VIBRATION ANALYSIS AND DAMPING CHARACTERISTICS OF Hybrid COMPOSITE PLATE USING FINITE ELEMENT ANALYSIS

Transcription

1 VIBRION NLYSIS ND DMPING CHRCERISICS OF Hybrd COMPOSIE PLE USING FINIE ELEMEN NLYSIS hss submttd n partal fulfllmnt of th Rqurmnts for th dgr of Mastr of chnology In Mchancal Engnrng Spcalzaton: Machn Dsgn and nalyss By SHIVPRSD BD Roll No. : ME7 Dpartmnt of Mchancal Engnrng Natonal Insttut of chnology Rourkla Rourkla, Odsha, , Inda Jun 04

2 VIBRION NLYSIS ND DMPING CHRCERISICS OF Hybrd COMPOSIE PLE USING FINIE ELEMEN NLYSIS hss submttd n partal fulfllmnt of th Rqurmnts for th dgr of Mastr of chnology In Mchancal Engnrng Spcalzaton: Machn Dsgn and nalyss By Shvaprasad Baad Roll No. : ME7 Undr th Gudanc of Prof. arapada Roy Dpartmnt of Mchancal Engnrng Natonal Insttut of chnology Rourkla Rourkla, Odsha, , Inda Jun 04

3 Ddcatd to, My parnts, my sstr, My matrnal aunt and uncls

4 DEP. OF MECHNICL ENGINEERING NIONL INSIUE OF ECHNOLOGY, ROURKEL ROURKEL , ODISH, INDI Crtfcat hs s to crtfy that th work n th thss nttld VIBRION NLYSIS ND DMPING CHRCERISICS OF HYBRID COMPOSIE PLE USING FINIE ELEMEN NLYSIS by Shvaprasad Baad s a rcord of an orgnal rsarch work carrd out by hr durng undr my suprvson and gudanc n partal fulfllmnt of th rqurmnts for th award of th dgr of Mastr of chnology n Mchancal Engnrng (Machn Dsgn and nalyss), Natonal Insttut of chnology, Rourkla. Nthr ths thss nor any part of t, to th bst of my knowldg, has bn submttd for any dgr or dploma lswhr. Plac: NI Rourkla Dat: 0 Jun 04 Profssor Dr. arapada Roy

5 DEP. OF MECHNICL ENGINEERING NIONL INSIUE OF ECHNOLOGY, ROURKEL ROURKEL , ODISH, INDI I crtfy that Dclaraton a) h work contand n th thss s orgnal and has bn don by myslf undr th gnral suprvson of my suprvsor. b) h work has not bn submttd to any othr Insttut for any dgr or dploma. c) I hav followd th gudlns provdd by th Insttut n wrtng th thss. d) Whnvr I hav usd matrals (data, thortcal analyss, and txt) from othr sourcs, I hav gvn du crdt to thm by ctng thm n th txt of th thss and gvng thr dtals n th rfrncs. ) Whnvr I hav quotd wrttn matrals from othr sourcs, I hav put thm undr quotaton marks and gvn du crdt to th sourcs by ctng thm and gvng rqurd dtals n th rfrncs. Shvaprasad Baad nd Jun 04

6 CKNOWLEDGEMENS It s my mmns plasur to aval ths opportunty to xprss my grattud, rgards and hartflt rspct to Prof. arapada Roy, Dpartmnt of Mchancal Engnrng, NI Rourkla for hs ndlss and valuabl gudanc pror to, durng and byond th tnur of th projct work. Hs prclss advcs hav always lghtd up my path whnvr I hav struck a dad nd n my work. It has bn a rwardng xprnc workng undr hs suprvson as h has always dlvrd th corrct proporton of apprcaton and crtcsm to hlp m xcl n my fld of rsarch. I also xprss my sncr grattud to Prof. (Dr.) K. P. Maty, Had of th Dpartmnt of Mchancal Engnrng for valuabl dpartmntal faclts. I would lk to mak a spcal mnton of th slflss support and gudanc I rcvd from my snor shrbad Swan, Ph.D. scholar, Dpartmnt of Mchancal Engnrng, NI Rourkla durng my projct work. Last but not th last; I would lk to xprss my lov, rspct and grattud to my parnts, youngr sstr and my matrnal aunt and uncls, who hav always supportd m n vry dcson I hav mad, gudd m n vry turn of my lf, blvd n m and my potntal and wthout whom I would hav nvr bn abl to achv whatsovr I could hav tll dat. SHIVPRSD BD shvaprasadbaad@gmal.com

7 BSRC Hybrd compost s a compost whch conssts of nanopartcls to nhanc th strngth as compard to convntonal composts. modl has bn proposd to dtrmn th lastc proprts of hybrd compost. h hybrd compost conssts of convntonal fbr and nanocompost as matrx. h frst stp hr s to dtrmn th proprts of nanocompost whch s don by usng Mor anaka mthod. h CNs ar consdrd as cylndrcal nclusons n polymr matrx n Mor anaka mthod. ssumng prfct bondng btwn carbon fbrs and nanocompost matrx, th ffctv proprts of th hybrd compost has bn valuatd usng mchancs of matrals approach. n 8 nodd shll lmnt has bn usd for th fnt lmnt analyss havng 5 dgrs of frdom ach nod u, v, w, x, y. 0 0 fnt lmnt msh has bn usd to modl th shll lmnt. h shll coordnats whch ar n Cartsan form ar convrtd nto paramtrc form usng two paramtrs,. hs paramtrs ar agan mappd nto soparamtrc form,. 6 layrd lamnat wth stackng squnc [ ] S has bn usd for vbraton analyss of smply supportd shll lmnt. h dynamc quatons of shll ar drvd usng Hamlton s prncpl. s th dampng charactrs of th dynamc systm ar not avalabl, for furthr nvstgaton dampng rato of frst mod and last actv mod ar assumd. Usng Raylgh dampng th dampng ratos of ntrmdat mods can b calculatd. h tm dcay of th systm from maxmum ampltud to 5% of th maxmum ampltud has bn usd as a paramtr to study varous shll structurs by varyng th volum fracton of CNs n nanocompost and by varyng carbon fbr volum fracton.

