THE MODELISATION OF CONSTRAINED DAMPING
LAYER TREATMENTS USING THE FINITE ELEMENT
METHOD: SPATIAL MODEL AND VISCOELASTIC
BEHAVIOUR
Rui Mo ei a 1 José Dias Rod igues 2
1 Depa amen o de Engenha ia Mecânica
Uni e sidade de A ei o,Campus San iago, 3810-193 A ei o – Po ugal
2 Faculdade de Engenha ia da Uni e sidade do Po o - DEMEGI
R. D . Robe o F ias, 4200-465 Po o – Po ugal
SUMMARY: Su ace and in eg a ed damping ea men s wi h iscoelas ic laye s play an impo an posi ion
among he passi e damping ea men s o ligh and lexible s uc u es unde ib a ion. Applica ion simplici y,
low cos , educed s uc u al modi ica ion and educed addi ional mass, along wi h an inhe en high e iciency,
a e he main easons o i success ul usage.
Howe e , he design p ocess o hese ea men s is no simple and equi es a eliable ool o adequa e designing
and analysis.
The ini e elemen me hod can be used o his pu pose. Howe e some conside a ions and special ca e a e
necessa y o he spa ial modelisa ion o he ea men and wi h he iscoelas ic ma e ial p ope ies
cha ac e isa ion.
In his wo k, a ini e elemen comme cial so wa e (MSC/Nas an) was used o simula e he cons ained and he
in eg a ed iscoelas ic ea men s applied on aluminium pla es.
The spa ial modeling o he ea men is de eloped using a laye ed scheme o pla e/b ick con en ional ini e
elemen s. The dynamic p ope ies o he iscoelas ic ma e ial a e aken in o accoun in he nume ical simula ion
using he complex modulus app oach.
The nume ical esul s a e co ela ed wi h expe imen al da a ob ained in ou ea ed specimens by di ec
compa ison o he equency esponse unc ions and by using some FRF-based co ela ion indica o s.
KEYWORDS: Damping Laye T ea men s, Viscoelas ic Complex Modulus, FRF-Based Co ela ion
INTRODUCTION
The applica ion o iscoelas ic laye s on ligh s uc u es can p o ide a simple and eliable passi e damping
mechanism, pa icula ly e icien unde speci ic condi ions o ib a ion [1,2,3].
The in oduced damping is capable o con ol and educe dynamic e ec s, such as high ib a ion le els and noise
emission, and o ex end wo king li e o pa s unde cyclic loading o impac .
The damping ea men s wi h iscoelas ic laye s can be applied on he su ace o he ib a ing s uc u e, wi h o
wi hou a cons aining laye , o in eg a ed in he s uc u e cons i u ing a sandwich ma e ial, Fig.1, 2 and 3.
H1
H2
H3
a1
b1
Fig.1: Su ace ea men Fig.2: In eg a ed ea men Fig.3: Damping ea men
con igu a ion
Home
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In he in eg a ed (ILD) and cons ained (CLD) laye damping ea men s, he iscoelas ic laye is s ongly
de o med in shea due o he e ec o he cons aining laye p esen in he cons ained con igu a ion o due o he
adjacen skins o he in eg a ed con igu a ion.
This cons aining e ec is esponsible by he la ge dissipa ion o he ib a ion ene gy ha occu s wi hin hese
ea men s, hus being possible o ha e e y e ec i e ea men s e en wi h e y hin damping laye s ha
minimise he addi ional mass and he s uc u al modi ica ion.
The su ace ea men s can be applied locally in speci ic and in e es ing a eas o he s uc u e, minimising he
cos and he mass o he ea men , main aining howe e he ea men e ec i eness o some mode shapes o
equency ange [3,4].
These ea men s a e widely used in he ae onau ical and ae ospace indus y, whe e a e he p ime solu ion o
passi e damping ea men s o ligh and la ge s uc u es.
1. FINITE ELEMENT MODELISATION
The damping e ec o he iscoelas ic ea men s can and should be p edic ed, and he ea men s ailo ed, e en
du ing he design s age o he a ge s uc u e.
