Free Vibration Analysis of Viscoelastic Sandwich Beam using Euler Bernoulli Theory

Transcription

1 Fr Vibratio Aalysis o Viscolastic Sadwich Bam usig Eulr Broulli hory Ch. Rajsh 1 Dpt o Mchaical Egirig, Faculty o Scic ad chology, IFHE Uivrsity, Hydrabad -5010, Idia Abstract h wid varity o dampig tchiqus ar usd to rduc th vibratios, out o thm costraid layr dampig (CLD) tratmt is usd or studyig th r vibratio bhaviour o sadwich bams. h sadwich modl is ormd by placig th viscolastic matrial bodd btw th ac ad bas layr. Viscolastic sadwich bam cosists o thr layrs with viscolastic matrial as a cor layr, th ac layrs ar isotropic ad liar lastic matrial. I this papr r vibratios o ixd r sadwich bam with dirt coiguratios ar ivstigatd aalytically. Aalytical solutio is carrid out usig Eulr-Broulli bam thory to id th atural rqucis with dirt coiguratios o th bam usig MALAB. h aalytical rsults o th sadwich bams ar to b compard with availabl litratur valus. Kywords Fr vibratio; sadwich bam; atural rqucy; CLD. I. INRODUCION A bam is a sldr horizotal structural mmbr that rsists latral loads by bdig. his is a importat lmt i girig structurs such as supportig mmbrs i high-ris buildigs, railways, log-spa bridgs, lxibl satllits, gu barrls, robot arms, airpla wigs, tc. A sadwich bam is a layrd assmbly mad rom two thi, strog ad sti ac shts bodd to a lightwight cor matrial to crat light wight ad strog structural lmt. h dvlopmt o sadwich structurs will cotiu to b dmad or th d o light wight, to carry both i-pla ad out-o-pla loads, good stability udr comprssio, providig xcllt stiss to wight ad strgth to wight charactristics. May typs o dampig mchaism hav b dvlopd to supprss th vibratio lvl. Passiv dampig usig viscolastic matrials is widly usd i both commrcial ad arospac applicatios. Viscolastic matrials ar lastomric whos log- chai molculs caus thm to covrt mchaical rgy ito hat wh thy ar dormd. Barj [1] carrid th r vibratio aalysis o thr layrd symmtric sadwich bams usig th dyamic stiss mthod. H usd th Hamilto s pricipl to driv th govrig partial dirtial quatios o motio i r atural vibratio that ar coupld both i axial ad bdig dormatio. Comput th atural rqucis ad mod shaps o a sadwich bam usig Wittrick-Williams algorithm by him. Mohammad Sdighi ad Hiwa Hossii [] ivstigatd r vibratio o stppd imoshko shat with isigiicat dampig usig FEM with uiorm ad ouiorm status. I cas o o-uiorm imoshko shat th J. Sursh Kumar Dpt o Mchaical Egirig, JNU Collg o Egirig, Hydrabad , Idia irst ad third part will b uiorm whil th middl part is i coical orm. hy calculatd atural rqucis ad corrspodig mod shap or two cass o boudary coditios amly clampd r ad simply supportd. hy compard th obtaid valus with thos o classical Eulr Broulli thory rsults. Mohammadi ad Hassairad [] itroducd two vibratio masurig istrumts that is Acclromtr ad RFID. hy rcogizd RFID is mor asir ad simpl to aalyz i ordr to masur r vibratio o a catilvr bam. Dawid Ckus [] usd Lagrag multiplir mthod to id th solutio o th r vibratio problm o th catilvr taprd bam. H carrid out sampl umrical calculatios or catilvr taprd bam ad compard with xprimtal valus. Mohammad Zaoc [5] studid vibratio o thi ilm catilvr bams ad drivd quatios o quilibrium. H drivd quatios o rqucy ad mod shaps to solv or dirt catilvr bams or various matrials. A Mat-Lab programm is dvlopd by him to id atural rqucis ad corrspodig mod shaps or o layr ad two layr bams with dirt dimsios ad proprtis. Rajsh ad Sursh Kumar [6] studid th vibratio aalysis o ixd r structur with dirt cor matrial. hy oud th dyamic ct o cor matrial i dirt positio o sadwich structur with sam volum o viscolastic ad structur matrials. hy oud that positioig o th cor matrial at th ctr has a grat iluc i cotrollig vibratio o structurs. Abdl