8 CONENS CKNOWLEDGEMENS... I BSRC... II CONENS... III NOMENCLURE... V LIS OF FIGURES... VII LIS OF BLES... VIII INRODUCION... LIERURE REVIEW ND MOIVION Ltratur Rvw Matral proprty Hybrd compost Matral Proprty Hybrd compost Vbraton analyss of plat Dampng n composts Motvaton Objctv MERIL MODELING CN bass nanocompost modlng Hybrd Compost modlng... 4 FINIE ELEMEN FORMULION... 4

9 4. Gomtry of md-surfac of shll Isoparamtrc mappng ransformaton matrx usd for soparamtrc mappng Stran dsplacmnt rlatons In-plan/bndng stran-dsplacmnt matrx ransvrs stran dsplacmnt matrx BBD matrx Equaton of moton Statc fnt quatons Dynamc fnt quatons Calculaton of non-dmnsonal frquncs Stat spac mthod for mpuls rspons RESULS ND DISCUSSION CONCLUSION REFERENCES: v

10 NOMENCLURE CN SWCN DWCN MWCN nc, m, hc C p, C p, C p, C p3, C p55 Carbon nanotubs Sngl walld carbon nanotubs Doubl walld carbon nanotubs Mult walld carbon nanotubs Nanocompost, matrx, hybrd compost Elastc proprts of consttunt phas nr, kr, pr, lr, m Hlls lastc constant r,, I S K G E Eulr angls for CN orntaton Strss and stran Stran concntraton tnsor Idntty matrx Eshlby tnsor Bulk modulus Shar modulus Young s modulus v Volum fracton of CN CN v m v f, s Volum fracton of matrx Volum fracton of fbr Curvlnar coordnats, s Isoparamtrc curvs r, r, R, R R a N angnt to soparamtrc curvs Lam s paramtrs Normal curvaturs of shll md-surfac wst curvaturs of shll md-surfac Lngth of shll along x-axs Shap functon u0, v0, w 0 Dsplacmnts of th nod n,, drctons Isoparamtrc coordnats and z v

11 J k Bb Bs NM, B D z h Q K U W L M uu Kuu d d t ** Jacoban matrx Shar stran Curvatur of th md-surfac In-plan stran dsplacmnt matrx ransvrs stran dsplacmnt matrx Rsultant forc and momnt pr unt lngth In-plan stffnss matrx Couplng stffnss matrx Bndng stffnss matrx hcknss of lamna hcknss of lamnat Rducd stffnss matrx Shar corrcton factor otal potntal nrgy Stran nrgy Kntc nrgy Work don by xtrnal forc Lagrangan Elmntal mass matrx Elmntal stffnss matrx Dsplacmnt vctor cclraton vctor m Dnsty Natural frquncy Non-dmnsonal natural frquncy Modal coordnats for th dgr of frdom Modal dampng rato... Stat vctor n v

12 LIS OF FIGURES Fgur 3- Rf: hshmat and yas (0)... 8 Fgur 3- Hxagonal RVE... Fgur 4- Gomtry of shll structur n Cartsan coordnats... 4 Fgur 4- Isoparamtrc mappng... 6 Fgur 4-3 Stackng squnc n lamnat... Fgur 5- Varaton of C w.r.t varaton of carbon fbr and CN volum fracton Fgur 5- Varaton of C w.r.t varaton of carbon fbr and CN volum fracton... 3 Fgur 5-3 Varaton of C 3 w.r.t varaton of carbon fbr and CN volum fracton... 3 Fgur 5-4 Varaton of C w.r.t varaton of carbon fbr and CN volum fracton... 3 Fgur 5-5 Varaton of C 55 w.r.t varaton of carbon fbr and CN volum fracton... 3 Fgur 5-6 mpuls rspons of cfrp compost for thck plat Fgur 5-7 Impuls rspons of cfrp compost for thn plat Fgur 5-8 Dcay tm for thck plat by varyng th cnt volum fractons for dffrnt volum fractons of carbon fbr Fgur 5-9 Dcay tm for thn plat by varyng th cnt volum fractons for dffrnt volum fractons of carbon fbr v

13 LIS OF BLES abl 5- Non-dmnsonal frquncy [Rf:3]... 9 abl 5- Non-dmnsonal frquncy for th prsnt formulaton... 9 abl 5-3 Matral proprts of varous consttunts n hybrd compost v

14 INRODUCION s long as thr s dvlopmnt n th fld of arospac, automobl, halthcar, lctroncs and consumr ndustry th dmand for nw matrals wll nvr sz. h dmand for nw matrals has ld to contnuous rsarch and dvlopmnt of nw tchnqus to satsfy th nds... Nanocomposts Nanocomposts consst of rnforcmnts of nanoscal sprad vnly or randomly n polymr matrx. h commonly usd polymrc matrx matrals ar: Epoxy Polystyrn Nylon Polymd PEEK Polythr thr kton h commonly usd nano fllrs ar: Carbon nanotubs SWCNs and Nanottanum oxd MWCNs Nanoslca Nanoalumnum oxd h rnforcmnts can b partcls or fbrs of sz of fw nanomtrs. h nanocompost has a wd rang of matrals from 3-D mtal matrx composts, -D lamnatd composts and nano-wrs of small dmnson rprsntng varatons of nano rnforcmnts. Usng nanoscal rnforcmnts was ntroducd by Usuk t al [] who

