FLEXOELECTRIC ACTUATION AND VIBRATION CONTROL OF RING SHELLS

Transcription

1 Proceedings of the ASME 5 Interntionl Design Engineering Technicl Conferences & Computers nd Informtion in Engineering Conference IDETEC/CIE 5 August -5, 5, Boston, Msschusetts, USA DETC FLEXOELECTRIC ACTUATION AND VIBRATION CONTROL OF RING SHELLS Bolei DENG StrucTronics nd Control Lb School of Aeronutics nd Astronutics, Zhejing University Hngzhou, Zhejing 7, P.R. Chin Emil: dengbolei@zju.edu.cn Huiyu LI The Stte Key Lb of Fluid Power Trnsmission nd Control Deprtment of Mechnicl Engineering, Zhejing University, Hngzhou, Zhejing 7, P.R. Chin Emil: vicy.cn@gmil.com Hornsen TZOU StrucTronics nd Control Lb School of Aeronutics nd Astronutics, Zhejing University Hngzhou, Zhejing 7, P.R. Chin Emil: hstzou@zju.edu.cn ABSTRACT The converse flexoelectric effect tht the grdient of polriztion (or electric field) induces internl stress (or strin) cn be utilized to control the vibrtion of flexible structures. This study focuses on the microscopic ctution behvior nd effectiveness of flexoelectric ctutor ptch on n elstic ring. An tomic force microscope (AFM) probe is plced on the upper surfce of the ptch to implement the inhomogeneous electric field inducing stresses inside the ctution ptch. The flexoelectric membrne force nd bending moment, in turn, ctute the ring vibrtion nd its ctution effect is studied. Actutor s influence in the trnsverse nd circumferentil directions is respectively evluted. For the trnsverse direction, the grdient of the electric field decys quicly long the ring thicness, resulting in nonuniform trnsverse distribution of the induced stress nd such distribution is not influenced by the ptch thicness. The flexoelectric induced circumferentil membrne force nd bending moment resembles the Dirc delt function t the AFM contct point. The influence of the ctutor cn be regrded s drstic bending on the ring. To evlute the ctution effect, dynmic response of controllble displcements of the elstic ring under flexoelectric ctution is nlyzed by djusting the geometric prmeters, such s the thicness of flexoelectric ptch, AFM probe rdius, ring thicness nd ring rdius. This study represents thorough understnding of the flexoelectric ctution behvior nd serves s foundtion of the flexoelectricity bsed vibrtion control. INTRODUCTION Precision ctution nd vibrtion control hs been constnt quest over the yers. The phenomenon tht the polriztion (or electric field) grdient induces internl stress (or strin) is clled the converse flexoelectric effect [-]. In the lst decde, both theoreticl studies [4-6] nd experimentl mesurements of the flexoelectric coefficients [7,8] hve drwn much ttentions nd the wor hs been crried out on different mterils [9-]. The signl nlysis nd dynmic responses of the flexoelectric mterils hve been reported recently nd their pplictions on rings nd cylindricl shells were studied [-5]. Sttic flexoelectric ctution effects of cntilever bem using the AFM probe were lso evluted [6]. Ring shells re widely used in engineering pplictions, such s stiffeners, gers, structurl sensors, motors, etc. Dynmic behviors, e.g., nturl frequencies nd mode shpes, of free floting rings hve been thoroughly studied [7]. Piezoelectric sensors nd ctutors nd their modl sensing nd ctution effects were nlyzed [8]. However, dynmic flexoelectric ctution chrcteristics nd vibrtion control of shell structures hve not been reported previously. This study is to investigte flexoelectric microscopic ctution nd control behviors of ring structures. An inhomogeneous electric field generted by AFM probe is used to ctute the flexoelectric ptch lminted on the ring structure. With the flexoelectric membrne force nd bending moment, stedy-stte dynmic response or controllble displcement of the ring is determined. Distribution of the electric field grdient over the trnsverse direction of different ptch thicness is nlyzed, followed by ctution effectiveness of the ctutor ptch thicness, AFM probe rdius, ring thicness nd ring rdius. Copyright 5 by ASME

