Vibration control of the axisymmetric spherical pendulum by dynamic vibration absorber moving in radial direction

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1 Joural of Mechaical Sciece a echology 5 (7) (0) 703~709 DOI 0.007/ Vibratio cotrol of the axiymmetric pherical peulum by yamic vibratio aborber movig i raial irectio L. D. Viet,* a Yougji Park Ititute of Mechaic, 64 Doi Ca, Haoi, Vietam Korea Avace Ititute of Sciece & echology, Sciece ow, Daejeo, , Korea (Maucript Receive December, 00; Revie February 6, 0; Accepte March, 0) Abtract I the curret paper, a yamic vibratio aborber (DVA) i propoe to uppre the vibratio of the axiymmetric pherical peulum. he propoe DVA ca reuce the bi-irectioal vibratio of the peulum by oe ma, which move i the raial irectio (up a ow). he global tability of the ytem i aalyze uig the Lyapuov fuctio. he propoe DVA i prove effective, epecially for the uppreio of large vibratio. he DVA atural frequecy houl be tue to twice a that of the axiymmetric pherical peulum. he effectivee of the propoe DVA i emotrate through a example of liqui lohig reuctio i a cylirical tak. Keywor: Vibratio cotrol; Dyamic vibratio aborber; Spherical peulum; Noliear mechaic; Liqui lohig Itrouctio hi paper wa recommee for publicatio i revie form by Eitor Yeo Jue Kag * Correpoig author. el.: , Fax.: are: laviet80@yahoo.com, lviet@imech.ac.v KSME & Spriger 0 A tue-ma amper, or a yamic vibratio aborber (DVA), i a well-kow vibratio cotrol evice. he DVA coit of a movig ma attache to the mai tructure through prig a amper. hi primary tructure i ofte moele a a prig-ma ytem, although other moel are alo wiely applie i reearch a egieerig. I particular, peulum-type ytem with a oli boy a a fixe fulcrum poit ca be ue to illutrate everal tructure type, uch a ropeway goola, crae, or floatig tructure (e.g., hip or teio leg platform). Several previou tuie have ue DVA to reuce the wig of peulum tructure. he DVA ha two mai type. he firt type, DVA movig i a circumferece irectio (left a right) (Fig. ), wa theoretically ivetigate by Matuhia []. A total of 0 ropeway i Japa have bee italle with DVA ice the firt itallatio i ropeway chair lift i 995 []. A more geeral tuy o the DVA italle i iverte peulum-type ytem wa preete by Ah et al. [3]. Matuhia et al. [4] alo propoe the eco type, i.e., DVA movig i the raial irectio (up a ow) (Fig. ). A Corioli ampig force i prouce by thi DVA type [4, 5]. However, i may practical ituatio, the plaar peulum houl be replace by a pherical peulum for a more accurate moel of the tructure. Dyamic a cotrol of the geeral rigi peulum are tuie i Ref. [6-9]. I Ref. [7], the equatio of motio of the geeral peulum, referre to a the triaxial attitue cotrol tetbe, are erive. Dyamic of ome actuatio mechaim icorporate ito the geeral peulum are alo coiere i Ref. [7]. Moreover, the feeback cotrol problem of the geeral peulum are icue i Ref. [8, 9]. I peulum tructure, the gravitatioal force oe ot geerate a momet arou the vertical axi. hu, chagig the vertical compoet of the agular mometum by ay cotrol actio uig iteral force, either through active or paive mea, i impoible [8, 9]. he active or paive DVA ca oly prouce iteral force. Although it ha lower Fig.. Aborber i a plaar peulum tructure: DVA movig i a circumferece irectio; DVA movig i a raial irectio.

