Piezoceramics-based Devices for Active Balancing of Flexible Shafts

Transcription

1 Piezoceamic-baed Device fo Active Balancing of Flexible Shaft P. J. Sloetje, A. de Boe Univeity of Twente, Faculty of Engineeing Technology, Applied Mechanic Section Dienelolaan 5, 7500 AE, Enchede, The Netheland Abtact Thi pape focue on vibation contol of flexible haft by mean of oto-fixed piezoelectic mateial. The taget i to ealize compact olution fo the uppeion of poblematic eonant vibation at o-called flexual citical peed. Fo analyi, paametic finite element model of flexible oto with piezoceamic heet and tain o diplacement eno ae developed, whee the numbe of degee of feedom i kept low. Seveal mechanim which can detabilize flexible oto ae uantiized, uch a oto mateial damping, diipation of cuent induced in oto-fixed piezoceamic and active feedback contol popotional to oto tain ate. The effectivene of low feuency feedback and feedfowad contol fo the uppeion of the unbalance epone i demontated uing analytic and expeimental eult. Emphai i on the inteaction between the dynamic of the oto and that of the connected electonic cicuit. The expeimental etup which i ued fo validation i a flexible haft euipped with piezoceamic heet and tain eno. A liping aembly i ued to implify meauement with, and contol of, the eno and actuato on the haft and to facilitate the development of compact dive electonic. 1 Intoduction Fatly otating flexible oto may exhibit evee bending vibation and may induce vibation, wea and noie in and nea thei uppot tuctue. Thee poblem can often be olved by impoving the oto balance and uppot damping by paive mean. In cae whee thee mean ae exhauted, active contol method fo oto balancing and damping povide a olution. Device fo active vibation contol of otating machiney have become moe compact and veatile in the pat decade. A elatively new appoach to oto vibation contol i baed on the ue of piezoceamic mateial which ae bonded to the uface of a flexible oto. Thi vibation contol olution wa invetigated by eveal autho in ecent yea (e.g. [2], [3]). They conideed the following application: uppeion of the eonant epone to unbalance, uppeion of foced vibation caued by a diving moto, and tabilization of intable vibation. The compactne and low powe conumption of piezoceamic actuato (and the confomability of fibe compoite actuato to cuved uface) make thei ue attactive in thee application. Howeve, in both invetigation [2] and [3], expeiment wee pefomed uing cotly and wea enitive liping aemblie fo powe tanmiion between tato and oto. In addition, the application of piezoceamic mateial to flexible oto ha cetain dawback which have not been analyzed thooughly yet. Thee emain theefoe ome wok to be done befoe thi contol olution eache matuity. It i focued in thi document on device fo active balancing of flexible oto at peed nea o-called flexual citical peed. (A flexual citical peed i defined a a peed at which the bending vibation epone of a oto to unbalance eache a maximum. Unbalance i defined a the ditibution of the cente of ma of the oto co ection time thei ditance fom the nominal otation axi. Flexual 543

2 544 PROCEEDINGS OF ISMA2006 citical peed ae often diffeent fom the feuencie of tuctual bending mode due to o-called gyocopic effect, which couple the thee-dimenional otational motion of oto co-ection). Fo a flexible oto with attached piezoceamic, the amplitude of bending vibation at a citical peed depend mainly on: the amount of unbalance exciting the epective bending mode, the amount of mechanical diipation in beaing and uppot, and the bending moment induced by active contol of oto-fixed piezoceamic. Active contol of oto-fixed piezoceamic o a to ealize vitual tato damping, modify the oto unbalance ditibution, o both, may effectively educe unbalance induced vibation at citical peed. In addition to foced vibation due to unbalance, flexible oto may alo exhibit intable vibation. (Diipative mechanim in elatic media which ae otating with epect to an inetial fame have nondiipative ma acceleation effect which can lead to intability. Such effect often limit the opeating peed of otating machiney.) Mounting piezoelectic mateial on flexible oto fo the pupoe of active vibation contol may give ie to the following detabilizing mechanim: mechanical diipation (hyteei) in piezoelectic mateial, electic diipation (eitive lo) of cuent induced in the piezoceamic, and bending moment induced by feedback contol popotional to oto tain/diplacement ate. In ode to avoid intable vibation at peed exceeding flexual citical peed, the magnitude of thee mechanim hould be detemined and be limited by caeful deign if neceay. 2 Finite element model 2.1 Two oto cae Two diffeent oto ytem ae conideed (Figue 1, 2), the main popetie of which ae ummed up in Table 1. Roto 1 i a hollow aluminium haft of length one mete which i connected by flexible coupling to hot haft that otate in ball beaing. Thi ytem wa developed a a down-caled model of a compoite helicopte tail dive haft and i ued alo in expeiment. The flexible coupling at the end of thi oto lead to low feuency bending mode with vitually no beaing motion and hence vey little damping, which i patly compenated by tiffne eduction and damping augmentation of the beaing uppot. The haft i euipped with piezoceamic heet and tain eno. Roto 2 i uppoted by ball beaing at it end and contain two heavy dik connected by a hollow haft. The model of thi oto i ued to invetigate heavy oto ytem with inetia concentated at a few poition. The ytem contain oto-fixed piezoceamic heet and ditance eno. The low feuency bending mode of thi oto ae lightly damped becaue the beaing ae aumed to povide no otational tiffne and damping. a) b) Figue 1. Co-ection of finite element model (axe not to cale). a) Roto 1. b) Roto 2.

