Floquet Theory Analysis of Rotor Blade Aeroelastic Stability at High Advance Ratios

Abstract

A compound rotorcraft operating at high forward speed may slow its rotor, resulting ina high advance ratio (μ). While this mitigates compressibility effects, it creates stability challenges, which are explored in the present work. The aeroelastic stability characteristics of rotor blades are analyzed in the rotating frame using Floquet theory for high-μ flight. Linearized coupled equations of motion are derived for the flap, lag, and torsion degrees of freedom for an isolated rotor system, with the assistance of symbolic manipulation software. Periodic coefficients, reverse flow, unsteady aerodynamics, pitch-flap/ pitch-lag/ structural coupling, chordwise center of gravity, aerodynamic center, and other structural and aerodynamic properties are considered. The flap trim state is found through periodic shooting, and a stall model provides bounding for stability predictions. Through numerical integration of the state transition matrix, the resultant eigenvalues and eigenvector motion allow for the determination of the type of response to perturbations from trim. A novel eigenvalue trajectory tracking method using a modal analysis of the periodic eigenvectors allows for a consistent identification of roots as parameters are changed. Damping and frequency response behaviors are evaluated, and rotor stability boundaries are presented for numerous configurations, to include various articulated and hingeless blade types. Advance ratios in excess of μ=3 are explored, with evaluations of both physical dynamic stability and numerical stability. Trim and stability predictions are validated using historical analyses and experimental results, a fundamental understanding of associated phenomena such as parametric resonance is achieved, and recommendations regarding stable configurations are provided.