Keywords- Active vibration control, cantilever composite beam, Newmark-β method

Transcription

1 Pratik K. Gandhi, J. R. Mvada / Intrnational Journal of Enginring Rsarch and Applications (IJERA) ISSN: Vol., Issu, May-Jun, pp.9-95 A Finit Elmnt Modl And Activ Vibration Control Of Composit Bams With Distributd Pizolctrics Using hird Ordr hory Pratik K. Gandhi*, J. R. Mvada** *Studnt, M.ch, (Dpartmnt of Mchanical & Mchatronics Enginring, Ganpat Univrsity, Khrva) ** Associat Prof., (Dpartmnt of Mchanical & Mchatronics Enginring, Ganpat Univrsity, Khrva) Abstract In this study, A finit lmnt modl basd on th third ordr thory is prsntd for th activ vibration control of composit bams with distributd pizolctric snsors and actuators. For calculating th total charg on th snsor, th dirct pizolctric quation is usd and th actuators provid a damping ffct on th composit bam by coupling a ngativ vlocity fdback control algorithm in a closd control loop. A modal suprposition tchniqu and a Nwmark-β mthod ar usd in th numrical analysis to comput th dynamic rspons of composit bams. Algorithm for calculating th various valus of variabls is carrid out using Matlab tool Kywords- Activ vibration control, cantilvr composit bam, Nwmark-β mthod. Introduction Enginring structurs work frquntly undr dynamic xcitations. h typ of xcitation may vary but th rsults of ths xcitations ar shown gnrally in th form of th vibrations. Vibrations can b attributd as an unwantd for many nginring structurs du to prcision losss, nois, wast of nrgy, tc. and should b kpt undr control for lightwight structurs. Attmpts at solving ths problms hav rcntly stimulatd xtnsiv rsarch into smart structurs and systms. A smart structur can b dfind as a structur with bondd or mbddd snsors and actuators with an associatd control systm, which nabls th structur to rspond simultanously to xtrnal stimuli xrtd on it and thn supprss undsird ffcts. Smart structurs hav found application in monitoring and controlling th dformation of th structurs in a varity of nginring systms. Advancs in smart matrials tchnology hav producd much smallr actuators and snsors high intgrity in structurs and an incras in th application of smart matrials for passiv and activ structural damping. Svral invstigators hav dvlopd analytical and numrical, linar and non-linar modls for th rspons of intgratd pizolctric structurs. hs modls provid platform for xploring activ vibration control in smart structurs. h xprimntal work of Baily and Hubbard, 985 [] is usually citd as th first application of pizolctric matrials as actuators. hy succssfully usd pizolctric snsors and actuators in th vibration control of isotropic cantilvr bams. P. R. Hyligr and J. N. Rddy, 988 [] had prsntd highr ordr bam thory for static and dynamic bhaviour of rctangular bams using finit lmnt quations. Ha, Kilrs and Chang, 99 [] dvlopd a modal basd on th classical laminatd plat thory for th dynamic and static rspons of laminatd composits with distributd pizolctric. H.S. zou and C. I. sng, 99 [] had prsntd finit lmnt formulation of distributd vibration control and idntification of coupld pizolctric systms. Chandrashkhara and Varadarajan, 997 [5] gav a finit lmnt modl basd on highr ordr shar dformation thory for laminatd composit bams with intgratd pizolctric actuators. Woo-sok Hwang, Woonbong Hwang and Hyun Chul Park, 99 [6] had drivd vibration control of laminatd composit plat with pizolctric snsor/actuator using activ and passiv control mthods. X. Q. Png, K. Y. Lam and G. R. Liu,998 [7] gav finit lmnt modl basd on third ordr thory for th activ position and vibration control of composit bams with distributd pizolctric snsors and actuartors. Bohua Sun and Da Huang, [8] had drivd analytical solution for activ vibration control and supprssion of smart laminatd composit bam with pizolctric damping layr. V. Balamuragan and S. Narayanan, [9] proposd finit lmnt formulation and activ vibration control on bams using smart constraind layr damping tratmnt (SCLD). S. Narayanan and V. Balamuragan, [] gav finit lmnt modal of pizolaminatd smart structurs for activ vibration control with distributd snsors and actuators.. Mathmatical modlling of bam. Pizolctric quations Assuming that a bam consists of a numbr of layrs (including pizolctric layrs) and ach layr posssss a plan of matrial 9 P a g