15 bult a nanocompost usng polymd and organophlc clay. h nanocompost formd had twc th tnsl modulus as compard to nat polymd wth just % volum fracton of nano rnforcmnt. Nanocomposts hav gand a wd popularty among rsarchrs. Rsarchrs hav dscovrd that th proprts of th nanocompost ar bttr whn compard to th ndvdual componnts of th compost. Proprts such as ncrasd tnsl strngth, ncrasd thrmal conductvty ar obsrvd... Hybrd Composts h mportant proprts that ar dsrd from any compost ar strngth, stffnss, ductlty, toughnss, dampng, nrgy absorpton, thrmal stablty and low wght. Wth convntonal matrals t s not possbl to gt all th dsrd proprts, but wth compost matrals w can talor th proprts of matral as pr our nds. By usng rnforcmnts of nanoscal n polymr composts thr has bn trmndous ncras n mchancal proprts as compard to nat polymr matrx. Hybrd composts ar nw typ of thr phas composts whch hav rnforcmnts of nanoscal n addton to convntonal rnforcng fbr n matrx or by growng rnforcmnts of nanoscal on th surfac of fbr. R.C.L. Dutra t al [] dfnd hybrd composts as composts consstng of dffrnt fbrs. h man purpos of usng hybrd composts s t ncrass th matrx domnatd proprts..3. Classfcaton of Plats [3] Plats can b classfd nto two typs: hn plats hck plats thn plat can furthr b classfd as: Plat wth small dflcton Plats wth larg dflcton

16 .3... Plats wth small dflcton If th dflcton of a plat whn subjctd to loadng s lss than or narly qual to thcknss, thn th plat s sad to hav small dflcton. h ncssary assumptons for dvlopng thory for plats havng small dflcton ar: h mddl plan dos not dform on loadng. Pont on th plat ntally normal to mddl plan rman normal to mddl plan vn aftr loadng. h strsss n th thcknss drcton can b nglctd hn plats wth larg dflcton If th dflctons of th plat ar larg whn subjctd to latral loadng as compard to thcknss, thn th plat s sad to hav larg dflcton. Whn plat s loadd md plan strans ar dvlopd. In plats wth small dflcton t s normally nglctd, as a rsult th strsss ar also nglctd. But f dflctons ar larg as compard to thcknss, th strans dvlopd ar larg. So strsss cannot b nglctd. In ths cas w obtan nonlnar quatons and analyss bcoms complcatd..3.. hck plat h abov approxmatons for thn plats ar not applcabl. hck plat thory must b usd. h thck plat thory nvolvs 3-D thory of lastcty and calculaton of strsss s qut complcatd Dffrnc btwn plat and shll h major dffrnc btwn plats and shlls can b obsrvd undr th acton of loadng. Whn a plat mmbr s subjctd to latral load, qulbrum s possbl by th acton of bndng and twstng momnts. In shlls, whn t s subjctd to latral loadng, qulbrum s possbl by mmbran strsss whch act paralll to tangntal plan at a pont on mddl surfac and ar dstrbutd unformly ovr th thcknss of shll. Plats ar plan mmbr and shlls ar curvd structural mmbrs..4 Dampng n composts Dampng s a vry mportant paramtr for vbraton control, nos rducton, stablty of systm, fatgu and mpact rsstanc [4]. h dampng n fbr rnforcd composts s dffrnt from that of mtals. h varous forms of nrgy dsspaton ar [5]: Dampng bhavour of matrx matral. h major contrbuton to dampng s from matrx, th dampng of fbr must also b ncludd for calculaton of dampng. Dampng bhavour of ntrphas Intrphas s th rgon btwn matrx and fbr. h typ of ntrphas plays an mportant rol n dampng. h ntrphas can b wak or strong. 3

17 Dampng du to damag a) Frctonal dampng du to dlamnaton. b) Dampng du to nrgy dsspaton of brokn fbrs or cracks n matrx. Vscoplastc dampng t hghr ampltuds of strsss, thr s non-lnar dampng du to prsnc of hgh strss and stran..5. Impuls rspons of lnar tm nvarant systm (LIS) Impuls forc s a forc whch acts on th systm for vry short amount of tm. Knowng th mpuls rspons of LIS w can obtan by suprposton th rspons of th sam systm to any nput provdd th nput condtons ar zro n all cass. Unt mpuls nput has vry short ntrvals of tm but vry larg ampltud and hnc th ffct of th bhavour of th systm undr study s not nglgbl. Ex: Ball httng th crckt bat, th ball s actd upon by vry larg forc for a short duraton of tm. 4

18 LIERURE REVIEW ND MOIVION. Ltratur Rvw Jan Png Lu [6] Elastc proprts of SWCNs, MWCNs and nanorops ar nvstgatd usng forc constant modl... Matral proprty Hybrd compost Raf t al [7] stmatd mchancal proprts of poxy basd nanocompost wth SWCN, MWCN and graphn platlts wr compard for wght fractons of 0.%. h matral proprts masurd wr Young s modulus, fractur toughnss, ultmat tnsl strngth. h tnsl strngth of graphn basd nanocomposts showd bttr proprts as compard to CN basd nanocomposts. F.H. Gojny t al [8] obsrvd mchancal proprts rsultd n an ncras n Young s modulus, strngth at wght fractons of 0.%. hr was good agrmnt btwn xprmntal obsrvd data and rsults from modfd Halpn-sa rlaton. Floran H [9] proposd choosng approprat typ of CNs (SWCNs or DWCNs or MWCNs) has bn a problm vr snc thy ar bng usd n composts. hy valuatd th proprts of nanocompost for dffrnt nano fllrs. h nanocomposts xhbtd gratr strngth, stffnss and fractur toughnss. hy found that DWCN basd nanocompost xhbtd gratr fractur toughnss. Sdl t al [0] stmatd ffctv lastc proprts of composts consstng of algnd SWCNs or MWCNs usng Mor-anaka mthod. h ffcts of an ntrphas layr btwn CNs and th polymr s also nvstgatd usng a mult-layr compost cylndrs approach. Lu and Chn [] stmatd ffctv lastc proprts of th nanocompost ar valuatd usng contnuum modllng and fnt lmnt mthod. h xtndd rul of mxtur s usd to dtrmn th proprts of th contnuum modl. 5

