Design optimization of a damped hybrid vibration absorber

Transcription

1 This is the Pe-Published Vesion. Design otimiztion of dmed hybid vibtion bsobe Y. L. CHEUNG, W. O. WONG, L. CHENG Detment of Mechnicl Engineeing, The Hong Kong Polytechnic Univesity, Hung Hom, Hong Kong SAR, Chin Abstct In this ticle, the H otimiztion design of hybid vibtion bsobe (HVA), including both ssive nd ctive elements, fo the minimiztion of the esonnt vibtion mlitude of single degee-of-feedom (SDO) vibting stuctue is deived by using the fied-oints theoy. The otimum tuning metes e the feedbck gin, the tuning fequency, dming nd mss tios of the bsobe. The effects of these metes on the vibtion eduction of the imy stuctue e eveled bsed on the nlyticl model. Design metes of both ssive nd ctive elements of the HVA e otimized fo the minimiztion of the esonnt vibtion mlitude of the imy system. One of the inheent limittions of the tditionl ssive vibtion bsobe is tht its vibtion bsotion is low if the mss tio between the bsobe mss nd the mss of the imy stuctue is low. The oosed HVA ovecomes this limittion nd ovides vey good vibtion eduction efomnce even t low mss tio. The oosed otimized HVA is comed to ecently ublished HVA designed fo simil oose nd it shows tht the esent design equies less enegy fo the ctive element of the HVA thn the comed design. Keywods: Vibtion bsobe, tuned mss dme, hybid contol.

2 . Intoduction The tditionl ssive vibtion bsobe (PVA) is n uiliy mss-sing-dme system which, when coectly tuned nd ttched to vibting system subject to hmonic ecittion, cuses to cese the stedy-stte motion t the oint to which it is ttched. The fist esech conducted t the beginning of the twentieth centuy consideed n undmed PVA tuned to the fequency of the distubing foce []. Such n bsobe is now-bnd device s it is unble to eliminte stuctul vibtion fte chnge in the distubing fequency. inding the otimum metes of viscous fiction PVA in SDO system dew the ttention of mny schols. One of the otimiztion methods is H otimiztion. Omondoyd nd Den Htog [] oosed the otimiztion incile of the dmed PVA in tems of minimizing the mimum mlitude esonse of the imy system, which is clled H otimiztion of PVA. ollowing this incile, Hhnkmm [3] deived the eessions fo the otimum tuning of PVA used in the SDO system. Block [] develoed the oimted otimum dming. The otimum design method of the dynmic vibtion bsobe is clled ied-oints theoy, which ws well documented in the tetbook by Den Htog [5]. The ect solution of the H otimiztion of PVA ttched to undmed imy system ws deived by Nishih nd Asmi [6]. Howeve, it ws found tht the minimum esonnt vibtion mlitude of the imy system ttched with the PVA deends on the mss tio [7]. When the mss tio is fied, the efomnce of the PVA is lso limited. In ode to imove the vibtion suession efomnce of the PVA, some eseches incoote n ctive ctuto to PVA to fom hybid vibtion bsobe (HVA). Vious methods wee oosed to contol the ctive element of the HVA including neul netwok [8], delyed esonto [9], line mti inequlities [], modl feedbck contol [-5] nd

3 closed-loo oles by modl feedbck [6-7]. Howeve, the contol methods of HVA found in litetue e vey comlicted nd most of the esech eoted in litetue focused on the imovement of the ctive contolle design the thn the otimiztion of both the ctive nd ssive comonents of the HVA. In this ticle, H otiml design of dmed hybid vibtion bsobe is oosed fo the minimiztion of the esonnt vibtion mlitude of SDO system. Both the ctive nd ssive elements e otimized. The oosed otimized tuning of the HVA cn lso minimize the ctution foce of the ctuto. Comisons with the esult of H otiml PD contol of HVA by Chttejee [8] show much bette esults of ou otimum design. inlly, we ly the oosed otimized HVA to bem stuctue nd come its vibtion suession efomnce to tht of the otimized PVA [8,9] nd lso to the otiml PD contol of HVA by Chttejee [8].. Theoy A HVA couled with imy system is shown s ig., whee, M nd K denote, esectively, dislcement, mss nd stiffness of the imy system; nd, m nd k e those of the bsobe. c is the dming coefficient of the bsobe. The equtions of motion of the imy mss M nd the bsobe mss m my be witten s M K k c m k c f f () whee is distubnce nd f is the ctive foce lied by the ctuto s illustted in ig.. Tking Llce tnsfomtion of Eq. (), the tnsfe function of the imy 3

