A study on the influence of design parameters on the power output characteristic of piezoelectric vibration energy harvester

Transcription

1 A udy on he influence of deign arameer on he ower ouu characeriic of iezoelecric vibraion energy harveer N. Aboulfooh, J. Twiefel Iniue of Dynamic and Vibraion eearch Leibniz Univeriä Hannover Aelr. 67 Hannover, Germany Abrac Thi conribuion derive a model of a vibraion energy harveer in he form of a canilever beam coniing of wo iezoelecric layer, a ingle non-iezoelecric layer in-beween and carrying a i ma. The derived model dicreize he infinie-degree-of-freedom yem ino an equivalen ingle-degree-offreedom one (-DOF) uilizing he ayleigh-iz mehod. The ouu ower i obained a a funcion of he deign arameer, including dimenion and maerial roerie. A he reonance oeraion condiion, he maximum ower and he oimal load reiance formula are obained. Boh are deenden on he deign arameer and hu, for an oimal deign of he harveer, he load reiance hould be conidered. Inroducion Energy harveing ha araced he inere of many reearcher over he la decade. I can uly ower for mall orable elecronic device and wirele enor nework. Vibraion i an available ource for he energy harveer a i exi in almo all indurial yem. Piezoelecric randucer i a ignifican mechanim o conver vibraion energy ino elecrical energy []. The iezoelecric randucer rovide relaively high ower deniy, in addiion o i alicabiliy o MEMS device []. William and Yae [] inroduced a general model of kineic energy harveing: They inroduced a generaor of a eimic ma aached o a ring. The harveed ower from hi dicree model wa obained from he roduc of he force on he ma and i velociy. Twiefel e al. [4] rereened he vibraion harveer by an equivalen elecromechanical circui. The volage acro he load reiance and he ouu ower were obained. The reul were verified exerimenally. Canilevered vibraion harveer i he focu of reearch due o i imliciy and alicabiliy o he MEMS device [5]. Several auhor have udied he modeling of a canilevered iezoelecric energy harveer under bae exciaion. Lumed-arameer modeling wa rooed in [6] where lumedarameer equaion were ued o obain he eady-ae vibraion reone, volage ouu and ower ouu. Sodano e al. [7] ued he generalized Hamilon rincile of elecromechanical yem for modeling a canilevered iezoelecric energy harveer baed on he Euler Bernoulli beam heory. The ayleigh iz mehod wa ued o give a dicreized model of he diribued-arameer yem. Princile of iezoelecric energy harveing, modeling rocedure, ye of iezoelecric harveing device and iezoelecric maerial were reviewed in [8]. I wa concluded ha here wa a need o develo iezoelecric maerial wih high couling coefficien o imrove he erformance of he generaor. In addiion, he iezoelecric maerial hould be flexible and reilien o harh vibraion and hock. In [9], variou deign and exerimenal reul of vibraion MEMS iezoelecric energy harveer 5

