Aeroelastic Systems with Softening Nonlinearity

Abstract

Aeroelastic systems with softening nonlinearity are not fully understood. The literature is not definitive, contains partial explanations, and even conflicting results. This problem is addressed for the case of a simple pitch-flap wing, with softening cubic nonlinearity in the pitch stiffness. Complex dynamical behavior is revealed. The study is carried out in three parts, the first of which is the identification and stability analysis of limit cycles in the frequency domain using describing functions combined with the Sherman–Morrison formula. Numerical integration of the nondimensionalized governing equations in the time domain is then carried out and, in addition to confirming the frequency domain results, new behavior is revealed, including asymmetric limit cycles and chaos. Finally, bifurcation analysis is undertaken using numerical continuation methods to reveal Hopf, symmetry breaking (pitchfork), fold and period doubling (flip) bifurcations. The effects of initial conditions and the proximity of limit cycle oscillation and chaos to static divergence are considered. Sections of the basin of attraction are presented for different wing configurations to show that the boundaries separating the different regions of dynamic behavior may be simple or nonsimple. It is found that some regions may appear where predicted stable limit cycle oscillation is free from the destabilizing effect of softening nonlinearity.