A method to measure the damping introduced by linear guides in large milling machines

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A METHOD TO MEASURE THE DAMPING INTRODUCED BY LINEAR GUIDES
IN LARGE MILLING MACHINES
I. Oleagaa,*,J.J. Zulaikaa,, F.J. Campab and J. He nandoc.
a Tecnalia, Paseo Mikele egi 7, 20009 Donos ia-San Sebas ian, Spain
{ibone.oleaga, juanjo.zulaika}@ ecnalia.com
b Uni e si y o he Basque Coun y, Depa men o Mechanical Enginee ing
Alameda de U quijo s/n, 48013 Bilbao, Spain
[email protected]
c Nicolás Co ea, calle Alcalde Ma ín Cobos, 16ª, 09007 Bu gos, Spain
j.he nando@co ea.es
Abs ac . Simula ion o he dynamic beha io o a milling machine equi es accu a e s i ness,
ine ia, and damping alues. Unlike he p ope ies o s i ness and ine ia, damping alues a e no
gene ally a ailable in he bibliog aphy, which con ains only e e ence alues. In he p esen wo k,
a me hod is p oposed o calcula e and o iden i y he damping in oduced by he linea guides
assembled in la ge-scale milling machines. One immedia e ad an age o his me hod would be o
imp o e he dynamic equency esponses o milling machines, and as a consequence, hei
p oduc i i y, whe e e he condi ions o cha e a ise ha limi p oduc i i y. The me hod is based
on a combina ion o expe imen al equency esponses and heo e ical modes ob ained om a
ini e elemen model, whe e damping is ep esen ed by means o ideal disc e e dampe s. Hence,
ene gy dissipa ion can be accu a ely ep esen ed by associa ing he damping o each linea guide
in a disc e e way, a he speci ic poin s whe e he dissipa ion o ene gy occu s. The me hod has
been sa is ac o ily alida ed on a eal la ge machine.
Keywo ds: Machine ool dynamics; damping; linea guides; modeling.
*Co esponding au ho . Tel.: +34 946 430 850; ax: +34 946 460 900.
This is he accep ed manusc ip o he a icle ha appea ed in inal o m in Mechanical Sys ems and Signal P ocessing 171 : (2022) // A icle
ID 108908, which has been published in inal o m a h ps://doi.o g/10.1016/j.ymssp.2022.108908. © 2022 Else ie unde CC BY-NC-ND
license (h p://c ea i ecommons.o g/licenses/by-nc-nd/4.0/)
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NOMENCLATURE
B Numbe o nodes o ini e elemen model
c Damping cons an
cm Modal damping cons an
ckl Damping alue o a k node in he s uc u e in he l di ec ion
ck lq Damping alue o an l node in he q di ec ion when a k node is exi ed in he
di ec ion
[𝐶𝐶] Damping ma ix
[𝐶𝐶𝑚𝑚] Modal damping ma ix
{𝐹𝐹(𝑡𝑡)} Ex e nal o ce ec o
{Fm} Ex e nal o ce ec o in modal coo dina es
[I] Iden i y ma ix
k Sp ing cons an s i ness
[𝐾𝐾] S i ness ma ix
m Mass
𝑚𝑚𝑖𝑖𝑖𝑖 Mass alue o j deg ees o eedom in i mode
𝑚𝑚𝑚𝑚𝑖𝑖𝑖𝑖 Modal mass alue o j deg ees o eedom in i mode
[𝑀𝑀] Mass ma ix
[𝑀𝑀𝑚𝑚] Modal mass ma ix
N Numbe o deg ees o eedom o ini e elemen model
{p},{ 𝑝𝑝󰇗},{𝑝𝑝󰇘} Displacemen , eloci y, and accele a ion ec o s in modal coo dina es.
P Ex e nal o ce in he Laplace domain
P0 Maximum o ce
𝑃𝑃0𝑚𝑚 Modal maximum o ce
s Laplace a iable
x, x󰇗, x󰇘 Displacemen , eloci y, and accele a ion
{x},{ x󰇗},{x󰇘} Displacemen , eloci y and accele a ion ec o in na u al coo dina es
X, X
󰇗 Displacemen , eloci y in Laplace domain
X0 Maximum displacemen ampli ude
𝑋𝑋0𝑚𝑚 Modal maximum displacemen
[𝑋𝑋] O hono mal modal ec o s
ξ
ij Modal damping coe icien in he i di ec ion in j mode.
