Uncharted Stable Peninsula for Multivariable Milling Tools by High-Order Homotopy Perturbation Method

applied
sciences
A icle
Uncha ed S able Peninsula o Mul i a iable Milling
Tools by High-O de Homo opy Pe u ba ion Me hod
Jose de la Luz Sosa 1, Daniel Ol e a-T ejo 1,* , Go ka U bikain 2,* , Osca Ma inez-Rome o 1,
Alex Elías-Zúñiga 1and Luis No be o López de Lacalle 2
1Tecnológico de Mon e ey, Escuela de Ingenie ía y Ciencias, A . Eugenio Ga za Sada 2501, Mon e ey,
Nue o León 64849, Mexico; [email p o ec ed] (J.d.l.L.S.); osca [email p o ec ed] (O.M.-R.);
[email p o ec ed] (A.E.-Z.)
2Depa men o Mechanical Enginee ing, Uni e si y o he Basque Coun y, Alameda de U quijo s/n,
48013 Bilbao, Spain; [email p o ec ed]
*Co espondence: daniel.ol e a. [email p o ec ed] (D.O.-T.); [email p o ec ed] (G.U.)
Recei ed: 9 Oc obe 2020; Accep ed: 3 No embe 2020; Published: 6 No embe 2020


Abs ac :
In his wo k, a new me hod o sol ing a delay di e en ial equa ion (DDE) wi h mul iple
delays is p esen ed by using second- and hi d-o de polynomials o app oxima e he delayed e ms
using he enhanced homo opy pe u ba ion me hod (EMHPM). To s udy he p oposed me hod
pe o mance in e ms o con e gency and compu a ional cos in compa ison wi h he i s -o de
EMHPM, semi-disc e iza ion and ull-disc e iza ion me hods, a delay di e en ial equa ion ha model
he cu ing milling ope a ion p ocess was used. To u he assess he accu acy o he p oposed me hod,
a milling p ocess wi h a mul i a iable cu e is examined in o de o ind he s abili y bounda ies.
Then, heo e ical p edic ions a e compu ed om he co esponding DDE inding uncha ed s able
zones a high axial dep hs o cu . Time-domain simula ions based on con inuous wa ele ans o m
(CWT) scalog ams, powe spec al densi y (PSD) cha s and Poinca
é
maps (PM) we e employed o
alida e he s abili y lobes ound by using he hi d-o de EMHPM o he mul i a iable ool.
Keywo ds: cha e ; mul i a iable ool; s able peninsula; homo opy pe u ba ion me hod
1. In oduc ion
The e a e many phenomena in di e en ields o science and enginee ing whe e he physical
esponse o a a iable in ol es no only he alue a ime
bu also he e ec s ha occu in an
ea lie s a e
−τ
. Thus, delay sys ems appea in many enginee ing p oblems, such as in he shimmy
e ec (wheel ib a ion) [
1
], ehicle a ic models [
2
], eedback s abiliza ion p oblems [
3
] and in he
egene a i e ib a ion o machine- ools be e known as cha e [
4
]. In cases whe e he ne o ce
depends on he cu en alues and some pas alues (his o y) such as posi ion and speed, he sys em
dynamic beha io can be modeled using a di e en ial delay equa ion (DDE).
I is well-known ha du ing a milling p ocess, uns able ib a ions also known as sel -exci ed
ib a ion o cha e may occu . Cha e educes he machining e iciency due o low ma e ial emo al
a e by educing he wo kload and a ec s su ace quali y, sho ens ool li e and accele a es ool
wea . Resea che s a e s udying se e al ways o o e come his limi a ion. Kuljanic e al. [
5
] s udied
he inco po a ion o a cha e de ec ion sys em based on mul iple senso s o milling ope a ions o
indus ial condi ions, Zhuo e al. [
6
] used a me hod based on ac al dimension o he lank milling o
a hin-walled blade, which can e lec he cha e se e i y le el h ough he mo phological change
in signal. Paul and Mo ales [
7
], o mi iga e cha e , p esen ed an ac i e con olle based on he
echnique o disc e e ime sliding mode con ol (DSMC) blended wi h he ype-2 uzzy logic sys em.
Mo eo e , Peng e al. [
8
] p esen ed a me hod based on a dynamic cu ing o ce simula ion model and a
Appl. Sci. 2020,10, 7869; doi:10.3390/app10217869 www.mdpi.com/jou nal/applsci
Appl. Sci. 2020,10, 7869 2 o 22
machine lea ning app oach based on s a is ical lea ning heo y o p edic and a oid he cu ing cha e .
In addi ion, o con ol and supp ess cha e ib a ions, he use o piezoelec ic ac ua o s embedded
in he ool holde [
9
], elec omagne ic ac ua o s in eg a ed in o he spindle sys em [
10
] and unable
clamping able [
11
] has been analyzed. In he milling p ocess, he use o a iable pi ch cu e s has
demons a ed o imp o e p oduc i i y [
12
]. Di e en om he uni o m pi ch cu e , when a a iable
pi ch cu e is used he dynamics model o cu ing ib a ion changes om DDEs wi h a single delay
o DDEs wi h mul iple delays [
13
]. A common echnique o line o p edic uns able ib a ions is he
so-called s abili y lobes o he DDE based on Floque heo y [
14
], in which a cu e desc ibes he limi
o s able ib a ion unde easible ange alues o cu ing pa ame e s.
The s abili y analysis o he milling p ocess wi h mul iple delays has been s udied h ough
di e en me hods. Among all hese me hods, hose wi h a iable pi ch ools play a c i ically impo an
ole [
15
]. Sla icek [
16
] was he one who i s demons a ed he e ec i eness o a iable pi ch cu e s in
supp essing ib a ions in he milling p ocess, he assumed a ec ilinea ool mo ion o cu ing ee h,
and applied he heo y o o hogonal s abili y o he i egula pi ch o he oo h, by assuming an
al e na ing s ep a ia ion hen, he ob ained an exp ession o he s abili y limi as a unc ion o he s ep
angle a ia ion. Budak [
17
,
18
] p oposed an analy ical me hod o noncons an pi ch milling cu e s
om a design poin o iew, showing o some applica ions how his a iable e ec helps o educe
sel -exci ed ib a ions, so he ound ha cha e s abili y can be imp o ed signi ican ly e en a slow
cu ing speeds by p ope ly designing he pi ch angles. Al in as e al. [19] used he equency domain
me hod o analyze he milling s abili y o he a iable pi ch cu e and in oduced a me hod o selec
he op imal pi ch angles. Olgac and Sipahi p oposed a ma hema ical app oach, he clus e ea men o
cha ac e is ic oo s (CTCR), which op imizes he design o a iable pi ch cu e s [
20
]. Jin e al. [
21
]
p esen ed an imp o ed semi-disc e iza ion algo i hm o p edic he s abili y lobes o a iable pi ch
cu e s, which we e e i ied and compa ed wi h p e ious wo ks such as he Al in as analy ical me hod
(ze o-o de me hod) [
19
]. Comak and Budak [
22
] showed he op imal design o a ool o milling
ope a ions wi h a iable geome y o widen he s abili y zones using he semi-disc e iza ion me hod,
alida ing i expe imen ally. They also used a design me hodology o de e mine he op imal pi ch
angle geome y o a gi en cu ing condi ion, allowing inc eased s abili y.
Za a ain e al. [
23
] ex ended he mul i equency solu ion p oposed by Budak and Al in as [
24
]
o include he helix e ec , hey poin ed ou ha he a ia ion o he helix angle plays an impo an
ole in s abili y g aphs due o epe i i e ib a ions d i en by impac ( lip), hey ound ha he lip
lobes became closed cu es ha a e sepa a ed by ho izon al lines whe e he dep h o cu is equal o
a mul iple o he helix pi ch. A simila phenomenon was con i med using he semi-disc e iza ion
me hod (SDM) in [
25
], meanwhile, B.R. Pa el e al. [
26
] conside ed he in luence o he helix angle o
he ool o ob ain an analy ical o ce model, hey ound ha isola ed islands o ins abili y can occu in
he milling p ocesses, which a e induced by he helix angle o he ool and lead o sepa a e egions o
pe iod-doubling and quasi-pe iod beha io . Sims e al. [
27
] by using an adap ed and ime-a e aged
e sion o he SDM analyzed bo h he in luence on he a ia ion o he helix angle and he pi ch angle
o he ool o imp o e he p edic ion o ib a ions and es ima e p edic ions o su ace e o s. They used
he semi-disc e iza ion me hod, he ime-a e aged semi-disc e iza ion me hod and he empo al ini e
elemen me hod o p edic ib a ion s abili y o a iable helix and a iable pi ch milling ools. Tu ne
e al. [
28
] modeled and compa ed s abili y o a iable pi ch and helix angle cu e s, demons a ing
ha a iable helix angle ools can ha e highe s abili y and p oduc i i y.
Yuso and Sims in [
29
] combined SDM wi h di e en ial e olu ion o op imize a iable helix milling
ools o minimize ib a ion, hei analysis p edic ed o al ib a ion mi iga ion using he op imized
a iable helix milling ool a low adial imme sion. Fu he mo e, Dombo a i and S epan [
30
] in oduced
a gene al mechanical model based on SDM o p edic he linea s abili y o special y cu e s wi h
op ional con inuous a ia ion o he helix angle. Using an ex ended second-o de SDM,
Zhan e al.
[
15
]
p edic ed he s abili y lobe diag ams o ools wi h a iable pi ch angles. Meanwhile,
Huang e al.
[
31
]
conduc ed a s abili y analysis o milling ope a ions wi h a iable pi ch mills a a iable speed,
Appl. Sci. 2020,10, 7869 3 o 22
while Cai e al. [
32
] p oposed an in eg a ed p ocess machine model based on he compu e g aphics
me hod o simula e he milling p ocess o a a iable pi ch cu e .
On he o he hand, Ol e a and El
í
as-Zuñiga in [
33
] led o he de elopmen o he enhanced
mul is age homo opy pe u ba ion me hod (EMHPM) o sol e di e en ial delay equa ions (DDEs)
wi h cons an and a iable coe icien s and hen his EMHPM was applied o p edic he s abili y o
a mul i a ia e milling ool in which hey conside he helix angle and he pi ch angle a ia ion o
he cu ing edges [
34
]. Based on he Laplace o mula ion, Sims [
35
] s udied he s abili y o milling
ope a ions wi h a a iable helix angle. Using he mul i- equency solu ion, O o e al. [
36
] de i ed a
dynamic p ocess model whe e he non-linea shea o ce and he unou e ec a e included o milling
wi h non-uni o m pi ch and a iable helix ools. Niu e al. [
37
] ound ha unou can signi ican ly
inc ease he s abili y limi s ega dless o spindle speed anges, while Ol e a e al. [
38
] in a s udy o a
hin-walled wo kpiece demons a ed ha by conside ing he e ec s o he unou , he helix angle and
cha ac e iza ion dependen on he cu ing speed, mo e p ecise s abili y bounda ies a e achie ed.
To demons a e ha one o he e ec i e ways o supp ess ib a ion in milling ope a ions is o use
ools wi h a iable pi ch and helix angle, Wang e al. [
12
] p oposed an imp o ed semi-disc e iza ion
me hod based on Floque
0
s heo y. Since he delay be ween each cu ing edge a ies along wi h he
axial dep h o he ool in milling, hey disc e ized he cu ing ool in some axial laye s o simpli y
he calcula ion. Iglesias e al. [
39
] p esen ed a me hod o ind he op imal angles be ween he inse s,
and he s abili y diag ams we e ob ained h ough he i e a i e b u e o ce (BF) me hod, which consis s
o an i e a i e maximiza ion o s abili y h ough he semi-disc e iza ion me hod. They conclude ha ,
i an op imal selec ion o he angle be ween he inse s is possible hen, he ma e ial emo al a e can be
imp o ed up o h ee imes. Gou e al. [
40
] p oposed an e ec i e op imiza ion me hod o he a iable
helical cu e in oducing an index called “supp ession ac o ” o measu e s abili y quan i a i ely.
The e o e, in he p esen wo k, he EMHPM de eloped in [
33
] and ex ended o analysis o
mul i a iable ools in [
34
], is now expanded o sol e he dynamics o he machining p ocess in milling
in which he app oxima ion o he delay is pe o med wi h polynomials o deg ee wo and h ee.
In o de o s udy he p oposed me hod pe o mance in e ms o con e gency and compu a ional cos ,
a mul i a iable milling ool wi h a a iable pi ch cu e and helix angle is used o de e mine milling
p ocess in s abili y domains.
This pape is summa ized as ollows. Sec ion 2 ocuses on he de elopmen o second- and
hi d-o de EMHPM o s abili y analysis o DDE. Sec ion 3s udies he applica ion o he second- and
hi d-o de EMHPM on he milling equa ion o demons a e i s imp o emen in he con e gence a e.
Sec ion 4is ocused on he use o he hi d-o de EMHPM o compu e he s abili y analysis in milling o
mul i a iable ools, and heo e ical p edic ions wi h ime-domain simula ions a e pe o med. Finally,
some conclusions a e d awn.
2. Enhanced Mul is age Homo opy Pe u ba ion Me hod
2.1. Second-O de EMHPM
Ol e a e al. enhanced in [
33
] he mul is age homo opy pe u ba ion me hod (MHPM) p oposed
by Hashim [
41
]. The EMHPM conside s he gene al case in which he nonlinea equa ion con ains
e ms o he independen a iable. This me hod is also use ul o sol e an n-dimensional DDE in he
s a e-space o m .
x( )=A( )x+B( )x( −τ)(1)
whe e
A( +τ)=A( )
,
B( +τ)=B( )
,
x( )
, is he s a e ec o , and
τ
is he ime delay. Equa ion (1)
can be w i en equi alen ly as: .
xi(T)−A xi(T)≈B xiτ(T)(2)
whe e
xi(T)
indica es he
m
-o de solu ion o Equa ion (1) ha sa is ies he ini ial condi ions
xi(0)=
xi−1
,
A
and
B
a e he pe iodic ma ix whose alues a y wi h ime
. In [
42
], Puma e al. applied
Appl. Sci. 2020,10, 7869 4 o 22
he i s -o de EMHPM o es ima e he delayed e m
xτ
i(T)
in Equa ion (2), in which he pe iod
[ 0−τ, 0]
was disc e ized in
N
equally spaced disc e e s a e alues, and he unc ion ha desc ibes he
delayed e m
xτ
i(T)
in he delayed in e al
[ i−N, i−N+1]
was app oxima ed as a i s -o de polynomial
ep esen a ion. De ining xi≡xi(T) o simpli y he no a ion, Equa ion (2) can be w i en as
.
xi(T)=A xi(T) + B xi−N+N−1
τ(xi−N+1−xi−N)T(3)
Figu e 1a shows he ep esen a ion o he app oxima ion o he delayed e m wi h he i s -o de
polynomial. In he second-o de EMHPM, o app oxima e he unc ion ha desc ibes he delayed
e m
xτ
i(T)
in Equa ion (2), he Lag ange equa ion is used, making use o he disc e e alues
xi−N,xi−N+1,xi−N+2as ollows:
n(x)=
n
X
i=0
Li(x) (xi),Li(x)=
n
Y
i=0,i,k
x−xi
xk−xi
(4)
o achie e a second-deg ee polynomial app oxima ion, we ha e om Equa ion (4) ha
P2(x)=(x−∆ )(x−2∆ )
(0−∆ )(0−2∆ ) (xi−N)+(x−0)(x−2∆ )
(∆ −0)(∆ −2∆ ) (xi−N+1)+(x−0)(x−∆ )
(2∆ −0)(2∆ −∆ ) (xi−N+2)(5)
Appl. Sci. 2020, 10, x 4 o 23
() () ()
i i i
TT T
τ
−≈xAx Bx
 (2)
whe e 𝐱(𝑇) indica es he 𝑚-o de solu ion o Equa ion (1) ha sa is ies he ini ial condi ions
𝐱(0) = 𝐱, 𝐀 and 𝐁 a e he pe iodic ma ix whose alues a y wi h ime 𝑡. In [42], Puma e al.
applied he i s -o de EMHPM o es ima e he delayed e m 𝐱
(𝑇) in Equa ion (2), in which he
pe iod 󰇟𝑡−𝜏,𝑡
󰇠 was disc e ized in 𝑁 equally spaced disc e e s a e alues, and he unc ion ha
desc ibes he delayed e m 𝐱
(𝑇) in he delayed in e al 󰇟𝑡,𝑡󰇠 was app oxima ed as a i s -
o de polynomial ep esen a ion. De ining 𝐱≡𝐱
(𝑇) o simpli y he no a ion, Equa ion (2) can be
w i en as
() ()
1
1
()xAxBx xx
τ
−−+−
−