2 6 POCEEDINGS OF ISMA6 INCLUDING USD6 were reviewed. I wa concluded ha he delivered ower ouu of hoe MEMS device wa ill inadequae for ue a he ower uly for mobile elecronic device a well a remoe enor. A major inere of reearch in energy harveing i o maximize he ouu ower. Mo of he inroduced vibraion harveer were deigned o oerae a reonance condiion []. However, here i a large number of deign arameer e o une and generae reonance o a given value, even for a imle canilever. Thu, udying deendence of he ouu ower on he deign arameer i a neceary o oimize he deign of he vibraion harveer. In order o deermine hi deendence, a ower formula a funcion of he deign arameer including dimenion, maerial roerie, i ma, load reiance and boundary condiion i required. In hi conribuion, a model of a canilevered iezoelecric energy harveer i derived and lead o a formula for he ouu ower and he oimal load reiance a funcion of deign arameer. Modeling of iezoelecric energy harveer The vibraion harveer conidered for he modeling i a canilever of wo iezoelecric layer, a ingle non-iezoelecric layer in-beween, and carrying a i ma a hown in figure -a. The modeling i baed on he dicreizaion of he coninuou Euler-Bernoulli beam uilizing he ayleigh-iz mehod. Figure - b how he equivalen -DOF yem, where m eq and k eq are he equivalen ma and he equivalen iffne of he vibraion harveer, reecively, Γ and C are he equivalen couling facor and he caaciance of he wo iezoelecric layer, reecively and c eq i he mechanical daming coefficien. (a) (b) Figure : (a) A chemaic diagram of he inveigaed vibraion harveer (b) Equivalen -DOF yem Fir he non-iezoelecric canilever i conidered for he modeling, ee figure. Then, he model i exended by adding he wo iezoelecric layer. The non-iezoelecric canilever ha Young modulu E, momen of ineria I, deniy ρ, lengh l, hickne h, widh b, ma er uni lengh µ (x) and hold a i ma m. In hi conribuion, he ubcri leer refer o he non-iezoelecric him layer while he ubcri leer refer o he iezoelecric layer. The i ma i aumed o be a oin ma a i dimenion are mall comared o he canilever lengh. The canilever i aumed o have a rigid fixaion and o be ubjeced o force er uni lengh F(x,). Figure : Non-iezoelecric canilever For he inveigaed canilever, he econd eigenfrequency i ignificanly higher han he fir one. Thu, he funcion of he fir mode hae i aumed o decribe he deflecion hae ufficienly. The ranveral dilacemen z(x,) of any oiion x along he beam i exreed a: z( x, ) ( x / l ) z ( ) ()

3 FP7 ANTAES: ENEGY EFFICIENT SMAT STUCTUES 7 where z () i he generalized modal coordinae normalized on he dilacemen a he i and ϕ (x/l ) rereen he fir mode hae funcion of a canilever wih a i ma, decribed in [] a: x x co( ( )) coh( ( )) x x ( x / l) c co( ( ) ) coh( ( ) ) (in( ( ) ) inh( ( ) )) l l in( ( )) inh( ( )) l l where c i an arbirary conan aumed o be, α i he ma raio (o be exlained laer in deail), and λ i he fir roo of he characeriic equaion of a canilever wih a i ma, given a: co( ( ))coh( ( )) () By uing erie exanion of he rigonomeric and hyerbolic funcion from [], he hae funcion i aroximaed a decribed in [] and exreed a: x x ( x/ l) ( ) ( ) l l The hae funcion a he oiion (x=l ) i obained a: ( x/ l ) xl (5) The dilacemen of vibraion in he -DOF yem i aumed o equal he dilacemen a i of he canilever. By ubiuing equaion 5 in equaion, he vibraion dilacemen of he -DOF yem a well a he dilacemen a i of he canilever i exreed a: z ( l, ) z ( ) (6) To obain he equivalen arameer of he -DOF yem, he kineic energy, oenial energy (rain energy), and work done by exernal force in he -DOF yem are equaed o hoe in he infiniedegree-of-freedom yem [4]. The kineic energy will be equaed beween he wo yem o obain he equivalen ma m eq. I will be obained in wo main e: he equivalen ma of he non-iezoelecric canilever carrying a i ma will be obained fir, and hen he equivalen ma of he wo iezoelecric layer will be obained. The kineic energy of he non-iezoelecric canilever carrying a i ma i exreed a: l ( ) (, ) (, ) E x z x dx m z l (7) elace μ (x) by ρ b h and ubiue equaion and 6 in equaion 7 o ge: Subiue equaion 4 and 5 in equaion 8 o ge: l x x x l l l E b h ( ) z ( ) dx m ( ) z ( ) (8) l 4 x 6 x 5 x l 9 l l E b h ( ( ) ( ) ( ) ) z ( ) dx m z ( ) (9) Execuing he inegraion in equaion 9, he kineic energy of he non-iezoelecric canilever carrying a i ma i exreed a: 4 4 E (.57) b hl z ( ) m z ( ) () 9 9 The kineic energy of he equivalen -DOF yem i exreed a: E DOF mz ( l, ) () () (4)