3 ACTIVE VIBRATION CONTROL AND SMART STRUCTURES 545 Main popetie oto ytem Roto 1 Roto 2 Roto: mateial type, tiffne, lo facto: Alum., , Steel, Roto: total length (m), total ma (kg): 1.052, , 42.6 Additional inetia: tan.(kg); ot.(kg.m 2 ), {node}: - 20; 0.15, {3, 8} Piezoelectic: mateial, tiffne (N/m), lo facto: PZT5H, , PZT5H, , Piezoelectic: thickne (m), width (m), ma (kg): , , , , Piezoelectic: total cap.(f), piezo voltage cont. d 31 : , , Piezoelectic actuato placement {node1,node2}: {7,8},{11,13},{16,17} {5,6} Seno placement {node()}: {9,10}, {14,15} {5} Beaing tiffne: tan. (N/m), ot. (N/m.ad): , , 0 Beaing damping: tan. (N/m.), ot. (N/m..ad): , , 0 Fit thee bending mode: feuencie (Hz): 26, 107, , 254, 461 Table 1. Main popetie of the conideed oto ytem. a) b) Figue 2. Fit thee mode hape at zeo peed. a) Roto 1. b) Roto Finite element model code Fo analyi, a oto finite element code i implemented in Matlab. Fit, a et of N n node i defined at the oto cente line o z-axi. Only tanvee bending and tanlation ae conideed, hence each node i aigned only fou degee of feedom, two tanlational (u x and u y ) and two otational (φ xz and φ yz ). The degee of feedom ae eodeed in a vecto x a 2N n complex pai: x 2n 1 =u x,n +iu y,n, x 2n =φ xz,n iφ yz,n. It i aumed that the oto peed ω (angula velocity dφ xy /dt) i pecibed and vaie only lowly. The oto and tato popetie ae aumed to be iotopic with epect to the otation axi, uch that the ytem popetie can be pecified fo both the x- and y-diection uing eal [2N n x2n n ] matice. Stato tiffne and damping matice K and C ae obtained by pecifying tiffne and damping coefficient at the beaing node. Ma and tiffne matice M and K fo the oto including attached mateial ae computed uing Timohenko beam element code (ee [1]). To take into account additional oto-fixed component o flexible coupling element, inetia and tiffne coefficient ae added to M and K whee neceay. A gyocopic matix G i computed fom the nodal and beam element otational inetia. A hyteetic damping matix H of the oto i computed by multiplying the beam element tiffne matice by thei epective mateial lo facto η. In addition, a oto vicou damping matix C i defined fo completene. Fom the oto ma ditibution, a nodal ma load vecto m i computed, which i multiplied by the gavitational acceleation g tanvee to the oto length axi in ode to obtain the oto ma loading. A ditibution of nodal eccenticitie of the cente of ma of the oto co-ection i aumed in ode to obtain a nodal unbalance vecto e. Thi vecto i multiplied by the uae of the angula velocity to obtain the eulting unbalance excitation. Roto-fixed tain gauge ae included in the model by auming that thee eno meaue uface tain which ae linealy dependent on the oto co-ection otation at the z-poition of the eno end. Placing fou eual tain eno at element n and connecting thee to electode pai k give ie to a eno matix S ([N x2n n ]) with nonzeo coefficient S k,2n = S k, 2(n+1) 0. A ditance eno on the tato at node n give ie to a coefficient S k,2n 1 0.

