Title. Author(s)Sato, Takahiro; Watanabe, Kota; Igarashi, Hajime. CitationIEEE Transactions on Magnetics, 50(2): Issue Date

Transcription

1 itle Coupled alysis of Electroagetic Vibratio Eergy uthor(s)sato, aahiro; Wataabe, Kota; garashi, Hajie CitatioEEE rasactios o Magetics, 5(): 776 ssue Date - Doc URL Rights EEE. Persoal use of this aterial is perit ay curret or future edia, icludig repritig/re creatig ew collective wors, for resale or redistr this wor i other wors. ype article (author versio) File foratio copuag_tsato.pdf structios for use Hoaido Uiversity Collectio of Scholarly ad ca

2 CMP-53 Coupled alysis of Electroagetic Vibratio Eergy Harvester with Noliear Oscillatio aahiro Sato, Kota Wataabe, ad Hajie garashi, Meber, EEE Graduate school of foratio sciece ad techology, Hoaido Uiversity, Kita, Nishi 9, Kita-u, Sapporo, 6-8, JPN Murora stitute of echology his paper presets a coupled aalysis of electroagetic vibratio eergy harvester (VEH) based o the ocoforig voel FEM. the preset aalysis ethod, the otio, Maewell, ad circuit equatios are alteratively solved usig the staggered ethod. he FE odels ca be geerated i the preset ethod with low coputatioal costs. Usig the coupled aalysis ethod, the characteristics of a VEH odel, which cotais the agetic aterial i the coil, is aalyed chagig the geoetric paraeters. t is show fro the aalysis results that the VEH odel has chaotic ad liear oscillatios. t is also foud that output power ca be geerated for a wide frequecy rage i the chaotic ode, ad i the liear ode, large output power is obtaied at the resoace. de ers Coupled proble, electroagetic iductio, eergy harvestig.. NRODUCON HE electroagetic vibratio eergy harvesters (VEHs), which trasfor vibratio to electrical eergy through agetic iductio []-[], have attracted attetio for ew sall ad aiteace-free power source. Usig VEH, it is possible to realie wireless etwor systes which are free fro battery replaceet [5]. order to develop practical VEH systes, the output power ad badwidth of VEH devices ust be iproved. o iprove the output, oe should desig the flu path to icrease the agetic flu across the coil. he use of agetic aterials is effective to for the flu path. Whe usig the agetic aterial, the strog agetic force acts o the oscillator. he strog agetic force gives rise to oliearity i the VEH devices. t has bee show that the badwidth ca be epaded by the oliear oscillatios [6], [7]. However, owig to coplicated otio i the VEH devices uder strog agetic force, the output power caot be accurately evaluated based o a siple sprig-daper odel [8]. o accurately aalye the behavior ad output of such devices, the couplig of otio, electroagetic field, ad electrical circuit ust be cosidered. this paper, a ew coupled aalysis ethod for VEH devices will be itroduced, i which Mawell, otio, ad circuit equatios are solved alteratively usig the staggered ethod [9]. Moreover, the electroagetic field is aalyed usig the ocoforig voel FEM (voel-fem) [] i which oe ca geerate the FE odels for ay device shapes with low coputatioal cost ad ovig objects are effectively odeled through the ocoforig techique. this wor, the VEH odel cotaiig agetic aterial i the coil is aalyed chagig the geoetric paraeters. t is show fro the coputatioal results that the VEH odel with agetic aterial i the coil has chaotic ad liear oscillatio odes. t is foud that output power ca be geerated for a wide frequecy rage i the chaotic ode. Mauscript received Jue 6, 3. Correspodig author: aahiro Sato (e-ail: tsato@e-si.eg.houdai.ac.jp). Digital Object detifier iserted by EEE Furtherore, i the liear ode, we ca obtai large output power at the resoace.. VBRON ENERGY HRVESERS We cosider a VEH device which cosists of a coil ad agets fied to the catilever bea. he VEH device would be placed o the vibratig objects such as egies, otors, ad bridges. he abiet vibratio aes siusoidal displaceets of the agets relative to the coil. s a result, the agetic flu across the coil chages i tie ad cosequetly electrootive force (EMF) is iduced. additio, the curret flowig i the coil geerates wea agetic force actig o the agets. Because the wea agetic force ca be assued as a electric daper, the vibratioal behavior of the VEH devices ca be odeled as the sprig daper syste [8], fro which the output ca be estiated. Whe the VEH device is assued to have liearity, it has resoaces. Relatively large output power ca be obtaied at the resoat frequecies. this paper, we aalye the VEH odel show i Fig.. he ovelty i this device is that the agetic aterial is itroduced i the coil by which the flu path is desiged to icrease the flu across the coil. Siultaeously, the strog agetic force acts o the agets, owig to which it is epected that the catilever has oliear oscillatio. Whe the VEH devices have coplicated otio, the output caot be evaluated fro the siple sprig daper odel. For this reaso, we itroduce a coupled aalysis Catilever bea Base Maget Magetic aterial Fig.. arget vibratio eergy harvester odel. Coil cotaiig agetic aterial

