-ingdire tionproje tedintheplane

(x, y)

,onlyenters into expressions

cos

2

_(α)

and

sin

2

_(α)

. It anthus be

asserted that all other parameters being equal, the

value of theee tiveradius isthesame for values

α

and

α + π

. This is equivalent to saying that for a given point,thevaluesoftheee tiveradius arethe

samewhetherupmillingor limbmillingina

diamet-ri allyoppositedire tion. Thisresultisunsurprising

insofarasthestudy oftheee tive radius isbased

on a proje tion ina planenormal to thema hining

dire tion.

Moreover,analysisoftherelation(30)showsthat

for

α = ±

π

2

,

R

ef f

= r

obtains, whi h onstitutes theminimumvalueoftheee tiveradius foratorus

milling utter. Itsmaximumvalue,whi his

theoret-i ally innite (horizontal ma hining) is approa hed

when

α

tends towards

0

and when

S

tends towards

0

.

3.2 Studyintolimitsofvalidityofthe

ex-pression of the ee tive radius

Relation(30)aordsananalyti al al ulationofthe

ee tiveradiusofthe utteratthe utter/workpie e

point of onta t whenma hining afree-form surfa e

with a torus milling utter. It should, however, be

re alled here what pre isely is theframework of

va-lidity for this relation. Firstly, this relation is only

valid at the utter/workpie e point of onta t.

In-deed,manyrelationsestablishedduring omputation

bounded by the hypotheses adopted during

al u-lation. The most restri tive of these hypotheses is

thatthefeedve tor

F

is onstantatanypointofthe utter. Asstated previously,thismeans thatlo ally

at least, the utter is moved by simple translation.

Appli ation-wise, this is always true on 3-axis NC

ma hines. For 4-and 5-axisNCma hines,thismay

be true for portions of the paths but this relation

annot be used systemati ally. In parti ular, when

the axes of rotation of the ma hine are a tivated,

therelativemovementofthe utterinrelationtothe

workpie e omprisesatranslationandarotation. In

this ase, thefeedrate annotberepresentedbythe

same ve tor

F

forall pointsofthe utter.

Duringthedemonstration,restri tionswerestated

as to the value of omponents

b

and

c

of the feed ve tor

F

(se tion 2.2.3), on the value of parameter

v

(se tion 2.2.4) and the values of angles

S

,

α

and

ψ

(se tion 2.4). Thus, relation (30) is only demon-strable if

b 6= 0

,

c 6= 0

,

v 6= −

π

2

,

S 6= 0

,

α 6= ±

π

2

,

ψ 6= 0

and

ψ 6=

π

2

. Analysis of the mathemati al and te hnologi al ontext shows that these

dier-ent ex eptional ases overlap. Indeed, these spe ial

ases orrespondtoquitespe i ma hining

ongu-rations. Ea hofthesespe ial ongurationswillnow

beanalysed,bearinginmindthatinall ases,itwas

possible,duringthedemonstration,topostulatethat

a = 0

withoutlosing ingenerality (se tion2.2.3):

ˆ Ma hiningofalo allyplanesurfa eatthe

ut-ter/workpie epoint of onta t(point

C

c

): in this instan e, ve tor

F

is parallel to theplane

(x, y)

andpoint

C

c

islo atedonthelowerlimit of the torus part of the utter. Then

c = 0

,

v = −

π

2

,

S = 0

(Fig. 3) and

ψ = 0

(Fig. 4) obtain. Proje tionofthe utterenvelope urve

inaplanenormaltofeedisthenastraightline

parallel to the plane

(x, y)

, orresponding to a null urvature. In all the other ma hining

ongurations,

v 6= −

π

2

and

S 6= 0

ne essarily apply.

c 6= 0

and

ψ 6= 0

also apply in all the otherma hining ongurations,ex ept for the

ase of milling perpendi ular to the dire tion

ofthegreatest slope (

α = ±

π

2

).

ˆ Ma hiningperpendi ulartothedire tionofthe

greatest slope. In this instan e

c = 0

and

α = ±

π

2

apply (see Fig. 3 with a ma hining dire tionperpendi ularto theplane

(D, z)

for

ne angle

α

). Proje tion in a planenormal to the feed of the utter envelope urve is then

anar of ir le orrespondingtothetoruspart

of the utter and the ee tive ma hining

ra-diusisequal tothetorusradius

r

. Inallother ma hining ongurations,

α 6= ±

π

2

ne essarily obtains.

