Finite element simulation of piezoelectric vibrator gyroscopes

Transcription

1 Enginring Elctrical Enginring filds Okayama Univrsity Yar 1996 Finit lmnt simulation of pizolctric vibrator gyroscops Yukio Kagawa Okayama Univrsity Toshikazu Kawashima Okayama Univrsity Takao Tsuchiya Okayama Univrsity This papr is postd at : Okayama Univrsity Digital Information Rpository. nginring/104

2 IEEE TRANSACTIONS ON UI.TRASONICS, FERROELECTRICS, AND FREQUENCY CONTROL, VOL. 43, NO. 4, JULY Finit Elmnt Simulation of Pizolctric Vibrator Gyroscops Yukio KaLgawa, Fllow, IEEE, Takao Tsuchiya, and Toshikazu Kawashima Abstruct- A finit llmnt approach to th simulation of pizolctric vibrator gyroscops is prsntd for charactristic prdiction. Th formulation its givn including th ffct of Coriolis forc du to rotation for a pizolctric thin plat, which is considrd to b two-dimnsional in plan vibration. For numrical xampls, th gyroscops of a thin squar plat, and a cross-bar and a ring built in th plat ar considrd, which pav th way for th dvlopmnt of th gyroscops of monolithic configuration. Th ffct of th rotation on th modal shaps, th rsonant frquncis, and th transmission charactristics ar discussd dmonstrating th snsing capability against th rotation Diffrntial vctor of intrpolation function NOMENCLATURE S Strain vctor {SI S2 s6) = uy,y u,,~ + u~,~}, whr SI and Sz ar strains in th x and y dirction, rspctivly, and S6 is a shar strain in th x-y plan. T Strain vctor {TI T2 T,}, whr TI and T2 ar strsss in th x and y dirction, rspctivly, and T6 is th shar strss in th x-y plan. se Elastic constant tnsor (E = 0). d Pizolctric constant tnsor {d31 $32 d36). 8 Gyro matrix [-; b]. E, Nodal displacmnt vctor of lmnts Ara coordinats whr ai = xj y, bi = yj -?/m ci = 2, - xj - x,yj (i, j, and m prmut in th ordr of nod 1, 2, and 3, and xi and y; ar th coordinat valus corrsponding to nod i). Diffrntial cofficint matrix for th z dirction 9, Nodal potntial vctor of lmnts N Scond-ordr intrpolation function vctor Manuscript rcivd Jun 19, 199.5; rtwisd Fbruary 14, This work was supportd in part by Grant in Aid for Scintific Rsarch (C), , th Ministry of Education, Scinc, ancl Cultur. Th contnts of this papr wr prsntd in part at th Tchnical Mting of Th Institut of Elctronics, Information, and Communication Enginrs, Japan [12]. Th authors ar with th Dpartmnt of Elctrical and Elctronic Enginring, Faculty of Enginring, Okayama Univrsity, Tsushima-naka, Okayarna 700, Japan (-mail: kagawa@ calc.lc.okayama-u.ac.jp). Publishr Itm Idntifir S (96) Diffrntial cofficint matrix for th y dirction, in which bi in B is rplacd by ci. Unit matrix of n x n. Matrix associatd with rotation O I [-I, 061. Natural frquncy of longitudinal vibration of a bar with lngth ffi [= l/(2ffi. I ) & 2x.f~~. Normalizd frquncy (= f/f~~). Normalizd angular vlocity (= R/2.rrf~~) /96$ IEEE

3 510 IEEE TRANSACTIONS ON ULTRASONICS, FERROELECTRICS, AND FREQUENCY CONTROL, VOL. 43, NO. 4, JULY 1996 I. INTRODUCTION IBRATOR gyroscops ar usd as th snsors for dtct- V ing angl and angular vlocity lik any othr gyroscop. Th pizolctric vibrator gyroscops provid pizolctric transducrs for xcitation and dtction. Thir opration is basd on th principl that a scondary vibration is gnratd in th dirction latral to th original vibration of th vibrator in rotation du to Coriolis forc. Th angular vlocity of th rotation can thus b snsd by masuring th amplitud of this latral vibration inducd or transmittd from th xciting vibration [ 11. For th analysis of th oprational charactristic of th pizolctric