4 mss M my be witten s X H () K whee s, n m, M K n M, k m,, n c nd mk. K The tnsfe function of the bsobe mss m my be witten s X K /. (b) Since f K, the tnsfe function of the ctive foce in the bsobe my be witten s H (c) whee is the Llce tnsfomtion of f. Accoding to Eqs. () (b) nd (c), the chcteistic eqution of the combined system my be witten s 3 (3) nd,,, R. To ly the Routh s stbility citeion, the y of coefficient my be witten s

5 3 () The system is stble if the el ts of ll oles e negtive. Since ll coefficients of the y in Eq. () e ositive if, the contol system is stble ccoding to the Routh s stbility citeion. Tht mens the oosed HVA contol system is licble in incile if. The fequency esonse function of mss M cn be obtined by elcing in Eq. () by j whee nd j = -. The fequency esonse function of mss M my be witten s n H j j. (5) The fequency esonse mlitude of the imy mss M, H, is clculted ccoding to Eq. (5) with thee diffeent dming tios nd the esults e lotted in ig. fo illusttion. It cn be seen in ig. tht the fequency esonse mlitudes of mss M t nd b e indeendent of the dming tio nd these two oints e clled fied oints. Consideing H H, we my wite. (6) The two oots of Eq. (6) e nd b whee b. The mlitudes of the fequency esonse t nd b my be witten s 5

6 H, nd (7) H b (7b) b At ny dming tio, the fequency must ss though these two fied oints. So the otimum condition should obey the following eqution: b m H, H, H min m H, H b (8), Accoding to the fied-oints theoy [] oiginlly develoed fo the design of the ssive dynmic vibtion bsobe, the otimum condition of the dynmic vibtion bsobe cn be chieved by djusting the fequency tio such tht the vibtion mlitude esonses t nd b e the sme, nd then finding the dming so tht the two fied oints become the mimum oints on the esonse cuves. A simil ocedue is lied to the otimiztion of the oosed HVA, i.e. H H. Using Eqs. (7) nd (7b) nd b noting tht H nd H e in oosite hses, we my wite Solving Eq. (6) fo nd b b b nd substituting them into Eq. (9), the tuning fequency tio leding to the sme esonse mlitude t the fied oints cn be found nd witten s (9) ot () om Eq. (), eists if ot, i.e. () Substituting Eq. () into Eq. (6), the esulting eqution my be witten s 6

7 7 () The oots of Eq. () my be witten s (3) nd b () The esonse mlitude t the fied oints H nd b H e clculted using Eqs. (7), (7b), (3) nd () with =. nd =.5 nd lotted in ig. 3fo illusttion. It cn be seen tht, when the ecittion fequency inceses, H inceses while b H deceses. The coesonding fequency tio t the intesection oint of the cuves in ig. 3 is the otimum fequency tio of the HVA such tht b H H. Substituting Eq. (3) into Eq. (7) o Eq. () into Eq. (7b), the esonse mlitude t the fied oints my be witten s b H H (5) The otimum dming is the dming vlue which cuses the fied oints to become the eks on the esonse cuve H nd theefoe we my conside b H H. (6) The dming equied leding to mimum vibtion mlitude t nd b my be solved using Eqs. (5), (3) () nd (6) nd witten s

8 , b (7) H would becomes the ek vlue of the esonse function H if nd H would becomes the ek vlue of the esonse function H if b. A b convenient oimte vlue of the otimum dming my be chosen s b ot (8) 8 Using Eqs. (3), () nd (8), the oot locus of the contol system s shown in ig. is clculted with mss tio =. nd feedbck gin vies fom to, nd the esults e lotted in ig. fo illusttion. All fou oles of the contol system hve negtive el ts with one ole oching the oigin when oches the limiting vlue. If the otimum tuning fequency ot nd the otimum dming ot e lied to the oosed HVA, using Eq. (5), the mimum vibtion mlitude of the imy mss M my be witten s X G (9) K m The fequency esonse mlitude of the imy mss M, H, t. 5 with the oosed otimum fequency nd dming tios e clculted ccoding to Eqs. (5), () nd (8) with =,. nd.5 esectively nd the esults e lotted in ig. 5 fo 8