19 .. Matral Proprty Hybrd compost Dutra t al [], mad a hybrd compost consstng of carbon fbr and Polypropyln fbr and mrcapto-modfd polypropyln blnd fbr (PPEVSH). hy found that hybrd composts showd bttr mpact rsstanc than CFRP compost. Mathur t al [] CNs wr grown on undrctonal carbon fbr. hs fbrs wr usd as rnforcmnts n matrx matral. hy found that th mchancal proprts mprovd wth ncras n amount of CN dposton as compard to nat CFRP compost. Garca t al [3] CNs wr grown on alumna fbr cloth. hs fbrs wr usd as rnforcmnts n matrx matral. h growth of CNs ld to an ncras n ntr-lamnar shar proprts of th ordr of 69% as compard to alumna cloth compost. Kundalwal and Ray [4], thy valuatd th lastc proprts of FFRC (Fuzzy fbr rnforcd compost) usng mchancs of matrals approach and Mor-anaka mthod consdrng wth and wthout th ntrphas btwn CN and polymr...3 Vbraton analyss of plat Roy and Chakraborty [5] formulatd layrd shll fnt lmnt modl for coupld lctromchancal analyss of curvd smart compost structur...4 Dampng n composts Gbson t al [4] usd vbraton usd modal vbraton rspons masurmnts to charactrz th mchancal proprts of lamnatd structurs. hy showd that vbraton n thr frst mod or multpl mods can b usd to dtrmn th lastc proprts and dampng ratos. Modal tstng was don by mpuls xctaton mthods. Kyrazoglou and Guld [7] found dampng rato usng xprmntal mthods and by FEM. h FEM uss Raylgh dampng mthod and partcularly mass proportonal dampng. R. Vrdjoan t al. [8] found that vn a small volum fracton of CN can ncras th sound absorpton capablts.. Motvaton Hybrd composts ar nw typ of thr phas composts whch ncras th matrx domnatd proprts. h hybrd compost that s to b modld hr conssts of nanocompost matrx and contnuous long carbon fbrs. h nanocompost s mad up of randomly dstrbutd CNs and polymr matrx. h nanocompost s modld usng Mor-anaka mthod. h hybrd compost can b modld usng mchancs of matrals approach. s th moblty of systm gos on ncrasng, modlng dampng for such systms bcoms complcatd. Raylgh dampng modl has bn usd to modl such mult dgr of frdom systms. Furthr nvstgaton has bn carrd out by assumng sutabl dampng ratos for frst mod and last sgnfcant mod whr 6

20 mass s proportonal to dampng. Impuls rspons of th systm has bn carrd out and a comparatv study has bn mad to know ffct of dampng n systms by varyng th volum fractons of carbon fbr and CNs..3 Objctv Matral modllng and matral charactrzaton. Nanocompost has bn modlld usng Mor anaka mthod.. Hybrd compost consstng of carbon fbr and nanocompost matrx has bn modlld usng mchancs of matrals approach. 8 nodd shll lmnt formulaton. Mndln thory of plats and shlls has bn usd to modl shll. Modllng dampng and Impuls rspons. Raylgh dampng has bn usd to modl th dampng of MDOF systm.. Impuls rspons of th systm has bn carrd out usng th stat spac modl. 7

21 3 MERIL MODELING h matral modlng s dvdd n two phass: CN basd nanocompost modlng. Hybrd compost modlng. h nanocompost conssts of randomly dstrbutd straght MWCN as rnforcmnts and poxy as matrx. s th CNs ar randomly dstrbutd th nanocompost can b modld as sotropc matral. h proprty of nanocompost s valuatd usng th Mor anaka mthod. ssumng prfct bondng btwn fbr and nanocompost th hybrd compost can b modld smlar to convntonal compost usng mchancs of matrals approach. 3. CN bass nanocompost modlng FIGURE 3- REF: HESHMI ND YS [8] Fg. 3. shows a RVE of randomly dstrbutd CN n poxy matrx. h Mor anaka mthod was usd to stmat th lastc proprts of th randomly dstrbutd MWCN n matrx. h procdur to dtrmn th sotropc proprts of randomly orntd MWCNs dsprsd n poxy matrx s as follows: 8

22 h Hll s lastc constants for th MWCN can b obtand by quatng th strss stran matrx of MWCN wth th Hll s lastc matrx. C CN CN CN CN C C C CN CN CN C C C CN CN CN C3 C3 C CN C CN C C CN 66 () C CN nr lr lr lr kr mr kr m r lr kr mr kr mr mr pr pr Eqn () and qn. () ar th strss stran matrx of MWCN and Hll s lastc matrx. h orntaton of th CN can b spcfd by two Eulr angls and. h bas vctors and of th global 0 x x x and local co-ordnats can b rlatd by th rlaton, j j 3 g (3) () Whr g j cos cos sn sn sn sn cos cos sn cos 0 sn cos s th CNs ar randomly dstrbutd n matrx t can b charactrzd by two Eulr angls and. h orntaton dstrbuton of CNs n th compost s charactrzd by th probablty dnsty functon p, satsfyng th normalzaton condton and s gvn by th quaton. / 0 0 p, sn ( d)( d ) (4) For compltly randomly orntd CNs, p, (5) From Mor anaka mthod on can rlat strss, and stran, th strss n th matrx m by,, C, C, C (6) CN CN CN m CN m m nd CN of th CN to 9

23 ,,, C (7) CN m m m Whr stran concntraton tnsor s gvn by, ( m) ( CN m) (8) I S C C C Whr S corrsponds to Eshlby nsor and s gvn by L and Dunn [9] for cylndrcal ncluson. h avrag strss and stran for th randomly orntd CNs s gvn by, / CN p, CCN, C m sndd m (9) 0 0 / CN p,, sndd m (0) 0 0 Usng rul of mxtur on can gt th strsss and strans n th nanocompost. s th CNs ar randomly dstrbutd n th matrx, th nanocompost bhavs lk an sotropc matral. h bulk modulus, shar modulus, Young s modulus of th nanocompost [0] ar gvn by, K G E nc nc nc Whr, CN CN K G m m vcn ( CN 3 KmCN ) 3 v v 9KncGnc 3K G nc nc m CN CN vcn ( CN GmCN ) v v m CN CN 3( K G ) k l 3( G k ) k l 3K G l m m m m m m 4 G m kcn l G K G G K G CN G m 5 3( G k ) G p G 3K G m 3K 7G n l 3 53 m m CN CN m CN CN CN n CN CN CN v CN CN m CN m CN m m m CN m m l CN CN m m CN G m k CN () 8G m p G m K G k l G l CN G p 3K m G G 7m G 3 G k m CN m m CN CN m CN m CN m CN m m CN m m CN () v (3) m v CN and v m ar volum fractons of CN and matrx matral, nc rprsnts nanocompost. 0