=++ −


i i iN iN iN
N
TT T
(3)
Figu e 1a shows he ep esen a ion o he app oxima ion o he delayed e m wi h he i s -o de
polynomial. In he second-o de EMHPM, o app oxima e he unc ion ha desc ibes he delayed
e m 𝐱
(𝑇) in Equa ion (2), he Lag ange equa ion is used, making use o he disc e e alues
𝐱,𝐱,𝐱 as ollows:
() () ( ) ()
00,
,
n
n
i
nii
iiik
ki
x
x
x Lix x Lx
x
x
==≠
−
==
−
∏ (4)
o achie e a second-deg ee polynomial app oxima ion, we ha e om Equa ion (4) ha
() ()( )
()( )
()
()( )
()( )
()
()( )
()( )
()
212
202 0
002 02 202
xx x
−−+ −+
−Δ − Δ − − Δ − −Δ
=+ +
−Δ − Δ Δ− Δ− Δ Δ− Δ−Δ
iN iN iN
x x x x x x
Px
(5)
Subs i u ing 𝑥=𝑇 and ∆𝑡 = (𝑁 − 1)/𝜏, we ob ain he unc ion ha desc ibes he delayed
in e al as:
() ()
22
11212
13 1 1
22
22 2
iN iN iN iN iN iN iN iN
NNT
TT
ττ
−+ − − −+ −+ − −+ −+
−−
 
≈+ − + − + − +
 
 
xx xxx xxx
(6)
When he delay is app oxima ed by a second-deg ee polynomial i is called second-o de
EMHPM and should no be con used wi h he o de o solu ion 𝑚 and which is de e mined by he
las de o ma ion aking in o accoun he app oxima ed solu ion. No ice ha a polynomial o he
second-deg ee equi es h ee poin s. Likewise ou poin s in he case o a hi d-deg ee polynomial as
shown in Figu e 1.
(a) (b)
Figu e 1. (a) Scheme o he app oxima ion o he delayed e m by a i s -o de (solid black line),
second-o de (dashed blue line) and hi d-o de (do ed ed line) polynomial: (b) zoom in he ime
in e al [𝑡,𝑡].
x
xi-N
i-N+1
i-N i-1 i
= (N-1)
xi-N+3
xi-N+2
xi-N+1
xi( )
xi+1( )xi+2( )
Figu e 1.
(
a
) Scheme o he app oxima ion o he delayed e m by a i s -o de (solid black line),
second-o de (dashed blue line) and hi d-o de (do ed ed line) polynomial: (
b
) zoom in he ime
in e al [ i−N+1, i−N+3].
Subs i u ing
x=T
and
∆ =(N−1)/τ
, we ob ain he unc ion ha desc ibes he delayed
in e al as:
xi−N+1(T)≈xi−N+N−1
τT−3
2xi−N+2xi−N+1−1
2xi−N+2+N−1
τ2T2
2(xi−N−2xi−N+1+xi−N+2)(6)
When he delay is app oxima ed by a second-deg ee polynomial i is called second-o de EMHPM
and should no be con used wi h he o de o solu ion
m
and which is de e mined by he las de o ma ion
aking in o accoun he app oxima ed solu ion. No ice ha a polynomial o he second-deg ee equi es
h ee poin s. Likewise ou poin s in he case o a hi d-deg ee polynomial as shown in Figu e 1.
The p ocedu e o calcula e he second-o de EMHPM solu ion is based on he EMHPM p ocedu e
desc ibed in [33]. The solu ion o second-o de EMHPM is ecu si ely exp essed o Xik(T)as
Xik =Xa
ik +Xb
ik +Xc
ik,k=1, 2, 3 . . . . (7)
whe e
Xa
i0=xi−1, Xb
i0=Xc
i0=0 (8)
Appl. Sci. 2020,10, 7869 5 o 22
and
Xa
ik =T
kA Xa
i(k−1)+g(k)B xi−N
Xb
ik =T
k+1A Xb
i(k−1)+g(k)N−1
τB T−3
2xi−N+2xi−N+1−1
2xi−N+2
Xc
ik =T
k+2A Xc
i(k−1)+g(k)N−1
τ2B T2
2(xi−N−2xi−N+1+xi−N+2)
(9)
So, he solu ion o Equa ion (1) is ob ained by adding each o he app oxima ions
Xik
o Equa ion (7).
xi(T)≈
m
X
k=0
Xik(T)(10)
2.2. Thi d-O de EMHPM Solu ion
Fo he polynomial ep esen a ion o he hi d-deg ee, he unc ion ha desc ibes he delayed e m
xτ
i(T)
is app oxima ed by a polynomial o o de h ee, hen Equa ion (4) o he Lag ange in e pola o
is used acco dingly. In his case, i is necessa y o employ he
xi−N
,
xi−N+1
,
xi−N+2
,
xi−N+3
disc e e
alues. Following he same p ocedu e desc ibed in Sec ion 2.1, he unc ion ha desc ibes he delayed
in e al is gi en as:
xτ
i(T)=xi−N+1(T)≈xi−N+N−1
τT−11
6xi−N+3xi−N+1−3
2xi−N+2+1
3xi−N+3+
N−1
τ2T2
2(2xi−N−5xi−N+1+4xi−N+2−xi−N+3)+N−1
τ3T3
6(−xi−N+3xi−N+1−3xi−N+2+xi−N+3)
(11)
Following he EMHPM p ocedu e, he ecu si e solu ion o Equa ion (1) Xik(T)is exp essed as
Xik =Xa
ik +Xb
ik +Xc
ik +Xd
ik,k=1, 2, 3 . . . . (12)
whe e
Xa
i0=xi−1,Xb
i0=Xc
i0=Xd
i0=0 (13)
and
Xa
ik =T
kA Xa
i(k−1)+g(k)B xi−N
Xb
ik =T
k+1A Xb
i(k−1)+g(k)N−1
τB T−11
6xi−N+3xi−N+1−3
2xi−N+2+1
3xi−N+3
Xc
ik =T
k+2A Xc
i(k−1)+g(k)N−1
τ2B T2
2(2xi−N−5xi−N+1+4xi−N+2−xi−N+3))
Xd
ik =T
k+3A Xd
i(k−1)+g(k)N−1
τ3B T3
6(−xi−N+3xi−N+1−3xi−N+2+xi−N+3))
(14)
The app oxima e solu ion o Equa ion (1) can be ob ained by subs i u ing Equa ion (12) in o
Equa ion (10) adding each o he app oxima ions Xik.
2.3. S abili y Analysis
To calcula e he s abili y o he di e en ial Equa ion (1) using he second-o de EMHPM, he
solu ion o Equa ion (10) o second-o de EMHPM mus be ew i en by g ouping each o he disc e e
alues xi,xi−N+2,xi−N+1,xi−N, esul ing in
xi(T)≈Pi(T)xi−1+Qi
0(T)xi−N+2+Qi(T)xi−N+1+Ri(T)xi−N(15)