2 704 L. D. Viet a Y. Park / Joural of Mechaical Sciece a echology 5 (7) (0) 703~709 poitio of the peulum ma (x, y, z) a the DVA ma (x, y, z ) are eaily obtaie: x = lcoφ i θ; y = li φ; z = lcoφcoθ () x = ( l + q) coφ i θ, y = ( l + q) i φ, () z = l + q coφco θ. ( ) he kietic eergy a potetial eergy U the have the followig form: Fig.. Geometrical ecriptio of the axiymmetric pherical peulum attache with the DVA; two wig agle. performace tha the active mea, the paive mea eure ytem tability a oe ot require exteral eergy or complex meauremet ytem. Comparig the two type of DVA i Fig., the eco type (Fig. ) ha a biirectioal vibratio cotrol ature, wherea the firt type ca oly reuce plaar vibratio. he curret paper how the avatage a iavatage of the DVA movig i a raial irectio of the peulum orbit. he tructure of thi paper i a follow: Firt, the oliear motio equatio i preete i a o-imeioal form. he global tability of the ytem i the aalyze uig the Lyapuov fuctio. Next, the optimal DVA parameter are tuie. Fially, the effectivee of the DVA i emotrate uig the liqui lohig reuctio problem i a cylirical tak.. Propoe DVA a equatio of motio. Geometrical ecriptio I Fig., coier a axiymmetric pherical peulum upporte by a fixe pivot O a italle with a DVA. he itace from the peulum ceter of ma to the pivot i eote by l. he iertial referece frame {I} = {x, y, z} ha it origi at pivot O. he x a y axe lie horizotally a are perpeicular to each other, wherea the z-axi i vertical to the irectio of gravity. he peulum ma i eote by m. he rotatioal agle (meaure i the xz plae) i eote by θ, wherea the agle meaure from the xz plae i φ. he otatio l eote the itace betwee the pivot a the DVA i the tatic poitio. he DVA ma i eote by m. he DVA ha a tralatioal motio with the iplacemet q meaure from the tatic poitio. It i upporte by the combiatio of prig a amper. Deote k a c a the prig cotat a ampig coefficiet, repectively. he tructural ampig eote by c i aume ietical i all irectio.. Lagragia expreio Coierig the cooriate ytem how i Fig., the = m( x& + y& + z& ) + m ( x& + y& + z& ) (3) U = mg( l z) + mg( l z q) + kq (4) where g i the acceleratio of gravity. Hece, the Lagragia expreio i give by L = U. Combiig Eq. ()-(4) how that the Lagragia i expree i term of three egree of freeom, i.e., φ, θ, a q, a their firt erivative..3 Equatio of motio he eergy iipatio fuctio F i give by F = c ( x& + y& + z& ) + cq&. (5) he Lagrage motio equatio become L L F + = M t ( & of ) ( of ) ( & of ) (6) = φθ,, q of where M i the vector of exteral (o-gravitatioal) momet actig o the peulum. he motio equatio ca be writte i the o-imeioal form by itroucig the followig parameter: ml + m l g c ; ; ; e = ω = = ml + ml le lemω l l Mφ Mθ τ = ωt; e= ; M = ; M = ; le mωle mωle m k/ m l q c μ = ; γ = ; u = ; α = ; = ; m l l ω m ω e e where l e i the legth of the equivalet peulum; ω a are the atural frequecy a the ampig of the peulum attache with the DVA, repectively; τ i the ormalize time with time cale ω - ; M φ a M θ are the compoet of exteral momet vector M, a M a M are the ormalize exteral momet, repectively; μ i the ma ratio; γ i the locatio parameter pecifyig the DVA poitio; u i the ormalize (7)

3 L. D. Viet a Y. Park / Joural of Mechaical Sciece a echology 5 (7) (0) 703~ form of the aborber iplacemet; α i the DVA frequecy ratio; a i the DVA ampig ratio. he motio equatio are implifie a rearrage ito the followig oimeioal form: ( + ( + + ))( + ) = ( ( )) i co ( )& ( ( ))( ) ( μ( γ )) iθ μ( γ )& & θcoφ e μ γ γu u && φ & θ iφcoφ e & φ e+ μ γ + u φ θ μ γ + u u& φ + M e+ μ γ + γu+ u && θ φ & φθ& φ = e & θ φ co i co e+ + u + u u + M u&& = α u u& + coφcoθ + γ + u & φ + & θ co φ. ( )( ) (8) (9) (0) he kietic eergy (3), potetial eergy (4), a eergy iipatio fuctio (5) ca alo be ormalize i term of the o-imeioal parameter: = = mω le () ( e+ μ ( γ + γu+ u ))( & φ + & θ co φ) + μu & U U = = ( e+ μ ( γ + u) ) coφcoθ mω le () + ( e+ μγ ) + μα u F F = = e ( & φ + & θ co φ) + μu& (3) mω l e where, U, a F are the ormalize form obtaie by iviig, U, a F, repectively, by the cotat mω l /. 