3 CMP-53 ethod for VEH, by which the behavior ad output of ay VEH devices ca accurately be aalyed.. Coupled aalysis ethod for vibratio eergy harvesters Let us cosider the goverig equatio of VEH. he catilever otio is odeled as sprig daper. he eteral circuit coected to the coil is represeted by a resistive load for siplicity. hus, the goverig equatios are give by p c p q p q F, () v J v M, () DΦ R, Dt where is ass of objects o the tip of the catilever, c ad are effective dapig ad stiffess coefficiets of the catilever, p is agets positio, ad q is coil positio which is equal to the iput vibratio, aely, q=qsi(ωt). additio,, M, v ad v are vector potetial, agetiatio vector, reluctivity of agetic aterial ad vacuu, respectively. Moreover, F() ad J are the agetic force actig o the agets ad the curret desity, respectively, ad R is the load resistace. he curret i (3) is obtaied by itegratig J o the cross-sectio of the electric circuit. additio, D/Dt is Lagrage derivative to cosider EMF due to the coil oveet []. the preset ethod, the voel-fem is eployed to solve (). this ethod, the ocoforig voels for the device shapes are autoatically geerated with low coputatioal cost []. his ethod is useful for desigig ew devices lie VEH i which we ust geerate ay FE eshes. Moreover, to treat the relative displaceet betwee the coil ad agets, two voel eshes are coected usig the ocoforig techique [] as show i Fig., i which esh ad esh correspod to the coil ad agets, respectively. Nodal positios i esh chage correspodig to their relative displaceet q-p. he, uows alog edges i esh o the iterface betwee two eshes are iterpolated by those i esh. the ocoforig techique, the uows assiged to the slave edges are epressed i ters of liear cobiatio of the uows assiged to the aster edges. Naely, the uow, s, assiged to a slave edge, e s, is iterpolated by the aster edges, i, as follows: s Nas i es i N (3) as N dl c, () i i i, s i where N i, l, N as are the iterpolatio fuctio of edge i, tagetial vector alog edge e s, the uber of the aster edges, respectively, ad c i, s represets the relatio betwee the uows of edge i ad s. he ocoforig edges i each voel esh are also iterpolated i the siilar way. Note that the aster edges for a slave are easily foud because edges are regularly placed i the voel eshes. s a result of the iterpolatio, all the uows, all, ca be epressed i ters of the uows alog the aster edges,, i a for all=c, where C is (N M) (N M+N S) atri which epresses iterpolatio relatios whose copoets cosists of c i, s, where N M ad N S are the uber of aster ad slave edges, respectively []. Here, the Lagrage coordiate syste is eployed to describe the otio of the esh. Hece, Lagrage derivative i (3) ca siply be coputed fro DΦ b u C, (5) Dt Δt where (b u) i=(j, N i) is the diesioless vector which cotais the curret desity for uit curret, j = J /. additio,, - are the variables defied i () whose diesio is Wb at tie steps, -, respectively. he coupled aalysis is perfored usig the staggered ethod [9]. Naely, (), (), ad (3) are solved alteratively util covergece at each tie step. his iteratio is here called subcycle. this wor, () is solved by the Ruge-Kutta ethod. he discretied goverig equatios are give by c F p q p q, p (6) C KC C b, (7) RΔt b C, (8) where is the step uber i the subcycle, b() is right had vector i FEM for. Whe the all solutios coverge i the subcycle, that is, p p,,, (9) p the is icreeted, ad ew subcycle starts. he flow of the coupled aalysis is show i Fig. 3. Mesh Mesh y. SMULONS OF VEH MODEL. Noraliatio he VEH odel show i Fig. is aalyed usig the coupled aalysis ethod. o cosider the characteristic which is idepedet of diesio, the syste is oralied. Here, legth, tie, weight ad agetic flu desity are oralied by L, S, M, ad B, respectively. Here, L ad B are set to the gap betwee agets, l g, ad the reaet flu desity of Neodyiu aget,., respectively. additio, S, M are set to.5s, g, respectively. he oralied VEH odel for the field aalysis is show i Fig.. fter the oraliatio, equatios (6)-(8) result i Nocoforig iterface Fig.. wo ocoforig voels for aalysis of ovig object. start solve (6) solve (7) solve (8) coverge Fig. 3. Flow of coupled aalysis ethod. yes o + +