vibrator gyroscops, both quivalnt circuit and analytical approachs hav bn usd which ar limitd to th gyroscops of simpl configuration [2],[3]. Thy ar difficult to apply to th gyroscops of gnral shap for dvloping th bst possibl configuration and tmpratur charactristics. Numrical approachs should b sought for this purpos. Pizolctric matrials involv th nrgy xchang btwn th mchanical and lctrical nrgy, for which th finit lmnt mthod can ffctivly b utilizd as it is basd on th nrgy principl. Sinc its first introduction to th analysis of pizolctric vibrators [4], it has xtnsivly bn usd for th dsign of vibrators, snsors, and transducrs, for which is now wll stablishd [5],[6]. Th application of th finit lmnt mthod to th analysis of rotational snsors was rcntly trid but no succssful xampl was givn [7]. In th prsnt papr, th finit lmnt formulation for a pizolctric vibrator in rotation is dvlopd including ffcts both of pizolctricity and Coriolis forc. A thin plat in plan vibration rotating in that plan is considrd, for which th dvlopmnt of th monolithic typ bing fabricatd with a lithographic procss is intndd. For numrical xampls, th pizolctric vibrator gyroscops consisting of a thin squar plat, a cross-bar, and a ring built in th plat ar considrd. Th ffct of th rotation on th modal shaps and th rsonant frquncis, and th transmission charactristics ar discussd dmonstrating th snsing capability against th rotation. 11. VIBRATORY SYSTEM IN ROTATION As shown in Fig. 1, a thin pizolctric plat vibrats in its x-y plan with constant angular frquncy w, which also rotats around th z axis with constant angular vlocity R. Equations of motion for a particl p in stady-stat motion ar givn as follows: fx = F, pw2u, + j2prwuy pr2x } (1) fy = Fy + PW U, - j2prwuz + pr2y whr f stands for total forc acting on th particl, p is mass of th particl, F is xtrnal forc, U is particl displacmnt, and x, y ar th coordinats of th particl. Th scond trms on th right-hand sid of (1) ar th inrtia forcs du to th vibration, th third trms ar th Coriolis forcs, and th fourth trms ar th cntrifugal forcs. Whn th plat rotats around th z axis, th Coriolis forc is gnratd in th dirction latral to th displacmnt of th vibration. Coriolis forcs and cntrifugal forcs both act in th 3;--y plan. If w >> 0, Fig. 1. Coordinats and Coriolis forc in rotation Fig. 2. ve% Z t lctrods Pizolctric thin plat in plan motion (cross-sctional viw). yt pizolctric thin plat Fig. 3. Pizolctric fild and triangular lmnt (scond-ordr polynominal intrpolation function, plain viw). th ffcts of th cntrifugal forcs or th last trms could b ignord FINITE ELEMENT FORMULATION PIEZOELECTRIC VIBRATORY SYSTEM IN ROTATION Finit lmnt analysis is a mthod of transforming a continuum systm to its quivalnt discrtizd systm in which th systm is dividd into lmnts. Hr, a thin pizolctric plat is shown in Fig. 2, in which th plat is dividd into triangular lmnts with th scond-ordr polynomial function intrpolatd as shown in Fig. 3. Pairs of partial lctrods ar placd on both surfacs of th pizolctric thin plat. As th thin plat vibrats in its z-y plan, it is assumd that th strain is indpndnt of thicknss or z dirction, and th lctric fild is paralll to th x dirction and constant. Th pizolctric constitutiv xprssions of d-typ in th prsnt cas ar whr se is th lastic modulus tnsor (E3 = 0), T is th strss vctor, d is th pizolctric constant vctor, E3, 0 3 is th lctric fild and lctric displacmnt in th z dirction, &T3 is th dilctric constant in th z dirction (T = 0), and ( )* is th transposition of th matrix or vctor.