2 Pratik K. Gandhi, J. R. Mvada / Intrnational Journal of Enginring Rsarch and Applications (IJERA) ISSN: Vol., Issu, May-Jun, pp.9-95 symmtrically paralll to th x-y plan and a linar pizolctric coupling btwn th lastic fild and th lctric fild th constitutiv quations for th layr can b writtn as, [] D E D () 5 5 k k k E k k Q E Q E () 5 k 55 k 5 k 5 k k h thrmal ffcts ar not considrd in th analysis. can b h pizolctric constant matrix xprssd in trms of pizolctric strain constant d as matrix dq Whr, d () d d5 (). Displacmnt fild of third ordr thory h displacmnt fild basd on th third ordr bam thory of Rddy [] givn by w u x, z, t u x, t zx x, t z x x (5) w x, z, t w x, t (6) Whr, and h is th total thicknss of th h bam. h displacmnt functions ar approximatd ovr ach finit lmnt by [7], i i i u x t u t x (7) x, t t x (8) x i i i, i i i w x t t x (9) Whr, ar th linar Lagrangian intrpolation i polynomials and th i ar th cubic Hrmit intrpolation polynomials. and rprsnt nodal valus of w, whras and rprsnt w nodal valus of x. W dfin u u w u u u () Using finit lmnt quations (5) and (6) can b xprssd as u N u () h strain-displacmnt rlations ar givn by 5 Whr, N Bu N and zz z x z x zz z x z x B ar as follows () () x z z z x z z x x x z z x x x B x z z z x z z x x x z z x x x (). Dynamic Equations h dynamic quations of a pizolctric structur can b drivd by Hamilton s principl [7]. t t U W dt (5) 9 P a g

3 Pratik K. Gandhi, J. R. Mvada / Intrnational Journal of Enginring Rsarch and Applications (IJERA) ISSN: Vol., Issu, May-Jun, pp.9-95 Whr is th kintic nrgy, U is th strain nrgy, and W is th work don by th applid forcs. In this, th kintic nrgy at lmntal lvl is u u dv (6) V Whr, V is th volum of th bam lmnt. h strain nrgy at lmntal lvl is U dv (7) V And work don by th xtrnal forcs is W u f dv V b u f ds s u f c S (8) Whr, f b is th body forc, S is th surfac ara of th bam lmnt, f s is th surfac forc and f is th concntratd load. c C a M bk (5) Whr, a andb ar th constants can b dtrmind from xprimnts []. Assmbling all lmntal quations givs global dynamic quation is as follows: M u C u K u F F V (6) Whr, Fis th xtrnal mchanical forc vctor and F is th lctrical forc vctor. V F K V v (7) uv.5 Snsor quations Sinc no xtrnal lctrical fild is applid to th snsor layr and a charg is collctd only in th thicknss dirction, only th lctric displacmnt is of intrst and can b writtn as [7] D (8). Equation of motion o dvlop th quation of motion of th systm, considr th dynamic bhavior of th systm. hs quations also provid coupling btwn lctrical and mchanical trms. h quation of motion in th matrix form can b writtn as, [7] uv M u K u F K V (9) Whr, V M N N dv () V K B Q B dv () K B B dv uv v () V F N fbdv N fsds N fc () V S Assuming that th snsor patch covrs svral lmnts, th total charg dvlopd on th snsor surfac is j Ns qt B B z zk z zk ds u j S j (9) Whr, B is th first row ofb. h distributd snsor gnrats a voltag whn th structur is oscillating; and this signal is fd back into distributd actuator using a control algorithm, as shown in fig.. h actuating voltag undr a constant gain control algorithm can b xprssd as, dq () V GV i s GiGc dt In ordr to includ th damping ffcts, Rayligh damping is assumd. So quation (9) modifid as, M u C u K u F K V () uv Whr, C is th damping matrix, which is Fig. Block diagram of fdback control systm h systm actuating voltags can b writtn as 9 P a g

