FLEXOELECTRIC SIGNALS ON RINGS IN TRANSVERSE MOTIONS

Proceeding of the ASME 011 International Deign Engineering Technical Conference & Computer and Information in Engineering Conference IDETC/CIE 011 Augut 8-31, 011, Wahington, DC, USA DETC011-4819 FLEXOELECTRIC SIGNALS ON RINGS IN TRANSVERSE MOTIONS S.D. Hu, H. Li, H.S. Tzou StrucTronic Sytem and Control Lab. School of Aeronautic and Atronautic, Zhejiang Univerity Hangzhou, Zhejiang 31007, P.R. China Email: hundihu@zju.edu.cn; Lhlihua@gmail.com; htzou@zju.edu.cn ABSTRACT Dynamic ening i eential to effective cloed-loop vibration control of preciion tructure and ytem. In a centroymmetric crytal ubjected to inhomogeneou deformation, where piezoelectricity i abent, only the train gradient contribute to the polarization known a the flexoelectricity. In thi tudy, a flexoelectric layer i laminated on ring hell to monitor the natural modal ignal ditribution. Due to the train gradient, only the bending train component contribute to the output ignal; the total ignal ha two component repectively induced by the tranvere modal ocillation and the circumferential modal ocillation. Analogue to the ignal analyi, the enitivity can alo be defined in two form: a tranvere enitivity induced by the tranvere modal ocillation and a tranvere enitivity induced by the circumferential modal ocillation. Analyi data ugget that the tranvere modal ocillation dominante the flexoelectric ignal generation and it magnitude/ditribution how nearly the ame a the total ignal. Furthermore, voltage ignal and ignal enitivitie are evaluated with repect to ring mode, enor egment ize, ring thickne, and ring radiu in cae tudie. The total ignal increae with mode number and enor thicknee, decreae with enor egment ize and ring radiu, and remain the ame with different ring thicknee. Thee data erve a deign guideline to elect proper parameter for practical engineering application uing flexoelectric tranducer. INTRODUCTION Flexoelectricity[1-4] exhibit two unique electromechanical coupling effect. One i the direct flexoelectric effect that mechanical train gradient induce an electric polarization; the other i the convere flexoelectric effect that tranform the polarization gradient into the mechanical tre or train. Initial theoretical analyi [4] and experimental meaurement [5] revealed the flexoelectric coefficient of the order of 10-11 -10-10 C/m. Although the flexoelectric effect i very mall in macro cale, large flexoelectricity ha been tudied in low dimenional ytem, uch a nanographitic ytem, epecially with high-dielectric material [6-10] and thi i very important to the nanotructure baed technologie, uch a application of ening and energy harveting. For example, energy-harveting can be dramatically enhanced in nanotructure [11] and thi mechanim provide a cheme for engineering application. Although flexoelectricity ha been becoming important, the tudy of flexoelectric behavior i till carce, epecially in dynamic ignal repone of hell tructure. Furthermore, in a centroymmetric crytal ubjected to inhomogeneou deformation, where piezoelectricity i abent, only the train gradient contribute to the polarization. Thu, due to the abence of piezoelectricity, flexoelectricity play a major role in charge (or polarization) generation and ignal repone. Although ditributed ening of variou hell, e.g., paraboloidal, cylindrical, pherical, conical, toroidal, etc. hell, with piezoelectric enor ha been evaluated over the year [1-19], flexoelectric ening repone of hell have not been reported in the pat. A an axiymmetric, imple and veratile tructure compared with other hell, the circular ring model ha been ued in many engineering application, uch a ring tiffener, gear and rate-enor. Free vibration behavior including nature frequencie and mode hape of free-floating ring hell have been well tudied before [0-]. Baed on the haping technique, patially haped orthogonal piezoelectric enor and actuator were deigned and their modal ening and actuation effect to ring were evaluated [3,4]. Recently, the average and cancellation effect of egmented piezoelectric enor applied to macrocopic elatic ring hell wa evaluated [5]. 1 Copyright 011 by ASME

