PIEZO-TRANSDUCER MODELLING WITH A SWITCHED OUTPUT VOLTAGE: APPLICATION TO ENERGY HARVESTING AND SELF-POWERED VIBRATION CONTROL

Transcription

1 19h INTERNATIONAL CONGRESS ON ACOUSTICS MADRID, 2-7 SEPTEMBER 27 PIEZO-TRANSDUCER MODELLING WITH A SWITCHED OUTPUT VOLTAGE: APPLICATION TO ENERGY HARVESTING AND SELF-POWERED VIBRATION CONTROL PACS: 43.4.Tm Guyomar, Danel; Lallar, Mckaël LGEF, INSA-Lyon, EA 682, 69621, France ; danel.guyomar@nsa-lyon.fr ABSTRACT I has been shown ha, for a ransducer bonded on a vbrang srucure, he oupu volage swchng ncreases sgnfcanly he performances n erms of mechancal o elecrcal energy converson. Consequenly n such a echnque he energy converson s a hghly non-lnear process. Ths energy ransfer s opmal f he swch occurs a mes ha are relaed o he srucure dsplacemen self. Whle he energy converson process has been suded by he frs harmonc mehod on he oupu volage, hs communcaon nroduces he modellng of he ransducer response n he swchng volage regme usng generalzed funcons. Thus he non-lnear naure of he process s mananed. Ths modellng leads o he volage/curren and energy oupus and s appled o sprng/mass and beam srucures. Based on he modellng resuls, he opmal swchng me sequence for narrow and broadband npu sgnals wll be dscussed. INTRODUCTION Pezoelecrc elemens are commonly used maerals for elecromechancal converson. Ther effecveness had made hem parcularly suable for vbraon dampng, wave ransmsson reducon and energy harvesng. In order o unfy he advanages of passve and acve echnques, new echnques have been developed hese las years ([1-4]. Parcularly, he Synchronzed Swch Dampng (SSD mehod ([5-9] allows a very powerful and sraghforward conrol. Ths mehod consss n swchng nermenly a pezoelemen on an elecrcal nework (ypcally an nducor - SSDI mehod as shown n Fgure 1. Ths echnque can also be used for energy harvesng, leadng o he so-called Synchronzed Swch Harvesng (SSH and Synchronous Elecrc Charge Exracon (SECE mehods ([1]. Alhough SSD and SECE mehods have been well-suded for monochromac excaon a he resonan frequency, deermnng an opmal swchng sequence for arbrary sgnals remans an ssue. As well, frs harmonc approxmaon s classcally made for he heorecal developmen, leadng o approxmaed expressons. Recenly, Anderson e al. proposed a modal analyss n [11]. Fgure 1: SSD echnque prncple Fgure 2: Suded srucure examples