2 DYNAMICS OF THE RING Figure illustrtes n elstic ring shell lminted with flexoelectric ctutor ptch, where nd coordintes respectively define the trnsverse nd the circumferentil directions; R is the neutrl surfce rdius of the ring; h is the ring thicness nd b is the ring width. Elstic ring b R F h Flexoelectric ptch (Not to scle) Figure. Schemtic digrm of the elstic ring The flexoelectric ptch perfectly lminted on the ring surfce is used s n ctutor nd its stiffness/mss effect to the ring dynmics is neglected since it is much thinner thn the ring thicness. The membrne control force nd bending control moment induced by the ctutor (i.e., flexoelectric ptch) is written s N nd M respectively, where the superscript denotes the ctutor induced component. The dynmic eqution with the mechnicl force nd ctution force of the ring cn be simplified from the dynmic equtions of the doublecurvture shell [8]. h m m u N M N M - t R R R R m m u N M N M h F t R R R R, (), () where is the mss-density of the ring; u is the trnsverse displcement; u is the circumferentil displcement which is ssumed to be smll compring with u ; F is the distributed m m mechnicl force in the trnsverse direction; N nd M re the mechniclly induced membrne force nd bending moment per unit width, where the superscript m denotes the mechnicl component. Combining the ctutor induced components respectively in the circumferentil nd trnsverse direction in ring dynmic equtions, Love s control opertors cn be expressed s [8] N M L R R, () N M L R R where L nd L re Love s ctution or control opertors, which denote the flexoelectric ctutor contributions to the dynmic equtions. Note tht only the trnsverse vibrtion is considered here. According to the modl expnsion method, ring s trnsverse displcement u cn be expressed by dding ll prticipting nturl modes multiplied by their respective modl prticiption fctors [7] u t t U t,,, where is the mode number in the trnsverse direction; is the modl prticiption fctor; U is the trnsverse mode shpe function for the -th mode. denotes the temporl contribution of ech mode nd U denotes the sptil distribution. It is observed tht = is brething mode; = is rigid body mode, nd set of trnsverse nd circumferentil modes occur when [7]. When the ring is freely floting in spce, the mode shpe function of the circumferentil direction nd the trnsverse direction cn be expressed s [7] U A sin( -) ; U B cos( -), (6,7) where U is the circumferentil mode shpe function; A nd B re the constnts of the mode shpe functions; is n rbitrry phse ngle tht must be included since the ring does not show preference for the orienttion of its modes. Modl vibrtion mplitudes of the trnsverse component modes nd the circumferentil component modes re coupled t lower mode numbers [8]. By substituting the modl expnsion expression of trnsverse displcement, i.e., Eq.(5) into dynmic equtions of ring, ting the dvntge of the orthogonlity of the mode shpe functions nd introducing the modl dmping rtio yields the independent modl eqution [7] ˆ F, (4) (5), (8) where is the -th nturl frequency; represents the modl dmping rtio of the ring, which depends on the equivlent dmping constnt c nd the nturl frequency, i.e., =c/(h ); ˆ F is the modl force for the -th mode. The modl prticiption fctor eqution, i.e., Eq.(8) is n nonhomogeneous second-order ordinry differentil eqution of the -th modl prticiption fctor. When ring model is considered, for ech vlue of, there re two component nturl frequencies, i.e., nd. The component frequency denotes the trnsverse vibrtion, nd denotes the circumferentil vibrtion. For the oscilltion t, mplitude Copyright 5 by ASME