ϕij Mass no malized modal ec o a poin i, in he j di ec ion
ωn Na u al equency
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1. INTRODUCTION
The p oduc i i y o milling ope a ions is equen ly es ic ed by he gene a ion o sel -exci ed
ib a ions, especially egene a i e cha e ib a ions [1, 2]. The modes exci ed in cha e
ib a ions can be om he s uc u al componen s o he machine [3, 4], om he spindle/ ool-
holde / ool sys em [5], o om he wo kpiece, when i has lexible hin walls o loo s [6, 7].
Cha e is nowadays a majo limi a ion o highe p oduc i i y le els in many machining p ocesses
[8]. Fo online de ec ion, one al e na i e is he es ima ion o he dominan cha e equency by
designing algo i hm-based adap i e il e s [9]. A e de ec ion, se e al au ho s ha e p oposed
me hods as he supp ession o he ib a ion h ough he combina ion o online a ia ion o he
spindle speed changing he pe iodici y o oo h impac s [10, 11, 12, 13], he use o a wo-link
obo ic a m [14], he use o uned mass dampe s (TMD) [15,16, 17, 18], o by using ac i e
damping de ices (ADD) [19, 20]. Fo o line a oidance, o e he pas i y yea s, a ious au ho s
ha e been wo king o unde s and, model and p edic he onse o machine ool cha e by
calcula ing he s abili y lobe diag ams [21, 22, 23].
In machining, he ine ia, s i ness and damping o he whole machining sys em, machine-
ool-wo kpiece- ooling, de e mines he dynamic pe o mance o he p ocess, hus a ec ing he
dimensional accu acy and su ace inish o he pa and he in eg i y o he ool and machine ool
componen s [24-25]. The p oblem unde s udy becomes e en mo e complex when ce ain e ec s
a e conside ed, such as mul i-poin con ac , p ocess damping, and he a ia ion o he modal
pa ame e s, due o machine ool mo ion and ma e ial emo al om a wo kpiece [27-29]. F om
he poin o iew o he machine ool builde , he iden i ica ion o he mos limi ing modes
ega ding cha e and hei modal pa ame e s helps o imp o e bo h he machine design and i s
p oduc i i y. To do so, a modal analysis using ini e elemen s mus be pe o med o calcula e he
in luence o he modes a he Tool Cen e Poin (TCP), bu he damping da a is also equi ed and
de e minan o a good p edic ion o he machine dynamic s abili y.
Milling machine damping can be classi ied in wo main ypes: inhe en damping ha occu s
na u ally, such as ma e ial damping o join -induced damping in a unc ional machine, and
addi ional damping, which esul s om specially cons uc ed de ices added o he machine
s uc u e as TMDs o ADDs [33]. In his pape , only inhe en damping is conside ed. The concep
o damping in ol es he physical mechanisms ha dissipa e mechanical ene gy [8, 30], which, in
u n, enhance he dynamic s abili y o mechanical sys ems. A p esen , he bibliog aphy con ains
only e e ence alues. Schlech [31] p o ided an o e iew o ypical damping alues o olling
bea ings, anging be ween 2 N s mm-1 and 1000 N s mm-1, o bea ings wi h inne diame e s o
55 mm and 160 mm, espec i ely. G oche and Ho mann [32] ga e an o e iew o ypical damping
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alues o linea olle and ball guideways om 4 N s mm-1 o linea guides o 35 mm in size o
10.5 N s mm-1 o linea guides o 45 mm in size.
E en i he damping in a machine ool canno be p edic ed accu a ely, acco ding o Beads
[34] and Weck [35], app oxima ely 90% o he o al ib a o y ene gy dissipa ion o he en i e
machine ool s uc u e o igina es a he join s. The emaining 10%, known as ma e ial damping,
o igina es in ma e ial du ing cyclic de o ma ion. In gene al, as long as he s ess a igue limi , he
ma e ial mel ing poin empe a u e, and he audio equency ange a e no exceeded, he ma e ial
damping will be independen o empe a u e, ib a ion ampli ude, and equency [36]. Wi hin
machine ool join s, linea guides can be highligh ed because o hei high capaci y o dissipa e
ene gy due o hei la ge con ac su ace. I is he e o e o in e es o ocus on he s udy o linea
guide damping and ma e ial damping o de elop accu a e models o machine ool damping.