Appl. Sci. 2020,10, 7869 6 o 22
whe e
Pi(T) = m
P
k=0
1
k!Ak
Tk,
Q0
i(T) = m
P
k=11
(k+2)!N−1
τ2Ak−1
B Tk+2−1
2(k+1)!N−1
τAk−1
B Tk+1
Qi(T) = m
P
k=1
1
(k+1)!N−1
τAk−1
B Tk+1−2Q0
i
Ri(T) = m
P
k=1
1
k!Ak−1
B Tk−Q0
i−Qi
(16)
Simila ly, o compu e he s abili y lobes o he hi d-o de EMHPM, he solu ion o he di e en ial
Equa ion (1) o hi d-o de EMHPM is ew i en as
xi(T)≈Pi(T)xi−1+Q”i(T)xi−N+3+Q0
i(T)xi−N+2+Qi(T)xi−N+1+Ri(T)xi−N(17)
whe e
Pi(T) = m
P
k=0
1
k!Ak
Tk,
Q”i(T) = m
P
k=11
(k+1)!N−1
τAk−1
B Tk+11
3−1
(k+2)!N−1
τ2Ak−1
B Tk+2+1
(k+3)!N−1
τ3Ak−1
B Tk+3
Q0
i(T) = m
P
k=1

1
(k+1)!N−1
τAk−1
B Tk+1+1
(k+2)!N−1
τ2Ak−1
B Tk+2−7
2
+1
(k+3)!N−1
τ3Ak−1
B Tk+39
2
−15
2Q0 0
i
Qi(T) = m
P
k=11
(k+1)!N−1
τAk−1
B Tk+1−3Q0 0
i−2Q0
i
Ri(T) = m
P
k=1
1
k!Ak−1
B Tk−Q0 0
i−Q0
i−Qi
(18)
The app oxima e solu ion ob ained om Equa ion (17) was used o de ine a disc e e map ollowing
he p ocedu e desc ibed in [43]:
wi=Diwi−1(19)
whe e
wi−1
is a ec o wi h dimension equal o he o al numbe o s a es (displacemen and eloci y)
o all Ndisc e e in e als:
wi−1= [x(i−1),.
x(i−1),x(i−2),. . . ,x(i−N)]T(20)
Diis a coe icien ma ix and o he hi d-o de EMHPM i has he o m:
Di=

P0 0 0 · · · 0Q00 iQ0iQiRi
I0 0 0 · · · 0 0 0 0 0
0I0 0 · · · 0 0 0 0 0
0 0 I0· · · 0 0 0 0 0
.
.
..
.
..
.
.....
.
..
.
..
.
..
.
..
.
..
.
.
0 0 0 0 ...0 0 0 0 0
0 0 0 0 · · · I0 0 0 0
0 0 0 0 · · · 0I0 0 0
0 0 0 0 · · · 0 0 I0 0
0 0 0 0 · · · 0 0 0 I0