9 illusttion. All the esonse cuves in ig. 5 show the tyicl double eks in the esonse sect of the imy mss M. When the feedbck gin =, the HVA becomes the tditionl ssive vibtion bsobe (PVA) nd the esonnt vibtion mlitude is bout times of the sttic deflection of mss M. When the ctive element is deloyed with =., the esonnt vibtion mlitude dos to bout. times of the sttic deflection of mss M. When the feedbck gin incese to.5, the esonnt vibtion mlitude dos futhe to bout. times of the sttic deflection of mss M. These esults show tht the oosed HVA is vey effective in suessing the esonnt vibtion mlitude of the imy vibting system in comison to the tditionl ssive vibtion bsotion when the mss tio is low such s the cses of using vibtion bsobes to suess oscilltions of tll buildings nd bidges. Since f K, the mimum ctive foce equied fo the HVA my be witten using Eq. (9) s m G G K () When, it cn be shown tht the esonse mlitudes t the fied oint is lwys highe thn tht t fied oint, i.e. H H. H nd H b b e clculted using Eqs. (6), (7) nd (7b) with =.5 nd =. nd they e lotted in ig. 6 fo illusttion. b The fequency of the fied oint cn be found by solving Eq. (6) nd witten s. () 9

10 The esonse mlitude of mss M t fequency my be found by substituting Eq. () into Eq. (5) nd witten s H. () The otimum dming is the dming vlue which cuses the fied oint to become the ek on the esonse cuve H, i.e. H, nd it cn be deived using Eq. (5) nd witten s H S S S S 3 8 (3) whee S, S, S nd S. 3 The coesonding tuning tio cn be deived using Eq. (3) nd witten s H G G G () whee G is the mimum mlitude esonse of H. To illustte the diffeence of H between the cses of using the low feedbck gin

11 nd the high gin, the fequency mlitude esonse H of both cses e clculted ccoding to Eq. (5) nd the coesonding otimum fequency nd dming tios nd the esult e lotted in ig. 7. igue 7 shows double eks in the fequency sectum with low gin with nd single ek with high gin. Since the ctive foce equied in the HVA is ootionl to the gin, it is theefoe ecommend tht low feedbck gin with should be used wheneve ossible. In ctice, the mimum fequency esonse G is often design constint. If is ssumed nd using Eq. (9), the nge of G my be witten s G (5) The coesonding feedbck gin cn be obtined fom Eq. (9) nd witten s G G (6) The otimum tuning fequency nd dming tios of the HVA cn then be detemined using Eqs. () nd (8) esectively. Since the use of multile feedbck signls is common in moden contol theoy [], feedbck signls fom both the imy nd bsobe msses e consideed in the following fo the ctive contol of the HVA nd comed to the oosed method which uses only the feedbck signl fom the imy mss. Assuming the ctive foce of the HVA is function of both the dislcements of imy nd bsobe msses witten s f b. The

12 ctive foce in the HVA my be ewitten s f k (7) e whee k e b nd b. k e my be conside to be n dded stiffness nd to be the contol gin to the HVA. Eq. () my be ewitten s M K k k c e m k k e c (8) The fequency esonse function of the mss M my be witten ccoding to Eq. (5) s H X K j j, (9) nd nd the fequency esonse of the bsobe mss m my be witten s X K / j j (3) whee m, M K n M, k k e, m, n c, mk k e nd n. K Coming Eqs. (5) nd (9), the two equtions e the sme ecet the bsobe s fequency, the dming tio nd the contol gin e diffeent in the two cses. The

13 otimum tuning fequency nd dming tios of the HVA in this cse e still eessed s Eqs. () nd (8) esectively with. K Accoding to Eq. (9), the mimum vibtion mlitude of the imy mss M my be witten s X G (3) K m whee K. Coming Eqs. (9) nd (3), the mimum vibtion mlitude of the imy mss M cnnot be futhe educed by using feedbck signls fom both the imy nd bsobe msses with the contol lw of Eq. (7) fo the ctive contol of the HVA. Since f b, the ctive foce equied in the HVA my be witten s X bx. (3) K K X if b is zeo nd theefoe the sectum of K K will hve two eks of equl height simil to the sectum of X K s shown in ig. 5. If b is not zeo, one ek of the sectum will ise while the othe ek will fll s illustted in ig.3 fo 3

