Mechancal Engneerng Research; Vol. 6, No. ; 206 ISSN 927-0607 E-ISSN 927-065 Publshe by Canaan Center of Scence an Eucaton Vbraton Isolaton of Lumpe Masses Supporte on Beam by Imposng Noes Usng Multple Vbraton Absorbers Sushl S Patl & Praeep J Awasare Mechancal Engneerng Department, Snhga College of Engneerng, Savtrba Phule Pune Unversty, Pune, Maharashtra state, Ina Corresponence: Sushl S Patl, Research Scholar, Mechancal Engneerng Department, Snhga College of Engneerng, Savtrba Phule Pune Unversty, Pune, Maharashtra state, Ina. E-mal: malsushl200@yahoo.co.n Receve: Aprl 5, 206 Accepte: May 8, 206 Onlne Publshe: May, 206 o:0.559/mer.v6np88 URL: http://x.o.org/0.559/mer.v6np88 Abstract In ths paper, varable stffness ampe absorbers are use to solate the substructures of Euler-Bernoull beam, moelle as lumpe masses, from vbratons. he novel algorthm s evelope that can be use to etermne the requre absorber masses an resonance frequences to mpose noes at selecte locatons on beam wth the constrant of vbraton ampltue of absorber mass. Numercal smulatons are performe to show the effectveness of the propose algorthm. Expermental test s conucte on a cantlever beam wth two absorbers to verfy the numercal results. Keywors: vbraton absorber, noe, cantlever beam, harmonc force, vbraton solaton. Introucton he tune vbraton absorber (VA) was nvente by (Frahm, 9), snce then, has been s an mportant engneerng tool for vbraton suppresson. he frst classcal paper on ynamc vbraton absorber was by (Ormonroy & Den, 928). her vbraton absorber consste of a tune sprng-mass use to suppress the response of a harmonc oscllator. An excellent survey of passve, sem-actve an actve ynamc vbraton absorbers was prepare by (Sun et al., 995). Sprng-mass systems whch are use as vbraton absorbers to mnmze excesses vbratons n contnuous structure have receve conserable nterest n recent years. (Young, 928) was the frst to conser the applcaton of absorber to control the vbratons of a contnuous structure.e. cantlever beam, at the absorber attachment pont wth the absorber tune to the frst natural frequency of the beam. (Mankanahally & Crocker, 99) employe vbraton absorbers to suppress any number of sgnfcant moes. he metho was successfully apple to a space structure moele as a mass-loae free free beam subjecte to localze harmonc exctaton. (Esmalzaeh & Jall, 998) presente a proceure n esgnng DVA for a structurally ampe beam system subjecte to strbute force exctaton. he absorber s moele as sprng-mass-amper system an the optmum tunng an ampng ratos are etermne to mnmze the beam ynamc response at the resonance frequency at whch they operate. Research on suppressng vbraton n a regon or the partcular span of an elastc structure by usng the sprng mass vbraton absorber has been reporte recently. (Cha, 200, 2005) employe sprng-mass vbraton absorber to reuce vbratons at esre locatons by mposng noe technque (Hao et al., 20). Suppresse the han-arm vbraton n electrc grass trmmer by nstallng tunable vbraton absorber at optmum locaton. he focus of (Cha & Rnker, 202) was on enforcng noes at esre locaton of ampe Euler-Bernoull beam urng force harmonc exctatons usng ampe vbraton absorbers. An effcent metho s evelope whch etermnes the restorng force exerte by the ampe absorbers usng Gaussan elmnaton an then the forces are use to etermne the parameters of oscllators. (Hao & Rpn, 20) apple mposng noe technque to acheve very low vbraton at hanle locaton usng two tunable vbraton absorbers. he grass trmmer system s smplfe as an arbtrary, supporte beam constrane by lumpe masses. he Matlab routne global search s utlze to obtan the tunng frequences of the absorbers. (Patl, & Awasare, 206) evelope an teratve proceure to fn the requre resonance frequences of absorbers to mpose noe at selecte locatons on beam. In the metho propose by Cha to nuce multple noes, a set of nonlnear algebrac equatons nee to be solve smultaneously. Numercally, the soluton of these equatons s very computatonally ntensve because the convergence s often very slow. he lmtatons of the proceure evelope by Patl & Awasare are that, the 88