Thi tudy focue on application of the direct flexoelectric effect to ditributed ignal ening on circular ring. Thi tudy aim to evaluate the ening characteritic and ignal generation capability of flexoelectric (cubic m3m centroymmetry crytal) patche laminated on flexible circular ring. Coupled dynamic equation of elatic ring with free vibration are preented firt, followed by invetigation of mode hape, modal train and ocillatory characteritic of ring. The flexoelectric modal ignal of a flexoelectric layer laminated on the inner urface of the ring are tudied in detail. Since the modal ignal depend on ring thickne, enor thickne and ring radiu, flexoelectric ignal with different deign parameter are evaluated and plotted in cae tudie. Baed on the egmentation technique, ignal with repect to the flexoelectric egment ize and ring mode are alo conidered. Thee figure and analyi data erve a deign guideline of flexoelectric tranducer in practical application. SIGNAL ANALYSIS OF SEGMENTED SENSORS A generic elatic ring with a thin flexoelectric enor layer on the inner urface i hown in Fig. 1, where R i the neutral urface radiu of the ring; h i the elatic ring thickne; b i the ring width. The flexoelectric enor layer i often egmented into a number of eparated patche with each egment covering the ring urface from 1 to (= - 1) in the circumferential direction, with the enor layer radiu R =Rh/-h / R, and the effective area of each enor patch S =br. D=Yh 3 /[1(1- )] i the bending tiffne. Eq.(1,) clearly indicate that the circumferential and tranvere diplacement are cloely coupled. Auming that all point on the ring ocillate harmonically at the natural frequency, one can write the diplacement in the modal expanion form:, u t t U, where n (t) are the modal i n in n participation factor and U in () are the mode hape function. The circumferential and tranvere mode hape function U ψn and U 3n can be defined a [1] in U n Anm n (3a) U3n Bnm co n (3b) where A nm i the circumferential ocillatory amplitude and B nm i the tranvere ocillatory amplitude; n i the mode number; the frequency component m=1, denote the tranvere and the circumferential component, repectively; and i an arbitrary phae angle that i aumed zero in the following analyi. Furthermore, there are two component natural frequencie for each mode number n for the free vibration ring tructure. Generally, the low component frequency n1 denote the tranvere ocillation, and the high component frequency n denote the circumferential ocillation. For the lower mode number n (i.e., n<10), the amplitude relationhip of tranvere mode B nm and circumferential mode A nm are approximately Segmented flexoelectric layer B n B n1 A, n n1 An 1 (4a,4b) n h R Elatic ring hell b (Not to cale) 3 Figure 1. Ring with a flexoelectric layer (left) and a egmented enor patch (right). Ring Dynamic The free vibration equation of ring are written a D u u R R 3 u3 K u3 hu 0 4 3 D u u R R 3 4 u3 K 4 u 3 4 3hu3 0 (1) () where u i i the diplacement in the i-th direction where i=x, (Fig. 1); i the ring ma denity; i Poion ratio; Y i the Young modulu; K=Yh/(1- ) i the membrane tiffne; which indicate that the circumferential ocillatory amplitude A n1 i le than the tranvere ocillatory amplitude B n1 at n1, while the circumferential ocillatory amplitude A n i greater than the tranvere ocillatory amplitude B n at n. The voltage ignal of flexoelectric enor layer i dicued next. Flexoelectric Signal A dicued previouly, the direct flexoelectric effect convert the mechanical train gradient to electric charge or voltage, which can be ued a the tranduction mechanim of enor. For the flexoelectric material with a centroymmetric point group like m3m, the direct flexoelectric effect can be generally defined with a matrix notation a G ˆ ij j ij j D μ S ε E (5) where {D j } i the electric diplacement; the gradient operator G ˆ in the hell coordinate ytem i defined a matrix Copyright 011 by ASME