14 X K. To illustte the effect of b on the ctive foce mlitude, the dimensionless ctive foce mlitudes e clculted using Eqs. (9), (3) nd (3) with. nd fou diffeent set of / K, b / K= (., ), (., -), (-.3,.5) nd (.,.) such tht b/ K. in ll fou cses nd the esults e lotted in ig. 8 fo illusttion. Accoding to Eq. (3), the mimum vibtion mlitude of the imy mss M, G cn be found to be.8 fo ll the fou cses being consideed but the mimum ctive foce equied in the fist cse with b = is smlle thn the othe thee cses with b not equl to zeo. This shows tht in the fist cse whee the HVA use only the feedbck signl fom the imy mss equies smlle ctive foces nd hence owe fo otimum efomnce thn the othe cses whee the HVA use both the feedbck signls b fom the imy nd the bsobe msses. 3. Simultion esults nd discussion The oosed HVA is comed to simil design of Chttejee [8] eoted ecently, in which the dislcement of the bsobe mss in HVA without dming ws used s feedbck signl. Chttejee oosed H otimum PD contol fo the minimiztion of esonnt vibtion mlitude of SDO system with the ctive foce of the HVA being f b nd the fequency esonse function of the imy nd bsobe msses my be witten esectively s X K j j (33)

15 X K whee nd K j b n. K The otimum tuning fequency nd dming tios of the bsobe cn be witten esectively s [8] (3), nd (35) b 3 (35b) The esonnt vibtion mlitude of the imy mss my be witten s [8] X G PD (36) K The feedbck gin my be witten using Eq. (36) s m PD PD G (37) G The dimensionless foce functions e defined s PD X j (38) K To come the oosed HVA to the one by Chttejee [8], the mlitude esonse of the oosed HVA,, H is clculted ccoding to Eqs. (5), () nd (8) nd the ot, ot mlitude esonse of the HVA by Chttejee is clculted ccoding to Eqs. (33), (35) nd (35b) nd the esults e lotted in ig. 9. G =. nd =. in both cses.. 5( in 5

16 ef. [8]) in the second cse. In ig. 9, both mlitude esonse cuves hve simil she but slightly diffeent esonnt fequencies. The ctive foce of the oosed HVA is clculted ccoding to Eqs. (c), () nd (8) nd tht of Chttejee [8] ccoding to Eqs. (6), (35), (35b) nd (38) nd the esults e lotted in ig. fo comison. The mimum ctution foce of the PD contol is.6 times of tht of the esent P contol with the otimized ssive dming of the bsobe. As shown in ig., the oosed contol with the oosed otimized metes cn educe the ctution foce while mintining the vibtion suession efomnce. The oosed HVA is tested numeiclly on simly-suoted bem simil to the one studied by Chttejee [8], s shown in ig., with unifomly distibuted foce. The men sque dislcement of the whole bem is evluted. The length of the bem is L m nd HVA is ttched t.5m. The dimension of the coss section is.5m.5m. The mss tio of the HVA is.5. The mteil of the bem is luminium with 3 7kgm nd E 6.9GP. The bem is ssumed to be n Eule-Benoulli bem nd its eqution of motion my be witten s w w A EI tg t h (39) t Hee it hs been ssumed tht the etenlly lied focing function cn be eessed s t g, whee t is function of time nd g is deteministic function of. h is the foce ecited by the HVA. The fequency esonse function of the bem cn be deived s shown in the Aendi A nd witten s 6

17 7 j j, q q q q n b L Lb A P W () whee is the th eigenvecto of the bem, nd b e ouie coefficients s descibed in the Aendi A; A EI n, A EI, n, AL m, nd EIL. Bsed on Eq. (), the men sque motion of the bem my be witten s j j, q q q q n L b L Lb A d P W L () The otimum fequency nd dming of the HVA my be ewitten [] in tem of, i.e. when, ot_hva, nd ()