4 differece (%) CMP-53 c d p q p q, d p F dt dt () C KC C b, () C, b () u RΔt S FL B F LM, M, (3) c S c M, S M, where overbar deotes oralied values, oralied agetic force, ad c dapig, sprig coefficiets, respectively. Maget, B _ r=, H _ c=75..5 N S r h r (l g ) S N Magetic Magetic aterial, aterial, μ r = μ r = Coil, tur Fig.. Noralied VEH odel for field aalysis., he behavior of this syste highly depeds o F are are oralied F i (). herefore the coupled aalysis is perfored by chagig the coil radius ad agets height, r ad h. this aalysis,,, R, ad iput aplitude are fied to., 5.,., ad.5, respectively. this wor, ie coditios with differet r ad h suaried i able are cosidered i which the aget sie _ is set to, 3, 5 whe h is 5, 7.5, respectively. B. Perforace of voel-fem We discuss here the perforace of the preset voel-fem. Fig. 5(a) shows the differece of coputatioal results betwee the voel-fem ad the covetioal FEM with tetrahedral eleets. he ubers of eleets i forer ad later ethods are 7,3 ad,973,, respectively. Fro Fig. 5(a), we ca see that the aiu differece betwee two ethods is less tha 5%. Fig. 5(b) shows the coputatioal tie for esh geeratio i the voel-fem. Fro Fig. 5(b) it ca see that the coputatioal cost for esh geeratio i the voel-fem is of order of. Note that the coputatioal cost for covetioal tetrahedral esh is of order of log [3]. Fro these results, we coclude that the voel-fem is suitable for coputatio of ovig objects as well as optiiatio probles because of its high coputatioal efficiecy. C. Nuerical results uit: (p.u.).75h he coupled aalysis is perfored for ie cases show i able. these aalyses, two iput frequecies, f =, f =, are cosidered. Figure 6 shows the trajectories o the phase plae of the relative positio = p q ad the relative velocity, v. Moreover, Fig. 7 shows the trasitios of ad EMF, V, correspodig to the trajectories show i Figs. 6(b) ad 6(d). Fro these results, it ca be see that the behavior draatically chages i depedece o r ad h. (), chaotic h y c oliear oscillatios are observed at f ad f as show i Figs. 6(a) ad 6(b). (E), the catilever has oliear oscillatio at f, whereas it has liear oscillatio at f. these cases, the aplitude of is relatively large for the coputed frequecy rage. his suggests that output ca be geerated for the wide rage of frequecy through the oliear oscillatio. O the other had, i (D), the catilever does ot have oscillatios at f, while liear oscillatio is observed fro f. his idicates that agetic force does ot give oliearity whe r is relatively large. t is foud fro the aalysis results that the VEH odel has chaotic ad liear oscillatio odes which vary by r ad h. o show the copleity of the catilever otio i each coditio, the Lyapuov epoet, λ, for the ie coditios is coputed usig the Rosestei s ethod []. Positive λ correspods to chaotic oscillatio, i cotrast, egative λ correspods to liear or liit cycle oscillatio. able suaries λ for the ie coditios at f ad f, where. eas that the catilever is ot oscillated. Fro able, we ca see that there is o oscillatio i (B), (C), ad (F) at f while positive λ is obtaied at f. these coditios, the agetic force coil positio (p.u.) (a): coputatioal accuracy Fig. 5. Perforace of voel-fem. BLE NLYSS CONDONS cod. () (B) (C) (D) (E) (F) (G) (H) () r h (a): (,,.6) (b): (,, 8.6) (c): (D,, ) (d): (D,, -3.7) EMF 5 5 uber of odes (e): (E,, 33.) (f): (E,, -3.) Fig. 6. rajectories for (cod., f, λ), where C d, f, λ represets coditio show i able, oralied frequecy, Lyapuov epoet, respectively. tie (s) 8 voel O() O(log) log (b): esh geeratio tie