ˆ ˆ G1 0 0 0 G 3 ˆ 0 ˆ G G ˆ 0 0 G1 (6) 3 0 0 Gˆ ˆ G1 0 3 where ˆ 1 Gi and i=1, and A i are Lamé 3 i Ai 1 Ri parameter; R i are hell radii of curvature. The flexoelectric contant matrix [6] i 11 1 1 0 0 0 1 11 1 0 0 0 0 0 0 μ (7) 0 0 0 0 44 0 0 0 0 0 0 44 1 1 11 ij 0 0 0 44 0 0 The dielectric permittivity contant matrix i 11 0 0 0 11 0 0 0 The train {S ij }=[S 11 S S S 3 S 31 S 1 ] T and the electric field {E j }=[E 1 E E 3 ] T. By electing the ring parameter A 1 =A =R, A =A y =1, R 1 =R =R, and R =R y =, neglecting the in-plane twiting hear train S 1, the tranvere hear train S 13 and S 3, and only conidering the tranvere electric field E 3, one can define the flexoelectric effect for the thin ring cae (Fig. 1) a S yy S D E3 1 3 3 where S and S yy are the normal train in the and y direction, repectively. According to Maxwell equation, the voltage acro the enor layer can be obtained by integrating the electric field E 3 over the thickne [1], i.e., S S 1 yy Ed 3 3 1 D d 3 3 3 3 3 (8) (9) (10) Again, when etting D=0 [1,6], integrating Eq.(10) over the flexoelectric enor urface S yield an open-circuit condition voltage Rb S 1 S yy d 3d S 1 3 3 3 (11) With the linear aumption, the normal train can be defined by a membrane train component and a bending train component: o o S S 3k, Syy Syy 3kyy (1a,1b) o where S ii are the membrane train; k ii are the change-incurvature term ( 3 k ii are the bending train) reulting from the diplacement of the thin hell [0]. k 1 u, t o S R u3, t,, 1 u t u3 t R (13a) (13b) o S k 0 (14) yy Subtituting Eq.(1-14) into (11) yield the flexoelectric output voltage ignal Rh b1 S hb1 S R 1 1 k d u yy, t u, t 3 d (15) where h i the flexoelectric enor thickne; R i the neutral urface radiu of the elatic ring. Note that there only exit the bending tain in the ignal expreion of the flexoelectric enor. Thi behavior i totally different from the conventional ditributed piezoelectric enor [1], due to the train gradient behavior in circular ring. Since the bending train are primarily contributed by tranvere component mode, tranvere ocillation mode are only conidered. Subtituting the modal expanion equation and mode hape function Eq.(3) with m=1, and auming the phae hift in mode hape function to zero yield hb1 S R n t An 1nBn1n con d (16) n where A n1 and B n1 are the modal amplitude of the tranvere ocillation component mode at n1. Since n=0 i a breathing mode; n=1 i a rigid body mode, the tranvere component mode only occur when n. The patially ditributed flexoelectric modal voltage i obtained by etting the modal participation factor to be unity, i.e., 1 3 Copyright 011 by ASME

hb A 1 B 1n in n in n1 1 n n n S R (17) Uing the modal amplitude relationhip of Eq.(4a), one can define the modal voltage ignal a hb1 1 n n B n1 in in n S R n n 1 (18) Furthermore, the n th (n) modal ignal can be divided into a circumferential bending component and a tranvere bending component: hb S R 1 A n 1 in n in n1 hb1 Bn 1 in n in n1 S R n 1 Bn 1n in n in n1 3 hb S R (19) (0) The firt ignal component indicate the flexoelectric ignal generated from the bending train induced by the circumferential ocillation; the econd component indicate the ignal from the bending train induced by the tranvere ocillation. Note that the tranvere component i (-n ) time of the circumferential component. The tranvere component in the output ignal ignificantly dominate and it increae with the mode number n. Senitivitie of Flexoelectric Senor Senitivity i an eential performance indicator of enor in practical application. When an output ignal i meaured, the tranvere or the circumferential ocillatory amplitude for each ring mode can be etimated uing the enitivity. The tranvere ocillatory amplitude B n1 i larger than the circumferential amplitude A n1 in ring tranvere ocillation. The output enitivity of a egmented flexoelectric enor can be defined in two form: one i S hb 1 n inn inn 1 n 1 c An 1 S R (1) which i defined by the circumferential ocillatory amplitude of the tranvere ocillation mode; the other i S hb 1 n i n inn1 n () n 1 t n Bn 1 S R which i defined by the tranvere ocillatory amplitude of the tranvere ocillation mode. Thee two enitivity form, a well a flexoelectric modal ignal, will be evaluated with repect to virou deign parameter (e.g., ring radiu, enor thickne, ring thickne, and enor egment ize) in cae tudie preented next. CASE STUDIES Flexoelectric reponed of variou ring cae with three enor egment ize, four ring thicknee, two enor thicknee and four ring radii are evaluated in cae tudie. Geometry and material propertie of the ring hell and the flexoelectric material are ummarized in Table 1 where b i the ring width; R i the ring radiu; h i the ring thickne; h i the enor