4 546 PROCEEDINGS OF ISMA2006 Roto-fixed piezoceamic heet ae included in the model by auming that pai of oppoitely poled heet ae mounted at oppoite ide of the oto, uch that electically chaging a pai of heet effectively induce bending moment on the oto at the heet end. Placing fou eual heet at fou ide of the oto at element n (ee Figue 3) and connecting thee to electode pai k give ie to an actuation matix Q [2N n xn ] with nonzeo coefficient Q 2n,k= Q 2(n+1),k. Thee coefficient ae function of the heet popetie (width w, length l, thickne t, aveage ditance fom otation axi, Young modulu E 11, piezoelectic voltage contant d 31 and dielectic contant e 33 ) and ae given by: Q D 33 2w nnd 31,n E 11,n 2w nlne n = 1..N n n 2 n,k = Q2(n 1),k = C v kk = with Ckk n= k Nn t = N n 1 + (1) The vecto of actuato chage i denoted = x +i y (coeponding to actuation voltage v=v x +iv y ). Matix Q i multiplied by the actuato chage in ode to obtain the actuation foce vecto. The capacitance of et of paallel connected actuato heet give ie to a diagonal capacitance matix C =diag(c k). Each actuato et i aumed to be connected to a eie cicuit of a eitance R k and an inductance L k. Thi give ie to electic ytem matice R =diag(r k) and L =diag(l k) ([N v xn ]). Fo the oto conideed in thi document, all actuato heet at two oppoite ide of the oto ae connected in paallel, hence only one pai of electode i peent (N v =1), and v ae ingle complex numbe and C, R and L ae ingle cala. (Note that the model doe not contain the 'dielectic tiffening' effect which aie becaue tain in piezoceamic give ie to dielectic diplacement which on thei tun give ie to oppoing tain, becaue it influence on the total oto tiffne i conideed negligible). φ φ y xz,m a) -yz,m b) w y u h c) φ x,m y,m xz,n z φ x -yz,n u y,n m x,n x n y x : poling diection Figue 3. Piezoceamic actuato: a) degee of feedom, b) geomety and c) wiing in y-plane. v y 2.3 Euation of motion The following tanfomation elate vectoial uantitie in the tationay ( ) and otating ( ) fame: x iω t iω t 2 iω t = x e x& = ( x& + iω x ) e && x = (&& x + 2iω x& ω x ) e (2) The euation of motion of the oto ytem in the tationay (inetial) efeence fame ae given by ([1]): 2 iω t iω t & x + ( C + C iω G ) x& + ( K + K ± i H iω C ) x = eω e + m g Q e (3) M + The coect ign in font of the hyteetic damping matix in Euation 3 can be detemined only if the oto exhibit cicula motion of which the diection in the otating fame i known. Euation 3 can be tanfomed to the otating (non-inetial) efeence fame to yield: 2 2 iωt M & x + ( C + C + i ω(2m G )) x& + ( K + K ± ih + iωc ω ( M G )) x = eω + mge + Q (4) Note that, if een fom the otating fame, the unbalance excitation i contant in diection and the piezoelectic foce ae contant if the actuato chage ae contant, wheea gavity lead to an excitation which otate backwad. The oto dynamic model i extended with a chage balance euation which decibe the dynamic of the piezoceamic actuato et with attached eie eitance and eie inductance: 1 L & + R & + C Q x = v (5) T