4 8 POCEEDINGS OF ISMA6 INCLUDING USD6 Subiue equaion 6 in equaion, he kineic energy of he -DOF yem i obained a: 4 E DOF m z () () 9 Equaing equaion o equaion, he equivalen ma of he non-iezoelecric canilever carrying a i ma i obained and exreed a: m.57 b h l m () The inroduced harveer i aumed o be a hin beam. Each iezoelecric layer ha he ame lengh and widh a he non-iezoelecric layer. The wo iezoelecric layer have he ame deflecion hae a he non-iezoelecric layer. Thu, he equivalen ma of each iezoelecric layer will be obained imilarly o he equivalen ma of he non-iezoelecric layer. From equaion, he equivalen ma of he noniezoelecric layer i.57ρ b h l. Thu, he equivalen ma of he wo iezoelecric layer i exreed a: m (.57 b h l ) (4) By adding equaion o equaion 4, he equivalen ma of he canilevered vibraion harveer carrying a i ma i exreed a: m.57( b h l b h l ) m (5) eq The i ma m will be normalized o he ma of he harveer beam (one non-iezoelecric layer and wo iezoelecric layer) by he raio α. The i ma i exreed a: m ( b h l b h l ) (6) where α i he raio beween he i ma and ma of he harveer beam. Subiuing equaion 6 in equaion 5, he equivalen ma of he canilevered vibraion harveer wih a i ma i exreed a: m (.57 )( b h l b h l ) (7) eq To obain he equivalen iffne, he oenial energy (rain energy) of he coninuou yem will be equaed o he oenial energy (rain energy) of he -DOF yem. The flexural rain energy ored in a bending beam i deermined a he work exered by he bending momen M(x) undergoing angular dilacemen dθ [4]. The rain energy in he non-iezoelecric canilever i given a: U l M ( x ) d (8) The angular dilacemen dθ i obained from he formula of he flexural curvaure of a beam a following: d z ( x, ) d M ( x ) (9) dx dx EI Subiuing equaion 9 in equaion 8, he rain energy in he non-iezoelecric canilever can be exreed a: l d z ( x, ) dx Subiuing equaion in equaion reul in he following equaion: U E I dx () l ( / ) ( ) U E I x l z dx ()

5 FP7 ANTAES: ENEGY EFFICIENT SMAT STUCTUES 9 Obaining he econd derivaive of equaion 4 wih reec o oiion and ubiuing i in equaion reul in he following equaion: l 4 4 x x U E I 4 ( ) ( ) ( ) z dx l l l () Execuing he inegraion in equaion, he rain energy in he non-iezoelecric layer i obained a: U The rain energy in he equivalen -DOF yem i given by: 4 E I z () l () U DOF k z ( l, ) (4) By ubiuing equaion 6 in equaion 4, he rain energy in he equivalen -DOF yem i obained a: 4 U DOF k z () (5) 9 By equaing equaion o equaion 5, equivalen iffne of he non-iezoelecric layer i obained a: k EI l The momen of ineria I of he non-iezoelecric layer i exreed a: (6) bh I (7) By ubiuing equaion 7 in equaion 6, he equivalen iffne of he non-iezoelecric layer i obained a: k b h E (8) 4 l The rain energy in he wo iezoelecric layer i exreed a: l d z ( x, ) dx U E I dx (9) The wo iezoelecric layer have he ame deflecion hae a he non-iezoelecric layer. Thu, he equivalen iffne of each iezoelecric layer will be obained imilarly o he equivalen iffne of he non-iezoelecric layer. The equivalen iffne of he wo iezoelecric layer i obained a: k EI ( ) () l The wo iezoelecric layer are ymmerical. The cenerline of each iezoelecric layer i a a diance of (h /+h /) away from he cenerline of he non-iezoelecric layer. The momen of ineria for each iezoelecric layer i given a: Equaion i o be arranged a following: bh h h I bh () I bh (8h hh 6 h ) () 4