(21)
Appl. Sci. 2020,10, 7869 7 o 22
I is impo an o poin ou ha in he case o he second-o de EMHPM, he ma ix
Di
is like he
ma ix o he hi d-o de EMHPM wi hou he ma ix Q00
i.
Then, he Floque ansi ion ma ix
Φ
is calcula ed o e he main pe iod
τ=(N−1)/∆
, coupling
each o he disc e e maps Di,i=1, 2, . . . ,(N−1), o ob ain:
Φ=DN−1DN−2. . . D2D1(22)
Thus, he s abili y o Equa ion (1) is de e mined by calcula ing he eigen alues o he ansi ion
ma ix gi en by Equa ion (22). The eigen alues o he ansi ion ma ix a e ac ually he Floque
mul iplie s which a e he exponen s o each complex exponen ial unc ions ha desc ibe he mo ion o
Equa ion (1). I he modulus o g ea es magni ude is g ea e han o equal o one, i implies ha he
sys em will beha e in an uns able way and he ampli ude o he ib a ion will inc ease exponen ially,
o he wise i will ha e a s able beha io .
3. Nume ic Solu ion o he Milling Equa ion
3.1. Dynamic Model o he Milling Equa ion
To alida e he p oposed EMHPM me hods, he nume ical solu ion o he delay di e en ial
equa ion analyzed by Ol e a e al., in [
33
] was calcula ed, which desc ibes he dynamic model o he
milling p ocess in one deg ee o eedom (DOF):
..
x( )+2ζωn
.
x( )+ω2
nx( ) = −
aphxx( )
mm
(x( )−x( −τ)) (23)
whe e
ζ
is he modal damping a io,
ωn
is he na u al equency o he wo kpiece,
ap
is he axial dep h
o cu ,
mm
is he modal mass,
τ
ep esen s he ime delay co esponding o he hi ing pe iod be ween
each oo h o he ool and
hxx( )
is he speci ic cu ing o ce in he x-di ec ion due o lexibili y in
x-di ec ion, which was calcula ed depending on he posi ion o he ool
hxx( ) =
zn
X
iz=1
g(φiz( ))sinφiz( )(K c cos φiz( ) + Knc sin φiz( ))(24)
zn
is he numbe o edges o he ool,
K c
and
Knc
a e he a e age speci ic cu coe icien s in he angen ial
and no mal di ec ion, espec i ely, and φiz( )is he angula posi ion o each le edge desc ibed by
φiz =(2πn/60) +2πiz/zn(25)
whe e
n
is he spindle speed in e olu ion pe minu e ( pm). The unc ion
g(φiz( ))
is a window
unc ion, which has he alue o one when he cu en edge
iz
is cu ing ma e ial, o he wise i akes he
alue ze o.
In up-milling
φs =
0 and
φex =cos−1(1−2ad)
, con e sely in down-milling
φs =cos−1(2ad−1)
and
φex =π
,
ad
is he adial imme sion a io o he cu and
φs
and
φex
a e he angula posi ions whe e
each edge en e s and lea es he wo kpiece.
The second- and hi d-o de EMHPM is applied o ob ain he solu ion o Equa ion (23) and i is
compa ed wi h he solu ion gi en by he i s -o de EMHPM [
33
]; o a egula ool he ma ix
A
and
B
a e ep esen ed as:
A =
0 1
−ω2
n−aphxx( )
mm
−2ζωn,B =
0 0
aphxx( )
mm0(26)
A
and
B
co espond o he pe iodic ma ix e alua ed a ime
. Fo demons a ion pu poses,
ime-domain simula ions we e compu ed o a ull-imme sion down-milling ope a ion. We used
he pa ame e s employed by Inspe ge e al., in [
43
] whe e he s abili y lobes we e also calcula ed.
Appl. Sci. 2020,10, 7869 8 o 22
The modal pa ame e s
n
=922 Hz,
ωn=
5793 ad/s,
ζ=
0.011 and
mm=
0.03993 kg co esponds o
a single deg ee o eedom. The angen ial and no mal cu ing coe icien s a e
K c =
6
×
10
8
N/
m2
and
Knc =
2
×
10
8
N/
m2
espec i ely o an end-mill wi h
zn=
2. The ime-domain solu ion was
compu ed using he EMHPM conside ing
N=
76 disc e e in e als and
m=
7. Two se s o cu ing
condi ions we e chosen o a ixed spindle speed alue o
n=
12,000 pm whe e he axial dep h o cu o
ap=1.5 mm
co esponds o a s able cu ing ope a ion while ha o an uns able ope a ion
ap=3 mm
was chosen. In Figu e 2we plo he second- and hi d-o de EMHPM solu ions and compa e i wi h
he i s -o de EMHPM and he dde23 ou ine in Ma lab, which is used o in eg a e DDE.
Appl. Sci. 2020, 10, x 10 o 25
(a)
(b)
Figu e 2. Nume ical compa ison o he enhanced homo opy pe u ba ion me hod (EMHPM)
solu ions o he milling equa ion, Equa ion E o ! Re e ence sou ce no ound., wi h he dde23
MATLAB ou ine. (a) S able milling ope a ion wi h 𝑎=1.5 mm, 𝑎=1 and 𝑛 = 12000 pm and
(b) uns able milling ope a ion wi h 𝑎=3 mm, 𝑎=1 and 𝑛 = 12000 pm.
3.2. Nume ical Compa ison be ween Me hods
In o de o obse e he a e o con e gence o he i s -, second- and hi d-o de EMHPM, we
chose he s able case wi h cu ing condi ions 𝑎=1.5 mm, 𝑎=1 and 𝑛 = 12000 pm p esen ed
in Figu e 3a, and he uns able case wi h cu ing condi ions 𝑎=3 mm, 𝑎=1 and 𝑛 = 12000 pm
showed in Figu e 3b. The a e o con e gence was analyzed by compu ing he absolu e e o be ween
he solu ion wi h N disc e e in e als and a con e ged solu ion. All me hods we e compa ed agains
i sel using he solu ion p o ided wi h N = 200 disc e e in e als, which a e conside ed he con e ged
solu ion. In Figu e 3a i is obse ed ha he con e gence is be e o he second- and hi d-o de han
he i s -o de , howe e , he di e ence o con e gence be ween second- and hi d-o de wi h he
pa ame e s used was negligible. On he o he hand, Figu e 3b shows ha o ew disc e e in e als
he hi d-o de EMHPM had he as es con e gence in compa ison wi h he second- and he i s -
o de EMHPM. Howe e , he second-o de and hi d-o de cu es beha ed e y simila ly a e N =
50 disc e e in e als. I is impo an o men ion ha o a ypical s abili y solu ion in he anges o
spindle speed 5000–10000 pm, N = 40 disc e e in e als will be enough o ha e accu a e p edic ions.
0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02
Time
(
s
)
-1
-0.5
0
0.5
1
Displacemen (m)
10-3
Figu e 2.
Nume ical compa ison o he enhanced homo opy pe u ba ion me hod (EMHPM) solu ions
o he milling equa ion, Equa ion (23), wi h he dde23 MATLAB ou ine. (
a
) S able milling ope a ion
wi h
ap=
1.5 mm,
ad=
1 and
n=
12000 pm and (
b
) uns able milling ope a ion wi h
ap=
3 mm,
ad=1 and n=12000 pm.
3.2. Nume ical Compa ison be ween Me hods
In o de o obse e he a e o con e gence o he i s -, second- and hi d-o de EMHPM, we chose
he s able case wi h cu ing condi ions
ap=
1.5 mm,
ad=
1 and
n=
12,000 pm p esen ed in Figu e 3a,
and he uns able case wi h cu ing condi ions
ap=
3 mm,
ad=
1 and
n=
12,000 pm showed in
Figu e 3b. The a e o con e gence was analyzed by compu ing he absolu e e o be ween he solu ion
wi h Ndisc e e in e als and a con e ged solu ion. All me hods we e compa ed agains i sel using
he solu ion p o ided wi h N=200 disc e e in e als, which a e conside ed he con e ged solu ion. In
Figu e 3a i is obse ed ha he con e gence is be e o he second- and hi d-o de han he i s -o de ,
howe e , he di e ence o con e gence be ween second- and hi d-o de wi h he pa ame e s used was
negligible. On he o he hand, Figu e 3b shows ha o ew disc e e in e als he hi d-o de EMHPM
had he as es con e gence in compa ison wi h he second- and he i s -o de EMHPM. Howe e ,
he second-o de and hi d-o de cu es beha ed e y simila ly a e N=50 disc e e in e als. I is
impo an o men ion ha o a ypical s abili y solu ion in he anges o spindle speed 5000–10,000
pm, N=40 disc e e in e als will be enough o ha e accu a e p edic ions.
Appl. Sci. 2020,10, 7869 9 o 22
Appl. Sci. 2020, 10, x 11 o 25
(a)
(b)
Figu e 3. Con e gence a e o absolu e e o be ween i s -, second- and hi d-o de EMHPM o
down-milling ope a ion. Cu ing pa ame e s o (a) 𝑎= 1.5 mm,𝑎=1 and 𝑛 = 12000 pm and
(b) 𝑎= 3 mm,𝑎=1 and 𝑛 = 12000 pm.
Since he a e o con e gence was p o ed o ime-domain simula ions, we nex explo ed he
con e gence o he me hods applied o he s abili y analysis. The s abili y lobes compu ed wi h he
second- and hi d-o de EMHPM o egula milling ools we e compa ed wi h i s p edecesso o
adial imme sion alue o 𝑎 =1 and he o he pa ame e s indica ed abo e as i was used in [44].
Figu e 4 shows he s abili y diag ams o spindle speed in he ange 2000–3000 e /min whe e he
p ecision o he me hod was comp omised due o he highe alue o he ime delay. While he
shaded g ay a ea ep esen s he s abili y lobes compu ed wi h N = 200 disc e e in e als in all
sub igu es, in each sub igu e solid black lines d aw he s abili y on ie o a speci ic disc e e in e al
and using he i s -, second- o hi d-o de EMHPM. In Figu e 4 he i s , second and hi d column
ep esen s he solu ion o he i s -, he second- and he hi d-o de EMHPM espec i ely, while he
i s and he second ow was o N = 60 and N = 100 disc e e in e als, espec i ely. I is obse ed ha
he e o achie ed in he hi d-o de EMHPM was less han hose a ained o he i s -o de and
second-o de EMHPM solu ions. This con i ms ha he hi d-o de EMHPM had he highes a e o
con e gence.
Absolu e E o
Figu e 3.
Con e gence a e o absolu e e o be ween i s -, second- and hi d-o de EMHPM o
down-milling ope a ion. Cu ing pa ame e s o (
a
)
ap=
1.5
mm
,
ad=
1 and
n=12,000 pm
and
(b)ap=3 mm, ad=1 and n=12,000 pm.
Since he a e o con e gence was p o ed o ime-domain simula ions, we nex explo ed he
con e gence o he me hods applied o he s abili y analysis. The s abili y lobes compu ed wi h he
second- and hi d-o de EMHPM o egula milling ools we e compa ed wi h i s p edecesso o
adial imme sion alue o
ad
=1 and he o he pa ame e s indica ed abo e as i was used in [
44
].
Figu e 4shows he s abili y diag ams o spindle speed in he ange 2000–3000 e /min whe e he
p ecision o he me hod was comp omised due o he highe alue o he ime delay. While he shaded
g ay a ea ep esen s he s abili y lobes compu ed wi h N=200 disc e e in e als in all sub igu es,
in each sub igu e solid black lines d aw he s abili y on ie o a speci ic disc e e in e al and using
he i s -, second- o hi d-o de EMHPM. In Figu e 4 he i s , second and hi d column ep esen s he
solu ion o he i s -, he second- and he hi d-o de EMHPM espec i ely, while he i s and he
second ow was o N=60 and N=100 disc e e in e als, espec i ely. I is obse ed ha he e o
achie ed in he hi d-o de EMHPM was less han hose a ained o he i s -o de and second-o de
EMHPM solu ions. This con i ms ha he hi d-o de EMHPM had he highes a e o con e gence.
Appl. Sci. 2020,10, 7869 16 o 22
oo h 0.05, 0.10, 0.015 and 0.20, so he esul ing cu ing coe icien s
K c
o he oo h 1, 2, 3 and 4 we e
1215
×
10
6
, 1369
×
10
6
, 897
×
10
6
and 1799
×
10
6
N/
m2
espec i ely, while ha he coe icien s
Knc
o
he oo h 1, 2, 3 and 4 esul ed 272 ×106, 520 ×106, 801 ×106and 859 ×106N/m2 espec i ely.
Table 2. Main geome ic pa ame e s o mul i a iable ool.
Appl. Sci. 2020, 10, x 17 o 25
4.2. Expe imen al Cha ac e iza ion o One Deg ee o F eedom Milling Equa ion and Cu ing Fo ce Model
4.2.1. Expe imen al Modal Analysis
An expe imen al wo kpiece was assembled wi h a 7075T6 aluminum block o 101 mm × 172 mm
suppo ed by wo hin pla es (walls) wi h a hickness o 4.5 mm. This assembly mimics a DOF as
desc ibed in Equa ion E o ! Re e ence sou ce no ound.. The wo kpiece assembly was igidly ixed
o he wo kbench o a Makino F3 machining cen e . Fo modal analysis, ap es ing was pe o med
using a 352C68 PCB Piezo onics accele ome e and an impac hamme model 9722A500. The signals
we e acqui ed wi h a Poly ec VIB-E-220 da a acquisi ion ca d and p ocessed wi h VibSo signal
analyze so wa e as shown in Figu e 5a. Using he Cu P o 8 so wa e, he modal pa ame e s we e
i ed esul ing he alues 𝜁 = 0.068, 𝑚=3.8 kg, 𝑓
 = 132 Hz and 𝜔= 829 ad/s.
4.2.2. Expe imen al De e mina ion o Cu ing Coe icien s
The o ce model in Equa ion E o ! Re e ence sou ce no ound. was used o p edic he cu ing
o ce magni ude o a gi en dep h o cu . I is based on a mechanis ic app oach ha assumes a
ela ionship be ween o ces and he uncu chip hickness by means o he cu ing coe icien s. The
cu ing o ce model was es ablished by in oducing cu ing (shea ing) and edge coe icien s o he
angen ial and no mal di ec ions o he milling ool. The cha ac e iza ion p ocedu e assumed he
linea ela ionship be ween he a e aged expe imen al cu ing o ces 𝐹
 and he eed a e z in x- and
y- di ec ions. This ela ionship is es ablished as ollows:
=+

z
ce
F FF
(42)
He e, 𝐹

 and 𝐹

 a e he cu ing shea and edge componen s, espec i ely. The expe imen al
o ces a each eed a e a e measu ed, and he cu ing-edge componen s 𝐹