18 ot_hva b (b) The HVA is tuned fo the dimensionless fequency esonse t G. The feedbck gin is detemined to be.58 while the otimum tuning tio nd the otimum dming tios e detemined using Eqs. () nd (b) s.7569 nd.63 esectively. The oosed otimum HVA is fistly comed to the PVA countet with =. The otimum tuning fequency nd dming tio my be witten esectively s [9] ot_dva, nd (3) 3 ot_dva (3b) 8 Dimensionless men sque dislcement of the bem with ssive vibtion bsobe is clculted with Eq. () when =, ot_dva nd ot_dva using Eqs. (3) nd (3b), esectively, nd the esult is lotted with the cse of the using the oosed HVA in ig.. ig. shows tht the mimum men sque motion of the imy mss using the oosed HVA is 6% lowe thn the one using the otimized PVA. Suession of the men sque motion of the imy mss using the oosed HVA t the highe modes is lso bette thn using the PVA. Secondly the oosed otimum HVA is lso comed to the otimized PD contolled HVA oosed by Chttejee [8]. Simil to Eq. (), the fequency esonse function of the bem using the otimized PD contolled HVA [7] my be deived nd witten s 8

19 9 j, q q q q n b L Lb A P W () whee A EI n, A EI, n, AL m,, EIL nd EIL b n. Simil to Eq. (), the men sque motion ove the whole domin of the bem using the otimized PD contolled HVA [8] my be deived nd witten s j, q q q q n L b L Lb A d P W L (5) The men sque motion esonse of the whole bem is clculted ccoding to Eqs. () nd (5) nd the esults e lotted in ig. 3. The coesonding ctive foce sect e lotted in ig.. Coming the oosed otimum HVA to tht of Chttejee [8]. The ctive foce equied by the oosed otimum HVA is much smlle thn tht equied by the one oosed by Chttejee [8].

20 Thee e some eoted design methods of HVA such s the zeo-ole lcement method [6] which is ble to educe vibtion eks while keeing the bsotion di simultneously in the fequency esonse of the closed-loo imy system. Howeve, it is shown in the Aendi B tht even though the zeo-ole lcement method cn oduce gete vibtion eduction of the vibting stuctue thn the oosed method but the ctive foce equied in tht method is vey much lge thn the oosed method. The oosed design method otimizes the dming effect using the ssive elements nd theefoe the ctive foce in the HVA cn be vey much educed even though the eduction of vibtion mlitude of the imy mss is not s good s the zeo-ole lcement method. The oosed design method would be good otion if the ctive foce comonent in the HVA cnnot be too lge. 8. Conclusion In this e, the H otimiztion design of hybid vibtion bsobe (HVA) fo the minimiztion of the esonnt vibtion mlitude of single degee-of-feedom (SDO) vibting stuctue is deived by using the fied-oints theoy. A genel design fmewok is estblished bsed on n nlyticl model. The otimum tuning metes e the feedbck gin, the tuning fequency tio, the dming tio nd the mss tio of the bsobe. The effects of these metes on the vibtion bsotion of the imy stuctue e systemticlly eveled. Design metes of both ssive nd ctive elements in the HVA e otimized. The inheent limittion of the tditionl ssive vibtion bsobe equiing eltively lge mss tio to chieve tgeted vibtion suession level is byssed using the oosed design, such fcilitting the liction of the technique in lictions involving lge stuctues such s buildings nd bidges. Comed to othe eisting HVA

21 designs fo simil oose, the esently oosed design equies smlle ctive foce nd hence less enegy fo the ctive element.

22 Aendi A Conside the motion of the bem s shown in ig. ecited by unifomly distibuted foce locted between nd L. A dmed hybid vibtion bsobe is ttched t. The length of the bem is L, nd mss e unit length is A with bending stiffness EI. The dded mss nd the stiffness of HVA e m nd k esectively. The boundy conditions my be inned, clmed o fee end. The oblem is descibed by the Benoulli Eule eqution fo smll motions of slende bems nd the following conditions. w w A EI tg t h (A) t c f m k (Ab) h t m (Ac) Hee it hs been ssumed tht the etenlly lied focing function is t is function of time nd g is deteministic function of. t t g, whee h is the foce lied to the bem fom the HVA. f is the ctive foce fom the HVA. The solution to this oblem my be ended in ouie seies witten s [],, t q t w (A) L i whee d L, whee i N (A3) Similly, the stil t of the focing function cn be ended s g (A) And the deivtive of Dic delt functions cn lso be ended s

