Practical flutter speed formulas for flexible structures considering all torsional-to-vertical frequency ratios

Developing a simple formula for predicting flutter speed has been a long-standing objective due to its high prediction efficiency and ability to clarify key influence mechanisms. Although Selberg's formula has been widely employed, its failure under close vertical and torsional frequency conditions has long been disturbing, especially as more engineering structures exhibit this feature, e.g., super-long-span bridges, cable-supported photovoltaic systems, and bundled conductors. Based on theoretical derivations, this study proposes simple practical formulas of flutter speed across the whole frequency ratio range and incorporates structural damping, with the validity examined through numerical examples under various parameters. The formulas comprise three parts: two formulas similar to Selberg's formula for vertical-branch flutter and torsional-branch flutter, and a minimum flutter speed formula for branch-switching flutter. Two frequency ratio formulas are provided to specify their boundaries, serving as the criteria of flutter type identification. The minimum flutter speed formula provides a lower-limit prediction across the whole frequency ratio range, making it practical for designing flutter-prone structures. The damping factor formulas enable easy quantification of both the amplification effect of structural damping on flutter speed and damping uncertainty impact in flutter analyses. The flutter speed sensitivity to damping of a close-frequency structure is much higher than that of a separated-frequency structure. Flutter patterns in different frequency ratio ranges are summarized for structural sections with various flutter derivatives, along with the criteria to predict whether flutter could occur. For the large air-to-structure mass ratio case, e.g., cable-supported photovoltaic systems, decreasing frequency ratio is a promising measure to control flutter. For the flutter-based energy harvesting system, the flutter speed is less sensitive to frequency ratio under high damping; the optimal frequency ratio for the lowest flutter speed increases with the equivalent damping and mass ratio.