Nonlinear aeroelastic analysis of airfoils: bifurcation and chaos

Different types of structural and aerodynamic nonlinearities commonly encou ntered in aeronautical engineering are discussed. The equations of motion o f a two-dimensional airfoil oscillating in pitch and plunge are derived for a structural nonlinearity using subsonic aerodynamics theory. Three classi cal nonlinearities. namely, cubic, freeplay and hysteresis are investigated in some detail. The governing equations are reduced to a set of ordinary d ifferential equations suitable for numerical simulations and analytical inv estigation of the system stability. The onset of Hopf-bifurcation, and ampl itudes and frequencies of limit cycle oscillations are investigated, with e xamples given for a cubic hardening spring. For various geometries of the f reeplay, bifurcations and chaos are discussed via the phase plane, Poincare maps, and Lyapunov spectrum. The route to chaos is investigated from bifur cation diagrams, and for the freeplay nonlinearity it is shown that frequen cy doubling is the most commonly observed route. Examples of aerodynamic no nlinearities arising from transonic flow and dynamic stall are discussed, a nd special attention is paid to numerical simulation results for dynamic st all using a time-synthesized method for the unsteady aerodynamics. The assu mption of uniform flow is usually not met in practice since perturbations i n velocities are encountered in flight. Longitudinal atmospheric turbulence is introduced to show its effect on both the flutter boundary and the onse t of Hopf-bifurcation for a cubic restoring force. (C) 1999 Elsevier Scienc e Ltd. All rights reserved.