ˆ Ma hining along axis

z

.

b = 0

and

ψ =

π

2

will then apply. This instan e ould possibly

arise when milling verti ally with a round

in-sert utter. The utter/workpie epointof

on-ta twouldthenbe lo ated onthe upper limit

ofthetoruspart(

v = 0

)andinthis asethe ef-fe tiveradiusofthe utter ouldbe onsidered

to be equal to its outsideradius

R

. Neverthe-less, su h ma hining onditions are extremely

unfavourable in terms of utting quality and

utter lifetime and are onsequently never

ap-plied industrially. However, in all the other

ma hining ongurations,

b 6= 0

and

ψ 6=

π

2

willne essarilyapply.

The hypothesesadopted during al ulation thus

orrespondto borderline ases that an be managed

regardlessof the ee tive radius. Whileit is

appro-priate to take them into a ount when developing

tools based on relation (30), this should not be an

obsta leto implementation.

4 Example of an appli ation

4.1 Introdu tion

Determining theee tive radius of the utter atthe

utter/workpie e point of onta t through a simple

analyti alformulaaswiththeone establishedin

re-lation (30) oersmany advantages. Indeed, despite

theimposedlimitsestablishedinse tion3.2,this

re-sult oers the perspe tive of multiple appli ations

thatwill befurther developed in forth oming

publi- ations. Using an analyti al formula is always

rapi-der than a numeri al pro edure. Cal ulation of the

ee tive radius by an analyti al formula instead of

the numeri al pro edures generally used means

ap-pli ationsthatwerehitherto onsideredtobehardto

ontemplate an bedeveloped. For example,thanks

totherelationestablishedin(30),anappli ationwas

almostinstantaneously.

This mapping tool was then used to ondu t a

studyinto the omparativeee tivenessofaballend

milland atorus milling utter withthesame radius

when ma hining a free-form surfa e from an

indus-trialenvironment ona 3-axisNC ma hinetool.

Thissurfa e,relatingtoaboatpropeller

measur-ing393 mm indiameter (Fig. 5), is theextrados of

theblade(Fig. 6).

Figure5: Boat propeller

Figure6: Extradosof ablade

Whateverthetoolpath planning strategy

envis-aged(parallelplanes,isoparametri s,iso-s allop)the

step over distan e (dened in se tion 4.2) must

re-spe tthemaximums allopheight.

Firstly, it should be re alled that, the step over

distan e at a point is dire tly related to the utter

distan ehasasigni antimpa tonprodu tivity. In

what follows, it will be shown how, ompared with

the results obtained using a ball-end mill, using a

torus end utter an be advantageous insome areas

of the workpie e and disadvantageous in others. It

willalsobeseenhowthesimpli ityoftheexpression

established in (30) allows this analysis to be

on-du tedinan extremely shorttime.

4.2 Relation between step over distan e

and ee tive radius

Whatever the tool path planning strategy used, in

ordertopositionthe uttersoastorespe tthe

max-imums allopheight atagivenpointofthetoolpath,

the distan e

d

dening itspositionintheplane per-pendi ularto thefeed dire tionmustrsthavebeen

al ulated (Fig. 7). Subsequently,thestep over

dis-tan e

s

od

an be readily determined asit isdire tly relatedto

d

byangle

γ

hara terising thelo al in li-nation of the surfa e in a plane normal to the

ma- hining dire tion. Showing that the step over

dis-tan eisdire tlyrelatedto the utteree tive radius

isthusequivalenttoshowingthatdistan e

d

depends dire tlyonthat ee tive radius.

̺

O

R

ef f

R

ef f

C

A

H

D

s

od

γ

β

d

_/

₂

s

h

Figure 7: Cal ulating thestep over distan e

To al ulate thevalueof

d

itisassumedthatthe urvature of surfa e

̺

( onsidered in a plane nor-mal to the feed dire tion) and the utter ee tive

radius

R

ef f

are onstant lo ally. Thetriangle made by the entre of urvature of the surfa e alled

O

, andpoints

C

and

H

(Fig. 7) is onsidered. For this

triangle,thefollowing an be stated:

R

2

_{ef f}

= (R

ef f

+ ̺)

2

+ (̺ + s

h

)

2

− 2(R

ef f

+ ̺)(̺ + s

h

) cos(β)

where

β

is theanglebetween ve tors

OC

and

OH

.