thickne; S i the area of the enor patch; i the dielectric contant; 1 i the flexoelectric contant. (It i worth noting again that baed on the experimental reult 1 wa calculated to be about ~10-6 C/m [9-1] four order of magnitude higher than the earlier general obervation ~10-11 - 10-10 C/m.) Since the flexoelectric contant varie with different condition and material, for generality, the contant 1 i normalized to be unity in later parametric analyi. In thi way, flexoelectric ignal can be inferred once pecific material contant are elected in practical deign application. The modal amplitude B n1 i alo aumed to be unity. Table 1. Propertie of the ring and the flexoelectric enor Propertie Ring Flexoelectric layer Unit b 0.01 0.01 m R 0.05, 0.05, 0.075, 0.1 / m h, h 0.0005, 0.001, 0.003, 0.005.5 10-5, 4.0 10-5 m S / br m / 1, 30, 60 deg. / / 1(aumed) C/F Recall that the modal ignal of flexoelectric enor patche laminated on the inner urface of the elatic ring hell are induced only by the bending train gradient dominated by tranvere ocillation mode with amplitude A n1 and B n1, while the membrane train component i eliminated due to the train gradient behavior the intrinic flexoelectric behavior. The flexoelectric modal ignal can be further divided into two component ignal: one i a bending ignal ( ) 3 induced by the tranvere ocillation component and the other i a bending ignal ( ) induced by the circumferential ocillation component at n1. Knowing the component contribution, one can further ue thi data to electively deign flexoelectric enor to repond to pecific train behavior and ring mode. Thoe two component ignal and the total modal ignal, a well a enitivitie, are evaluated in cae tudie. Again, the purpoe of the parametric tudy i to evaluate deign parameter and to erve a deign guideline in practical application. 4 Copyright 011 by ASME

Flexoelectric Modal Signal A dicued previouly, there are three enor egment ize, four ring thicknee, two enor thicknee and four ring radii to be evaluated in cae tudie. Flexoelectric ignal reulting from ring modal ocillation of thee cae are preented and compared in thi ection. Evaluation of flexoelectric enor enitivity i preented later. Table. The ditribution of modal ignal with three enor egment ize. mode n = 1 = 30 = 60 3 4 n = 5 90 ( ) ( ) 3 150 30 5 -.4 0.0.4 10 0 70 6 5 Copyright 011 by ASME

Senor egment ize There are three enor egment ize conidered here: =1, =30 and =60. The =1 cae would provide a near-prefect and mooth ignal ditribution. However, becaue thi fine egmentation i impractical, ignal generation and ditribution of the =30 and =60 cae are further evaluated. 1) Senor egment =1 An elatic ring tructure with radiu 50mm, width 10mm, thickne 1mm and flexoelectric layer thickne 5m i applied a a reference model in the flexoelectric modal ignal analyi. Applying the egmentation technique and making each enor egment =1 give 360 egment uniformly laminated on the ring hell. The flexoelectric ignal ditribution, including the circumferential ignal component ( ), the tranvere ignal component ( ) 3 and the total ignal ( ), are repectively plotted and ummarized in Table (left column). From thee mooth ignal ditribution, one can oberve that the bending component ignal induced by the tranvere ocillation component ( ) 3 i the dominant component to the total flexoelectric ignal and the total ignal i lightly le than the tranvere component ignal becaue of the ignal cancellation with the addition of the circumferential component. Thee plot alo reveal that the maximal modal ignal change with mode number when the modal participation factor are all unitie, which will be further invetigated in later cae tudie. However, ince the ultra-fine egmentation of =30 i impractical, enor egment of =30 and =60 are dicued next. ) Senor egment =30 and =60 For practical conideration, enor egment ize of =30 (1 egment) and =60 (6 egment) are evaluated here. Similar to the =1 (360 egment) cae, modal ignal reulting from enor egment of the =30 cae and the =60 cae with n=-6 ring mode are repectively ummarized in the center and right column in Table. Note that in thee two cae, averaged flexoelectric ignal at egment center are joined together with traight line. The ame concluion i evident that the ignal induced by the tranvere