2 The purposes of hs paper are hreefold. Frs proposes a me-based model of he swch effec. Then from hs model s derved he deermnaon of opmal swchng sequence consderng several crera. Nex he exac expresson of he aenuaon n harmonc case wll be nvesgaed. MODELLING Sysem modellng I s proposed here a smple 1D model for srucures as shown n Fgure 2. Applyng he fundamenal dynamc equaon usng he pezoelecrcy equaons (1 (E s he elecrc feld, D he nducon, S he sran and T he sress yelds (2, akng no accoun a vscous frcon coeffcen C and an exernal force F (S s he cross-secon of he elemen and assumng consan sran and elecrcal feld, wh V he pezovolage and l he lengh of he elemen. K E and α are defned as he shor-crcu sffness and he force facor respecvely. The expresson of he pezocurren s gven by he Maxwell s equaon as (3. C s aken as he clamped capacance of he pezoelemen. T = c E S e E D = ɛ S E + es (1 Mü = S c E l u S e V C u + F = K E u αv C u + F Mü + C u + K E u = F αv (2 l I = js = ḊS = S ɛ S V S e u = C V + α u l l (3 C V + α u = δ( (4 As shown by Fgure 3, he effec of he SSD conrol s a fas volage nverson and a curren characerzed by a sum of dela funcons. From 4 he volage can hus be defned as (5, wh V con he conrol volage correspondng o a pecewse funcon. Ths conrol volage wll be explced n he hrd sub-secon. Mergng hs equaon wh he moon equaon (2 yelds (6, wh K D defned as he open-crcu sffness. Consequenly s possble o defne he sysem usng he mpulse (7 or sep (8 responses. V = α C u + V con (5 ( α Mü + C u + K E u = F α u + V con Mü + C u + K D u = F αv con (6 C 1/2 ( h δ = (K D M C2 e C K 2M D sn 4 M C2 4M 2 (7 { h H = 1 1/2 [ K D 1 (1 C2 4K D e C K 2M D sn M M C2 4M2 + Arcan ( ] } 4KD M C 2 1 (8 Takng no accoun mulmodal srucures for broadband conrol s acheved usng he combnaon of several monomodal cases. The dsplacemen u s herefore decomposed n several dsplacemens u modelled as exposed n he prevous sub-secon, and he resulng dsplacemen s gven by he sum of each componen. Consequenly, he mpulse (resp. sep response h δ (resp. h H s gven by he sum of he mpulse (resp. sep responses of each sprng-mass model h δ (resp. h H. In he case of mulmodal srucures however, he conrol volage s common o all he modes, yeldng (9. ( 1 V = α u + V con (9 C 2 19h INTERNATIONAL CONGRESS ON ACOUSTICS - ICA27MADRID

3 Fgure 3: SSDI conrolled volage Pecewse funcon modellng In he SSDI echnque, he swch on he nducor allows a fas nverson of he volage, as shown by Fgure 3. However hs nverson s no perfec due o nernal losses, represened by an elecrcal qualy facor Q, leadng o an nverson facor γ = e π 2Q. I s possble o express he pecewse funcon V con from Fgure 3. Frs, each swch ha changes he value of he pezovolage from V p( o γv p( s equvalen o applyng a sep volage (1 + γv p(. Consequenly, he conrol volage V con can be expressed as (1 1, wh H( he Heavysde funcon. As well, hs volage beng consan beween wo swch nsans yelds (11, wh u ( = and V sw ( =. The response of he sysem s herefore gven by (12. I s hus possble o see he effec of swch as a seres of sep responses. V con = (1 + γ n V sw (nh( n (1 V sw (n = γv sw (n 1 + α (u (n u (n 1 C (11 u sw ( = F h δ + (1 + γ ( α V sw (nh H ( n n (12 OPTIMAL SWITCHING SEQUENCE DETERMINATION Excaon reducon The frs proposed creron s o reduce he drvng force F. Consderng he moon equaons developed n he prevous secon, one can observe ha he sysem s drven by an equvalen force gven as (13. Mnmzng hs equvalen hus leads o he nex swchng me k+1 fulfllng (14. F eq ( = F( αv = F( ( α 2 u ( + (1 + γα V sw (nh( n (13 C n F eq ( k+1 = F( k+1 α 2 C u ( k+1 = (14 Magnude reducon Assumng ha he conrolled and unconrolled dsplacemens are n phase, s proposed here o conrol he vbraon usng he creron gven as (15, leadng o he nex swchng me (16. (u ( 2 sw ( u2 un ( d = u ( un( u un( 2αV con( h H ( d (15 1 For clary reasons, he nsans n are shorened as n. 3 19h INTERNATIONAL CONGRESS ON ACOUSTICS - ICA27MADRID