5 ACTIVE VIBRATION CONTROL AND SMART STRUCTURES Roto dynamic tability analyi 3.1 Influence of mateial damping The dynamic decibed by the homogeneou Euation 3, with C =[0], i expeed in the tate pace a: ± x& z& = A z z = A = M ( C iω G ) M ( K K i H ) (6) x I 0 The 4N n eigenvalue λ n of matix A can be detemined fo both ign in font of H. The ign of the imaginay pat I(λ n ) i eual to the diection of cicula bending motion in the tationay fame (fowad o backwad, whee the pin peed ω i alway fowad). Fo each one of the eigenvalue, the coect ign in font of H i theefoe ign(i(λ n ) ω), the diection of cicula bending motion in the otating fame. Uing thi ule, the coect eigenvalue can be detemined fo any value of the peed ω ([4]). Fo the two conideed oto ytem, the feuencie of fowad and backwad bending vibation and thei epective decay ate a function of the pin peed ae hown in Figue 3 in o-called Campbell diagam. Note that the feuencie ae nealy contant function of the peed fo Roto 1, wheea they change conideably fo Roto 2 due it lage otational inetia. The citical peed can be ead fom the coing of the feuency tajectoie I(λ n ) with the diagonal line λ=ω. At thee peed, the coeponding decay ate can be een to change abuptly. Thi i due to the defomation of the oto changing diection, uch that the effect of oto hyteetic damping at one become detabilizing. Although both oto ae found to be table in the elected peed ange, it can be een that the fit fowad mode of Roto 1 ha only vey little damping at peed above the fit citical peed. To uantify the elative influence of the hyteetic damping in the piezoceamic attached to both oto ytem, the change in decay ate R(λ) at the citical peed i compaed with cae whee the lo facto of the piezoceamic i et to zeo. It i found that the hyteei in the piezoceamic aie the oto hyteetic damping at the fit two citical peed by 115% and 135% pecent fo Roto 1 and by 3.7% and 1.3% fo Roto 2, epectively. The lage detabilizing effect of hyteei in the piezoceamic in cae of Roto 1 i imply due to the elatively lage amount of piezoceamic mateial which i added to thi oto. Fo Roto 2, the change in decay ate due to added piezoceamic mateial i negligible. a) b) Figue 4. Campbell diagam howing influence of mateial damping. a) Roto 1. b) Roto 2.

6 548 PROCEEDINGS OF ISMA Influence of electic diipation Connecting eitive and/o inductive element in eie with the electode of piezoceamic lead to cuent diipation upon actuato taining. Thi diipation can be maximized fo a tuctual eonance by electing the eonance feuency of the connected electic cicuit eual to that of the tuctue [4]. Howeve, vey lage inducto ae euied to ealize low eonance feuencie in the cae of lowcapacitance actuato. In the following analyi, it i theefoe aumed that only eito ae connected to the actuato electode (L =[0]). The euation of motion of the oto with eitive damping ae obtained by tanfoming euation 4 to the tationay fame uing fictitiou chage tate, a follow: iω t iω t 1 1 T & & & (7) = e = ( iω )e = (iω ( R C The dynamic matix A including otating eitive damping ead: 1 x& A 11 A12 M Q z& = A z z = x A = A 21 A 22 0 (8) R Q T iω ( R C ) The lowe ight cell of the dynamic matix indicate that the tabilizing and detabilizing effect of electic cuent diipation depend on the feuencie (R C ) 1 and on the oto peed. Fo tuctue, maximum damping i obtained by electing eitance R kk=(1 k 2 31) 0.5 /(ω n C kk), with ω n the feuency of the bending mode to be damped and k 31 the electomechanical coupling contant of the piezoceamic ([4]). Fo the oto ytem unde conideation, with k 31 =0.38, the eitance fo maximum damping of the fit bending mode at tandtill ae found to be 42kΩ and 2.8kΩ, epectively. The eulting eitive diipation ha almot no effect in the cae of Roto 2. In contat, it can detabilize the fit fowad bending mode of Roto 1 at peed exceeding the fit citical peed (26 p (evolution pe econd)). Figue 5 how the decay ate fo the fit two bending mode of Roto 1 fo fou value of R, in the abence of hyteetic damping (H =[0]). The following i noted: 1) The econd bending mode how no change in decay ate becaue it i othonomal to the actuato ditibution. 2) Simila to vicou oto damping, eitive damping ha no effect on the decay ate of the fit bending mode at a peed eual to the fit citical peed. 3) Fo R =42kΩ, eitive damping peak at tandtill and at a peed twice the fit citical peed. 4) Reitive damping educe to zeo fo eitance appoaching zeo o infinity. a) b) ) ) + R Q x c) d) Figue 5. Influence of eitive diipation on decay ate fo Roto 1 (no hyteetic damping). a) R =4.2kΩ. b) R =42kΩ. c) R =420kΩ. d) R =4200kΩ.

