A Higher-Order FEM for Vibration Control of Composite Plates with Distributed Piezoelectric Sensors and Actuators

Transcription

1 A Highr-Ordr FEM fr Vibratin Cntrl f Cmpsit Plats with Distributd Pizltri Snsrs and Atuatrs Xialin Chn ), Hnging Hua ) and Yijun Liu ) ) Dpartmnt f Mhanial Enginring Univrsit f Cininnati Cininnati, OH 5-007, USA ) Stat K Labratr f Vibratin, Shk and Nis Shanghai Jiatng Univrsit Shanghai 0000, China Summar Th ativ vibratin ntrl f laminatd mpsit plats using distributd pizltri snsrs and atuatrs (Fig. ) is studid numriall with a highr-rdr finit lmnt mthd (FEM). Rdd s simplifid highr-rdr thr fr transvrs shar dfrmatin, whih is fund t b muh mr aurat fr bth thin and thik mpsit plats as mpard with thr thris, is mpld fr th finit lmnt mdls f th mpsit plats and pizltri dvis. On th thr hand, th finit lmnts ar basd n highr-rdr Lagrang intrplatins in rdr t prvid highr aura and ffiin in th FEM prss. Th analsis f th sstm f mpsit plats with pizltri snsrs and atuatrs is frmulatd in suh a wa that th atuating ffts ar intrdud int th dnami quatins thrugh th damping ffiint matri. Th vibratin ntrl f a rtangular mpsit plat with th pizltri dvis is studid t validat th dvlpd FEM apprah. Th minimal numbr and ptimal latins f th pizltri snsrs and atuatrs ar invstigatd intnsivl thrugh tn ass. Th numrial rsults shw that b using th pizltri dvis, th dnami rspns f th plat an rdu as muh as 0% and als da at a fastr rat. It is fund that th bst latin fr th pizltri atuatrs is in th high strain aras f th strutur, and that, in this as, mr smallr, sgmntd pizltri atuatrs an hav bttr ntrl ffts than a fw largr, ntinuus ns. Z b Y Sgmntd atuatr O Plat a X Sgmntd snsr Figur. Ativ vibratin ntrl f a mpsit plat with pizltri snsrs and atuatrs.

2 Ovrviw f th Finit Elmnt Frmulatin Th finit lmnt mthd has bn applid t th vibratin analsis f pizltri slids fr quit sm tim sin th wrk b Allik and Hughs using -D lmnts []. Fr th dnami analsis f plats and shlls, it is in gnral mr ffiint t mpl -D finit lmnts basd n plat and shll thris, sin th pizltri snsrs and atuatrs bndd n th surfas f th plats r shlls ar in gnral vr thin. Sm f th rnt wrk in this ara an b fund in [-] fr th vibratin ntrl r masurmnt f thin, istrpi plats and shlls with distributd pizltri snsrs and atuatrs vring th whl uppr and lwr surfas; in [5, 6] fr th sam tpis but with sgmntd pizltri films vring nl part f th surfas; and in [7-9] fr th vibratin ntrl analsis f anistrpi pizltri laminatd plats. Cmpsit plats with distributd pizltri snsrs and atuatrs was studid rntl in [0] with ightnd quadrati lmnts. In Rf. [0], th mpsit plat lmnt is basd n Rdd s highr-rdr thr fr laminatd mpsit plats [, ]. Hwvr, th distributd pizltri snsrs and atuatrs mdld in [0] vr th whl tp and bttm surfas f th mpsit plat, whih ma nt b th mst ffiint wa t mpl th pizltri lmnts n mpsit plats, as shwn in th urrnt stud. In th urrnt stud a Lagrang tp f lmnt, whih an prvid arbitraril highr-rdr plnmials in th shap funtins, is dvlpd fr th mpsit plats with distributd pizltri snsrs and atuatrs. Ths pizltri dvis ar sgmntd and bndd n parts f th tp and bttm surfas f th mpsit plat, whih prvids th flibilit in studing th ptimal latins f ths pizltri dvis in th ativ vibratin ntrl. Rdd s simplifid highr-rdr thr fr laminatd mpsit plats [, ] is fund during this stud t b mr aurat and ffiint fr th vibratin analsis f bth thin and thik mpsit plats, mpard t th mdls basd n Kirhhff thin plat, Mindlin-Rissnr thik plat and thr highr-rdr shar dfrmatin (suh as LCW) thris. In Rdd s simplifid highr-rdr thr th displamnt fild is rprsntd b u(, z, = u v(, z, = v w(, z, = w (, + zϕ (, + z φ (,, (, + zϕ (, + z φ (,, (,, () whr u, v and w ar th displamnts f a pint (, ) n th midplan; ϕ and ϕ ar th rtatins f th nrmal t th midplan abut th and as, rsptivl; φ and φ ar highr-rdr trms t b dtrmind [, ]. In this stud th inplan displamnts ar nt nsidrd ( u v = 0 ). = At a tpial nd i f a finit lmnt fr th mpsit plat, th gnralizd displamnt vtr is dfind as d { } T i ( = w, ϕ, ϕ, φ, φ. Th gnralizd displamnt vtr at a pint insid th lmnt is givn b N = d( N i ( ξ, η) d (, () i= i in whih N is th numbr f nds n th lmnt and N i th shap funtin in lal rdinats ( ξ, η) fr nd i. In this stud th fllwing Lagrang intrplatin funtins ar mpld as th shap funtins:

