Generalized Den Hartog tuned mass damper system for control of vibrations in structures

Transcription

1 Earhqua Rssan Engnrng Sruurs VII 85 Gnralzd Dn Harog und ass dapr sys for onrol of vbraons n sruurs I. M. Abubaar B. J. M. ard Dparn of Cvl Engnrng, auly of Engnrng, Alahad Unvrsy, Sr, Lbya Absra Th Dn Harog hod s on of old, bu ffn hods for h dsgn of und ass dapr TMD syss for passv onrol vbraons n buldngs subjd o arhqua aons. Howvr, Dn Harog, n hs odl, dd no nlud h an ass sruur dapng n drvng h quaons for opung h opu dsgn parars for h TMD sys dapng and frquny raos. In hs wor an ap s ad o gnralz h Dn Harog hod o nlud h ff of h sruur dapng n h drvd quaons. Th dyna quaons for a TMD sys aahd o SDO sruur wh dapng ar drvd. Thn, usng h opl nubrs hnqu, h quaon for au dsplan of h sruur whn subjd o haron aon s drvd. Th nural opzaon hnqu s usd o opu h opu valus of TMD parars ha wll nz h au dsplan of h sruur. nally, h urv fng produr wh ral funons s usd o oban quaons for opung h opu frquny rao and opu dapng rao for a TMD sys. Kywords: und ass dapr, Dn Harog odl, sruural dapng, arhqua ngnrng, haron aon. Inroduon Th und ass dapr TMD sys s a passv nrgy absorbng dv onssng of a ranslaonal ass, ranslaonal sprng and ranslaonal vsous dapr aahd o a vbrang sruur o rdu undsrabl vbraons. Thr has bn a onsdrabl aoun of rsarh don on how bs o dsgn h ass daprs for us n h passv onrol of sruurs undr aon fors suh WIT Transaons on Th Bul Envronn, Vol 4, 9 WIT Prss ISSN on-ln do:.495/eres97

2 86 Earhqua Rssan Engnrng Sruurs VII as wnd and arhqua. Mos of h rsarhrs agrd ha h prforan of TMDs s snsv o h auray of unng h naural frquny of h TMD o h naural frquny of h sruur. Dn Harog [] has drvd h forula for h opu valus of h TMD parars for an undapd SDO sruur subjd o a haron aon. Warburon and Ayornd [] and Warburon [3] drvd losd for prssons for obanng TMD parars for an undapd SDO sys for bas haron and wh nos rando aon. Tsa and Ln [4] sudd a SDO wh TMD sys whr a dapng n h an ass s onsdrd wh haron aons a h bas. Rana and Soong [5] n hr papr prsnd a nural sudy for h prforan of TMD syss. Sad al. [6] suggsd a hod for sang h dsgn parars of TMDs aahd o a dapd SDO sruur for ss applaons, h rron usd o oban h opu parars was o sl, for a gvn ass rao, h frquny and dapng raos ha would rsul n qual and larg odal dapng n h frs wo ods of vbraon. Abubaar [7] proposd a Translaonal-Roaonal TMD sys nludng h sruur dapng. In h prsn wor, h Dn Harog opzaon produr for h TMD parars wh haron loadng appld o an undapd SDO sruur s ndd o onsdr h dapng of h an sruur. Mahaal odl of h probl gur shows h odl usd n h prsn wor. I onsss of a SDO sruur wh proprs, and. Th sruur s aahd o a TMD sys wh proprs, and. Th rsulng sys s wo dgrs of frdo sys. Ths odl s slar o h Dn Harog Modl p ha Dn Harog ngld h sruur dapng.. n hs sudy. Whn hs sys s subjd o dyna fors and ang on h sruur and TMD rspvly, as shown n gur, h dyna qulbru of h wo asss an b rprsnd by h followng dffrnal quaons: or ass : or or ass : or WIT Transaons on Th Bul Envronn, Vol 4, ISSN on-ln 9 WIT Prss

3 gur : A TMD aahd o SDO sruur. Th abov wo quaons an b wrn n a ar for as follows: 3 Now onsdr h probl whn h sruur s subjd o haron for gvn n opl for as o whr s h frquny of h haron aon for, and no load appld on h TMD. Thn, quaon 3 an b wrn as: o 4 Th soluon of h probl an b oband usng opl nubrs hnqu and assung h rsponss ar n h for of haron funons as follows: 9 WIT Prss WIT Transaons on Th Bul Envronn, Vol 4, ISSN on-ln Earhqua Rssan Engnrng Sruurs VII 87

4 5 Subsung h abov rlaons no quaon 4 ylds 6 or 7 Solvng for and rarrangng, ylds d b a 8 whr { } [ ] { } [ ] 3 4, d b a Dfnng h followng non-dnsonal parars for h sruur and TMD sys Naural frquny of h sruur ξ Dapng rao of h sruur 9 WIT Prss WIT Transaons on Th Bul Envronn, Vol 4, ISSN on-ln 88 Earhqua Rssan Engnrng Sruurs VII