3 rtio of trnsverse modes B is proportionl to circumferentil mode s mplitude A s B A. (9) rctn. (5) For typicl rings >>, thus the circumferentil vibrtion is not considered nd only the trnsverse remins s used lter [8]. Apprently, the solution is influenced by the modl force, which cn be divided into two components: the modl force induced by the mechnicl force nd the modl force induced by the flexoelectric ctutor, i.e., ˆ ˆ m ˆ F F F, () where F ˆ m is the modl force induced by the mechnicl force; F ˆ is the modl force induced by the ctutor. Let the externl mechnicl force be expressed s F, Fig.. F ˆ m nd F ˆ cn be respectively expressed s [8] where ˆ F m = F U Rd π hn, F = L U L U Rd, ˆ hn π N U U Rd. () () () Note tht the ptch is lmented from to. When the mechnicl force F nd the ctution voltge re hrmonic with the mechnicl force frequency, i.e., F e j t = F jt, ( t ) = e, the modl response will lso be hrmonic nd ˆ ˆ F = e j t F, where F ˆ indictes the mgnitude of the -th modl force. The modl response (modl prticiption fctor ) cn be solved from the modl prticiption fctor eqution, i.e., Eq.(8) s Fˆ ( ) e t - + j jt = ˆ F e j( t ), (4) With the modl prticiption fctor (Eq.(4)) nd the mode expnsion expression (Eq.(5)), the trnsverse deflection induced by the mechnicl force nd flexoelectric ctutor cn be obtined s u (, t) ( t) U (, t) ˆ m ˆ F F U jt U (, t)e (, t) U M R hn N M N R R R U Rd F URd. (6) The trnsverse vibrtion of the ring is ffected by the flexoelectric ctution membrne force nd the bending control moment. For the flexoelectric mteril, the ctution behvior depends on the specific form of the inhomogeneous electric field. The derivtion of the ctution force inside the flexoelectric ptch from the electric field of the AFM probe is introduced next. FLEXOELECTRIC ACTUATION An AFM probe is used to ctute the flexoelectric ptch. The ring model of the flexoelectric ctutor is shown in Fig. where r is the rdius of the AFM probe nd h is the thicness of the flexoelectric ptch. According to the converse flexoelectric effect, n inhomogeneous electric field is needed to drive the flexoelectric ptch. This electric field is induced by n AFM probe locting t. The AFM probe tip rdius r is fr less thn the flexoelectric ptch thicness nd the ptch thicness is fr less thn the ring thicness, i.e., r h h. where is the phse ngle expressed s Copyright 5 by ASME

4 h R Conductive AFM probe r h (Not to scle) Figure. A ring model of flexoelectric ctution To generte the inhomogeneous electric field, n ctution voltge is pplied between the AFM probe nd the electrode under the flexoelectric ptch. The electric potentil inside the flexoelectric ptch cn be expressed bsed on Abplnlp s pproximte []. By trnsforming the rectngulr coordintes to the polr coordintes, the potentil ner the AFM probe cn be expressed s r, (7) h R sin - r + +h - ( R)cos( ) R where is the AFM probe loction. In this study, the net electric field in the direction is lmost zero [6], thus only the trnsverse electric field is considered here. The trnsverse electric field cn be obtined by differentiting the potentil expression in the opposite trnsverse direction. E r R sin h r h R cos R cos - h R sin r h R cos R (8) where E is the electric field in the trnsverse direction. Note tht the trnsverse electric field (Eq.(8)) only indictes the electric field ner the AFM probe. However, for the points wy from the AFM probe, both the ctully electric field nd the vlue of Eq.(8) re lmost zero []. Thus it is ccurte enough to ssume tht Eq.(8) cn be used to describe the electric field for the whole plne. The circumferentil stress induced by the electric field cn be derived from the generl converse flexoelectric equtions [6] nd expressed s T E π r h R sin r h R cos R π r h R sin r h R cos R h R sin r h R cos R h cos R sin r h R cos R cos where 5 (9) T denotes the circumferentil stress in the flexoelectric ptch. The membrne force induced by the flexoelectric ctutor is the overll effect of the stress expressed in Eq.