The ini e elemen me hod can p o ide a eliable ool in he design p ocess o s uc u es ha inco po a e his
kind o ea men s. Howe e , he e a e wo main aspec s ha should be conside ed du ing he applica ion o his
nume ical ool. One is ela ed o he spa ial modelling o he iscoelas ic laye , in o de o ge a ealis ic
desc ip ion o he high shea de o ma ion pa e n de eloped in his laye du ing he s uc u e ib a ion mo ion.
The o he is ela ed o he empe a u e and equency dependence o he iscoelas ic ma e ial p ope ies and o
he high loss ac o ha usually is exhibi ed by hese ma e ials.
1.1. SPATIAL MODEL OF THE TREATMENT
The damping mechanism o hese ea men s is closely ela ed o he high shea de o ma ion ha occu s in he
iscoelas ic laye as a esul o he es ain e ec o he adjacen laye s. Thus, i is e y impo an o desc ibe
co ec ly he de o ma ion o he dissipa i e laye . The Classical Lamina e Pla e Theo y is no adequa e o
accu a ely desc ibe he shea de o ma ion o he iscoelas ic laye , hus being necessa y o use a di e en
model [5,6].
All he h ee used models in his nume ical s udy sha e a common ep esen a ion o he iscoelas ic laye using
solid b ick elemen s (HEXA8). The base pla e and he cons aining laye o he su ace ea men s, o he skin
pla es o he in eg a ed laye con igu a ion, a e bo h modelled by ei he pla e elemen s (QUAD4) o b ick
elemen s (HEXA8). The h ee used models in he nume ical s udy a e ep esen ed in Fig.4.
QUAD4
HEXA8
QUAD4
RBE
RBE
QUAD4
HEXA8
+OFFSET
+OFFSET
QUAD4
HEXA8
HEXA8
HEXA8
Model 1 Model 2 Model 3
Fig.4: FEM models o he iscoelas ic ea men s
The i s and second FEM models a e qui e simila . In he i s , he pla e’s deg ees o eedom a e connec ed
wi h he b ick ones by means o igid links (RBE) [7]. Using his model, he mos complex one, i is possible o
simula e bonding ailu es be ween he iscoelas ic laye and he adjacen pla es simply by emo ing hose links
in speci ic nodes o he FEM mesh.
Cons aining laye o
uppe skin
Viscoelas ic laye
Base pla e o
lowe skin
In he second model, he pla e elemen nodes a e o se , by hal o he pla e hickness, o he plane in con ac
wi h he solid elemen , ins ead o he s anda d mid-plane. This esul s in coinciden nodes and ansla ional
deg ees o eedom be ween he pla e and he adjacen ace o he solid elemen . The las model uses solid
elemen s o desc ibe all he laye s.
The h ee models conside ed ha e exac ly he same o al numbe o deg ees o eedom. As hey include solid
b ick elemen s, he spa ial disc e iza ion mus be e ined enough o a oid shea -locking p oblems ela ed o high
a ea/ hickness a io.
The nume ical esul s ob ained a e iden ical, independen ly o he model used. Ne e heless, model gene a ion
e o and ime consuming a e less impo an using model 2.
1.2. VISCOELASTIC MATERIAL MODELLING
The iscoelas ic ma e ials a e cha ac e ised by a complex shea o ex ensional modulus exhibi ing a la ge loss
ac o which is esponsible o he dissipa ion e ec , specially wi hin he ansi ion empe a u e ange. Many
au ho s [8,9,10] ha e been s udying he modelling o he iscoelas ic ma e ial cha ac e iza ion. Some o hese
ha e de eloped, based on he heological models, o mula ions in ime and equency domains ha equi e ex a
deg ees o eedom o desc ibe he ma e ial modulus beha iou wi h equency.
Conside ing single ha monic exci a ion i is possible o use he complex modulus app oach o desc ibe he
ma e ial beha iou in he equency domain.
() () ()
()
,
,,
1T
TT
j
EE ω
ωω η
′+
=⋅ (1)
Thus, he iscoelas ic ma e ial is conside ed as an elas ic ma e ial wi h a complex modulus o elas ici y, whe e
()
,T
Eω
′ ep esen s he s o age modulus and
()
,Tω
η is he loss modulus o he iscoelas ic ma e ial.