4 KAGAWA t al.: PIEZOELECTRIC VIBRATOR GYROSCOPES 511 It is assumd that th dirction of polarization is paralll to T, = th z dirction. (Th concrt dfinition for th componnts is givn in th Nomnclatur.) Suppos that th plat is dividd into triangular l- - mnts. Within an lmnt, displacmnt vctor U, = {uz(x, Y), uy(x, Y)> and th potntial p = p(x, 9) ar, rspctivly, dfind and xprssd in trms of th linar combination of nodal displacmnt vctor [ and nodal potntial vctor 4 dfind at th nodal coordinats (xi, y;) (i = 1-6) as whr [, is th nodal displacmnt, 4, is th nodal potntial vctor, and N is th intrpolation function vctor. Th strain vctor S, which is th drivativ of th displacmnt with rspct to th coordinats, is xprssd as whr A, is th ara of th triangular lmnt, n is th vctor associatd with th drivativ of th intrpolation function, and B, C, ar th cofficint matrics associatd with th drivativ of th intrpolation function with rspct to th x or y dirction. Th z dirctional lctric fild E3, in th lmnt is xprssd with th nodal potntial vctor 4, as for th pizolctric plat of thicknss 2t3. Th nrgy functional for a pizolctric vibrator consists of strain nrgy, kintic nrgy, lctrostatic nrgy, and work don by xtrnal forcs. [n th finit lmnt mthod, th nrgy functional is minimizd applying th variational principl. Howvr, whn th systm dissipats or rotats, th nrgy functional bcoms complx. Hr, an adjoint systm is introducd to stablish that th functional is ral [8], in which cas th physical maning of rninimization is much clarr. Strain nrgy U,, lctrostatic nrgy H, and luntic nrgy T, ar thus dfind for th lmnt as follows: U, = /// (S$ 2, + %r-t S:T + S$T) dx dy dz (3) (6) SJJ (3 2, + G;f + 6; f) dz dy dz whr K, is th stiffnss matrix, P, is th lctromchanical coupling matrix, G, is th lctrostatic matrix, 6 is th matrix of rotation, M is th mass matrix, and I, is th w x w unit matrix. (-) rfrs to th adjoint systm and (-) to th complx conjugat. No mchanical work xtrnally applid is considrd. As th systm is mchanically and lctrically dampd du to structural and dilctric loss, both mchanical (Rm) and lctrical (Rg) nrgy losss must b includd, which ar givn as follows: + S:Tf - 3:Tf) dx dy dz - I. t3 1 =J --(-t;kg, + [:Kg) 2 &m R,, = j tan S ///(-E;E+E:E (9) (10) + l3:dt - &DT) dx dy dz t =j 2 tan S (-4fGL + 6:G$) (11) whr Qm is th mchanical quality factor of th matrial, TE is th strss vctor (E3 = 0), tan 6, is th tangnt of dilctric loss factor, and DT is th dilctric flux dnsity (T = 0). Th dilctric loss is takn into account only ovr th rgion whr th lctrods ar providd. Extrnal work W is only lctrically mad through th lctrods A, du to th chargs applid, which is givn as follows: = 2 (4*$ + $*$ + gq) (12) 2 - whr D, is th lctric (displacmnt) flux dnsity normal to th boundary, and q is th nodal charg vctor corrsponding to th lctrods. ( ) rfrs to known valu. Lagrangian L for th whol systm is thus givn with th compatibility implid on th conncting nods as follows: L = (U, - H - T, - R, - Rg) - W

5 512 IEEE TRANSACTIONS ON ULTRASONICS, FERROELECTRICS, AND FREQUENCY CONTROL, VOL. 43, NO. 4, JULY 1996 Th application of th variational principl (6C = 0) to (13) yilds four sts of th discrtizd quations of motion for th whol systm; th on corrsponding to th ral systm is givn in (14) at th bottom of th pag. All th notations rfr to th sam dfinition, but for th whol systm this tim mad of simpl pizolctric matrial. is th nodal potntial vctor inducd ovr th lctrods du to th applid charg vctor ij for xcitation. Subscript 1 rfrs to th nods associatd with th lctrods and subscript 2 to th nods of th rgion without lctrods. Th mchanical loss is du to th structural damping, for which th stiffnss is simply valuatd as of complx valu. On th othr hand, n/w in (14) xprsss th ffct of rotation, with which th mass is now also valuatd as of complx valu, and providing phas lag in th motion. Th trm for th rotation in th systm matrix in (14) appars to caus additional damping. It should b notd, howvr, that th dirction of th displacmnt which is includd by Coriolis forc du to rotation is