BEHAVIOUR OF THE ELECTROMECHANICAL COUPLING FACTOR OF CYLINDER SHAPED PIEZOCERAMICS WITH DIFFERENT ASPECT RATIOS.

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1 BEHAVIOUR OF THE ELECTROMECHANICAL COUPLING FACTOR OF CYLINER SHAPE PIEZOCERAMICS ITH IFFERENT ASPECT RATIOS. Pacs: A Ia, Antonio; Lambti, Nicoa; Paaado, Massimo iatimnto di inggnia ttonica - Univsità Roma T Via da Vasca Nava ROMA - ITALY T Fax Emai: ia@nioma3.it Univsità di Sano Via Pont don Mio 8484 Fisciano (SA) - ITALY ABSTRACT Th IEEE Standad on iocticity dfins th ctomchanica coing facto () in static conditions as th atio btn th ngy convtd fom ctic to mchanic fom o vic vsa. Rcnty, th athos hav shon that this dfinition can b xtndd aso to dynamic cass, by anaysing a od ioctic mnt by mans of a distibtd mod. In this o, th dynamic coing facto is comtd fo cyind shad iocamic mnts of diffnt asct atios by sing cassica mods and a 3 anaytica mod. It is shon that it can b coatd to th ffctiv ctomchanica coing facto ( ff ) fo any asct atio. INTROUCTION Th ctomchanica coing facto is a vy sf aamt of ioctic matias bcas it synthticay chaactis th convsion fom mchanica ngy to ctica ngy and vic vsa. Sva diffnt dfinition of h facto hav bn givn in itat. Th ij, aso cad th matia coing facto ( mat ) [], fs to a fndamntay on dimnsiona gomty of th iocamic mnt and it is asiy atd to th ngis invovd in a dynamic tansfomation cyc. Th ffctiv coing facto ff fs to an osciating scimn of any gomty, bt it is not ca in itat ho it is atd in a dynamic tansfomation cyc. Anoth dfinition of th coing facto as givn by Bincot t a. in thi fndamnta o on iocamics [].This dfinition can b aid in static and dynamic sitation and it can b comtd, in inci, fo any gomty of th iocamic mnt bt, it dos not a cis hysica maning and its ation ith ff, hich among a factos is th ony on that can b masd, is not ca. In a cnt o [3] th athos hav shon that i in static sitation aso in dynamic sitation it is ossib to assign to th facto th hysica maning of sqa oot of atio of ngis. In this o, th dynamic coing facto is comtd fo cyind shad iocamic mnts of diffnt asct atios by sing a 3 aoximatd anaytica mod [4]. EFINITIONS OF THE K FACTOR Th IEEE Standad on iocticity [5] dfins th static ioctic coing facto as th sqa oot of th atio btn th ngy dnsity convtd fom an ctica to a mchanica fom c and th tota ngy dnsity invovd in a tansfom cyc of a ioctic mnt:

2 c = () Th tota ngy dnsity stod nit vom by th mnt can b xssd by [6]: SiTi + iei = () In th gna cas this xssion is ath comicat; anyay if on dimnsiona gomtis a concnd, th constittiv qations a simifid and this dfinition aos to giv a hysica maning to th facto. Bincot dfind th coing facto as []: U m = (3) U U h U, U d and U m a th astic, dictic and mta ngy dnsitis, sctivy. Evn if it is ossib to giv a hysica maning to th qantity in (3), this ngy dnsity a not coatd to any ngy convsion cyc; as a consqnc th hysica maning of this facto is diffnt fom that givn in qation (). Hov in th static cas and fo on dimnsiona gomtis th Bincot dfinition givs th sam st of th facto dfind by (), and it can b aid aso in th dynamic cas, vn if it bcoms fqncy dndnt. Th ffctiv ctomchanica coing facto as dfind as []: ff = f h f s and f a th sis and aa fqncis of th qivant md constants cicit hich dscibs th iocamic scimn at ach sonanc fqncy. Fo o ff vas th ation btn ff and mat is : π 8 f d f mat ff s = (5) hi fo high ff vas this ation is ottd in [,fig.9] fo th main vibation mods. Th dfinition of th facto as a sqa oot of atio of ngis as xtndd to dynamic cass in [3]; fo a ossss ioctic scimn insatd both cticay and mchanicay th dynamic coing facto is dfind as: = (6) This dfinition is fomay idntica to qation () bcas th tota ngy in a cyc coincids ith th intic ngy and th convtd ngy is th ctic ngy. Fo distibtd systms both and a fqncy dndnt and th dynamic coing facto mst b vaatd at th aa sonanc fqncy: MOELS ω = im ω ω ω Fig shos a iocamic od (a) and a iocmic dis (b), both od in -diction and ith th fat sfacs ctodd. Ths gomtis can b sd fo dscibing th imotant on dimnsiona vibation mods: th RO mod (fig.a), th THICKNESS and th RAIAL mod(fig.b). In th fooing th dynamic coing facto i b comtd and comad ith th ff fo ach of ths mods. Rod Mod Th constittiv qations fo th od mod a []: (4) (7)