The complex modulus is usually ep esen ed as a unc ion o empe a u e and equency by he Reduced
F equency Nomog am [3]. The nomog am o he ma e ial 3M ISD112 [11] used in his s udy is ep esen ed in
Fig.5.
1.3. FINITE ELEMENT SPATIAL MODEL
The ini e elemen spa ial model, de ining he equa ions o mo ion o he sys em in ma ix o m, can be w i en
as:
[
]
()
{
}
()
[
]
()
{
}
()
{
}
+=
 ,TMx K x ω (2)
whe e
[
]
M is he mass ma ix,
()
[
]
,TKω is he o al s i ness ma ix and
()
{}
x and
()
{
}
a e, espec i ely,
he o ced esponse and he exci a ion ec o s.
Fig.5.:Reduced- equency nomog am o 3M ISD 112 [11]
The s i ness ma ix
()
[]
,TKω con ains he s i ness ma ix o he base pla e and o he cons aining laye ,
which is a eal en i y, plus a complex s i ness ma ix due o he iscoelas ic laye :
()
[
]
[
]
()
[
]
e
,,TTKKKωω=+ (3)
The iscoelas ic s i ness ma ix is, he e o e, a complex ma ix whose e ms depend on he empe a u e and
equency.
Conside ing an ha monic exci a ion o equency ωas:
()
{
}
{}
e
j
Fω
= (4)
hen, he s eady s a e esponse o he sys em can be w i en as:
()
{}
{
}
e
j
x Xω
= (5)
whe e
{
}
Xis a complex ec o . Subs i u ion o Eqn 4 and Eqn 5 and i s app op ia e de i a i es in o Eqn 2 yields
he algeb aic se o equa ions:
()
[]
[]
[
]
{
}
{}
2
,TF
KX
M
ωω=
− (6)
om which he ec o
{
}
X, which depends on ω and he sys em pa ame e s, can be ob ained.
1.4. RESPONSE MODEL
The ecep ance equency esponse unc ions o a e e ence k (inpu deg ee o eedom) a e de ined as:
()
()
1, ,
0,
i
j
jk in
kFik
X
F
ω
αω =




=
≠






=



 … (7)
These unc ions can be gene a ed di ec ly om he spa ial model sol ing he Eqn 6 o di e en alues o he
exci a ion equency ω:
()
[]
[]
[
]
()
{
}
{}
=
−2k
k
,TF
KX
M
ωω
ω (8)
whe e all he e ms o he exci a ion ec o o he sys em
{
}
k
Fa e equal o ze o, excep he one co esponding
o he exci a ion deg ee o eedom.
I he empe a u e a iable is conside ed as a cons an en i y, hen he esponse model can be gene a ed by a
equency sweep in which he complex s i ness ma ix o he iscoelas ic laye is ecalcula ed a each equency
alue, as ep esen ed in Fig.6.
1N
ωω ω=
()
[
]
[
]
()
[
]
ie i
KKKωω=+
()
[
]
[]
[
]
()
{}
{}
2
ii i k
k
KMXFωω ω−=
()
()
j
jk
k
X
F
ω
αω=
i
ωω=
Fig.6: Response model gene a ion diag am
2. EXPERIMENTAL STUDY
To alida e he nume ical models some equency esponse unc ions we e measu ed on ou aluminium pla es
wi h CLD and ILD ea men s. Table 1 p esen s he cha ac e is ics o he specimens used in his expe imen al
s udy.
Table 1- Specimens used in he expe imen al s udy
Specimen Dimensions a
x b [mm] H1[mm] H2[mm] H3[mm]
1 298x197 2.0 0.125 0.250
2 297x197 2.0 0.125 0.200
3 298x198 1.0 0.250 1.0
4 298x198 1.0 0.125 1.0
The expe imen al specimens we e suppo ed by ubbe bands o simula e ee bounda y condi ions. A measu ing
mesh o 25 poin s was de ined on each specimen, as ep esen ed in Fig. 7.
The pla e was exci ed a poin numbe 17 ( he e e ence deg ee o eedom o his s udy) by a shake d i en by a
andom signal in he equency band o [0,400 Hz]. The esponse eloci y a each one o he mesh poin s was
measu ed by a non-con ac lase dopple measu ing de ice ( ib ome e ). The expe imen al se -up is ep esen ed
in Fig. 8.