orthogonal to th dirction of th vibratory displacmnt (no diagonal trm prsnts in th matrix of rotation), so that no dissipation taks plac. Th rotational ffct incrass as th vibratory angular frquncy dcrass. Whn th systm is stationary (0 = 0), (14) arrivs at th sam conclusion as found in th usual finit lmnt xprssion for pizolctric vibrations. In th prsnt cas, as th matrix of rotation 8 is nonsymmtric. th systm matrix in (14) is complx and nonsymmtric. Th ignvalus ar thrfor complx, but whn no damping is assumd to xist, th ignvalus ar ral, as th systm matrix is Hrmitian. IV. NUMERICAL EXAMPLES Equation (14) is implmntd on a computr cod to simulat th bhavior of th pizolctric vibrator gyroscops. Th gyroscops considrd hr ar mad of a thin plat and hav pairs of lctrods on both thir facs for driving and dtcting. Hr, th plat mad of lctrostrictiv cramics is considrd. Th matrial proprtis of th pizolctric cramics (NEPEC 6, TOKIN Corp.) ar givn in Tabl I. For th calculation of th natural frquncis of th fr vibration, both th LU dcomposition mthod and dtrminant sarch mthod ar usd, and th invrs powr mthod is also incorporatd for th natural mod valuation. For th forcd rsponss, th LU dcomposition mthod is usd. All of th calculations ar mad for complx quantitis with doubl prcision. A. A Squar Thin Plat W first considr th cas of th simplst monolithic gyroscop mad of a thin pizolctric squar plat. Th plat provids pairs of th partial lctrods. Elctrod arrangmnt and th finit lmnt division ar shown in Fig. 4. Th plat TABLE I MATERIAL PROPERTIES OF PIEZOELECTRIC CERAMICS (NEPEC 6) SE x (m /N) svm d I { x lo- (C/N) 1050 P I i730 (kg/m3) Qm I 1500 driving lctrods 0.3 (%) t3= a 200 Fig. 4. Pizolctric thin plat gyroscop-lctrod arrangmnt and finit lmnt division. is drivn by th pairs of four lctrods distributd around its cntr and th lctrods for dtction ar placd in its four comrs. 1) Natural Frquncis and Mods: Th cas in which th thin plat is simply lastic without pizolctricity and damping IS xamnd first. Th bhaviors of th pizolctric thin plat in plan motion hav bn wll xamind [9]. To obsrv th ffct of th rotation, th natural frquncis and mods of fr vibration ar compard with thos in rotation. Th natural mods without rotation ar shown in Fig. 5, in which 7, ar th natural frquncis of a squar plat of a x a, normalizd with rspct to ~ L[= B 1/(2a&)], th natural frquncy of longitudinal vibration of a thin bar of lngth a. Th scond, sixth, and ighth mods (f2, f6, fs) ar dgnratd mods. Whn th pizolctric ffct is includd, th natural frquncis incras as much as a fw prcnt, though thy dpnd also on th condition at th lctrical trminals, and

6 KAGAWA t al.: PIEZOELECTRIC VIBRKrOR GYROSCOPES 513 1st x= nd %= (dgnratd mod) 3rd = th 3 = th = th fb= (dgnratd mod) 7th &= th %= (dgnratd mod) Fig. 5. Modal pattrns and corrsponding natural frquncis (no rotation, 7,: frquncy normalizd with rspct to ~ LB) th ordr of th natural frquncis may intrchang btwn th scond and third mods, as wll as btwn th fifth and sixth mods. Th ffct of th rotation is thn considrd. Th first and scond mods ar shown in Fig. 6. Th normalizd rotational angular frquncy Ti(= Cl/2rf~~) is chosn to b 6.3 x 10W4. Th mods bcom complx in rotation as xpctd, and it should b notd that dgnration is rsolvd for th scond mod (dgnratd without rotation) which sparats into two mods, th lowr mod (f,,) and uppr mod (TZU). This also occui*s for th othr dgnratd mods. Th sparation of th dgnratd mods is causd du to th apparanc of th mass anisotropy in th plat bcaus th Coriolis forc is prpndicular to th dirction of th inrtia forc (s th matrix of rotation 8 in th Nomnclatur). Th modal shaps of th imaginary part ar th sam as th ons of th ral part xcpt that thy ar only rotatd clockwis or anticlockwis by 90". Fig. 7 shows th chang of th normalizd natural frquncis of th scond mods against th normalizd rotational angular frquncy. Th natural frquncis incras or dcras in proportion to th rotational angular frquncy for << 1. Th chang of th natural frquncis may occur for othr nondgnratd mods whn th plat rotats, but th rat of th chang of thir natural frquncis is vry small. 