(9). The flexoelectric membrne control force induced by the electric field cn be obtined by integrting the stress long the ctutor thicness. h h h N T d r h R h sin cos cos r R h h h R h sin r R h cos h h h R sin r h R cos( ) cos h h R sin r h R cos () The flexoelectric control moment cn be clculted by integrting the product of the ctution stress nd the corresponding moment rm (the distnce between the locl point nd ring s neutrl lyer). Considering tht the ptch thicness is fr less thn the ring thicness, the bending moment cn be pproximtely obtined by multiplying the norml force by the distnce between ring s neutrl lyer nd the ptch s 4 Copyright 5 by ASME

5 h+ h h h M N T d h +h h h h r h R h sin r R h cos cos h h R h sin r R h cos h h h R sin cos( ) cos r h R h h R sin r h R cos () Actutor voltge, (V) Strting position of ptch, (rd) Ending position of ptch, (rd) () Distribution of the Trnsverse Electric Field With the prmeters in Tble., the trnsverse distribution of the electric field grdient defined in Eqs.(7,8) is nlyzed first. Note tht by flexoelectric stress eqution, i.e., Eq.(9) the induced stress in the flexoelectric ctutor is proportionl to the grdient of electric field in the direction. Figure shows the electric field grdient vrition in the trnsverse direction right under the contct point with the AFM probe. Note tht the verticl coordinte is set to strt on the top surfce of the flexoelectric ptch nd points downwrd to the bottom surfce, i.e., 5µm in this cse. With the explicit expression of the membrne force nd bending moment induced by the flexoelectric ctutor, the flexoelectric ctution response of the ring cn be obtined by the expression of trnsverse displcement, i.e., Eq.(6). Bsed on the formultions of the flexoelectric ctuted vibrtion, microscopic ctution behviors nd specific modl control chrcteristics re nlyzed nd evluted in cse studies. CASE STUDIES With the geometric nd mteril prmeters listed in Tble., microscopic ctution behvior of the flexoelectric ptch nd the modl control effects of the ring shell re evluted. Since the converse flexoelectric effect depends on the electric field grdient, this grdient is nlyzed first, followed by detiled evlution of the flexoelectric membrne control force nd control moments. Stedy-stte mximl controllble displcements for vrious ring modes re nlyzed with respect to ctutor thicness, AFM probe rdius, ring thicness nd rdius when the ctution voltge is set t volts. Tble. Prmeters nd properties of the ring model. Properties vlues Ring rdius R, (m).5 Ring width b, (m). Ring thicness, h (m). Young s modulus of ring, Y (N/m ) Flexoelectric ptch thicness, h (m) 5 AFM probe tip rdius, r (nm) 5 AFM probe tip loction, (rd) Ring mss density, (g/m ) Poisson s rtio,. Flexoelectric constnt, (C/m) Figure. The trnsverse distribution of the electric field grdient Figure revels tht the electric field grdient is extremely high ner the AFM probe (or thicness equls to zero). The grdient in the lower prt of the ptch is reltively smll compring with tht of the upper surfce. More precisely, the mgnitude of the electric grdient t the hlf thicness of the ptch (thicness equl to 5m) is only % of the top surfce (thicness equl to zero). Eq.(9) shows tht the induced stress T is proportionl to the electric field grdient, thus the stress in the upper prt of the flexoelectric ptch is much lrger thn the lower prt, which leds to the upper prt of the ptch n dominnt component of the induced membrne control force N which is clculted by integrting the induced stress over the thicness of the ptch. Further nlysis indictes tht the grdient distribution of vrious ctutor thicnesses is similr. The top portions of vrious ctutor thicnesses ner the AMF probe re the sme nd the distribution follows the trend extending down s the ctutor becomes thicer. Recll tht the membrne force induced by upper prt of the ptch domintes the overll flexoelectric membrne control force. Hence the ctution effect increses little s the ptch thicens nd this behvior will be vlidted lter. () Microscopic Actution Behvior of The Flexoelectric Actutor Ptch 5 Copyright 5 by ASME