270
174
47.5
80
55
1
y
x
5
25 21
17
Fig. 7: Measu ing mesh Fig.8: Expe imen al se -up
A dynamic signal analyse was used o acqui e he exci a ion and he esponse signals. The eloci y esponse
signal was di e en ia ed and he accele ance equency esponse unc ions e alua ed a each measu ing poin .
Since he iscoelas ic ma e ial p ope ies a e empe a u e dependen , he measu emen s we e ca ied on nea
iso he mal condi ions, wi h empe a u e acquisi ion using a empe a u e p obe.
3. EXPERIMENTAL RESULTS AND CORRELATION
3.1. TEMPERATURE EFFECTS
All he specimens we e es ed unde iso he mal condi ions wi h a oom empe a u e close o 17.5ºC. Specimens 1
and 3 we e also es ed a a lowe empe a u e (11.5ºC).
The di ec accele ance equency esponse unc ions measu ed in specimen 3 a empe a u es o 11.5ºC and
17.5ºC a e ep esen ed in Fig.9. The g aphic shows ha i is e y impo an , conce ning he design p ocess, o
know he empe a u e condi ions o he applica ion.

050 100 150 200 250 300 350 400
-pi
-½pi
0
½pi
pi
Hz
Phase
10
-3
10
-2
10
-1
10
0
10
1
10
2
ABS (acc/F)
Specimen 3 - empe a u e e ec s
Exp. 11.5ºC
Exp. 17.5ºC
Fig.9: F equency esponse unc ions o specimen 3 a empe a u es 11.5ºC and 17.5ºC
3.2. CORRELATION OF NUMERICAL AND EXPERIMENTAL RESULTS
O e laying he di ec equency esponse unc ions (magni ude and phase cu es) measu ed on he ou
specimens wi h he nume ically gene a ed unc ions, i is possible o e alua e, by isual inspec ion, he o e all
le el o co ela ion. I may be men ioned ha o all specimens he e a e, as expec ed, a weak co ela ion in he
low equency ange, due o he p esence o he igid body modes o he expe imen al se -up. Ne e heless, he
i s s uc u al equency is well abo e ha equency ange.
In he nex igu es (Fig.10 o Fig.15) i is ep esen ed he compa ison be ween he expe imen al and he
nume ical di ec equency esponse unc ions (accele ance). The nume ical ones we e gene a ed by he ini e
elemen me hod using he model 2.
050 100 150 200 250 300 350 400
-pi
-½pi
0
½pi
pi
Hz
Phase
10
-3
10
-2
10
-1
10
0
10
1
10
2
ABS (acc/F)
Specimen 1 - 11.5ºC FRF17-17 **
Exp. 11.5ºC
FEM 11.5ºC
050 100 150 200 250 300 350 400
-pi
-½pi
0
½pi
pi
Hz
Phase
10
-2
10
-1
10
0
10
1
10
2
ABS (acc/F)
Specimen 1 - 17.5ºC FRF17-17 **
Exp. 17.5ºC
FEM 17.5ºC
Fig.10: F equency esponse unc ions o specimen 1 a
empe a u e 11.5ºC | expe imen al s. nume ical
Fig.11: F equency esponse unc ions o specimen 1 a
empe a u e 17.5ºC | expe imen al s. nume ical
050 100 150 200 250 300 350 400
-pi
-½pi
0
½pi
pi
Hz
Phase
10
-3
10
-2
10
-1
10
0
10
1
10
2
ABS (acc/F)
Specimen 2 - 17.5ºC FRF17-17 **
Exp. 17.5ºC
FEM 17.5ºC
050 100 150 200 250 300 350 400
-pi
-½pi
0
½pi
pi
Hz
Phase
10
-3
10
-2
10
-1
10
0
10
1
10
2
ABS (acc/F)
Specimen 3 - 11.5ºC FRF17-17 **
Exp. 11.5ºC
FEM 11.5ºC
Fig.12: F equency esponse unc ions o specimen 2 a
empe a u e 17.5ºC | expe imen al s. nume ical
Fig.13: F equency esponse unc ions o specimen 3 a
empe a u e 11.5ºC | expe imen al s. nume ical