2) Gyroscopic Charactristic: Th squar plat gyroscop is thn considrd, in which both mchanical and dilctric damping proprtis ar includd (l/qm = 6.67 x lo-*, tan 6 = 0.3%). Whn pairs of th four lctrods placd around th cntr of th plat ar drivn by a voltag of =tl V at frquncy f as shown in Fig. 4, th driving lctrical input admittanc charactristic at a pair of th lctrods nar th scond mod rsonant frquncy is shown in Fig. 8. Th input admittanc is normalizd with rspct to th admittanc of dampd capacitanc at frquncy flb. As th lctrods ar arrangd symmtrically, it is th sam for ach pair of driving lctrods. And, to obsrv th ffct of rotation, th opn-circuit voltag that appars in th dtcting lctrods providd at th comrs is shown in Fig. 9. Fig. 9(a) shows th chang of th output voltag in Trminal 1 (T1) and Fig. 9(b) shows th voltag in Trminal 2 (T2). For both cass th output is complx indicating th phas shift in th transmission. Solid lins ar th cass without rotation and th brokn lins ar th cass with rotation of 2 = 6.3 x lop4. In T1, th voltag appars in th vicinity of th rsonant frquncy which, howvr, is not much ffctd by rotation. In T2, th voltag appars only whn th rotation taks plac. This maks it possibl to dtct th rotation. Th output voltags in T3 and T4 ar th sam as th voltag in T2 and T1, xcpt th sign is rvrsd. Th sign of th output voltags is also rvrsd for th rotation rvrsd. Fig. 10 shows th chang of output voltag against th normalizd rotational angular frquncy 2 for th input voltag of on volt on th driving lctrods givn. As xpctd from Fig. 9, only th output voltags in T2 and T3 ar snsitiv to th rotation, and thir changs ar proportional to th rotational angular frquncy. W dfin hr th normalizd figur of mrit T/ as

7 514 IEEE TRANSACTIONS ON ULTRASONICS, FERROELECTRICS, AND FREQUENCY CONTROL, VOL. 43, NO. 4, JULY No rotation --- a= 6.3~10~.B conductanc suscptanc -10 I I ral pan - imaginary part Normalizd frqncy 7 (a) (fi ) Fig. 8. Effct of th rotation on th input admittanc at th lctrical trminals, scond mod. Both mchanical and dilctric losss ar includd. imaginary part ral part - -maginary pan fi~= Normalizd frquncy? (4 3r - 21 imaginary pan \ - No rotation --- a= 6.3~10~ ral pan - imaginary part f2u (b) Fig, 6. Modal pattrns of th plat in rotation (a = 0 2 / w = ~ 6.3 ~ x lor4), (a) first and (b) scond mods. No damping is includd ; Normalizd frquncy f (b) Fig. 9. Effct of th rotation on th output voltag, scond mod, (a) at T1 and (b) at T2. is 0.82, which is not satisfactory for practical applications, though th snsitivity could b doubld with th diffrntial connction of th lctrods in T2 and T3. This diffrntial opration could b usd to stablish th tmpratur stability, as th ffct of th tmpratur on th chang of th natural frquncy must b th sam for ach trminal. Fig. 7. rotation. Normalizd rotational angular frquncy Chang or sparation of th natural frquncy as th incras of whr V, and V, ar input and output voltag, rspctivly. Th normalizd figur of mrit q of th prsnt squar plat B. Cross-Bar Gyroscop Equation (14) suggsts that th natural frquncy must b chosn as low as possibl for ralizing th high snsitivity. Hr, w mploy flxural mods in plan. Th configuration mad of crossd bars shown in Fig. 11 is a candidat in which two bars ar coupld at thir cntr, supportd at som placs. This could b mad of a pizolctric thin plat with th tching procss rmoving th shadd portions. In th prsnt

8 KAGAWA t al.: PIEZOELECTRIC VIBRATOR GYROSCOPES 515 'I= - T L T A04 Normalizd rotational angular frquncy a ral part - imaginary part fi~= Th normalizd figur of mrit q = 0.82 Fig. 10. Chang of th output voltag against th rotation (input: 1 V). ral pan - imaginary part fiu Fig. 12. Mods of cross-bar gyroscop in rotation (n = bl/w~,~ = 6.3 x loss is nglctd). E7 : driving lctrods : dtcting lctrods : rmovd parts Fig. 11. Cross-bar gyroscop O.Cl2U Numbr of lmnts : 512 Numbr of frdom :1695 Band width of th systm matrix : 761 Fig ' I x10-6 Normailzd rotational angular frquncy Th normalizd figur of mrit q = 8.1 