ON THE BEAT PHENOMENON IN COUPLED SYSTEMS

Transcription

1 8 h ASCE Specialy Conference on Probabilisic Mechanics and Srucural Reliabiliy PMC-38 ON THE BEAT PHENOMENON IN COUPLED SYSTEMS S. K. Yalla, Suden Member ASCE and A. Kareem, M. ASCE NaHaz Modeling Laboraory, Universiy of Nore Dame, Nore Dame, IN syalla@nd.edu, kareem@nd.edu Absrac The classical bea phenomenon has been observed in mos coupled srucure-damper sysems. The focus of his paper is o provide a beer undersanding of his phenomenon, which is caused by he coupling ha is inroduced hrough he mass marix of he combined sysem. However, beyond a cerain level of damping in he secondary sysem, he bea phenomenon ceases o exis. This is due o coalescing of he modal frequencies of he combined sysem o a common frequency beyond a cerain level of damping in he secondary sysem. Numerical and experimenal resuls are presened in his paper o elucidae he bea phenomenon in combined srucure-damper sysems. Alhough his papers focusses primarily on coupled sysems wih liquid dampers, his is applicable o any ype of coupled sysem, for e.g., a linear uned mass damper or oher such vibraion absorber. Inroducion The effeciveness of liquid dampers in conrolling srucural moions under wind and earhquake loadings has been demonsraed in heory and pracice. The mos commonly used liquid dampers are Tuned Liquid Dampers (TLDs) and Tuned Liquid Column Dampers (TLCDs). The TLCD is a special ype of TLD ha insead of sloshing relies on he oscillaions of a column of liquid in a ube-like conainer o cancel he forces acing on he primary srucure (Sakai and Takaeda, 989). Damping in he TLCD is inroduced by providing an orifice o dampen he oscillaions of he liquid column. Experimenal sudies involving a TLCD combined wih a simple srucure have provided insighful undersanding of he behavior of liquid damper sysems. The moivaion of his paper is porayed in Figs. (a) and (b), which show he free vibraion decay of a combined srucure-tld and - TLCD obained by experimens. The conrolled response exhibis he classical bea phenomenon characerized by a modulaed insead of an exponenial decay in he signaure. The bea phenomenon has been discussed in many classical exs on vibraion (e.g., Den Harog, 956). There is a ransfer of energy beween he coupled sysem, similar o he coupled penduli problem. The focus of his paper is o beer undersand his phenomenon for he combined srucure-tlcd sysem..5 Response of Srucure Unconrolled Conrolled wih TLD Response of Srucure.5.5 Unconrolled conrolled wih TLCD (sec) ime (sec) Figure. Unconrolled and Conrolled srucural response wih (a) TLD (b) TLCD. Yalla and Kareem

2 8 h ASCE Specialy Conference on Probabilisic Mechanics and Srucural Reliabiliy The equaions of moion of he combined single degree of freedom srucure (primary sysem) and a TLCD (secondary sysem) shown in Fig. (a) are given by, m + m αm ẋ αm m c ẋ k x + + c ẋ ẋ k () where and are he displacemens of primary sysem and he secondary sysem, respecively; m mass of fluid in he ube ρal; c nonlinear damping of he liquid damper/ρaξ; k siffness of he liquid column ρag;m, k,c mass, siffness and damping in he srucure; ρ densiy of liquid; A cross secional area of he ube;ξ headloss coefficien; α is he lengh raio b/l l oal lengh of he waer column; and b horizonal lengh of he column. Deails of his sysem can be found in Yalla, e al In he following secions, differen cases of his general combined sysem are discussed. Case : Undamped Combined Sysem The coupled equaions of moion wihou damping in he primary and secondary sysem can be obained from Eq. by dropping c and c, + µ αµ α ẋ + ω ω () where µ is he mass raiom /m ; and ω and ω are he naural frequencies of he srucure and damper respecively. Figure (b) shows he ime hisories of he displacemen of he undamped primary sysem for α and.6. As expeced, when coupling is presen beween he wo sysems, he displacemen signaure is ampliude modulaed. The modal frequencies of his sysem are given by: ω ϖ, + ω ( + µ ) ± Π ( + µ α µ ) (3) where Π ( ω ω ( + µ )) + 4ωωα µ. To undersand his phenomenon beer, one can consider he soluion of he sysem of equaions given in Eq.. Afer some mahemaical manipulaion he displacemen of he primary sysem for he iniial condiions, ( ) x ; ( ) ; ẋ ( ) and ẋ ( ), is given by: () x cos( ω B ) cos( ω A ) (4)