6 4 POCEEDINGS OF ISMA6 INCLUDING USD6 Subiuing equaion in equaion, he equivalen iffne of he wo iezoelecric layer i exreed a: E b h k 8h hh 6h () 4 l The equivalen iffne of he non-iezoelecric layer i in arallel connecion wih he equivalen iffne of he wo iezoelecric layer. Thu, adding equaion 8 o equaion, he equivalen iffne o he whole canilevered vibraion harveer i exreed a: The reonance frequency i exreed a: E b h k h hh h (4) Ebh eq l 4 l k eq n (5) meq Subiuing equaion 7 and 4 in equaion 5, he reonance frequency i obained a: E h 8h h h 6h E h n l ( h h )(.57 ) (6) The work erformed by he alied force in he infinie-degree-of-freedom yem equal he work erformed by he equivalen force in he -DOF yem. From hi rincile, he equivalen driving force of he -DOF yem i obained. Aume ha a virual dilacemen δz(l,) i alied o he infiniedegree-of-freedom yem. The work erformed on he non-iezoelecric canilever carrying a i ma due o he virual dilacemen i given by: where u i he driving acceleraion. l W F ( x, ) z ( x, ) dx m uz ( l, ) (7) Subiue equaion 4 and 5 in equaion 7. Then, relace he force er uni lengh F(x,) by he roduc of μ (x) and, where μ (x)=ρ b h. The work exered on he infinie-degree-of-freedom yem i exreed a: l x x l l W b h u ( ) ( ) z ( ) dx m u ( ) z ( ) (8) The work exered by he equivalen force due o a virual dilacemen δz(l,) in he -DOF yem i exreed a: W (, ) DOF F z l (9) where F i he driving force of he -DOF yem equivalen o he driving force of he non-iezoelecric canilever carrying a i ma. Subiuing equaion 6 in equaion 9, he work exered by he equivalen force in he -DOF yem i obained a: W DOF F ( ) z ( ) (4) By execuing he inegraion in equaion 8 hen equaing i o equaion 4, he driving force of he -DOF yem equivalen o he driving force of he non-iezoelecric canilever carrying a i ma i obained a: F b h l (.75) u( ) m u( ) (4)

7 FP7 ANTAES: ENEGY EFFICIENT SMAT STUCTUES 4 Since he wo iezoelecric layer have he ame deflecion hae a he non-iezoelecric layer, hu, he equivalen driving force o he wo iezoelecric layer will be obained imilarly o he equivalen driving force o he non-iezoelecric layer. The equivalen driving force o he wo iezoelecric layer i exreed a: F (.75)( b h l ) u (4) By adding equaion 4 o equaion 4 and relacing m wih i value in equaion 6, he equivalen driving force for a canilevered vibraion harveer wih a i ma i obained a: F (.75 )( b h l b h l ) u (4) eq If aumed ha he equivalen driving force of he canilevered vibraion harveer wih a i ma i exreed in erm of a ma force mulilied by he driving acceleraion a: F eq m u() (44), hen he equivalen driving ma force for he canilevered harveer wih a i ma i obained a: f m (.75 )( b h l b h l ) (45) f To fully decribe he mechanical ar of he vibraion harveer, he mechanical daming will be eimaed. A imle way o eimae he equivalen daming coefficien i from he following equaion: where ζ i he mechanical daming raio. c m (46) eq eq n The mechanical daming raio ζ can be obained exerimenally by alying he conce of logarihmic decremen. The logarihmic decremen i obained from he naural log of he raio beween he amliude of any wo ucceive eak [5]. The logarihmic decremen i exreed a: A ( ) ln n A ( n ) where A() i he amliude of vibraion a ime and A(+nτ) i he amliude afer any n ucceive oiive eak and τ i he eriodic ime. The daming raio i calculaed from he following relaion: 4 The model will be exended o include he iezoelecric effec and obain he elecrical arameer Γ and C a decribed in [7]. The equivalen elecromechanical couling facor Γ of he wo iezoelecric layer i exreed a: h h l h l x x h h l h h l h (47) (48) e b y ( ) dxdy e b y ( ) dxdy (49) E E where e i he iezoelecric couling coefficien ( e d / ), conan elecric field and d i he iezoelecric charge conan. i he elaic comliance under By obaining he econd derivaive of equaion 4 wih reec o oiion and ubiuing i in equaion 49 hen execuing he inegraion, he equivalen elecromechanical couling facor i obained a: d b ( h h ) (5) E l The caaciance C of he wo iezoelecric layer in arallel connecion i exreed by:

8 4 POCEEDINGS OF ISMA6 INCLUDING USD6 h h h S S ( ) ( ) h h h h h (5) C b l dy b l dy S where i he ermiiviy of he iezoelecric maerial a conan rain. By execuing he inegraion in equaion 5, he caaciance of he wo iezoelecric layer i obained a: C bl S h (5) A S T ( / E T d ) where i he ermiiviy of he iezoelecric maerial a conan re, hen he caaciance C i exreed a: C bl T d ( ) (5) h The governing equaion of hi elecromechanical model are exreed a following: E m z ( l, ) c z ( l, ) k z ( l, ) v ( ) m u (54) eq eq eq f z ( l, ) C v ( ) I ( ) (55) where v i he ouu volage and I i he curren which can be exreed by: where i he load reiance. I v (56) By ubiuing equaion 56 in equaion 55, hen erforming Lalace ranform, he following equaion i obained: v ( j) jz ( l, j ) C jv ( j ) (57) where Ω i he driving frequency and j i he imaginary number. By rearranging equaion 57, he ouu volage i obained a: v z ( l, j ) j ( j) ( C j) By erforming Lalace ranform o equaion 54, and relacing he ouu volage v by i value exreed in equaion 58, he ranfer funcion beween he ouu dilacemen a he i z(l,ωj) and he inu dilacemen u(ωj) i obained a: z ( l, j ) mf H u( j) j meq ceqj k eq ( C j) The ranfer funcion i rearranged o be a following: m f H m eq n C n j j meq ( C ) ( ) meq C (58) (59) (6)

9 FP7 ANTAES: ENEGY EFFICIENT SMAT STUCTUES 4 The ower i exreed a: v P (6) By ubiuing equaion 58 in equaion 6, and uing he relaion z ( l, j ) H u obained from equaion 59, he value of he real ower i obained a: P H u ( C j ) By ubiuing equaion 6 in equaion 6 and calculaing he abolue value of he magniude, he real ower i obained a: f u m eq P C n n C meq ( C ) ( ) meq C The harveer i uoed o oerae a reonance. The daming raio i aumed o be low o obain higher ouu ower. The oimal reiance ha give he maximum ower i obained by equaing he denominaor of he ower formula (equaion 6) by zero. The oimal load reiance ha give he maximum ower a reonance oeraion condiion i obained a: oimal m m n 4 m C 6 4 n eq eq n (6) (6) (64) Simulaion and analyi The maerial roerie of he iezoelecric layer and he non-iezoelecric layer conidered for he re of he conribuion are included in able. The harveer ha a lengh of 4 mm and a widh of 7. mm. The hickne of he non-iezoelecric layer i. mm and he hickne of each iezoelecric layer i. mm. The ma raio α i zero. The fir eigenfrequency for hoe roerie i 9.9 Hz. The mechanical daming raio i aumed o be.. The caaciance C i calculaed and found o be 6.48 nf and he equivalen elecromechanical couling facor Γ i 8.4 mn/v. The oimal reiance for hee arameer i calculaed and found o be 44.9 Ω a he reonance oeraion condiion. Mechanical daa Piezoelecric daa Piezoelecric E =.6e- m /N, E = 4.e- m /N, ρ = 8 kg/m d =5e- C/N, T =.985e-8 F/m Table : Maerial roerie Non-iezoelecric E = e N/m, ρ = 78 kg/m