www.ccsenet.org/mer Mechancal Engneerng Research Vol. 6, No. ; 206 maxmum allowable absorber ampltues were not consere whle fnng the resonance frequences of the absorbers. he purpose of ths paper s ) o evelop mathematcal moel of the beam carryng multple ampe absorbers an to formulate the equaton for mposng noes at esre locatons on the beam 2) o evelop an algorthm to fn the requre absorber parameters to mpose noes at selecte locatons on the beam wth constrant of the tolerable vbraton ampltues of the absorber mass.) o perform numercal smulaton on beam to show the utlty of the propose algorthm ) o carry the expermental test to valate the numercal results. 2. heory Fgure shows an arbtrarly supporte, Euler-Bernoull beam wth n tunable vbraton absorbers attache at x. he absorber moele as sngle egree of freeom sprng-mass-amper system havng mass m, stffness k an ampng coeffcent c of the th absorber. he lumpe masses are supporte at locatons x m on the beam. he external harmonc force j () et f t = Fe ω s apple to the structure at x f, where F represents the forcng ampltue, ω enotes the exctaton frequency, an j = e Fgure. Beam wth varable stffness ampe multple absorbers subjecte to localze harmonc exctatons Usng the assume-moes metho the eflecton of the beam at any pont x along the structure s gven by Leonar Merovtch (2007). N = φ η () = wxt (, ) ( x) ( t), where N s the number of moes use n the assume-moes expanson, φ ( x) are the egenfunctons of the unampe beam an η ( t) are corresponng generalze co-ornates. Applyng Lagrange s equatons an assumng smple harmonc moton wth same response frequency as the exctaton frequency, the followng equatons of moton are obtane 2 [ M ] 0 [ C] [ Rc] [ K] [ Rk] Fφ( xf ) ωe jω η, 0 [ m] + e [ Rc] [ c] + [ Rk] [ k] z = 0 (2) where η = [ η η2 η ] N, z = [ z z2 z ] n, an the matrces [m] [c] an [k] of sze n n are agonal, whose th elements are gven by m, c an k respectvely. he N N [M] [C] an [K] matrces of equaton (2) are L [ M ] = [ M ] + mlφ( xm) φ ( xm), = n [ C] = [ C ] + cφ( x) φ ( x), = n [ K] [ K ] kφ( x ) φ ( x ) = = + () where [ M ],[ C ] an [ K ] are agonal matrces whose th elements are M, C, an K are the generalze masses, ampng an stfnesses of beam. Vector of the egenfunctons of the beam an the matrces [ Rc ] an [ R k ] 89
www.ccsenet.org/mer Mechancal Engneerng Research Vol. 6, No. ; 206 of sze N n are gven by ( x) = [ ( x) 2( x) N( x)], φ xf = φ xf φ2 xf φn xf φ φ φ φ ( ) [ ( ) ( ) ( )], φ φ φ φ ( xm ) = [ ( xm ) 2( xm ) N ( xm )] [ Rc] = [ cφ( x) cφ( x) cnφ( xn)] [ Rk] = [ kφ( x) kφ( x) knφ( xn)] (5) Usng secon equaton of equaton (2), the z are foun to be In equaton (6) resonance frequency 2 ω + jωec m 2 2 e j ec m z = φ ( x) η, ω ω + ω =, n (6) ω an ampng coeffcent ω = k m c of th absorber are gven by () c = 2ζ ω, (7) where ζ s the ampng rato of the th absorber Equaton (6) s substtute nto the frst equaton of equaton (2) an then solvng for η to obtan In equaton (8) S 2 e M j e C K x x F xf = η = ω [ ] + ω [ ] + [ ] + σ φ( ) φ ( ) φ( ) (8) 2 2 ( mω + jcωe) ωe 2 2 e jc m e σ = ω ω ω Substtutng equaton (8) n to equaton (), followng equaton s formulate, the soluton of whch gves the absorber parameters requre to mpose noe at esre locatons along the beam n 2 2 2 ( mω + jcωe) ω e W( xnr ) = φ ( xnr ) ωe [ M] + jωe[ C ] + [ K ] + φ( x ) ( ) ( ) 0 2 2 φ x Fφ xf = = ωe ω jc mωe x nr =, n (0) Once the beam wth ts bounary contons are specfe, absorbers attachment locatons x, exctaton frequency ω e an the exctaton locaton x f are known, equaton (0) can be use to