23 b (A5) whee the ouie coefficients i nd b i e esectively i L g L i d nd b i i (A6) L Hee i deend only on the stil distibution of the focing function g. If the Eqs. (A), (A3), (A), (A5) nd (A6) e substituted into Eq. (A) nd the Llce tnsfom is tken with esect to time, the esult is set of lgebic equtions s EI Q s Ps b s As Q, whee i N (A7) i i i i i h If this is solved fo the genelized co odintes Q i s the esult is Q i s P s b s i i h (A8) As EI i Then if P s nd s h wee known then the s domin motion of ny oint on the bem could be given s W, s P s As b h EI s (A9) whee W, s is the Llce tnsfom of t w, with esect to time. Let the ctive foce be f. By Eqs. (Ab) nd (Ac), the eltions between the motion of the oint of ttchment nd the foce tnsmitted to the bem t the oint of ttchment is h s W s, ms cs k ms cs k (A) 3

24 s W, cn be obtined by Eqs. (Ad), (A9) nd (A), i.e.,, EI As k cs ms k cs ms s W b s s W (A) By Eq. (A), s W, cn be obtined when, i.e.,, EI As k cs ms k cs ms b EI As s s W (A) Substitute Eq. (A) into (A), the tnsfe function of the bem is, EI As EI As b k cs ms k cs ms EI As b s s W (A3) Relcing the comle vible s in Eq. (A3) by j, the fequency esonse function of the bem my be witten in dimensionless fom s j j, q q q q n b L Lb A W (A) whee A EI n, A EI, n, AL m, nd

25 5 EIL The eigenfunctions of the bem obey the othogonlity eltions nd the othogonlity eltions cn be witten s, d j L i, if j i (A5) L d j L i, if j i (A5b) Conside the othogonlity eltions nd the eqution s s W s s W s s W,,,, the men sque motion ove the whole domin of the bem cn be witten s, L d P W L, j j q q q q n b L Lb A (A6)

26 Aendi B Conside c = in ig., the equtions of motion my be witten using Eq. () s M K k m k f f (B) whee is distubnce nd f n ctution foce. Llce tnsfomtion is tken with esect to time, the esult is set of lgebic eqution witten s Ms X ms X KX k k X X X X (B) ollowing the och of [6], the ctive foce my be witten s s s s X (B3) whee,, nd e the feedbck gins. The tnsfe function of the imy mss my be solved using Eq. (B) nd witten s X mms m K ms k k km s m s m kk (B) Relcing s by j in Eq. (B), the fequency esonse function of the imy mss my be ewitten in dimensionless fom s X K j 3 j (B5) 6

27 7 whee M K n, m k, n, n, M m, K n, K, n K, n K nd j. Similly, the fequency esonse function of the ctive foce of the HVA my be ewitten in dimensionless fom s K X j j (B6) A numeicl emle of the HVA design using zeo-ole ssignment method [6] is esented in the following. Assume nd in Eq. (B5), the fequency esonse of the imy stuctue with dmed HVA using the zeo-ole ssignment method my be witten s 6 3 j j K X (B7) whee.9 6. Eq. (B5) is lotted togethe with fequency esonse function of the imy mss using the oosed design method of the HVA in ig. B fo comison. As shown in ig. B, the vibtion mlitude of the imy mss using the zeo-ole ssignment method is smlle thn using the oosed method. The zeo-ole ssignment method [6] is ble to educe vibtion eks while keeing the bsotion di simultneously in the fequency esonse of the closed-loo imy system. Howeve, s shown in ig. B the lots of Eq. (B6)

28 togethe with the fequency esonse function of the ctive foce using the oosed design method of the HVA, the highest fequency esonse of the ctive foce of HVA using the zeo-ole ssignment method is foty two times highe thn using the oosed design method. Coming the es unde the fequency esonse cuves of the ctive foce of HVA using the two diffeent design methods s shown in ig. B, the e using zeo-ole ssignment method is found to be one hunded thity five times highe thn using the oosed design method. 8