24 3. Hybrd Compost modlng FIGURE 3- HEXGONL RVE Fg. 3. shows a hxagonal RVE of hybrd compost consstng of carbon fbrs dstrbutd n nanocompost matrx. Usng th abov calculatd nanocompost proprts, th proprts of th transvrsly sotropc hybrd compost can b valuatd by th formulaton of Kundalwal and Ray for fuzzy fbr [4]. ssumng prfct bondng btwn carbon fbr and nanocompost, th normal strans n hybrd compost, carbon fbr and nanocompost ar qual along th fbr drcton and th transvrs strsss n th sam phas ar qual along th drcton transvrs to th fbr from sofld condtons. Usng ruls of mxtur on can xprss th longtudnal and transvrs strsss and strans n trms of volum fractons of nanocompost and carbon fbr. Usng sofld condtons and rul of mxtur w can wrt, v f v (4) NC Whr v f and v NC ar volum fractons of carbon fbr and nanocompost. f NC HC f NC HC f NC HC f NC HC f NC HC f NC HC HC rprsnts hybrd compost. v f f NC HC f NC HC f NC HC v f NC NC HC f NC HC f NC HC (5) (6)

25 hr th strss and stran n th hybrd compost lamna s gvn by, HC C f C NC (7) HC V f V NC (8) But from so-fld condtons, f NC C C (9) By solvng abov quatons w gt, HC HC HC C C HC gvn by, (0) s th ffctv lastc matrx of th proposd Carbon fbr s rnforcd polymr and s HC C C V3 C V4 () h varous matrcs apparng n th abov quatons ar gvn blow. C f f f C C C vf () C C 3 NC NC NC vncc vncc vncc NC NC NC C C C NC NC NC C C C NC C NC C44 0 NC C f f f C C C f f f C3 C3 C p C p C55 0 p C55 (3) (4)

26 C V NC NC NC C C C NC NC NC C C C NC C NC C44 0 NC C v f 0 0 v f v f v f v f (5) (6) V v NC 0 0 v NC vnc vnc vnc V V V C C (7) (8) V V V C C (9)

27 4 FINIE ELEMEN FORMULION 4. Gomtry of md-surfac of shll FIGURE 4- GEOMERY OF SHELL SRUCURE IN CRESIN COORDINES h shll gomtry usd n th prsnt formulaton has bn dvlopd usng an orthogonal curvlnar coordnat systm wth th md-plan of th shll assumd to b th rfrnc surfac as shown n Fg.4.. h shll md-surfac n th Cartsan rctangular coordnat systm has bn frst mappd nto a paramtrc doman through th sutabl xact paramtrzaton. wo ndpndnt coordnats (, ) n th paramtrc spac hav bn consdrd as th md-surfac curvlnar coordnats of th shll. h normal drcton coordnat to th mddl surfac of th shll has bn rprsntd by z. h rfrnc surfac or th shll md-surfac can b dscrbd n th global Cartsan coordnats n trms of th poston vctor as, r(, ) X (, ) Y(, ) j Z(, ) k (30) Whr,, j and k ar unt vctors along th X, Y and Z axs, rspctvly. h tangnt to th soparamtrc curvs s and s rspctvly ar r r ; r r (3) h vctor jonng two ponts on th mddl surfac (, ) and ( d, d) s gvn as 4

28 Scalar product of ds, ds rd r d (3) ds. ds ( r. r ) d ( r. r ) d (33) Lam s paramtrs can b dfnd as, r. r r. r (34) Eqn. (4) can b wrttn as ds d d (35) Snc th and ar ndpndnt coordnats ds ds ds (36) Whr, ds ds d d h unt tangnt vctors to th soparamtrc curv s and s can b xprssd rspctvly as, r r (37) ; h unt normal vctor to th tangnt plan of any pont on th rfrnc surfac can b xprssd as n r r r r h normal curvaturs and twst curvaturs of th md-surfac of shll can b xprssd as: (38). r R n R R. r n n. r (39) Whr R, R ar th normal curvaturs of th mdsurfac of th shll and R s th twst curvatur of th md-surfac of th shll. 5

29 4.. Isoparamtrc mappng FIGURE 4- ISOPRMERIC MPPING Fg.4. shows Cartsan coordnats ar convrtd nto curvlnar coordnats and t s mappd nto soparamtrc form. h shll mdsurfac n th rctangular cartsan coordnat systm has bn mappd nto th paramtrc spac (, ) and th mdsurfac n th paramtrc spac has bn dvdd nto rqurd numbr of quadrlatral lmnts or sub-domans. h rfrnc coordnats (, ) map th quadrlatral lmnt n th curvlnar coordnats (, ) nto th rfrnc coordnats that s a squar as shown n Fg. 4.. ny pont wthn an lmnt n th paramtrc spac has bn approxmatd by th soparamtrc mappng. Hnc th curvlnar coordnats (, ) of any pont wthn an lmnt may b xprssd as nd nd N N (, ) s th coordnat of mdsurfac at (40) th nod n curvlnar coordnat systm. u0, v0 and w 0 ar th dflcton of mdsurfac at th nod n, and z drctons rspctvly. s th rotaton of normal at th nod about axs and s th rotaton of normal at th nod about axs. h dsplacmnt componnts on th shll mdsurfac at any pont wthn an lmnt may b xprssd as 6