6 Microscopic behvior of the flexoelectric ptch on ring ctution is nlyzed here. Due to the shrpness of the AFM probe induced electric field, the circumferentil distributions of the membrne force is very shrp spie. Figure 4 shows the circumferentil distribution (i.e., from π to +π) of the induced flexoelectric membrne force nd control moment respectively mred on different scles. Note tht the AFM probe is plced t Bending moment (Nm) Figure 4. Circumferentil distribution of the ctution Figure 4 indictes tht the sptil distribution of the membrn force is extremely inhomogeneous. Note tht the insert pictue in Fig.4 plots the force distribution from -5-6 rd to 5-6 rd, which shows tht the membrne force concentrtes ner the AFM loction. According to Eq.(), the bending moment cn be obtined by multiplying the membrne force by the moment rm. The circumferentil distribution of bending moment is lso observed in Fig.4 with the moment scle on the left. The moment rm is constnt in the ring model, thus the control moment is propotionl to the membrne control force nd their distributions re similr. The ctution effect, s indicted by the insert picture of Fig.4, is limited in very smll re, specificlly, round.5m. The following nlysis focuses on this microscopic rigeon nd evlutes its microscopic ctution chrcteristic. Two flexoelectric ctution terms L nd L, i.e., the Love opertors respectively expressed in Eqs.(,4), in the totl ctution ˆ F = LU LU Rd hn, Eq.(), re nlyzed to evlute the flexoelectric ctution behvior ner the point contcting with the AFM probe. Both the flexoelectric membrne force N nd bending moment M influence the ring vibrtion. Compring Eq.() with Eq.(), the trnsverse Love opertor L cn be regrd s the trnsverse loding nd the circumferentil Love opertor L be induced circumferentil loding induced by the flexoelectric ctutor. The chrcteristics of these two induced loding re nlyzed. Flexoelectric Circumferentil Actution Specificlly, L hs two components: one is induced by the flexoelectric membrne force nd the other by the control bending moment. The circumferentil distributions of these two components nd their totl effect is drwn in Fig.5. Induced circumferentil Figure 5. The distribution of the circumferentil loding induced by ctutor As shown in Fig.5, in the re on the left of the line, the circumferentil loding points rightwrd nd the right points leftwrd. In nother word, the circumferentil ctution points to where the AFM probe plced. Figure 5 lso shows tht the membrne force component domintes the totl induced circumferentil ctution. Such result is resonble becuse the membrne force minly influences the circumferentil vibrtions, while the bending moment minly influences the trnsverse ones. The induced microscopic circumferentil ctution behvior is shown in Fig.6. Note tht the dimemsion considered here is very smll, thus the ring is nerly stright s bem in Fig.6. The ppernce of the AFM probe s electric field results in circumferentil control ctution drging the mteril to where the probe loctes. Circumferentil loding Elstic ring Flexoelectric lyer (Not to scle) Figure 6. Distribution of the induced circumferentil loding Note tht usully the trnsverse oscilltion is dominting nd more importnt. Thus, only the trnsverse vibrtion is considered, which is minly influened by the ctutor induced trnsverse Love opertor L. The distribution of the two components of the induced trnsverse ctution, i.e., the membrne force one nd bending moment one, together with their totl effect, re plotted in Fig.7. As shown in Fig. 7, for the re closed to the AFM probe the induced ctution is downwrd; the re wy from the AFM probe the ctution is upwrd nd the re frther wy is lmost zero. 6 Copyright 5 by ASME