5 CMP-53 Output (W) aget sie is sufficietly large, as a result, the agetic force is so strog that the relative positio betwee the coil ad agets is fied whe the iput frequecy is relatively low. (D), (G), (H), ad (), λ is egative at f, whereas the catilever has o oscillatio at f. hese results suggest that the catilever has liear oscillatio whe r/h is large. he reaso why the catilever is oscillated oly at high frequecy is that the resoat frequecy for the fied echaical paraeters is about 5. () ad (E), λ is positive ad the catilever ca oscillate over the whole frequecy rage V tie (p.u.) - 3 V (p.u.) BLE LYPUNOV EXPONEN FOR 9 CONDONS () (B) (C) (D) (E) (F) (G) (H) () f f D. Output power agaist frequecy he catilever has relatively large oscillatio aplitude at f ad f i () ad (E). his fact idicates that we ca have wide frequecy rage for power geeratio i the chaotic ode. t is also show that the catilever has liear oscillatio whe r/h is large. such case, it is epected that large output would be obtaied at the resoace. o clarify the frequecy characteristics, the output power agaist frequecy is coputed usig the coupled aalysis ethod for () ad (D) is coputed. Moreover, to reveal the effect o the agetic aterial i the coil, the output of a VEH odel without agetic aterial i the coil whe r ad h are.5 ad 5, respectively, is also coputed. his coditio is here called (L). these aalyses, the output is ot oralied ad l g is set to.. he outputs agaist the iput frequecy for (), (D), ad (L) are show i Fig. 8 where the output uder μw is ot plotted. t is clearly observed that output power over the whole rage is larger tha.w i (). Hece, we ca have wide badwidth for power geeratio through the chaotic oscillatio. t ca be also see fro Fig. 8 that large output power is obtaied i (D) whe the iput frequecy is H. t the resoace, the output power i (D) is uch grater tha that of (L). his is due to the fact that the agetic flu across the coil is effectively icreased by the agetic aterial tie (p.u.) (a): (, ) (b): (E, ) Fig. 7. rasitios of ad V i coditios (cod., f) frequecy (H) Fig. 8. Output power agaist frequecy. () (D) (L) V -8 3 V (p.u.) Fro these results, we coclude that the VEH device havig chaotic oscillatio is suitable for the operatio i the eviroet where the abiet frequecy is ot predicted. cotrast, that havig liear oscillatio is effective whe the iput vibratio is predictable. V. CONCLUSON he coupled aalysis ethod for VEH has bee preseted based o the voel-fem, by which the ovig echaical syste ca effectively be odeled without tie-cosuig esh geeratios. Usig the preset ethod, the VEH odel cotaiig agetic aterial i the coil has bee aalyed. he aalysis results show that the VEH device has chaotic oscillatio which realies wide frequecy rage for power geeratio. t is also show that that has liear oscillatio whe the radius of the coil is large. the liear oscillatio, we have large output at the resoat frequecy. For the future wors, the aalysis results will be copared with easureet data. CKNOWLEDGMEN his study was supported by research grats fro Japa Power cadey ad a Grat-i-id for Scietific Research (37). REFERENCES [] S. P. Beeby, et. al., icro electroagetic geerator for vibratio eergy harvestig, J. Microech. Microeg. vol.7, pp , 7. [] D. Zhu, et. al., Geeral odel with eperietal validatio of electrical resoat frequecy tuig of electroagetic vibratio eergy harvesters, Sart Mater. Struct. vol., pp. -,. [3] Y. Sag, el. al., Vibratio-Based Hybrid Eergy Harvester for Wireless Sesor Systes, EEE ras. Mag., vol.8, o., pp.95-98,. [] Y. Zhu, et. al., Magetoelectric Geerator for Eergy Harvestig Fro the Vibratio of Magetic Levitatio,, EEE ras. Mag., vol.8, o., pp ,. [5]. Galchev, et. al., Vibratio Harvestig Syste for Bridge Health Moitorig pplicatios, i PowerMEMS, pp. 79-8,. [6] L. G. W. vedt, et. al., Noliear Behavior of a Electrostatic Eergy Harvester Uder Wide- ad Narrowbad Ecitatio, Joural of Microelectroechaical systes, vol.9, o., pp ,. [7] M. Ferrari, et. al., proved eergy harvestig fro widebad vibratios by oliear pieoelectric coverters, Sesors ad actuators, vol.6, pp. 5-3,. [8] N.G. Stephe, O eergy harvestig fro abiet vibratio, Joural of Soud ad Vibratio, vol.93, pp. 9 5, 6. [9]. Niho, et. al., Nuerical stability of Magetic Dapig Proble of Elastic Plate, EEE ras. Mag., vol.36, o., pp ,. []. Sato, et. al., 3D Optiiatio of Ferrite ductor Cosiderig Hysteresis Loss, EEE ras. Mag., vol.9, o.5, pp. 9-3, 3. [] K. Muraatsu, et. al., hree-diesioal Steady-State Eddy-Curret alysis of Movig Coductor Usig Edge Eleets ad Movig- Coordiate Syste, EEE ras. Mag., vol.38, o., pp ,. [] H. Koetai, S. Saabe,. Kaeari, 3-D aalysis of iductio otor with sewed slots usig regular couplig esh, EEE ras. Mag., vol.36, o., pp ,. [3] S. W. Sloa, fast algorith for costructig Delauay triagulatios i the plae, dvaces i Egieerig Software, vol.9, o., pp.3-55, 987. [] M. Rosestei, et. al., practical ethod for calculatig largest Lyapuov epoets fro sall data sets, Physica D, vol.65, pp. 7-3, 993.