050 100 150 200 250 300 350 400
-pi
-½pi
0
½pi
pi
Hz
Phase
10
-3
10
-2
10
-1
10
0
10
1
10
2
ABS (acc/F)
Specimen 3 - 17.5ºC FRF17-17 **
Exp. 17.5ºC
FEM 17.5ºC
050 100 150 200 250 300 350 400
-pi
-½pi
0
½pi
pi
Hz
Phase
10
-2
10
-1
10
0
10
1
10
2
ABS (acc/F)
Specimen 4 - 17.5ºC FRF17-17 **
Exp. 17.5ºC
FEM 17.5ºC
Fig.14: F equency esponse unc ions o specimen 3 a
empe a u e 17.5ºC | expe imen al s. nume ical
Fig.15: F equency esponse unc ions o specimen 4 a
empe a u e 17.5ºC | expe imen al s. nume ical
The isual compa ison o he o e laid cu es p o ides a global idea o he co ela ion be ween he nume ical
esul s and he expe imen al da a. The abo e ep esen ed unc ions, as well as he o he unc ions o he esponse
model, show a globally sa is ac o y o e all co ela ion. Howe e , such compa ison does no quan i y he le el o
he co ela ion and only p o ide a subjec i e and quali a i e idea o i .
3.3 FRF-BASED CORRELATION INDICATORS
In o de o ob ain a quan i a i e measu emen o he co ela ion be ween he nume ical esul s ob ained by he
ini e elemen model 2 and he expe imen al da a, se e al equency esponse unc ions co ela ion c i e ia
a ailable [12,13,14 ] we e used.
The F equency Response Assu ance C i e ion (FRAC) and he F equency Ampli ude Assu ance C i e ion
(FAAC) p o ide a global co ela ion quali y measu emen o each deg ee o eedom o e he whole equency
ange.
()
{}
()
{}
()
{}
()
{}
()
()
{}
()
{}
()
2
H
ii
XA
jk jk
jk HH
ii ii
XX AA
j
kjk jkjk
HH
FRAC HH HH
ωω
ωω ωω
= (9)
()
{}
()
{}
()
{}
()
{}
()
()
{}
()
{}
()
2H
ii
XA
jk jk
jk HH
ii ii
XX AA
j
kjk jkjk
HH
FAAC HH HH
ωω
ωω ωω
=+ (10)
In he abo e exp essions,
()
{
}
XijkHω and
()
{
}
AijkHωs and o , espec i ely, he expe imen al and nume ical
equency esponse unc ions be ween he deg ees o eedom j and k.
These indica o s clea ly ou line he con ibu ion o each deg ee o eedom on he o e all esponse model
co ela ion le el. Fo specimens 1 and 3, es ed and simula ed a 17.5ºC, hese co ela ion indica o s a e
ep esen ed in he Fig.16 and Fig.17, espec i ely.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
0
0.2
0.4
0.6
0.8
1
F equency Response Assu ance C i e ion
FRAC
DOF
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
0
0.2
0.4
0.6
0.8
1
F equency Ampli ude Assu ance C i e ion
FAAC
DOF
12345678910 11 12 13 14 15 16 17 1819 20 21 22 23 24 25
0
0.2
0.4
0.6
0.8
1
F equency Response Assu ance C i e ion
FRAC
DOF
12345678910 11 12 13 14 15 16 17 1819 20 21 22 23 24 25
0
0.2
0.4
0.6
0.8
1
F equency Ampli ude Assu ance C i e ion
FAAC
DOF
Fig.16: FRAC and FAAC co ela ion indica o s o
specimen 1 a empe a u e 17.5ºC
Fig.17: FRAC and FAAC co ela ion indica o s o
specimen 3 a empe a u e 17.5ºC
On he o he hand, he Global Shape C i e ion (GSC) and he Global Ampli ude C i e ion (GAC) [14] quan i y,
as a unc ion o equency, he o e all ag eemen , shape-based and ampli ude-based, espec i ely, be ween he
nume ical esul s and he expe imen al da a.