 and 𝐹

 we e e alua ed
4, 4
yc
x
c
c nc
np np
F
F
KK
za za
==−

(43)
A mul i a iable cu e p o ided by a local oolmake was cha ac e ized by using Equa ion
E o ! Re e ence sou ce no ound. and he expe imen al se up shown in Figu e 5b. Table 2
summa izes he main geome ic cha ac e is ics o he mul i a iable ool. A o al o i e cu ings we e
pe o med o ull adial imme sion in aluminum 7075T6 du ing d y machining. The o ces we e
eco ded by using a dynamome e 9257B Kis le and he spindle speed was se a 3000 pm based on
he dynamome e ’s na u al equency o a oid he ampli ica ion o milling o ces. The o ce signals
we e acqui ed using a VibSo -20 acquisi ion ca d a a sample a e o 48 kHz and p ocessed in a
cus om-made MATLAB app o emo e d i and noise. Cu ing o ces da a we e collec ed o he
axial dep h o cu o 2 mm and ou alues o eed pe oo h 0.05, 0.10, 0.015 and 0.20, so he esul ing
cu ing coe icien s 𝐾 o he oo h 1, 2, 3 and 4 we e 1215 × 10, 1369 × 10, 897 × 10 and
1799 × 10 N/m espec i ely, while ha he coe icien s 𝐾 o he oo h 1, 2, 3 and 4 esul ed
272 × 10, 520 × 10, 801 × 10 and 859 × 10 N/m espec i ely.
Table 2. Main geome ic pa ame e s o mul i a iable ool.
Diame e 12.7 mm
Cu ing leng h 25 mm
Coa ing ype Uncoa ed
Numbe o ee h 4
Helix angles 39°, 37°, 39°, 41°
Pi ch angles 80°, 100°, 70°, 110°
Diame e 12.7 mm
Cu ing leng h 25 mm
Coa ing ype Uncoa ed
Numbe o ee h 4
Helix angles 39◦, 37◦, 39◦, 41◦
Pi ch angles 80◦, 100◦, 70◦, 110◦
4.3. S abili y Analysis o 1 DOF Milling wi h a Mul i a iable Tool
The s abili y lobes compu ed o he mul i a iable ool using he hi d-o de EMHPM wi h a
mesh o 400
×
200 (
n×ap
) a e shown in Figu e 6 oge he wi h s abili y lobes o a egula ool (angles
o 90
◦
and helix angles o 30
◦
o all lu es). An app oxima ion o o de
m=
7 was used wi h N=241
and
ad=
1 mm. No ice om Figu e 6 ha he s able zone ob ained o he mul i a iable ool was
signi ican ly la ge , meaning ha he c i ical dep h o cu was highe in mos spindle speeds, which
allowed ha ing mo e global p oduc i i y. I is also obse ed in he ange o spindle speed be ween
2000 and 3000 pm, a s able peninsula o med wi h axial dep h anging om 11 o 20 mm o highe
alues o c i ical dep h o cu . Fo ins ance, o he mul i a iable cu e a 2500 pm, he c i ical dep h
o cu
ap
was 2.17 mm, howe e i became s able again as shown in Figu e 6 o he in e al alues
be ween 11 and 20 mm. To alida e his unexpec ed beha io , we pe o med se e al ime-domain
simula ions using he hi d-o de EMHPM solu ion desc ibed by Equa ion (37).
Appl. Sci. 2020, 10, x 18 o 25
(a)
(b)
Figu e 5. Scheme o he expe imen al se up o (a) he modal analysis and (b) cu ing o ces
cha ac e iza ion.
4.3. S abili y Analysis o 1 DOF Milling wi h a Mul i a iable Tool
The s abili y lobes compu ed o he mul i a iable ool using he hi d-o de EMHPM wi h a
mesh o 400 ×200 (𝑛 × 𝑎𝑝) a e shown in Figu e 6 oge he wi h s abili y lobes o a egula ool
(angles o 90° and helix angles o 30° o all lu es). An app oxima ion o o de 𝑚 = 7 was used wi h
N=241 and 𝑎𝑑= 1 mm. No ice om Figu e 6 ha he s able zone ob ained o he mul i a iable ool
was signi ican ly la ge , meaning ha he c i ical dep h o cu was highe in mos spindle speeds,
which allowed ha ing mo e global p oduc i i y. I is also obse ed in he ange o spindle speed
be ween 2000 and 3000 pm, a s able peninsula o med wi h axial dep h anging om 11 o 20 mm
o highe alues o c i ical dep h o cu . Fo ins ance, o he mul i a iable cu e a 2500 pm, he
c i ical dep h o cu 𝑎𝑝 was 2.17 mm, howe e i became s able again as shown in Figu e 6 o he
in e al alues be ween 11 and 20 mm. To alida e his unexpec ed beha io , we pe o med se e al
ime-domain simula ions using he hi d-o de EMHPM solu ion desc ibed by Equa ion
E o ! Re e ence sou ce no ound..
(a)
(b)
Figu e 6. (a) Compa ison o s abili y lobes o egula (black solid line) and mul i a iable ( ed solid
line) cu e s by using he hi d-o de EMHPM and (b) zoom in on chosen cu ing condi ions o ime-
domain simula ions. The selec ed poin s a e ma ked as ollows: uns able (c oss ma k), s able (ci cle
ma k) and ansi ion (plus ma k) cu ing condi ions.
Fu he mo e, he simula ed ib a ions o he chosen cu ing condi ions we e analyzed using
he con inuous wa ele ans o m (CWT), he powe spec al densi y (PSD) and Poinca é maps (PM).
The CWT is a ime- equency ep esen a ion o a signal ha o e s he capabili y o obse e how
equencies e ol e in ime. The scalog ams display he absolu e alue o CWT o he simula ed
ib a ion and he e o e, hey we e used o de ec cha e phenomena ha appea ed when milling
wi h a mul i a iable ool. The PSD is based on he Fou ie ans o m ha p o ides he ans o ma ion
om he ime-domain o he equency-domain. Addi ionally, PSD is de ined as he squa ed alue o
he signal and desc ibes he powe o a signal o ime se ies dis ibu ed o e di e en equencies
[46]. Mo eo e , a PM ep esen s poin s in phase space, which a e sampled e e y spindle o a ion [47].
The equencies 𝑓 o he CWT and PM we e no malized 𝑓
𝑛= 𝑓 𝑓ℎ
⁄ acco ding o he spindle
Figu e 6.
(
a
) Compa ison o s abili y lobes o egula (black solid line) and mul i a iable ( ed solid line)
cu e s by using he hi d-o de EMHPM and (
b
) zoom in on chosen cu ing condi ions o ime-domain
simula ions. The selec ed poin s a e ma ked as ollows: uns able (c oss ma k), s able (ci cle ma k) and
ansi ion (plus ma k) cu ing condi ions.
Fu he mo e, he simula ed ib a ions o he chosen cu ing condi ions we e analyzed using
he con inuous wa ele ans o m (CWT), he powe spec al densi y (PSD) and Poinca
é
maps (PM).
The CWT is a ime- equency ep esen a ion o a signal ha o e s he capabili y o obse e how
equencies e ol e in ime. The scalog ams display he absolu e alue o CWT o he simula ed
ib a ion and he e o e, hey we e used o de ec cha e phenomena ha appea ed when milling wi h
a mul i a iable ool. The PSD is based on he Fou ie ans o m ha p o ides he ans o ma ion om
he ime-domain o he equency-domain. Addi ionally, PSD is de ined as he squa ed alue o he
signal and desc ibes he powe o a signal o ime se ies dis ibu ed o e di e en equencies [
46
].
Mo eo e , a PM ep esen s poin s in phase space, which a e sampled e e y spindle o a ion [
47
].