5 Earhqua Rssan Engnrng Sruurs VII 89 Naural frquny of h T-R TMD alon ξ Translaonal dapng rao of T-R TMD q Rao of T-R TMD frquny o sruur frquny Rao of h ass of h T-R TMD o h ass of h sruur µ r Rao of ng for frquny o h frquny of sruur whr Thn quaon 8 an b wrn n non-dnsonal for as follows: 4 a b 9 d a q b ξ r rq r r q µ 4ξ ξ q q d rqξ rq ξ r ξ ξqµ ξq In h Dn Harog hod, h opu parars of h TMD sys ξ and q for a spf ass rao µ and sruur dapng rao ξ, ar dfnd as hos parars whh nz h au dsplan of h sruur whn subjd o haron aon. or hs purpos, a spal nural opzaon algorh usng MATLAB nvronn [8] s dvlopd o sarh for h opu valus of ξ op and q op for a gvn valu of ass rao µ and sruur dapng rao ξ. Th algorh s basd on varyng h varabls r, q and ξ and sarhng for h valus ha wll nz h sruur dsplan. Ths MATLAB sofwar was hn usd o fnd q op and ξ op for valus of ass rao µ bwn.5 o. and sruur dapng rao bwn. and.5. gur shows h ff of ass rao µ and ξ on h opu valu of dapng rao ξ op fro whh s sn ha ξ op nrass wh h nras of µ and ξ. Howvr, h aoun of nras s sall for hghr valus of ξ. gur 3 shows h ff of µ and ξ on h opu valu of frquny rao q op. I s shown ha q op drass wh h nras of µ and ξ. gur 4 shows h varaon of h au nondnsonal dsplan of sruur wh h ass rao µ. or dffrn valus of ξ, an b sn ha au dsplan of h sruur s rdud wh h nras of µ and ξ as pd. 3 WIT Transaons on Th Bul Envronn, Vol 4, ISSN on-ln 9 WIT Prss

6 9 Earhqua Rssan Engnrng Sruurs VII Dapng rao ξ op Mass Rao µ gur : Varaon of dapng rao ξ op wh ass rao µ usng h prsn sudy rsuls. 3 Praal dsgn quaons or Dsgn purposs, ay b or onvnn o prsn h opu TMD parars by spl dsgn quaons rahr han fgurs. or hs purpos, h urv fng Tool Bo whn h MATLAB sofwar [8] was usd o arry ou nsv urv fng rals usng h nural daa oband as dsrbd abov. Th followng suggsd quaons wr found o gv a vry los approaon o h valus of ξ op and q op shown n fgurs abov urv fng orrlaon offn was hghr han µ.66ξ op 8 µ µ ξ WIT Transaons on Th Bul Envronn, Vol 4, ISSN on-ln 9 WIT Prss

7 Earhqua Rssan Engnrng Sruurs VII 9.98 gur 3: rquny rao q op Mass Rao µ Varaon of frquny rao q op wh ass rao µ usng h prsn sudy rsuls. 5 gur 4: Mau non-dnonal sruur / o Mass Rao µ Varaon of au non-dnsonal sruur dsplan wh ass rao µ for varous valus of ξ. WIT Transaons on Th Bul Envronn, Vol 4, ISSN on-ln 9 WIT Prss

8 9 Earhqua Rssan Engnrng Sruurs VII..8 Dapng rao op Nural Rsuls Equaon Mass Rao µ gur 5: Varaon of dapng rao ξ op wh ass rao µ for ξ rquny rao q op Nural Rsuls Equaon Mass Rao µ gur 6: Varaon of frquny rao q op wh ass rao µ.4. WIT Transaons on Th Bul Envronn, Vol 4, ISSN on-ln 9 WIT Prss

9 Earhqua Rssan Engnrng Sruurs VII 93 µ q op.596 ξ µ µ Ths quaons wr found o gv a au rror wh rsp o nural rsuls of approaly.5 pr n n ξ op and q op. I should b nod ha hs quaons ar suggsd for h rang of µ bwn.5 up o.. Th rsuls oband usng h abov quaons ar opard wh hos of nural analyss n gur 5 and 6 rspvly. Th fgurs show h los agrn bwn h nural rsuls and h quaons rsuls. Equaons and ar rdud o h Dn Harog quaons [] for ξ.. 4 Conlusons Th Dn Harog odl for obanng opu dsgn parars for a TMD sys aahd o undapd SDO sruur has bn ndd o nlud h sruur dapng. Nural rsuls wr oband usng nural opzaon. nally, spl dsgn quaons wr suggsd for h drnaon of h opu dsgn parars of TMD syss. Rfrns [] Dn Harog, J. P. Mhanal Vbraon, MGraw-Hll, Nw Yor. N.Y., 956. [] Warburon, G. B. and Ayornd, E. O., Opu absorbr parars for spl syss, Earhq. Engng Sru. Dyna, 8, pp 97-7, 98. [3] Warburon, G. B., Opal absorbr parars for varous obnaons of rspons and aon parars, Earhq. Engng Sru. Dyna,, pp 38-4, 98. [4] Tsa, H. C. and Ln, G. C., Opu und ass daprs for nzng sady-sa rspons of suppor d and dapd syss, Earhq. Engng Sru. Dyna.,, pp , 993. [5] Rahul Rana and T. T. Soong, Parar, sudy and splfd dsgn of und ass daprs, Engnrng Sruurs, 3, pp. 93-4, 998. [6] Sad,., Mohraz, B., Taylor, A. and Chung, R., A Mhod of Esang Th Parars Of Tund Mass Daprs or Ss Applaons, Earhq. Engng Sru. Dyna, 6, pp , 997. [7] Abubaar, I. M., Conrol of RC Buldngs Vbraons usng Tund Mass Daprs sys, PhD Thss Subd To Th auly of Engnrng, Alandra Unvrsy, Alandra, Egyp, 6. [8] MATLAB Vrson 7., Th Languag of Thnal Copung, Th Mah Wors, In. 5. WIT Transaons on Th Bul Envronn, Vol 4, ISSN on-ln 9 WIT Prss