3. Stability propertie I thi ectio, the ytem tability i aalyze uig the Lyapuov fuctio. Coier the ytem without tructural ampig a exteral momet ( = M = M = 0) a well a the caiate Lyapuov fuctio V give by the um of ormalize kietic eergy a potetial eergy: e V = + U (4) where a U are etermie by Eq. () a (), repectively. No exteral eergy i applie; thu, the coervatio of eergy give V + U + Fτ = cot or = F (5) τ where F i the ormalize iipatio fuctio calculate from Eq. (3), a the itegral i Eq. (5) efie the iipate eergy. Uig Eq. (3) yiel the followig: Let u fi the vibratio tate where: V/τ = 0. (7) If 0, applyig coitio Eq. (7) a (6) provie two equatio: V = + U = V = cot (8) 0 u & = 0 or u = u0 = cot. (9) Subtitutig Eq. (9) ito Eq. (0) a (8), a ytem of two liear equatio i obtaie with the followig form: Av = b (0) where γ + u0 A = ; ( e+ μ( γ + u0) ) e+ μ( γ + γu0 + u0) α u0 + b = V0 μα u0 ( e+ μγ ) () v = coθ coφ & φ + & θ co φ. () he coitio γ > 0 a u 0 > 0 are obtaie if the DVA i locate below the pivot poit. herefore, et(a) 0. he matrix liear Eq. (0) ha a uique olutio that oe ot epe o time, i.e., coθ co φ = cot. (3) Subtitutig Eq. (3) ito Eq. () give the followig cocluio. he erivative of the Lyapuov fuctio i maller tha zero. hi erivative i equal to zero if a oly if the peulum height i cotat or if the peulum move i circle at the horizotal plae. A icue i the itrouctio, thi coitio i caue by the coerve vertical compoet of the agular mometum. I particular, if the iitial vertical compoet of the agular mometum i equal to zero, the the ytem i aymptotically table i the global ee. 4. DVA characteritic 4. Optimal DVA frequecy ratio Several implificatio houl be mae to oberve clearly the iteractio betwee the DVA a the pherical peulum. - Firt, aume i each perio that the DVA oe ot igificatly chage the peulum vibratio agle. hu, the peulum vibratio agle ca be writte i the followig form: V = F = 4μ u 0. τ & (6) = ; θ pφ co( τ ϕ) φ φ coτ m = + ; (4) m

4 706 L. D. Viet a Y. Park / Joural of Mechaical Sciece a echology 5 (7) (0) 703~709 where φ m i the vibratio amplitue of the agle φ ;p i the ratio betwee the vibratio amplitue of θ a φ ;a ϕ i the phae hift betwee two agle. Note that φ m, p, a ϕ are cotat i each perio. - Seco, the vibratio agle are aume to be mall eough to elimiate the term with orer larger tha i the motio equatio. Uig Eq. (4) a after ome maipulatio, the eco-orer approximate equatio of (0) become γ u&& = αtu tu& + ( p + ) φm 4 γ + co i τ τ ϕ φm. ( p ( )) (5) Several importat characteritic of the propoe DVA ca be raw from Eq. (5), uch a the followig: - he DVA motio i proportioal to the eco orer of the peulum vibratio amplitue φ m. hi pheomeo i ifferet from the covetioal liear DVA, where the motio i proportioal to the firt orer of the peulum vibratio amplitue. herefore, the propoe DVA effectively reuce the large peulum vibratio amplitue. - he yamic term co(τ) a i(τ+ϕ) actig o the DVA motio have a frequecy of. herefore, the optimal frequecy ratio α houl be equal to to prouce the reoace a amplify the DVA motio. - I a pecial cae, whe the peulum move circularly at the horizotal plae, two vibratio agle ha the ame magitue (p = ), a the phae ifferece i equal to 90 o [co(ϕ) = ]. he yamic term i Eq. (5) the iappear, a the DVA motio caot be excite. hi cae i how i the cocluio of Sectio Optimal DVA ampig ratio he atural frequecy ratio α houl be tue to, a icue i the previou ubectio. I thi ubectio, the umerical calculatio i ue to ivetigate the effect of DVA ampig ratio. For implicity, oly plaar-free vibratio with a iitial agle i coiere. he followig aumptio are applie to Eq. (8)-(0): ( 0 ) = φ0; ( ) ( ) ( ) ( ) ( ) φ & φ 0 = θ 0 = & θ 0 = u 0 = u& 0 = M = M = 0. (6) he optimal DVA ampig ratio ca alo work i the threeimeioal cae ue to the homogeou ature of the pherical peulum. he oliear Eq. (8)-(0), with the iitial coitio of Eq. (6), are umerically olve. he mea value of the o-imeioal tore eergy V i Eq. (4) i coiere miimize. he performace iex i f f J = Vτ ( U) τ = + f 0 f 0 where f i the total ormalize time of the imulatio, which (c) () Fig. 3. J a a fuctio of ; (I), (II), (III): α =.9,,., repectively; other parameter: μ = %, γ =, φ 0 = 0 o ; J a a fuctio of ; (I), (II), (III): φ 0 = 5 o, 0 o, 5 o, repectively; other parameter: μ = 5%, γ =, α = ; (c) J a a fuctio of ; (I), (II), (III): μ = %, 4%, 7%, repectively; other parameter: α =, γ =, φ 0 = 0 o ; () J a a fuctio of ; (I), (II), (III): γ = 0.5,,.5, repectively; other parameter: μ = 3%, φ 0 = 0 o, α =. i take from 00. he DVA parameter μ, γ, a α a well a the iitial agle φ 0 are varie to tuy their effect. he plot of the performace iex veru the DVA ampig ratio for variou parameter are how i Fig. 3.