Salam ad Bodok [7] dvlopd sadwich bams to calculat th lxural rigidity ad dyamic charactristics. hy ivstigatd dirt cass o sadwich bams with multi layr cors, multi clls, dirt shap ad oritatio o hols i its cors. Rsults o atural rqucis ad mod shaps ad th static dlctio o th sadwich bams wr obtaid by thm usig ANSYS ad compard with aalytical valus. Iiods t al. [8] prstd th mthods to prdict th atural rqucis ad mod shaps o composit bams. hy solvd th quatios o motio i two ways i.., iit lmt mthod ad xact itgratio mthod. Natural rqucis ad mod shaps or a varity o ply layups ar obtaid by thm. I. M. Dail t al. [9] dvlopd simpl modls i ordr to xplai th bhaviour o composit sadwich as a uctio o matrial, gomtric ad loadig paramtrs. hy usd dirct obsrvatio ad odstructiv valuatio to study th mchaical charactrizatio o th sadwich costitut matrials. Liar variatio i strai through thickss was dtrmid by thm i both cor ad ski thickss. Jih L [10] mployd th psudospctral mthod to o-idal boudary imoshko bam modl or r vibratio aalysis whr as aalytical mthod was carrid out or Eulr 566

2 Broulli bam. H oud that wighig actor icrass as atural rqucy dcrass or o-idal at o d ad r boudary coditios at th othr d. Natural rqucis o a imoshko bam ar computd or various thickss to lgth ratio with th o-idal clampd boudary coditio at th lt d ad idal r boudary coditio at th right d o th bam. Baki Ozturk ad Saa Bozkurt Cosku [11] obtaid th aalytical rsults or r vibratio aalysis o bam o lastic oudatio or dirt support coditios hy validatd th aalytical solutios with th variatioal itratio mthod ad homotopy prturbatio mthod rsults ad oud i clos agrmt. Zhig Liu t al. [1] studid dirtial trasorm simulatio mthod to solv th r vibratio problms o uiorm Eulr Broulli bam. hy coductd mod xprimtal mthod to obtai xprimtal data or aalysis by sigal procssig. From rsults thy oud modiid dirtial trasorm mthod ca achiv good valus i prdictig solutio o sadwich problms. Prokic t al. [1] prstd a umrical mthod or solutio o r vibratio o bams govrd by a st o scod-ordr ordiary dirtial quatios with dirt boudary coditios. hy dmostratd th accuracy o proposd mthod by comparig th calculatd rsults with litratur valus. hy showd that good accuracy valus ca b obtaid or rlativly small umbr o ods. Ahmd [1] ivstigatd th lxural vibratio charactristics o curvd sadwich bams usig iit lmt displacmt mthod. H studid various displacmt modls to ivstigat thir ct o th atural rqucis o curvd ixd-ixd sadwich bams. h ct o shar dormatio o th irst thr mods o th ixd-ixd curvd sadwich bams is vry small was oud by him. II. MAHEMAICAL MODELLING A solid lastic ixd r bam with uiorm cross sctio o lgth (L), width (b), thickss (h), mass dsity (ρ) ad modulus o lasticity (E) is cosidrd or aalysis. h quatio o motio or th trasvrs vibratio o a bam with uiorm cross sctio ad homogous matrial at ay poit x ad tim t is dotd by w( x, ad trasvrs orc pr uit lgth by ( x, usig Eulr-Broulli thory ad th rotatio o th lmt is isigiicat. Eulr- Broulli thory is valid i th ratio btw th lgth o th bam ad its hight is rlativly larg (mor tha 10). h ct o shar dormatio ad rotary irtia ar igord. h dirtial quatio o motio o th bam is [1]: w( x, x w( x, A ( x, (1) t Assumig th xtral xcitatio is zro or r vibratio i.., ( x, 0, th Eq. () ca b rwritt as w x w 0 t Whr, I is th ara momt o irtia o th bam cross sctio, th mass pr uit lgth is A, lxural rigidity is ad bam iit lmt o lgth dx. () h solutio o th Eq. () is solvd by th mthod o sparatio o variabl, o is dpdig o th positio ad th othr is dpdig o tim, as ollows w( x, ( x) ( () Whr, ( x) ad ( ar idpdt o tim t ad positio x, rspctivly. Substitutig Eq. () ito Eq. () IV II () Variabls hav b sparatd ad ach sid o abov quatio must qual to a costat. IV II I th tim variabl t is sparatd rom