7 ACTIVE VIBRATION CONTROL AND SMART STRUCTURES Influence of active velocity feedback The following tate pace model i ued fo analyi of active feedback contol: z& = A z B = 1 v i M Q e ω t + B 0 0 T = K y i S e Y = 0 ω t y = Y z iω S e i S e with A the oto dynamic matix a given in Euation 8, B the piezoelectic input gain matix, Y the output gain matix elating oto tain meauement y to tationay tate z (o elating tationay meauement to thei tanfomed value y in the otating fame) and K the feedback gain matix. It follow diectly fo the cloed loop ytem matix A k : iω t k k k z& = A z = ( A + A ) z with A = B K Y (10) A common method to educe vibation of tuctue with attached piezoceamic i to impoe actuato chage popotional to tain o diplacement ate. Fo the conideed oto ytem with ingle pai of actuato and eno, a ingle gain k can be ued to define the feedback matix K fo active oto damping. The feedback ytem matix A k i expeed in tem of an 'active damping matix' C k : A k = B K Y 1 = M C 0 k M 1 iω C 0 k with ω t K C k 0 0 [ k 0 ] = = Q k Note that thi gain matix ha the ame detabilizing effect a oto vicou damping. Fo tabilization, the gain matix hould intead be choen o a to obtain vitual vicou tato damping a follow: A k = B K Y 1 = M C 0 k 0 0 with K = C k [ k iω k ] = Q k Figue 6 how the decay ate fo the fit thee mode of Roto 1 and Roto 2 fo Euation 11 and 12, whee modeate feedback gain ae ued. The model alo contain hyteetic oto damping and eie eitance R =42kΩ and R =2.8kΩ. Figue 6a and 6b how that active damping in the otating fame (Euation 11) ha a lage detabilizing effect on mot of the fowad mode of both oto ytem. Fom figue 6c and 6d, it can be concluded that vitual tato damping (Euation 12) ha the deied tabilizing effect on the fit fowad and backwad mode of both oto ytem. (It alo ha a detabilizing influence on the thid fowad and backwad mode of Roto 1 due to non-collocation of it eno and actuato). a) b) S S (9) (11) (12) c) d) Figue 6. Decay ate. Euation 11: a) Roto 1, b) Roto 2. Euation 12: c) Roto 1, d) Roto 2.

8 550 PROCEEDINGS OF ISMA Analyi of unbalance epone eduction 4.1 Active damping uing low feuency feedback contol The main tability iue having been analyzed in the peviou ection, it i focued in thi ection on uppeion of unbalance induced vibation at citical peed. The teady tate epone at peed ω can be olved fom Euation 4 if the oto damping, ma load and tate deivative ae et to zeo: 2 2 ( K + K + i ω C ω ( M G )) x = eω + Q (13) The gyocopic effect may inceae the tiffne o a to avoid the actual occuence of citical peed ([1]). At low peed, howeve, the gyocopic effect often ha only a mall influence and the epone at a low citical peed ω=ω c i detemined mainly by the tato damping and actuation foce: K + K 2 2 ω ( M G ) i ω C x eω + Q (14) c The inability of paive tato damping C to limit the unbalance epone of (flexible) oto at citical peed i the main motivation fo the active olution conideed hee. On thei tun, due to thei limited elatic tiffne, oto-fixed piezoceamic actuato can inceae the dynamic tiffne (the matix at the left-hand ide of Euation 13) only ignificantly unde eonance condition. The dynamic tiffne i theefoe uually dominated by inetia at high peed, by tiffne at low peed and by active athe than paive damping at citical peed. Analog to the effect of tationay damping, the effective component of vitual active tato damping (K =[k, iωk ]) in uppeing the unbalance epone at citical peed i the othogonal tiffne ealized by the imaginay pat of the feedback gain matix ([0, iω k ]). If the ate of change in the oto peed i mall, uch a i the cae with oto ytem which acceleate not too fat fom tandtill to thei nominal peed and back, the ate of change in the epone to unbalance i mall a well. Since the contol ytem opeate on the oto-fixed uantitie y and, low pa filte with cone feuencie ignificantly lowe than the fit citical peed can be placed at the input (o altenatively output) of the contol ytem. The advantage of uch an aangement i that the contolle can no moe detabilize backwad mode o non-collocated high feuency mode. Fo analyi of the contolled ytem including low pa filte, the tate pace euation i extended with the complex tate j (and it euivalent tationay tate j ) to denote the low-pa filteed meauement time the feedback matix, whee the pai of low-pa filte i paametized by eitance R j and capacitance C j : z& j j = A z j & j j j = ( iω ( R C ) z j x& = x j 1 ) j A j j R = k 1 k A A R S( x& + iωx ) j 1 S c iω k A 12 A22 j 1 R S c 1 M Q 0 j j iω ( R C ) Figue 7 how the decay ate fo Roto 1 and 2 fo a low pa filte time contant (R j C j ) 1 =1. Fo Roto 1, the decay ate of the fit bending mode i ignificantly aied at the citical peed, while the detabilizing influence of active vitual tato damping on the thid fowad and backwad mode ha become negligible in the conideed peed ange. Fo Roto 2, the decay ate of the fit and econd fowad mode ae ignificantly aied at the epective citical peed a well. The epone to unbalance at citical peed i lagely inveely popotional to the decay ate. Figue 8 how the magnitude of the diplacement at node 10 and 5 in epone to the aumed unbalance ditibution fo Roto 1 and 2, epectively (ee alo Table 2). Fo the flexible Roto 1, a poblematic diplacement in the ode of eveal mm i educed to the ode of two hunded µm. Fo the heavy Roto 2, a diplacement in the ode of hunded µm i educed to the ode of eveal µm, which hould lead to acceptable beaing load and tato vibation. (In ode to uppe the econd fowad mode moe effectively, the feedback gain hould be made a function of the peed in the cae of Roto 2, wheea the actuato would have to be eaanged o wied diffeently in the cae of Roto 1). 1 (15)