7 Induced trnsverse When the ring ctuted by the flexoelectric ctutor driven by n AFM probe, trnsverse deformtion is induced s illustrted bove. As the deformtion occurs, the mteril is drgged to where the AFM probe loctes. While the ring would resist such drg nd results in circumferentil ctution. Thus, the overll ctution effect of the flexoelectric ptch on the ring cn be regrded s drstic shrp bending effect within very smll region. Stedy-stte dynmic responses of the ring ctuted by flexoelectric ctutor with vrious design prmeters re nlyzed next. Figure 7. The distribution of the trnsverse loding induced by ctutor The schemtic digrm of induced microscopic trnsverse ctution behvior is shown in Fig.8. Agin, Fig.8 is drwn t very smll region which is bout.5m. The bending control moment plys dominnt role in the trnsverse ctution, becuse the trnsverse deformtion cn be considered s drstic bending resembling Dirc delt function t the AFM probe s contct point, shown in Fig.8. Note tht this shrp bending hs been describled s the bucling chrcteristic in n erlier reserch on stitic deformtion control of the cntilever bem bsed on AFM probe [6]. Trnsverse loding Flexoelectric lyer () Flexoelectric Ptch Thicness Recll tht the microscopic behvior of the flexoelectric ctutor shows tht the ctution effect only exists in very smll region, i.e., ners the AFM contct point. Thus, the ptch size hs little influence on the overll ctution effect. A reltively smll ptch, covers from -/4 to /4, is chosen for displcement control of rings nd other mteril nd geometric properties re summrized in Tble. The influence of flexoelectric ptch thicness on control of ring oscilltions is nlyzed here. The flexoelectric induced mgnitudes (or the mximl controllble displcements) driven t V t the stedy re used to compenste ring s vibrtion mplitudes, i.e., the lrger the ctutor induced mgnitudes, the better the vibrtion control effects on rings. The mximl flexoelectric induced displcements of -6 modes re evluted with respect to ptch thicness of 5m, 5m, 75m nd m in Fig.. Note tht stedy-stte controllble mgnitudes of -6 modes re clculted when the ctution frequency is set to its nturl frequency respectively. Elstic ring (Not to scle) Figure 8. Distribution of the induced trnsverse loding To nlyze the chrcteristic of the flexoelectric ptch under the control of the AFM probe, L nd L should be considered s whole. The schemtic digrm of the totl distributed flexoelectric ctution on the ring is shown in Fig.9. Circumferentil loding Elstic ring Trnsverse loding Flexoelectric lyer (Not to scle) Figure 9. Circumferentil distribution of the totl induced flexoelectric ctution ner the AFM probe Figure. Mximl controllble displcement (=-6 ring modes) with flexoelectric ptch thicness Figure shows tht when the thicness of the flexoelectric ptch increses, the mximl induced displcement of =-6 increses. The membrne control force in the flexoelectric ptch is enhnced s the ptch thicness increses. Additionlly, the induced control moment goes up becuse the moment rm increses s the flexoelectric ctutor thicens. Thus, the ctution effect is strengthened s the flexoelectric ptch thicness increses. Recll tht the upper prt of the flexoelectric ptch contributes dominnt component to the overll membrne force of the ptch nd different thicness ptches shre the sme distribution of grdient t the re close 7 Copyright 5 by ASME