Moreover, in triangle

(OAC)

, the following ap-plies:

d

2

= (̺ + R

ef f

) sin(β)

Stating

t = tan

2



β

2



,thefollowing is obtained:

(

R

2

_{ef f}

= (R

ef f

+ ̺)

2

+ (̺ + s

h

)

2

− 2(R

ef f

+ ̺)(̺ + s

h

)

1−t

2

1+t

2

d

2

= (̺ + R

ef f

)

1+t

2 t

2

Resolution ofthis systemof equations gives:

d =

r



4 R

2

ef f

+ 4 ̺ R

ef f

− 2 s

h

̺ − s

h

2



(2 ̺ + s

h

) s

h

̺ + s

h

(31)

This expression shows that the distan e

d

is in-deeddire tlyrelatedtothe ee tiveradius

R

ef f

, es-pe ially onsidering that

s

h

an be negle ted in re-lation to the other magnitudes. The step over

dis-tan e

s

od

thus depends dire tly on the ee tive ra-dius

R

ef f

. Now,foragivens allopheightvalue(the a eptable toleran eon the surfa e),the in rease in

stepoverdistan eallows forsigni antgains in

pro-du tivity. Consequently, it an be said that this

in- reaseintheee tiveradiusofthe utterhasadire t

impa ton produ tivity.

4.3 Comparison methodology

A seen previously (se tion 3.1), the ee tive radius

value anvarybetweenthetorusradius

r

inthe ase ofapathperpendi ulartothedire tion ofthe

great-estslope, and a value that tendsto innity for

hor-izontal milling. Given the relation between ee tive

radius utter and step overdistan e, itis lear that

where the ee tive radius equals

r

,using a ball-end utterinsteadof atorus milling utterwill allowfor

greater produ tivity. However, the loser the path

be omes to being horizontal in the dire tion of the

greatestslope,themore thetorusmilling utter will

prove to be more ee tive as ompared witha

ball-end mill of the same diameter. It therefore seems

useful to be able to determine, for a given surfa e,

thezones where the torus milling utter is more

ef-fe tive than the ball-end utter and vi e-versa.

For theball-end utter, the ee tiveradius is

al-ways equal to its nominal radius

R

, whatever the feed dire tionand slopeof thesurfa e.

For the torus milling utter, al ulation of the

ee tive radius with theformulaestablished in(30)

requiresknowledgeof theslope ofthesurfa e atthe

point onsidered and theangle formedbythe

dire -tionof thegreatest slope and thefeed dire tion. To

pursuethisanalysis,ama hiningdire tionrstneeds

to be dened that will be parameterized by the

an-gle

θ

its proje tion makes in the plane

(X, Y)

with the axis

X

of the ma hine. Then a meshing of the parametri spa e omprising 256 x 256 tiles is

on-sidered. In the entre of ea h tile thus onstituted,

theee tiveradiusofthetorusmilling utter anbe

readily al ulated using relation (30). The value of

theee tive radius thus obtained is thenasso iated

with a orresponding olour from a s ale of olours

varyinglinearly from

r

(blue) to

2R

(red). Thegrid of olours is then applied as texture to the 3D

rep-resentation ofthesurfa etogenerategraphi images

like thoseshowningures 8 to11.

The entire pro edure ( al ulation and

visualisa-tion)wasdevelopedusingtheJavaprogramming

lan-guage.

4.4 Results

Here, the results for two utters with outer radius

R = 5mm

are presented. One utter is a ball-end mill while the other is a torus milling utter whose

torus radius is

r = 2mm

. The surfa e onsidered is theextrados ofaboat propellerasshownpreviously

(se tion4.1).

Applyingthe methodology dened inse tion4.3

to a number of representative ma hining dire tions

ˆ forama hining dire tiondened by

θ = −45

°, gure8is obtained

ˆ for a ma hining dire tion dened by

θ = 0

°, gure9is obtained

ˆ for a ma hining dire tion dened by

θ = 45

°, gure10 isobtained

ˆ for a ma hining dire tion dened by

θ = 90

°, gure11 isobtained

Figure8: Visualisationoftheee tiveradiusfor

θ =

−45

°

Figure9: Visualisationoftheee tiveradiusfor

θ =

0

°

On these gures, the white urves represent the

limit between the zones, that is the points where

Figure 10: Visualisation of the ee tive radius for

θ = 45

°

Figure 11: Visualisation of the ee tive radius for

θ = 90

°

R

ef f

= R = 5mm

. In the regions that are pre-dominantly blue,

R

ef f

< R

applies. It an thus be said that in these regions the ball-end mill is more

ee tivethanthetorusmilling utter. Conversely,in

mainly redandgreen regions, theee tive radius of

the torus milling utter is greater than thenominal

radiusoftheball-end utter (

R

ef f

> R

);asaresult, it an besaidthatintheseregions, thetorusmilling

utter ismore ee tivethan the ball-end utter.

The al ulationtimeneededto ondu ttheentire

analysispro edure( omputation anddisplay) forall

the tests ondu ted always took less than one

se -ond. This rapidity in al ulation is essentially due

(relation 30).