9 ACTIVE VIBRATION CONTROL AND SMART STRUCTURES 551 a) b) Figue 7. Decay ate fo fit mode fo Euation 15 with (R j C j ) 1 =1, modeate gain. No oto hyteei no cuent diipation. a) Roto 1, b) Roto 2. a) b) Figue 8. Repone to unbalance. No feedback:. Feedback:. a) Roto 1, b) Roto 2. Roto ytem: Roto 1 Roto 2 Eccenticity ditibution (µm) {nodal diection}: 100 {0,0,0,1,1,1,1,i,i,1,1,1, 1,1, i, i,1,1,1,0,0,0,0} 2 {0,0,1,0,0,1,0,i,0,0} Mode: Modal unbalance e (10-8 ): i i 18+69i 13 3i Actuato modal foce (10-4 /V): Voltage compenating unbalance at citical peed (V): Table 2. Modal unbalance and actuation voltage euied fo unbalance compenation. 4.2 Active balancing uing feedfowad contol The compenation of unbalance in off-eonance condition i nomally pohibited by the diffeence in the ditibution of unbalance and the diplacement that can be ealized by actuation. (An exception being the cae whee unbalance can be compenated by contolling vey little degee of feedom, i.e. the eccenticity and oientation of a ingle heavy dik on a flexible haft). Nea citical peed, howeve, the oto epone to unbalance i dominated by a ingle bending mode, uch that a modal unbalance excitation e can be compenated by a ingle modal actuation foce (whee both e and ae complex numbe in the notation employed). Table 2 how fo Roto 1 and 2 the ditibution of nodal coection eccenticity, and, fo the fit thee mode, the modal unbalance e and modal actuation foce pe volt (whee the mode hape of the undamped ytem at zeo peed ae employed fo computation of thee modal paamete). Note that the eccenticitie of the light-weight Roto 1 ae athe lage along the haft length, while the much malle eccenticitie of Roto 2 ae located mainly at the cente of the dik.