8 to the probe. Thus the induced membrne force increses little s the ptch thicness grows, which in turn leds to the limited increses of the mximl controllble displcement. (4) AFM Probe Rdius Erlier nlysis indictes tht the AFM probe rdius is ey fctor influencing electric field grdient, membrne control force nd control moments, consequently vibrtion control effectiveness. The influence of the AFM probe rdius on the ctution effect is evluted here. The mximl controllble displcement of =-6 ring modes re evluted with respect to probe rdius of 5 nm, 5 nm, 75nm nd nm in Fig.. As shown in Fig., the mximl ring displcement increses s the ring thicness decreses. Keeping other prmeters unchnged, it is obvious tht s the ring thicness increses, the ring becomes stiffer resulting in higher nturl frequencies. Generlly, it is resonble tht to ctute thin ring is esier thn thic ring nd the ctution effect decreses s the ring thicness increses. For specific ring mode, Fig. shows tht the reltionship between the mximl controllble displcement nd ring thicness is qudric. The bending stiffness is proportionl to the cubic of ring thicness nd the induced bending moment is proportionl to the ring thicness. Thus, the ctution effect, which is influenced by both fctors, is proportionl to the squre of ring thicness. (6) Ring Rdius The influence of ring rdius on the ctution effect is nlyzed here. Mximl controllble displcements of modes - 6 with respect to ring rdius of 5mm, 5mm, 75mm nd mm re plotted in Fig.. Figure. Mximl controllble displcement (=-6 ring modes) with the rdius of AFM probe As shown in Fig., the mximl displcement decreses s the rdius of AFM probe increses from 5nm to nm. Although the circumferentil domin re of the electric field grows with probe rdius, the mximl electric field grdient decreses, which leds to the decrese of the mximl membrne force in the flexoelectric ptch. The ltter decrese outweighs the former increses, s result the ctution effect decreses s the AFM probe rdius increses from 5nm to nm. (5) Ring Thicness After nlyzing flexoelectric ctutor chrcteristics (e.g., the AFM probe nd the flexoelectric ptch), the influence of the structurl chrcteristics of the elstic ring is evluted here. The mximl controllble displcements of =-6 modes re clculted with respect to ring thicness of.5 mm,.mm,.5mm nd. mm in Fig.. Figure. Mximl controllble displcement (=-6 ring modes) with the ring thicness Figure. Mximl controllble displcement (=-6 ring modes) with the ring rdius Figure indictes tht the mximl displcement for ech mode increses with the ring rdius. With other prmeters fixed, ring with lrger rdius is softer thn smller rdius one. Furthermore, s the ring rdius increses the nturl frequency decreses, the ring become reltively esier to ctute. Thus, the ctution effect increses with the ring rdius. The mximl controllble displcement, for ech mode respectively, is proportionl to the ring rdius s plotted in Fig.. For two rings with different rdius, they shre the sme bending stiffness nd ctution force. Thus, it s resonble to deduce tht the reltionship between ctution effect nd the ring size is liner. CONCLUSIONS This study focused on the converse flexoelectricity bsed ctution chrcteristics of elstic rings. The converse flexoelectric effect depends on n inhomogeneous electric filed implemented by n AFM probe on flexoelectric ctutor ptch. The distribution of the induced stress in the flexoelectric ctutor ptch is extremely nonuniform. The top surfce crries most of the induced membrne control force. Furthermore, the trnsverse distribution of the electric grdient does not chnge 8 Copyright 5 by ASME

9 with the ctutor thicness. In the circumferentil direction, the distribution of ctution forces resembles the Dirc delt function due to the inhomogeneous electric field grdient induced by the AFM probe. The microscopic behviors of ctutor induced circumferentil nd trnsverse control ctions were evluted nd the induced flexoelectric ctutions were pplied to drive ring shell. With the bsic dynmics of elstic rings, the vibrtion chrcteristics driven by the flexoelectric ctutor ptch were studied nd the mximl controllble displcements for five ring modes, i.e., =-6, were evluted with respect