30 u0 v u0 v (4) 0 nd 0 w N w Whr, nd s th numbr of nods n an lmnt, nod and shap functons of 8 nodd srndpty lmnt ar gvn blow N s th shap functon corrspondng to th th N N N N N N N N ( )( )( ) 4 ( )( ) ( )( )( ) 4 ( )( ) ( )( )( ) 4 ( )( ) ( )( )( ) 4 ( )( ) (4) 4.. ransformaton matrx usd for soparamtrc mappng Snc th ntgraton s to b don n natural coordnats (, ), th lmnt s mappd nto th soparamtrc spac (, ) usng th soparamtrc shap functons. h transformaton matrx usd s gvn blow. h rlaton btwn th shap functon drvatvs n paramtrc spac (, ) and n soparamtrc spac (, ) ar gvn as N N J N N (43) 7

31 h jacoban matrx can b xprssd as J (44) J * a J (45) d J dd (46) 4. Stran dsplacmnt rlatons Nglctng normal stran componnt n th thcknss drcton, th fv stran componnts of a doubly curvd shll may b xprss as 0 xx xx kxx 0 yy k yy yy 0 xy z k xy xy 0 yz 0 yz 0 xz 0 xz Whr, and s th n-plan strans of th mdsurfac n th cartsan coordnat systm xx yy xy and kxx, kyy and k xy ar th bndng strans (curvaturs) of th mdsurfac n th cartsan coordnats systm. ftr ncorporatng th ffct of transvrs stan n Kotr s shll thory, nplan and transvrs stran-dsplacmnt rlatons may b xprssd as dscrbd n th followng subsctons. 4.. In-plan/bndng stran-dsplacmnt matrx h stran componnts on th mdsurfac of shll lmnt ar xx yy xy xx yy xy (47) k k k (48) By usng soparamtrc 8-nodd shll lmnt, th dsplacmnt componnt on th shll mdsurfac at any pont wthn an lmnt can b xprssd as u0 v 0 w Nd (49) 8

32 h md-surfac strans and curvaturs from Kotr s shll thory ar: u v w 0 xx R v u w 0 yy R v u u v w 0 xy R (50) (5) (5) k xx v u u v R (53) k k yy xy v u u v (54) R (55) v u u v R R By usng 8-nodd soparamtrc shap functons from Eqn. (3), th stran componnts at any pont on th shll mdsurfac can b xprss 8 N N N 0 0 R N N N 0 0 R N N N N N u0 0 0 R v 0 N N N N N N 0 w R R N N N N N 0 N R R N N N N N N N N C 0 0 C 0 B b d (56) (57) B b s th lmnt n plan stan- dsplacmnt matrx and C o R R 9

33 4.. ransvrs stran dsplacmnt matrx ccordng to th FSD, th transvrs shar stran vctor of a doubly curvd shll lmnt may b xprssd as w u v yz R R (58) xz w u v R R nd hnc th transvrs shar stran at any pont on th shll md surfac can b xprssd as u 0 N N N 0 N v nd 0 yz R R w xz k N N N N 0 R R (59) B d yz s xz (60) B s s th lmnt transvrs stan- dsplacmnt matrx BBD matrx h rsultant forc pr unt wdth s xx yy xy N N N N (6) h rsultant momnt pr unt wdth s xx yy xy M M M M (6) In plan strans of md-surfac s xx yy xy (63) Curvaturs of md-surfac s xx yy xy k k k k (64) h stran at any pont on an lmnt ( ) 0 xy zk (65) h rsultant forc pr unt wdth can b xprssd as h/ N h/ ( xy) dz 0 N Bk (66) 0

34 h rsultant momnt pr unt wdth h/ ( xy) zdz h/ (67) M M B D k o Eqs. (37) and (38) can b xprssd n matrx form as 0 N B M B D k B B D (68) (69) Q Q Qb k Q Q 0 0 Q Q Q E 0 0 Q 66 FIGURE 4-3 (7) E (7) E E (73) SCKING SEQUENCE IN LMINE E (75) E (76) (77) (70) Q G (74) 66 h transformaton matrx sn cos sn cos cos sn cos sn sn cos R sn cos sn cos k h transformd lastcty matrx for bndng Q R Q R b k b k k k (78) (79)

35 Q S KG KG Whr K dnots shar corrcton factor h transformaton matrx R s cos sn 3 sn cos h transformd lastcty matrx for transvrs shar Qs R Q R s S s (80) (8) (8) n D Q ( Z Z ) (83) s s k k k h xtnsonal stffnss matrx n Q Z Z (84) k b k k k h xtnsonal-bndng couplng stffnss matrx (85) n B Q b Zk Zk k h bndng stffnss matrx n 3 3 D Q b Zk Zk k k 3 (86) k 4.3 Equaton of moton h lastc fnt lmnt formulaton has bn drvd for statc and dynamc analyss Statc fnt quatons h total potntal nrgy of th lmnt s gvn by U W (87) U and W ar th stran nrgy of th ntr structur and work don by th xtrnal forc. Stran nrgy of th shll structur s gvn by, U V U C dv V U d B C B dv d u u U d K d V dv uu Work don by xtrnal forc s gvn by, (88)

36 s, W d f x y d s, W d N f x y d (89) W d F Substtutng th valus of U and W n total potntal nrgy Eqn.58, w hav K d F uu (90) Eqn. (6) s th lastc quaton for on lmnt. Whr, K uu s th lmnt structural stffnss matrx s gvn by uu u u V K B C B dv (9) Kuu Kbb Kss (9) 0 B b Bu 0 B s C Db 0 0 D s D s th n-plan/bndng consttutv matrx and D b s (93) (94) s th transvrs shar consttutv matrx. K bb, s th lmnt n plan/bndng stffnss matrx. bb b b b V K B D B dv B B D K bb B b B b d (95) F K ss, s th lmnt transvrs shar stffnss matrx ss s s s V K B D B dv ss s s s K B D B d, s th lmnt xtrnal mchancal forc vctor s, F N f x y d (97) Hnc all th lmnt stffnss matrcs can b xprssd n ts fnal form as (96) 3

