Novel Flutter Speed Formulas for Classical Flutter

The evolution of flutter speed in airfoils has long attracted interest, yet certain observed patterns remain theoretically unexplained, owing to the complicated interactions among numerous structural and aerodynamic parameters. This study presents novel analytical formulas for flutter speed and critical parameters using a two-degree-of-freedom (plunge–torsion) airfoil model. These formulas are derived from an explicit eigenvalue solution and validated through numerical examples. The flutter speed formula contains a quadratic function (elliptic paraboloid) of frequency ratio (Formula: see text) and mass center position (Formula: see text), predicting a minimum flutter speed at a critical Formula: see text and Formula: see text; the decreasing-then-increasing trend of flutter speed is explicitly clarified. The formulas for critical Formula: see text and Formula: see text provide quick estimates and explicitly predict their relationships with aerodynamic and structural parameters. Equivalently, the flutter speed formula contains a quadratic function of mass ratio (Formula: see text), predicting a minimum flutter speed at a critical Formula: see text, and the no-flutter phenomenon at very small Formula: see text is explained. Compared with existing flutter formulas, the proposed ones incorporate damping, aerodynamic stiffness, and mass unbalance while providing simple expressions for non-monotonic relationships with varying parameters. The clearer understanding of flutter and formula-based predictions can support design decisions with far less computational effort in the preliminary design stage. These formulas unify observed flutter behaviors within a coherent theoretical framework and facilitate practical application across a wide range of aeroelastic configurations.