Eq. (5) (5) II 0 (6) h gral solutio o th Eq. (6) is (7) ( C Si t C Cost (8) 5 6 whr C 5 ad C 6 ar costats. Similarly i th positio variabl x is sparatd rom Eq. (5) IV 0 (9) h gral solutio o th Eq. (9) is w ( x) C Cosh x C Sih x C Cos x C Si x (10) whr C C ar costats, hyprbolic uctios. Sih ad Cosh By multiplyig Eq. (8) with Eq. (10) upo solvig ad substitutig iitial ad boudary coditios, six combid costats ar obtaid. h atural rqucy o th bam is calculatd rom Eq. (7) as, (11) O substitutig th ixd r boudary coditios i Eq. (10), th ollowig quatios ar obtaid ad rwritt i matrix orm as: Sihl Sil Coshl Cosl Coshl Cosl Sihl Sil C C 1 ar 0 (1)

3 h o-trivial solutio o th dtrmiat o th coicit matrix is as ollows: Cos l Coshl 1 (1) which is th charactristic quatio o a ixd r Eulr- Broulli solid bam. h Eig valus ar obtaid by calculatig th irst thr roots o Eq. (1) usig Nwto-Raphso mthod. 1 l whr 1,,... (1) 1 1 whr 1,,... (15) l For a symmtric sadwich structur, thickss ad matrial proprtis o all layrs ar sam as show i Figur 1. Fig. 1: Sadwich Bam h lxural rigidity o sadwich structur with sigl cor is giv as [7]: D E bh Ecbhc ( h h ) (16) c 1 1 E bh whr, D is th bdig stiss (or) lxural rigidity, E ad E c is th youg s modulus o ac matrial ad cor matrial rspctivly, h ad h c ar th thickss o ac ad cor matrial rspctivly, b is th width o structur, h = otal bam hight (h +h c) Dyamic charactristics o a ixd r bam ar xprsss as [7]: i. h quivalt mass, m b h h ) (17) ( c c ii. h quivalt stiss, * D K L (18) iii. h quivalt atural rqucy, 1 K (.5m ) 0.5 (19) h rqucy o dirt mods ca b calculatd accordig to th ollowig quatio i i Whr i K (.5m ) 0.5 (0) is a costat dpd o th boudary coditios ad i = 1,,,..., ar th rqucy corrspods to ach od. III. RESULS AND DISCUSSION h matrials cosidrd or sadwich bam ar Alumiium as ac matrials ad atural rubbr as cor matrial is cosidrd or r vibratio aalysis. h rqucis o various sadwich bams which ar mad with dirt ac ad cor matrials ar calculatd usig Mat lab. Various ivstigatios ar carrid out or r-vibratio bhaviors o rctagular cross-sctiod alumiium sad bams with dirt gomtric charactristics udr ixd r boudary coditio. Matrial proprtis o alumiium ad polyurtha rigid oam ar [1]: Modulus o lasticity o ac matrial (E )= N/m ; modulus o lasticity (E c)= Gpa; thickss o acs (h ) = 0.57 mm ; thickss o cor h c = 1.7 mm ; dsity o cor matrial (ρ c)=.8 kg/m ; dsity o ac matrial (ρ )= 680 kg/m ; Width o bam b = mm; Lgth o th bam L = 0.91 m. Matrial proprtis o alumiium ad atural rubbr ar [6]: Modulus o lasticity o ac matrial (E )= N/m ; modulus o lasticity (E c)= Gpa; thickss o acs (h ) = mm ; thickss o cor h c = 1 mm ; dsity o cor matrial (ρ c)= 950 kg/m ; dsity o ac matrial (ρ )= 680 kg/m ; Width o bam b = 0 mm; Lgth o th bam L = 00 mm. Fig.: Natural rqucis o ixd r sadwich bam with ac matrial as alumiium ad cor as polyurtha rigid oam 568

4 Fig.: Natural rqucis o ixd r sadwich bam with ac matrial as alumiium ad cor as atural rubbr with rspct to lgth to width ratio Natural rqucis o ixd r sadwich bam with ac matrial as alumiium ad cor as polyurtha rigid oam is show i Fig.. h matrial proprtis o alumiium ad polyurtha rigid oam ar cosidrd or r vibratio bhaviour o sadwich bam. From this igur, it is oticd that as mod umbr is icrasig atural rqucis ar icrasig. It is du to o dimsioal umbr icrass or corrspodig mod umbr icras. Fig. rprsts th atural rqucis o a rctagular cross sctio sadwich bam with ac as alumiium ad cor as atural rubbr or various lgths to width ratios. From this igur, it is obsrvd as l/b ratio icrass atural rqucy dcrass or various mod umbrs. his may b bcaus o atural rqucy is ivrsly proportioal to th lgth paramtr. Natural rqucis o a rctagular cross sctio sadwich bam with ac as alumiium ad cor as atural