3 N i m ( ξ ξ n j ) ( η ηk ) ( ξ, η) =, () j i ( ξi ξ j ) k i ( ηi ηk ) whr m and n ar th numbrs f nds in th ξ and η dirtins, rsptivl. Th advantags f using th Lagrang intrplatins ar th flibilit f inrasing th rdr f plnmials and th high aura in th analsis f mpsit plats, dspit th disadvantag that sm intrnal nds ar ndd n ths lmnts. Basd n th assumptins in () and th intrplatins () and (), th finit lmnt quatin fr th dnami analsis f a mpsit plat an b writtn as: [ M ] d + [ C ] d + [ K]{ d } = { F }, () fllwing th standard FEM prdur fr dnami prblms, whr { d } lasti displamnt, vlit and alratin vtrs, rsptivl; [ ] [ ] [ ] and stiffnss matris, rsptivl; and { } F th fring vtr., d and d ar th ndal M, C and K th mass, damping Whn th pizltri lmnts ar applid n th tp and bttm f th mpsit plat as atuatrs and snsrs, rsptivl th FE quatin an b shwn t hav th fllwing frm: [ ] d + [ C ] + [] C ( ) d + [ K]{ d } { F } M = s whr [] C = [ K ] GG [ K ] T s with [ ] [ ] ua, (5) ua K K and bing tw stiffnss-lik matris assiatd with th lastiltri upling, and G and G ar th gains f th amplifir (ntrllr) and th harg amplifir, rsptivl fr th atuatr in th fdbak ntrl. Drivatins f Eq. (5) is rathr lngth and th dtails, as wll as th ntrl mhanism, an b fund in th wrk [,, 0]. Frm Eq. (5), n an nlud that th frs ating n th mpsit plat b atuatrs ar quivalnt t th fft f additinal damping n th strutur. Th nw damping matri is mpsd f tw parts, n frm th riginal strutural damping and n frm th fdbak ntrl frs f th atuatrs. Th appliatin f th atuatrs an fftivl inras th damping f th nw sstm and thus supprss th vibratin f th strutur. Numrial Eampls Th dvlpd highr-rdr Lagrang lmnts ar first tstd n a mpsit plat, as shwn in Fig. (withut th pizltri lmnts), t mput th natural frqunis f th plat. In this as, th rsults an b validatd using thr mthds. Th Lagrang plat lmnts basd n Rdd s thr wr fund t b trml aurat fr mpsit plats with bth small and larg thiknsss (.g., using nl n lmnt with 7 7 nds t mdl th whl pla.

4 T vrif th dvlpd finit lmnt frmulatin and th ntrl mhanism fr ativ vibratin ntrl, th rtangular lampd mpsit (unidirtinal glass-p) plat with pizltri (PVDF) snsrs and atuatrs, as shwn in Fig., is analzd. Th plat is disrtizd using 6 nin-nd ( ) Lagrang lmnts and a shadd lmnt indiats that th pizltri snsr and atuatr ar applid n th tp and bttm surfas f th plat n that lmnt latin. Th transint rspns f th plat du t an initial vlit spifid at th fr dg f th plat was analzd. Tn distributin pattrns fr th pizltri lmnts wr tstd and th rsults fr fur f ths ass (pattrns I-IV) ar prsntd in Figs. -6. Th displamnt rspnss at th uppr-right rnr nd f th plat (n lmnt 6) ar plttd fr th unntrlld and ntrlld nditins (with tw diffrnt sts f amplifing fatr G fr th fdbak ntrl). Fig. shws that whn th pizltri lmnts ar plad in th rlativl lw strain ara (middl f th pla, th ntrl fft n th rspns is insignifiant. Whn th pizltri lmnts ar plad nar th fid nd f th plat (Figs. and 5), whr rlativl high strains ur, th ntrl ffts imprv dramatiall. Th bst r ptimal ntrl rsults in this stud (with abut 0% rdutin in th displamnt rspns) ar ahivd b plaing fur pizltri lmnts alng th fid nd (Fig. 6). If th pizltri lmnts ar applid all vr th tp and bttm surfas f th plat, as shwn in Fig., th rsults ar almst idntial t ths in Fig. 6. This suggsts that abut thr quartrs f th pizltri lmnts ar usd infftivl and thus wastd, if th ar applid n th whl surfas Dimnsins f th plat: 5 Lngth = 00 mm; Width = 6.5 mm; Thiknss = 5 mm; Thiknss f th pizltri films = 0 µm. Figur. Th FE mdl f a mpsit plat with pizltri snsrs and atuatrs displamnt (m) Unntrlld Cntrlld (G=E, G=500) tim (s) Figur. Pizltri lmnt distributin pattrn I and th displamnt rspns at th rnr nd.

