Curve Veering of Eigenvalue Loci of Bridges with Aeroelastic Effects Xinzhong Chen 1 and Ahsan Kareem 2 Abstract: The eigenvalues of bridges with aeroelastic effects are commonly portrayed in terms of a family of frequency and damping loci as a function of mean wind velocity. Depending on the structural dynamic and aerodynamic characteristics of the bridge, when two frequencies approach one another over a range of wind velocities, their loci tend to repel, thus avoiding an intersection, whereas the mode shapes associated with these two frequencies are exchanged in a rapid but continuous way as if the curves had intersected. This behavior is referred to as the curve veering phenomenon. In this paper, the curve veering of cable-stayed and suspension bridge frequency loci is studied. A perturbation series solution is utilized to estimate the variations of the complex eigenvalues due to small changes in the system parameters and establish the condition under which frequency loci veer, quantified in terms of the difference between adjacent eigenvalues and the level of mode interaction. Prior to the discussion of bridge frequency loci, the curve veering of a two-degree-of-freedom system comprised of a primary structure and tuned mass damper is discussed, which not only provides new insight into the dynamics of this system, but also helps in understanding the veering of bridge frequency loci. To study this more complicated dynamic system, a closed-form solution of a two-degree-of-freedom coupled flutter is obtained, and the underlying physics associated with the heaving branch flutter is discussed in light of the veering of frequency loci. It is demonstrated that the concept of curve veering in bridge frequency loci provides a correct explanation of multimode coupled flutter analysis results for long span bridges and helps to improve understanding of the underlying physics of their aeroelastic behavior. DOI: 10.1061/ASCE0733-93992003129:2146 CE Database keywords: Eigenvalues; Bridges, span; Aeroelasticity. Introduction Wind–bridge interaction results in the generation of self-excited forces, which provide additional aerodynamic damping and stiff- ness to that already present in the structure. In addition, these self-excited forces induce aerodynamic coupling of structural modes, changing the eigenmodes of the bridge. Therefore, the real-valued structural modes are only observed when the mean wind velocity is zero, while complex modes are present under wind excitation. To avoid confusion, these complex modes are referred to as complex mode branches. The eigenvalues associ- ated with complex mode branches can be estimated utilizing a complex eigenvalue analysis e.g., Katsuchi et al. 1999; Chen et al. 2000. These eigenvalues are commonly portrayed in terms of a family of frequency and damping loci as a function of mean wind velocity. The behavior of these loci has interesting ramifications for the bridge flutter problem. Depending on the structural dynamic and aerodynamic characteristics of the bridge, two adjacent frequency loci may approach each other over a range of wind velocities. When this occurs, the curves may intersect or repel each other. However, even in the case where the curves repel each other, the eigenmodes eigenvectors associated with these two eigenvalues are exchanged continuously as if the curves had intersected Chen et al. 2001. This behavior has been termed the ‘‘curve veering phenomenon.’’ Because of this ambiguous behavior of frequency loci, tradi- tional flutter analysis that employs an iterative calculation proce- dure, based on frequency-dependent state-space equation, has proven to be computationally cumbersome. In this approach, the target mode identification has to be done iteratively, which may not permit complete automation of the analysis procedure Chen et al. 2000. This behavior of frequency loci may also result in coupled multimode flutter, involving more than two structural modes, which may initiate from a complex mode branch that is different from the commonly observed torsional mode branch. For ex- ample, in multimode bridge flutter analyses, it has been shown that a suspension bridge coupled flutter initiated from a lateral mode branch Miyata and Yamada 1988; Chen et al. 2000; Chen et al. 2001, and a cable-stayed bridge coupled flutter initiated from a mode branch associated with a tower bending mode as the wind velocity increased Chen et al. 2001. This behavior may give the impression that the physics of the multimode coupled flutter is different from the general understanding of coupled flut- ter in which two fundamental structural modes of the bridge deck, i.e., heaving and torsional fundamental structural modes, are most important. This general understanding of coupled flutter has been the foundation of both the analytical bimodal flutter predictions and wind tunnel based spring-supported section model studies. 1 Postdoctoral Research Associate, Dept. of Civil Engineering and Geological Sciences, Univ. of Notre Dame, Notre Dame, IN 46556. E-mail: xchen@nd.edu 2 Professor and Chair, Dept. of Civil Engineering and Geological Sci- ences, Univ. of Notre Dame, Notre Dame, IN 46556. E-mail: kareem@nd.edu Note. Associate Editor: Roger G. Ghanem. Discussion open until July 1, 2003. Separate discussions must be submitted for individual papers. To extend the closing date by one month, a written request must be filed with the ASCE Managing Editor. The manuscript for this paper was submitted for review and possible publication on February 28, 2002; approved on May 20, 2002. This paper is part of the Journal of Engineering Mechan- ics, Vol. 129, No. 2, February 1, 2003. ©ASCE, ISSN 0733-9399/2003/2- 146 –159/$18.00. 146 / JOURNAL OF ENGINEERING MECHANICS / FEBRUARY 2003