Thissimpli ityofexpressionmeansthattoolsfor

analysis, like the one introdu ed here, an be

de-ned toprovide pre ioushelpin hoosinga

ma hin-ingstrategy.

5 Con lusions and perspe tives

Whenend milling of free-formsurfa es witha torus

milling utter, the ee tive radius on ept is

essen-tial to analyse the ma hining pro edure in purely

geometri alterms.

The study presented in the present publi ation

enabledthe ee tive radius ofa torusmilling utter

millingwithatranslationmovementtobeexpressed.

Adopting this an original approa h, this expression

was determined in analyti al form without having

to resort to geometri approximation. This relation

wasalso analysed and the limits to its validitywere

studied.

Astheexpressionwasrelativelysimple,itshould

pavethewayforappli ationsthatitwouldbe

impos-sibleto implement ina reasonable timeframe using

a numeri al approa h. As an example, a tool for

numeri al analysis was presented that ould prove

formma hining using parallel planes.

The possibilities oered by the relation

estab-lished in the present study are, however, far from

being limited to theexample adopted here. Due to

it being so easy to implement, the analyti al

for-mula dened here to ompute the ee tive radius

may readily be integrated into re ently developed

"intelligent CAM"pro esses [24,25℄.

In forth oming publi ations it will be seen how

analyti al expression of the utter ee tive radius

an nd many dierent appli ations in studies into

thema hiningoffree-formsurfa eswithatorusmilling

utter thatadopt ageometri approa h.

A Cal ulation detail

A.1 Identi ation of terms in

r

in

equa-tion (9) From equation(13):

c cos (t)

pb

2

_sin

2

_{(t) + c}

2

=

s

1 −

c

2

_sin

2

_(v)

b

2

_cos

2

_(v)

cos (v)

it an bededu ed su essively that:

c

2

cos

2

(t) =



1 −

c

2

b

2

tan

2

_(v)



cos

2

(v) b

2

sin

2

(t) + c

2

b

2

c

2

cos

2

(t) 1 + tan

2

(v)
= b

2

_{− c}

2

_tan

2

_(v)

c

2

+ b

2

sin

2

(t)

= b

2

c

2

+ b

4

sin

2

_{(t) − c}

4

tan

2

_{(v) − b}

2

c

2

tan

2

_{(v) 1 − cos}

2

(t)

b

2

c

2

cos

2

(t) = b

2

c

2

+ b

4

_{1 − cos}

2

(t)

_{− c}

4

tan

2

_{(v) − b}

2

c

2

tan

2

(v)

b

2

cos

2

(t) b

2

+ c

2

= b

2

_c

2

_{+ b}

4

_{− c}

2

_tan

2

_{(v) b}

2

_{+ c}

2

cos

2

(t) = c

2

+ b

2

₋

c

2

b

2

tan

2

_(v)

and this leadsto the following equation(12) :

cos(t) =

s

1 −

c

2

_sin

2

_(v)

b

2

_cos

2

_(v)

A.2 Identi ation of terms in

r

in

equa-tion (10) From equation(14) :

c sin(t)

pb

2

_sin

2

_{(t) + c}

2

= −

 c

3

b

sin(v) + b c sin(v)



itfollows that:

c

2

sin

2

(t)

b

2

_sin

2

_{(t) + c}

2

=

 c

3

_{+ b}

2

_c

b



2

sin

2

(v)

=

c

2

+ b

2

2

c

2

b

2

sin

2

_(v)

b

2

c

2

sin

2

(t) = c

2

sin

2

(v) b

2

sin

2

(t) + c

2

b

2

c

2

sin

2

_{(t) 1 − sin}

2

(v)

= c

4

sin

2

(v)

b

2

sin

2

(t) cos

2

(v) = c

2

sin

2

(v)

1 − cos

2

(t) =

c

2

_sin

2

_(v)

b

2

_cos

2

_(v)

from whi h follows equation(12):

cos(t) =

s

1 −

c

2

sin

2

_(v)

b

2

_cos

2

_(v)

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end-millsfor urvedsurfa e ma hining. Journal

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the ASME, 111(1):2226,Fév 1989.

[3℄ W.L.R.IpandM.Loftus.Cuspgeometry

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2711, Nov 1992.

[4℄ H.D. Cho, Y.T. Jun, and M.Y. Yang. 5-axis

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Toroidal versus ball nose and at bottom

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332, 1997.

[7℄ C.G.JensenandD.C.Anderson.A uratetool

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ma hining. JournalofDesignandManufa ture,

3:251261, 1993.

[8℄ Jean-Max Redonnet, Walter Rubio, Frédéri

Monies, and Gilles Dessein. Optimising tool

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