10 552 PROCEEDINGS OF ISMA2006 The actuation voltage euied fo compenation of modal unbalance (ee Table 2) ae found to be vey eaonable, a voltage in the ±200V ange can afely be ued with piezoceamic heet of 500µm thickne and piezoceamic fibe actuato of 200µm thickne. The magnitude of modal unbalance can uually be conideed contant duing a un. An effective appoach to the uppeion of unbalance induced vibation at citical peed i theefoe to etimate unbalance at peed not too fa no too nea to citical peed, in ode to ubeuently chage the actuato to contant voltage o a to compenate the unbalance excitation of the epective mode. Algoithm fo thi appoach wee developed and analyzed uing time domain imulation and a educed modal model of Roto 1. The appoach and imulation eult ae not futhe detailed in thi document. Intead, an example of the eult i given in ection 5.2 fo the coeponding expeiment. 4.3 Active balancing uing elf-poweed actuato An inteeting uetion i whethe the mall contant actuation chage which ae euied fo active balancing could be geneated by toing chage induced by hamonic taining of the piezoceamic due to the oto deflection unde it own weight. The extaction of electic powe fom cyclically tained piezoelectic element i known a powe haveting (ee [5]). The maximum amount of powe P which can be geneated fom bending of the oto unde it own weight i appoximately eual to: 1 2 g E P = ω k (16) with E g i the maximum tain enegy in the piezoceamic mateial due to the gavity load and k 31 the electomechanical coupling contant of the piezoceamic. The tain enegy E g i eaily computed uing the finite element model. It i found that at a peed of 60 p, 0.56mW and 0.96mW of electic powe can be extacted fom Roto 1 and Roto 2, epectively. The magnitude of thee value indicate that the deflection of flexible oto unde gavity i not an ideal tain ouce fo powe haveting, the feuency and magnitude of peiodic taining uually being athe mall. Yet, at the conideed peed of 60 p, in the ideal cae of powe haveting in the abence of eitive loe, the actuato on both haft could in pinciple all be chaged to 200V within 9.6 and 21.5 econd fo Roto 1 and Roto 2, epectively. Hence, active balancing uing elf-poweing actuato might be feaible in cetain cae. Fo example, the eno and contol ytem could be placed on the tato with an extenal powe ouce, while the chage tate of the oto-fixed actuato could be optimized fo minimum unbalance by egulating it though low powe optical o electomagnetic intefeence mechanim. Expeimental wok on powe haveting and high voltage geneation i decibed in Section 5.3. A a lat point, it hould be noted that powe haveting ha a a dawback a detabilizing effect on fowad mode which i at mot eual to that of eitive diipation optimized fo the ame peed (ee ection 3.2). Powe haveting i theefoe not adviable in the cae of lightly damped oto with a elatively lage amount of attached piezoelectic mateial, a it may ignificantly educe the maximum peed of table opeation. 5 Expeiment 5.1 Expeimental etup The expeimental etup which i ued to invetigate active balancing and powe haveting fo Roto 1 i hown in Figue 9. The hollow aluminium haft i diven by an electic moto and dive a liping aembly. The piezoelectic actuato and tain gauge bidge on the haft ae connected by mean of the liping aembly to high voltage amplifie and tain gauge amplifie, epectively, which on thei tun ae connected to a dspace contol ytem. Fo efeence meauement and calibation of the tain eno, lae ditance eno ae placed at midhaft to meaue the oto diplacement. The moto peed i contolled fom dspace uch that any peed pofile can be geneated.