Appl. Sci. 2020,10, 7869 17 o 22
The equencies
o he CWT and PM we e no malized
n= / h
acco ding o he spindle equency
h
. When milling wi h a egula milling ool he exci a ion equency
e
is equal o
zn
imes equencies
o he spindle speed
h
bu in a mul i a iable ool, he e a e se e al exci a ion equencies since he
angula spacing be ween ee h change as a unc ion o he axial dep h o cu .
Figu e 7illus a es he CWT, PSD and PM o simula ed ib a ions using he mul i a iable ool
wi h di e en axial dep hs deno ed as cu ing condi ions A, B and C o he axial dep hs o cu o 1.0,
1.7 and 1.7 mm espec i ely. Figu e 7a–c e e s o he ib a ions o he cu ing condi ions A ma ked
in Figu e 6, using a egula ool. The scalog am in Figu e 7a iden i ies poin A as a s able cu ing
since no malized cu ing equencies p esen a dominan alue o
n
=3.2, which co esponds o he
na u al equency
m=
132 Hz. This is also con i med by he PSD analysis shown in Figu e 7b. The
PM illus a ed in Figu e 7c shows a ib a ion ha dec eased wi h ime and sampled da a concen a ed
in he cen e con i med a ypical s able case. When he axial dep h o cu was inc eased o 1.7 mm,
he s abili y diag am p edic ed uns able cu ing condi ions acco ding o he s abili y lobes o he
egula ool. This case is deno ed wi h cu ing condi ions B and he co esponding scalog am (shown
in Figu e 7d) illus a ed how he in ensi y o he dominan equency inc eased wi h ime e en when
he exci a ion equency was he same as he case in A.
Appl. Sci. 2020, 10, x 19 o 25
equency 𝑓ℎ. When milling wi h a egula milling ool he exci a ion equency 𝑓
𝑒 is equal o 𝑧𝑛
imes equencies o he spindle speed 𝑓ℎ bu in a mul i a iable ool, he e a e se e al exci a ion
equencies since he angula spacing be ween ee h change as a unc ion o he axial dep h o cu .
Figu e 7 illus a es he CWT, PSD and PM o simula ed ib a ions using he mul i a iable ool
wi h di e en axial dep hs deno ed as cu ing condi ions A, B and C o he axial dep hs o cu o 1.0,
1.7 and 1.7 mm espec i ely. Figu e 7a–c e e s o he ib a ions o he cu ing condi ions A ma ked
in Figu e 6, using a egula ool. The scalog am in Figu e 7a iden i ies poin A as a s able cu ing since
no malized cu ing equencies p esen a dominan alue o 𝑓
𝑛 = 3.2, which co esponds o he
na u al equency 𝑓
𝑚=132 Hz. This is also con i med by he PSD analysis shown in Figu e 7b. The
PM illus a ed in Figu e 7c shows a ib a ion ha dec eased wi h ime and sampled da a
concen a ed in he cen e con i med a ypical s able case. When he axial dep h o cu was inc eased
o 1.7 mm, he s abili y diag am p edic ed uns able cu ing condi ions acco ding o he s abili y lobes
o he egula ool. This case is deno ed wi h cu ing condi ions B and he co esponding scalog am
(shown in Figu e 7d) illus a ed how he in ensi y o he dominan equency inc eased wi h ime
e en when he exci a ion equency was he same as he case in A.
(a)
(b)
(c)
(d)
(e)
( )
(g)
(h)
(i)
Figu e 7. Analysis o cu ing condi ions A, B and C. Con inuous wa ele ans o m (CWT) scalog ams:
(a,d,g); powe spec al densi y (PSD): (b,e,h) and Poinca é maps (PM): (c, ,i) co esponds o he
cu ing condi ions A, B and C espec i ely.
Figu e 7.
Analysis o cu ing condi ions A, B and C. Con inuous wa ele ans o m (CWT) scalog ams:
(
a
,
d
,
g
); powe spec al densi y (PSD): (
b
,
e
,
h
) and Poinca
é
maps (PM): (
c
,
,
i
) co esponds o he cu ing
condi ions A, B and C espec i ely.
Appl. Sci. 2020,10, 7869 18 o 22
The PM diag am shown in Figu e 7 exhibi ed a ib a ion a om ze o. In ac , he PM diag am
shows ha he ib a ion ampli ude g ows exponen ially because ou equa ion o mo ion did no
conside nonlinea e ec s such as hose ha appea ed when he ool los con ac wi h he wo kpiece.
Bo h cu ing condi ions A and B ag eed wi h he s abili y bounda ies in Figu e 6. Now, he cu ing
condi ions B we e used bu wi h a mul i a iable ool, which was e e ed o as cu ing condi ions
C. The CWT plo ed in Figu e 7g desc ibed comple ely di e en esul s since he e we e no single
dominan equencies in compa ison wi h cu ing condi ions A, bu appea ed se e al equencies
a ound
n
=3.2 and close o
n
=1 ha educed in in ensi y wi h ime, sugges ing a s able cu ing.
Figu e 7i illus a es how he ib a ion ampli ude app oached o ze o when using a mul i a iable ool
in con as o he PM ob ained o he egula ool and exhibi ed in Figu e 7 . This can be explained by
obse ing ha he e we e se e al exci a ion equencies due o he i egula pi ch and helix angles ha
b eak a single exci a ion equency a oiding egene a i e cha e phenomena.
Figu e 8illus a es he CWT, PSD and PM o simula ed ib a ions using he mul i a iable ool
wi h di e en axial dep hs deno ed as cu ing condi ions D, E, F and G o he axial dep hs o cu
o 2.3, 3.0, 8.55 and 18 mm espec i ely. No ice ha a s able case C was al eady alida ed when he
axial dep h was 1.7 mm in Figu e 7g–i ha co esponded o cu ing condi ions unde he s abili y
bounda ies shown in Figu e 6. Fo case D, a ansien cu ing condi ion was chosen e y close o he
c i ical axial dep h o he cu . I is in e es ing o poin ou ha ansi ion cu ing condi ions in he CWT
scalog am shown in Figu e 8a no only shows equencies wi h highe in ensi y in compa ison wi h
he s able case B, bu also p esen s shi ed equencies ha a ied in in ensi y e e y single e olu ion.
This shi ing sugges s a ma ginally s able cu ing condi ion ha was con i med by he PM illus a ed
in Figu e 8c, whe e ci cula ajec o ies we e desc ibed close o he cen e poin .
Appl. Sci. 2020, 10, x 20 o 25
The PM diag am shown in Figu e 7 exhibi ed a ib a ion a om ze o. In ac , he PM diag am
shows ha he ib a ion ampli ude g ows exponen ially because ou equa ion o mo ion did no
conside nonlinea e ec s such as hose ha appea ed when he ool los con ac wi h he wo kpiece.
Bo h cu ing condi ions A and B ag eed wi h he s abili y bounda ies in Figu e 6. Now, he cu ing
condi ions B we e used bu wi h a mul i a iable ool, which was e e ed o as cu ing condi ions C.
The CWT plo ed in Figu e 7g desc ibed comple ely di e en esul s since he e we e no single
dominan equencies in compa ison wi h cu ing condi ions A, bu appea ed se e al equencies
a ound 𝑓
𝑛 = 3.2 and close o 𝑓
𝑛 = 1 ha educed in in ensi y wi h ime, sugges ing a s able cu ing.
Figu e 7i illus a es how he ib a ion ampli ude app oached o ze o when using a mul i a iable ool
in con as o he PM ob ained o he egula ool and exhibi ed in Figu e 7 . This can be explained
by obse ing ha he e we e se e al exci a ion equencies due o he i egula pi ch and helix angles
ha b eak a single exci a ion equency a oiding egene a i e cha e phenomena.
Figu e 8 illus a es he CWT, PSD and PM o simula ed ib a ions using he mul i a iable ool
wi h di e en axial dep hs deno ed as cu ing condi ions D, E, F and G o he axial dep hs o cu o
2.3, 3.0, 8.55 and 18 mm espec i ely. No ice ha a s able case C was al eady alida ed when he axial
dep h was 1.7 mm in Figu e 7g–i ha co esponded o cu ing condi ions unde he s abili y
bounda ies shown in Figu e 6. Fo case D, a ansien cu ing condi ion was chosen e y close o he
c i ical axial dep h o he cu . I is in e es ing o poin ou ha ansi ion cu ing condi ions in he
CWT scalog am shown in Figu e 8a no only shows equencies wi h highe in ensi y in compa ison
wi h he s able case B, bu also p esen s shi ed equencies ha a ied in in ensi y e e y single
e olu ion. This shi ing sugges s a ma ginally s able cu ing condi ion ha was con i med by he PM
illus a ed in Figu e 8c, whe e ci cula ajec o ies we e desc ibed close o he cen e poin .
(a)
(b)
(c)
(d)
(e)
( )
Figu e 8. Con .
Appl. Sci. 2020,10, 7869 19 o 22
Appl. Sci. 2020, 10, x 21 o 25
(g)
(h)
(i)
(j)
(k)
(l)
Figu e 8. Analysis o cu ing condi ions D, E, F and G. CWT scalog am:s (a,d,g,j); PSD: (b,e,h,k) and
PM: (c, ,i,l) co esponds o he cu ing condi ions D, E, F and G espec i ely.
Uns able ib a ions ha appea ed o case E we e because o he in ensi y o equencies
inc eased exponen ially wi h ime, see Figu e 8d. No ice ha o he equencies a ose wi h ime close
o he alues o 𝑓
𝑛 = 0.5 and 𝑓
𝑛 = 1.5. These enquencies also occu ed o cu ing condi ions D,
which is an indica ion o he appea ance o cha e phenomena. In con as o Figu e 7i o a s able
case, Figu e 8 exhibi ed ew ajec o ies because he ib a ion ampli ude was ou o he ange
selec ed (±1 mm). The quali a i e and quan i a i e dynamic beha iou due o cu ing condi ions F,
and illus a ed in Figu e 8g-i, we e classi ied as ansi ion cu ing beha iou . He e, a mo e se e e
shi ing in equencies was obse ed in he scalog am (Figu e 8g). F om Figu e 8g, i is seen ha
d as ic shi ing occu ed in he ime domain in he ange o no malized equencies om 3.5 o 6. I
was also e iden in he PM showed in Figu e 8i, ha he ampli ude o ib a ion emained below 1
mm du ing se e al e olu ions o he ool bu he ampli ude o ib a ion ne e ap oached o he
cen e poin , in con as o he s able cu ing condi ion C shown in Figu e 7i in which he oscilla ion
apli udes ap oached he cen e .
An in e es ing dynamic beha iou was obse ed in he milling cu ing p ocess when he cu ing
condi ions we e selec ed in he middle o he s able peninsula, abo e uns able cu ing condi ions such
as E cu ing condi ions. The axial dep h o he cu was inc eased om he uns able axial dep h o cu
o 3–18 mm, 6 imes highe o he s able cu ing condi ion C and 2 imes highe han he uns able
condi ion E. Since he ib a ion quickly dec eased in a ew e olu ions no dominan equencies
appea ed in he CWT and PSD ailed o clea ly iden i y a dominan equency since he ib a ion
ampli ude dec eased o ze o a e ew e olu ions, as con i med by he PM shown in Figu e 8l.
Figu e 9 shows he no malized exci a ion equencies ha he mul i a iable ool p oduced o a
ixed spindle speed o 2500 pm. The o al numbe o disks o 50 μm o hickness was g ouped in se s
o each millime e in he axial di ec ion. The wa e all plo in Figu e 9 explains ha a s able peninsula
was o med abo e 11 mm because he wo kpiece was exci ed wi h se e al equencies
simul aneously. Fo ins ance, o a milling ope a ion wi h he axial dep h o cu o 1 mm (s able
cu ing), 80 disc e e disks we e cu wi h ou no malized exci a ion equencies alues (3.3, 3.6, 4.5
and 5.1). On he o he hand, when milling a 18 mm (s able cu ing), he e we e 14 no malized
Appl. Sci. 2020, 10, x 21 o 25
(g)
(h)
(i)
(j)
(k)
(l)
Figu e 8. Analysis o cu ing condi ions D, E, F and G. CWT scalog am:s (a,d,g,j); PSD: (b,e,h,k) and
PM: (c, ,i,l) co esponds o he cu ing condi ions D, E, F and G espec i ely.
Uns able ib a ions ha appea ed o case E we e because o he in ensi y o equencies
inc eased exponen ially wi h ime, see Figu e 8d. No ice ha o he equencies a ose wi h ime close
o he alues o 𝑓
𝑛 = 0.5 and 𝑓
𝑛 = 1.5. These enquencies also occu ed o cu ing condi ions D,
which is an indica ion o he appea ance o cha e phenomena. In con as o Figu e 7i o a s able
case, Figu e 8 exhibi ed ew ajec o ies because he ib a ion ampli ude was ou o he ange
selec ed (±1 mm). The quali a i e and quan i a i e dynamic beha iou due o cu ing condi ions F,
and illus a ed in Figu e 8g-i, we e classi ied as ansi ion cu ing beha iou . He e, a mo e se e e
shi ing in equencies was obse ed in he scalog am (Figu e 8g). F om Figu e 8g, i is seen ha
d as ic shi ing occu ed in he ime domain in he ange o no malized equencies om 3.5 o 6. I
was also e iden in he PM showed in Figu e 8i, ha he ampli ude o ib a ion emained below 1
mm du ing se e al e olu ions o he ool bu he ampli ude o ib a ion ne e ap oached o he
cen e poin , in con as o he s able cu ing condi ion C shown in Figu e 7i in which he oscilla ion
apli udes ap oached he cen e .
An in e es ing dynamic beha iou was obse ed in he milling cu ing p ocess when he cu ing
condi ions we e selec ed in he middle o he s able peninsula, abo e uns able cu ing condi ions such
as E cu ing condi ions. The axial dep h o he cu was inc eased om he uns able axial dep h o cu
o 3–18 mm, 6 imes highe o he s able cu ing condi ion C and 2 imes highe han he uns able
condi ion E. Since he ib a ion quickly dec eased in a ew e olu ions no dominan equencies
appea ed in he CWT and PSD ailed o clea ly iden i y a dominan equency since he ib a ion
ampli ude dec eased o ze o a e ew e olu ions, as con i med by he PM shown in Figu e 8l.
Figu e 9 shows he no malized exci a ion equencies ha he mul i a iable ool p oduced o a
ixed spindle speed o 2500 pm. The o al numbe o disks o 50 μm o hickness was g ouped in se s
o each millime e in he axial di ec ion. The wa e all plo in Figu e 9 explains ha a s able peninsula
was o med abo e 11 mm because he wo kpiece was exci ed wi h se e al equencies
simul aneously. Fo ins ance, o a milling ope a ion wi h he axial dep h o cu o 1 mm (s able
cu ing), 80 disc e e disks we e cu wi h ou no malized exci a ion equencies alues (3.3, 3.6, 4.5
and 5.1). On he o he hand, when milling a 18 mm (s able cu ing), he e we e 14 no malized
Figu e 8.
Analysis o cu ing condi ions D, E, F and G. CWT scalog am:s (
a
,
d
,
g
,
j
); PSD: (
b
,
e
,
h
,
k
) and
PM: (c, ,i,l) co esponds o he cu ing condi ions D, E, F and G espec i ely.
Uns able ib a ions ha appea ed o case E we e because o he in ensi y o equencies inc eased
exponen ially wi h ime, see Figu e 8d. No ice ha o he equencies a ose wi h ime close o he
