Misconceptions and generalizations of the Den Hartog galloping criterion

Classical quasi-steady galloping analysis deals exclusively with cases of across-wind vibrations, leaving aside the more general situation where the wind and motion may not be normal. This can arise in many circumstances, such as in the motion of a power transmission cable about its equilibrium configuration that is swayed from the vertical plane as a result of the mean wind or in a tall slender structure in a skewed wind. Furthermore, the generalization to such situations, when this had been made, has only considered special issues. In this paper, the correct equations for the quasi-steady aerodynamic damping coefficients for a rotated system or wind are derived, and the differences from other variants are highlighted. Motion in two orthogonal structural planes is considered, potentially giving coupled translational galloping, for which previous analysis has often been limited or has even arrived at erroneous conclusions. For the two-degree-of-freedom case, the behavior is dependent on the structural as well as the aerodynamic parameters, in particular the orientation of the principal structural axes and the relative natural frequencies in the two planes. For the first time, the differences in the aerodynamic damping and zones of galloping instability are quantified between solutions from the correct perfectly tuned, well detuned, and classical Den Hartog equations (and also an incorrect generalization of the latter) for a variety of typical cross-sectional shapes. It is found that although the Den Hartog summation often gives a reasonable estimate for the actual aerodynamic damping, even in the rotated situation, in some circumstances the differences can be quite large.