Machine damping in ini e elemen modelling (FEM) can be ep esen ed in di e en o ms.
Modal damping consis s o associa ing a modal damping coe icien wi h each mode o na u al
equency. When damping is cha ac e ized in ha way, a global alue in a con inuous o m is
associa ed wi h he whole s uc u e o each na u al equency and a damping coe icien o 0.02
is usually conside ed [37]. Di e en modal damping coe icien measu emen me hods can be
ound in he bibliog aphy. Fo example, Mon al ão [38] p esen ed a no el me hod o he
iden i ica ion o he modal damping coe icien s o each na u al equency om FRFs, based on
dissipa ion ene gy pe ib a ion cycle. Zhang [39] analyzed he dynamic cha ac e is ics o a
guideway and p oposed a sys ema ic p ocedu e o p edic he dynamic beha io o a whole
machine- ool s uc u e wi h espec o na u al equencies. Howe e , no absolu e eal damping
alues a e associa ed wi h he machine elemen s in e ms o modal damping. Al hough i is a good
way o analyzing he global beha io o a machine, his me hod canno iden i y he sou ces o
damping in a gi en machine, and in he case o a edesign, i canno help iden i ying he s uc u al
join s whe e a change in damping will ha e mos in luence.
Ano he way o model damping by FEM is o ep esen he iscous damping o he linea
guide h ough a dis ibu ion o disc e e dampe s oge he wi h ma e ial hys e e ic damping. The e
a e many mechanisms whe eby ib a ional ene gy can be dissipa ed wi hin he olume o a
ma e ial elemen as i is cyclically de o med [40]. Such mechanisms a e associa ed wi h in e nal
econs uc ions o he mic o and mac o s uc u e. Accu a e ma e ial damping alues can be ound
in he bibliog aphy [41], as he majo i y o published in o ma ion is o empi ical na u e. On he
o he hand, he dynamic cha ac e is ics o linea guides in a machine- ool s uc u e a e de e mined
by many ac o s [39], such as he size o he linea guide, exci a ion o ce equencies and na u al
machine equencies, he alue o modal ec o s o linea guides a hose equencies, dis ibu ed
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p essu e on he join su aces, he ypes o guides, he guide ma e ial, lub ica ion on he linea
guide su aces, and he su ace inishing me hod, among o he s. A ew e e ence alues o he
disc e e damping can be ound in he bibliog aphy. Se e al wo ks ha e been comple ed o he
iden i ica ion o uni e sal damping pa ame e s o olle linea guide elemen s. Deng [42], Popo
[43], De Vicen e [44], and Al-Bende [45] s udied he in e ace beha io a olling con ac . A
sou ce o unce ain y o hese s udies is ha he elemen s a e s udied in isola ion om he o he
elemen s, a he han when assembled in he milling machine. B eche , Fey and Bäume [46]
desc ibed a me hodology o iden i y disc e e dampe alues o machine ool componen s o linea
axes. To do so, hey used a specially designed es bench. Albe [47], Neugebaue [48],
Ross eusche [49] and Wu [50] also measu ed linea guide damping, al hough hei measu emen s
we e also done on linea guides assembled on a es bench. Howe e , he posi ion o he linea
guide in a machine and he loads i ca ies will change om one machine o ano he , which means
di e en damping alues o each machine and posi ion. In he expe ience o he au ho s, e en
guides wi h he same cha ac e is ics on he same machine, bu joining di e en componen s, can
dissipa e di e en amoun s o ene gy. The e o e, he es s pe o med on a guide assembled on a
es bench will no always be ep esen a i e, as can easily be seen when analyzing a machine wi h
he same linea guides joining di e en componen s [51]. I hese linea guides a e eplaced one
by one wi h highe damping linea guides, he in luence o he new linea guide a he TCP will
no be always he same. Hence, he e is a need o de elop a me hod ha can calcula e he damping
in oduced by a guide once assembled in he machine.