5 L. D. Viet a Y. Park / Joural of Mechaical Sciece a echology 5 (7) (0) 703~ Fig. 4. Phyical moel; peulum moel of the liqui lohig. I Fig. 3, J a J u eote the performace iexe with a without DVA, repectively. Some remark ca be raw a follow: - he reult i Fig. 3 agai verify that the optimal value of the atural frequecy ratio α houl be equal to. - he optimal value of i withi %-5%. Whe γ, μ, or φ 0 ecreae, the optimal ampig ratio alo lightly ecreae. - he reult i Fig. 3 a 3() how that the DVA effectivee icreae whe the poitio parameter γ or the iitial agle φ 0 icreae. hee fiig prove that the propoe DVA ha a goo effect i cae of a large vibratio. - he icremet of ma ratio reuce the peulum repoe better. However, it ca alo icreae the DVA tore eergy. A how i Fig. 3(c), the overly large ma ratio μ oe ot ofte achieve the eire reuctio of performace iex becaue the total tore eergy, icluig the DVA tore eergy, i coiere. 5. Numerical emotratio 5. Peulum moel of liqui lateral lohig Liqui lateral lohig ca occur i the fuel tak o a movig vehicle or i the water tak o the top of tructure ubject to wi or wave. he mai yamic effect of lateral lohig i a horizotal ocillatio of the liqui ceter of ma relative to the tak. hi effect ca be equally well repreete by a peulum moel [0]. he liqui lohig i a circular cylirical tak ca be moele a a pherical peulum [0], a how i Fig. 4. If oly the firt lohig frequecy i coiere, the the equivalet peulum moel cotai a rigi ma m 0 a a peulum ma m that i free to ocillate. If the tak iameter a the liqui level are eote by a h, repectively, the the parameter of the peulum moel ca be etermie a follow [0]: ( ξh ) ( ) h tah / m= mliq ; ξ ξ l = ; m0 = mliq m; ξtah / ( ξh ) where m liq i the total liqui ma, a ξ =.84. A DVA Fig. 5. Propoe DVA to reuce liqui lohig. ak velocitie (m/) movig i a raial irectio i propoe to reuce lohig, a how i Fig. 5. he DVA i hug by a buoy plate o the liqui urface a move through a cylirical guie. he flui yamic houl be aalyze to cofirm the DVA effect. However, thi problem i too complex to be tuie i the preet article. hu, the peulum moel i aume till vali eve though the DVA i attache. hi aumptio i reaoable becaue the DVA ma i oly 0.45% of the total liqui ma i the umerical example. he umerical value are choe a follow: tak iameter = m; liqui level h = m; liqui ma eity = 000 kg/m 3 ; DVA ma m = 8.53 kg; itace from the DVA to the liqui urface l =.0439 m; DVA frequecy ratio α = ; a DVA ampig ratio = 5%. he tructural ampig ratio i aume = 0.5%. 5. Free vibratio he RECURDYN oftware [] i ue to imulate the oliear motio of the peulum. he tak i aume to move with the tep velocitie i two irectio, a how i Fig. 6. he tep velocity patter ca caue free lohig. he time hitorie of two peulum wig agle are how i Fig. 7, a the DVA repoe i how i Fig. 8. he DVA i how to reuce free lohig i both irectio. 5.3 Raom vibratio X-velocity Fig. 6. ak velocitie i two irectio. Y-velocity he propoe DVA alo ha a poitive effect i cae the