Chang of th output voltag against th rotation (input: 1 V). modl, only th crossd bars including supporting bars which ar fixd at th othr nds ar considrd for th analysis. NEPEC 6 is again assumd for th pizolctric thin plat. Th natural mods of vibration ar shown in Fig. 12 for th lowst dgnratd mods in rotation (n = 6.3 x Pizolctric ffct is includd, but not damping. Th mods ar complx in which dgnration is rsolvd, so that th mod sparats into two mods (fzl, Tzu). Fig. 13 shows th chang of th opn-circuit output voltag against th normalizd rotational angular frquncy for th input voltag of on volt on th driving lctrods. In this calculation, both dampings of th matrial ar includd. Th output voltag is qual in T1 and T3, and also in T2 and T4. Th normalizd figur of mrit 77 is 8.1. Th snsitivity or figur of mrit is now improvd as much as about tn tims compard with that of th squar plat gyroscop. C. Ring-Plat Gyroscop Th vibrator gyroscop first dvlopd was th on making us of th bnding mods associatd with a bll-shapd configuration. This is a plat vrsion of th original configuration mad of a pizolctric thin plat which is shown in Fig. 14. A ring-plat is partly supportd with th shadd portions rmovd. Only th ring including th supporting bars is considrd for th analysis, and NEPEC 6 is again assumd for th matrial. 1) Ring-Plat Without Support: W first considr th cas without supporting bars. Th finit lmnt division and th lctrod arrangmnt ar shown in Fig. 15. Th natural mods of vibration ar shown in Fig. 16 for th scond mods in rotation (1 = 6.3 x 10W6), which includ th pizolctric ffct and th lctrods but not damping. Th mods ar originally dgnratd without rotation. Th mods sparating

9 516 IEEE TRANSACTIONS ON ULTRASONICS, FERROELECTRICS, AND FREQUENCY CONTROL, VOL. 41, NO. 4, JULY 1996 ral part f2l= imaginary part Fig. 14. Ring-plat typ gyroscop. L a ral pan -f2u= imaginary part Fig. 16. Scond mods of ring-plat gyroscop in rotation (E = S2/wr,~ = 6.3 x lop6. loss is nglctd), without support. driving lctrods dtcting lctrods Numbr of lmnts : 128 Numbr of frdom : 904 Band width of th systm mamx : 333 Fig, 15. Ring-plat pizolctric vibrator gyroscop, lctrod arrangmnt. and finit lmnt division. into two mods (fzl, TZU) ar complx du to th ffct of th rotation as th dgnration is rsolvd. Fig. 17 shows th chang of th opn-circuit output voltag against th normalizd rotational angular frquncy 2 for th input voltag of on volt on th driving lctrods. Th dampings ar includd. Th output voltags ar qual in T2 and T4 in th amplitud, but rvrsd in phas. Th phas shift is obsrvd, as th driving frquncy dos not compltly mt th rsonant frquncy in th prsnt cas. (Th output voltags ar not snsitiv against rotation both in T1 and T3.) Th normalizd figur of mrit 7 is Th snsitivity is now improvd as much as 9.3 tims compard with that of th cross-bar gyroscop. 2) Ring-Plat with Support: W nxt considr th cas whn th supporting systm is includd. Th lmnt division and th lctrod arrangmnt is shown in Fig. 18. Supporting bars ar addd to th ring-plat. Th natural mods of vibration,& m - Y 0 Y & Y 6,I, /,, I,,*',- \imaginary ral part I t, I ~10-~ Normalizd rotational angular frquncy fi Th normalizd figur of mrit 7 = 75.4 Fig. 17. Chang of th output voltag against rotation at T4 (input voltag: 1 V), without supporting bars. ar shown in Fig. 19 for th scond mods in rotation (2 = 6.3 x for th width w = 2.56 x low4a. Lik th ring-plat without support, th mods ar again complx sparating into two mods (f,,, T,,), which corrspond to th scond mods of th ring-plat without support. Th modal shaps ar not much affctd by th prsnc of th supporting bars which ar vry thin. Fig. 20 shows th chang of th opn-circuit output voltag against th normalizd rotational angular frquncy 2 for th input voltag of on volt on th driving lctrods. Th dampings ar includd. Th output voltags ar qual in T2 and T4 in amplitud, but rvrsd in phas. Th output voltag is not snsitiv to rotation both in TI and T3. Th normalizd figur of mrit is Th snsitivity is slightly dcrasd compard with th cas without support.