4 Pratik K. Gandhi, J. R. Mvada / Intrnational Journal of Enginring Rsarch and Applications (IJERA) ISSN: Vol., Issu, May-Jun, pp.9-95 V G K u () x M u () v Whr, G G G. i c G is th control gain matrix and In th fdback control, th lctrical forc vctor F v can b rgardd as a fdback forc. Substituting quation () into quation (7) givs, Fv Kuv GKv u (). Matrial proprtis of cantilvr composit bam A cantilvr composit bam with both uppr and lowr surfacs symmtrically bondd by pizolctric cramics, shown in fig. W dfin, C Kuv GKv () hus, th systm quation of motion quation (6) bcoms M u C C u Ku F () As shown in quation (), th voltag control algorithm quation () has a damping ffct on th vibration supprssion of a distributd systm. For obtaining a dynamic rspons undr a givn xtrnal loading condition, a modal analysis is usd, and th nodal displacmnt uis rprsntd by [] u x (5) Whr x ar rfrrd to as th gnralizd displacmnts. is th modal matrix and has th orthogonal proprtis as: K, M I (6) Whr, is a diagonal matrix that stors th squars of th natural frquncis. Substituting quation (5) into () and thn multiplying quation () givs M x C C x Kx F (7) Substituting quation (6) into (7) givs x C x x F (8) h initial condition on x is x M u (9) i Fig. A cantilvr composit bam h bam is mad of /976 graphit/poxy composits and th pizocramic is PZ G95N. h adhsiv layrs ar nglctd. h matrial proprtis ar givn in tabl. h stacking squnc of composit bam is 5 / 5 s. h total thicknss of th composit bam is 9.6 mm and ach layr has th sam thicknss (. mm). and th thicknss of ach pizo-layr is. mm. h lowr pizocramics srv as snsors and th uppr ons as actuators. h rlativ snsors and actuators form snsor/actuator pairs through closd control loops. abl. Matrial proprtis of PZ and /976 graphit/poxy composit Proprtis Symbol PZ /976 Young moduli E (GPa) E =E Poisson s ratio v =v v.... Shar moduli G =G. 7. (GPa) G..5 Dnsity (Kg/mg ) Ρ 76 6 Pizolctric d =d 5 - constants (m/v) Elctrical prmittivity (F/m) ε =ε ε Activ vibration supprssion Hr siz of th bam is considrd as 6 mm long, mm width and 9.6 mm thick, also thicknss of ach pizo layr is. mm. In th analysis, th composit bam is dividd into lmnts. h pizolctric composit bam givn in figur is considrd to 9 P a g

5 ip Displacmnt (m) ip Displacmnt (m) ip Displacmnt (m) ip Displacmnt (m) ip Displacmnt (m) ip Displacmnt (m) Pratik K. Gandhi, J. R. Mvada / Intrnational Journal of Enginring Rsarch and Applications (IJERA) ISSN: Vol., Issu, May-Jun, pp.9-95 simulat th activ vibration supprssion through a simpl S/A activ control algorithm (ngativ vlocity fdback). h pizocramics on th lowr surfac srv as snsors,and on th uppr surfac ar actuators. In th analysis th bam is dividd into lmnts so ach S/A pairs covrs six lmnts. First, modal suprposition tchniqu is usd to dcras th siz of th problm. th diffrnt mods ar usd in th modal spac analysis and transint rspons of th cantilvr bam is computd by th Nwmark dirct intgration mthod []. h paramtrs α and β ar from Rayligh mthod. Rsults obtain aftr ths mthods ar as undr..5 ip Displacmnt of cantilvr bam Fig. 5 h ffct of ngativ vlocity gain on th bam subjctd to scond mod vibration. (Gain= im (sc) Fig. 6 h ffct of ngativ vlocity gain on th bam subjctd to scond mod vibration. (Gain=-5 Figur 5 and 6 shows th vibration control prformanc for PZ fully covrd bam with control gain - V/A and -5 V/A for th scond valu of alpha and bta of Rayligh damping...5 ip Displacmnt of cantilvr bam ip Displacmnt of cantilvr bam im (sc) Fig. h ffct of ngativ vlocity gain on th bam subjctd to first mod vibration. (Gain=-. ip Displacmnt of cantilvr bam im (sc) Fig. 7 h ffct of ngativ vlocity gain on th bam subjctd to third mod vibration. (Gain=-.5. ip Displacmnt of cantilvr bam im (sc) Fig. h ffct of ngativ vlocity gain on th bam subjctd to first mod vibration. (Gain=-5 Figur and shows th vibration control prformanc for PZ fully covrd bam with control gain - V/A and -5 V/A for first valu of alpha and bta of Rayligh damping ip Displacmnt of cantilvr bam im (sc) Fig. 8 h ffct of ngativ vlocity gain on th bam subjctd to third mod vibration. (Gain=-5 Figur 7 and 8 shows th vibration control prformanc for PZ fully covrd bam with control gain - V/A and -5 V/A for th scond valu of alpha and bta of Rayligh damping. As shown in figurs, th vibrations dcay mor quickly whn highr control gains ar applid. Howvr, it must b notd that th gains should b limitd for th sak of th brakdown voltag of th pizolctrics im (sc) 9 P a g