The s udy p esen ed in his pape will in oduce a heo e ical-expe imen al me hodology o
measu e absolu e damping alues o linea guides assembled in he milling machine. The
ma hema ical basis o he me hod will be i s p esen ed. Then, i will be applied and alida ed in
an ope a ional machine and, inally, he conclusions will be p esen ed.
2. PROCEDURE TO MEASURE THE DAMPING OF THE ASSEMBLED LINEAR
GUIDES
Linea guide damping in a milling machine can be disc e ely modeled using FEM so wa e
o unde s and i s e ec and p edic he machine dynamic beha io . Bu o do so, a me hod is
needed o measu e damping once he guides ha e been assembled in he milling machine, as hei
damping capabili y depends on hei posi ion along he machine ool s uc u e, hus iden i ying
he main sou ces o damping in he machine.
In his s udy, linea guide damping was calcula ed om expe imen al da a ga he ed om
expe imen al impac es s a he TCP. To do so, wo poin s a ound each linea guide we e es ed:

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one on he s uc u e be o e he linea guide, nea he ca iage, and he o he on he s uc u e on
he o he side o he linea guide, see Figu e 1. The ed base o Figu e 1 is a magne ic base o he
accele ome e . Magne ic bases can in oduce damping in he measu emen s, bu as i is used on
bo h sides o he linea guide, when he di e ence o bo h poin s is calcula ed, ha e ec should
cancel i sel ou . Mo e poin s can be es ed o each linea guide, al hough i he poin s a e loca ed
nea he ca iages, hen simila in o ma ion is ob ained in he expe ience o he au ho s. Using he
expe imen ally measu ed equency esponses, he damping a each poin can be calcula ed, and
he di e ence be ween bo h poin s will be associa ed wi h he damping in oduced by he linea
guide assembled in he s uc u e.
Fig. 1 Measu emen s poin s be o e and a e a linea guide assembled in he es machine.
In he case o a model wi h one deg ee o eedom (do ), he esonance measu emen me hod
can be used o measu e damping om expe imen al da a [38, 52]. La ge-scale milling machines
equi e N do sys ems o ep esen he milling machine, and he esonance measu emen me hod
canno be applied. Howe e , i a change o modal coo dina es is applied o he N do sys em,
ins ead o an N do sys em, he sum o N sys ems o 1 do is ob ained by means o modal
supe posi ion, and he esonance me hod can hen be applied o each 1 do sys em. Hence, he
damping alue o he poin o each do can be ob ained in modal coo dina es. Finally, he modal
coo dina es change can be undone and he damping o he poin can be ob ained in na u al
coo dina es, in hei a ious di ec ions: Cxx, Cyy, Czz, Cxy, Cxz, Cyx, Cyz, Czx, and Czy. This app oach
is explained s ep by s ep below.
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2.1 Resonance measu emen me hod in 1 do sys em
The equa ion o mo ion o a o ced 1 do sys em wi h iscous damping is:
𝑚𝑚𝑥𝑥󰇘+𝑐𝑐𝑥𝑥󰇗+𝑘𝑘𝑥𝑥=𝐹𝐹(𝑡𝑡) (1)
whe e x, x󰇗,and x󰇘 ep esen he displacemen , eloci y, and accele a ion o he do
espec i ely; m, c, and k ep esen he mass, damping, and s i ness; and, F is he o ce. Assuming
a ha monic solu ion:
𝐹𝐹(𝑡𝑡)=𝐹𝐹0 𝑒𝑒𝑖𝑖𝑖𝑖𝑖𝑖 (2)
x = 𝑋𝑋0 𝑒𝑒𝑖𝑖𝑖𝑖𝑖𝑖, 𝑥𝑥󰇗 =𝑖𝑖𝑖𝑖𝑋𝑋0 𝑒𝑒𝑖𝑖𝑖𝑖𝑖𝑖,𝑥𝑥󰇘 =−𝑖𝑖2𝑋𝑋0 𝑒𝑒𝑖𝑖𝑖𝑖𝑖𝑖 (3)
Subs i u ing equa ions (2) and (3) in equa ion (1), and applying esonance condi ion 𝑖𝑖1=
�𝑘𝑘
𝑚𝑚 , ine ial and s i ness e ms a e cancelled:
−𝑖𝑖1
2𝑚𝑚𝑋𝑋0 + 𝑐𝑐𝑖𝑖𝑖𝑖1𝑋𝑋0 + 𝑘𝑘𝑋𝑋0 = 𝐹𝐹0 (4)
−𝑘𝑘
𝑚𝑚𝑚𝑚𝑋𝑋0 + 𝑐𝑐𝑖𝑖𝑖𝑖1𝑋𝑋0 + 𝑘𝑘𝑋𝑋0 = 𝐹𝐹0 (5)
𝑐𝑐𝑖𝑖𝑖𝑖1𝑋𝑋0 = 𝐹𝐹0 (6)
F om equa ion (6), i is concluded ha a esonance, he ex e nal o ce F0 will be equal o he
damping o ce. Wi h hamme impac expe imen al es , he damping o a one do sys em can be
easily ob ained using he alue o he measu ed mobili y a he na u al equency.