train gradient i the dominant component of the total ignal and the total ignal i nearly the ame a the tranvere ocillation component. (In fact, it i little le, due to the cancellation of the tranvere and circumferential component.) However, the =30 cae and =60 cae are dicued together here ince both exhibit imilar behavior and obviou meaurement deficiencie. Table how that the ignal magnitude reduce when egment ize enlarged in the =30 cae and the =60 cae. In extreme cae, the output ignal of n=6 mode in =30 cae and n=3, 6 mode in =60 cae are all reduced to zero, due to the complete cancellation of poitive and negative ignal on thee egment. Figure illutrate the maximal flexoelectric ignal change with different enor egment ize and mode number. It further indicate that for the ame ring mode, the maximal output voltage ignal decreae with larger egment ize. The tranvere component ignal ( ) 3 i little larger than the total ignal due to the ignal cancellation of membrane component ignal ( ). Figure. Maximal voltage (n=-6 ring mode) veru enor egment ize (Upper-left: ( ) 3, upper-right: ( ), bottom: ). Accordingly, the egmentation of =30 i preferred for the further experimental tudie and application ince it reult are more rational and feaible. Due to meaurement deficiencie of large enor egment, parametric tudie of four ring radii, two enor thicknee and four ring thickne with egment ize =1 are tudied next. Ring thickne The maximal flexoelectric output ignal of n=-6 ring mode are evaluated with repect to variou ring thicknee of 0.5mm, 1mm, 3mm and 5mm in Fig. 3. The purpoe of thi parameter tudy i to erve a deign guideline to elect optimal ring thickne for tranducer application. Figure 3 indicate that the output ignal remain the ame when the ring thicken, becaue the train gradient (i.e., the change-in-curvature term k ii ) i independent of the ring thickne with the linear deformation aumption. Furthermore, the total output ignal and the tranvere component increae while the circumferential component remain unchanged with the mode number n. The reaon i that the total train gradient and the tranvere component train gradient are enhanced with higher ring mode when the ocillatory amplitude i normalized. Senor thickne Then, the maximal flexoelectric output ignal of n=-6 ring mode are evaluated with repect to variou enor thicknee of 5m and 40m in Fig. 4. Since the flexoelectric effect i enhanced when the flexoelectric 6 Copyright 011 by ASME

tranducer thicken, thu the output ignal increae with the enor thickne a evident in Fig. 4. Ring radiu Lat, the maximal flexoelectric output ignal are evaluated with repect to ring radii (i.e., the neutral urface radiu of ring) of 5mm, 50mm, 75mm and 100mm in Fig. 5. The reult indicate that the output voltage ignal decreae when the ring radiu increae, due to reduced train gradient of the ring. Figure 5. Maximal voltage (n=-6 ring mode) veru neutral urface ring radiu (Upper-left: ( ) 3, upper-right: ( ), bottom: ). Figure 3. Maximal voltage (n=-6 ring mode) veru ring thickne (Upper-left: ( ) 3, upper-right: ( ), bottom: ). Senitivity Analyi Baed on the modal characteritic and ignal generation of egmented flexoelectric enor of the ring, there are two form of the enitivitie: 1) a tranvere enitivity S c induced by the circumferential modal ocillation and ) a tranvere enitivity S t induced by the tranvere modal ocillation. Thee two enitivitie are evaluated, again, with repect to deign parameter, i.e., ring thickne, enor thickne, ring radiu and egment ize. Ring thickne The two flexoelectric enitivitie are evaluated with repect to four ring thicknee of 0.5mm, 1mm, 3mm and 5mm and the enitivitie veru ring mode are plotted in Fig. 6. It can be inferred from thee two enitivitie that the tranvere modal amplitude i larger than the circumferential modal amplitude. The tranvere and circumferential modal enitivitie both remain unchanged with the ring thickne ince the train gradient i independent of the ring thickne. It indicate that the ring thickne doen t affect the enitivity of flexoelectric effect baed on ring tructure. Figure 4. Maximal voltage (n=-6 ring mode) veru enor thickne (Upper-left: ( ) 3, upper-right: ( ), bottom: ). Senor thickne Similarly, the tranvere enitivitie S t and S c defined at the tranvere component frequency n1 induced by the tranvere and circumferential modal ocillation are evaluated with repect to two enor thicknee of 5m and 40m and thee enitivitie are plotted in Fig. 7. Thee plot indicate that both enitivitie increae with the thickne of flexoelectric enor layer and thu, a thicker flexoelectric layer enhance the enor enitivity. 7 Copyright 011 by ASME