6 Pratik K. Gandhi, J. R. Mvada / Intrnational Journal of Enginring Rsarch and Applications (IJERA) ISSN: Vol., Issu, May-Jun, pp Conclusion A finit lmnt modl and computr cods in Matlab, basd on third ordr laminat thory, ar dvlopd for th cantilvr composit bam with distributd pizolctric cramics. Equation of motion of composit bam is drivd by using Hamilton s principl. h S/A pairs must b placd in high strain rgions and away from th ara of low strain rgions for maximum ffctivnss. Mor S/A pairs can gnrally induc mor fficincy on th activ vibration supprssion. So th numbr of S/A pairs has a grat ffct on th prformanc of smart structurs. As th fdback gain incrass, th damping ratios ar larg and as th fdback gain dcrass, th damping ratios ar smallr. Changs in stiffnss and th location and th numbr of S/A pairs also affct th dynamic rspons of th composit structur. So, basd on this work, activ vibration control with finit lmnt modl is rliabl and fail-saf. Rfrncs [] Baily. and Hubbard J.E., Distributd pizolctric-polymr activ control of a cantilvr bam, Journal of Guidanc, Control and Dynamics, vol. 8, pp. 65-6, (985). [] P. R. Hyligr and J. N. Rddy, A highr ordr bam finit lmnt for bnding and vibration problm, Journal of Sound and Vibration 6, 9-6 (988). [] S. K. Ha, C. Kilrs and F. K. Chang, Finit lmnt analysis of composit structurs containing distributd pizocramics snsors and actuators, Amrican Institut of Aronautics and Astronautics Journal, pp (99). [] H. S. zou and C. I. sng, Distributd pizolctric snsor/actuator dsign for dynamic masurmnt/control of distributd paramtr systm: a pizolctric finit lmnt approach, Journal of Sound and Vibration 8,7- (99). [5] K. Chandrashkhara and S. Varadarajan, Adaptiv shap control of composit bams with pizolctric actuators, Journal of Intllignt Matrial Systms and Structurs, vol. 8, pp. -, (997). [6] Woo-Sok Hwang, Woonbong Hwang and Hyun Chul Park, Vibration control of laminatd composit plats with pizolctric snsor/actuator: Activ and passiv control mthods, Mchanical Systms and Signal Procssing, vol. 8(5), 57-58, (99). [7] X. Q. Png, K. Y. Lam and G. R. Liu, Activ vibration control of composit bams with pizolctrics, Journal of Sound and Vibration, 9, 65-65, (998). [8] Bohua Sun, Da Huang, Vibration supprssion of laminatd composit bams with pizolctric damping layr, Composit Structurs vol. 5, 7-7, (). [9] V. Balamurugan and S. Narayanan, Finit lmnt formulation and activ vibration control study on bam using SCLD tratmnt, Journal of Sound and Vibration, vol. 9(), 7-5, (). [] S. Narayanan and V. Balamurugan, Finit lmnt modling of pizolaminatd smart structurs for activ vibration control with distributd snsors and actuators, Journal of Sound and Vibration, vol. 6, 59-56, (). [] K. J. Bath, Finit lmnt procdurs in nginring analysis. Englwood Cliffs, Nw Jrsy: Prntic- Hall. [] IEEE Standard on Pizolctricity ANSI- IEEE Std 76-87, (987) 95 P a g