3 and th av qation is: g T = S s s g E = S + β S s ρ = t s By imosing stss-f bonday conditions and assming a sinsoida ctica xcitation j t = ω, th sotion of th av qation is: h v = g [ sin( ) tan( θ /)cos( ) ] () ω v = / ρs is th av oagation aong th diction, = v/ ω and θ = xssions of th intic ngy dnsity and of th ctic ngy dnsity a [3]: v sin ρ ρ θ θ = v d = g 4 θ θ cos S g = C V = β tan( θ/) S A β θs ω ω (i.. θ π Th imit of th atio btn ths to ngis hn maniations is: U ω ω S S ϑ π π sε π sβ (8) (9).Th () () ) aft som 8 g 8 g = im = im = = = ( ω ) (3) Fig a shos th bhavio of and vss fqncy fo a PZT5A (Mogan-Matoc) scimn [7]. As it can b sn at th aa fqncy f both ths ngy dnsitis tnd to infinity. Th atio btn ths to ngis, as as th coing facto U vss fqncy is ottd in fig b: vn if th bhavios of ths fnctions a comty diffnt, thy assm th sam va at th aa sonanc fqncy f = f. This va sighty diffs th ff va in agmnt ith th ot givn by Bincot in [,fig.9]. Thicnss Mod Th constittiv qations fo th thicnss mod a []: T = c S h E = h S + β S Ths qations a fomay idntica to qs.8 and th av qation is idntica to q.9. Thfo th sts i diff fom th vios ons ony fo som constants. hav: h v sin = ρ = 4 c θ ρ θ θ v d θ cos h S = C V = β S A β θc Th imit of th atio btn ths to ngis hn maniations is: tan( θ/) ω ω (i.. θ π (4) (6) (7) ) aft som

4 8 h 8 = im = im = = = ( ω ) (8) U ω ω S S ϑ π π cβ π εc Fig 3 shos th bhavio of and (a), and thi atio and U (b). Th sam considations as thos mad fo fig hod. Radia Mod Th constittiv qations fo th adia mod a [5]: T T θθ = c = = c,, (, + c + c + / + / ) ε / + By imosing stss-f bonday conditions and assming a sinsoida ctica xcitation j t = ω, th sotion of th av qation is: h J ( ) E E E (9) jωt = () jω J( a) ( ) πaε aj ( a) J( a) = c + c + J( a) ε Th xssion of th votag V is: h, ω/ = v, c / ρ v =. B V = () aj ( a) J( a) B = j c + c. ωa J( a) Th intic and th ctic ngy dnsitis a comtd by mans of th foo qations: a = ρ v d a () ε = V (3) to th snc of th Bss fnctions th ngy dnsitis and hav no sim anaytica xssion; thy a ottd vss fqncy in fig 4a. Aso fo th adia mod both th ngis tnd to infinity at th aa fqncy f. Thi atio is ottd in fig 4b; th va obtaind is ss than th ff va. 3 MOEL In od to vaat th dynamic coing facto fo a cyind shad iocamic of any asct atio, an aoximatd 3_ mod cnty oosd by th athos [4] as sd. This mod as divd by assming as sotion of th av qation systm to othogona av fnctions, i.., th coodinat axs and a mod oagation dictions ( = () and = (). As a consqnc of this choic, th bonday conditions a not satisfid in vy oint of th xtna sfacs bt ony in an intga ay. Th comaison ith sts fom on dimnsiona mods hav shon a vy good agmnt fo od and thicnss mod, and an o of abot 5% fo th sonanc fqncy of th adia mod. By imosing stss-f bonday conditions and assming J ( ) j t = ω, hav: v = v (4) J( a)