29 Refeences [] H. hm, Device fo Dming Vibtions of Bodies, U.S. Ptent, No. 989, 958, 9, [] J. Omondoyd, J. P. Den Htog, The Theoy of the Dynmic Vibtion Absobe, ASME Jounl of Alied Mechnics 5(7) (98) 9-. [3] E. Hhnkmm, Die Dmfung von undmentschwingungen bei vendeliche Eegegequenz, Ingenieu Achiv (93) 9- (in Gemn). [] J. E. Bock, A note on the Dmed Vibtion Absobe, ASME Jounl of Alied Mechnics 3() (96) 8. [5] J.P. Den Htog, Mechnicl Vibtions, Dove Publictions Inc., 985. [6] O. Nishih, H. Mtsuhis, Design nd Tuning of Vibtion Contol Devices vi Stbility Citeion, Pe. Of Jn Soc. Mech. Eng., No (997) (in Jnese) [7] B.G. Koenev, L.M. Reznikov, Dynmic Vibtion Absobes: Theoy nd Technicl Alictions, Wiley, New Yok, 993. [8] R.P. M nd A. Sinh, A neul-netwok-bsed ctive vibtion bsobe with stte-feedbck contol, Jounl of Sound nd Vibtion 9 (996) -8. [9] N. Olgc nd B. Holm-Hnsen, Tunble ctive vibtion bsobe: the delyed esonto, Tnsctions of the Ameicn Society of Mechnicl Enginees Jounl of Dynmic Systems, Mesuement nd Contol 7 (995) [] D. V. Blndin nd I. A. edotov, LMI-Bsed Synthesis of Dynmic Vibtion Dmes, Jounl of Comute nd Systems Sciences Intentionl 8 (9) [] R. J. Ngem, S.I. Mdnshetty, G. Medhi, An electomechnicl vibtion bsobe, Jounl of Sound nd Vibtion (997) [] M. Ysud, R. Gu, O. Nishih, H. Mtsuhis, K. Uki M. Kondo, Develoment of nti-esonnce enfoced ctive vibtion bsobe system, JSME intentionl jounl, Seies C, Dynmics, contol obotics, design nd mnufctuing 39 (996) [3] A. M. Nonmi, Distubnce cncelltion contol fo vibtion of multi-degee feedom systems, JSME Intentionl Jounl (99) [] G.J. Lee-Gluse, Otiml ctive vibtion bsobe: design nd eeimentl esults, Tnsctions of Ameicn Society of Mechnicl Enginees Jounl of Vibtion nd Acoustics 7 (995)

30 [5] G. J. Lee-Gluse, Integted ssive-ctive vibtion bsobe fo multi-stoy buildings, Jounl of Stuctul Engineeing 3 (997) [6] J. Yun, Hybid vibtion bsotion by zeo/ole-ssignment, Jounl of Vibtion nd Acoustics () [7] J. Yun, Multi-oint hybid vibtion bsotion in fleible stuctue, Jounl of Sound nd Vibtion () [8] S. Chttejee, Otiml ctive bsobe with intenl stte feedbck fo contolling esonnt nd tnsient vibtion Jounl of Sound nd Vibtion 39 () [9] Y.L. Cheung, W.O. Wong, H nd H otimiztions of dynmic vibtion bsobe fo suessing vibtions in ltes, Jounl of Sound nd Vibtion 3 (9) 9-. [] Y.L. Cheung, Otimiztions of dynmic vibtion bsobes fo suessing vibtions in stuctues, PhD Thesis, The Hong Kong Polytechnic Univesity, 9. [] K. Ogt, Moden Contol Engineeing, Pentice-Hill, 997. [] R. G. Jcquot, The stil vege men sque motion s n objective function fo otimizing dming in dming in dmed modified systems, Jounl of Sound nd Vibtion 59 () (3)

31 igue Ctions ig.. Schemtic digm of the oosed hybid vibtion bsobe (m-k-c-f system) ttched to the imy (M-K) system. ig.. The fequency esonse of the imy mss M with HVA t. nd.. =, =., =. ig. 3. The mlitude esonse t the fied oints vesus tuning tio t. nd.5. H, H. b ig.. Root locus of the SDO imy system with the oosed HVA in ig. with. nd,. Root, oot, oot 3, oot of Eq. (3). ig. 5. The fequency esonse of the imy mss M with HVA t...5,., (PVA). ig. 6. The mlitude esonse t the fied oints vesus tuning tio t. nd. H, H. b ig. 7. The fequency esonse of the imy mss M with HVA tuned to the otimum tuning t. nd G. 5. H,, ot, ot ; H, H, H,. ig. 8. Dimensionless ctive foce of the HVA in ig. whee X bx with. nd G.. =.K, b = ; =.K, b = - K; = -.3 K, b =.5 K; =. K, b =. K. ig. 9. The fequency esonse t. nd G.. esent theoy; otimum contol by Chttejee [8]. ig.. Dimensionless ctive foce of the HVA with. nd G.. esent theoy; otimum contol by Chttejee [8]. 3