alues o
n
=0.5 and
n
=1.5. These enquencies also occu ed o cu ing condi ions D, which is an
indica ion o he appea ance o cha e phenomena. In con as o Figu e 7i o a s able case, Figu e 8
exhibi ed ew ajec o ies because he ib a ion ampli ude was ou o he ange selec ed (
±
1 mm).
The quali a i e and quan i a i e dynamic beha iou due o cu ing condi ions F, and illus a ed in
Figu e 8g-i, we e classi ied as ansi ion cu ing beha iou . He e, a mo e se e e shi ing in equencies
was obse ed in he scalog am (Figu e 8g). F om Figu e 8g, i is seen ha d as ic shi ing occu ed in
he ime domain in he ange o no malized equencies om 3.5 o 6. I was also e iden in he PM
showed in Figu e 8i, ha he ampli ude o ib a ion emained below 1 mm du ing se e al e olu ions
o he ool bu he ampli ude o ib a ion ne e ap oached o he cen e poin , in con as o he s able
cu ing condi ion C shown in Figu e 7i in which he oscilla ion apli udes ap oached he cen e .
An in e es ing dynamic beha iou was obse ed in he milling cu ing p ocess when he cu ing
condi ions we e selec ed in he middle o he s able peninsula, abo e uns able cu ing condi ions such
as E cu ing condi ions. The axial dep h o he cu was inc eased om he uns able axial dep h o cu o
3–18 mm, 6 imes highe o he s able cu ing condi ion C and 2 imes highe han he uns able condi ion
E. Since he ib a ion quickly dec eased in a ew e olu ions no dominan equencies appea ed in he
CWT and PSD ailed o clea ly iden i y a dominan equency since he ib a ion ampli ude dec eased
o ze o a e ew e olu ions, as con i med by he PM shown in Figu e 8l.
Figu e 9shows he no malized exci a ion equencies ha he mul i a iable ool p oduced o a
ixed spindle speed o 2500 pm. The o al numbe o disks o 50
µ
m o hickness was g ouped in se s
o each millime e in he axial di ec ion. The wa e all plo in Figu e 9explains ha a s able peninsula
was o med abo e 11 mm because he wo kpiece was exci ed wi h se e al equencies simul aneously.
Fo ins ance, o a milling ope a ion wi h he axial dep h o cu o 1 mm (s able cu ing), 80 disc e e
disks we e cu wi h ou no malized exci a ion equencies alues (3.3, 3.6, 4.5 and 5.1). On he o he
hand, when milling a 18 mm (s able cu ing), he e we e 14 no malized exci a ion equencies (3.30,
Appl. Sci. 2020,10, 7869 20 o 22
3.35, 3.39, 3.44, 3.49, 3.54, 3.60, 4.55, 4.64, 4.73, 4.82, 4.92, 5.02 and 5.13), mos o hem wi h a leas 115
disc e e disks.
Appl. Sci. 2020, 10, x 23 o 25
Figu e 9. The numbe o disc e e disks and disc e e exci a ion equencies as a unc ion o he axial
dep h o cu o he mul i a iable ool.
5. Conclusions
In his wo k, quad a ic and cubic polynomials we e used o app oxima e he delayed e ms o
delay di e en ial equa ions. Nume ical simula ions showed ha using second- and hi d-o de
EMHPM imp o ed he con e gence a e and equi ed less compu a ional ime when compa ed o
he i s -o de EMHPM, and o semi-disc e iza ion and ull-disc e iza ion me hods, since ewe
app oxima ions o less disc e e in e als we e needed o educe he compu a ion ime.
To u he assess he applicabili y o he p oposed me hod, he hi d-o de EMHPM was used
o de e mining he s abili y bounds in one-deg ee-o - eedom milling ope a ion wi h a mul i a iable
ool, demons a ing ha he s abili y zone imp o ed in compa ison wi h a egula ool. Fo ins ance,
a 2500 pm he c i ical axial dep h o he cu was 1.3 mm using he egula milling ool. Howe e ,
using he mul i a iable ool, he c i ical axial dep h o he cu was inc eased un il 2.17 mm bu mo e
in e es ing, a s able zone appea ed abo e 8.55 mm.
The CWT scalog ams, PSD cha s and PM we e employed o alida e he s abili y lobes ound
by using he hi d-o de EMHPM o he mul i a iable ool. Nume ical solu ions con i med he
sys em dynamics beha io p edic ed by he hi d-o de EMHPM.
Based on he abo e esul s, his pape p o ided e idence he hi d-o de EMHPM could be used
o s udy dynamic phenomena ha appea ed a highe axial dep hs o cu due o he mul i a iable
design o he ool, which b oke he exci a ion equencies a a lowe dep h o cu .
Au ho Con ibu ions: Concep ualiza ion, J.d.l.L.S. and D.O.-T.; Me hodology, J.d.l.L.S.; Resou ces, O.M.-R.,
A.E.-Z. and L.N.L.d.L.; Supe ision, D.O.-T., A.E.-Z. and L.N.L.d.L.; Valida ion, J.d.l.L.S., D.O.-T. and G.U.;
W i ing—o iginal d a , J.d.l.L.S. and D.O.-T.; W i ing— e iew and edi ing, O.M.-R., G.U. and A.E.-Z. All
au ho s ha e ead and ag eed o he published e sion o he manusc ip .
Funding: This esea ch was unded by Tecnológico de Mon e ey h ough he Resea ch G oup o
Nano echnology o De ices Design, and by he Consejo Nacional de Ciencia y Tecnología de México (Conacy ),
P ojec Numbe s 242269, 255837, 296176, and Na ional Lab in Addi i e Manu ac u ing, 3D Digi izing and
Compu ed Tomog aphy (MADiT) LN299129.
Acknowledgmen s: The au ho s acknowledge he echnical assis ance om Osca Escale a a Tecnologico de
Mon e ey.
Con lic s o In e es : The au ho s decla e no con lic o in e es . The ounding sponso s had no ole in he design
o he s udy; in he collec ion, analyses, o in e p e a ion o da a; in he w i ing o he manusc ip , and in he
decision o publish he esul s.
Re e ences
1. Mi, T.; Chen, N.; S epan, G.; Takacs, D. Ene gy dis ibu ion o a ehicle shimmy sys em wi h he delayed
y e model. IFAC-Pape sOnLine 2018, 51, 7–12, doi:10.1016/j.i acol.2018.07.190.
2. O osz, G.; S épán, G. Subc i ical hop bi u ca ions in a ca - ollowing model wi h eac ion- ime delay. P oc.
R. Soc. A Ma h. Phys. Eng. Sci. 2006, 462, 2643–2670, doi:10.1098/ spa.2006.1660.
3. Hu, H.; Zaihua, W. Dynamics o Con olled Mechanical Sys ems wi h Delayed Feedback; Sp inge Science &
Business Media: Be lin, Ge many, 2002; ISBN 9783642078392.
Figu e 9.
The numbe o disc e e disks and disc e e exci a ion equencies as a unc ion o he axial
dep h o cu o he mul i a iable ool.
5. Conclusions
In his wo k, quad a ic and cubic polynomials we e used o app oxima e he delayed e ms
o delay di e en ial equa ions. Nume ical simula ions showed ha using second- and hi d-o de
EMHPM imp o ed he con e gence a e and equi ed less compu a ional ime when compa ed
o he i s -o de EMHPM, and o semi-disc e iza ion and ull-disc e iza ion me hods, since ewe
app oxima ions o less disc e e in e als we e needed o educe he compu a ion ime.
To u he assess he applicabili y o he p oposed me hod, he hi d-o de EMHPM was used o
de e mining he s abili y bounds in one-deg ee-o - eedom milling ope a ion wi h a mul i a iable
ool, demons a ing ha he s abili y zone imp o ed in compa ison wi h a egula ool. Fo ins ance,
a 2500 pm he c i ical axial dep h o he cu was 1.3 mm using he egula milling ool. Howe e ,
using he mul i a iable ool, he c i ical axial dep h o he cu was inc eased un il 2.17 mm bu mo e
in e es ing, a s able zone appea ed abo e 8.55 mm.
The CWT scalog ams, PSD cha s and PM we e employed o alida e he s abili y lobes ound by
using he hi d-o de EMHPM o he mul i a iable ool. Nume ical solu ions con i med he sys em
dynamics beha io p edic ed by he hi d-o de EMHPM.
Based on he abo e esul s, his pape p o ided e idence he hi d-o de EMHPM could be used
o s udy dynamic phenomena ha appea ed a highe axial dep hs o cu due o he mul i a iable
design o he ool, which b oke he exci a ion equencies a a lowe dep h o cu .
Au ho Con ibu ions:
Concep ualiza ion, J.d.l.L.S. and D.O.-T.; Me hodology, J.d.l.L.S.; Resou ces, O.M.-R.,
A.E.-Z. and L.N.L.d.L.; Supe ision, D.O.-T., A.E.-Z. and L.N.L.d.L.; Valida ion, J.d.l.L.S., D.O.-T. and G.U.;
W i ing—o iginal d a , J.d.l.L.S. and D.O.-T.; W i ing— e iew and edi ing, O.M.-R., G.U. and A.E.-Z. All au ho s
ha e ead and ag eed o he published e sion o he manusc ip .
Funding:
This esea ch was unded by Tecnol
ó
gico de Mon e ey h ough he Resea ch G oup o Nano echnology
o De ices Design, and by he Consejo Nacional de Ciencia y Tecnolog
í
a de M
é
xico (Conacy ), P ojec Numbe s
242269, 255837, 296176, and Na ional Lab in Addi i e Manu ac u ing, 3D Digi izing and Compu ed Tomog aphy
(MADiT) LN299129.
Acknowledgmen s:
The au ho s acknowledge he echnical assis ance om Osca Escale a a Tecnologico
de Mon e ey.
Con lic s o In e es :
The au ho s decla e no con lic o in e es . The ounding sponso s had no ole in he design
o he s udy; in he collec ion, analyses, o in e p e a ion o da a; in he w i ing o he manusc ip , and in he
decision o publish he esul s.
Re e ences
1.
Mi, T.; Chen, N.; S epan, G.; Takacs, D. Ene gy dis ibu ion o a ehicle shimmy sys em wi h he delayed y e
model. IFAC-Pape sOnLine 2018,51, 7–12. [C ossRe ]
2.
O osz, G.; S
é
p
á
n, G. Subc i ical hop bi u ca ions in a ca - ollowing model wi h eac ion- ime delay. P oc. R.
Soc. A Ma h. Phys. Eng. Sci. 2006,462, 2643–2670. [C ossRe ]
Appl. Sci. 2020,10, 7869 21 o 22
3.
Hu, H.; Zaihua, W. Dynamics o Con olled Mechanical Sys ems wi h Delayed Feedback; Sp inge Science &
Business Media: Be lin, Ge many, 2002; ISBN 9783642078392.
4.
Al in as, Y. Manu ac u ing Au oma ion. Me al Cu ing Mechanics, Machine Tool Vib a ions, and CNC Design;
Camb idge Uni e si y P ess: New Yo k, NY, USA, 2012.
5.
Kuljanic, E.; To is, G.; So ino, M. De elopmen o an in elligen mul isenso cha e de ec ion sys em in
milling. Mech. Sys . Signal P ocess. 2009,23, 1704–1718. [C ossRe ]
6.
Zhuo, Y.; Jin, H.; Han, Z. Cha e iden i ica ion in lank milling o hin-walled blade based on ac al
dimension. P ocedia Manu . 2020,49, 150–154. [C ossRe ]
7.
Paul, S.; Mo ales-Menendez, R. Cha e mi iga ion in milling p ocess using disc e e ime sliding mode con ol
wi h ype 2- uzzy logic sys em. Appl. Sci. 2019,9, 4380. [C ossRe ]
8.
Peng, C.; Wang, L.; Liao, T.W. A new me hod o he p edic ion o cha e s abili y lobes based on dynamic
cu ing o ce simula ion model and suppo ec o machine. J. Sound Vib. 2015,354, 118–131. [C ossRe ]
9.
Li, D.; Cao, H.; Chen, X. Fuzzy con ol o milling cha e wi h piezoelec ic ac ua o s embedded o he ool
holde . Mech. Sys . Signal P ocess. 2020,148, 107190. [C ossRe ]
10.
Wan, S.; Li, X.; Su, W.; Yuan, J.; Hong, J.; Jin, X. Ac i e damping o milling cha e ib a ion ia a no el
spindle sys em wi h an in eg a ed elec omagne ic ac ua o . P ecis. Eng. 2019,57, 203–210. [C ossRe ]
11.
Munoa, J.; Sanz-Calle, M.; Dombo a i, Z.; Iglesias, A.; Pena-Ba io, J.; S epan, G. Tuneable clamping able o
cha e a oidance in hin-walled pa milling. CIRP Ann. 2020,69, 313–316. [C ossRe ]
12.
Wang, Y.; Wang, T.; Yu, Z.; Zhang, Y.; Wang, Y.; Liu, H. Cha e p edic ion o a iable pi ch and a iable
helix milling. Shock Vib. 2015,2015. [C ossRe ]
13.
Mei, Y.; Mo, R.; Sun, H.; He, B.; Bu, K. S abili y analysis o milling p ocess wi h mul iple delays. Appl. Sci.
2020,10, 3646. [C ossRe ]
14.
Wan, M.; Zhang, W.H.; Dang, J.W.; Yang, Y. A uni ied s abili y p edic ion me hod o milling p ocess wi h
mul iple delays. In . J. Mach. Tools Manu . 2010,50, 29–41. [C ossRe ]
15.
Zhan, D.; Jiang, S.; Niu, J.; Sun, Y. Dynamics modeling and s abili y analysis o i e-axis ball-end milling
sys em wi h a iable pi ch ools. In . J. Mech. Sci. 2020,182, 105774. [C ossRe ]
16. Sla icek, J. The e ec o i egula oo h pi ch on s abili y o milling. P oc. 6 h MTDR Con . 1965,1, 15–22.
17.
Budak, E. An analy ical design me hod o milling cu e s wi h noncons an pi ch o inc ease s abili y, Pa I:
Theo y. J. Manu . Sci. Eng. T ans. ASME 2003,125, 29–34. [C ossRe ]
18.
Budak, E. An analy ical design me hod o milling cu e s wi h noncons an pi ch o inc ease s abili y, Pa 2:
Applica ion. J. Manu . Sci. Eng. T ans. ASME 2003,125, 35–38. [C ossRe ]
19.
Al in as, Y.; Engin, S.; Budak, E. Analy ical s abili y p edic ion and design o a iable pi ch cu e s. J. Manu .
Sci. Eng. T ans. ASME 1999,121, 173–178. [C ossRe ]
20.
Olgac, N.; Sipahi, R. Dynamics and s abili y o a iable-pi ch milling. J. Vib. Con ol
2007
,13, 1031–1043.
[C ossRe ]
21.
Jin, G.; Zhang, Q.; Hao, S.; Xie, Q. S abili y p edic ion o milling p ocess wi h a iable pi ch cu e . Ma h.
P obl. Eng. 2013,2013. [C ossRe ]
22.
Comak, A.; Budak, E. Modeling dynamics and s abili y o a iable pi ch and helix milling ools o
de elopmen o a design me hod o maximize cha e s abili y. P ecis. Eng. 2017,47, 459–468. [C ossRe ]
23.
Za a ain, M.; Muñoa, J.; Peigne, G.; Inspe ge , T. Analysis o he in luence o mill helix angle on cha e
s abili y. CIRP Ann. Manu . Technol. 2006,55, 365–368. [C ossRe ]
24.
Budak, E.; Al in as, Y. Analy ical p edic ion o cha e s abili y in milling—Pa I: Gene al o mula ion.
Am. Soc. Mech. Eng. Dyn. Sys . Meas. Con ol 1998,120, 22–30. [C ossRe ]
25.
Inspe ge , T.; Muñoa, J.; Za a ain, M.; Peign
é
, G. Uns able islands in he s abili y cha o milling p ocesses
due o he helix angle. In P oceedings o he CIRP 2nd In e na ional Con e ence High Pe o mance Cu ing,
Vancou e , BC, Canada, 12–13 June 2006.
26.
Pa el, B.R.; Mann, B.P.; Young, K.A. Uncha ed islands o cha e ins abili y in milling. In . J. Mach. Tools Manu .
2008,48, 124–134. [C ossRe ]
27.
Sims, N.D.; Mann, B.; Huyanan, S. Analy ical p edic ion o cha e s abili y o a iable pi ch and a iable
helix milling ools. J. Sound Vib. 2008,317, 664–686. [C ossRe ]
28.
Tu ne , S.; Me dol, D.; Al in as, Y.; Ridgway, K. Modelling o he s abili y o a iable helix end mills. In . J.
Mach. Tools Manu . 2007,47, 1410–1416. [C ossRe ]