fn the absorber parameters, mass of the absorber m an resonance frequences ω, for gven absorber ampng rato ζ at whch splacements of beam W( x nr ) becomes zero to mpose noes at x nr. An algorthm s evelope, whch s base on fnng the resonance frequency of absorber ω at whch W( x n ) s less than W( xnp ) an W( xnn ).e the absolute value of the splacement of the beam at noe locaton s less than the absolute value of the splacement at prevous an next to noe locaton as shown n Fgure 2. he conton W( xn) < W( xnp) an W( xn) < W( xnn ), gves the absorber frequency ω for gven mass m necessary to mpose noe. he proceure to fn the absorber masses m an m 2 an frequences of the absorbers ω an ω2 to mpose two noes s as follows Algorthm to fn the masses an corresponng resonance frequences of the absorbers to mpose two noes () Assume the lower value for the absorber masses m an m 2. (2) Set ntal frequences of the absorbers ω < ωe anω 2 < ω e. () Determne σ an σ 2 from Equaton (9). () Compute W( x n) W( xnp ) an W( xnn ) usng Equaton (0). (5) If W( xn) < W( xnp) an W( xn) < W( xnn), s the absorber mass an ω (9) 90
www.ccsenet.org/mer Mechancal Engneerng Research Vol. 6, No. ; 206 (6) s the absorber frequency requre to mpose noe. Else ncrease the frequency of absorber ω n steps, compute W( xn), W( xnp), W( x nn), tll above conton s acheve. (7) Increase the mass m of frst absorber an repeat steps () to (5) tll, z zmax. Recor the corresponng resonance frequency ω. (8) Replace frequency an mass of frst absorber n σ wth new frequency ω an mass m obtan from step number (6). (9) Compute W( x n2 ) W( xn2p) an W( xn2n) usng Equaton (0). (0) If W ( xn2 ) < W ( xn2p ) an W ( xn 2) < W ( xn 2 N ), m2 s the absorber mass an ω2 s the absorber frequency requre to mpose noe. () Else ncrease the frequency of absorber ω 2 n steps, compute W( xn2), W( xn2p), W( x n2n), tll above conton s acheve. (2) Increase the mass m 2 of secon absorber an repeat steps (8) to (0) tll, z2 z2 max. Recor the corresponng resonance frequencyω 2. () Replace frequency an mass of secon absorber n σ 2 wth new frequency ω 2 an mass m 2 obtan from step number (). () Repeat proceure from step number () to (2) wth revse σ 2. Fgure 2. Illustraton of conton W( x ) < W( x ) an W( x ) < W( x ), use to fn frequency of absorber n np n nn. Numercal Results Because the assume-moe metho was use to formulate the equatons of motons, the propose proceure can be easly mplemente to mpose noe along any arbtrary supporte beam subjecte to harmonc exctatons. For cantlever beam, ts normalze (wth respect to mass per unt length, ρ, of the beam) egenfunctons φ ( x), generalse masses M an generalse stffnesses K are gven by sn βl snh βl φ( x) = cosβx cosh βx+ (snβx snh βx) ρl cos βl+ cosh βl () M = an = ( β ) ( ρ ) (2) K L EI L where β L satsfes the followng transcenental equaton cos βlcosh β L = () where E s Young s moulus, I s the moment of nerta of the cross-secton of the beam. In the followng example the frequences an an vbraton ampltues are non-mensonalse by vng by EI ( ρ L ) an F ( EI / L ) respectvely. Number of moes, N = 5 s use n the assume-moes expanson. he value of the absorber frequency ω an mass m are ncremente by 0.00 EI ( ρ L ) an 0.000ρ L, n each teraton respectvely. he absorber has a low ampng to obtan the greatest vbraton attenuaton at the ntene frequency. 9