Ring radiu Again, the two flexoelectric enitivitie S t and S c with repect to ring radii of 5mm, 50mm, 75mm and 100mm are plotted in Fig. 8. Generally, the train in a larger ring i lower than that in a maller ring, o doe the train gradient (i.e., the change-in-curvature term k ii ). Accordingly, flexoelectric enitivitie decreae when the ring radiu increae. Therefore, a maller ring (or radiu) i preferred to enhance the enitivity for application. Segment ize Flexoelectric enor enitivitie with repect to three egment ize =1, 30 and 60 are plotted in Fig. 9. Analogue to the enor ignal analyi, the modal obervation deficiencie and ignal cancellation for =30 and =60 cae can alo be explained through the enitivity analyi. Two enitivitie are mooth curve for the =1 cae, but they are reduced with larger egment ize due to the ignal average and cancellation effect. Particularly, thee two enitivitie alo how no ignal or enitivity in the =30 cae (or 1 enor egment) for all n=6 (or it multiple) ring mode and in the =60 cae (or 6 enor egment) for all n=3, 6 (or multiple of 3) ring mode. Figure 8. Senitivitie (n=-6 ring mode) veru ring radiu (Left: S t, right: S c ). Figure 9. Senitivitie (n=-6 ring mode) with variou egment ize (Left: S t, right: S c ). Figure 6. Senitivitie (n=-6 ring mode) veru ring thickne (Left: S t, right: S c ). Figure 7. Senitivitie (n=-6 ring mode) veru enor thickne (Left: S t, right: S c ). CONCLUSIONS Spatially ditributed flexoelectric ignal on flexible circular ring baed on the direct flexoelectric effect were invetigated in thi tudy. Due to the train gradient behavior (i.e., the partial derivative of train with repect to 3 ), only the bending train gradient dominated by tranvere component mode with ocillation amplitude A n1 and B n1 at n1 contribute to the output modal ignal. Thu, the flexoelectric ening characteritic of circular ring in tranvere motion wa only conidered. Two ignal component of the total output ignal were defined: (1) a bending ignal induced by the tranvere modal ocillation and () a bending ignal induced by the circumferential modal ocillation. Analyi reult ugget that the tranvere modal ocillation i dominant and it ignal component can nearly repreent the total ignal. The output enitivity of flexoelectric enor egment i alo defined in two form: (1) the tranvere enitivity defined by the tranvere modal ocillation and () the tranvere enitivity defined by the circumferential modal ocillation. With thee ignal enitivitie, the output voltage ignal can be ued to etimate the tranvere or the circumferential ocillatory amplitude for each ring mode. Spatial ditribution of flexoelectric voltage ignal and enitivitie around the ring were evaluated and analyzed in cae tudie with repect to enor egment ize, ring thickne, flexoelectric enor thickne and ring radiu. Beide, each cae wa dicued with different ring mode n=-6. Parametric 8 Copyright 011 by ASME

tudie indicate that the maximal flexoelectric ignal increae with the enor thickne, decreae with the ring radiu and the egment ize, and remain unchanged with the ring thickne. The bending ignal ( ) 3 induced by the tranvere ocillation component and the total ignal increae with the ring mode, while the bending ignal ( ) induced by the circumferential ocillation component remain unchanged. For the =1 cae (360 egment), the flexoelectric ignal of ring mode n=-6 exhibit mooth ditribution. However, the ignal magnitude reduce in the =30 cae (1 egment) and the =60 cae (6 egment) due to the ignal average and cancellation effect on each egment. In extreme cae, the output ignal vanihe in the =30 cae when the mode number n=6 (or it multiple) and in the =60 cae when the mode number n=3, 6 (or multiple of 3). Thee reult erve a deign guideline to elect proper egment and parameter for practical engineering application with flexoelectric tranducer. 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