𝑐𝑐 = 𝐹𝐹0
𝑖𝑖𝑖𝑖1𝑋𝑋0=F
𝑋𝑋󰇗0 (7)
The same is no so o he N do sys em. In he ollowing sec ion, his me hod will be ex ended
o an N do sys em wi h some coo dina e changes.
2.2 Resonance measu emen me hod in an N do sys em
In sys ems wi h N do , he equa ion o mo ion is a sys em wi h N coupled equa ions:
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[M]∗{x󰇘}+[C]∗{x󰇗}+[K]∗{x}={F( )} (10)
whe e [M] is he mass ma ix, [C] is he damping ma ix, [K] is he s i ness ma ix, {𝐹𝐹(𝑡𝑡)}
is he ex e nal o ce ec o , and {x} is he displacemen ec o in na u al coo dina es.
Milling machines can be ep esen ed by linea and p opo ional sys ems, so i a change o
modal coo dina es, {p}, is applied o he N do sys em p e-mul iplying by he anspose o he
modal ma ix [X]T, a decoupled sys em o equa ions o he model can be ob ained:
{x}=[X]∗{p} (11)
[X]T[M][X]∗{p󰇘}+[X]T[C][X]∗{p󰇗}+[X]T[K][X]∗{p}=[X]T{F( )} (12)
[Mm]∗{p󰇘}+[Cm]∗{p󰇗}+[Km]∗{p}={𝐹𝐹𝑚𝑚} (13)
whe e, [Mm] is he modal mass ma ix, [Cm] is he modal damping ma ix, [Km] is he modal
s i ness ma ix, and {𝐹𝐹𝑚𝑚} is he modal ex e nal o ce ec o . In his way, i is possible o ob ain
N decoupled equa ions o 1 do , and he damping o each equa ion can he e o e be ob ained in
modal coo dina es applying equa ion (7):
cm=𝐹𝐹0m
X0m∗𝑖𝑖n (14)
Now, as damping alues a e needed in na u al coo dina es o in oduce hem in he nume ical
model, he base coo dina e change mus be undone.
[C]= [[X]T]−1[Cm][X]−1 (15)
In a ini e elemen model, di e en kinds o elemen s can be used: mass, beams, dampe s,
bush, ia, quads, e as, e c. Mass, beam, dampe , bush, ia and quad elemen s nodes ha e 6 do ,
3 ansla ional do and a u he 3 o a ional do , and e a elemen s nodes only ha e 3 ansla ional
do . Some sol e s, such as NASTRAN, apply 6 do o e e y node, and hose ha a e no used,
like o a ional do o e a elemen s nodes, a e au oma ically g ounded. Thus, a B node ini e
elemen model equi es 6B *6B modal ma ix.
I is ad isable o a oid he in e se o his ma ix, because o he la ge dimensions o he modal
ec o ma ix, as a la ge-scale milling machine can be usually ep esen ed by be ween 300000
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nodes and 700000 nodes. Ins ead, [X]−1 and [[X]T]−1 can be ob ained o he base change wi hou
pe o ming a ma ix in e sion, in he ollowing way:
[X]T[M][X]=[X]T�mij�[X]=[Mm]=�mmij� (16)
whe e, mij and mmij a e he mass and he modal mass alues, espec i ely, o j do in he i h
mode.