www.ccsenet.org/mer Mechancal Engneerng Research Vol. 6, No. ; 206 Now conser the example of a unform cantlever beam. It s esre that two noes to be mpose, at xn = 0.L an xn2 = 0.6L, for ωe = 6 EI ( ρl ) at x f = L, for vbraton solaton of lumpe masses ml = 0.02ρL, ml2 = 0.0ρL an ml = 0.02ρL supporte at xm = 0.L, xm2 = 0.5L an xm = 0.6L respectvely. he two absorbers are attache at locaton x = 0.L an x 2 = 0.65L on the beam. he absorber parameters, resonance frequences an masses for gven ampng rato, requre to mpose noes obtane by usng the algorthm evelope are lste n able-. able. Summary of absorber parameters for the example of vbraton solaton of lumpe masses supporte on unform cantlever beam Absorber Dampng rato Attachment Locatons Mass Requre Frequency olerable Mass ampltue ξ x m m kg Frst 0.00 0.L 0.059 L ω ra/sec ρ z max m 68.52 EI ( ρ L ) 0.005 F ( EI / L ) Secon 0.00 0.6L 0.085 L ρ 6. EI ( ρ L ) 0.005 F ( EI / L ) Fgure, shows the steay state eforme shapes of cantlever beam wth noe at 0.L an 0.6 L. Note the regon on beam up to 0.6L experences less vbraton comparng to the beam wthout absorbers. Fgure. he absolute steay-state response of beam wth an wthout absorbers to mpose noes at 0.L an 0.6L, for the absorber parameters lste n able. Expermental est For mposng two noes the ual cantlevere mass absorbers were esgne an constructe as shown n Fgure 5. he resonance frequency of evce s ajuste by movng the masses towars or away from the base support, whch alters the effectve stffness n the system an alters ts resonance frequency. Note that the structural ampng of the absorbers use s consere as equvalent vscous ampng. Next n orer to verfy the numercal result, expermental test was conucte for the case of cantlever beam wth two absorbers as shown n Fgure 5. he two absorbers are attache at 0.L an 0.65L an the harmonc nput force s apple at the tp of the beam wth frequency 52 Hz.e ω = 6 EI ( ρl ). he force ampltue s kept constant at 5 N through the experment an use to e non-mensonalze the splacements of the beam by vng by F ( EI / L ). he system parameters an materal propertes use n expermental test are lste n able 2. he toltal weghts of absorber en masses attache at 0.L an 0.65L are 0.25 Kg an 0.2 Kg respectvely. he ampng ratos of the both absorber are ξ = ξ 2 = 0.00. he absorbers were tune by movng the en masses n or out such that the splacement at 0.L an 0.6L was mnmze. After tunng of the absorbers the vbraton ampltues were measure at twenty ponts on the beam s surface by the accelerometer an recore by vbraton analyzer to plot expermental steay state response as shown n Fgure 6. It s observe that the vbratons at the noe locatons of the beam are reuce to a mnmum level. Comparng Fgure, wth Fgure 6, t s observe that there s goo agreement between the numercal results an expermental results. 92
www.ccsenet.org/mer Mechancal Engneerng Research Vol. 6, No. ; 206 Fgure. Illustraton of expermental set up Fgure 5. Expermental test on cantlever beam supportng lumpe masses wth two absorbers tune to mpose noes able 2. he system parameters an materal propertes use n the expermental test Length of the beam L m hckness of the beam t 0.0 m Wth of the beam b 0.065 m Densty of the beam 780 Kg/m Young s moulus of the beam E 2. x 0 N/m 2 Mass per unt length of beam ρ 5 Kg/m Parameter EI / L use to nonmensonlse stffness of beam 7.5 N/m Parameter EI ( L ), ρ use to nonmensonlse frequences.95 N/m/Kg Parameter F EI / L use to nonmensonlse vbraton ampltues of beam an absorber masses 0.00 m Weght of lumpe masses ml 0. Kg = 0.02 ρ L ml 2 0.2Kg = 0.0 ρ L ml 0. Kg = 0.02 ρl otal weght of en masses of frst absorber m 0.25 Kg = 0.052 ρ L otal weght of en masses of secon absorber m 2 0.2 Kg = 0.085 ρ L Dampng rato of absorbers ξ = ξ2 0.00 9