3 8 h ASCE Specialy Conference on Probabilisic Mechanics and Srucural Reliabiliy where ω A ϖ + ϖ and ω B ϖ ϖ, which means ha he resuling funcion is an ampliude-modulaed harmonic funcion wih a frequency equal o ω B and he ampliude varying wih a frequency of. ω A.5. α α Figure. (a) Experimenal Se-up of coupled Srucure-TLCD sysem (b) hisories of primary sysem displacemen for α and.6 Case : Linearly Damped Srucure wih Undamped Secondary Sysem In his secion a linearly damped primary sysem wih undamped secondary sysem is considered. Accordingly, he equaions of moion are given by: + µ αµ α ẋ ẋ ω ζ ω + + ẋ ω (5) Figure 3(a) shows he effec of damping in he primary sysem on he response of he srucure. As he damping increases, he response dies ou in an exponenial decay. However, he bea phenomenon sill exiss. This poses difficuly in he esimaion of sysem damping from free vibraion response ime hisories. A his sage, he effec of decreasing he bea frequency on he response signal can be furher examined. Figure 3(b) shows ha as ω B approaches zero, T B (he ime period of he bea frequency) becomes very large. As a resul, due o he damping in he primary sysem, he response dies ou before he nex peak of he bea cycle arises. Therefore, he response resembles ha of a damped single degree of freedom (SDOF) sysem. 3

4 8 h ASCE Specialy Conference on Probabilisic Mechanics and Srucural Reliabiliy..5 X X Figure 3. (a) hisories for ζ.5 % and ζ 5% (b) Anaomy of he damped response Case 3: Damped Primary and Secondary Sysem In his secion, consider he sysem represened by Fig. (a), where now an orifice impars damping ino he sysem. Equaion is numerically simulaed for he free vibraion case a differen levels of he headloss coefficien (Fig. 4(a)). The figure shows an ineresing behavior of he liquid damper sysem. In he previous secion, he damping simply caused an exponenial decay of he bea response. However, in his case, he bea phenomenon disappears afer a cerain level of he headloss coefficien. Since an analyical soluion is no convenien for his equaion due o he quadraic nonlineariy in he damping marix, a linearized version of his sysem is generally considered. Therefore, Eq. is recas as: + µ αµ α ẋ ω ζ ẋ ω + + ω ζ ẋ ω (6) In order o furher validae he observaions made in his paper, a simple experimen was conduced using he experimenal seup shown in Fig. (a). The TLCD was designed wih a variable orifice, o effecively change he headloss coefficien. A Φ degrees, he valve is fully opened and he headloss is increased wih an increase in he angle of roaion, Φ. In Fig. 4(b), clearly a low headloss coefficiens, here is an obvious bea paern bu as he headloss coefficien is increased, he bea phenomenon disappears and an exponenially decaying signaure is obained. Figure 5 explains he disappearance of he bea phenomenon due o coalescing of he modal frequencies afer a cerain value of ξ is reached. The resuling bea frequency approaches zero and hence bea phenomenon ceases o exis. This is similar o a previous case where here was no bea phenomenon for coupling erm α, in which case he bea frequency was also zero. 4

5 8 h ASCE Specialy Conference on Probabilisic Mechanics and Srucural Reliabiliy..5.5 Φ degrees Φ5 degrees 5 5 Φ6 degrees Figure 4. (a) hisories of response for ξ., and 5 (b) Experimenal resuls for Φ, 5, 6 degrees.5. Conclusions Modal frequencies Modal damping raio Equivalen damping raio, ζ Equivalen damping raio, ζ Like coupled mechanical sysems, he combined srucure-liquid damper sysem exhibis he bea phenomenon due o he coupling in he mass marix of he combined sysem. The free vibraion response resembles an ampliude modulaed signal. The bea frequency of he modulaed signaure is given by he difference in he modal frequencies of he coupled sysem. However, beyond a cerain level of damping in Figure 5. Modal frequencies and damping raios he secondary sysem (liquid damper), he bea phenomenon ceases o exis. This is aribued o he coalescing of he modal frequencies of he combined sysem o a common frequency over ha range of damping in he secondary sysem. Acknowledgemens The auhors graefully acknowledge he suppor provided by NSF Gran CMS References Den Harog, J.P., (956), Mechanical Vibraions, 4h Ed, McGraw-Hill, New York. Sakai, F. and Takaeda, S., (989), Tuned Liquid Column Damper - New Type Device for Suppression of Building Vibraions, Proceedings Inernaional Conference on High Rise Buildings, Nanjing, China, March 5-7. Yalla, S.K., Kareem, A., and Kanor, J.C., (998), Semi-Acive Conrol Sraegies for Tuned Liquid Column Dampers o reduce Wind and Seismic Response of Srucures, Proceedings of Second World Conference on Srucural Conrol, Kyoo, John Wiley. 5