The mass ma ix used in he ini e elemen model is a lumped ma ix. As i is a diagonal
ma ix, only he diagonal componen s will ha e a alue di e en om ze o. This p ope y will
help o a oid calcula ing he in e se o he p e ious ma ix, because i modal ec o s a e
no malized o uni y mass, he p e ious equa ion can be ew i en as:
[X]T[M][X]=[X]T�mij�[X]=[I] (17)
The e o e, he in e se o [[X]T]−1 can be ob ained as:
[[X]T]−1 =[M][X] (18)
And he in e se o [X]−1 as:
[X]−1 =[X]T[M] (19)
Now, he alue o he mass ma ix wi h na u al coo dina es is needed o he subs i u ion o
he modal coo dina es in Eq. (18) and Eq. (19). I he modal ec o ma ix is no malized o uni y
mass and mul iplied by i s anspose, hen he in e se o he mass ma ix in na u al coo dina es
is ob ained:
[M]−1 =[X][X]T (20)
And as [M] is a diagonal ma ix, he in e se o he diagonal mass ma ix will also be a
diagonal ma ix wi h alues only in he diagonal componen s:
[M]−1 =�1
mij� (21)
F om Eq. (20), he alue o he in e se mass ma ix in na u al coo dina es is ob ained:
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o he TCP o each FRF measu emen , which we e hen a e aged o yield he inal esul . The
cohe ence be ween hese h ee measu emen s was moni o ed o ensu e good cohe ence especially
in na u al equencies. Each FRF measu ed 3200 lines.
Six poin s on he machine we e selec ed o measu e he linea -guide damping alues. The
machine was di ided in o di e en modules o he selec ion o hese poin s, joined by linea
guides: am, ame, column and bed. Figu e 8 shows hese six measu ed poin s in he ini e
elemen s model o he milling machine.
Fig. 8 The six measu ed poin s on he FEM model o he milling machine.
The damping gene a ed on he ca iages o each guide can be ob ained wi h he me hod
p esen ed in his s udy. The same ca iage wi h he same damping capaci ies, will dissipa e
di e en amoun o ene gy depending on he s uc u al join ha is measu ed, because he mass
and s i ness dis ibu ion o he join is di e en . The same happens wi h di e en modes o a
single join , because each mode will ha e di e en mass and s i ness dis ibu ions, which will
ha e a di ec in luence on he amoun o ib a o y ene gy ha he guide will dissipa e. The
mobili y o one poin is domina ed by mass and s i ness dis ibu ion, and damping. The na u al
equencies will depend on he dis ibu ion o bo h s i ness and mass, bu he ampli ude o he
modes a na u al equencies a e in luenced by s i ness, mass, and damping. In his me hod, i
was assumed ha he damping in oduced by all he ca iages o each linea guide was he same.
Figu e 9 shows he mobili ies in he Z di ec ion o poin s a bo h sides o he linea guide ha
joins he column o he bed, as shown in Figu e 8. In his igu e, he ca iage a ha s uc u al
join dissipa es ene gy in he i s mode a 13 Hz, bu dissipa es no ene gy in he ou h mode, a
27.8 Hz. This esul is because he dis ibu ed mass and s i ness o he poin s be o e and a e he
linea guide a o he damping o he linea guide a 13 Hz, bu he same is no ue a 27.8 Hz.
An analysis o he i s mode in Figu e 7 shows ha he modal shape pe mi s he wo poin s,
be o e and a e he linea guide, o mo e in di e en ways, which means ha hei con ibu ion
o linea guide damping is likely o be highe . The mode shape o he ou h mode in Figu e 7 a
27.8 Hz, shows ha bo h poin s mo e oge he , so he e will he e o e be no ene gy dissipa ion a

17
ha join . Hence, i can be concluded om Figu e 9 ha he ca iages ha join he column o he
bed dissipa e ene gy in he i s mode a 13 Hz, bu dissipa e no ene gy in he ou h mode a 27.8
Hz.
Fig. 9 Dynamic mobili ies be o e and a e he linea guide joining he column/bed in he Z
di ec ion. The mobili y axis is blind o con iden iali y easons.