rubbr agaist lgth to thickss ratios is show i Fig.. From this igur, it is oticd as l/h ratio icrass atural rqucy dcrass or various mod umbrs. his may b bcaus o atural rqucy is dirctly proportioal to thickss ad ivrsly proportioal to th lgth paramtr. Fig. 5 givs th rlatioship btw atural rqucis to various aras o cross sctio o sadwich bam. h matrial proprtis o alumiium ad atural rubbr ar cosidrd or studyig th r vibratio bhaviour o sadwich bam. It is s rom this igur as th cross sctio o sadwich bam varis atural rqucy icrass du to thickss icrass. Fig.: Natural rqucis o ixd r sadwich bam with ac matrial as alumiium ad cor as atural rubbr with rspct to lgth to thickss ratio IV. CONCLUSIONS I this work various studis ar prormd to ivstigat th r-vibratio bhaviour o rctagular cross-sctiod alumiium sadwich bams with dirt gomtric charactristics udr ixd r boudary coditio. h covtioal Eulr-Broulli bam thory was cosidrd or modllig th various sadwich bams, i.., alumiium, polyurtha rigid oam as ac ad cor matrial rspctivly i irst sadwich bam ad alumiium, atural rubbr as ac ad cor matrial rspctivly i scod sadwich bam. h atural rqucis ar obtaid ad discussd or th irst our mods icludig th cts o lgth to thickss, lgth to width ad ara o cross sctio o th sadwich bam with ixd r boudary coditio. h aalytical rsults obtaid usig Mat lab ar compard with litratur valus ad oud i good agrmt. REFERENCES Fig.5: Natural rqucis o ixd r sadwich bam with ac matrial as alumiium ad cor as atural rubbr with rspct to ara o cross sctio [1] JR Barj, Fr vibratio o sadwich bams usig th dyamic stiss mthod, Computrs ad structurs, Vol. 81,00, PP [] Mohammad Sdighi, Hiwa Hossii, Fr Vibratio Aalysis o Stppd imoshko Shats by th Fiit Elmt Mthod, h 1 st Itratioal Cogrss o Soud ad Vibratio, ICSV1, Bijig, Chia, July 01, PP [] S. Mohammadi, A. Hassairad, Applid ad hortical Catilvr Bam Fr Vibratio Aalysis, World Acadmy o Scic, Egirig ad chology, 61, 01, PP [] Dawid Ckus, Fr Vibratio O A Catilvr aprd imoshko Bam, Scitiic Rsarch o th Istitut o Mathmatics ad Computr Scic, Vol., Issu 11, 01, PP [5] Mohammad Zao, Fr Vibratio o hi Film Catilvr Bam, Itratioal Joural o Egirig ad chical Rsarch (IJER), ISSN: , Vol., Issu 11, Novmbr 01, PP

5 [6] Ch. Rajsh, J. Sursh Kumar, Aalysis o Sadwich Structurs usig Eulr Broulli hory, Itratioal Corc O Emrgig chologis I Mchaical Scics (ICEMS-01), Dcmbr 01,(ISBN Numbr: ) PP 7-5. [7] M. Abdl Salam & Nadia E. Bodok. Fr Vibratio Charactristics or Dirt Coiguratios o Sadwich Structurs, Itratioal Joural o Mchaical & Mchatroics Egirig, Vol.10, No., 010, PP 7-6. [8] D.H. Hodgs, Ali R. Atilga, Mark V. Fulto ad Lawrc W. Rhild, Fr-Vibratio Aalysis o Composit Bams, Joural o th Amrica Hlicoptr Socity, July 1991, PP 6-7. [9] I. M. Dail, J. L. Abot, ad K. A. Wag, stig Ad Aalysis o Composit Sadwich Bams. PP [10] J. L, Fr vibratio aalysis o bams with o-idal clampd boudary coditios, Joural o Mchaical Scic ad chology, Vol.7, No., 01, PP [11] Baki Ozturk ad Saa Bozkurt Cosku Aalytical Solutio or Fr Vibratio Aalysis o Bam o Elastic Foudatio with Dirt Support Coditios, Hidawi Publishig Corporatio Mathmatical Problms i Egirig, Vol. 01, PP 1-7. [1] Z. Liu, Y. Yi, F. Wag, Y. Zhao, L. Cai, Study o modiid dirtial trasorm mthod or r vibratio aalysis o uiorm Eulr-Broulli bam, Structural Egirig ad Mchaics, Vol. 8, No. 5, 01, PP [1] Prokic, M. Bsvic, D. Lukic, A umrical mthod or r vibratio aalysis o bams, Lati Amrica Joural o Solids ad Structurs, Vol. 11, 01, PP 1-1. [1] Ahmd KH, Fr vibratio o curvd sadwich bams by th mthod o iit lmts, Joural o Soud Vibratios, 18(1), 1971, PP [15] Mad DJ, Sivakumara S, h Stodola mthod applid to sadwich bam vibratio, I Procdigs o th Symposium o Numrical Mthods or Vibratio Problms, Uivrsity o Southampto, UK,