The modelisation of constrained damping layer treatments using the finite element method: spatial model and viscoelastic behaviour

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 Back o Index 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 1. Johnson, C.D., “Design o Passi e Damping Sys ems”, Special 50 h Anni e sa y Design Issue, T ansac ions o he ASME, Vol.117, 1995, pp.171-176. 2. Nashi , A.D., Jones, D.I.G., Hende son, J.P., “Vib a ion Damping”, John Wiley & Sons, 1985. 3. 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