As shown in Table 1, he damping alues ob ained o he ca iages a e be ween 0-128
N·s·mm-1. These alues a e o ca iages o size 55 and size 65, which a e he e o e bigge han
he e e ence ca iage alues o size 35 and size 45 p esen ed in he in oduc ion [31, 32];
ne e heless, hey a e o he same o de o magni ude.
Table 1. Measu ed damping alues a he linea guides in N·s·mm-1.
CXX
CYY
CZZ
CXY
CXZ
CYX
CYZ
CzX
CZY
Ram/F ame 4 4 12 8 0 0 0 4 16
F ame/Column
4
4
92
8 0 0 0 4 28
Column/Bed
76
4
16
56 128 0 12 4 24
The damping alues shown in Table 1 indica e which linea guide dissipa es g ea e ene gy.
The milling machine has he same ype and size o linea guides ha join he am o he ame and
he column o he bed, bu hey in oduce e y di e en damping alues: he bigges di e ence
be ween hese guides is when o ces a e in oduced in he X di ec ion, as he guides ha join he
column o he bed will dissipa e mo e ene gy han hose ha join he am o he ame, speci ically,
20 imes mo e in he X di ec ion and 7 imes mo e in he Y di ec ion. Howe e , he bigges
di e ence is when he TCP is also exci ed in he X di ec ion and he ene gy is dissipa ed in he Z
di ec ion: he ca iages joining he am o he ame dissipa e no ene gy as hei lexibili y on bo h
18
sides o he linea guides is he same, and ins ead, he ca iages joining he column o he bed
in oduce he bigges damping alue in he machine.
Ca iages and guides joining he ame o he column a e smalle han he o he linea guides,
al hough hey in oduce mo e damping in he Z di ec ion when he machine TCP is exci ed in he
Z di ec ion han he ca iages and guides joining he am o he ame and he column o he bed.
These poin s he e o e con i m he impo ance o measu ing damping alues unde eal wo king
condi ions.
As p e iously men ioned, he damping alues ob ained o each ca iage in Table 1 a e used
o de ine he dis ibu ed disc e e dampe s in he model. A ca iage is modeled wi h 2D elemen s
and 1D elemen s ep esen ing ca iage s i ness and damping. The s i ness and damping in each
ca iage a e ep esen ed by ou sp ings and ou dampe elemen s, one o each di ec ion. In
Figu e 10, a linea guide is ep esen ed: a 1D sp ing elemen ep esen s he s i ness and, in
pa allel, a 1D dampe ep esen s he damping in oduced in he milling machine.
Fig. 10 Fini e elemen model o slide way and ca iage.
3.2 Valida ion o damping alues a linea guides
The me hodology p esen ed in his pape has been alida ed by compa ing he dynamic
lexibili y o he machine a he TCP ob ained wi h he ini e elemen model and he expe imen al
dynamic lexibili ies measu ed a he TCP, see Figu e 11. I can be concluded ha he ini e
elemen model o he machine, wi h ma e ial damping and disc e e damping alues a he linea
guides, accu a ely ep esen s he eal milling machine o cha e p edic ion pu poses. These
igu es ep esen eal dynamic lexibili y cu es oge he wi h heo e ical dynamic lexibili y
cu es. The dynamic lexibili y cu e alues and shapes a esonance a e in bo h cases e y
simila , bea ing in mind he simpli ica ions done:
1D elemen s
de ining linea
guide s i ness and
damping
Ca iages
Guide
19
Fig. 11 Dynamic lexibili y a he TCP o he machine measu ed expe imen ally (EXP) and
calcula ed wi h FEM in he X, Y, and Z di ec ions. The lexibili y axis is blind o con iden iali y
easons.
The aim o he me hod was o measu e he damping in each guide o he milling machine o
ob ain accu a e equency esponse unc ions a he TCP o p edic he dynamic beha io o he
machine. In Figu e 11, he simila i y al hough no exac , may be seen be ween he expe imen ally
measu ed ampli ude alues and he ampli ude alues ob ained wi h FEM due o simpli ica ions
and model assump ions. Some sys ems equi ed o adjus FRF ou pu s. Fo example, in he s udy
o a o sional ib a ion dampe he ou pu s need o be adjus ed because iscoelas ic ma e ials a e
used, and hese a e he aim o he s udy, so a amewo k is implemen ed in he model o adjus
iscoelas ic ma e ials p ope ies [54].