5 displamnt (m) Cntrlld (G=E, G=50) Unntrlld Cntrlld (G=E, G=500) tim (s) Figur. Pizltri lmnt distributin pattrn II and th displamnt rspns at th rnr nd Unntrlld Cntrlld (G=.E, G=50) displamnt (m) Cntrlld (G=.E, G=500) tim (s) Figur 5. Pizltri lmnt distributin pattrn III and th displamnt rspns at th rnr nd displamnt (m) Unntrlld Cntrlld (G=.E, G=50) Cntrlld (G=.E, G=500) tim (s) Figur 6. Pizltri lmnt distributin pattrn IV and th displamnt rspns at th rnr nd. 5

6 Cnlusin Th ativ vibratin ntrl f laminatd mpsit plats using th ativ pizltri lmnts is studid in this papr using th Lagrang tp f finit lmnts basd n Rdd s simplifid mpsit plat thr. Th stud shws that th dnami rspns f a mpsit plat an b fftivl supprssd with th appliatin f th pizltri lmnts. Muh ffiin and hn savings an b ahivd in appling ths pizltri lmnts if th ar sgmntd and plad stratgiall vr th high strain aras f th plat, as mpard with th ptin f appling th ativ lmnts all vr th tw main surfas f th plat. Appliatin f th pizltri lmnts n ral struturs mad f mpsit plats fr ativ vibratin ntrl purpss sms quit prmising. Aknwldgmnt Th authrs wuld lik t aknwldg th rsarh supprt at bth th Shanghai Jiatng Univrsit in China and th Univrsit f Cininnati in USA. Rfrns. H. Allik and T. J. R. Hughs, "Finit lmnt mthd fr pizltri vibratin," Int. J. Numr. Mthds Engrg.,, 5-57 (970).. H. S. Tzu and C. I. Tsng, "Distributd pizltri snsr/atuatr dsign fr dnami masurmnt/ntrl f distributd paramtr sstms: a pizltri finit lmnt apprah," J. Sund Vib., 8, 7- (990).. H. S. Tzu, "Distributd mdal idntifiatin and vibratin ntrl f ntinua: thr and appliatins," Jurnal f Dnami Sstms, Masurmnt, and Cntrl,, 9-99 (99).. H. S. Tzu and C. I. Tsng, "Distributd mdal idntifiatin and vibratin ntrl f ntinua: pizltri finit lmnt frmulatin and analsis," Jurnal f Dnami Sstms, Masurmnt, and Cntrl,, (99). 5. H. S. Tzu and H. Q. Fu, "A stud f sgmntatin f distributd pizltri snsrs and atuatrs, part I: thrtial analsis," J. Sund Vib., 7, 7-59 (99). 6. H. S. Tzu and H. Q. Fu, "A stud f sgmntatin f distributd pizltri snsrs and atuatrs, part II: paramtri stud and ativ vibratin ntrls," J. Sund Vib., 7, 6-75 (99). 7. S. E. Millr, H. Abramvih, and Y. Oshman, "Ativ distributd vibratin ntrl f anistrpi pizltri laminatd plats," J. Sund Vib., 8, (995). 8. S. E. Millr, Y. Oshman, and H. Abramvih, "Mdal ntrl f pizlaminatd anistrpi rtangular plats Part : Mdal transdur thr" AIAA J.,, (996). 9. S. E. Millr, Y. Oshman, and H. Abramvih, "Mdal ntrl f pizlaminatd anistrpi rtangular plats Part : Cntrl thr" AIAA J.,, (996). 0. B. Samanta, M. C. Ra and R. Bhattahara, "Finit lmnt mdl fr ativ ntrl f intllignt struturs," AIAA J.,, (996).. J. N. Rdd "A simpl highr-rdr thr fr laminatd mpsit plats," J. Appl. Mh., 5, (98).. J. N. Rdd "A rfind nnlinar thr f plats with transvrs shar dfrmatin," Int. J. f Slids & Struturs, 0, (98). 6