An analytical solution for the galloping stability of a 3 degree-of-freedom system based on quasi-steady theory

Abstract

General rightsThis document is made available in accordance with publisher policies. Please cite only the published version using the reference above. Full terms of use are available: http://www.bristol.ac.uk/pure/about/ebr-terms An analytical solution for the galloping stability of a 3 degree-of-freedom system based on quasi-steady theory a b s t r a c tThe aerodynamic forces on a two-dimensional three-degree-of-freedom (3DOF-heave, sway and torsion) body of arbitrary cross-section are considered, for arbitrary wind direction relative to the principal structural axes. The full 3DOF aerodynamic damping matrix is derived, based on quasi-steady theory, using the commonly-used concept of an aerodynamic centre to represent the effect of the torsional velocity on the aerodynamic forces. The aerodynamic coefficients are assumed to be consistent functions of only the relative angle of attack. It is shown that the determinant of the quasi-steady aerodynamic damping matrix is always zero. The galloping stability of the aerodynamically coupled system is then addressed by formulating the eigenvalue problem, for which analytical solutions are derived for the case of perfectly tuned structural natural frequencies. The solutions define a non-dimensional effective aerodynamic damping coefficient, indicating how stable the system is. A trivial solution always exists, with zero effective aerodynamic damping, corresponding to rotation about the aerodynamic centre, and relatively simple exact closed-form solutions are derived for the other one or two solutions, the minimum solution defining the stability of the system. Example results are presented and discussed for square, rectangular (aspect ratio 3) and equilateral triangular sections and a lightly iced cable, and they are compared with results using previous solutions for 2DOF translational and 1DOF pure torsional galloping. For the shapes considered it is found that the stability of the 3DOF system is normally close to that of the 2DOF translational system, with a relatively small influence of the stability of the torsional degree of freedom, although in some instances, especially at the critical angles of attack, it can significantly affect the stability.

Aeroelastic stability of a 3DOF system based on quasi-steady theory with reference to inertial coupling

“…Recently, He and Macdonald (2016) presented a 2-dimensional 3DOF model and derived the aerodynamic damping matrix in a simple form, including all three degrees of freedom, based on quasi-steady theory. Inertial coupling was excluded, implying coincidence of the elastic centre (O) and mass centre (G), which is applicable for sections with symmetrical geometry and lightly iced sections with a negligible offset of the centre of mass.…”

“…The incorporation of all three degrees of freedom results in a difficulty of quasi-steady theory, namely suitable treatment of the rotational velocity. The approach employed follows the common approach in the literature (Slater, 1969;Blevins and Iwan, 1974;Nakamura and Mizota, 1975;Blevins, 1994;Gjelstrup and Georgakis, 2011;He and Macdonald, 2016), where an aerodynamic centre is defined to emulate the effect of the rotational velocity on the aerodynamic forces using the wind velocity relative to that point. Fig.…”

“…which, based on quasi-steady theory and for any wind direction and orientation of the body, has been shown to be (He and Macdonald, 2016):…”

“…More recent contributions on 3DOF galloping include work by Piccardo et al (2014) who presented the full aerodynamic damping matrix in a more general form, while Demartino and Ricciardelli (2015) compared various existing quasi-steady models for galloping, using wind tunnel measurements for bridge cables and hangers with ice accretion, shedding some light on the application of each model. He and Macdonald (2016) extended the work by Nikitas and Macdonald (2014) and rigorously studied 3DOF galloping of a system coupled only by aerodynamic damping and proposed a simple closed-form solution for galloping of a perfectly tuned 3DOF system. They also numerically investigated the effects of the tuning of the structural natural frequencies (He and Macdonald, 2015).…”

“…The aim of this paper is to extend the previous 3DOF analytical model by He and Macdonald (2016) to include inertial coupling. Firstly, the inertial coupling terms in the mass matrix are derived in the same way as in Gjelstrup and Georgakis (2011).…”