5 v v3 + v sin( 3) v3 v cos( 3) + sin( 3 / ) cos( 3 / ) h h h I V v + v + v3 + jωa jω jω jωc = (5) = (6) h th xssions of th v, v and v 3 can b fond in [4]. Th intic and ctic ngy dnsitis can b no comtd as: a ( v v )dd C V πa = + πρ (7) = (8) Th dynamic coing facto as comtd fo sva asct atios G=a/. Th sts a shon in fig 5, h aso th ff comtd ith th 3 mod as as ith a FEM simations, caid ot ith th cod ANSYS 5.7, is shon. As can b sn th th cvs snt a vy simia tnd, vn if fo high vas of G (adia mod) a gat diffnc btn and ff can b obsvd. CONCLUSION In this o th dynamic coing facto, dfind as th sqa oot of th atio btn th ctic and dynamic ngy dnsitis, has bn comtd and comad ith th ffctiv ctomchanica coing facto fo th main on dimnsiona gomtis of cyind shad iocamic mnts, and a stong coation btn thm as obsvd. Th comaison has bn thn xtndd to cyind shad iocamics ith diffnt asct atios by sing an aoximatd 3 mod. Aso in this cas a coation btn th to coing factos mgd, vn if, d to th aoximations at th basis of th 3 mod, thi vas a mch diffnt than thos obtaind ith mods fo high diamt to thicnss atios. BIBLIOGRAPHICAL REFERENCES []. A. Bincot, Pioctic cysta and camic.-in Utasonic Tansdc Matia. O.E. Mattiat (d.). Pnm Pss, N Yo, 97, []. A. Bincot. R. Can, H. Jaff, Pioctic and iomagntic matias and thi fnction in tansdcs.-in: Physica Acostic..P. Mason (d.). Acadmic Pss, N Yo, 964, vo., Cha. 3, [3] N. Lambti, A. Ia, M. Paaado, Th ctomchanica coing facto in static and dynamic conditions, ACUSTICA-acta acstica, vo. 85, 999, [4] A. Ia, N. Lambti, M. Paaado, An Aoximatd 3- Mod of Cyind-Shad Piocamic Emnts fo Tansdc sign, IEEE Tansaction on Utasonics, Foctics and Fqncy Conto, vo. 45, 998, [5] IEEE Standad on iocticity, ANSI-IEEE Std 76, 987. [6] H.F. Titsn, Lina ioctic at vibation.pnm Pss, N Yo, 969. [7] Fiv modn ioctic camics. Vniton B. 66/E, Rv. Janay 97. a a) b) Fig : Picamic cyind shad scimn: a) RO, b) ISK.

6 5 3 9 f =f =84.66,5,,5 ( / ) / K f s =37.7,,5 ff =.666 = U (f )= f [H], 3 4 f/f Fig : RO mod; a)engy dnsitis and, b) / and U f s =.943 f =f =.78,9,,,,3,4 f [MH] 3,,5,,5,,5 ( / ) / U ff =.45 = U (f )=.439, 3 4 f/f Fig 3: THICKNESS mod; a)engy dnsitis and, b) / and U 8 f =f =7.4 6,,5 4 ( / ) /, f s =99.478,5 ff =.5 = f [H],,,5,,5,,5 3, 3,5 4, f/f Fig 4: RAIAL mod; a)engy dnsitis and, b) /,8,76,7,68,64 RO ff 3 MOEL ff ANSYS ff MOELS 3 MOEL,6,56,5,564,53,48, RAIAL, G Fig 5: and ff vss G