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

• 3DOF aerodynamic damping matrix derived for arbitrary orientation of a 2D body.
• Determinant of the quasi-steady 3DOF aerodynamic damping matrix is always zero.
• Closed-form solution for galloping stability of perfectly-tuned 3DOF system.
• Example solutions for certain cross-sections.
• 3DOF solution normally close to 2DOF translational one with some effect of torsion.

The 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.