4 (1 + γ ( α k 1 n=1 V sw (nh δ ( k n = 2 u un ( k (16 Mechancal energy reducon / Harvesed energy opmzaon Anoher way of dampng s o consder he mechancal energy of he srucure. The prncple of such a conrol s o mnmze he nernal energy of he srucure (17. Such a conrol also allows he maxmum energy exracon. Ths energy could eher be dsspaed no he swchng nework (vbraon dampng, or sored on a capacor (energy harvesng. In hs las case, he man dea s o maxmze he sum of he squared volage k V (k2, as descrbed n [1]. As he basc operaon of he pezoelemen s o conver elasc energy no elecrc energy, he swch should occur when he pezoelemen energy s maxmum, leadng o he condon (18. E mech ( = 1 (M u ( 2 + K D 2 u ( 2 (17 ( k 1 K D α 4 u un ( k + α V sw (nh δ ( k n = (18 n=1 APPLICATION EXAMPLES The presen secon ams a nvesgang he performance of he prevously proposed crera versus he classcal SSDI echnque. One can remnd ha he purpose of he paper s o have a deeper undersandng of he swch process. Consequenly, he performance comparson s done usng smulaon (Runge-Kua 4 raher han expermenal se-up, allowng he access o all of he parameers. For expermenally verfed conrol law, he reader can refer for example o [8] or [9]. Resonan frequency excaon n monomodal case I he parcular case of resonan excaon appled o a monomodal srucure, he mpulse and sep responses h δ and h H have he same pseudo-frequency han he force and he speed. Consderng an excaon gven as F( = F M sn(2πf ( τ, he speed s cancelled accordng o (19, gvng he swchng me derved from he frs creron. A hs parcular me n he force and he speed are null and hus he hrd creron s also fulflled. For he second crera he frs swchng me 1 s obaned for he cancellaon of he speed u un (. n+1 can be obaned by nong ha, n he parcular case of resonan frequency excaon, he speed and he mpulse response have he same frequency. These consderaons hus lead o he same expresson of he swchng me (19. k = + 2k 1 4f (19 In hs case, s also possble o express exacly he aenuaon on he dsplacemen. Indeed, consderng he seady sae, V sw (k = V sw (k 1 and u(k = u(k 1 = U M, leadng o (2 from (11. Consequenly he pecewse conrol volage V con can be expressed as (21 (remndng ha he swch occurs for he speed cancellaon. I can be shown ha from he expressons (12 and (7 he exac aenuaon s herefore gven by (22. Consderng a very low dampng coeffcen (allowng neglgble hgher harmoncs, he classcal expresson for he aenuaon can be found. α V sw (k = 2 C (1 γ U M (2 (1 + γ ( V sw (n = α 1 + γ C n 1 γ U Msgn( u = α 1 + γ n C 1 γ U M H( H( (21 =2 πc 1 U sw = 1 + α2 1 + γ 1 e 4 K D M ( U un C 1 γ K D 1 C2 snh 4K D (22 M 4 πc 4 K D M 19h INTERNATIONAL CONGRESS ON ACOUSTICS - ICA27MADRID

5 Table I: Srucure parameers Common parameers frs mode Second mode M 35g K D 1 44 K D 2 4 C 25nF C 1.8 C 2.4 γ.781 α 1.5 α 2.15 Impulse excaon n monomodal case Here s assumed ha he sysem s exced by an mpulsonal force F = F M δ(. As before he frs creron gves he swchng me as he cancellaon of he speed, yeldng (23. As well, he second and hrd crera also gve hs parcular swchng me, for he same reasons as exposed n he prevous subsecon. n = + 2n 1 4f (23 Impulse excaon n mulmodal case In he case of mulmodal sysems, he bes creron for such sysems s he hrd one, ha focuses on he mos energec mode a a gven nsan. In hs sub-secon s proposed o apply hs creron o a bmodal srucure exced by a unary pulse force. The parameers of he srucure are presened n Table I. Resuls are presened on Fgure 4, clearly showng he performances of he creron over classcal SSDI echnque. The performance opmzaon s even more obvous for energy harvesng, gvng almos wce he resuls of SSDI echnque. V u thrd creron.25 SSDI Unconrolled 5 5 Mechancal energy Squared volage sum x Fgure 4: Impulse excaon n mulmodal case Whe nose excaon n mulmodal case I s proposed here he evaluaon of he hrd creron for Whe Gaussan Nose (WGN excaon of he srucure. The obaned resuls are shown n Fgure 5. As n he prevous case, s clear ha he proposed creron ouperforms he classcal SSDI. CONCLUSION Ths paper presens a new and orgnal explanaon of he SSD and SECE echnques. Ths explanaon, based on he emporal response of he sysem, hus allows a beer undersandng of he mechansms 5 19h INTERNATIONAL CONGRESS ON ACOUSTICS - ICA27MADRID