to the ptch (thicness), the probe (tip rdius) nd the ring (thicness nd rdius). The ctution effect is enhnced with thicer flexoelectric ctutor, smller AFM probe rdius, thinner ring shell or lrger ring rdius. Accordingly, this study provides bsic understnding of the microscopic flexoelectric ctution behvior nd serves s foundtion for the flexoelectricity bsed vibrtion control of flexible structures. ACKNOWLEDGEMENT The wor ws supported by the Ntionl Nturl Science Foundtion of Chin (76 nd 4) nd the ndphse of the 985 Project t Zhejing University. REFERENCES [] Kogn, S. M., 964, Piezoelectric Effect During Inhomogeneous Deformtion nd Acoustic Scttering of Crriers in Crystls, Soviet Physics-Solid Stte, 5(), pp [] Tgntsev, A. K., 986, Piezoelectricity nd Flexoelectricity in Crystlline Dielectrics, Physicl Review B, 4(8), pp [] Todorov, A. T., Petrov, A. G., nd Fendler, J. H., 994, First Observtion of the Converse Flexoelectric Effect in Bilyer Lipid Membrnes, The Journl of Chemicl Physics, 98(), pp [4] Mindlin, R. D., 969, Continuum nd Lttice Theories of Influence of Electromechnicl Coupling on Cpcitnce of Thin Dielectric Films, Interntionl Journl of Solids nd Structures, 5(), pp [5] Mindlin, R. D., nd Eshel, N. N., 968, On Fisrst Stringrdient Theories in Liner Elsticity, Interntionl Journl of Solids nd Structures, 4(), pp.9-4. [6] Shin, E., nd Dost, S., 988, A Strin-Grdients Theory of Elstic Dielectrics with Sptil-Dispersion, Interntionl Journl of Engineering Science, 6(), pp.-45. [7] Cross, L. E., 6, Flexoelectric Effects: Chrge Seprtion in Insulting Solids Subjected to Elstic Strin Grdients, Journl of Mterils Science, 4(), pp.5-6. [8] Bsrn, S., He, X., nd Chen, Q.,, Experimentl Studies on the Direct Flexoelectric Effect in -phse Polyvinylidene Fluoride Films, Applied Physics Letters, 98(4), p.49. [9] Fu, J. Y., Zhu, W., Li, N., Cross, L. E., 6, Experimentl Studies of the Converse Flexoelectric Effect Induced by Inhomogeneous Electric Field in Brium Strontium Titnte Composition, Journl of Applied Physics, (), pp.4. [] Agronin, A., Molotsii, M., Rosenws, Y., Rosenmn, G., Rodriguez, B. J., Kigon, A. I., nd Gruvermn, A., 6, Dynmics of Ferroelectric Domin Growth in the Field of Atomic Force Microscope, Journl of Applied Physics, 99(), p.4. [] Zubo, P., Ctln, G., Bucley, A., Welche, P. R. L., nd Scott, J. F., 7, Strin-Grdient-Induced Polriztion in SrTiO Single Crystls, Physicl Review Letters, 99, p.676. [] Hu, S.D., Tzou, H. S.,, Signl Anlysis of Flexoelectric Trnsducers on Rings, Proceedings of the Symposium on Piezoelectricity, Acoustic wves, nd Device Applictions, SPAWDA, pp.6-. [] Ro, Z., Hu, S.D., Tzou, H.S.,, Digonl Flexoelectric Sensor on Cylindricl Shell Substructure, Proceedings of the Symposium on Piezoelectricity, Acoustic wves, nd Device Applictions, SPAWDA, pp.6-. [4] Hu, S.D., Li, H., nd Tzou, H.S.,, Flexoelectric Responses of Circulr Rings, Journl of Vibrtion nd Acoustics, 5(), p.. [5] Hu, S.D., Li, H., nd Tzou, H.S., 4, Comprison of Flexoelectric nd Piezoelectric Dynmic Signl Responses on Flexible Rings, Journl of Intelligent Mteril Systems nd Structures, 5(7), pp [6] Hu, S.D., Li, H., nd Tzou, H.S.,, Sttic Nno- Control of Cntilever Bems Using the Inverse Flexoelectric Effect, Proceedings of the ASME IMECE, IMECE-65, pp [7] Soedel, W.,99, Vibrtions of Shells nd Pltes, Mrcel Deer, New Yor. [8] Tzou, H. S., 99, Piezoelectric Shells: Distributed Sensing & Control, Kluwer Acdemic Publishers, Dordrecht, Boston, London. [9] Abplnlp, M.,, Piezoresponse Scnning Force Microscopy of Ferroelectric Domins, Ph.D. Disserttion, Swiss Federl Institute of Technology, Zürich. [] Agronin, A., Molotsii, M., Rosenws, Y., Rosenmn, G., Rodriguez, B. J., Kingon, A. I., nd Gruvermn, A., 6, "Dynmics of Ferroelectric Domin Growth in the Field of Atomic Force Microscope," Journl of Applied Physics, 99(), p.4. 9 Copyright 5 by ASME