37 K Kh J d d (98) ftr assmblng th lmntal stffnss matrcs, th global st of lastc quatons s gvn by K d F uu (99) 4.3. Dynamc fnt quatons h dynamc fnt lmnt formulaton has bn drvd by usng Hamlton s prncpl as t t L W dt 0 (00) whr, th Lagrangan, L U s th kntc nrgy of th systm, U s th lastc stran nrgy, W s th xtrnal work don by th forc on th structur. t t U W dt 0 (0) whr t and t dfns th tm ntrval. Kntc nrgy s gvn by, d d dv (0) V s th mass dnsty. Indvdual parts of th Hamlton quaton can b wrttn as follows: Part : h kntc nrgy of th systm t t t t V dt d d dvdt t t t t V t dt d d dvdt dt t d N N dv d dt = t t V t dt d M uu d dt (03) t t t Part : h total ntrnal nrgy of th systm t t Udt V m dt Udt d K uu d dt (04) t t t t t t Part 3: h work don by th xtrnal forcs 4

38 t t t t t W dt d fs x, y ddt t W dt d N f x y ddt s, t t t Wdt d F (05) t Substtutng qns. (74), (75), (76) n q. (7) gvs t t uu uu d M d K d F dt 0 Snc d can b any arbtrary valus, d 0 uu (06) and thrfor th q. (43) s zro only f Muu d K d F (07) Eqn. (78) s dynamc fnt lmnt quatons of on lmnt. Whr th structural mass matrx s gvn by, Muu N N dv V (08) h dsplacmnt at any pont n th lamnat can b xprssd as u u0 z x v v0 z y w w (09) u v u N nd zn 0 v 0 N 0 0 zn w (0) w 0 0 N 0 0 x y u v N d w Muu N N dv V () () h M uu N N dzd h (3) 5

39 Mass matrx n ts fnal form can b xprssd as h/ Muu N Ndz J dd (4) h/ 4.4 Calculaton of non-dmnsonal frquncs In ordr to vrfy th abov formulaton for sphrcal shll havng msh0 0, th varous nondmnsonal frquncs of th prsnt formulaton ar compard wth xstng ons. ** a (5) h E ** Whr s th non-dmnsonal frquncy, s th natural frquncy, a s th lngth of shll structur along x-axs, h s th thcknss of th lamnat, s th dnsty of compost, E s th transvrs modulus of th lamna havng 0 0 orntaton. 4.5 Stat spac mthod for mpuls rspons h quaton of moton of systm consdrng dampng n modal form s gvn as: f (6) Whr,, and f ar th modal coordnats, modal dampng rato, natural frquncy and forc vctor n modal form for =,, 3., dof. For dampd systms whch ar modld usng Raylgh proportonal dampng, t s dffcult to dtrmn th Raylgh constants. Calculatng Raylgh dampng coffcnts for larg dgr of frdom systm has bn provdd wth dtal n Chowdhury and Dasgupta []. ssumng 3% of th total mods to b actv mods or whch partcpat n mass proportonal dampng. s thr s no data rgardng th dampng ratos for th abov formulatd hybrd compost w procd th nvstgaton by assumng th dampng rato of th frst mod to b 0.0 and dampng rato of th last actv mod to b h constants and can b calculatd usng th dampng ratos and th natural frquncs of th frst mod and th last actv mod. h dampng rato for th ntrmdat mod can b calculatd onc th valu of and ar known. hus (7) 6

40 h stat spac and output quatons n modal form for n numbr of mods can b wrttn as, 0 I 0 nn nn dof dof dof F dof n nn n n n n dof dof M dof dof n n n n n n I 0 dof dof Fdof n n Whr 0 n n s null matrx, matrcs,... n ar th stat vctors. I s th dntty matrx, n n n n n and (8) (9) n n n n ar dagonal 7

41 5 RESULS ND DISCUSSION Summary of th abov chaptrs Hybrd compost s a compost whch conssts of nanopartcls to nhanc th strngth as compard to convntonal composts. modl has bn proposd to dtrmn th lastc proprts of hybrd compost. h hybrd compost conssts of convntonal fbr and nanocompost as matrx. h frst stp hr s to dtrmn th proprts of nanocompost whch s don by usng Mor anaka mthod. h CNs ar consdrd as cylndrcal nclusons n polymr matrx n Mor anaka mthod. ssumng prfct bondng btwn carbon fbrs and nanocompost matrx, th ffctv proprts of th hybrd compost has bn valuatd usng mchancs of matrals approach. n 8 nodd shll lmnt has bn usd for th fnt lmnt analyss havng 5 dgrs of frdom ach nod u, v, w, x, y. 0 0 fnt lmnt msh has bn usd to modl th shll lmnt. h shll coordnats whch ar n Cartsan form ar convrtd nto paramtrc form usng two paramtrs. hs paramtrs ar agan mappd nto soparamtrc form,,. 6 layrd lamnat wth stackng squnc [ ] S has bn usd for vbraton analyss of smply supportd shll lmnt. h dynamc quatons of shll ar drvd usng Hamlton s prncpl. s th dampng charactrs of th dynamc systm ar not avalabl, for furthr nvstgaton dampng rato of frst mod and last actv mod ar assumd. Usng Raylgh dampng th dampng ratos of ntrmdat mods can b calculatd. h tm dcay of th systm from maxmum ampltud to 5% of th maxmum ampltud has bn usd as a paramtr to study varous shll structurs by varyng th volum fracton of CNs n nanocompost and by varyng carbon fbr volum fracton. 8

42 Valdaton of formulaton Fr vbraton analyss s don to valdat th abov formulaton., 3 and 4 layrd cross ply lamnat has bn usd to carry out fr vbraton analyss by varyng th a/h rato and R/a rato of th sphrcal shll. Rfrnc [3] Panl a/h R/a / / /90/ /90/ /90/90/ /90/90/ Prsnt Formulaton BLE 5- NON-DIMENSIONL FREQUENCY [REF:3] Panl a/h R/a / / /90/ /90/ /90/90/ /90/90/ BLE 5- NON-DIMENSIONL FREQUENCY FOR HE PRESEN FORMULION 9