Appl. Sci. 2020,10, 7869 22 o 22
29.
Yuso , A.R.; Sims, N.D. Op imisa ion o a iable helix ool geome y o egene a i e cha e mi iga ion.
In . J. Mach. Tools Manu . 2011,51, 133–141. [C ossRe ]
30.
Dombo a i, Z.; S epan, G. The e ec o helix angle a ia ion on milling s abili y. J. Manu . Sci. Eng.
T ans. ASME 2012,134. [C ossRe ]
31.
Huang, J.; Deng, P.; Li, H.; Wen, B. S abili y analysis o milling sys em wi h a iable pi ch cu e s unde
a iable speed. J. Vib oeng. 2019,21, 331–347. [C ossRe ]
32.
Cai, S.; Yao, B.; Feng, W.; Cai, Z.; Chen, B.; He, Z. Milling p ocess simula ion o he a iable pi ch cu e
based on an in eg a ed p ocess-machine model. In . J. Ad . Manu . Technol.
2020
,106, 2779–2791. [C ossRe ]
33.
Ol e a, D.; El
í
as-Z
ú
ñiga, A.; L
ó
pez De Lacalle, L.N.; Rod
í
guez, C.A. App oxima e solu ions o delay
di e en ial equa ions wi h cons an and a iable coe icien s by he enhanced mul is age homo opy
pe u ba ion me hod. Abs . Appl. Anal. 2015, 1–12. [C ossRe ]
34.
Compe
á
n, F.I.; Ol e a, D.; Campa, F.J.; L
ó
pez De Lacalle, L.N.; El
í
as-Z
ú
ñiga, A.; Rod
í
guez, C.A.
Cha ac e iza ion and s abili y analysis o a mul i a iable milling ool by he enhanced mul is age homo opy
pe u ba ion me hod. In . J. Mach. Tools Manu . 2012,57, 27–33. [C ossRe ]
35.
Sims, N.D. Fas cha e s abili y p edic ion o a iable helix milling ools. P oc. Ins . Mech. Eng. Pa C J.
Mech. Eng. Sci. 2016,230, 133–144. [C ossRe ]
36.
O o, A.; Rauh, S.; Ihlen eld , S.; Radons, G. S abili y o milling wi h non-uni o m pi ch and a iable helix
Tools. In . J. Ad . Manu . Technol. 2017,89, 2613–2625. [C ossRe ]
37.
Niu, J.; Ding, Y.; Zhu, L.M.; Ding, H. Mechanics and mul i- egene a i e s abili y o a iable pi ch and a iable
helix milling ools conside ing unou . In . J. Mach. Tools Manu . 2017,123, 129–145. [C ossRe ]
38.
Ol e a, D.; U bikain, G.; El
í
as-Zuñiga, A.; L
ó
pez de Lacalle, L.N. Imp o ing s abili y p edic ion in pe iphe al
milling o Al7075T6. Appl. Sci. 2018,8, 1316. [C ossRe ]
39.
Iglesias, A.; Dombo a i, Z.; Gonzalez, G.; Munoa, J.; S epan, G. Op imum selec ion o a iable pi ch o
cha e supp ession in ace milling ope a ions. Ma e ials 2018,12, 112. [C ossRe ] [PubMed]
40.
Guo, Y.; Lin, B.; Wang, W. Op imiza ion o a iable helix cu e o imp o ing cha e s abili y. In . J. Ad .
Manu . Technol. 2019,104, 2553–2565. [C ossRe ]
41.
Hashim, I.; Chowdhu y, M.S.H. Adap a ion o homo opy-pe u ba ion me hod o nume ic-analy ic solu ion
o sys em o ODEs. Phys. Le . Sec . A 2008,372, 470–481. [C ossRe ]
42.
Puma-A aujo, S.D.; Ol e a-T ejo, D.; Ma
í
nez-Rome o, O.; U bikain, G.; El
í
as-Z
ú
ñiga, A.; Lacalle, L.N.L. de
Semi-ac i e magne o heological dampe de ice o cha e mi iga ion du ing milling o hin- loo componen s.
Appl. Sci. 2020,10, 5313. [C ossRe ]
43.
Inspe ge , T.; S
é
p
á
n, G. Upda ed semi-disc e iza ion me hod o pe iodic delay-di e en ial equa ions wi h
disc e e delay. In . J. Nume . Me hods Eng. 2004,61, 117–141. [C ossRe ]
44.
Inspe ge , T. Full-disc e iza ion and semi-disc e iza ion o milling s abili y p edic ion: Some commen s.
In . J. Mach. Tools Manu . 2010,50, 658–662. [C ossRe ]
45.
Ding, Y.; Zhu, L.M.; Zhang, X.J.; Ding, H. A ull-disc e iza ion me hod o p edic ion o milling s abili y.
In . J. Mach. Tools Manu . 2010,50, 502–509. [C ossRe ]
46.
Lee, E.T.; Eun, H.C. S uc u al damage de ec ion by powe spec al densi y es ima ion using ou pu -only
measu emen . Shock Vib. 2016,2016. [C ossRe ]
47. F ies, R.H. Fundamen als o Vib a ions; Wa eland P ess: Long G o e, IL, USA, 2000; ISBN 9781482270372.
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