The di e ence in alues could be because any o he h ee is no exac ly ep esen ed due o
he simpli ica ions in he ini e elemen model, o because o he simpli ica ions. Fo example, no
clea ances ha e been modeled. Fo example, in FRFXX, 1 Hz di e ence can be seen in he i s
mode, and 3 Hz in he second mode. So, he s i ness idealiza ion will no only ha e an e ec on
he na u al equency alues, i will also ha e a di ec e ec on he compliance equency esponse
ampli ude alues.
On he o he hand, in Fig. 11, he wid h o he modes is e y simila . The e a e di e en
me hods o he nume ical calcula ion o mode modal damping om he wid hs o compliance
20
esponses. The hal -powe bandwid h me hod is a p ocedu e commonly employed o ex ac
damping a ios ξ om he FRF es ima es and has been p o ed o be su icien ly accu a e o a
numbe o p ac ical cases in which damping is less han 10% [55]. Table 2 shows each mode
modal damping in each di ec ion, calcula ed wi h he Hal -Powe Bandwid h Me hod. As he
wid h o he modes depends only on he damping, he alues in Table 2 sugges ha he es ima ion
o he damping o linea -guides assembled in he machine using he p oposed me hod yields good
esul s.
Table 2. Modal damping alues in each di ec ion ob ained wi h he expe imen al Hal -Powe
Bandwid h Me hod (EXP) and wi h ma e ial damping and calcula ed linea guide damping alues
(FEM).
Modal damping
alue
EXP (%) FEM (%)
ξx1
4.2
3.9
ξx2
3.0
2.4
ξy1
2.1
2.3
ξz1
2.0
2.0
ξz2
6.2
9.2
4. CONCLUSIONS
A heo e ical-expe imen al me hodology has been p esen ed in his pape o measu e absolu e
damping alues o linea guides assembled in a milling machine. These alues a e o g ea in e es
o p edic he machine dynamic beha io . He e, he damping is ob ained wi h he guides in eg a ed
in he machine because, as i has been seen in he case s udy, he same ype and size o linea
guides can dissipa e di e en amoun s o ib a o y ene gy when joined o di e en s uc u al
componen s. Consequen ly, modelling machine ool damping wi h disc e e dis ibu ed damping
alues is a basic equi emen , in o de o asce ain whe e damping o igina es in he machine ool.
In e ms o damping, linea -guide damping alues will help o milling machine edesign. These
alues will show whe e he ib a o y ene gy dissipa es and will help he milling machine
designe s o decide whe e linea guides should be eplaced wi h g ea e o lesse damping
capaci y. Fo example, a e a dynamic s abili y s udy, linea guides wi h highe damping alues
could be in oduced whe e hey ha e a g ea e e ec on cha e limi ing modes.
I is wo h highligh ing ha his me hod also allows o iden i y he linea guides ha in oduce
less damping han hei capaci ies in machine ools, o iden i y ce ain linea guides ha can be
eplaced by o he linea guides wi h less damping capaci ies, and he eby o op imize he milling
21
machine and i s cos a ios. In he machine o he case s udy, imp o emen s in he damping
capaci y o he linea guides be ween he am and he ame a e no ecommended, because hese
linea guides dissipa e less ib a o y ene gy han hei capaci y, while he same linea guides
be ween he column and he bed dissipa e mo e ene gy.
Finally, i mus be unde lined ha his heo e ical-expe imen al me hod is designed o imp o e
he ib a o y ene gy dissipa ion o ully ins alled and assembled machines. In his espec , a u u e
esea ch would be o analyze he damping alues o he linea guides calcula ed wi h he me hod
p oposed o es ablish hei dependence on he modal pa ame e s, damping capaci ies, and posi ion
in he kinema ic chain o he machine, so ha he damping alues o pa icula linea guides may
be p edic ed o he mechanical design o new machines.
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ACKNOWLEDGMENT
We a e g a e ul o he Basque Go e nmen o i s inancial suppo h ough he E o gai P og am
o he P ojec “ZERO Pla a o mas de p oducción en egimen de ele ada p oduc i idad y ce o
de ec os de piezas so is icadas de al o alo añadido”.