11 ACTIVE VIBRATION CONTROL AND SMART STRUCTURES 553 (2) (1) (4) (3) Figue 9. Expeimental etup fo Roto 1. (1) Roto. (2) Moto. (3) Sliping aembly. (4) Lae ditance eno. The fit natual feuency of the oto i 24.6Hz, which i nea to the pedicted value of 26Hz. The midhaft diplacement intoduced by the actuato i 1.15µm/V, whee the finite element model pedict 1.13µm/V. In the expeiment decibed in thi ection, the oto i lowly acceleated fom tandtill to a peed of 50 p in 30 econd. At the fit citical peed, the diplacement epone to unbalance at midhaft exceed 10mm and i theefoe limited by a catche beaing to 4mm (ee Figue 10a). (It i noted that catche beaing ae ued alo with the full cale helicopte tail dive haft. They can give ie to cutting damage of the thin-walled dive haft and hence to dangeou ituation, epecially if they ae impopely mounted. An active but afe olution could avoid the haft to hit the catche beaing at all.) 5.2 Suppeion of the unbalance epone A ingle expeiment with the uppeion of unbalance induced vibation at the fit citical peed i decibed in thi ection. The appoach ued i to: a) etimate the modal unbalance e fom the tain meauement uing an invee modal model while a cheduled gain δ e (ω) i nonzeo, b) apply feedback contol accoding to Euation 15 (low-feuency vitual tato damping) while a cheduled gain δ d (ω) i nonzeo and c) apply feedfowad contol (active modal balancing) on the bai of etimate e while a cheduled gain δ b (ω) i nonzeo. The eult ae hown in Figue 10b), with the midhaft diplacement at the bottom, the applied voltage at the top and the gain a a function of peed in the cente of the figue. The vibation epone at the citical peed i educed fom fa moe than 3800µm to le than 120µm: a eduction of 97%. Thi magnitude could not have been educed much futhe, becaue the midhaft diplacement coeponding to the etimated modally balanced tate wa 105µm. 5.3 Powe haveting and elf-chaging of the actuato To invetigate powe haveting, the cicuit hown in Figue 11a) i ued. While the haft i otating at a peed of 60 p, the voltage ove the toage capacito in thi cicuit i meaued fo eitance in a ange of 30kΩ to 80kΩ. Figue 11b) how that a eitance of 70kΩ maximize the diipated powe at 0.48mW. Thi coepond uite accuately to the 0.56mW computed in Section 4.3, conideing that no voltage dop ove the diode wa taken into account in the pediction. Figue 10c) how the unbalance epone in the peence of electic diipation. Note that the haft i only maginally table in the conideed peed ange. Slightly educing the tato damping by modifying the beaing uppot actually led to intability at peed exceeding 40 p, a ik which wa noted at the end of Section 4.3 Fo the ealization of elf-poweing active balancing ytem, voltage multiplie cicuit ae conideed to be pomiing component. The cicuit in Figue 11c i implemented uing ceamic capacito of 100nF. At a peed of 60 p, thi cicuit chage the actuato to ±90V within 20 econd. Duing an acceleation fom tandtill to 30 p within 15 econd, the actuato voltage each only ±40V. Futhe eeach i conducted to optimize thee cicuit and to combine them with eno olution.

12 554 PROCEEDINGS OF ISMA2006 a) b) c) Figue 10. Unbalance epone fo a) uncontolled oto, b) contolled oto and c) uncontolled oto with electic cuent diipation. a) b) c) Figue 11. a) Cicuit fo powe diipation meauement. b) Voltage and diipated powe a function of eitance. c) Voltage mulitplie cicuit fo elf-chaging of actuato.

13 ACTIVE VIBRATION CONTROL AND SMART STRUCTURES Concluion Two oto ytem with oto-fixed piezoceamic actuato ae analyzed, the fit oto being light-weight and flexible, the econd oto being heavy and moe tiff in bending. Numeical tability analyi i pefomed to detemine the elative impotance of hyteetic damping in the piezoceamic and of eitive diipation of electic chage induced in the piezoceamic. It i found that thee diipative mechanim have a ignificant effect on the fit oto ytem, while they have a negligible effect on the econd oto ytem. Next, tability analyi i pefomed fo the cae of active feedback popotional to oto tain o diplacement ate expeed in eithe the otating o tationay fame. It i found that active oto damping i detabilizing at peed highe than the fit citical peed, while active vitual tato damping i alway tabilizing fo ytem with collocated actuato and eno. Uing computation and expeiment, a combination of low-feuency feedback contol (vitual tato damping) and feedfowad contol (modal balancing) i demontated to be vey effective fo the uppeion o avoidance of unbalance induced vibation. Finally, computation and expeiment ae ued to detemine the amount of electic powe which can be geneated fom cyclic taining of the piezoceamic duing otation of the oto. It i concluded that elf-poweing ytem fo active balancing might be feaible in cetain cae. Refeence [1] G. Genta, Dynamic of Rotating Sytem, Politecnico di Toino, Spinge Science+Buine Media, Inc., (2005) ISBN [2] H.-G. Hot, H. P. Wölfel, Active vibation contol of a high-peed oto uing PZT patche on the haft uface, Jounal of Intelligent Mateial Sytem and Stuctue (2004), Vol. 15, pp [3] H. Kunze, M. Riedel, K. Schmidt, E. Bianchini, Vibation eduction on automotive haft uing piezoceamic, Poceeding of SPIE (2003), [4] J.T. Sawicki, G. Genta, Modal uncoupling of damped gyocopic ytem, Jounal of Sound and Vibation (2001), Vol. 244(3), pp [5] H. A. Sodano, D. J. Inman, G. Pak, A eview of powe haveting fom vibation uing piezoelectic mateial, The Shock and Vibation Diget (2004), Vol. 36(3):

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