6 V Mechancal energy u 2 x Thrd.35creron SSDI Unconrolled Squared volage sum x Fgure 5: Whe Gaussan Nose (WGN excaon n mulmodal case ha le behnd he synchronzed swchng mehod. Generally, hs echnque can be seen as a sum of sep responses perodcally mposed o he sysem. From hs observaon s consequenly possble o predc he response of he sysem o a gven swchng sequence. Then from hs explanaon have been derved several crera for opmal swchng sequence deermnaon. Parcularly, her applcaon n mulmodal cases shows her effcency and he mporance of he swchng sequence. However, he applcaon of he crera supposes he knowledge of he appled force. Thus, excep he non-neglgble conrbuon for he mechansm explanaon, he applcaon of such crera on real-lfe srucure seems chmercal (however esmaon heory can be use, e.g. probablsc approach - c.f. [9]. References:[1] L. R. Corr and W. W. Clark, Comparson of Low-Frequency Pezoelecrc Shun Technques for Srucural Dampng, Smar Maer. Sruc., vol. 11, pp , 22. [2] K. A. Cunefare, Sae-Swched Absorber for Vbraon Conrol of Pon-Exced Beams, J. Inell. Maer. Sys. Sruc., vol. 13, pp , 22. [3] L. R. Corr and W. W. Clark, A Novel Sem-Acve Mul-Modal Vbraon Conrol Law for a Pezoceramc Acuaor, J. of Vb. and Acous., vol. 152, pp , 23. [4] G. A. Leseure, G. K. Oman and H. F. Hofmann, Dampng as a resul of pezoelecrc energy harvesng, J. of Sound and Vbraon, vol. 269, pp , 24. [5] C. Rchard, D. Guyomar, D. Audger and G. Chng, Sem passve dampng usng connuous swchng of a pezoelecrc devce, Proc. SPIE SSMa. Conf., Passve Dampng and Isolaon, San Dego, vol 3672, p14, [6] C. Rchard, D. Guyomar, D. Audger and H. Bassaler, Enhanced sem passve dampng usng connuous swchng of a pezoelecrc devce on an nducor, Proc. SPIE SSMa. Conf., Passve Dampng and Isolaon, vol 3989, p288, 2. [7] D. Guyomar, A. Faz, L. Pe, and C. Rchard, Wave reflecon and ransmsson reducon usng a pezoelecrc sempassve nonlnear echnque, J. Acous. Soc. Am., Vol. 119 (1, pp , 26. [8] A. Badel, G. Sebald, D. Guyomar, M. Lallar, E. Lefeuvre, C. Rchard and J. Qu, Pezoelecrc vbraon conrol by synchronzed swchng on adapve volage sources: Towards wdeband sem-acve dampng, J. Acous. Soc. Am., Vol. 119 (5, pp , 26. [9] D. Guyomar and A. Badel, Nonlnear sem-passve mulmodal vbraon dampng: An effcen probablsc approach, J. of Sound and Vbraon, Volume 294, Issue 1-2, pp , 26. [1] E. Lefeuvre, A. Badel, C. Rchard, L. Pe and D. Guyomar, A comparson beween several vbraon-powered pezoelecrc generaors for sandalone sysems, Sensors and Acuaors, vol. 126, no2, pp , 26. [11] T. Aderson, U. Manubarh, G. Cor and M.nderson, Response Predcon of Swched Inducor/Pezoelecrc Vbraon Suppresson, Smar Maer. Sruc., vol. 17, pp , h INTERNATIONAL CONGRESS ON ACOUSTICS - ICA27MADRID