()
()
{}
()
{}
()
{}
()
{}
()
()
{}
()
{}
()
2
H
XA
HH
XX AA
HH
GSC HHHH
ωω
ωωωωω
= (11)
()
()
{}
()
{}
()
{}
()
{}
()
()
{}
()
{}
()
2H
XA
HH
XX AA
HH
GAC HH HH
ωω
ωωω ωω
=+ (12)
050 100 150 200 250 300 350 400
0
0.2
0.4
0.6
0.8
1
Global Shape C i e ion
GSC
w[Hz]
050 100 150 200 250 300 350 400
0
0.2
0.4
0.6
0.8
1
Global Ampli ude C i e ion
GAC
w[Hz]
050 100 150 200 250 300 350 400
0
0.2
0.4
0.6
0.8
1
Global Shape C i e ion
GSC
w[Hz]
050 100 150 200 250 300 350 400
0
0.2
0.4
0.6
0.8
1
Global Ampli ude C i e ion
GAC
w[Hz]
Fig.18: GSC and GAC co ela ion indica o s o
specimen 1 a empe a u e 17.5ºC
Fig.19: GSC and GAC co ela ion indica o s o
specimen 3 a empe a u e 17.5ºC
The equency dis ibu ion o hese indica o s o specimens 1 and 3 is ep esen ed in Fig.18 and Fig.19. Fo bo h
c i e ia, global shape and global ampli ude, a e y sa is ac o y ag eemen is e ealed be ween he expe imen al
and he nume ical esul s wi hin he bandwid h analysis. Mo eo e , wi h hese c i e ia, he abo e men ioned igid
body modes e ec in he low equency ange is well highligh ed.
Ano he co ela ion c i e ion, he Local Ampli ude C i e ion (LAC) [14], is a help ul ool in he way ha i
quan i ies he co ela ion be ween he nume ical esul s and he expe imen al da a as a equency unc ion o
each indi idual deg ee o eedom. Thus, i is possible o indi idually e alua e he equency co ela ion o each
equency esponse unc ion.
()
()
()
()
()
()
()
()
()
()
()
()
()
()
()
*
**
2Xjk Ajk
jk
XjkAjk XjkAjk
HH
LAC HH HH
ωω
ωωω ωω
+
= (13)
This indica o has been success ully applied in he iden i ica ion o he e o sou ce e i ied on he p e ious
global co ela ion indica o s. F om he esul s ob ained, oo ex ensi e o be p esen ed he e, i was clea ly
iden i ied which deg ees o eedom and co esponding equency anges con ibu e o he decay o he o e all
co ela ion.
CONCLUSIONS
The passi e damping ea men s using iscoelas ic ma e ial laye s can p o ide an e ec i e dynamic dissipa i e
mechanism ha can be applied wi h success in la ge and hin s uc u es. The e ec i eness o hese ea men s is
closely ela ed o he shea de o ma ion ene gy dissipa ed by he iscoelas ic laye .
This s udy has shown ha he cons ained and he in eg a ed laye ea men s can p o ide a simple, cos e ec i e
and eliable way o in oduce he damping necessa y o he dynamic con ol o esonan s uc u es, e en o low
equencies.
The dynamic beha iou o he ea men s he eby s udied can be e ec i ely simula ed using models based on he
ini e elemen me hod, hus making a ailable an analysis ool ha can be used in he design p ocess o op imize
he dynamic con ol o he ea men s, as well as he s uc u al beha iou o he applica ion pa s.
The h ee pu posed models, based on a h ee-dimensional solid ep esen a ion o he iscoelas ic laye , we e able
o cha ac e ise he shea de o ma ion pa e n ha occu s in i , leading o iden ical esul s.
Using he complex modulus app oach in a equency di ec sol ing scheme i has been possible o easily
in oduce he equency dependen iscoelas ic p ope ies in o he nume ical calcula ion p ocedu e o gene a e
he esponse model.
The alida ion o he ini e elemen models was based on he co ela ion be ween he nume ical and
expe imen al equency esponse unc ions o CLD and ILD pla e specimens.
The applica ion o equency esponse unc ions based co ela ion indica o s p o ided a co ela ion le el
e alua ion ha alida es he ini e elemen models used.
REFERENCES
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3. Jones, D.I.G.,”Handbook o Viscoelas ic Vib a ion Damping”, John Wiley & Sons, 2001.
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