32 ig.. ig.. Schemtics of simly suoted bem with hybid vibtion bsobe ecited by unifom distubed foce. The men sque motion esonse ig. with G. esent theoy using Eq. (); otimum PVA [9]. L L W, P d of the bem s shown in ig. 3. The men sque motion esonse L L W, P d of the bem s shown in ig. with G. esent theoy using Eq. (); otimum contol by Chttejee [8]. ig.. Active foce sect of the HVA in ig. with G. esent theoy, W o, using Eq. (); P otimum contol by Chttejee [8]. ig. B. ig. B The fequency esonse of the imy mss M in ig. with. nd G. 5 esent theoy using Eq. (B6); zeo-ole ssignment method [6]. ig. B Active foce sect of the HVA in ig. with. nd G. 5 esent theoy using Eq. (B6); zeo-ole ssignment method [6]. 3

33 sint M K/ k f f c K/ m ig.. Schemtic digm of the oosed hybid vibtion bsobe (m-k-c-f system) ttched to the imy (M-K) system. 33

34 H() 3 b equency tio, ig.. The fequency esonse of the imy mss M with HVA t. nd.. =, =., =. 3

35 H() nd H(b) The lgest height of the fied oint Tuning tio Tuning fequency tio, ig. 3. The mlitude esonse t the fied oints vesus tuning tio t. nd.5. H, H. b 35

36 Imginy is Rel is ig.. Root locus of the SDO imy system with the oosed HVA in ig. with. nd,. Root, oot, oot 3, oot of Eq. (3). 36

37 H() equency tio, ig. 5. The fequency esonse of the imy mss M with HVA t...5,., (PVA). 37

38 H() nd H(b) The height of the fied oint Tuning tio Tuning fequency tio, ig. 6. The mlitude esonse t the fied oints vesus tuning tio t. nd. H, H. b 38

39 equency esonse H() Dimensionless fequency equency tio, ig. 7. The fequency esonse of the imy mss M with HVA tuned to the otimum tuning t. nd G. 5. H,, ot, ot ; H, H, H,. 39

40 Actution foces Dimensionless fequency equency tio, ig. 8. Dimensionless ctive foce of the HVA in ig. whee X bx with. nd G.. =.K, b = ; =.K, b = -K; = -.3K, b =.5K; =.K, b =.K.

41 H() equency esonse Dimensionless fequency equency tio, ig. 9. The fequency esonse of the imy mss M with. nd G.. esent theoy; otimum contol by Chttejee [8].

42 Actution oce.5 equency tio,.5 - Dimensionless fequency equency tio, ig.. Dimensionless ctive foce of the HVA with. nd G.. esent theoy; otimum contol by Chttejee [8].

43 (t)g(,y) L ig.. Schemtics of simly suoted bem with hybid vibtion bsobe ecited by unifom distubed foce. 3

44 Dimensionless fequency esonse Dimensionless fequency equency tio, ig.. The men sque motion esonse ig. with G. L L W, P d of the bem s shown in esent theoy using Eq. (); otimum PVA [9].

45 Dimensionless fequency esonse 8 6 equency tio, Dimensionless fequency ig. 3. The men sque motion esonse ig. with G. L L W, P d of the bem s shown in esent theoy using Eq. (); otimum contol by Chttejee [8]. 5

46 Dimensionless ctution foce Dimensionless fequency equency tio, ig.. Active foce sect of the HVA in ig. with G. esent theoy, W o, using Eq. (); P otimum contol by Chttejee [8]. 6

47 H() equency tio, ig. B The fequency esonse of the imy mss M in ig. with. nd G. 5 esent theoy using Eq. (B6); zeo-ole ssignment method [6]. 7

48 equency tio, ig. B Active foce sect of the HVA in ig. with. nd G. 5 esent theoy using Eq. (B6); zeo-ole ssignment method [6]. 8