10 44 POCEEDINGS OF ISMA6 INCLUDING USD6 A conan-velociy exciaion of 7.5 mm/ i rovided for he analyi. The vibraing harveer i loaded by he oimal reiance 44.9 Ω. Boh he ranfer funcion and he ouu ower veru he exciaion frequency are dilayed in figure -a and figure -b, reecively. The analyi how ha a low daming raio, he ranfer funcion and he ouu ower have heir maximum value a he reonance frequency. (a) (b) Figure : (a) Tranfer funcion veru exciaion frequency (b) Ouu ower veru exciaion frequency The ouu ower veru he load reiance i dilayed in figure 4. The vibraion harveer oerae a reonance (9.9 Hz). The analyi how ha he maximum ouu ower occur a load reiance of 44.9 Ω. Thi value of he load reiance i he ame a he oimal reiance calculaed from equaion 64. Figure 4: Ouu ower veru load reiance, oeraing a reonance (9.9 Hz) The oimal reiance ha roduce he maximum ower a he reonance oeraion condiion i a funcion of he deign arameer a concluded from equaion 64. For an oimal deign of he harveer a he reonance condiion, he load reiance hould be conidered. To illurae hi conce, he influence of he e of ma raio and harveer lengh on he maximum ouu ower and oimal load reiance i analyzed and dilayed in figure 5. The lengh of he vibraion harveer i decreaed from 4 mm o mm and he correonding ma raio i e o kee he reonance frequency conan a 9.9 Hz. All he oher reviouly-aumed deign arameer are ke conan for hi analyi. A hown in figure 5, decreaing he harveer lengh (increaing he correonding ma raio) increae he maximum ouu ower a well a he oimal load reiance. There i an oimal load reiance ha roduce he maximum ower caabiliy for every e of he i ma and harveer lengh. A an examle, if he alied load reiance i kω, hen o ge he maximum ower, he ma raio hould be around and he harveer lengh hould be around 4. mm, a hown in figure 5. Any oher e of ma raio and harveer lengh will roduce le ower for hi load reiance.

11 FP7 ANTAES: ENEGY EFFICIENT SMAT STUCTUES 45 Figure 5: Influence of he e of harveer lengh and ma raio (reonance i conan a 9.9 Hz) on he maximum ower and oimal load reiance. 4 Concluion A model of a bimorh canilevered iezoelecric energy harveer i derived. Formula for he ouu ower a well a he oimal load reiance ha roduce he maximum ower are obained a reonance oeraion condiion. Boh he maximum ower and he oimal reiance are deenden on he deign arameer. Thu, when uning he vibraion harveer o a ecific reonance frequency, i i imoran o conider he load reiance. The derived formula of he ouu ower and he oimal reiance faciliae inveigaing he influence of variou deign arameer e on he ouu ower. In fuure reearch, he deendency of he maximum ower and he oimal load reiance on he deign arameer will be inveigaed and analyzed for he uroe of finding an oimal deign of he vibraion harveer under conrained volume. Acknowledgemen The auhor graefully acknowledge he Euroean Commiion for i uor of he Marie Curie rogram hrough he ITN ANTAES rojec (GA 6687). eference [] M. Umeda, K. Nakamura, S. Ueha, Analyi of he ranformaion of mechanical imac energy o elecric energy uing iezoelecric vibraor, Jaanee Journal of Alied Phyic, Vol. 5, No. 5S, (996), [] S. oundy, E. S. Leland, J. Baker, E. Carleon, E. eilly, B. Oi, J. M. abaey, P. K. Wrigh,V. Sundararajan, Imroving ower ouu for vibraion-baed energy cavenger, IEEE Pervaive comuing, Vol. 4, No., (5),. 8-6.

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