www.ccsenet.org/mer Mechancal Engneerng Research Vol. 6, No. ; 206 he frequency response plots for frst an secon tune absorbers are epcte n Fgure 7 (a) an Fgure 7 (b) respectvely. he frequency response plot shows the resonance frequency, of frst absorber = 6Hz 68.9 EI ( ρl ) ω an of secon absorber ω = 5Hz 6 EI ( ρl ), are n close match wth numercal soluton.e ω = 68.52 EI ( ρl ) an 2 ω = 6.2 EI ( ρl ). he frequency response plot for beam s as shown n Fgure 8(a) an peaks gve the frst three 2 natural frequences of the cantlever beam 8 Hz, 5 Hz an 5 Hz respectvely. he frequency response plot for beam wth absorbers s shown n Fgure 8(b), an observe that the response rops to a mnmum at exctaton frequency 52Hz 6 EI ( ρ L ) an peaks at 08Hz 7 EI ( ρ L ) an 62Hz 68 EI ( ρ L ). Fgure 6. Measure vbraton ampltue of beam wth absorber tune to mpose noes at 0.L an 0.6L an wthout absorber Fgure 7. Frequency response of a) absorbers attache at 0.L b) absorbers attache at 0.65L, by expermental moal analyss when absorbers tune to mpose noe at 0.L an 0.6L Fgure 8. Frequency response of a) beam wthout absorber b) beam wth absorbers when absorbers tune to mpose noe at 0.L an 0.6L, by expermental moal analyss 9
www.ccsenet.org/mer Mechancal Engneerng Research Vol. 6, No. ; 206 5. Conclusons hs nvestgaton presents novel algorthm to fn the absorber parameters to mpose noes at chosen locatons on beam. Once the generalze program s evelope, t can be use to etermne the feasble absorber parameters an can be easly mofe to accommoate beam wth fferent bounary contons. For mposng multple noes, the proceure gves fferent combnatons of two absorber masses an corresponng resonance frequences requre to mpose noes for gven ampng rato. he esgn constrant on maxmum allowable vbraton ampltues on absorber masses makes the propose proceure more practcal. he results show that by mposng noes at approprate locatons, the vbratons are suppresse for the segment of beam thus solatng the lumpe masses. he expermental results show goo agreement wth those obtane by numercal experments. References Frahm, H. (92). Devce for ampng vbraton of boes. US Patent No 989,958. Ormonroy, J., & Den, H. (928). he theory of ynamc vbraton absorbers. ransacton of ASME, 50, 9-22. Sun, J. Q., Jolly, M. R., & Norrs, M. A. (995). Passve, aaptve an actve tune vbraton absorber-a survey. Journal of Mechancal Desgn,, 9-508. Young, D. (952). heory of ynamc vbraton absorbers for beams. Proceengs of frst U.S. Natonal Congress of Apple Mechancs, 9-96. Mankanahally, D. N., & Crocker, M. J. (99). Expermental nvestgaton of mnmzaton of ynamc response of mass-loae beams usng vbraton absorber. Journal of Acoustcal Socety of Amerca, 9, 896-907 Esmalzaeh, E., & Jall, N. (998). Optmum esgn of vbraton absorbers for structurally ampe moshenko beams. ASME Journal of Vbraton an Acoustcs, 20(), 8-8. http://x.o.org/0.5/.289908. Cha, P. D. (200). Imposng noes at arbtrary locatons for general elastc structure urng harmonc exctatons. Journal of Soun an Vbraton, 272, 85-868. http://x.o.org/0.06/s0022-60x(0)0095- Cha, P. D. (2005). Enforcng noes at requre locatons n a harmoncally excte structure usng smple oscllators. Journal of Soun an Vbraton, 279, 799 86. http://x.o.org/0.06/j.jsv.200..067 Hao, K. Y., Me, L. X., & Rpn, Z. M. (20). une vbraton absorber for suppresson of han-arm vbraton n electrc grass trmmer. Internatonal Journal of Inustral Ergonomcs,, 9-508. http://x.o.org/0.06/ j.ergon.20.05.005 Cha, P. D., & Rnker, J. M. (202) Enforcng noes to suppress vbraton along a harmoncally force ampe Euler-Bernoull beam. ASME Journal of Vbraton an Acoustcs, (5), -0. http://x.o.org/0.5/.00 675 Hao, K. Y., & Rpn, Z. M. (20). Noal control of grass trmmer hanle vbraton. Internatonal Journal of Inustral Ergonomcs,, 8-0. http://x.o.org/0.06/j.ergon.202.0.007 Patl, S. S., & Awasare, P. J. (206). Vbraton reucton at esre locatons on a beam by creatng noes usng tunable vbraton neutralzers. Journal of vbraton an control, 22(), 205-22. http://x.o.org/0.77/ 0775652896 Leonar, M. (2007). Funamentals of Vbratons. ata McGraw-Hll, NewDelh, 2007. Copyrghts Copyrght for ths artcle s retane by the author(s), wth frst publcaton rghts grante to the journal. hs s an open-access artcle strbute uner the terms an contons of the Creatve Commons Attrbuton lcense (http://creatvecommons.org/lcenses/by/.0/). 95