6 708 L. D. Viet a Y. Park / Joural of Mechaical Sciece a echology 5 (7) (0) 703~709 Agle φ (ra) Agle θ (ra) Fig. 7. wo peulum agle v. time. DVA iplacemet (m) Fig. 8. DVA iplacemet meaure from the tatic poitio. liqui tak ha a raom motio. wo tak acceleratio i two irectio are moele a white oie with the ame iteity, which i (m/ 3 ). he Mote Carlo imulatio i ue to obtai the mea value, a how i Fig. 9 a 0. he total ample are 000. Mea value of φ (ra) Mea value of θ (ra) Fig. 9. Mea value of the peulum vibratio agle. Mea value of DVA iplacemet (m) Fig. 0. Mea value of the DVA iplacemet meaure from the tatic poitio. i a movig tak emotrate the effectivee of the propoe DVA. 6. Cocluio he DVA movig i a raial irectio reuce the vibratio of a axiymmetric pherical peulum. hi type of DVA ha the avatage of a biirectioal vibratio cotrol by oe ma. If the vertical compoet of the agular mometum i equal to zero, the the peulum attache with the propoe DVA i aymptotically table. he optimal DVA frequecy ratio houl be equal to, wherea the optimal DVA ampig ratio houl be withi %-5%. A reuctio of liqui lohig Ackowlegmet hi work wa upporte by the eco tage of the Brai Korea Project i 0. Referece [] H. Matuhia, R. Gu, Y. Wag, O. Nihihara a S. Sato, Vibratio cotrol of a ropeway carrier by paive yamic vibratio aborber, JSME Iteratioal Joural (Serie C),

7 L. D. Viet a Y. Park / Joural of Mechaical Sciece a echology 5 (7) (0) 703~ (995) [] H. Matuhia a M. Yaua, Locatio effect of yamic aborber o rollig tructure, Proc. of Aia-Pacific Vibratio Coferece (003) [3] N. D. Ah, H. Matuhia, L. D. Viet a M. Yaua, Vibratio cotrol of a iverte peulum type tructure by paive ma-prig-peulum yamic vibratio aborber, Joural of Sou a Vibratio, 307 (007) [4] H. Matuhia, H. Kitaura, M. Ioo, H. Utuo, J. G. Park a M. Yaua, A ew Corioli yamic aborber for reucig the wig of goola, Proc. of Aia-Pacific Vibratio Coferece (005) -5. [5] L. D. Viet, N. D. Ah a H. Matuhia, he effective ampig approach to eig a yamic vibratio aborber uig Corioli force, Joural of Sou a Vibratio, 330, Iue 9 (0) [6] J. She, A. K. Sayal, N. A. Chaturvei, D. Bertei a N. H. Mc-Clamroch, Dyamic a cotrol of a 3D peulum. Proc. 43r IEEE Cof. o Deciio a Cotrol (004) [7] S. Cho, J. She a N. H. Mc-Clamroch, Mathematical moel for the triaxial attitue cotrol tet be, Mathematical a Computer Moelig of Dyamical Sytem, 9: (003) [8] J. She, A. K. Sayal a N. H. McClamroch, Aymptotic Stability of Multiboy Attitue Sytem, Stability a Cotrol of Dyamical Sytem with Applicatio, Birkhauer, Boto (003) [9] S. Cho a N. H. McClamroch, Feeback cotrol of triaxial attitue cotrol tetbe actuate by two proof ma evice, Proc. of 4t IEEE Coferece o Deciio a Cotrol (00) [0] F.. Doge, he ew yamic behavior of liqui i movig cotaier, echical Report, Southwet Reearch Ititute, Sa Atoio, X (000). [] Fuctio Bay Ic., RECURDYN, (00). La Duc Viet receive hi B.S. a Ph.D. egree i Mechaic from the Vietam Natioal Uiverity, Haoi, Vietam, i 00 a 009, repectively. I 00, he worke a a pot-octoral reearcher at Korea Ititute of Sciece a echology (KAIS). He i curretly a reearcher at the Ititute of Mechaic, Haoi, Vietam. Hi reearch iteret iclue vibratio cotrol, tructural yamic, tructural cotrol, a tochatic mechaic. Yougji Park receive hi B.S. a M.S. egree i Mechaical Egieerig from the Seoul Natioal Uiverity, South Korea, i 980 a 98, repectively. He receive hi Ph.D. egree i Mechaical Egieerig from the Uiverity of Michiga, USA, i 987. From 987 to 988, he worke a a reearch fellow at the Uiverity of Michiga. He alo worke a a aitat profeor at New Jerey Ititute of echology, from 988 to 990. He joie the KAIS faculty i 990 a a profeor of Mechaical Egieerig. Hi reearch iteret iclue geeral cotrol theorie, virtual auio ythei, active cotrol of oie a vibratio, a ytem ietificatio.