10 KAGAWA t al.: PIEZOELECTRIC VIBRATOR GYROSCOPES 517 T U $ c- *.,, '.I'.,.,. --<imaginary,,.- ral pan I... I 1 I Normalizd rotational angular frquncy fi Numbr of lmnts : 192 Numbr of frdom : 1344 I3and width of th systm mamx : 321 Fig. 18. Finit lmnt division and lctrod arrangmnt whn th sup porting bars ar includd. Th normalizd figur of mrit q = 60.5 Fig. 20. Chang of th output voltag against rotation bars at T4 (input voltag: 1 V), with support. Width of supporting bars w/u Fig. 19. = 6.3 x ral part i ral part &=O f3u= imaginary part I imaginary part Scond mods of ring-plat gyroscop in rotation (2 = C~/WLB loss is nglctd), with support. Th ffcts of th rigidity of th supporting bars on th rsonant frquncy of th third mods ar shown in Fig. 21, in which th frquncy dviation against th width of th supporting bars is givn. For ths particular mods of vibration, th lctrods and th supporting bars may not b bst placd for snsing th rotation. This can b achivd through simulation procdurs. Th numrical xamplls ar shown only for th purpos of dmonstration. Fig. 21. Dviation of rsonant frquncy against th width of supporting bars rfrring to rsonant frquncy without support. v. CONCLUDING REMARKS Th us of th finit lmnt approach is proposd for th charactristic prdiction of pizolctric vibrator gyroscops. Th finit lmnt formulation including th ffct of Coriolis forc du to rotation is givn for a pizolctric thin plat in plan vibration, which is considrd as a two-dimnsional modl. Using th computr program dvlopd, th modal shaps, lctric input admittanc, th chang of th transmission charactristic, or th output voltag du to th rotation ar dmonstratd for thr typs of vibrators whr th dvlopmnt of th gyroscops of monolithic configuration is kpt in mind. Th following could b dducd from th xamination: 1) Th proprty of Coriolis forc is clarly rprsntd in th finit lmnt formulation, which shows that th vibratory frquncy must b chosn as low as possibl for high snsitivity to rotation. Th snsitivity also dpnds on th quality factor of th mods. 2) Whn flxural mods ar usd in low frquncy rang, out-of-plan motion may b involvd. In this cas, th assumption of th in-plan vibration may fall, so that thr-dimnsional analysis [ 101 will b rquird. 3) Th simulation shows that th two typs of monolithic configuration, cross-bar and ring-plat, making us of th flxural vibration in plan, can b promising for snsitiv snsing of th rotation.