43 Matral Proprts Proprts of varous consttunts of Hybrd compost Consttunt C (GPa) C (GPa) C (GPa) C 3 (GPa) C 55 (GPa) Dnsty(kg/m 3 ) Carbon [4] fbr[] (5,5), [5] walld CN[6] Epoxy[8] BLE 5-3 MERIL PROPERIES OF VRIOUS CONSIUENS IN HYBRID COMPOSIE C FIGURE 5- VRIION OF C W.R. VRIION OF CRBON FIBER ND CN VOLUME FRCION From Fg. 5., t can b sn that as th carbon fbr volum fracton ncrass th longtudnal lastc proprts ncras. t lowr volum fractons of carbon fbr (0%) t can b obsrvd that for CFRP compost th lastc modulus s around 35GPa, but wth th ncras n volum fracton of CN from % to 5% th lastc modulus has ncrasd from 38GPa to 47.3GPa. Wth th ncras n carbon fbr volum fracton, th volum fracton of nanocompost gos on dcrasng; as a rsult th lastc proprts almost convrg at hghr volum fractons of carbon fbr. 30

44 C and C3 FIGURE 5- VRIION OF C W.R. VRIION OF CRBON FIBER ND CN VOLUME FRCION FIGURE 5-3 VRIION OF C 3 W.R. VRIION OF CRBON FIBER ND CN VOLUME FRCION From Fg. 5. and Fg. 5.3, smlarly as C, th lastc proprts along - drcton also ncras wth th ncras wth n volum fracton of carbon fbr. t lowr volum fractons (0%) t can b obsrvd that CFRP has lastc modulus of 6Gpa, but wth ncras n CN volum fracton from % to 5%, th lastc proprts hav ncrasd from 6.8Gpa to 0Gpa. Wth th ncras n carbon fbr volum fracton, t can b obsrvd that composts havng lowr CN volum fractons show stp ncras n lastc proprts than compard to composts havng hghr CN volum fractons. 3

45 C FIGURE 5-4 VRIION OF C W.R. VRIION OF CRBON FIBER ND CN VOLUME FRCION From Fg. 5.4, t can b sn that th transvrs lastc proprts hav mprovd wth th addton of CN n matrx matral for lowr volum fractons of carbon fbr. hs ncras n lastc proprts can b attrbutd to th randomly dstrbutd CNs. Wth th ncras n carbon fbr volum fracton th composts havng lowr volum fractons of CN hav shown an ncras n transvrs lastc modulus, but for composts havng hghr volum fractons of CN wth th ncras n carbon fbr volum fracton thr s dcras n transvrs lastc modulus. C55 FIGURE 5-5 VRIION OF C 55 W.R. VRIION OF CRBON FIBER ND CN VOLUME FRCION From Fg. 5.5, t can b obsrvd that th n-plan shar proprts hav ncrasd wth ncras n volum fractons of CN and carbon fbr. Composts normally fal du to shar. So n ordr avod falur t s advantagous to us hybrd compost n plac of convntonal CFRP composts. 3

46 Impuls rspons FIGURE 5-6 IMPULSE RESPONSE OF CFRP COMPOSIE FOR HICK PLE FIGURE 5-7 IMPULSE RESPONSE OF CFRP COMPOSIE FOR HIN PLE Fg. 5.6 and Fg. 5.7 ndcat th rspons of thck plat and thn plat n modal coordnats for th frst mod of vbraton. 33

47 FIGURE 5-8 DECY IME FOR HICK PLE BY VRYING HE CN VOLUME FRCIONS FOR DIFFEREN VOLUME FRCIONS OF CRBON FIBER FIGURE 5-9 DECY IME FOR HIN PLE BY VRYING HE CN VOLUME FRCIONS FOR DIFFEREN VOLUME FRCIONS OF CRBON FIBER Fg. 5.8 and Fg. 5.9 show th dcay tm for thck and thn plats. t lowr volum fractons carbon fbr, th dcay tm of th systm gos on dcrasng wth ncras n CN volum fracton. s th volum fracton of carbon fbr ncrass th dcay tm also dcrass. 34

48 6 CONCLUSION hs chaptr prsnts mportant obsrvatons on th matral proprts, Raylgh dampng n compost matrals, mpuls rspons, and dcay tm. Conclusons h hybrd compost has bn modld usng Mor-anaka mthod and mchancs of matrals mthod. It s found that. h longtudnal proprts C of th hybrd compost ncras wth th ncras n volum fracton of CN at lowr volum fractons of carbon fbr. s th volum fracton of carbon fbr gos on ncrasng th longtudnal modulus tnds to convrg bcaus th volum fracton of CN gos on dcrasng.. hr s trmndous ncras n lastc proprts C and C3 of th hybrd compost wth th ncras n volum fracton of CN at lowr volum fractons of carbon fbr. s th volum fractons of carbon fbr gos on ncrasng thr s slow ncras n composts havng hghr volum fractons of CN as compard to composts havng lowr volum fractons of CNs. 3. h transvrs modulus C of th hybrd ncrass wth th ncras n CN volum fracton but as th volum fracton of carbon fbr ncrass th composts havng lowr CN volum fractons show ncras n transvrs modulus and composts havng hghr CN volum fracton show gradual dcras n transvrs modulus. 4. h n-plan shar modulus C 55 ncrass wth ncras n CN and carbon fbr volum fracton. 5. h ampltud gos on dcrasng wth ncras n dampng rato. 6. s th volum fracton of CN ncrass th dcay tm gos on dcrasng. 7. h dcay tm of thck plat s lss than th dcay tm of thn plat. 35

49 Futur Scop Estmat tmpratur dpndnt and hygrothrmal proprts. Bucklng analyss of hybrd compost lamnatd shll structur. ctv vbraton control of th lamnatd shll structur. Dlamnaton analyss. Nonlnar analyss of lamnatd shll structur. 36

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