11 518 IEEE TRANSACTIONS ON ULTRASONICS, FERROELECTRICS, AND FREQUENCY CONTROL, VOL. 43, NO. 4, JULY ) Th dgnratd mods sparat into two mods, th uppr and th lowr, du to th rotation. Ths mods can b advantagously utilizd to provid th tmpratur stability, as th ffct of th tmpratur chang is th sam on th rsolvd natural frquncis, which could b cancld. Th tmpratur charactristic is an important factor for th dvic dsign, which can b asily incorporatd into th program [ Th inclusion of th tmpratur charactristic for a crystal quartz plat is now undr way, which will b rportd in du cours of tim. Th nxt stp of this rsarch should includ th tmpratur ffcts. REFERENCES [I] W. D. Gats, Vibrating angular rat snsor may thratn th gyroscop, Elctron., vol. 10, pp , [2] M. Konno, S. Sugawara, S. Oyama, and H. Nakamura, Equivalnt circuit for pizolctric vibratory gyro of angular rat snsor, Trans. Inst. Elctron., Infiirm. Commun. Eng., vol. J70-A, no. 11, pp. 172L1727, 1987 (in Japans). [3] C. S. Chou, J. W. Yang, Y. C. Hwang, and H. J. Yang, Analysis on vibrating pizolctric bam gyroscop, lnr. J. Appl. Elctromagn. Matrials, vol. 2, no. 3, pp , Y. Kagawa and G. M. L. Gladwll, Application of a finit lmnt mthod to vibration problms in which lctrical and mchanical systms ar coupld-an analysis of flxur-typ vibrators with lctrostrictiv transducrs, IEEE Trans. Sonics Ultrason., vol. SU-17, no. 1, [5] Undrwatr Acoustics Group, Finit lmnts applid to sonar trausducrs, in Proc. Inst. Acoust.. Univ. of Birmingham, England, vol. 10. no. 9, Dc. 4, [6] Y.-K. Yong, Fundamntals of th finit lmnt mthod of a pizolctric rsonators, in Cours 6 Short Tutorial Cours IEEE ht. Ulti-uson. Symp., Canns, Franc, [7] G. M. Rs, E. L. Mark, and D. W. Lobitz, Thr dimnsional finit lmnt calculations of an xprimntal quartz rotation snsor, in IEEE Ultrason. Symp., 1989, pp [SI G. M. L. Gladwll, A variational formulation of dampd acoustostructural vibration problms, J. Sound Vib., vol. 4, no. 2, pp , [9] Y. Kagawa and T. Yamabuchi, Finit lmnt simulation of twodimnsional lctromchanical rsonators, IEEE Trans. Sonics Ultrason., vol. SU-21, no. 4, pp , [lo] Y. Kagawa, T. Tsuchiya, and T. Kataoka, Finit lmnt simulation of dynamic rspons of pizolctric actuators, J. Sound Vib., vol. 191, no. 4, pp , [l I] Y. Kagawa, H. Mada, and M. Nakazawa, Finit lmnt approach to frquncy-tmpratur charactristic prdiction of rotatd quartz crystal plat rsonators, ZEEE Trans. Sonics Ultrason., vol. SU-28, vol. 4, pp , [12] Y. Kagawa, T. Tsuchiya, T. Kawashima, and E. Ando, Finit lmnt analysis for pizolctric gyroscops, Inst. Elctron., Inform., Commun. Eng., Tch. Rp., vol. US93-70, pp , Nov (in Japans). Yukio Kagawa (SM 75-F 86) rcivd th B.Eng., M.Eng., and D.Eng. dgrs in 1958, 1960, and 1963, rspctivly, all from Tohoku Univrsity, Japan. In 1970, h was appointd Profssor of Elctrical Enginring, Toyama Univrsity, Japan. Currntly, h is a Profssor of Elctrical and Elctronic Enginring, Okayama Univrsity, Japan, and Profssor Emritus, Toyama Univrsity. His prsnt intrsts includ th application of th numrical approach to th simulation of acoustic and lctromagntic filds and systms, spcially invrs problms. Dr. Kagawa is a fllow of th Institut of Acoustics (U.K.) and appars in Who s Who in th World. Takao Tsuchiya rcivd th B.Eng., M.Eng., and D.Eng. dgrs in 1984, 1956, and 1989, rspctivly, all from Doshisha Univrsity, Japan. From 1989 to 1990, h was with th Dpartmnt of Elctrical and Information Enginring, Toyama Univrsity, Japan. Sinc 1990, h has bn with th Dpartmnt of Elctrical and Elctronic Enginring, Okayama Univrsity, Japan, and h is currntly a Lcturr working mainly on th numrical simulation of acoustic problms. Toshikazu Kawashima rcivd th B.Eng. and M.Eng. dgrs in 1993 and 1995, rspctivly, both from Okayama Univrsity, Japan. Sinc 1995, h has bn with Matsushita Elctric Works Co., Limitd.